(a) In the Solow model, the steady-state capital stock associated with the golden rule rate
of saving is a decreasing function of the depreciation rate of capital.
(b) In the Diamond-Mortensen-Pissarides model of the labour market, the fundamental
theorems of welfare do not apply.
(c) The equity premium puzzle would be less severe if, in the data, consumption growth and
excess returns of stocks over bonds were less strongly correlated, everything else equal.
(d) The Hodrick-Prescott filter can be used to take a trend with constant growth out of a
data series.
(e) Consider a labour market matching function m = µuγ
v
1−γ
, where m is the number of
new matches, u is the unemployment rate, v is the number of job vacancies, and γ ∈ (0, 1)
and µ > 0 are parameters. Given this matching function, the job finding rate is negatively
related to the vacancy filling rate.
(f) The Kaldor growth facts imply that, on average, the aggregate capital stock grows at
the same rate as aggregate output.
1
Question 2. Labour supply (25 points)
Consider a static model of a household with preferences given by:
U(c, h) = c
1−σ − 1
1 − σ
−
κ
1 + ψ
h
1+ψ
,
where σ, κ, ψ > 0 are preference parameters. The household chooses consumption (c) and
hours worked (h) to maximize U(c, h), subject to the following budget constraint:
c = wh,
where w is the wage rate per hour worked.
(a) Derive the first-order conditions for consumption and hours worked (labour supply).
(b) Derive the Frisch elasticity of labour supply in this model.
(c) Show that when σ = 1 (so that U(c, h) = ln c −
κ
1+ψ
h
1+ψ
), the number of hours worked
chosen by the household does not react to a change in the wage rate w.
(d) Suppose now that ψ = 1 and σ = 2. If the wage increases by 1 percent, by how much
does labour supply change? Decompose your answer into the underlying effects.
(e) Derive an analytical expression for elasticity of consumption with respect to the wage
(for general σ, κ, ψ > 0). Show that this elasticity strictly exceeds 1 if and only if σ < 1 and
explain intuitively why this is the case.
2
Question 3. Consumption of durables and non-durables (15 points)
Consider an infinitely-lived household which consumes both durables and non-durables. Expected discounted utility at time zero is given by:
E0
X∞
t=0
β
tu (ct
, dt)
where ct denotes consumption of non-durables in period t, dt
is the stock of durables owned
by the household in period t, β ∈ (0, 1) is the household’s subjective discount factor, and Et
is the expectations operator conditional upon information available in period t. The price of
both durables and non-durables is one. The budget constraint of the household in period t
is given by:
ct + dt = yt + (1 − δ) dt−1, t = 0, 1, 2, 3..
where δ ∈ (0, 1) is the depreciation rate of durables and yt > 0 is an exogenous income
variable, which follows a stochastic but stationary process. In each period t, the household
chooses ct and dt such as to maximize the expected present value of lifetime utility.
(a) Derive the first-order optimality conditions associated with the household’s decision
problem.
Suppose now that the utility function is given by u (ct
, dt) = ct + γ ln dt
.
2
(b) Derive an analytical expression for dt
in terms of the model parameters and show that
dt remains constant over time (given the parameters).
(c) Show that dt
is increasing in β and explain intuitively why this is the case.
2For simplicity, ignore any non-negativity constraint on ct.
3
Question 4. Pricing inflation risk (20 points)
Consider an infinitely-lived, representative household with preferences given by
U0 = E0
X∞
t=0
β
t

c
1−ρ
t − 1
1 − ρ

,
where ct denotes consumption at time t, β ∈ (0, 1) is the discount factor and ρ > 0 is the
coefficient of risk aversion. The household can invest in two types of one-period bonds: (i)
nominal bonds, denoted Bn
t
, which offer a nominal interest rate r
n
t
, (ii) inflation-linked
bonds, denoted Bi
t
, which offer a nominal interest rate r
i
t
(1 + πt+1), where πt+1 =
Pt+1
Pt
− 1 is
the inflation rate in period t+1, and where Pt
is the nominal price level in period t. Inflation
is uncertain, i.e. it evolves stochastically over time.
The budget constraint of the household in period t, in nominal terms, is given by:
Ptct + B
n
t + B
i
t = Ptyt +

1 + r
n
t−1


B
n
t−1 +

1 + r
i
t−1


(1 + πt) B
i
t−1
,
where yt
is an exogenous and stochastic income flow. In each period, the household chooses
ct
,Bn
t
and Bi
t
in order to maximize the utility objective given above, subject to the budget
constraint. Let b
n
t ≡ Bn
t
/Pt and b
i
t ≡ Bi
t
/Pt denote the real values of nominal and inflationlinked bonds, respectively.
(a) Re-write the budget constraint in real terms (as opposed to nominal terms).
(b) Derive the Euler equations for the two types of bonds. Which of the two bonds would
the household consider to be risk-free?
Suppose now that in equilibrium it holds that yt = ct and that the central bank sets a
monetary policy according to a rule which targets inflation as a function of output growth:
πt = γ

yt
yt−1
− 1

, where γ > 0 is a policy parameter.
(c) Consider the risk premium formula derived in class for the excess return of equity, but
apply it instead to the two bonds considered above. Which of the two bonds earns a higher
ex-ante expected return?
(d) Discuss the intuition for your answer under (c).