theoretical

Consider a two-period production economy that consists of a representative consumer, a
representative firm, and a government. In the current period the representative consumer
is endowed with h hours of time, to be allocated between leisure (`) and work (Ns
). If
the consumer works he earns a real wage w. In addition the consumer consumes goods
C, pays lump-sum taxes T to the government, and receives dividend income from the
representative firm π. The same is true in the future period (denote all future variables
with a prime). The consumer can save/lend from one period to the other at real interest
rate r. The objective of the consumer is to maximize utility from consumption and leisure
over the two periods. The representative firm produces output in the current period (Y )
according to the constant returns to scale production function Y = zF(K, N), where z
and K are current TFP and capital stock respectively which are exogenous. The same
production function with primes holds in the future period. The representative firm
chooses labor in each period (N, N0
) as well as investment (I), which determines future
capital (current capital depreciates at rate δ < 1). To undertake one unit of investment
the firm has to give up one unit of the current consumption good. The representative
firm makes these choices so as to maximize the present value of profits. Finally the
government purchases G units of the current consumption good and G0 of the future, but
can borrow by issuing bonds.
(a) Derive the household’s lifetime budget constraint and describe the three marginal
conditions that determine the consumer’s choice between consumption-leisure in the
current period, consumption-leisure in the future period, and between consumption
in the current to the future period.
(b) What is the capital accumulation equation? Describe the optimal investment rule
for the firm? Draw the firm’s optimal investment schedule.
(c) Suppose now that there is asymmetric information in the credit market that induces
financial institutions to charge each borrowing firm a default premium x on the
borrowing rate. How does this alter the firm’s optimal investment rule? Show it on
a graph relative to the case without credit market uncertainty.
1
2. Suppose the intertemporal economy described in Question 1 is in a competitive equilibrium in which the current labor and goods markets clear. Suppose that the government
decreases current taxes, holding government spending in the current and future period
the constant.
(a) Using diagrams determine the equilibrium effects on consumption, investment, the
real interest rate, aggregate output, employment, and the real wage. What is the
multiplier and how does it differ from the government expenditure multiplier?
(b) How would your answer to part (a) change if at the same time the government
reduced current government spending by the exact same amount as the decrease in
taxes?
(c) How would your answer change relative to part (b) if the intertemporal substitution
effect of the real interest rate on current leisure was stronger?
3. Consider now the full intertemporal model in which in addition to the goods and labor
markets (environment of Question 1), the money market also clears. The supply of money
is determined exogenously by the government. The demand for real money balances of
consumers and firms is given by Md
P = L(Y, R), which depends positively on real income
Y (transaction motive) and negatively on the nominal interest rate R (opportunity cost
of holding money). Suppose the economy is initially in equilibrium. Now, suppose that
there is a new technological innovation that is expected to come to production in the
future period.
(a) How will this affect the current equilibrium values of the price level, consumption,
investment, the real interest rate, aggregate output, employment, and the real wage.
(b) How would your answer to part (a) change if at the same time the marginal propensity to consume increases the responsiveness of the demand for real money balances
to interest rate changes was weaker.
4. Using the Fisher relation show that the nominal interest rate is the opportunity cost of
holding money.
5. Suppose that there is a two period exchange economy, in which the representative consumer has endowment income of y in the current period and y
0
in the future period, and
wishes to consume in both periods of life. The rate at which this consumer can borrow or
lend is r. Explain using equations and graphs what the permanent income hypothesis is.
Suppose that the representative consumer now faces a higher interest rate on borrowing