SUMMER TERM 2023
ONLINE EXAMINATION Advanced Macroeconomic Theory
Answer all THREE questions.
In cases where a student answers more questions than requested by the examination
rubric, the policy of the Economics Department is that the student’s first set of answers up
to the required number will be the ones that count (not the best answers). All remaining
answers will be ignored.
TURN OVER
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Question 1 - 20 Points
Consider the DMP model studied in class. Let u be the number of unemployed
workers and v the number of vacant firms. The number of new matches, m, is given by
an aggregate Cobb-Douglas, constant returns to scale, matching function:
m = Auαv1−α,
where A > 0 and α ∈ (0, 1).
Households are risk neutral, discount future consumption at rate β and are either
employed or unemployed. When unemployed, the household receives a compensation,
equivalent to b(z) unit of consumption, which can depend on productivity z. When em-
ployed, the worker receives a wage w. Wages are determined through Nash bargaining,
ϕ is the bargaining power of the worker.
On the production side, when a firm is matched with a worker, it produces output z y
– with z aggregate productivity – each period and receives profits π = z y − w. When a
firm is not matched with a worker it must pay a cost of κ for each period it is vacant and
looking for a worker. Every period, productive matches are destroyed with (exogenous)
probability δ. Assume free entry, which implies that the expected value of posting a new
vacancy is zero. Firms discount future profits at rate 1/(1 + r).
1.A Denote by ϵb(z) = z b′(z)/b(z) the elasticity of the compensation with respect to
productivity. Show that the elasticity of labor market tightness θ = v/u with
respect to productivity is given by:
z
θ
dθ
dz
= zy − ϵb(z)b(z)
zy − b(z)
ϕAθ1−α + δ + r
ϕAθ1−α + α (δ + r) .
1.B Discuss the "Shimer Puzzle" in the case ϵb(z) = 0, in the case ϵb(z) ̸= 0.
We now want to investigate how b varies with aggregate productivity. Our first step
is to build a microfoundation for b(z). All households are part of a large family which
pools resource and redistribute consumption among them. The family problem is defined
by a Bellman equation:
Vt(et, Zt, at) = sup
Cet ,C
u
t
etU(Cet , Nt) + (1− et)U(Cut , 0) + βEt (Vt+1(et+1, Zt+1, at+1))
s.t. etC
e
t + (1− et)Cut + at+1 = wtetNt + (1− et)Bt + (1 + rt)at +Πt
where Zt denotes aggregate productivity, et is the fraction of employed households, Cet ,
Cut consumption of employed and unemployed households, Bt the level of unemployment
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benefits, Πt firm profits, at is asset, rt the interest rate and Nt hours per worker. Note
that households take Nt as given. Employment evolves as:
et+1 = (1− δt)et + Aθ1−αt (1− et).
1.C Derive the first order conditions for Cet and Cut . Define St(et, Zt, at) = ∂eVt(et, Zt, at)/
∂cU(Cet , Nt), the marginal benefit – in consumption units – of having one more em-
ployed worker in the family. Show that:
St(et, Zt, at) =wtNt −
(
Bt + Cet − Cut +
U(Cut , 0)− U(Cet , Nt)
∂cU(Cet , Nt)
)
+
(
1− δt − Aθ1−αt
)
Et
(
β∂cU(Cet+1, Nt+1)
∂cU(Cet , Nt)
St+1(et+1, Zt+1, at+1)
)
.
1.D How would you define the compensation b in this model? Assume that:
U(C,N) = ln(C)− χ 11 + ψN
1+ψ, χ, ψ ≥ 0.
How does the compensation vary with Bt, Cet , Cut , Nt? Do you expect the compen-
sation to be pro-cyclical (larger when Zt is larger)?
1.E Using this model, explain how you would measure b and ϵb in the data. You are
able to measure ϵb(z) = 1.01, write a short essay (one or two paragraphs) discussing
the implications of this result.
