Advanced marcoeconomics test
ECON 401
Advanced Macroeconomics
Midterm Exam
Fabio Ghironi University of Washington
April 29, 2022
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.
Problem 1: The RBC Model with Endogenous Labor Supply (50 Points)
The figure in the next page shows the responses to a one-percent innovation to technology at time 0
in the basic RBC model with fixed labor supply with the following parameter values: g = 0.005,β =
0.99,γ = σ = 1,r = − logβ +γg,α = 0.667,δ = 0.025, and φ = 0.95 (this value of persistence may
seem very high to you, but it is actually a conventional value that is often used in quantitative
applications of models with technology shocks).
The log-linearized version of the RBC model with endogenous labor supply in the slides boils
down to the following equations:
kt+1 ≈ λ1kt + λ2 (at + nt)+(1− λ1 − λ2)ct,
Et (ct+1 − ct) ≈ λ3Et (at+1 + nt+1 − kt+1) ,
nt ≈ µ [(1− α)kt + αat − ct] ,
at = φat−1 + εt,
where all the variables, parameters, and coeffi cients are defined in the slides. These four equa-
tions allow us to solve for the dynamics of capital, consumption, and employment in response to
technology shocks. The solution takes the form:
kt+1 = ηkkkt + ηkaat,
ct = ηckkt + ηcaat,
nt = ηnkkt + ηnaat,
with:
at = φat−1 + ε,
where the expressions of the η’s are in the slides.
Given the solution for capital, consumption, and employment, we can solve for other variables
of interest: output:
yt = α(at + nt)+(1− α)kt,
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Basic RBC Model Impulse Response Example
a k c y r i
the return to investment:
rt+1 ≈ λ3 (at+1 + nt+1 − kt+1) ,
investment:
i = r + δ
(1− α)(g + δ) y −
[ r + δ
(1− α)(g + δ) −1 ] c,
and the real wage:
ωt = αat +(1− α)(kt − nt) .
• Use the solution and the equations for other variables of interest to set up an
Excel file that allows you to compute and show in a figure the impulse responses of
capital, consumption, employment, output, the return to investment, investment,
and the real wage to a one-percent innovation to technology at time 0. Assume
that the economy was at trend up to and including period −1 and use the same
values of parameters as those indicated above for the model with fixed labor
supply. In addition, assume that γn = σn = 1 and _ N = 1/3 (these assumptions are
suffi cient for you to calculate the value of µ). Submit your Excel file in addition
to your written answers to this problem and those below.
• Explain your intuitions for the responses in the figure that your file generates
and for any noticeable difference in the responses relative to those in the figure
for the model with fixed labor supply.
Problem 2: Indifference Across Assets (20 Points)
Suppose the representative consumer can invest in three assets: nominal bonds, stocks, and physical
capital. The consumer supplies one unit of labor inelastically in each period, lives in a world of
perfect foresight, and wants to maximize the intertemporal utility function:
∞∑ s=t
βs−tu(Cs) ,
where Cs is consumption in period s and β is a discount factor strictly between 0 and 1.
The budget constraint in period t is:
PtCt + PtIt + Bt+1 + Vtxt+1 = Wt + RtKt +(1+ it)Bt +(Vt + Dt)xt.
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Going right to left in this equation: The consumer begins period t with stock holdings xt, receives
nominal dividends (Dt) and the value of selling the stocks at the nominal price Vt; the consumer
begins the period with nominal bond holdings Bt and receives the nominal interest rate it on these
bonds; the consumer begins the period with capital Kt and receives income from renting this capital
to firms at the nominal rental rate Rt; and the consumer receives nominal wage income Wt. The
consumer then uses these resources to buy stocks and bonds to be carried into t+1 (xt+1 and Bt+1),
to buy investment goods (It), and consumption (Ct). Pt is the nominal price of consumption and
investment (the underlying assumption here is that the same good or bundle of goods can be used
for consumption or investment). There is one such budget constraint in each period.
Capital evolves according to the law of motion:
Kt+1 = (1− δ)Kt + It,
where δ is a depreciation rate between 0 and 1. A similar law of motion holds in each period.
• Find the Euler equations for the consumer’s optimal decisions about capital ac-
cumulation, bond holdings, and stock holdings. Use your favorite method (La-
grangian or constraint substitution), but do the math.
