PS5

Submit before the final if you want the bonus

1. The point of this problem set is to help you solidify newer content before

the final.

2. For that reason, doing the problem set is optional, but you will receive a

single bonus point (equivalent to one additional point on a problem set)

if you complete it.

3. Some of the algebra here is messier than I would put on an exam. As one

of my colleagues likes to say, you train harder in the gym to make the

competition feel easier!

1 Problems

1. Suppose MC(y) = ayb, where a, b > 0. Suppose that demand is given by

y(p) =

c

pd

,

where c > 0 and d > 1.

(a) Compute the (price) elasticity of demand.

(b) Compute the level of output that maximizes monopoly profits. Com-

pute the price at this level of output.

(c) Compute the markup.

(d) How does the markup depend on elasticity of demand? What’s your

intuition for this?

(e) Compute the level of output consistent with price-taking. Compute

the price at this level of output.

(f) Set a = c = 1 and b = 0 (to simplify the calculations), and then

compute the deadweight loss of monopoly as a function of d. Hint:

remember that deadweight loss is the difference between the demand

curve and marginal cost between monopoly output and price-taking

output. That is,

DWL =

∫ yprice-taking

ymono

p(y)−MC(y)dy.

1

(g) How does the deadweight loss depend on the elasticity of demand?

(Hint: you will probably want to use a computer to plot the dead-

weight loss you computed.) What’s your intuition for this?

2. Suppose that each firm in an industry has production function

f(K,L) =

√

KL.

Let w denote the price of labor, and r denote the price of capital, and p

the price of output.

(a) Find the bundle of inputs that minimize the cost of producing y units

of output. (Your answers should be functions of y, w, r.)

(b) What is each firm’s marginal cost function? (Hint: first compute the

minimum cost of producing y units of output.)

(c) Suppose that each firm behaves competitively (take prices as given).

What must p be for each firm’s output-choice problem to have a

nonzero solution? (Hint: notice that the firm has constant returns

to scale.)

(d) Suppose that the firm sells to consumers with utility

u(x1, x2) = ln(x1) + x2,

where x1 is the good produced by the firms and x2 is money (so it has

a price of 1). Suppose that each consumer has income m > 1. If x1

is sold at the price from (c), how much will each consumer demand?

(e) How does consumer demand depend on r and w?

(f) Suppose that the firm cannot adjust its level of capital in the short

run. When capital is fixed at K¯, what is the firm’s marginal cost

function? (Hint: first find the short-run cost of producing y units of

output.)

(g) Show that the short- and long-run marginal cost functions are equal

when K¯ happens to equal the level of capital you found in (a).

(h) Draw the firm’s short-run and long-run supply curves on the same

diagram (make the diagram large, as you will need to add to it). Is

supply more elastic in the short run or the long run?

(i) Add the consumers’ demand curve to your diagram, assuming that

the market is currently in long-run equilibrium.

(j) On your diagram, illustrate the short- and long-run effects of a quan-

tity tax. Is the effect on quantity greater in the short run or the long

run?

3. Suppose that supply is given by

S(p) = apb

2

and that demand is given by

D(p) =

c

pd

where a, b, c > 0 and d > 1.

(a) What is the elasticity of supply? What is the elasticity of demand?

(b) Suppose that the government decides to impose a value tax, meaning

that the price paid by buyers is 1 + t times the price paid to sellers.

(This is more tractable than a quantity tax in this case.) Compute

the equilibrium quantity and price paid to suppliers.

(c) Compute the ratio of seller price with tax to price without tax. How

do the elasticities of supply and demand affect this ratio? What is

your intuition for this?

(d) Compute the ratio of quantity with tax to quantity without tax. How

do the elasticities of supply and demand affect this ratio? What is

your intuition for this?

(e) Compute the tax rate that maximizes tax revenue. How do the elas-

ticities of supply and demand affect this rate? What is your intuition

for this?

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