A monopolist can produce at constant average (and marginal) costs of \(A C=M
C=5 .\) The firm faces a market demand curve given by
\\[
Q=53-P
\\]
a. Calculate the profit-maximizing price-quantity combination for this
monopolist. Also calculate the monopolist's profits.
b. Suppose a second firm enters the market. Let \(q_{1}\) be the output of firm
1 and \(q_{2}\) the output of firm 2. Market demand now is given by
\\[
q_{1}+q_{2}=53-P
\\]
Assuming firm 2 has the same costs as firm \(1,\) calculate the profits of firms
1 and 2 as functions of \(q_{1}\) and \(q_{2}\)
c. Suppose (after Cournot) each of these two firms chooses its level of output
so as to maximize profits on the assumption that the other's output is fixed.
Calculate each firm's "reaction function," which expresses desired output of
one firm as a function of the other's output.
d. On the assumption in part (c), what is the only level for \(q_{1}\) and
\(q_{2}\) with which both firms will be satisfied (what \(q_{1}, q_{2}\)
combination satisfies both reaction curves)?
e. With \(q_{1}\) and \(q_{2}\) at the equilibrium level specified in part (d),
what will be the market price, the profits for each firm, and the total
profits earned?
f. Suppose now there are \(n\) identical firms in the industry. If each firm
adopts the Cournot strategy toward all its rivals, what will be the profit-
maximizing output level for each firm? What will be the market price? What
will be the total profits earned in the industry? (All these will depend on \(n
.\) )
g. Show that when \(n\) approaches infinity, the output levels, market price,
and profits approach those that would "prevail" in perfect competition.