• Show that when all three Euler equations hold the consumer is indifferent between
the three assets (capital, bonds, stocks).
Problem 3: Tax Policy and Monetary Policy in the New Keynesian Model (30
Points)
Suppose the representative consumer maximizes the intertemporal utility function:
Et
∞∑ s=t
βs−tu(Cs,1− Ns) ,
where Et denotes the expectation conditional on information available at time t, Cs is consumption
in period s, Ns is labor effort supplied in the same period, the duration of the period has been
normalized to 1, and β is a discount factor strictly between 0 and 1.
The budget constraint in period t is:
PtCt + Bt+1 = (1− τt)WtNt +(1+ it)Bt + Tt,
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where Pt is the consumer price level, Wt is the nominal wage, Bt denotes nominal bonds at the
beginning of period t, it is the nominal interest rate, τt is the rate of taxation of labor income, and
Tt is a lump-sum transfer with which the government rebates the revenue from taxation back to
the consumer. The tax rate τt and the transfer Tt are the only instruments of government policy in
this exercise. Note that τt can be negative, in which case τt is a subsidy rate and Tt is a lump-sum
tax that finances the subsidy. Both the tax (or subsidy) rate τt and the lump-sum transfer (or
tax) Tt are taken as given by the consumer. There is one budget constraint like that above in each
period.
• Write the Euler equation for the consumer’s choice between consumption and
bond accumulation and the optimality condition for labor supply. Explain the
reasoning that led you to write these equations as you did.
Assume that consumption consists of a bundle of differentiated goods produced by firms that
operate under monopolistic competition:
Ct =
[∫ 1 0 ct (j)
θ−1 θ dj
] θ θ−1
,
where ct (j) is consumption of the differentiated product j (produced by firm j), there is a continuum
between 0 and 1 of such firms, and θ > 1 is the elasticity of substitution between their products.
• Let pt (j) denote the price of product j. Given the amount of consumption Ct determined by the Euler equation, what is the expression for the consumer’s
demand of product j in period t? Explain this expression intuitively.
Suppose that firm j’s production function is:
yt (j) = ZtNt (j) ,
where yt (j) is the output of good j, Nt (j) is the amount of labor employed by firm j, and Zt is
exogenous labor productivity.
Given this production function, firm j’s marginal cost is wt/Zt, where wt is the real wage
(Wt/Pt).
• Why is wt/Zt the expression for marginal cost?
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Suppose that firms set prices subject to a cost of price adjustment equal to:
ψ
2
( pt (j)
pt−1 (j) −1 )2
,
with ψ ≥ 0. This cost is measured in units of the consumption basket. It implies that the price of
product j will be set as a time-varying markup over (nominal) marginal cost:
pt (j) = µt (j)Pt wt Zt ,
where µt (j) > 1. If prices are flexible, the markup is constant and equal to θ/(θ −1).
Imposing symmetry across firms in equilibrium, pt (j) = Pt and µt (j) = µt. It follows that:
wt = 1
µt Zt.
There is a wedge between the real wage and labor productivity equal to the reciprocal of the
markup.
• Use the last equation and the first-order condition for the optimal choice of labor
supply to find the level of the labor income tax rate τt that removes the impact of
monopoly power and nominal rigidity on the labor market (i.e., the labor income
tax rate that delivers labor market effi ciency).
• What is the expression of this tax rate if prices are flexible?
• If you did things right, you found that the labor income tax rate that removes
the effects of distortions on the labor market is negative. This means that, in
order to restore labor market effi ciency, the government must actually subsidize
labor (instead of taxing it). In other words, the government must tax leisure.
What is your intuition for this result?
• Suppose that the government is indeed setting the labor income tax rate at the
level that implies labor market effi ciency in each period. Does this mean that
there is nothing left to do for monetary policy to improve the outcome of markets?
Why? If there is anything left for the central bank to do, what should it do?
• Whereas monetary policy can be adjusted frequently, it is diffi cult for govern-
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ments to adjust tax rates frequently because it involves a legislative process.
Suppose now that this is the case here and that the government sets the tax rate
in every period at the constant level that delivers labor market effi ciency under
flexible prices (in other words, the tax rate cannot deviate from this level even if
the economy is outside the steady state and all the other variables are changing
over time). What inflation rate should the central bank set? Why?
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