ECON0057: Advanced Macroeconomic Theory

An unemployed worker receives each period an offer to work for wage w. This initial
wage offer is drawn from a distribution F(w) that is constant over time (entry-level wages
are stationary); successive drawings across periods are independently and identically
distributed. In the first period, an accepted job pays wt = w. The wage then grows
according to wt = φ
tw after t periods in the job. Assume φ > 1, that is, wages increase
with tenure. The worker’s objective function is to maximize
E
X∞
t=0
β
t
yt
, 0 < β < 1,
and yt = wt
if the worker is employed and yt = c if the worker is unemployed, where c
is unemployment compensation. Let v(w) be the optimal value of the objective function
for an unemployed worker who has offer w in hand. Assume that φβ < 1.
1.A Write down the value of the objective function if the worker accepts wage w.
1.B Write down the Bellman equation representing the worker’s problem.
1.C Show that the optimal policy is to set a reservation wage. Graphically illustrate
the reservation wage and provide intuition for how it is determined.
1.D Characterize the reservation wage as a function of β,φ, and F(w). Hint: The
condition should not feature v(w).
1.E Argue that, if two economies differ only in the growth rate of wages of employed
workers, say φ1 > φ2 , the economy with the higher growth rate has the smaller
reservation wage. Assume that φiβ < 1, i = 1, 2.
1.F Write a brief note (3 -4 paragraphs) summarizing the implications for the design of
unemployment insurance. Your intended audience should be a government minister
(unlikely to have studied economics at a graduate level).
Question 2
Consider the following incomplete markets economy. The economy is populated by
a mass 1 of households. In each period, household i receives an endowment e
i
t
. e
i
t
takes
two values eh > el > 0 and follows a Markov process defined by the transition probability
2π(ej
| ei) with:
π(eh | eh) > π(el
| eh) > 0 and π(eh | eh) + π(el
| eh) = 1
π(el
| el) > π(eh | el) > 0 and π(eh | el) + π(el
| el) = 1.
Households may save/borrow using a single asset denoted ai,t, which are in zero net
supply. Assume an ad-hoc borrowing constraint ai,t ≥ −φ.
Households streams of consumption according to
E0
X∞
t=0
β
tu(c
i
t
)
!
,
s.t. ai
t+1 = R(a
i
t + e
i
t − c
i
t
)
a
i
t ≥ −φ
where β ∈ (0, 1) is the discount factor; R is the gross interest rate to be determined in
equilibrium and u(c) is a CRRA utility function u(c) = c
1−γ/(1 − γ), with γ > 0.
The current state of a household is described by their asset holdings and endowment
(a, s). Define the unconditional stationary distribution of (a, s) pairs over households as
λ
∗
(a, s).
2.A Write down the Bellman equation that describes the household’s problem and derive
the Euler equation. (Be explicit about the expectations that household form and
use the relevant Markov transition matrices.)
2.B Derive the “natural borrowing constraint” φ
∗
(R) for a household in this economy
when the interest rate is R. How should you choose φ for the problem to be well
defined? What is the intuition behind the “natural borrowing constraint”?
In the rest of the problem, I denote V (a, ei), c(a, ei) and a(a, ei) the value function,
optimal consumption and optimal savings of the agent (with a(a, ei) = R(a+ei−c(a, ei)).
You can assume without proof that V is increasing, strictly concave, bounded and
differentiable in a and that consumption and savings are continuous in a.
2.C Show that c(a, ei) and a(a, ei) are non-decreasing in a.
2.D Show that ∂aV (a, eh) ≤ ∂aV (a, el) and that c(a, eh) ≥ c(a, el).
(Hint. Start from V0(a, ei) increasing, concave and continuous in a with ∂aV0(a, eh) ≤
3∂aV0(a, el). Show by induction that ∂aVn(a, eh) ≤ ∂aVn(a, el) where Vn+1 = T Vn,
where T is the operator defined by your Bellman equation in 2A. Call an(a, ei) the
optimal saving function when the continuation value is Vn and use Theorem 9.9 in
SLP to conclude.)
2.E Fix R such that βR < 1 and φ < φ∗
(R). Show that there exists a
∗
l with a
∗
l > −φ
such that a(a, el) = −φ if and only if a ≤ a
∗
l
. Show that a(a, el) < a for all a > −φ.
2.F Show that for all R with βR < 1 there exists a
∗
such that a(a
∗
, eh) = a
∗
(Hint. Note
that if this is not the case we have a(a, eh) > a for all a, use the previous question
to then show that this implies a(a, eh) > a(a, el) and the proof in lecture 4 to derive
a contradiction ). In addition show that there exists R∗
h
such that for R ≥ R∗
h
,
a(a, eh) > −φ for all a ≥ −φ.
