Macro assignment
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NeoclassicalGrowthandRealBusinessCycles.pdf
Macroeconomic Theory
1 Neoclassical Growth
The Neoclassical Growth Model is a variant of our in�nite-period framework that is a applied
to study the determinants of long-run growth in economic output. The main take-away is the
neoclassical growth model asserts that long-run economic growth is driven by increases in `tech-
nological capabilities' and population growth.
1.1 The �Long-Run"
• Steady State: a situation in the in�nite-period framework where all real variables stop �uctuating (or grow at a constant rate) over time absent external disturbances.
→ For any variable xt, in a steady state xt = xt+1 = xt+2 = ... = x̄ ⇒ Why does the model converge to a steady state? → Consider the economy's intertemporal optimality condition (i.e. �nancial market equilib-
rium condition):
∂u/∂ct ∂u/∂ct+1
= β
( ∂f
∂kt+1 + 1 −δ
) (1)
• Case 1: Growing Economy → ct < ct+1 < ... and kt+1 < kt+2 < ... → This implies that the left-hand side (LHS) of Equation 1 is greater than 1, which itself
implies that the right-hand side (RHS) is also greater than 1.
→ if k is growing, then ∂f/∂k is shrinking over time −→ RHS decreases towards 1, which implies that LHS is also decreasing towards 1.
• Case 2: Shrinking Economy → ct > ct+1 > ... and kt+1 > kt+2 > ... → This implies that LHS of Equation 1 is less than 1, which itself implies that RHS is also
less than 1.
→ if k is shrinking, then ∂f/∂k is growing over time −→ RHS increases towards 1, which implies that LHS is also increasing towards 1.
⇒ Putting together the implications from Cases 1 and 2, the positive but diminishing marginal product of capital implies that the economy settles at a steady state.
1.2 Technological Progress in Neoclassical Growth Model
• Technological Progress: increases in the productivity of the factors of production → more output for a given amount of inputs to the production function.
• Labor-Augmenting Technological Progress: Technological progress is modeled as in- creases in `e�ective' labor input. (required for existence of steady state)
1
→ Let Zt be the level of technology embodied by the labor input, nt. Then the `e�ective' labor input is Ztnt, and total output is given by f(kt,Ztnt).
→ Assume that Zt grows by a rate of γz in the steady state such that dZ̄/dt
Z̄ = γz
1.
1.3 Neoclassical Growth Model
→ For simplicity, we will assume that labor supply is exogenous, n̄, but grows at a rate of γn in
steady state such that dn̄/dt
n̄ = γn. (Loosely thought of as population growth)
→ Exogenous labor means that leisure is also exogenous, l̄.
Households:
max {ct+s,at+s}∞s=0
V = ∞∑ s=0
βsu(ct+s, l̄)
subject to:
ct + at − (1 + rt)at−1 −wt(1 − l̄) = 0 for all t = 1, 2, 3, ...
Firms:
max {kt+1+s}∞s=0
Profit = ∞∑ s=0
(1 + rt+s) −s{f(kt+s,Zt+sn̄) − (kt+1+s − (1 − δ)kt+s) −wt+sn̄}
⇒ Since the First Welfare Theorem will hold, the economy's optimality conditions can be obtained either by solving for general equilibrium in a decentralized fashion using the house-
holds' and �rms' problems one at a time, or by using the Social Planner's framework.
→ With exogenous labor, we need two equilibrium conditions for the economy:
Intertemporal Optimality Condition (Financial Market Equilibrium Condition):
∂u/∂ct β∂u/∂ct+1
=
( ∂f(kt+1,Zt+1n̄)
∂kt+1 + 1 −δ
) (2)
Aggregate Resource Constraint (Goods Market Equilibrium Condition):
f(kt,Ztn̄) = ct + (kt+1 − (1 −δ)kt) (3)
→With endogenous labor we would also need the economy's intratemporal optimality condition (i.e. labor market equilibrium condition)
1 Since for any variable, x,
dx̄/dt
x̄ is the instantaneous rate of growth, this tells us that steady capital grows
continuously at the rate of technological progress plus the rate of labor supply growth.
2
→ With a functional form for the production function, we can solve directly for the steady state capital stock, output, and consumption.
EXAMPLE: Suppose f(kt,Ztn̄) = k α t (Ztn̄)
1−α. Solve for steady state capital and output.
→ Impose steady state on intertemporal optimality condition:
∂u/∂ct β∂u/∂ct+1
= ( αkα−1t (Ztn̄)
1−α + 1 −δ )
∂u/∂c̄
β∂u/∂c̄ = ( αk̄α−1(Z̄n̄)1−α + 1 −δ
) 1
β = ( αk̄α−1(Z̄n̄)1−α + 1 −δ
)
→ Solve for k̄:
1
β − 1 + δ = αk̄α−1(Z̄n̄)1−α
1
β − 1 + δ
α
=
( k̄
Z̄n̄
)α−1
1
β − 1 + δ
α
1/(α−1)
= k̄
Z̄n̄
⇒ k̄∗ = Z̄n̄
1
β − 1 + δ
α
1/(α−1)
(4)
→ Use expression for k̄ into the production function:
f(k̄∗, Z̄n̄) =
Z̄n̄
1
β − 1 + δ
α
1/(α−1) α
(Z̄n̄)1−α
f(k̄∗, Z̄n̄) = (Z̄n̄)α(Z̄n̄)1−α
1
β − 1 + δ
α
α/(α−1)
⇒ f(k̄∗, Z̄n̄) = Z̄n̄
1
β − 1 + δ
α
α/(α−1)
(5)
3
→ Equations (4) and (5) are the expressions for steady state capital and output. Using these expressions into the resource constraint evaluated at the steady state, one can also obtain an
expression for steady state consumption:
f(k̄, Z̄n̄) = c̄ + (k̄ − (1 −δ)k̄)
f(k̄, Z̄n̄) = c̄ + δk̄
⇒ c̄∗ = f(k̄∗, Z̄n̄) −δk̄∗
1.3.1 Technological Progress and Economic Growth
What does technological progress (and population growth) imply for steady state growth of capital
stock and output?
