ECON 214 - ADVANCED PHILLIPS CURVE ANALYSIS AND
INFLATION-UNEMPLOYMENT TRADE-OFF
INSTRUCTIONS
Solve all problems. Show your work, including all mathematical steps. Each question
carries equal weight.
1. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
2. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
3. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
4. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
5. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
6. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
7. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
8. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
9. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
10. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
11. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
12. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
13. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
14. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
15. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
16. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
17. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
18. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
19. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
20. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
21. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
22. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
23. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
24. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
25. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
26. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
27. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
28. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
29. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
30. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
31. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
32. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
33. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
34. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
35. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
36. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
37. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
38. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
39. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
40. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
41. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
42. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
43. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
44. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
45. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
46. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
47. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
48. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
49. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
50. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
51. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
52. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
53. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
54. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
55. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
56. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
57. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
58. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
59. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
60. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
61. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
62. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
63. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
64. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
65. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
66. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
67. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
68. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
69. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
70. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
71. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
72. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
73. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
74. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
75. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
76. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
77. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
78. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
79. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
80. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
81. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
82. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
83. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
84. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
85. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
86. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
87. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
88. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
89. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
90. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
91. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
92. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
93. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
94. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
95. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
96. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
97. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
98. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
99. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
100. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
101. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
102. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
103. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
104. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
105. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
106. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
107. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
108. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
109. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
110. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
111. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
112. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
113. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
114. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
115. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
116. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
117. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
118. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
119. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
120. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
121. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
122. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
123. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
124. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
125. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
126. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
127. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
128. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
129. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
130. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
131. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
132. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
133. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
134. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
135. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
136. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
137. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
138. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
139. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
140. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
141. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
142. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
143. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
144. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
145. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
146. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
147. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
148. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
149. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
150. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
151. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
152. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
153. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
154. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
155. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
156. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
157. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
158. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
159. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
160. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
161. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
162. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
163. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
164. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
165. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
166. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
167. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
168. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
169. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
170. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
171. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
172. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
173. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
174. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
175. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
176. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
177. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
178. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
179. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
180. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
181. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
182. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
183. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
184. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
185. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
186. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
187. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
188. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
189. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
190. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
191. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
192. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
193. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
194. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
195. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
196. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
197. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
198. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
199. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
200. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
201. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
202. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
203. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
204. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
205. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
206. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
207. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
208. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
209. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
210. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
211. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
212. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
213. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
214. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
215. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
216. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
217. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
218. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
219. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
220. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
221. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
222. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
223. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
224. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
225. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
226. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
227. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
228. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
229. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
230. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
231. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
232. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
233. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
234. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
235. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
236. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
237. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
238. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
239. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
240. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
241. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
242. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
243. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
244. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
245. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
246. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
247. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
248. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
249. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
250. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
251. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
252. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
253. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
254. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
255. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
256. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
257. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
258. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
259. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
260. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
261. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
262. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
263. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
264. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
265. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
266. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
267. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
268. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
269. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
270. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
271. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
272. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
273. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
274. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
275. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
276. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
277. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
278. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
279. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
280. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
281. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
282. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
283. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
284. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
285. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
286. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
287. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
288. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
289. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
290. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
291. Consider an economy described by the following equations:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
𝑦𝑡= 𝑦𝑡
∗+ 𝛽(𝑚𝑡− 𝑝𝑡− 𝑦𝑡
∗)+ 𝜂𝑡
𝑚𝑡− 𝑝𝑡= 𝑦𝑡− 𝛾𝑖𝑡+ 𝜈𝑡
292. where 𝜋𝑡 is inflation, 𝜋𝑡
𝑒 is expected inflation, 𝑦𝑡 is output, 𝑦𝑡
∗ is potential output,
𝑚𝑡 is money supply, 𝑝𝑡 is price level, 𝑖𝑡 is interest rate, and 𝜖𝑡, 𝜂𝑡, and 𝜈𝑡 are
shock terms.
a) Derive the Phillips curve equation relating inflation to unemployment.
b) Assuming rational expectations, solve for the equilibrium inflation rate and
output gap.
c) How does the trade-off between inflation and unemployment change if
expectations are adaptive rather than rational?
