(Edgeworth duopoly)** There are two identical firms producing a homogeneous good whose demand…
(Edgeworth duopoly)** There are two identical firms producing a homogeneous good whose demand curve is q = 100 — p. Firms simultaneously choose prices. Each firm has a capacity constraint of K. If the firms choose the same price they share the market equally. If the prices arc unequal, pi <>j, the low-price firm, i, sells min(100 — pi, K) and the high-price firm, j, sells min[max(0,100 —pi — K), K]. (There are many possible rationing rules, depending on the distribution of consumers' preferences and on how consumers are allocated to firms. If the aggregate demand represents a group of consumers each of whom buys one unit if the price pi is less than his reservation price of r, and buys no units otherwise. and the consumer's reservation prices arc uniformly distributed on [0.1001 the above rationing rule says that the high-value consumers arc allowed to purchase at price pi before lower-value consumers are.) The cost of production is 10 per unit.
(a) Show that firm l's payoff function is
(b) Suppose 30 < k="">< 45.="" (note="" that="" these="" inequalities="" are="" strict.)="" show="" that="" this="" game="" does="" not="" have="" a="" pure-strategy="" nash="" equilibrium="" by="" proving="" the="" following="" sequence="" of="" claims:="">
(i) If (p1, p2) is a pure-strategy Nash equilibrium, then p1 = p2. (Hint: If p1 ≠ p2, then the higher-price firm has customers (Why?) and so the lower-price firm's capacity constraint is strictly binding. What happens if this firm charges a slightly higher price?)
(ii) If (p. p) is a pure-strategy Nash equilibrium, then p > 10.
(iii) If (p. p) is a pure-strategy Nash equilibrium, then p satisfies p ≤100 2K.
(iv) If (p.p) is a pure-strategy Nash equilibrium, then p = 100 — 2K. (Hint: If p < 100="" —="" 2k.="" is="" a="" deviation="" to="" a="" price="" between="" p="" and="" 100="" —="" 2k="" profitable="" for="" either="" firm?)="">
(v) Since K > 30. there exists b > 0 such that a price of 100 2K + earns a firm a higher profit than 100 – 2K when the other firm charges 100 — 2K.
Note: The Edgeworth duopoly game does satisfy the assumptions of theorem 1.3 (restrict prices to the set [0, 1001) and so has a mixed-strategy equilibrium.