Question 2 - 60 Points
Consider an economy populated by a mass 1 of households indexed by i, a government
and a representative firm. Households value streams of consumption and labor supply
according to:
E
( ∞∑
t=0
βtu(ci,t, ℓi,t)
)
,
with β ∈ (0, 1) is the discount factor and
u(c, ℓ) = (c− ℓ
2/2)1−γ
1− γ ,
with γ > 0. Assume that there is a maximum number of hours ℓ¯ that households can
supply (ℓt ≤ ℓ¯). Households may save using government bonds denoted bt or claims on
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capital at and have to pay a tax τk,t on interest rate payments (capital tax) and an income
tax τw,t on labor income. The household’s consumption is constrained by
ci,t + ki,t+1 + bi,t+1 = (1 + (1− τk,t)rbt )bi,t + (1 + (1− τk,t)rt)ai,t + (1− τw,t)wt si,tℓi,t,
where rbt is the interest rate on bonds, rt is the interest rate on capital claims and wt the
competitive wage to be determined in equilibrium. The household’s labor productivity,
st ∈ {0 < s1 < ... < sn}, is iid with Pr(si,t = sj) = π(sj) for all t ≥ 0. Assume that∑
s2iπ(si) = 1. Finally, assume an ad hoc borrowing constraint on bonds and capital
claims: bi,t ≥ 0, ai,t ≥ 0.
On the supply side, there is an aggregate production function whose arguments are
the average levels of capital and efficiency unit of labor:
F (Kt, Nt) = Kαt N1−αt ,
with α ∈ (0, 1). Capital depreciates at rate δ ∈ (0, 1).
Finally, the government consumes an amount Gt of goods in period t. The flow of
public consumption is valued according to:
∞∑
t=0
βtU(Gt),
with U(·) increasing and concave. The government issues new debt in every period,
receives the proceeds of the tax and pays for public consumption. The government
budget constraint is
Gt + rbtBt = Bt+1 −Bt + τk,t
(
rtKt + rbtBt
)
+ τw,twtNt,
with Bt the amount of government debt in period t.
2.A Argue that if Bt > 0, Kt > 0 we have rbt = rt in equilibrium and that the problem
of the agent can be written in terms of a single asset a˜i,t = ai,t + bi,t. From now on
we assume rbt = rt.
2.B Assume rt, wt τw,t and τk,t are constant, given by r, w τw and τk. Write down the
Bellman equation that describes the household’s problem. Derive the labor supply
decision of households in terms of w, τw and labor productivity. Derive the Euler
equation.
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2.C Derive the “natural debt limit” for a household in this economy. What is the
intuition behind the “natural debt limit”?
2.D Assume that taxes τk, τw and level of public consumption G are exogenously given.
Derive the steady state level of efficient labor demanded by firms and the steady
state level of efficient labor supplied by households. Derive an expression for the
steady state wage and equilibrium level of efficient labor as a function of K and
1− τw.
2.E Assume that taxes τk, τw and level of public consumption G are exogenously given.
Derive the steady state level of assetsK+B supplied by the firm and the government
as a function of r, δ, τk, τw and G.
2.F The taxes τk, τw and level of public consumption G are exogenously given. Define
a “stationary competitive equilibrium” for this economy.
2.G Outline a pseudo code for computing this equilibrium (Recall that pseudo code is
an algorithm describing precisely how to implement the solution on the computer.
It should be unambiguous to a human reader, but is not necessarily written in a
specific computer programming language.)
Now assume that the government chooses Gt, Bt, τw,t and τk,t to maximize:
∫ 1
0
E
( ∞∑
t=0
βtu(ci,t, ℓi,t)
)
di+
∞∑
t=0
βtU(Gt).
The government still has to satisfy its budget constraint, households and firms optimize
given the realized prices {rt, wt}t≥0 and taxes and markets clear at all t. We assume that
an optimum of the tax problem exists.
