Advanced marcoeconomics test
Will.Luke
AdvancedUMacroSampleMidtermExamAnswers.pdf
ECON 401
Advanced Macroeconomics
Midterm Exam Answers
Fabio Ghironi University of Washington
May 5, 2020
Important: My suggested answers include a lot more than I expected you could know and
write. I am including the material I am including so you can continue learn from this exam about
things we do not have time to discuss in class.
Problem 1: Government Spending in the RBC Model (50 Points)
Consider the RBC model with government spending that you studied in Homework 1.
The representative household maximizes:
Et
∞∑ s=t
βs−t C 1−γ s
1 −γ ,
where 0 < β < 1 and γ > 0, subject to the constraint:
Ct + It + Xt = r̃tKt + wt
in each period. In this constraint, Xt is exogenous lump-sum taxation, which we assume is equal
to government spending. The rest of the notation is as in the slides.
The law of motion for capital is:
Kt+1 = (1 −δ) Kt + It, 0 < δ < 1,
in each period.
The production function is:
Yt = A α t K
1−α t , 0 < α < 1.
You found in the homework that the solution for consumption and capital in the log-linearized
model in which there is no deviation of the exogenous productivity At from its trend path (i.e., it
is at+1 = at = at−1 = 0) and there are only the government spending shocks xt = φxt−1 + εt is
given by:
ct = ηckkt + ηcxxt,
kt+1 = ηkkkt + ηkxxt,
in each period, where the η’s are elasticities that depend on the parameters of the model and the
ratio X̄t/Ȳt, which I told you to treat as exogenously given.
• Use the method of undetermined coeffi cients described in the slides to find the
expressions for ηck, ηcx, ηkk, and ηkx as functions of the underlying parameters
and of X̄t/Ȳt.
Answer
Substituting ct+1 = ηckkt+1 + ηcxxt+1, ct = ηckkt + ηcxxt, and kt+1 = ηkkkt + ηkxxt in the log-linear
Euler equation that you obtained in Homework 1, and using Et (xt+1) = φxt, yields:
ηck(ηkkkt + ηkxxt) + ηcxφxt −ηckkt −ηcxxt = −σλ3 (ηkkkt + ηkxxt) ,
or, after rearranging:
ηck (ηkk − 1) kt + [ηcx (φ− 1) + ηckηkx] xt = −σλ3ηkkkt −σλ3ηkxxt.
Equating coeffi cients on xt in the left- and right-hand side of the equation, we have:
ηcx (φ− 1) + ηckηkx = −σλ3ηkx, (1)
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and equating coeffi cients on kt:
ηck (ηkk − 1) = −σλ3ηkk. (2)
Now recall that you also obtained the following equation in Homework 1:
kt+1 = λ1kt + λ4xt + (1 −λ1 −λ2 −λ4) ct.
Hence, substituting ct = ηckkt + ηcxxt into this equation, yields:
kt+1 = λ1kt + λ4xt + (1 −λ1 −λ2 −λ4) (ηckkt + ηcxxt),
or:
kt+1 = [λ1 + (1 −λ1 −λ2 −λ4) ηck] kt + [λ4 + (1 −λ1 −λ2 −λ4) ηcx] xt.
It follows that the elasticities ηkk and ηkx are given by:
ηkk = λ1 + (1 −λ1 −λ2 −λ4) ηck,
ηkx = λ4 + (1 −λ1 −λ2 −λ4) ηcx.
Substituting these expressions into equations (1) and (2) yields, respectively:
ηcx (φ− 1) + ηck [λ4 + (1 −λ1 −λ2 −λ4) ηcx] = −σλ3 [λ4 + (1 −λ1 −λ2 −λ4) ηcx] (3)
and:
ηck [λ1 + (1 −λ1 −λ2 −λ4) ηck] −ηck = −σλ3 [λ1 + (1 −λ1 −λ2 −λ4) ηck] . (4)
Therefore, from equation (4):
(1 −λ1 −λ2 −λ4) η2ck + [λ1 − 1 + σλ3 (1 −λ1 −λ2 −λ4)] ηck + σλ1λ3 = 0,
or:
Q2η 2 ck + Q1ηck + Q0 = 0,
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where we defined:
Q0 ≡ σλ1λ3,
Q1 ≡ λ1 − 1 + σλ3 (1 −λ1 −λ2 −λ4) ,
Q2 ≡ 1 −λ1 −λ2 −λ4.
As in the slides, the correct solution for ηck is:
ηck = 1
2Q2
( −Q1 −
√ Q21 − 4Q0Q2
)
since the other root would imply unstable dynamics.