Solution: a) To derive the Phillips curve:
– Assume Okun’s law: 𝑢𝑡− 𝑢𝑡
∗= −𝛿(𝑦𝑡− 𝑦𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This is the expectations-augmented Phillips curve
b) With rational expectations:
– 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]
– Taking expectations of the first equation: 𝐸[𝜋𝑡]= 𝜋𝑡
𝑒+𝛼𝐸[𝑦𝑡− 𝑦𝑡
∗]
– In equilibrium, 𝐸[𝑦𝑡− 𝑦𝑡
∗]= 0
– Therefore, 𝜋𝑡
𝑒= 𝐸[𝜋𝑡]= 𝜋𝑡 (absent shocks)
– The equilibrium output gap is zero: 𝑦𝑡− 𝑦𝑡
∗= 0
c) With adaptive expectations:
– Assume 𝜋𝑡
𝑒= 𝜋𝑡−1
– The Phillips curve becomes: 𝜋𝑡= 𝜋𝑡−1 −𝛼
𝛿(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies a short-run trade-off between inflation and unemployment
– In the long run, expectations adjust, eliminating the trade-off
293. Develop a dynamic model of the Phillips curve incorporating both backward-
looking and forward-looking expectations:
𝜋𝑡= 𝜃𝜋𝑡−1 +(1 − 𝜃)𝐸𝑡[𝜋𝑡+1]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
294. where 0 ≤ 𝜃 ≤ 1.
a) Interpret the parameters of this model.
b) Solve for inflation as a function of the output gap, assuming rational
expectations and that the output gap follows an AR(1) process: 𝑦𝑡− 𝑦𝑡
∗=
𝜌(𝑦𝑡−1 − 𝑦𝑡−1
∗)+ 𝜈𝑡.
c) How does the persistence of inflation change as 𝜃 varies from 0 to 1?
Solution: a) Interpretation of parameters:
– 𝜃: weight on backward-looking expectations
– 1 − 𝜃: weight on forward-looking expectations
– 𝜅: sensitivity of inflation to the output gap
– 𝜖𝑡: cost-push shock
b) Solving for inflation:
– Guess a solution of the form: 𝜋𝑡= 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡
– Substitute into the original equation: 𝑎(𝑦𝑡− 𝑦𝑡
∗)+ 𝑏𝜖𝑡= 𝜃[𝑎(𝑦𝑡−1 − 𝑦𝑡−1
∗)+
𝑏𝜖𝑡−1]+(1 − 𝜃)[𝑎𝜌(𝑦𝑡− 𝑦𝑡
∗)] + 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
– Equate coefficients: 𝑎 = 𝜅
1−(1−𝜃)𝜌, 𝑏 = 1
– Final solution: 𝜋𝑡=𝜅
1−(1−𝜃)𝜌(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
c) Inflation persistence:
– As 𝜃 increases (more backward-looking), persistence increases
– When 𝜃 = 0 (purely forward-looking), inflation depends only on current
and expected future output gaps
– When 𝜃 = 1 (purely backward-looking), inflation has high persistence
295. Consider a New Keynesian Phillips Curve with indexation:
𝜋𝑡− 𝛾𝜋𝑡−1 = 𝛽𝐸𝑡[𝜋𝑡+1 − 𝛾𝜋𝑡]+ 𝜅(𝑦𝑡− 𝑦𝑡
∗)+ 𝜖𝑡
296. where 𝛾 is the indexation parameter.
a) Derive the implications of this model for the relationship between inflation and
the output gap in the long run.
b) How does the presence of indexation (𝛾 > 0) affect the sacrifice ratio (the
cumulative output loss needed to reduce inflation)?
c) Solve for the optimal monetary policy rule under discretion, assuming the
central bank’s loss function is 𝐿𝑡=1
2[(𝜋𝑡− 𝜋∗)2+ 𝜆(𝑦𝑡− 𝑦𝑡
∗)2].
Solution: a) Long-run implications:
– In steady state, 𝜋𝑡= 𝜋𝑡−1 = 𝜋𝑡+1 = 𝜋‾
– Substituting into the Phillips curve: (1 − 𝛾)𝜋‾ = (1 − 𝛾)𝛽𝜋‾ + 𝜅(𝑦‾ − 𝑦∗)
– If 𝛽 = 1, then 𝑦‾ = 𝑦∗ (no long-run trade-off)
– If 𝛽 < 1, there’s a small long-run trade-off
b) Effect on sacrifice ratio:
– Higher 𝛾 increases inflation inertia
– This makes inflation less responsive to output gaps
– Therefore, the sacrifice ratio increases with 𝛾
c) Optimal discretionary policy:
– Central bank minimizes 𝐿𝑡 subject to the Phillips curve
– First-order condition: (𝜋𝑡− 𝜋∗)+ 𝜙𝑡= 0 𝜆(𝑦𝑡− 𝑦𝑡
∗)− 𝜅𝜙𝑡= 0 where 𝜙𝑡 is
the Lagrange multiplier
– Combining these with the Phillips curve: (𝜋𝑡− 𝜋∗)= − 𝜆
𝜅(𝑦𝑡− 𝑦𝑡
∗)
– This is the optimal targeting rule under discretion
297. Analyze the implications of a time-varying NAIRU (Non-Accelerating Inflation
Rate of Unemployment) for the Phillips curve. Consider the model:
𝜋𝑡= 𝜋𝑡
𝑒− 𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
𝑢𝑡
∗= 𝑢𝑡−1
∗+ 𝜈𝑡
298. where 𝑢𝑡
∗ is the time-varying NAIRU.
