Download MACROECONOMIC THEORY I Martin Boileau Final Examination 1

ECON7020: MACROECONOMIC THEORY I
Martin Boileau
Final Examination
1. Consider the following economy populated by a representative consumer, a representative ¯rm, and a government. The representative consumer chooses consumption c and
leisure 1 ¡ n to maximize her lifetime utility
1
X
t=0
¯ t [ln(ct ) + µ ln(1 ¡ nt )]:
The representative ¯rm chooses labor n to maximize pro¯ts. The production technology
is
yt = Ant :
The government issues bonds and collects tax revenues to provide a ¯xed amount of government expenditures g in every period and service its outstanding debt. Its period budget
constraint is
g + (1 + rt )bt = bt+1 + Tt :
a) Find the government's intertemporal budget constraint.
b) Assume that taxes are levied in a lump-sum manner from the consumer. Solve for a
competitive equilibrium.
c) To stimulate output and employment, the government enacts a one-period tax break
to the consumer by issuing more bond: dTt < 0 and dbt+1 > 0. Discuss the impact of
this policy on output and employment.
d) Assume that taxes are levied as income tax, Tt = ¿t wt nt where ¿t is the tax rate and
wt is the real wage rate. Solve for a competitive equilibrium.
e) To stimulate output and employment, the government enacts a one-period tax break
to the consumer by issuing more bond: d¿t < 0 and dbt+1 > 0. Discuss the impact of
this policy on output and employment.
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2. Consider the following 2-period asset-pricing problem in general equilibrium. The
economy is populated by a representative consumer who wishes to solve the following
problem:
( 1
)
X h
a 2i
t
max E0
¯ ct ¡ ct
2
t=0
subject to
c0 + p0 a1 + q0 s1 = y0
c1 = d1 a1 + b1 s1 :
That is, the consumer can buy claims to one-period lived trees. There are two such trees.
In period 0, she purchases an amount a1 at price p0 of claims to the period 1 stochastic
output d1 of the ¯rst tree and an amount s1 at price q0 of claims to the period 1 stochastic
output b1 of the second tree.
a) Is the consumer risk averse? Does she have a precautionnary savings motive?
b) Find the pricing equation for both trees.
c) Assume that d1 is distributed with mean d¹ and variance ¾2 and that b1 = °d1 , ° < 1.
Show that p0 > q0 . Explain. (Hint: in general equilibrium, c1 = d1 + b1 .)
d) Assume that d1 is distributed with mean d¹ and variance ¾2 and that b1 is distributed
wiht mean ¹b = d¹ and variance ´¾2 , ´ > 1. Finally, assume that the correlation between
d1 and b1 is Corr(d1 ; b1 ) = 0. Show that p0 > q0 . Explain
e) Assume that the second period budget constraint now is
c1 = y1 + d1 a1 + b1 s1 ;
such that c1 = y1 +d1 +b1 in equilibrium. Assume that the endowment y1 is distributed
with mean 0 and variance ¾2 . Also assume that both d1 and b1 are distributed with
mean d¹ and variance ¾2 . Finally, assume that the correlations between y1 , d1 , and b1
are as follows: Corr(y1 ; d1 ) = ¡1, Corr(y1 ; b1 ) = 0, and Corr(d1 ; b1 ) = 0. Show that
p0 > q0 . Explain.
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3. Imagine the following two-period overlapping generations model with no population
growth. Consider the choice of an agent. When young, the agent inelastically supplies one
unit of labor for which she receives a wage rate of wt . The agent also chooses consumption
c1t and savings st to maximize her expected lifetime utility:
½
¾
ln(c1t ) + ¯Et ln(c2t+1 ) :
When old, the agent consumes c2t+1 ¯nanced from her savings. The technology to produce
goods is
yt = kt® (zt nt )1¡® ;
where kt is the stock of capital, zt is the level of technology, and nt is the labor input. The
level of technology is
zt = (1 + g)t exp(²t );
where g is the rate of technology growth and ²t is iidN (0; ¾2 ). Finally, capital accumulation
is
kt+1 = xt + (1 ¡ ±)kt ;
where xt is investment and where there is full depreciation (± = 1).
a) What is the deterministic rate of growth of aggregate output?
b) Solve for a competitive equilibrium.
c) Using your solution, discuss the business cycle implications of this economy. These
implications refer to the variances and covariances of the deviations of per capita
aggregate variables from their trend growth.
d) Imagine an alternate economy that is identical, except that it is populated by a representative consumer who supplies labor inelastically and has the following expected
lifetime utility:
(1
)
X
E0
¯ t ln(ct ) :
t=0
Solve for a competitive equilibrium.
e) Discuss the business cycle implications of this alternate economy and compare these
implications to your previous results.
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