Once we have the solution for ηck, we can compute ηcx from equation (3) as:
ηcx = − λ4 (σλ3 + ηck)
(1 −λ1 −λ2 −λ4) (σλ3 + ηck) + φ− 1 .
The solutions for output and return to investment follow immediately from the production
function and the log-linear investment return that you obtained in Homework 1:
yt = ηykkt with ηyk ≡ 1 −α,
and
rt = −λ3kt.
To find the solution for investment and compute its dynamics, note that
It = Yt −Ct −Xt.
Hence, log-linearizing this equation yields:
it = Ȳt K̄t
K̄t Īt yt −
C̄t K̄t
K̄t Īt ct −
X̄t Ȳt
Ȳt K̄t
K̄t Īt xt.
You know from Homework 1 that: Ȳt K̄t
= r + δ
1 −α ,
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and C̄t K̄t
=
( r + δ
1 −α
)( 1 −
X̄t Ȳt
) −g −δ.
Moreover, the law of motion for capital (Kt+1 = (1 −δ) Kt + It) immediately implies:
K̄t Īt
= 1
g + δ .
Therefore, it follows that:
it = r + δ
(1 −α) (g + δ) yt −
[ r + δ
(1 −α) (g + δ)
( 1 −
X̄t Ȳt
) − 1 ] ct −
X̄t Ȳt
[ r + δ
(1 −α) (g + δ)
] xt,
which you can use to compute the path of investment given the solutions for yt and ct and the
exogenous path of xt.
• Modify the Excel file posted in the Files section of the course Canvas appro-
priately to trace the responses of ct, kt+1, yt, rt, and it to a 1 percent positive
innovation to government spending at time t = 0 with persistence φ = 0.9 and no
other innovations in the following periods. Set the value of X̄t/Ȳt to 0.2. Include
your modified file with your answers. Highlight in red the modifications you make
to my original file.
Answer
See the Excel file included with this document.
• What is your intuition for how the variables respond to the government spending
shock? Explain the responses as clearly as you can.
Answer
The increase in government spending results in an immediate contraction of investment and con-
sumption, which fall already in period 0, and a gradual contraction of capital and output, which
begin falling in period 1, before all variables gradually return to their trend levels. Even if the
return to capital accumulation rises from period 1 on (consistent with the higher marginal prod-
uct of capital implied by a lower capital stock) before returning to trend, this is not suffi cient to
stimulate higher investment in response to the expansion of government spending. The response
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of consumption is smoother over time than that of investment because of the household’s desire to
smooth consumption fluctuations across periods captured by the Euler equation. Falling investment
and declining capital stock imply that government spending expansion results in a contraction of
output below trend until the economy returns to its long-run path.
Why do consumption and investment fall? Remember: Government spending coincides with
taxation in our exercise. If we denote government spending with GOVt, it is true that Yt =
Ct + It + GOVt, from which you may expect an expansionary impact of higher GOVt, but the
government’s budget constraint implies that GOVt = Xt, the lump-sum tax that household’s must
pay to finance government spending. Higher taxes reduce household wealth and induce the repre-
sentative household to respond by lowering its consumption and investment, ultimately resulting
in a contraction of the economy.
For your “fun,”notice what happens if you make the increase in government spending permanent
by setting φ = 1: In that case, a permanent, downward adjustment of consumption is the only thing
that happens, with no change in investment, capital, and output. Why? When the shock is not
permanent, the household is smoothing its negative consumption effect over time by reducing
investment, making it possible to sustain a smoother consumption profile with declining capital.
When the shock is permanent, it implies an immediate, permanent reduction of household wealth
with a consumption effect that cannot be smoothed by adjusting investment. The best thing to
do is simply to adjust consumption immediately and permanently downward, leaving investment,
capital, and output unchanged.
• Do you think the response of consumption would be different if the government
did not finance the increase in government spending today (taken to be time 0)
with the lump-sum tax, but instead financed by issuing debt to be repaid by
taxation in the future? What is your intuition for your answer?
Answer
No. The response would not be different. The reason is that, in this model, shifting taxation to
the future by using government debt would not alter household behavior because households would
recognize anyway that higher government spending today (at time 0) would mean higher taxes in
the future. This would induce the household to reduce its consumption (and investment) today in
response to the implied wealth reduction (i.e., it does not matter that taxes are increased in the
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future instead of today).