a) How does this model differ from the standard Phillips curve?
b) Derive the implications for inflation dynamics if agents have to learn about
changes in the NAIRU.
c) Discuss the challenges this model poses for monetary policy.
Solution: a) Differences from standard Phillips curve:
– NAIRU (𝑢𝑡
∗) varies over time instead of being constant
– This introduces additional uncertainty in the inflation-unemployment
relationship
– The long-run trade-off becomes more complex and potentially time-
varying
b) Implications with learning:
– Assume agents form expectations: 𝜋𝑡
𝑒= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)
– Substituting into the Phillips curve: 𝜋𝑡= 𝜋𝑡−1 + 𝛿(𝑢𝑡−1
∗− 𝑢𝑡−1)−
𝛽(𝑢𝑡− 𝑢𝑡
∗)+ 𝜖𝑡
– This implies persistent forecast errors and more complex inflation
dynamics
– Inflation can be more volatile as agents learn about NAIRU changes
c) Challenges for monetary policy:
– Difficulty in estimating the current NAIRU
– Increased uncertainty in forecasting inflation
– Potential for policy mistakes due to misperceiving the NAIRU
– Need for more adaptive and flexible policy frameworks
– Importance of robust policy rules that perform well under NAIRU
uncertainty
299. Develop a model of the Phillips curve in an open economy:
𝜋𝑡= 𝜋𝑡
𝑒+ 𝛼(𝑦𝑡− 𝑦𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
𝑦𝑡− 𝑦𝑡
∗= −𝛽(𝑖𝑡− 𝜋𝑡
𝑒)+ 𝛿(𝑒𝑡− 𝑒‾)+ 𝜂𝑡
𝑒𝑡= 𝑒𝑡+1
𝑒+ (𝑖𝑡
𝑓− 𝑖𝑡) + 𝜈𝑡
300. where 𝑒𝑡 is the log exchange rate, 𝑒‾ is the long-run equilibrium exchange rate, 𝑖𝑡
𝑓
is the foreign interest rate, and 𝑒𝑡+1
𝑒 is the expected future exchange rate.
a) Interpret each equation in this system.
b) Derive the open-economy Phillips curve relating inflation to unemployment and
exchange rate changes.
c) How does exchange rate pass-through affect the inflation-unemployment
trade-off?
d) Discuss the implications of this model for monetary policy in small open
economies.
Solution: a) Interpretation of equations:
– Equation 1: Open-economy Phillips curve with exchange rate pass-
through
– Equation 2: IS curve with exchange rate effects on output
– Equation 3: Uncovered interest parity condition
b) Open-economy Phillips curve:
– Assume Okun’s law: 𝑦𝑡− 𝑦𝑡
∗= −𝜆(𝑢𝑡− 𝑢𝑡
∗)
– Substitute into the first equation: 𝜋𝑡= 𝜋𝑡
𝑒−𝛼
𝜆(𝑢𝑡− 𝑢𝑡
∗)+ 𝛾(𝑒𝑡− 𝑒𝑡−1)+ 𝜖𝑡
c) Exchange rate pass-through effects:
– Higher 𝛾 implies greater pass-through
– This makes inflation more sensitive to exchange rate movements
– The trade-off becomes more complex, as unemployment and exchange
rates both affect inflation
– Policy may need to balance internal (unemployment) and external
(exchange rate) objectives
d) Implications for monetary policy:
– Increased complexity in managing inflation due to external factors
– Potential conflicts between internal and external balance
– Importance of considering exchange rate movements in policy decisions
– Possible need for alternative policy frameworks (e.g., inflation targeting
with exchange rate considerations)
– Challenges in small open economies due to greater vulnerability to
external shocks
