Assignments for Advanced Macroeconomics
– ECONOMIC FLUCTUATIONS –
summer semester 2017, this version: June 8, 2017
1 Different shocks in the RBC model
1.1 TFP shocks
Consider the following baseline RBC model:
max E0
{Ct ,Lt }
∞
X
β t [ln Ct + η(1 − Lt )]
t=0
s.t. Kt+1 = Bt Ktα Lt1−α + (1 − δ)Kt − Ct
γ
Bt = Bt−1
et
K0 = given, B0 = 1
Kt ≥ 0, lim β t E0 (µt Kt ) = 0
t→∞
a) Derive the first-order conditions and provide an economic interpretation.
One of the main problems of the standard RBC approach lies in the idea that business
cycle fluctuations are driven predominantly by shocks to productivity, implying that there
are both periods of technical progress (t > 0) as well as periods of technical regress
(t < 0) (King and Rebelo, 1995, p. 930). Therefore, the next two exercises will introduce
two alternative sources of economic fluctuations within the RBC setup.
1.2 Fiscal policy shocks
Instead of TFP-shocks consider fiscal policy shocks in the form of an exogenous and
stochastic tax rate:
max E0
{Ct ,Lt }
∞
X
β t [ln Ct + η(1 − Lt )]
t=0
1
s.t. Kt+1 = (1 − τt )Ktα L1−α
+ (1 − δ)Kt − Ct + Tt
t
(1 + τt − τ̄ ) = (1 + τt−1 − τ̄ )γ et
K0 = given
Kt ≥ 0, lim β t E0 (µt Kt ) = 0,
t→∞
is a lump sum transfer, viewed
where τt is a linear income tax rate and Tt = τt Ktα L1−α
t
as being exogenous from the perspective of the individual agent. The innovation term
is specified as t ∼ N (0, σ). It is assumed that the realization of the innovation term
is always sufficiently close to zero such that it always holds that 0 ≤ τt ≤ 1. It can be
verified that the steady state tax rate is equal to the model parameter 0 ≤ τ̄ ≤ 1.
a) Derive the first-order conditions and provide an economic interpretation.
1.3 Oil price shocks
Consider now a small-open, producer-consumer economy described by the following economic structure:
max E0
{Ct ,Lt }
∞
X
β t [ln Ct + η(1 − Lt )]
t=0
s.t. Kt+1 = Ktα1 Lαt 2 Mt1−α1 −α2 + (1 − δ)Kt − Ct − Pt Mt
γ
Pt = Pt−1
et
K0 = given, P0 = given
Kt ≥ 0, lim β t E0 (µt Kt ) = 0,
t→∞
where Mt is crude oil, Pt the exogenously given price of oil, 0 < γ < 1, t ∼ N (0, σ ), and
α1 , α2 ∈ (0, 1), α1 + α2 < 1.
a) Derive the first-order conditions and provide an economic interpretation.
2 Price setting a la Calvo (1983) - 2-Period Example
Consider a firm that exists for two periods only (period t = 0 and t = 1) and acts
under monopolistic competition. It produces a differentiated consumption good cj,t . The
2
demand schedule and the production technology are as follows
Demand Schedule:
Production technology:
cj,t =
pj,t
Pt
−θ
Yt
(1)
cj,t = At Lj,t
(2)
where cj,t denotes the amount of consumption good j at time t ∈ {0, 1} (i.e. time is
discrete), pj,t denotes the price of cj,t , Pt a price index (and the price of the numeraire
good Yt ), θ > 1 a constant parameter, Yt household income, At a stochastic technology
parameter, and Lj,t the amount of labor employed by firm j.
We assume that there are nominal price rigidities according to Calvo (1983). That is, each
period there is a constant probability 0 ≤ ω ≤ 1 that the firm is not allowed to adjust its
goods price. Assuming that the firm can reset its goods price at t = 0, the firm’s problem
may then be written as
maxE0
{pj,0 }
pj,0 cj,0
− M C0 cj,0 + βω
P0
pj,0 cj,1
− M C1 cj,1
P1
s.t. eq. (1), eq. (2),
where 0 < β < 1 denotes the discount factor, M Ct the real marginal (equal to average)
costs at time t, and E0 expectations conditional on information available at time t = 0.
a) Let
Wt
Pt
denote the real wage. Describe the real marginal costs of cj,t -production in
terms of the numeraire good Yt .
b) Assume the firm is allowed to set its goods price at t = 0. Determine this optimal
goods price pj,0 .
c) Discuss the similarities and the differences between the Calvo (1983) approach and
the ”convex price adjustment cost setup”.