This result is a manifestation of something that is known as Ricardian Equivalence, which
happens when we have infinite-horizon, intertemporally optimizing, identical households subject
to lump-sum taxation. (The concept of Ricardian Equivalence was so named by Robert Barro of
Harvard University, who formalized thoughts by David Ricardo and Antonio de Viti de Marco.) In
this environment, as you learned from Homework 1, the household’s Euler equation is not affected
by taxation. Moreover, the household’s and the government’s intertemporal budget constraints,
and the aggregate resource constraint of the economy, are not affected by the timing of taxation
versus government debt. This implies that changes in the timing of taxation versus debt do not
matter. (Another way of saying this is the statement that government debt is not net wealth.)
How do we see all this? What follows is obviously material I did not expect that you would
know anything about, but that you should study as part of your work for this course.
The resource constraint of the economy is the statement that the economy’s production (Yt)
has to be equal to the economy’s absorption of resources (Ct + It + GOVt). You already saw that,
in absence of government debt, combining the household’s budget constraint
Ct + It + Xt = r̃tKt + wt
with the government’s budget constraint GOVt = Xt and the fact that total income must equal the
total of payments to factors of production (Yt = r̃tKt + wt) implies Yt = Ct + It + GOVt.
Now suppose that the government can finance excesses of spending over taxation by issuing
bonds that pay an interest rate rDt . Let Dt denote the government’s debt at the beginning of
period t. It follows that the government’s budget constraint becomes:
Dt+1 = ( 1 + rDt
) Dt + GOVt −Xt. (5)
The government begins the period with debt Dt, it pays the interest burden of this debt (rDt Dt), it
spends GOVt, and it receives the revenue from taxation Xt. This determines the debt with which
the government will begin period t + 1. When GOVt > Xt, the government is running a deficit–
often referred to as primary deficit because it does not include interest expenditure for previously
accumulated debt, rDt Dt. When GOVt < Xt, the government is running a (primary) surplus.
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The household’s budget constraint now is:
Ct + It + Xt + Dt+1 = (1 + r D t )Dt + r̃tKt + wt. (6)
In addition to labor income and capital income, the household now also holds the debt issued by
the government and receives its interest payments. Among the uses of the household’s resources,
we have the fact that the household buys the bonds that the government issues during period t,
i.e., Dt+1.
Now notice that we still have Yt = r̃tKt + wt. Hence, the household’s budget constraint (6) can
be rewritten as:
Ct + It + Xt + Dt+1 = (1 + r D t )Dt + Yt. (7)
Notice also that the government’s budget constraint (5) implies:
Xt = ( 1 + rDt
) Dt + GOVt −Dt+1. (8)
Substituting (8) into (7) immediately implies Yt = Ct+It+GOVt, proving thathaving introduced
government debt does not affect the economy’s aggregate resource constraint.
This is not suffi cient, however, to prove that the timing of taxation versus debt does not matter.
For that, we need to look at the government’s and the household’s intertemporal budget constraints.
What are these? The budget constraints we have introduced are period budget constraints. They
describe how the household’s and the government’s resources and their uses are connected within
any given period, determining next period’s starting position based on this period’s starting posi-
tion, resources, and uses. The intertemporal budget constraint captures how an agent’s resources
and their uses are connected (and constrained) over time, across the span of the agent’s lifetime.
To understand this concept, let us begin with the government’s budget constraint. In order to
simplify the algebra, I am going to assume that the interest rate on government bonds is constant:
rDt = r D in all periods. Allowing for time variation of the interest rate on government bonds would
not change the key conclusions, but it would make the algebra more complicated. With a constant
interest rate, the government period budget constraint becomes:
Dt+1 = ( 1 + rD
) Dt + GOVt −Xt. (9)
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Note that the same constraint must hold also in period t + 1, so that:
Dt+2 = ( 1 + rD
) Dt+1 + GOVt+1 −Xt+1. (10)
We can solve this equation for Dt+1 to obtain:
Dt+1 = Dt+2
1 + rD − GOVt+1 1 + rD
+ Xt+1
1 + rD . (11)
If we substitute this into (9) for Dt+1 and we rearrange, we have:
Dt+2 1 + rD
= ( 1 + rD
) Dt + GOVt +
GOVt+1 1 + rD
−Xt − Xt+1
1 + rD . (12)
Now, notice that equation (9) must hold also in t + 2, implying:
Dt+3 = ( 1 + rD
) Dt+2 + GOVt+2 −Xt+2,
or:
Dt+2 = Dt+3
1 + rD − GOVt+2 1 + rD
+ Xt+2
1 + rD .
