Question

Hotelling's model of competition on a linear beach is used widely in many applications, but one application that is difficult to study in the model is free entry. Free entry is easiest to study in a model with symmetric firms, but more than two firms on a line cannot be symmetric because those located nearest the endpoints will have only one neighboring rival, whereas those located nearer the middle will have two. To avoid this problem, Steven Salop introduced competition on a circle. \(^{18}\) As in the Hotelling model, demanders are located at each point, and each demands one unit of the good. A consumer's surplus equals \(v\) (the value of consuming the good) minus the price paid for the good as well as the cost of having to travel to buy from the firm. Let this travel cost be \(t d\), where \(t\) is a parameter measuring how burdensome travel is and \(d\) is the distance traveled (note that we are here assuming a linear rather than a quadratic travel-cost function, in contrast to Example 15.5 . Initially, we take as given that there are \(n\) firms in the market and that each has the same cost function \(C_{i}=K+c q_{i}\) where \(K\) is the sunk cost required to enter the market [this will come into play in part (e) of the question, where we consider free entry] and \(c\) is the constant marginal cost of production. For simplicity, assume that the circumference of the circle equals 1 and that the \(n\) firms are located evenly around the circle at intervals of \(1 / n\). The \(n\) firms choose prices \(p_{i}\) simultancously. a. Each firm \(i\) is free to choose its own price \(\left(p_{i}\right)\) 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 $$p+t x=p^{*}+t[(1 / n)-x]$$ b. Given the pricing decision analyzed in part (a), firm \(i\) sells \(q_{i}=2 x\) 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 \(t / n^{2}-K\) 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 overdifferentiation.

Short answer

Answer: The symmetric equilibrium price will increase as the cost of production (c) increases. This is because in a symmetric equilibrium, every firm's price is equal to p*. When calculating the symmetric equilibrium price as a function of p*, c, t, and n, the cost of production, c, will have a positive effect on the price level. Therefore, if the cost of production increases, the symmetric equilibrium price will also increase.

Step-by-step solution

1. Deriving the equation for the extent of a firm's market

In a symmetric equilibrium, each firm i will have the same price, \(p^{*}\), and therefore the same market extent on either side, \(x\). In a consumer's perspective, the total cost incurred in purchasing from firm i is the price paid plus the travel cost, \(p_{i} + t x\). A consumer located at \(x\) is indifferent between buying from firm i and its neighbor if the total cost of buying from both firms is equal. This gives the equation: \[p_{i} + t x = p^{*} + t[(1 / n)-x]\]

2. Calculating the profit-maximizing price for a firm

By symmetry, the extent of a firm's market on both sides is equal. Thus, the firm's total quantity sold, \(q_{i}\), is given by \(q_{i} = 2x\). The firm's revenue is the product of its price and quantity sold, \(R_{i} = p_{i}q_{i}\). Its costs are given by the cost function, \(C_{i} = K + cq_{i}\). The firm wants to maximize its profit by choosing its price \(p_{i}\), so we have: \(\Pi_{i}(p_{i}) = R_{i} - C_{i}\) Using the \(p_{i} + tx = p^{*} + t[(1/n) - x]\) equation, solve for \(p^{*}\) and plug it in to find the profit maximizing price as a function of \(p^{*}, c, t,\) and \(n\).

3. Determining the symmetric equilibrium price

In a symmetric equilibrium, every firm's price is equal to \(p^{*}\). Substitute \(p_{i} = p^{*}\) into the profit-maximizing price function obtained in part 2, and solve for the symmetric equilibrium price in terms of \(c, t,\) and \(n\).

4. Finding firm's profit in equilibrium

Use the symmetric equilibrium price and the cost function to find a firm's profit in equilibrium. Subtract the total cost of the firm from its revenue.

5. Calculating the number of firms in long-run equilibrium

In the long run, firms will enter the market until profit is zero. Using the equilibrium profit found in part 4, find the number of firms, \(n^{*}\), such that profit is zero.

6. Analyzing the optimal level of differentiation

To find the socially optimal level of differentiation (the number of firms that minimizes the sum of production costs and demander travel costs), calculate the total cost function including its two components. Minimize this total cost function with respect to the number of firms, and compare the result to \(n^{*}\) found in part 5 to analyze the possibility of overdifferentiation.