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Chapter 15: Problem 10

Use the first-order condition (Equation 15.2 ) for a Cournot firm to show that the usual inverse elasticity rule from Chapter 11 holds under Cournot competition (where the elasticity is associated with an individual firm's residual demand, the demand left after all rivals sell their output on the market). Manipulate Equation 15.2 in a different way to obtain an equivalent version of the inverse elasticity rule: \\[ \frac{P-M C}{P}=-\frac{s_{i}}{e_{Q, P}} \\] where \(s_{i}=q_{i} / Q\) is firm i's market share and \(e_{Q, p}\) is the elasticity of market demand. Compare this version of the inverse elasticity rule with that for a monopolist from the previous chapter.

Short Answer

Expert verified
Answer: The inverse elasticity rule for a Cournot firm can be written as: \\[ \frac{P-MC(q_i)}{P} = -\frac{s_i}{e_{Q,P}} \\] Here, \(P\) is the price, \(MC(q_i)\) is the marginal cost function for firm i, \(s_i\) is firm i's market share, and \(e_{Q,P}\) is the elasticity of market demand. This formula shows how the markup of a firm is related to its market share and the price elasticity of demand. Comparing this formula to the inverse elasticity rule for a monopolist: \\[ \frac{P-MC(q)}{P} = -\frac{1}{e_Q} \\] we can see that they have the same structure. The main difference lies in the fact that the Cournot firm's rule uses residual demand elasticity (\(e_{Q,P}\)) and the firm's market share (\(s_i\)), while the monopolist's rule uses the total demand elasticity (\(e_Q\)).

Step by step solution

01

Write down Equation 15.2 for the first-order condition of a Cournot firm

Given Equation 15.2 as the first-order condition for a Cournot firm: \\[ MR(q_i) = MC(q_i) \\] where \(MR(q_i)\) is the marginal revenue for firm i, and \(MC(q_i)\) is the marginal cost function for firm i.
02

Write the marginal revenue function as a function of the price, quantity, and market demand

The marginal revenue function for a Cournot firm can be written in terms of price, output of firm i (\(q_i\)), and elasticity of market demand: \\[ MR(q_i) = P\left(1+\frac{1}{e_{Q,P}}\right) \\]
03

Substitute Equation 2 into Equation 1 and solve for the inverse elasticity rule

Substituting the marginal revenue function equation into the first-order condition equation and then solving for the inverse elasticity rule give us: \\[ P\left(1+\frac{1}{e_{Q,P}}\right) = MC(q_i) \\] Now, divide both sides by price: \\[ 1+\frac{1}{e_{Q,P}} = \frac{MC(q_i)}{P} \\] Subtract one from both sides: \\[ \frac{1}{e_{Q,P}} = \frac{MC(q_i)-P}{P} \\] Now, we can define firm i's market share (\(s_i\)) as \(s_i=\frac{q_i}{Q}\), and the inverse elasticity rule becomes: \\[ \frac{P-MC(q_i)}{P} = -\frac{s_i}{e_{Q,P}} \\]
04

Compare this version of the inverse elasticity rule with that for a monopolist

In the case of a monopolist, the inverse elasticity rule developed in the previous chapters would be: \\[ \frac{P-MC(q)}{P} = -\frac{1}{e_Q} \\] where \(MC(q)\) is the monopolist's marginal cost and \(e_Q\) is the elasticity of demand. Comparing both the Cournot and the monopolist inverse elasticity rules, we can see that they have the same structure, with the only difference being that for Cournot competition, we use a firm's residual demand (\(e_{Q,P}\)) and its market share (\(s_i\)), while in the monopoly case we use total demand elasticity (\(e_Q\)).

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Most popular questions from this chapter

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.

Recall the Hotelling model of competition on a linear beach from Example \(15.5 .\) Suppose for simplicity that ice cream stands can locate only at the two ends of the line segment (zoning prohibits commercial development in the middle of the beach). This question asks you to analyze an entry-deterring strategy involving product proliferation. a. Consider the subgame in which firm \(A\) has two ice cream stands, one at each end of the beach, and \(B\) locates along with \(A\) at the right endpoint. What is the Nash equilibrium of this subgame? Hint: Bertrand competition ensues at the right endpoint. b. If \(B\) must sink an entry cost \(K_{B}\), would it choose to enter given that firm \(A\) is in both ends of the market and remains there after entry? c. Is \(A\) 's product proliferation strategy credible? Or would \(A\) exit the right end of the market after \(B\) enters? To answer these questions, compare \(A\) 's profits for the case in which it has a stand on the left side and both it and \(B\) have stands on the right to the case in which \(A\) has one stand on the left end and \(B\) has one stand on the right end (so \(B\) 's entry has driven \(A\) out of the right side of the market).

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).

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

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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