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Problem Set 3
Professor: Kenza Benhima
Assistant: Martina Insam
Business Cycles, Spring 2012
Due May 2nd
1. Nominal rigidities and the neutrality of money
Consider an economy with three goods -money, output and labor- and
three agents: an aggregate firm, an aggregate household and the government.
There are two markets where output and labor are exchanged against money
at price P and wage W , respectively. We assume that transactions are
equal to the minimum of supply and demand. Y and N are respectively the
output and labor transactions. Y = F (N ), with F strictly concave. The
firm maximizes profits Π = P Y − W N , which are then redistributed to the
household.
The household has an initial endowment of money M . It consumes C,
works N , saves M and its budget constraint is:
PC + M = WN + Π + M − PT
where T are the real taxes levied by the government. The household has
the following utility function:
M
− V (N )
α log(C) + (1 − α) log
P
V (N ) is the disutility of labor, with V 0 > 0 and V 00 > 0. The government
issues money through transfers to the household, so its budget constraint is:
M − M = −P T
1.1. The Walrasian Equilibrium.
1.1.1. Derive the labor demand N d and output supply Y s by the firm as a
function of the real wage W/P .
1.1.2. Derive the output demand C by the household as a function of M
and P and the labor supply N s as a function of C and the real wage W/P .
1.1.3. The Walrasian equilibrium is defined by N = N d = N s and Y =
Y s = C. Show that this equilibrium can be characterized by the following
equations:
αF 0 (N ) = F (N )V 0 (N )
Y = F (N )
P =
αM
(1 − α)Y + αT
W
= F 0 (N )
P
1.1.4. Show that money is neutral.
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1.2. The fixprice-fixwage Equilibria. We will now deviate from the Walrasian equilibrium by assuming that the prices P and W are completely rigid
and can differ from their market-clearing values derived above. This economy has three possible regimes:
• Keynesian unemployment, with excess supply of both output and
labor
• Classical unemployment, with excess supply of labor and excess demand for goods
• Suppressed inflation, with excess demand for both labor and output
We focus successively on the Keynesian unemployment and the Classical
unemployment regimes (we will not consider suppressed inflation).
1.2.1. Keynesian unemployment. In this regime, the labor market is in excess supply, which means that N s > N d . In that case, the transaction is
equal to the minimum, which is N d . This implies that the household faces a
d
constraint on his labor transaction N = N . It therefore solves the following
program:
M
d
max α log(C) + (1 − α) log
− V (N )
C,M
P
d
PC + M = WN + Π + M − PT
e as a function of M/P and, using
(a) Derive the demand for output C
the fact that profits are redistributed to the household and the government
budget constraint, show that:
M
e
+Y
C=α
P
e transactions Y are
Because of excess supply on the good market (Y s > C),
e
equal to output demand C:
M
e
Y =C=α
+Y
P
(b) Derive the equilibrium transaction on the good market:
α M
Y =
1−α P
The good market is in excess supply, so the firm faces the constraint Y =
e Labor demand therefore does not solve for the firm’s profit maximization,
C.
but must satisfy:
e=Y
F (N d ) = C
(c) Is money neutral? Describe the effect of money on output, labor and
consumption. Explain the mechanism.
1.2.2. Classical unemployment. Now we study Classical unemployment, where
there is an excess supply of labor and an excess demand for goods.
(a) Does the firm deviate from its Walrasian plan? Explain why. The
labor demand and output supply by firms. What are the equilibrium transactions?
(b) Do nominal rigidities generate monetary non-neutrality in this regime?
Why?
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2. Imperfect competition
Here we take the same framework and introduce monopolistic competition. For this, we suppose that there is not one good but a continuum of
goods of length 1. These goods are indexed by i ∈ [0, 1]. Each good i is
produced by a different firm, also indexed by i, which redistribute
its profits
R1
to the household. Aggregate profits are now: Π = 0 πi di. C is now a
consumption basket that depends on the consumptions Ci of goods i:
γ
Z 1 γ−1 γ−1
γ
C=
Ci di
0
where γ is the elasticity of substitution between goods. Denote by X the
amount of resources that households devote to consumption. It should satisfy:
Z
1
X=
Pi Ci di
0
where Pi is the price of good i.
We first derive the optimal demand for good i, given total resources devoted to consumption P C, then solve for the rest of household’s program.
2.1. The household maximizes:
Z
max
Ci
1
γ−1
γ
Ci
0
γ
γ−1
di
subject to:
Z
1
Pi Ci di
X=
0
Denote by λ the multiplier of the constraint. Derive the demand function
for Ci in terms of Pi , C and the multiplier λ.
2.2. Show how the multiplier λ is related to the following aggregate price
index
1
Z 1
1−γ
1−γ
P =
Pi
0
2.3. Write the demand function in terms of the price of good i; Pi , the
aggregate price level, P ; and the aggregate consumption C.
2.4. Show that X = P C.
The rest of the household’s program can now be expressed as follows:
M
max α log(C) + (1 − α) log
− V (N )
C,M
P
PC + M = WN + Π + M − PT
which is similar to the problem we solved in 1.1, except that P is now the
general price index defined above.
2.5. Using the results of question 1.1.2, express the demand for good i by
the household as a function of M , P and Pi .