And we can substitute this into (12) to obtain:
Dt+3
(1 + rD) 2
= ( 1 + rD
) Dt + GOVt +
GOVt+1 1 + rD
+ GOVt+2
(1 + rD) 2 −Xt −
Xt+1 1 + rD
− Xt+2
(1 + rD) 2 . (13)
If we do the same substitution again and again, until some time t + T , we get:
Dt+T
(1 + rD) T−1 =
( 1 + rD
) Dt + GOVt +
GOVt+1 1 + rD
+ GOVt+2
(1 + rD) 2
+ ... + GOVt+T−1
(1 + rD) T−1
−Xt − Xt+1
1 + rD −
Xt+2
(1 + rD) 2 − ...−
Xt+T−1
(1 + rD) T−1 ,
or, in compact form,:
Dt+T
(1 + rD) T−1 =
( 1 + rD
) Dt +
t+T−1∑ s=t
( 1
1 + rD
)s−t GOVs −
t+T−1∑ s=t
( 1
1 + rD
)s−t Xs. (14)
Now, our government has an infinite horizon. Hence, let us take the limit of both sides of (14)
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for T →∞:
lim T→∞
Dt+T
(1 + rD) T−1 =
( 1 + rD
) Dt +
∞∑ s=t
( 1
1 + rD
)s−t GOVs −
∞∑ s=t
( 1
1 + rD
)s−t Xs. (15)
Consider the limitat the left-handsideof this equation. Because, 1/ ( 1 + rD
) < 1, 1/
( 1 + rD
)T−1 must tend to 0 as T goes to infinite. It follows that, unless Dt+T (government debt) is exploding
to infinite at a rate faster than the interest rate, it must be:
lim T→∞
Dt+T
(1 + rD) T−1 = 0.
Hence, unless the government is letting its debt explode, it must be:
0 = ( 1 + rD
) Dt +
∞∑ s=t
( 1
1 + rD
)s−t GOVs −
∞∑ s=t
( 1
1 + rD
)s−t Xs. (16)
This is the government’s intertemporal budget constraint: The government’s initial debt plus
interestplus thegovernment’s totaldiscountedspendingover timemustbebalancedbythe totaldis-
countedrevenue fromtaxation. Note that this is equivalent to the statement that limT→∞Dt+T/ ( 1 + rD
)T−1 =
0, i.e., to the statement that debt is not exploding. (In an economy with growth, we would want
to think about debt as a ratio to GDP, and to spending and revenue from taxation as ratios to
GDP. Once we do that, it is the difference between interest rate and growth rate that matters for
intertemporal debt sustainability, but I am leaving growth out of the picture for now. Suggested
exercise for you: Re-do the algebra above thinking in terms of ratios to GDP, assuming that GDP
is growing at a constant rate g so that Yt+1 = (1 + g) Yt.) Government behavior must be such that
both the period budget constraint (9) and the intertemporal budget constraint (16) are satisfied in
each period.
Notice an implication of equation (16): It is not the timing of government spending versus
taxation that matters for the sustainability of government finances over time. It is the discounted
total of spending and taxation.
Now consider the household’s budget constraint, taking into account that Yt = r̃tKt +wt. With
constant interest rate on government debt, it is:
Ct + It + Xt + Dt+1 = (1 + r D)Dt + Yt. (17)
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This constraint implies:
Dt+1 = (1 + r D)Dt + Yt − (Ct + It + Xt) .
But this equation is similar to (9)! We can proceed exactly as we did for the government’s budget
constraint and obtain the household’s intertemporal budget constraint:
0 = ( 1 + rD
) Dt +
∞∑ s=t
( 1
1 + rD
)s−t Ys −
∞∑ s=t
( 1
1 + rD
)s−t (Cs + Is + Xs) . (18)
The household’s initial asset position (the portfolio of government bonds it begins period t with)
plus interest plus the total discounted income stream over the household’s infinite lifetime must
balance the sum of the total discounted uses of household resources for consumption, investment,
and tax payments. (In obtaining (18), we use limT→∞Dt+T/ ( 1 + rD
)T−1 = 0.) Note that also
here the timing of taxation does not matter: It is the discounted total over the household’s lifetime
that matters for the intertemporal constraint that must be satisfied in each period along side the
period constraint (18).
One more step: Equation (16) implies:
( 1 + rD
) Dt =
∞∑ s=t
( 1
1 + rD
)s−t Xs −
∞∑ s=t
( 1
1 + rD
)s−t GOVs.
Hence, substituting this into (18) yields:
0 =
∞∑ s=t
( 1
1 + rD
)s−t Xs −
∞∑ s=t
( 1
1 + rD
)s−t GOVs
+ ∞∑ s=t
( 1
1 + rD
)s−t Ys −
∞∑ s=t
( 1
1 + rD
)s−t (Cs + Is + Xs) .
