Multiple equilibrium with menu costs. (Ball and D. Roomer, 1991.) Consider an economy consisting of many imperfectly competitive firms. The profits that a firm loses relative to what it obtains with pi = p∗ are K(pi − p∗) 2, K > 0. As usual, p∗ = p +φy and y = m − p. Each firm faces a fixed cost Z of changing its nominal price.

                Initially m is 0 and the economy is at its flexible-price equilibrium, which is y = 0 and p = m= 0. Now suppose m changes to m .

(a) Suppose that fraction f of firms change their prices. Since the firms that change their prices charge p∗ and the firms that do not charge 0, this implies p = f p∗. Use this fact to find p, y, and p∗ as functions of m and f .

(b) Plot a firm’s incentive to adjust its price, K(0− p∗) 2 = Kp∗2 , as a function of f . Be sure to distinguish the cases φ< 1="" and="" φ=""> 1.

(c) A firm adjusts its price if the benefit exceeds Z, does not adjust if the benefit is less than Z, and is indifferent if the benefit is exactly Z. Given this, can there be a situation where both adjustment by all firms and adjustment by no firms are equilibrium? Can there be a situation where neither adjustment by all firms nor adjustment by no firms is equilibrium?