Midterm Preparation, Microeconomics

March 17, 2017

1. One question from the first question cluster and:

(a) Let f(x1; x2) = 2x0 1:5x0 2:3 + 0:0001x2 1x2 2 Find the short run average

cost curve for x2 = 1; 1:5; 2; 2:5; 3 Find the long run average cost

curve.

(b) Let c(y1; y2) = y1 + y2 + (y1y2) 1 3 Does this cost function have

economies of scale for y1? What about economies of scope for any

strictly positive y1 and y2. Hint, economies of scale exist if for a

positive set of y1 and y2, c(y1; y2) > c(y1; 0) + c(0; y2)

(c) Let y = Axalpha. Suppose that we see when p = 2,w = 1, x = 4

and y = 8 and when p = 1, w = 0:2, x = 6. Can we identify

the production function. Suppose that we know nothing of the

functional form and that in the second example y=10. Graph

what we know about the production function.

(d) Suppose a Cobb Douglass production function with two inputs

and exponents inside the production function y = xα 1 1xα 2 2 that are

less than one. Derive the profit maximizing choices of x1; x2; andy

for arbitrary prices.

(e) Let AC(y)=1/y+2+0.1y. Find the efficient point of production

for this firm. Suppose every firm in the industry has access to the

technology, find the region of economies of scope. Derive the long

run average cost for this function.

(f) Assume a market demand function of D(P)=100-P and a per firm

cost function of C(q) or 20q. Find the competitive equilibrium.

Assume that in order to boost wages, the local govt applies a minimum price on output of 30. What are the effect of this on output?

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Suppose instead that the regulation raises the cost of product. Repeat the analysis. Go back to the first part of the analysis with

the regulated price but not the additional cost. Suppose there is

an innovation which can lower the cost of providing the good to

5q. What is the value of this innovation under both an unregulated market and a regulated market to consumers, producers and

society as a whole (total surplus)?

(g) Use profit-maximization and revealed choices by firms to show

the law of supply. Be sure to define what is meant by the law of

supply?

(h) Assume an isoquant for a fixed level of output equal to ¯ y =

1=2x

121

x

132

Fix w1 = 1. Show how the profit bundle changes as

w2 moves from 1 to 2.

(i) Provide a historical situation in which periods of low inflation can

be used to estimate the effect of usury laws (a price ceiling on

interest rates) for similar values of real interest rates without low

inflation. How do mortgage lenders typically ration credit when

price ceilings bind.

(j) Lets suppose that a firm owns a proven oil reserve of k units

which cannot be transferred and a technology for producing oil

from reserves of y = k 13 . Suppose the price of oil is constant the

interest rate (1+r)=1.03 and the oil expires after 2 periods. Solve

for efficient use of oil over two periods. Suppose that an outside

policy maker wants to reduce current oil production. Can they do

that with a constant tax on oil production. What must be true

of any of tax profile which accomplishes the policy makers goal?

Suppose instead the cost function equals y = k 1 3 + 1 for any y > 0

but is equal to zero for no production. How do your answers to

the above change?

(k) Suppose there is an industry with fixed costs equal to 1 and an

average variable cost equal to 5 + 0:1y. Suppose that price in

this industry equals 10. Find the amount of production and the

producer surplus (short run). Be sure to define producer surplus

and show it graphically. Find also the short run and long run

break even prices. Suppose demand is sufficiently large. What is

the long run equilibrium price for this industry?

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(l) Assume the previous industry is such that there is an incumbent

firm which cannot recover the (already paid) fixed cost over any

horizon even if it exits the industry but firms that want to enter

still must pay it. What is the long run equilibrium price and

distribution of profits for this industry?

(m) Derive the isoquant for a firm that has two inputs which are perfect

compliments. Use this to solve for the firms cost minimizing input

bundle and in terms the cost function c(y). Repeat for a firm that

has a perfect substitute technology for producing output.

(n) Given our discussion in class about eyeglasses and correcting for

any typos in the slide, we can speculate about another similar historical event. In 1977, the Supreme Court ruled that individual

states could not ban lawyers from advertising either their availability or their prices. Unlike with eyeglasses, all states enacted

a ban. What do you think was the effect of this decision on legal

fees?

(o) Suppose the demand for product X = 10 – 2X + Y where Y is a

substitute that is not currently being produced. Assume X has a

marginal cost of one dollar. Entry is completely barred and there

is a unique monopolist that produces good X. Find the firms price

quantity and profit. [Hint: remember that MR(Q)=MC(Q) for

the function MR(Q) which is computed by taking the derivative

of the demand curve. The market price is taken directly from the

demand curve.]

i. Next suppose that the incumbent considers introducing good

Y . The demand for good Y is Py = 10–2Y –X. Assume that

there is a fixed cost of 4 dollars for introducing Y and a constant MC of 1 dollar for each additional unit of Y . Find the

optimal values for Px; Py; x; and y. Will this incumbent introduce Y ? Is it in societys best interest for Y to be introduced.

Compute TS to show this.

ii. Next allow entry to occur but that conditional on entry by

one firm into the market for Y , the two firms will split profits

evenly. Will the entrant have the incentive to introduce Y ?

iii. Finally assume that entry is allowable and that our initial

incumbent firm can prevent entry by introducing good Y first

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(this comes from the property that if the incumbent picks the

entire profit maximizing quantity, there arent enough profits

left over for the entrants to actually enter. Will the incumbent

introduce good Y .

iv. Repeat the above exercise with a fixed cost of 6. Is it still

optimal from the societal prospective for good Y to be introduced. Look at all the market structures (no entry, entry

with cooperation, entry with deterrence). Are there any market rule(s) that will reproduce the TS maximizing rule for the

decision to provide Y ? If so, why? If not why not?

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## Midterm Preparation, Microeconomics

28
Mar