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.