The terms involving taxation cancel, and this equation can be rewritten as:
∞∑ s=t
( 1
1 + rD
)s−t Ys =
∞∑ s=t
( 1
1 + rD
)s−t (Cs + Is + GOVs) . (19)
This is just the intertemporal version of the aggregate resource constraint Yt = Ct + It + GOVt!
Not surprisingly, total production and total absorption must balance not just within the period but
also intertemporally. And again, the timing of taxation is nowhere to be seen.
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So, we have a model in which taxes do not show up in the Euler equation that determines
consumption versus investment decisions and in which intertemporal constraints show no impact
of the timing of debt versus taxation. These things together ensure Ricardian Equivalence.
Things would be different if taxation distorted household decisions instead of being lump sum.
Suppose that government spending is financed by taxing capital income at the rate XPt as I men-
tioned in the Homework 1 answers. The government budget constraint would become:
Dt+1 = ( 1 + rD
) Dt + GOVt −XPt r̃tKt,
and the household’s budget constraint would become:
Ct + It + Dt+1 = (1 + r D)Dt + (1 −XPt )r̃tKt + wt.
We would still have Yt = Ct + It + GOVt and its intertemporal counterpart (19), and we would still
have intertemporal constraints for government and household that balance initial positions plus
total discounted resources with total discounted uses. But now the timing of taxation XPt versus
debt would matter (i.e., Ricardian Equivalence would not hold). Why? Because the tax rate XPt
shows up in the Euler equation, and this is going to affect the time profile of capital accumulation,
consumption, and output. In this case, the government’s choices of when to use debt versus taxes
to finance spending would matter!
Finally, one more thing: Ricardian Equivalence is obviously the implication of an extreme,
unrealistic scenario. Besides distortionary taxation, there are many other reasons for it to break. An
important one is heterogeneity across agents in the economy. For instance, Ricardian Equivalence
does not hold in overlapping generations (OLG) models in which the economy is populated by
agents of different ages, unless we assume that everyone cares about the welfare of offsprings
enough that bequests ensure that everyone has the same assets (and therefore the model behaves
like one with identical agents). Another simple way to break Ricardian Equivalence is to assume
that a fraction of the agents in the economy behaves as those we modeled in the RBC setup, but
the remaining fraction does not accumulate assets and just lives “paycheck to paycheck.”(In macro
models, those agents are often referred to as hand-to-mouth consumers.) This will break Ricardian
Equivalence (by implying different asset positions between the two groups, just like the no-bequests
OLG models) and, in standard models, it will also ensure that government spending expansion is
expansionary if the fraction of hand-to-mouth consumers is suffi ciently large. (This was studied by
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N. Gregory Mankiw of Harvard University.)
Problem 2: Markups, Distortions, and Optimal Inflation in the New Keynesian
Model (30 Points)
The optimality condition for price setting by wholesaler j in the New Keynesian model with sticky
prices that we studied can be written as:
1
1 −ε
( pt (j)
Pt
)− ε ε−1
Yt + ε
ε− 1
( pt (j)
Pt
)−2ε−1 ε−1
mct (j) Yt −ψ (
pt (j)
pt−1 (j) − 1 )
Pt pt−1 (j)
+ βψ
1 + πt+1
( pt+1 (j)
pt (j) − 1 ) Pt+1 pt (j)
pt+1 (j)
pt (j)
= 0,
where I am using pt (j) to denote the price set by wholesaler j instead of Pjt, mct (j) denotes
the wholesaler’s marginal cost of production, Yt (instead of yt) denotes the output of the final
retail bundle (which has price Pt), and πt+1 is the inflation rate between t and t + 1 (πt+1 ≡
(Pt+1 −Pt) /Pt); ε > 1 is the flexible-price markup, ψ ≥ 0 is the scale parameter for the cost of
adjusting prices, and β is the representative household’s discount factor.
Now let us make one more change of notation and set:
ε = θ
θ − 1 .
If you replace ε with θ/ (θ − 1) everywhere in the equation above, you get:
−(θ − 1) ( pt (j)
Pt
)−θ Yt + θ
( pt (j)
Pt
)−θ−1 mct (j) Yt −ψ
( pt (j)
pt−1 (j) − 1 )
Pt pt−1 (j)
+ βψ
1 + πt+1
( pt+1 (j)
pt (j) − 1 ) Pt+1 pt (j)
pt+1 (j)
pt (j)
= 0. (20)
• Note that ( pt (j)
Pt
)−θ Yt = yt (j)
by virtue of the demand function facing the wholesaler (where yt (j) denotes the
wholesaler’s output). Use this and algebra to show that equation (20) can be
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rewritten as:
pt (j)
Pt =
θ
θ − 1
1 + ψ(θ − 1) yt (j)
( pt(j) pt−1(j)
− 1 )
Pt pt−1(j)
− β 1+πt+1
( pt+1(j) pt(j)
− 1 ) Pt+1 pt(j)
pt+1(j) pt(j)
−1
mct (j) .
