Reference no: EM132396571

Problem

Given N firms

Eeach cost = C_i = k + cg_i [k is a sunn cost to enter market like breach sale lilense ] [c = constant manager cost ]
Assume circumference of circle = 1
N firm location at spacing of: 1/N (Every)
N Firms choose prices Pi simultaneously

a. Each firm i is free to choose its own price (p_i) but is constrained by the price charged by its nearest neighbor to either side. Let p* be the price these firms set in a symmetric equilibrium. Explain why the extent of any firm's market on either side (x) is given by the equation
p + tx = p* + t[(1/n) - x]

b. Given the pricing decision analyzed in part (a), firm i sells q_i = 2x because it has a market on both sides. Calculate the profit-maximizing price for this firm as a function of p*, c, t, and n.

c. Noting that in a symmetric equilibrium all firms' prices will be equal to p*, show that p_i = p* = c + t/n. Explain this result intuitively.

d. Show that a firm's profits are t/n² - K in equilibrium.

e. 'What will the number of firms n* be in long-run equilibrium in which firms can freely choose to enter?

f. Calculate the socially optimal level of differentiation in this model, defined as the number of firms (and products) that minimizes the sum of production costs plus demander travel costs.

Show that this number is precisely half the number calculated in part (e). Hence this model illustrates the possibility of over-differentiation.