2.G Draw the policy function a(a, eh) and a(a, el) when 1/β > R ≥ R∗
h
and discuss the
evolution of consumption.
2.H Define a “stationary competitive equilibrium” for this economy.
2.I Draw a diagram in the (r, A) space with demand for and supply of assets (label all
parts of the diagram and explain the behavior of asset demand as R goes to 1/β).
Mark the equilibrium and explain how it depends on the debt limit φ.
2.J Suppose for this question only that φ = 0. Explain why, in a stationary equilibrium,
we necessarily have a(a, ei) = 0. Show that in a stationary equilibrium we could
have any R ≤ R0 and determine R0
.
2.K 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.)
From now on suppose that there is an aggregate shock Z taking two values Z0 and Z1.
Z follows a Markov process defined by π
Z
(Zj
|Zi). When Z = Z0 the endowment evolves
according to π
0
(ej
| ei
, Z0) = π(ej
| ei) as defined above. When Z = Z1, the endowment
evolves according to π
1
(ej
| ei
, Z1) with π
1
(eh | eh, Z1) < π0
(eh | eh, Z0) and π
1
(eh | el
, Z1) <
π
0
(eh | el
, Z0), and π
1
(eh | eh, Z1) > π1
(el
| eh, Z1), π
1
(el
| el
, Z1) > π1
(eh | el
, Z1).
2.L Define a recursive equilibrium for this economy. Discuss the difficulty in computing
a solution to the value function once we add aggregate shocks. Would the method
proposed by Krussell and Smith to approximate the solution work in this case?
2.M Assume that φ = 0. What is the interest rate when Z = Z0? When Z = Z1?
Discuss.
4Question 3
Assume that utility is exponential, u(ct) = −exp(−θct) with θ > 0, and the discount
rate is equal to the interest rate, Rβ = 1. Consider the consumption-savings problem
E0
X
T
t=0
β
tu(ct)
s.t. : R(at + yt − ct) = at+1
aT +1 ≥ 0 a.s.
where at
is asset and income yt
follows the following process:
yt = y
p
t + ut
y
p
t = y
p
t−1 + vt
where {u0, …, uT , v0, …, vT } are independent and normally distributed: ut ∼ N (0, σu
t
) and
vt ∼ N (0, σv
t
). Note that the variance of ut and vt are time dependent. Remember that
for a normally distributed random variable x ∼ N (0, σ), E(exp(x)) = exp(σ
2/2).
3.A What is the role of the condition aT +1 ≥ 0? Derive consumption at T, cT , as a
function of aT and yT .
3.B Derive the Euler equation and explain the intuition behind it.
3.C Show that the consumption policy function is ct(at
, y
p
t
, ut) = y
p
t + αt(at + ut) − κt
where αt = ρt(R − 1)/R, ρt = (1 − 1/RT −t+1)
−1 and κt
is deterministic. Express κt
in terms of θ, R, σ
v
t+s
and σ
u
t+s
.
3.D Discuss how consumption responds to transitory and permanent shocks.
3.E How does consumption depend on at? Is it consistent with empirical evidence?
3.F Discuss how consumption depends on risk ({σ
v
t }0≤t≤T , {σ
u
t }0≤t≤T ).
Now, suppose that you have a dataset in which you observe the joint distribution of
income, consumption and household age in every year. As before, the stochastic process
for income for an individual i in birth cohort k:
yit = y
p
it + uit for i ∈ k,
y
p
it = y
p
i,t−1 + vit,
5where ui,t and vi,t are normally distributed with mean 0 and independent across individuals.
As before {u
i
0
, …, ui
T
, vi
0
, …, vi
T } are independent. All individuals live for T period and they
have exponential utility with a common parameter θ.
3.G Assume that the variances of the shocks are the same any period for all individuals
in any birth cohort k (but remember that these variances are not constant over
time and are not necessarily equal across cohorts). Define varkt(u) as the crosssection variance of the transitory shocks for cohort k in period t and varkt(v) the
corresponding permanent shocks. Derive an expression for conosumption growth
ct+1 − ct as a function of vt+1, ut+1, ut
, κt+1 and κt
. Then, derive expressions for
i. The growth in the cross-sectional variance of income for cohort k: ∆varkt(y)
ii. The growth in the cross-sectional variance of consumption for cohort k: ∆varkt(c)
iii. The growth in the cross-sectional covariance of consumption and income for
cohort k: ∆covkt(c, y)
3.H In this question only, assume that the parameter θ is not constant across individuals
(but is distributed independently from the permanent and transitory shocks across
individuals). What would be the growth in the cross-sectional variance of consumption
for cohort k in that case? Discuss.
3.I Write a brief note (3 to 4 paragraphs) summarizing what we learn from Figure 1.
Your intended audience should be a government minister (unlikely to have studied
economics at the graduate level).