Make sure to write all the steps in your answer.
Answer
Using (pt (j) /Pt) −θ Yt = yt (j), equation (20) becomes:
−(θ − 1) yt (j) + θ ( pt (j)
Pt
)−1 yt (j) mct (j) −ψ
( pt (j)
pt−1 (j) − 1 )
Pt pt−1 (j)
+ βψ
1 + πt+1
( pt+1 (j)
pt (j) − 1 ) Pt+1 pt (j)
pt+1 (j)
pt (j)
= 0,
which we can rearrange as:
θ
( pt (j)
Pt
)−1 yt (j) mct (j) = (θ − 1) yt (j) + ψ
( pt(j) pt−1(j)
− 1 )
Pt pt−1(j)
− β 1+πt+1
( pt+1(j) pt(j)
− 1 ) Pt+1 pt(j)
pt+1(j) pt(j)
,
or:
( pt (j)
Pt
)−1 =
( θ − 1 θ
) 1
mct (j) +
ψ
θyt (j) mct (j)
( pt(j) pt−1(j)
− 1 )
Pt pt−1(j)
− β 1+πt+1
( pt+1(j) pt(j)
− 1 ) Pt+1 pt(j)
pt+1(j) pt(j)
=
( θ − 1 θ
) 1
mct (j)
1 + ψ(θ − 1) yt (j)
( pt(j) pt−1(j)
− 1 )
Pt pt−1(j)
− β 1+πt+1
( pt+1(j) pt(j)
− 1 ) Pt+1 pt(j)
pt+1(j) pt(j)
.
Hence,
pt (j)
Pt =
θ
θ − 1
1 + ψ(θ − 1) yt (j)
( pt(j) pt−1(j)
− 1 )
Pt pt−1(j)
− β 1+πt+1
( pt+1(j) pt(j)
− 1 ) Pt+1 pt(j)
pt+1(j) pt(j)
−1
mct (j) .
This equation implies that the wholesaler sets the price of its good as a markup over marginal
cost. When prices are flexible (ψ = 0), the markup boils down to the constant θ/ (θ − 1) (or ε in
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the original notation). When prices are sticky (ψ > 0), the markup is time-varying and given by:
θ
θ − 1
{ 1 +
ψ
(θ − 1) yt (j)
[( pt (j)
pt−1 (j) − 1 )
Pt pt−1 (j)
− β
1 + πt+1
( pt+1 (j)
pt (j) − 1 ) Pt+1 pt (j)
pt+1 (j)
pt (j)
]}−1 .
This captures the fact that costs of price adjustment give the wholesalers an incentive to smooth
price changes across periods, absorbing the consequences of shocks in part by letting the markup
component of prices vary. Let us denote the expression for the markup charged by wholesaler j
with µt (j):
µt (j) ≡ θ
θ − 1
{ 1 +
ψ
(θ − 1) yt (j)
[( pt (j)
pt−1 (j) − 1 )
Pt pt−1 (j)
− β
1 + πt+1
( pt+1 (j)
pt (j) − 1 ) Pt+1 pt (j)
pt+1 (j)
pt (j)
]}−1 .
(21)
Therefore: pt (j)
Pt = µt (j) mct (j) .
Now, if labor is the only factor of production, and technology is such that yt(j) = Ztnt(j), the
wholesaler’s marginal cost is equal to the real wage (wt) divided by productivity:
mct (j) = wt Zt .
• Why is this the expression of marginal cost?
1 Answer
The production function yt(j) = Ztnt(j) implies that producing one unit of output (yt(j) = 1)
requires nt(j) = 1/Zt units of labor. Since each unit of labor is paid the real wage wt, this implies
that the cost of producing one unit of output (marginal cost) is wt/Zt.
Note that this implies that marginal cost is identical across all the wholesalers. Optimal price
setting becomes: pt (j)
Pt = µt (j)
wt Zt .
However, now consider that all the wholesalers in the economy are symmetric to each other. Other
than the fact that each one of them produces a wholesaler-specific good over which it has monopoly
power, they are all identical. Hence, they will all set the same markup µt (j) = µt and the same
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price pt (j) = pt. We thus have: pt Pt
= µtwt. (22)
Finally, consider the production function of the representative retailer:
Yt =
[∫ 1 0 yt (i)
θ−1 θ di
] θ θ−1
.
