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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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Elasticity Rule
The inverse elasticity rule provides a crucial insight into how firms price their products relative to their marginal costs. In the context of a Cournot competition, this rule highlights the relationship between price, marginal cost, market share, and the elasticity of demand. This rule can be expressed as:
\[\frac{P-MC}{P} = -\frac{s_i}{e_{Q,P}} \] where:
  • \(P\) is the price of the product,
  • \(MC\) is the marginal cost of producing one more unit,
  • \(s_i\) represents firm i's market share,
  • \(e_{Q,P}\) is the elasticity of market demand.
This formula shows that the markup over marginal cost is inversely proportional to the product of the market share and the elasticity of demand.
For a monopolist, this rule simplifies, as there is no consideration of market share; the firm sells all of the market output. However, in a Cournot setting, where firms compete in quantities, each firm's ability to mark up prices depends on both their share of the market and how sensitive the overall market is to price changes.
First-Order Condition
The first-order condition in the context of a Cournot firm is derived from maximizing its profit, which is the difference between total revenue and total costs. The condition can be simplified to state that marginal revenue should equal marginal cost:
\[MR(q_i) = MC(q_i) \]This means that the additional revenue earned from selling one more unit should match the additional cost of producing that unit.
In practical terms, if a firm's marginal revenue is higher than its marginal cost, it should increase production to boost profit. Conversely, if marginal cost exceeds marginal revenue, the firm should reduce production. This balancing condition ensures that a firm maximizes its profit by neither overproducing nor underproducing relative to its cost structure.
Marginal Revenue
Marginal revenue is a crucial concept in understanding firm behavior under Cournot competition. It represents the added income from selling an additional unit of a good. For a firm in Cournot competition, marginal revenue is affected by the elasticity of market demand, as expressed in the formula:
\[MR(q_i) = P\left(1+\frac{1}{e_{Q,P}}\right)\]This shows that marginal revenue is not simply the price, but includes an adjustment factor derived from the elasticity of demand.
The key point is that the more elastic the demand, the smaller the adjustment factor, resulting in lower marginal revenue for any given price. This reflects the competitive pressures that a firm faces; as market demand becomes more elastic, the ability of the firm to price above marginal cost is diminished. Understanding this relationship helps firms decide on the optimal level of output to maximize their profits.
Market Share
In the Cournot model, market share is a vital variable influencing the pricing and output decisions of a firm. Defined as the ratio of a firm’s output to the total market output:
\[s_i = \frac{q_i}{Q} \]where:\
  • \(q_i\) is the quantity produced by firm i,
  • \(Q\) is the total market output.
A firm’s market share reflects its power within the market.
Higher market shares imply more pricing power, allowing firms to sustain higher markups over marginal costs. This is captured in the inverse elasticity rule, where the firm's market share directly influences its ability to deviate from marginal cost pricing.
Understanding market share is vital for strategic business decisions as it ties into competitive dynamics, influencing how aggressively a firm might price to maintain or grow its position in the marketplace.

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Most popular questions from this chapter

Recall Example \(15.6,\) which covers tacit collusion. Suppose (as in the example) that a medical device is produced at constant average and marginal cost of \(\$ 10\) and that the demand for the device is given by \\[ Q=5,000-100 P \\] The market meets each period for an infinite number of periods. The discount factor is \(\delta\). a. Suppose that \(n\) firms engage in Bertrand competition each period. Suppose it takes two periods to discover a deviation because it takes two periods to observe rivals' prices. Compute the discount factor needed to sustain collusion in a subgame-perfect equilibrium using grim strategies. b. Now restore the assumption that, as in Example \(15.7,\) deviations are detected after just one period. Next, assume that \(n\) is not given but rather is determined by the number of firms that choose to enter the market in an initial stage in which entrants must sink a one-time cost \(K\) to participate in the market. Find an upper bound on \(n\). Hint: Two conditions are involved.

Let \(c_{i}\) be the constant marginal and average cost for firm \(i\) (so that firms may have different marginal costs). Suppose demand is given by \(P=1-Q\) a. Calculate the Nash equilibrium quantities assuming there are two firms in a Cournot market. Also compute market output, market price, firm profits, industry profits, consumer surplus, and total welfare. b. Represent the Nash equilibrium on a best-response function diagram. Show how a reduction in firm 1 's cost would change the equilibrium. Draw a representative isoprofit for firm 1

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.

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.

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.
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