- (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.
