(Repeated Bertrand duopoly) Consider Bertrand's model of duopoly (Section 3.2)
in the case that each firm's unit cost is constant, equal to \(c\). Let
\(\Pi(p)=(p-c) D(p)\) for any price \(p\), and assume that \(\Pi\) is continuous and
is uniquely maximized at the price \(p^{m}\) (the "monopoly price").
a. Let \(s\) be the strategy for the infinitely repeated game that charges
\(p^{m}\) in the first period and subsequently as long as the other firm
continues to charge \(p^{m}\), and punishes any deviation from \(p^{m}\) by the
other firm by choosing the price \(c\) for \(k\) periods, then reverting to
\(p^{m}\). Given any value of \(\delta\), for what values of \(k\) is the strategy
pair \((s, s)\) a Nash equilibrium of the infinitely repeated game?
b. Let \(s\) be the strategy for the infinitely repeated game defined as
follows:
\- in the first period charge the price \(p^{m}\)
\- in every subsequent period charge the lowest of all the prices charged by
the other firm in all previous periods.
Is the strategy pair \((s, s)\) a Nash equilibrium of the infinitely repeated
game for any discount factor less than 1 ?
