Macroeconomic Theory
Exam 2
Instructions: Write your answers to each question on separate paper. You must show your
work for all derivations and computations for full credit. The point value of each question is
indicated below. Partial credit is assigned based upon the completeness of the entire question
answered not on the proportion of sub-parts answered.
1. (36 points) General Equilibrium: Consider the representative household, who chooses
a path of consumption and leisure over an infinite horizon, {ct+s, lt+s}∞
s=0, to maximize
the following objective function:
V =
X∞
s=0
β
su(ct+s, lt+s)
where u(ct
, lt) is a well-behaved utility function, and β is a discount factor. The household
faces the following real budget constraint each period:
at = (1 + rt)at−1 + wtnt − ct − Tt
where at
is real wealth, rt
is the real interest rate, wt
is the real wage rate, nt
is labor
supply, and Tt
is a lump-sum tax. The household also faces a unitary time endowment
which holds each period:
1 = lt + nt
Also consider the representative firm, who chooses a path of capital and labor input over
an infinite horizon, {kt+1+s, nt+s}∞
s=0 to maximize the following real profit function:
P rof =
X∞
s=0
1
1 + rt+s
s
f(kt+s, nt+s) − invt+s − wt+snt+s
where f(kt
, nt) is a well-behaved production function, rt
is the real interest rate, wt
is the
real wage rate, and k0 is given. For any period t, net investment is defined as:
invt = kt+1 − (1 − δ)kt
where δ is the rate of capital depreciation.
Finally, each period the government purchases an amount of real goods and services equal
to real wage income tax revenue:
gt = Tt
so that government savings is always zero.
(a) Derive the household’s intertemporal and intratemporal optimality conditions in
terms of the general utility function u(ct
, lt).
(b) Derive the firm’s intertemporal and intratemporal optimality conditions in terms of
the general production function f(kt
, nt).
(c) Using the optimality conditions obtained from parts (a) and (b), derive the equilibrium conditions for the financial market, labor market, and goods market.
