## 17.2 Cournot Industry Performance

### Learning Objective

1. What happens to quantity-setting firms when there are fixed costs of entry?

How does the Cournot industry perform? Let us return to the more general model, which doesn’t require identical cost functions. We already have one answer to this question: the average price-cost margin is the HHI divided by the elasticity of demand. Thus, if we have an estimate of the demand elasticity, we know how much the price deviates from the perfect competition benchmark.

The general Cournot industry actually has two sources of inefficiency. First, price is above marginal cost, so there is the deadweight loss associated with unexploited gains from trade. Second, there is the inefficiency associated with different marginal costs. This is inefficient because a rearrangement of production, keeping total output the same, from the firm with high marginal cost to the firm with low marginal cost, would reduce the cost of production. That is, not only is too little output produced, but what output is produced is inefficiently produced, unless the firms are identical.

To assess the productive inefficiency, we let $c ′ 1$ be the lowest marginal cost. The average deviation from the lowest marginal cost, then, is

$χ= ∑ i=1 n s i ( c ′ i − c ′ 1 ) = ∑ i=1 n s i (p− c ′ 1 −(p− c ′ i )) =p− c ′ 1 − ∑ i=1 n s i (p− c ′ i )$ $=p− c ′ 1 −p ∑ i=1 n s i (p− c ′ i ) p =p− c ′ 1 − p ε ∑ i=1 n s i 2 =p− c ′ 1 − p ε HHI .$

Thus, while a large HHI means a large deviation from price equal to marginal cost and hence a large level of monopoly power (holding constant the elasticity of demand), a large HHI also tends to indicate greater productive efficiency—that is, less output produced by high-cost producers. Intuitively, a monopoly produces efficiently, even if it has a greater reduction in total output than other industry structures.

There are a number of caveats worth mentioning in the assessment of industry performance. First, the analysis has held constant the elasticity of demand, which could easily fail to be correct in an application. Second, fixed costs have not been considered. An industry with large economies of scale, relative to demand, must have very few firms to perform efficiently, and small numbers should not necessarily indicate the market performs poorly even if price-cost margins are high. Third, it could be that entry determines the number of firms and that the firms have no long-run market power, just short-run market power. Thus, entry and fixed costs could lead the firms to have approximately zero profits, in spite of price above marginal cost.

Using Exercise 1, suppose there is a fixed cost F that must be paid before a firm can enter a market. The number of firms n should be such that firms are able to cover their fixed costs, but add one more cost and they can’t. This gives us a condition determining the number of firms n:

$( 1−c n+1 ) 2 ≥F≥ ( 1−c n+2 ) 2 .$

Thus, each firm’s net profits are $( 1−c n+1 ) 2 −F≤ ( 1−c n+1 ) 2 − ( 1−c n+2 ) 2 = (2n+3) (1−c) 2 (n+1) 2 (n+2) 2 .$

Note that the monopoly profits πm are ¼ (1-c)2. Thus, with free entry, net profits are less than $(2n+3)4 (n+1) 2 (n+2) 2 π m ,$ and industry net profits are less than $n(2n+3)4 (n+1) 2 (n+2) 2 π m .$

Table 17.1 "Industry Profits as a Fraction of Monopoly Profits" shows the performance of the constant-cost, linear-demand Cournot industry when fixed costs are taken into account and when they aren’t. With two firms, gross industry profits are 8/9 of the monopoly profits, not substantially different from monopoly. But when fixed costs sufficient to ensure that only two firms enter are considered, the industry profits are at most 39% of the monopoly profits. This percentage—39%—is large because fixed costs could be “relatively” low, so that the third firm is just deterred from entering. That still leaves the two firms with significant profits, even though the third firm can’t profitably enter. As the number of firms increases, gross industry profits fall slowly toward zero. The net industry profits, on the other hand, fall dramatically and rapidly to zero. With 10 firms, the gross profits are still about a third of the monopoly level, but the net profits are only at most 5% of the monopoly level.

Table 17.1 Industry Profits as a Fraction of Monopoly Profits

Number of Firms Gross Industry Profits (%) Net Industry Profits (%)
2 88.9 39.0
3 75.0 27.0
4 64.0 19.6
5 55.6 14.7
10 33.1 5.3
15 23.4 2.7
20 18.1 1.6

The Cournot model gives a useful model of imperfect competition, a model that readily permits assessing the deviation from perfect competition. The Cournot model embodies two kinds of inefficiency: (a) the exercise of monopoly power and (b) technical inefficiency in production. In settings involving entry and fixed costs, care must be taken in applying the Cournot model.

### Key Takeaways

• The Cournot industry has two sources of inefficiency: too little output is produced, and what output is produced is inefficiently produced (unless the firms are identical).
• The HHI analysis has held constant the elasticity of demand, which could easily fail to be correct, and fixed costs have not been considered.
• Consideration of fixed costs reduces the apparent inefficiency of Cournot industry.

### Exercises

1. Suppose the inverse demand curve is p(Q) = 1 – Q, and that there are n Cournot firms, each with constant marginal cost c, selling in the market.

1. Show that the Cournot equilibrium quantity and price are $Q= n(1−c) n+1$ and $p(Q)= 1+nc n+1 .$
2. Show that each firm’s gross profits are $( 1−c n+1 ) 2 .$
2. Suppose the inverse demand curve is p(Q) = 1 – Q, and that there are n Cournot firms, each with marginal cost c selling in the market.

1. Find the Cournot equilibrium price and quantity.
2. Determine the gross profits for each firm.
3. What formula from the Cournot model is used in antitrust analysis? How is it used?
4. Consider n identical Cournot firms in equilibrium.

1. Show that the elasticity of market demand satisfies $ε>1/n.$
2. Is this consistent in the case when n = 1 (monopoly)?
5. The market for Satellite Radio consists of only two firms. Suppose the market demand is given by P = 250 – Q, where P is the price and Q is the total quantity, so Q = Q1 + Q2. Each firm has total costs given by C(Qi) = Qi2 + 5 Qi + 200.

1. What is the market price predicted by the Cournot duopoly model?
2. If the industry produces a total quantity X, what allocation of quantity (with X = Q1 + Q2) between the two companies minimizes total cost? (Your answer should express total cost as a function of X.)
3. If the firms merge with the cost found in b, what is the market price?