2.H Consider an equilibrium for the path of taxes, prices, government consumption,
households and firms decisions. We want to build a new equilibrium with a dif-
ferent sequence of government consumption. Consider the following deviation: at
t0 the government decreases Gt0 and Bt0+1 by ϵ (∆Gt0 = ∆Bt0+1 = −ϵ), Bt is left
unchanged at all other dates. The government adjusts taxes to absorb all deviations
in prices rt∆τk,t = (1− τk,t)∆rt, wt∆τw,t = (1− τw,t)∆wt.
(i) Argue that households’ decisions are left unchanged. (ii) How does capital
adjust to satisfy asset market clearing? How do prices rt, wt adjust? (iii) Con-
clude that Gt0+1 adjusts to satisfy the government budget constraint(and Gt is left
unchanged at t ̸= t0, t0 + 1) and the adjustment is given by:
∆Gt0+1 = (1+rt0+1)ϵ+Kαt0+1N
1−α
t0+1
α(1 + ϵ
Kt0+1
)α−1
+ (1− α)
(
1 + ϵ
Kt0+1
)α
− 1
 .
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2.I Using the previous question, show that the optimal path of government consumption
satisfies:
U ′(Gt) = (1 + rt+1)βU ′(Gt+1).
Assume that under the optimal sequence {Gt, τk,t, τw,t}t≥0, the economy converges
toward a a stationary equilibrium with G∗ > 0 in the long run. What is the value
of the pre-tax interest rate r∗ in the long run optimum? Comment.
2.J Suppose that in the optimum, the capital tax converges towards a long run value τ ∗k .
Using a result from Lecture 3, argue that we necessarily have 1+ (1− τ ∗k )r∗ < 1/β.
Comment on the optimal long run capital tax. Explain graphically how you expect
the long run capital tax to change when the borrowing constraints are relaxed.
Question 3 - 20 Points
Consider an economy populated by a mass one of dynasties indexed by i. In each
dynasty, households live for one period, inherit the wealth left by their parents and are
then replaced by their children. The problem of the nth household in dynasty i is given
by:
sup
ci,n,ai,n+1
ln(ci,n) + βln(ai,n+1)
s.t. ai,n+1 = Ri,n (ai,n + yi,n − ci,n)
ai,0 > 0,
with ai,n inherited wealth, yi,n > 0 income, ci,n consumption and Ri,n > 0 the gross rate
of return to wealth, 0 < β < 1.
3.A Show that the evolution of cash on hand zi,n = ai,n + yi,n is given by:
zi,n+1 =
β
1 + βRi,nzi,n + yi,n+1.
3.B Assume that Ri,n is constant, equal to R, across dynasties and generations, that yi,n
is iid across dynasties and generations with E(yi,n) < ∞. Argue that a stationary
distribution of cash-on-hands with finite mean can only exist if β/(1 + β)R < 1.
Assume that β/(1+β)R < 1, under what conditions can the stationary distribution
of cash-on-hands have power law tails?
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3.C Assume that both Ri,n and yi,n are iid across dynasties and generations with
E(yi,n) < ∞. Argue that a stationary distribution of cash-on-hands with finite
mean can only exist if β/(1 + β)E(Ri,n) < 1. Assume that β/(1 + β)E(Ri,n) < 1,
under what conditions can the stationary distribution of cash-on-hands have power
law tails?
3.D You collect data on the wealth and income of the top 20%, top 2% and top 0.2%
of households in this economy:
top 20% top 2% top 0.2%
Wealth (in millions) 10 7.40 5.48
Income (in thousands) 200 63 20
Is it more likely that the cash-on-hands accumulation process satisfies the assump-
tions of 3.B or 3.C?
3.E Suppose that the government imposes a non linear income tax: households’ income
is now given by yi,n−τ (yi,n). Discuss whether such a tax can be effective in reducing
upper tail inequality in wealth in this economy.