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

Expert verified
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

01

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: \[p_{i} + t x = p^{*} + t[(1 / n)-x]\]
02

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 \(n\).
03

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 \(n\).
04

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.
05

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.
06

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.

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Most popular questions from this chapter

Suppose that firms' marginal and average costs are constant and equal to \(c\) and that inverse market demand is given by \(P=a-b Q\) where \(a, b > 0\) a. Calculate the profit-maximizing price-quantity combination for a monopolist. Also calculate the monopolist's profit. b. Calculate the Nash equilibrium quantities for Cournot duopolists, which choose quantities for their identical products simultaneously. Also compute market output, market price, and firm and industry profits. c. Calculate the Nash equilibrium prices for Bertrand duopolists, which choose prices for their identical products simultaneously. Also compute firm and market output as well as firm and industry profits. depose now that there are \(n\) identical firms in a Cournot model. Compute the Nash equilibrium quantities as functions of \(n\). Also compute market output, market price, and firm and industry profits. e. Show that the monopoly outcome from part (a) can be reproduced in part (d) by setting \(n=1\), that the Cournot duopoly outcome from part (b) can be reproduced in part (d) by setting \(n=2\) in part (d), and that letting \(n\) approach infinity yields the same market price, output, and industry profit as in part (c).

Assume for simplicity that a monopolist has no costs of production and faces a demand curve given by \(Q=150-P\) a. Calculate the profit-maximizing price-quantity combination for this monopolist. Also calculate the monopolist's profit. b. Suppose instead that there are two firms in the market facing the demand and cost conditions just described for their identical products. Firms choose quantities simultaneously as in the Cournot model. Compute the outputs in the Nash equilibrium. Also compute market output, price, and firm profits. c. Suppose the two firms choose prices simultaneously as in the Bertrand model. Compute the prices in the Nash equilibrium. Also compute firm output and profit as well as market output. d. Graph the demand curve and indicate where the market price-quantity combinations from parts (a)-(c) appear on the curve.

Consider the following Bertrand game involving two firms producing differentiated products. Firms have no costs of production. Firm 1's demand is \\[ q_{1}=1-p_{1}+b p_{2} \\] where \(b > 0 .\) A symmetric equation holds for firm 2 's demand. a. Solve for the Nash equilibrium of the simultaneous price-choice game. b. Compute the firms' outputs and profits. c. Represent the equilibrium on a best-response function diagram. Show how an increase in \(b\) would change the equilibrium. Draw a representative isoprofit curve for firm 1

Assume as in Problem 15.1 that two firms with no production costs, facing demand \(Q=150-P\), choose quantities \(q_{1}\) and \(q_{2}\) a. Compute the subgame-perfect equilibrium of the Stackelberg version of the game in which firm 1 chooses \(q_{1}\) first and then firm 2 chooses \(q_{2}\) b. Now add an entry stage after firm 1 chooses \(q_{1}\). In this stage, firm 2 decides whether to enter. If it enters, then it must sink cost \(K_{2}\), after which it is allowed to choose \(q_{2}\). Compute the threshold value of \(K_{2}\) above which firm 1 prefers to deter firm \(2^{\prime}\) s entry c. Represent the Cournot, Stackelberg, and entry-deterrence outcomes on a best-response function diagram.

This question will explore signaling when entry deterrence is impossible; thus, the signaling firm accommodates its rival's entry. Assume deterrence is impossible because the two firms do not pay a sunk cost to enter or remain in the market. The setup of the model will follow Example \(15.4,\) so the calculations there will aid the solution of this problem. In particular, firm \(i\) 's demand is given by $$q_{i}=a_{i}-p_{i}+\frac{p_{j}}{2}$$ where \(a_{i}\) is product \(i\) 's attribute (say, quality). Production is costless. Firm 1's attribute can be one of two values: either \(a_{1}=1\) in which case we say firm 1 is the low type, or \(a_{1}=2,\) in which case we say it is the high type. Assume there is no discounting across periods for simplicity. a. Compute the Nash equilibrium of the game of complete information in which firm 1 is the high type and firm 2 knows that firm 1 is the high type. b. Compute the Nash equilibrium of the game in which firm 1 is the low type and firm 2 knows that firm 1 is the low type. c. Solve for the Bayesian-Nash equilibrium of the game of incomplete information in which firm 1 can be either type with equal probability. Firm 1 knows its type, but firm 2 only knows the probabilities. Because we did not spend time this chapter on Bayesian games, you may want to consult Chapter 8 (especially Example 8.7 ). d. Which of firm 1 's types gains from incomplete information? Which type would prefer complete information (and thus would have an incentive to signal its type if possible)? Does firm 2 earn more profit on average under complete information or under incomplete information? e. Consider a signaling variant of the model chat has two periods. Firms 1 and 2 choose prices in the first period, when firm 2 has incomplete information about firm 1 's type. Firm 2 observes firm 1 's price in this period and uses the information to update its beliefs about firm 1's type. Then firms engage in another period of price competition. Show that there is a separating equilibrium in which each type of firm 1 charges the same prices as computed in part (d). You may assume that, if firm 1 chooses an out-of-equilibrium price in the first period, then firm 2 believes that firm 1 is the low type with probability 1 . Hint: To prove the existence of a separating equilibrium, show that the loss to the low type from trying to pool in the first period exceeds the second-period gain from having convinced firm 2 that it is the high type. Use your answers from parts (a)-(d) where possible to aid in your solution.

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