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Pashigian, Chapter 10,
Exercise 3
Since marginal cost is zero, I assume each firm can
produce the entire market demand. This sounds to
me like a "winner take all bidding situation". The
demand curve for firm A for instance would be equal
to zero when its price was above that of firm B, and
equal to 60-P when its price was below B's. That
means a Nash Equilibrium at a price of zero, with
the firms splitting the market of 60; each firm would
produce 30. Each firm then loses $50, assuming
fixed costs of $50. If the firms merged, they would
act like a monopoly and force the price to 30, and
market demand to 30.
Now suppose you were working with
the FTC. If the merger does not go
through, neither firm will stay in the
business. No consumer surplus will be
generated. But if the merger goes
through, there will be consumer and
producer surplus generated. Thus I
would support the merger as the lesser
of two evils.
NB: some people would work this as a
Cournot problem, and get output of 20 and 20
for each firm. We gave full credit.
Pashigian, Chapter 10,
Exercise 4
If they build plants of fixed and limited production
capacity, then they can effectively prohibit their
customer from employing a winner take all strategy
against them. For example, suppose the competitive
solution would have production of 20 units and the
monopoly solution would have production of 10 units.
If each builds a plant capable of producing 5 units,
then they cannot be subject to a price war. Between
them, they will only build 10 units for that is all they
can produce.
Pashigian, Chapter 10,
Exercise 10
No just because you have a dominant strategy
does not mean that strategy will maximize
profits.
The demand curve for a particular product
is given by
Q = 630000-300p.
The marginal cost of producing the
product is $400. Two firms produce the
product, working as a Cournot Duopoly.
Plot their reaction functions. Given their
reaction functions, calculate the quantity
produced and the market price for the
product.
A
510,000
A
Market price is $93.33
255,000
B
Each firm produces
170,000
Accept a lot of round
off
255,000
510,000
B
The industry demand curve for widgets is
given by
Q = 240 - 10 P
Initially there are ten plants producing
widgets. Each plant belongs to a different
firm. (Indeed, there is a law restricting
each firm to one plant). Nine of the ten
plants have a cost function
16 + q2
The tenth plant (Acme) has an exemption
from environmental laws so that its cost
function is
9 + q2
•Assuming initially that only these ten
firm/plants may produce widgets,
determine the equilibrium price and
quantity of widgets, as well as the profits
of each firm, including Acme.
Each firm has a MC = 2q, so each firm’s supply is
q = P/2. Since there are 10 firms total industry
supply is 10(P/2) = 5P.
Since supply and demand must equal
240-10P = 5P
Solving P =16. Total quantity demanded is 80,
each firm produces 8. Profits. Each firm has
revenue of $144. Nine firms have costs of 16 + 82
=$80, or profits of $64. Acme’s profits are $71.
•Now assume that other firms may open a
(single) plant and produce widgets if they
wish. If they do, their cost function will be
the same as the nine plants. Determine the
equilibrium price of widgets, the number
of firms in the industry, the quantity of
widgets produced by each firm, and the
profits of each firm, including Acme.
Lets find the minimum of the AC function.
AC is 16/q + q, and MC = 2q. Since MC =
AC at the minimum of the AC function, we
solve 16/q +q = 2q, and get q = 4. At that
level of output MC = AC =8. That will be
the price.
Total demand is 240-10(8) =160.
Since each firm produces 4, there must be 40
firms. Each firm except Acme will have
zero profits; Acme will have profits of $7.
•Now suppose that Acme Widgets, the
owner of the first plant, is given a legal
monopoly to produce widgets, but is also
given the right to open as many other
plants as it wishes. Determine how many
plants Acme will operate, the number of
widgets it will produce at each plant, the
price it will charge for widgets, and its
profits. (Break on 1 plant)
LRMC = 8 (we see that from the previous problem).
We want to know MR. To get that, note that the
monopolist’s revenue function is
R = PQ = (24-0.1Q)Q = 24Q –0.1Q2
Thus
MR = 24 - 0.2Q
Since MR = MC for a monopolist, we can solve to
get Q = 80. It will sell at a price of $16.
To get profits, remember that revenues are
(80)($16)= $1440. Costs are $8 each or $640. So
profits are $800. EXCEPT for the $7 extra profit
Acme makes on its first plant, so profits turn out to
be $807.
•Now suppose that widgets are subject to a
$1 tax. What happens to Acme's supply
curve?
•Now suppose that widgets are subject to a
$1 tax. What happens to Acme's supply
curve?
Silly! A monopolist does not
have a supply curve