Since every wholesaler produces the same amount yt (i) = yt, it follows that Yt = yt (the amount
of the final bundle produced by the retailer is equal to the amount of output produced by every
wholesaler). Given the demand function for a wholesaler’s output:
yt =
( pt Pt
)−θ Yt,
it is immediate to prove formally that it has to be Pt = pt.
• Write this proof.
Answer
Imposing Yt = yt in the demand function yt = (pt/Pt) −θ Yt implies 1 = (pt/Pt)
−θ, from which it
follows immediately that it has to be Pt = pt.
Given Pt = pt, it follows that the optimal price setting equation (22) implies:
wt = Zt µt .
Now, the first-order condition for optimal labor supply implies:
U1−n (ct, 1 −nt) Uc (ct, 1 −nt)
= wt,
where ct is consumption in period t, and 1 −nt is leisure.
With flexible prices and perfect competition, it would be wt = Zt (the marginal product of
labor), and the amount of labor employed would be determined by:
U1−n (ct, 1 −nt) Uc (ct, 1 −nt)
= Zt.
In our economy with monopoly power and sticky prices, the real wage is lower than the marginal
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product of labor, and the amount of labor employed by the economy is such that:
U1−n (ct, 1 −nt) Uc (ct, 1 −nt)
= Zt µt ,
• What are the distortions that affect this condition?
Answer
There are two sources of distortion (two distortions) that impinge on this condition: One is
monopoly power. In and of itself, monopoly power implies the presence of a constant wedge
1/µ = (θ − 1) /θ in the condition that would otherwise equate the marginal rate of substitution
between leisure and consumption to the marginal product of labor. The other distortion is price
stickiness, which implies that the wedge is time varying and equal to 1/µt, with µt determined by
equation (21) once you have imposed symmetry in it:
µt = θ
θ − 1
{ 1 +
ψ
(θ − 1) yt
[( Pt Pt−1
− 1 )
Pt Pt−1
− β
1 + πt+1
( Pt+1 Pt − 1 ) Pt+1 Pt
Pt+1 Pt
]}−1 =
θ
θ − 1
{ 1 +
ψ
(θ − 1) yt [πt (1 + πt) −βπt+1 (1 + πt+1)]
}−1 , (23)
where we used the definition of the inflation rate πt ≡ (Pt −Pt−1) /Pt−1.
• Suppose you are the central banker and you can commit the economy to a choice
of inflation rate. What inflation rate would you choose and why?
Answer
The central banker would want to commit to a policy of zero inflation: πt = 0 in all periods.
Monetary policy cannot do anything directly about the distortion caused by monopoly power.
Given this, the best thing that the central bank can do is to remove completely the impact of
price stickiness by implementing a policy of zero inflation. When inflation is zero and prices are
constant, the economy behaves as if prices were flexible (because, when prices are constant, firms
do not incur the costs of adjusting prices). The sticky-price distortion still exists, but the central
bank undoes its impact by causing the economy to mimic the flexible price equilibrium: With
πt = πt+1 = 0, equation (23) immediately implies µ = θ/ (θ − 1) and marginal cost is constant at
the level wt/Zt = 1/µ = (θ − 1) /θ (or the real wage is determined by wt = (θ − 1) Zt/θ), which is
what happens with flexible prices.
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Problem 3: The New Keynesian Phillips Curve (20 Points)
Once you impose symmetry across wholesalers in the optimality condition for wholesaler price
setting in the New Keynesian model we studied, you have the following relation between real
marginal cost (mct), output (yt), and inflation in periods t and t + 1 (respectively, πt and πt+1):
1
1 −ε (1 −εmct) yt −ψπt (1 + πt) + βψπt+1 (1 + πt+1) = 0,
where the notation is the same as in Problem 2. Sanjay Chugh refers to this equation as the New
Keynesian Phillips Curve (NKPC). Most scholars think of the NKPC as the equation that you find
after log-linearizing this equation and imposing equilibrium conditions on mct. This question asks
you to explore the implications of this equation for the behavior of markups.
Start by recalling from Problem 2 that, in equilibrium, it has to be:
mct = 1
µt ,
where µt is the markup charged by every wholesaler. Therefore, the NKPC equation above can be
rewritten as: 1
1 −ε
( 1 −
ε
µt
) yt −ψπt (1 + πt) + βψπt+1 (1 + πt+1) = 0. (24)
Denote the steady-state levels of variables by dropping the time subscript.
• Suppose that the steady-state inflation rate is zero. Use equation (24) to show
that the steady-state markup µ is such that µ = ε.
