Download Homework Set #5 (Analysis of Market and Public Policy) I. Varian 22

Homework Set #5
(Analysis of Market and Public Policy)
I. Varian
22-1, 2, 3, 4, 5, 7, 8
23-1
24-1, 5, 7
(Sol) See the “solution to Varian.doc” file.
II. Nicholson (7th ed.)
14-1, 2, 3, 4, 8
15-2, 3, 6, 7
18-1, 2, 7, 8
19-2
(Sol) Solution file will be distributed in two parts. Problem 15-6, 7 will not be discussed.
III. Additional questions on oligopoly: Suppose there are two firms producing pumpkins in a
country, Carl and Simon. Market demand function in each period is given by Q = 3,200 - 1,600p.
Naturally, Q = QC + QS, where QC is the units of pumpkins produced by Carl and QS is the units of
pumpkins produced by Simon. Cost of producing pumpkins for either firm is $0.5 per pumpkin.
1. Suppose that every spring, the snow thaws off of Carl's pumpkin field one week earlier before
it thaws off of Simon's. Therefore, Carl can plant his pumpkins a week earlier than Simon.
(1)Suppose that Carl plants QC units. How many units will Simon plant?
(Sol) Answer can be calculated by deriving reaction function for Simon. By definition, reaction
function (best response) for firm A is the profit-maximizing output given opposing firm's
output. Since profit for Simon is
and
, profit function for
Simon becomes
of
pumpkins.
. Let's assume that Carl had produced
Then
Simon's
profit
function
is
only
dependent
amount
on
:
. Profit-maximizing output for Simon can be derived by using first
order condition:
therefore Simon will plant
units of pumpkins given Carl's output.
(2)How many units will Carl and Simon each plant?
(Sol) Now Carl knows that when he produces
units of pumpkins, Simon will produce
units of pumpkins. Using the advantage of being able to produce earlier than
Simon, Carl will maximize his profit
which
is
equal
to
where
.
Therefore
Carl's
profit
function
becomes
. Profit-maximizing output can be calculated by using first order
condition:
On the other hand, after observing that Carl will produce 1200 units, Simon will produce 600
units of pumpkins (see the reaction function for Simon at problem (1)).
(3)What price will prevail?
(Sol) Since demand function is
, when
, P will be 7/8=0.875.
(4)What will be the profits of each firm?
(Sol) For Carl:
other hand, for Simon:
, when
, profit for Carl will be 450. On the
, when
, profit for Simon will be 225.
Clearly Carl is better off than Simon for Carl (market leader) can produce earlier than Simon
(follower). This is called 'first-mover advantage'.
2. Suppose that due to a climate change that has to do with global warming, the snow thaws off
of the fields owned by Carl and Simon at the same time.
(1)What is the best response of Simon to an expected quantity produced by Carl?
(Sol) Reaction function of Simon is
(see problem 1-(1)).
(2)What is the best response of Carl to an expected quantity produced by Simon?
(Sol) Since profit for Carl is
and
becomes
Simon is denoted by
, profit function for Carl
. Let's assume that expected quantity produced by
. Profit-maximizing output for Simon can be derived by using first
order condition:
therefore Simon will plant
units of pumpkins given Carl's output. When
be variable, reaction function of Carl will be
is taken to
.
(3)How many units will each produce? How much profit will each make?
(Sol) Two reaction functions
and
must coincide with the same Qs
and Qc. Therefore substituting the latter reaction function into the former reaction function will
give optimal production equilibrium which is
price will be 1. Also
. Since
, market
.
3. Suppose that Carl and Simon decided to collude and maximize the joint profit that they will
share equally.
(1)How many units will each produce? What will be the price?
(Sol) Now the market has just one monopolistic firm with demand function P=2-Q/1600. Profit
for
monopolistic
firm
will
be
.
maximizing output can be calculated using first order condition:
Profit-
. Therefore
Q*=1200. When Q=1200, P will be 4/3=:1.33 and profit for monopolist will be 900. Since they
agree to share the total profit equally, Simon and Carl's profit will be 450 respectively. Since
Simon's profit function is
, to have 450 profit, Qs should be 600.
Likewise Carl will produce 450 units too.
(2)What will be the profit of each?
(Sol) 450 for all.
(3)Show that Carl has an incentive to cheat on the collusion
(Sol) When Carl believes that Simon will follow their agreement and produce 600 units of output,
since Carl's reaction function is
900 instead of 600. (Carl's profit when
, optimal response to Simon's output should be
is greater than profit under
: you may check it out if this really is the case) Therefore Carl has an
incentive to cheat. Note that the same argument can apply to Simon, too: that Simon has an
incentive to cheat on Carl, too.
4. Suppose that they compete in each of an infinite number of periods and that demand and cost
in each period are the same as in 2. Describe a collusion agreement that can be credible in
the sense that each firm has an incentive to adhere to the agreement. Derive a condition for
the collusion to be effective.
(Sol) Note that we have an infinite number of periods. When this is the case there might be the
possibility of maintaining collusion. For this: See Varian (Ch.27-11). It says that "We have
seen that a cartel is fundamentally unstable in the sense that it is always in the interest of
each of the firms to increase their production above that which maximizes aggregate profit. If
the cartel is to operate successfully, some way must be found to “stabilize” the behavior. One
way to do this is for firms to threaten to punish each other for cheating on the cartel
agreement. In this section, we investigate the size of punishments necessary to stabilize a
cartel.
Consider a duopoly composed of two identical firms. If each firm produces half the monopoly
amount of output, total profits will be maximized and each firm will get a payoff of, say, πm. In
an effort to make this outcome stable, one firm announces to the other: “If you stay at the
production level that maximizes joint industry projects, fine. But if I discover you cheating by
producing more than this amount, I will punish you by producing the Cournot level of output
forever.” This is known as a punishment strategy."
When will this sort of threat be adequate to stabilize the cartel? We have to look at the benefits
and costs of cheating as compared to those of cooperating. Suppose that cheating occurs, and
the punishment is carried out. Since the optimal response to Cournot behavior is Cournot
behavior (by definition), this results in each firm receiving a per-period profit of, say, πc. Of
course, the Cournot payoff, πc is less than the cartel payoff, πm. Let us suppose that the two
firms are each producing at the collusive, monopoly level of production. Put yourself in the
place of one of the firms trying to decide whether to continue to produce at your quota. If you
produce more output, deviating from your quota, you make profit πd, where πd > πm. This is
the standard temptation facing a cartel member described above: if each firm restricts output
and pushes the price up, then each firm has an incentive to capitalize on the high price by
increasing its production. But this isn’t the end of the story because of the punishment for
cheating. By producing at the cartel amount, each firm gets a steady stream of payments of
πm. The present value of this stream starting today is given by
Present value of cartel behavior =
If the firm produces more than the cartel amount, it gets a one-time benefit of profits πd, but
then has to live with the breakup of the cartel and the reversion to Cournot behavior:
Present value of cheating =
When will the present value of remaining at the cartel output be greater than the present value
of cheating on the cartel agreement? Obviously when
Numerically we have cournot equilibrium (when Simon and Carl decide the outputs
simultaneously) with
(See problem 2-(3)) while monopolistic equilibrium
(See problem 3-(2)). On the other hand quantities resulted from Carl cheating while Simon
doesn't is
and the market price in this case will be 2-900/1600= 23/16=
1.4375. Profit of Carl is therefore 1.4375*600-300=562.5 which is the
Therefore the condition for collusion to be effective is
.
.