3
3 A baseline New-Keynesian model
Consider a closed economy with mass one of identical households. Households supply
labor, purchase goods for consumption, and hold bonds. Intertemporal utility is given by
E0
∞
X
t=0
h
χ i
β t ln Ct − Nt2 ,
2
(3)
where β is the subjective discount factor, Ct is consumption, and χ is a utility parameter
reflecting the degree of disutility derived from working Nt . Households maximize eq. (3)
subject to the budget constraint
Ct +
Bt
Wt
Bt−1
≤
Nt + (1 + it−1 )
+ Πt ,
Pt
Pt
Pt
where Bt is the nominal holding of bonds, Pt is the aggregate price level, Wt is the nominal
wage, it is the nominal interest rate, and Πt are real profits from the intermediate goods
sector. The composite consumption good Ct is given by a CES index
Z
1
θ−1
θ
cjt dj
Ct =
θ
θ−1
,
0
where θ > 1.
a) Derive the demand for cjt .
b) Derive the Euler equation and the intra-temporal optimality condition for leisure
and consumption.
The firm sector consists of mass one of identical firms. Each firm hires labor to produce
a differentiated good that is sold in monopolistically competitive markets under the production function cjt = At Njt , where Njt is the labor demanded by firm j in period t and
At is a stochastic TFP variable with Et (At+1 ) = 1. Each firm sets the price of the good
it produces, but that is not possible in every period. The probability that a firm cannot
adjust its price in the next period is given by ω (Calvo pricing).
c) Derive the optimal price p∗jt .
d) Derive the New-Keynesian Phillips Curve.
4
4 Search and matching model
Exercises marked with * will not be discussed during the practice session, but solutions will be available
online.
Consider a closed economy. There are infinitely many identical households on the unit
interval which supply labor and capital and purchase goods for consumption. Further,
there are infinitely many identical firms on the unit interval which employ labor and
capital in order to produce a homogeneous good. Goods and capital markets are cleared
by a Walrasian auctioneer. Labor markets are determined by a search-and-matching
approach.
4.1 Model Derivation
a) Explain the basic intuition of a search-and-matching model. What are the main
differences to a RBC model with a Walrasian labor market?
Consider the representative household. Its decisions are governed by the following constrained optimization problem
max
Ct ,Nt ,Kt+1
E0
∞
X
β t log(Ct )
t=0
s.t. Ct + It = Wt Nt + (1 − Nt )b + Rt Kt + Dt
Nt = ρNt−1 + (1 − ρNt−1 )f (θt )
Kt+1 = (1 − δ)Kt + It ,
where β is a discount factor, Ct is consumption, It is investment into capital, Wt is the
wage rate, Nt is the employment rate, b are unemployment benefits, Rt is the gross interest
rate, Kt is the capital stock, Dt are firm dividends paid to households, ρ is the labor match
survival rate, f (θt ) is the job-finding rate, and δ is the capital depreciation rate.
b) Set up the household Lagrangian and derive the first-order conditions. Give an
economic interpretation of the first-order conditions and the shadow variables.
5
Consider the representative firm. Its decisions are governed by the following constrained
optimization problem
max
Nt ,Kt ,vt
E0
∞
X
t=0
βt
λt
[Yt − Wt Nt − Rt Kt − κvt ]
λ0
s.t. Yt = Zt Ktα Nt1−α
Nt = ρNt−1 + q(θt )vt ,
where λt is the shadow variable w.r.t. the household budget constraint, Yt is production,
κ are vacancy posting costs, vt is the vacancy rate, Zt is total factor productivity, α is a
production function parameter, and q(θt ) is the vacancy-filling rate.
*c) Set up the firm Lagrangian and derive the first-order conditions. Give an economic
interpretation of the first-order conditions and the shadow variables.
Due to the matching frictions there are both unemployed workers who would like to work
and unfilled vacancies. This is, M RS ≤ Wt ≤ M P L. Households and Firms bargain
over the wage in order to maximize their respective surplus. The maximization problem
is posed as a Nash bargain and is given by
Wt = arg max V (Wt )η ψ(Wt )1−η ,
where V (Wt ) is the household value function, Jt is the firm value function, and 0 < η < 1
is the bargaining weight of the household.
d) Derive the first-order condition of the wage bargaining problem. Solve for the wage
rate Wt by using the value functions of the firm and household. Explain the elements
that determine the wage rate.
If you have not solved 1.1 and 1.2, use the following value functions and free-entry condition:
Vt = Wt − b + ρEt βt,t+1 ((1 − f (θt+1 ))Vt+1 )
ψt = (1 − α)
Yt
− Wt + ρEt [βt,t+1 ψt+1 ]
Nt
κ
ψt =
q(θt )
*e) Under which assumption does the search-and-matching model shrink to a standard
6
RBC model? Hint: Employment Nt must become a jump variable.
4.2 Comparative Statics
a) Draw the job-creation and wage curve in a θ − W graph. Explain the economic intuition of the curves. Explain the change in equilibrium when total factor productivity
A increases. Draw the adjustments in the graph.
b) Draw the job-creation and Beveridge curve in a u − v graph. Explain the economic
intuition of the curves. How does the equilibrium change when the job-survival rate
ρ increases. Draw the adjustments in the graph.
7