Answer
The steady-state version of (24) is:
1
1 −ε
( 1 −
ε
µ
) y −ψπ (1 + π) (1 −β) = 0.
Setting π = 0 implies 1
1 −ε
( 1 −
ε
µ
) y = 0,
from which it follows immediately that it has to be µ = ε.
In addition to assuming that π = 0, assume also that the steady-state level of output is equal
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to 1: y = 1. Use hats to denote percentage deviations from the steady state, so that:
ŷt ≡ dyt y
= dyt,
µ̂t ≡ dµt µ
= dµt ε .
In the case of inflation, focus on gross inflation, but note, however, that our assumption that π = 0
implies:
π̂t ≡ d (1 + πt)
1 + π = d (1 + πt) = dπt.
• Log-linearize equation (24) and write the resulting equation with only µ̂t on the
left-hand side as a linear function of only π̂t and π̂t+1 on the right-hand side.
Answer
Applying the differential operator to (24) yields:
1
1 −ε
[ −d ( ε
µt
)] y +
1
1 −ε
( 1 −
ε
µ
) dyt −ψdπt (1 + π) −−ψπd (1 + πt)
+βψdπt+1 (1 + π) + βψπd (1 + πt+1)
= 0,
or:
1
1 −ε εdµt µ2
y − 1
1 −ε
( 1 −
ε
µ
) dyt −ψdπt (1 + π) −−ψπd (1 + πt)
+βψdπt+1 (1 + π) + βψπd (1 + πt+1)
= 0. (25)
Recalling that µ = ε, y = 1, π = 0, and using the definitions of µ̂t and π̂t, equation (25)
becomes: 1
1 −ε µ̂t −ψπ̂t + βψπ̂t+1 = 0,
or: 1
1 −ε µ̂t = ψ (π̂t −βπ̂t+1) .
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Hence,
µ̂t = −(ε− 1) ψ (π̂t −βπ̂t+1) .
• What happens to the markup if current inflation increases?
Answer
The markup decreases if current inflation increases.
This is what motivates central bankers acting under discretion (instead of commitment) to
tend to use monetary policy expansion and produce inflation: The presence of monopoly power
causes too little labor to be employed (which implies that the real wage is below the marginal
product of labor, as you saw in Problem 2). A monetary expansion that causes inflation erodes the
markup below its flexible-price level (θ/ (θ − 1)) causing more labor to be employed and output
to rise above its ineffi cient steady-state level . A central bank acting under discretion faces the
temptation to exploit the fact that price stickiness implies that monetary policy affects the markup
in order to try and ameliorate the impact of the monopoly power distortion in this fashion. The
problem is that agents would come to expect the central bank’s behavior, with consequences for
expected future inflation and, ultimately, for the inflation we would see arise in the equilibrium of
this interaction today. Policymaking under commitment (as we assumed at the end of Problem 2)
removes the temptation and its consequences. The consequences of commitment (and rules) versus
discretion were first studied by Finn Kydland (U.C. Santa Barbara) and Edward Prescott (Arizona
State University)– the same scholars who developed the RBC model– and Robert Barro (Harvard
University) and David Gordon (Clemson University).
• Given the observation that real wages in the United States are pro-cyclical or
a-cyclical in response to demand expansions (i.e., if aggregate demand rises, real
wages rise or stay flat), can you explain why a countercyclical markup was a de-
sirable feature of New Keynesian models? (Hint: Remember that the real wage,
wt, is equal to the marginal product of labor– MPL– under perfect competition,
but this is no longer the case when we have monopolistic competition, and, as you
reviewed in Problem 2, the wedge between wt and MPLt created by the markup
becomes time-varying when prices are sticky.)
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Answer
A model in which the real wage is tied to the marginal product of labor implies that the real wage
falls whenever the economy employs more labor and the marginal productivity of labor declines.
This is counterfactual, given the observation of pro-cyclical or a-cyclical real wages. A constant
markup does not resolve the problem because it simply implies
wt = 1
µ MPLt,
where MPLt is the marginal product of labor. The real wage is still tied to moving in the same
direction as MPLt. But a time-varying, counter-cyclical markup as implied by the New Keynesian
model helps:
wt = 1
µt MPLt.
Suppose there is an expansion (say, because of a monetary or fiscal policy expansion) that causes
more labor to be employed and MPLt to fall. The fact that µt declines (because the inflation
associated with the expansion of the economy causes the markup to do so) implies that 1/µt rises.
Hence, an expansion of the economy and of the amount of labor employed can coexist with a real
wage that does not move or moves upward in pro-cyclical fashion.
Having this result and mechanism in the model was among the key drivers of early New Key-
nesian model development.
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