Download Problem Set 5 - University of Notre Dame

Problem Set 5
Intermediate Macroeconomics, Fall 2009
The University of Notre Dame
Professor Sims
Instructions: You may work in groups of up to four students. Please make sure to include
the names of all group members, and please write legibly (better yet, type your answers!).
This problem set is due at the beginning of class on Wednesday October 28
(1) Investment and Taxes: Consider a …rm which operates for two periods. It does not
require any labor for production. It produces output each period according to the following
production function:
Y = zK
0<
<1
The current capital stock is exogenously given. The …rm can in‡uence its future capital
stock through investment. Capital accumulates according to:
K 0 = I + (1
d)K
The …rm liquidates itself (i.e. sells o¤ its remaining capital) at the end of the second period.
The …rm’s objective is to maximize its value, given by:
V =
+
0
1+r
denotes pro…ts, and the …rm takes the interest rate as given.
(a) Write down expressions for both current and future pro…ts,
and
0
.
(b) Write down the …rm’s optimization problem. What is/are its choice variables?
(c) Algebraically solve for the …rm’s optimal choice of investment, I.
(d) Now suppose that there is a proportional tax on …rm pro…ts, t, which is the same in both
periods (i.e. t = t0 ). Re-do the above, solving for the optimal investment rule. What is the
e¤ect of the tax rate on investment?
(e) Instead suppose that investment expenditures are not tax deductible. That is, after tax
…rm pro…ts in the …rst period are now (1 t)Y
I instead of (1 t)(Y
I). Redo the
problem. What is the e¤ect of the tax rate on investment? How does your answer compare
with your answer in part (d)?
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(2) Minimum Wages and Aggregate Supply: Suppose that we have an economy populated by households and …rms, both of whom live for two periods. Households demand
consumption and supply labor. Firms produce output and demand labor. There is no
capital. The production function of the representative …rm is:
Y = zf (N )
The function f (N ) has the “usual” properties – increasing and concave. Optimization by
households and …rms yields the following (qualitative) functions:
c = c(Y; Y 0 ; r)
N s = N s (w; Y 0 ; r)
N d = N d (w; z)
Consumption is assumed to be increasing in current income, increasing in future income
(which we can think of as being exogenous), and decreasing in the interest rate. Labor
supply is increasing in the real wage, decreasing in future income, and increasing in the
interest rate. Labor demand is decreasing in the real wage and increasing in current TFP.
(a) Suppose that there is a government imposed minimum real wage, w. The equilibrium
real wage cannot go below this minimum; that is, w w. Suppose that, at interest rate r0 ,
the labor supply curve intersects the labor demand curve at w. Entertain a hypothetical
interest rate increase to r1 > r0 . What will happen to employment and output at the higher
interest rate? Now entertain a hypothetical interest rate decrease from r0 to r2 , r2 < r0 < r1 .
What will happen to employment and output at the lower interest rate? Use these answers
to graphically derive the aggregate supply curve.
(b) Suppose that the aggregate demand curve intersects the aggregate supply curve you
derived in part (a) at some r > r0 . What would happen to output and the equilibrium real
interest rate if the minimum wage were eliminated?
(c) If there were investment and capital in the model, what would be the e¤ect of the policy
change in part (b) on investment? Explain.
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