Advanced marcoeconomics test

ECON 401

Advanced Macroeconomics

Midterm Exam

Fabio Ghironi University of Washington

April 29, 2020

Instructions:

You have 5 hours to work on this exam. It is worth 100 points, contributing to your overall

score for the course as described in the Syllabus. You may consult all course materials and standard

Internet resources while working on the exam, but your work must be original and you may not

solicit or obtain assistance from or provide assistance to other people for any specific content of the

exam. Activities considered cheating include copying or closely paraphrasing content from websites

and discussing exam questions with other students. All exams will be checked for originality and

copied content, and anyone found cheating will be assigned a zero score for the exam. Read

carefully each step of each problem before you jump into working on it and do not panic if you

cannot complete everything. The exam is intended also to stretch your knowledge by forcing you

to use the tools and information you have acquired to think about some things we have not talked

about in class. I want to see how you think about those things based on what you learned. In

Problems 2 and 3 below, if there is an item that you cannot complete, just take the result in that

item for granted and move on to the next one.

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

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and of X̄t/Ȳt.

• 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.

• What is your intuition for how the variables respond to the government spending

shock? Explain the responses as clearly as you can.

• 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?

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:

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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.

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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. (1)

• 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 (1) can be

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.

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

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 .

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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?

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

price pt (j) = pt. We thus have: pt Pt

= µtwt. (2)

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.

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Given Pt = pt, it follows that the optimal price setting equation (2) 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

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?

• 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?

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):

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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

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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. (3)

Denote the steady-state levels of variables by dropping the time subscript.

• Suppose that the steady-state inflation rate is zero. Use equation (3) to show

that the steady-state markup µ is such that µ = ε.

In addition to assuming that π = 0, assume also that the steady-state level of output is equal

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 (3) 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.

• What happens to the markup if current inflation increases?

• 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

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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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