## Midterm Preparation, Microeconomics

28 Mar

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 = 102Y 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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