(1) Numerical Problem 5, Chapter 3, page 101 for 8th edition, pages 104 for 9th edition, page 108 for
the 10th edition.
Consider an economy in which the marginal product of labor MP N is MP N = 309−2N, where N
is the amount of labor used. The amount of labor supplied, NS, is given by NS = 22 + 12w + 2T,
where w is the real wage and T is a lump-sum tax levied on individuals.
a. Use the concepts of income effect and substitution effect to explain why an increase in lump-sum
taxes will increase the amount of labor supplied.
b. Suppose that T = 35. What are the equilibrium values of employment and the real wage?
c. With T remaining equal to 35, the government passes minimum-wage legislation that requires
firms to pay a real wage greater than or equal to 7. What are the resulting values of employment
and the real wage?
Note: In the 8th, 10th editions, the MP N is MP N = 309 − 2N while in the 9th edition, the
MP N is MP N = 3095 − 2N. I will treat either expression as correct when I mark your homework.
However, I believe that the expression in the 8th edition makes more sense given the rest of the
problem.
(2) Numerical Problem 7, Chapter 4, pages 148-149 for 8th edition, pages 152 for 9th edition, page 156
for the 10th edition.
Suppose that the economywide expected future marginal product of capital is MPKf = 20−0.02K,
where K is the future capital stock. The depreciation rate of capital, d, is 20% per period. The
current capital stock is 900 units of capital. The price of a unit of capital is 1 unit of output.
Firms pay taxes equal to 15% of their output. The consumption function in the economy is C =
100+ 0.5Y −200r, where C is consumption, Y is output, and r is the real interest rate. Government
purchases equal 200, and full-employment output is 1000.

a. Suppose that the real interest rate is 10% per period. What are the values of the tax-adjusted
user cost of capital, the desired future capital stock, and the desired level of investment?
b. Now consider the real interest rate determined by goods market equilibrium. This part of the
problem will guide you to this interest rate.
i. Write the tax-adjusted user cost of capital as a function of the real interest rate r. Also
write the desired future capital stock and desired investment as functions of r.
ii. Use the investment function derived in Part (i) along with the consumption function and
government purchases, to calculate the real interest rate that clears the goods market.
What are the goods market-clearing values of consumption, saving, and investment? What
are the tax-adjusted user cost of capital and the desired capital stock in this equilibrium?

(1) Write down the consumers’ intertemporal budget constraint and indicate which terms stand for the
present value of lifetime income and lifetime consumption. State the full consumers’ maximization
problem. [Do not solve yet]
(2) Substitute the budget constraint in the utility function to turn the consumers’ problem into an
unconstrained maximization problem.
(3) Derive the first-order condition that characterizes the optimal consumption choice. Show that the
condition you obtain is the standard consumer Euler equation:
u
0
(c) = β(1 + r)u
0
(c
f
)
and substitute u
0
(·) with its actual value in this exercise.
(4) Incomes today and tomorrow are such that y = y
f = 50, 000. Using the Euler equation and the
intertemporal budget constraint, solve for consumption today, c, and tomorrow, c
f
(5) Assume now that today’s income increases by 10%. Compute the new optimal consumption choices
for today and tomorrow

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