→ Want steady state growth rate of capital in terms of steady state growth rates of technology and labor.
→ Consider the expression for the steady state capital stock:
k̄ = Z̄n̄
1
β − 1 + δ
α
1/(α−1)
→ Taking the natural log of both side yields:
log k̄ = log Z̄ + log n̄ + ((1/(α− 1)) log
1
β − 1 + δ
α
→ Next, taking the total derivative of the above equation with respect to time:
dk̄/dt
k̄ = dZ̄/dt
Z̄ + dn̄/dt
n̄ dk̄/dt
k̄ = γz + γn (6)
⇒ Steady state capital grows at the rate of technological progress plus population growth. → Intuition: The marginal product of capital increases as e�ective labor input exogenously
grows, which increases the desired future capital stock by the �rm thereby lowering the marginal
product of capital. In a steady state these two e�ects on the marginal product of capital cancel
each other out.
4
→ Repeating this process for our expression for steady state output:
f(k̄, Z̄n̄) = Z̄n̄
1
β − 1 + δ
α
α/(α−1)
log f(k̄, Z̄n̄) = log Z̄ + log n̄ + ((α/(α− 1)) log
1
β − 1 + δ
α
df̄/dt
f̄ = dZ̄/dt
Z̄ + dn̄/dt
n̄
df̄/dt
f̄ = γz + γn
⇒ Steady state growth in output is determined by growth rates in technology and population. → Intuition: Steady state output grows at same rate as each of the factors of production.
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Macroeconomic Theory
1 Chapter 14: Real Business Cycles
We utilize the in�nite-period general equilibrium framework to study a theory of business cycles.
• Real Business Cycle (RBC) Theory asserts that short-run �uctuations to economic ac- tivity are �e�cient� responses to temporary technological shocks.
1.1 Technological Shocks in Real Business Cycle Model
• Technological Shock: unexpected, temporary changes to `total factor productivity' • Total Factor Productivity (TFP): Portion of output not determined by capital or labor. → Let Θt be the level of TFP, then output is modeled as Θtf(kt,nt). → Graphically, TFP shocks look as follows:
1.2 Real Business Cycle Model
Households:
max {ct+s,at+s,lt+s}∞s=0
V =
∞∑ s=0
βsu(ct+s, lt+s)
subject to:
ct + at − (1 + rt)at−1 −wt(1 − lt+s) = 0 for all t = 1, 2, 3, ...
Firms:
max {kt+1+s,nt+s}∞s=0
Profit =
∞∑ s=0
(1 + rt+s) −s{Θt+sf(kt+s,nt+s)−(kt+1+s−(1−δ)kt+s)−wt+snt+s}
⇒ Since the First Welfare Theorem will hold, the economy's optimality conditions can be obtained either by solving for general equilibrium in a decentralized fashion using the house-
holds' and �rms' problems one at a time, or by using the Social Planner's framework.
1
→ We need three equilibrium conditions for the economy:
Intratemporal Optimality Condition (Labor Market Equilibrium Condition):
∂u/∂lt ∂u/∂ct
= ∂Θtf(kt,nt)
∂nt (1)
Intertemporal Optimality Condition (Financial Market Equilibrium Condition):
∂u/∂ct β∂u/∂ct+1
=
( ∂Θt+1f(kt+1,nt+1)
∂kt+1 + 1 −δ
) (2)
Aggregate Resource Constraint (Goods Market Equilibrium Condition):
Θtf(kt,nt) = ct + (kt+1 − (1 −δ)kt) (3)
→ Unlike the steady state of Neoclassical Growth Model, we cannot solve analytically for the equilibrium values of economic aggregates in the RBC model. However, we can use the economy's
optimality conditions to characterize what happens following a TFP shock.
⇒ How does a temporary increase in Θt and Θt+1 a�ect equilibrium in the model?
→ First, recall that �rms hire labor and invest into future capital until each respective marginal product equals its marginal cost:
∂Θtf(kt,nt)
∂nt = wt
∂Θt+1f(kt+1,nt+1)
∂kt+1 = rt+1 + δ
where Θ directly a�ects each marginal product. From the above �rm's optimality conditions:
→ Θt ↑ → MPn ↑ −→ �rm's demand for labor increases at wage rate → Θt+1 ↑ → MPk ↑ −→ �rm's demand for investment increases at real interest rate
→ Second, an increase in the marginal products of labor and capital will increase the RHS of the labor market and �nancial market equilibrium conditions.
→ MPN ↑→ return to labor increases −→ quantity of n supply t ↑ along given supply function,
which causes MRSl,c ↑ so that Equation (1) holds in equilibrium → MPK ↑ → return to capital increases −→ quantity of s
supply t ↑ along given supply func-
tion, which causes MRS1,2 ↑ so that Equation (2) holds in equilibrium
→ Third, an increase in nt,t+1,kt+1, and Θt,t+1 increases AS in t = 1, 2 → AD↑ (from increase in investment demand) so that Equation (3) holds in equilibrium.
2
→ Graphically this change looks as follows in the labor, �nancial, and goods markets:
⇒ This is considered to be an e�cient response to the technological shock because the social planner would choose the change to resource allocation that would be obtained from �rms and
households interacting in decentralized markets.
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