Introduction to Economic Analysis, v. 1.0

17.1 Cournot Oligopoly

The CournotAugustus Cournot (1801–1877). oligopoly model is the most popular model of imperfect competition. It is a model in which the number of firms matters, and it represents one way of thinking about what happens when the world is neither perfectly competitive nor a monopoly.

In the Cournot modelA model of imperfect competition where firms simultaneously set quantities., there are n firms, who simultaneously set quantities. We denote a typical firm as firm i and number the firms from i = 1 to i = n. Firm i chooses a quantity q_i ≥ 0 to sell, and this quantity costs c_i(q_i). The sum of the quantities produced is denoted by Q. The price that emerges from the competition among the firms is p(Q), and this is the same price for each firm. It is probably best to think of the quantity as really representing a capacity, and competition in prices by the firms determining a market price given the market capacity.

The profit that a firm i obtains is ${\pi }_i=p(Q)q_i-c_i(q_i)\text{.}$

Each firm chooses q_i to maximize profit. The first-order conditionsBear in mind that Q is the sum of the firms’ quantities, so that when firm i increases its output slightly, Q goes up by the same amount. give

$$0=\frac{\partial {\pi }_i}{\partial q_i}=p(Q)+p^{\prime }(Q)q_i-{c^{\prime }}_i(q_i)\text{.}$$

This equation holds with equality provided q_i > 0. A simple thing that can be done with the first-order conditions is to rewrite them to obtain the average value of the price-cost margin:

$$\frac{p(Q)-{c^{\prime }}_i(q_i)}{p(Q)}=-\frac{p^{\prime }(Q)q_i}{p(Q)}=-\frac{Q{p}^{\prime }(Q)}{p(Q)}\frac{q_i}{Q}=\frac{s_i}{\epsilon }\text{.}$$

Here $s_i=\frac{q_i}{Q}$ is firm i’s market share. Multiplying this equation by the market share and summing over all firms i = 1, …, n yields ${\displaystyle \sum _{i=1}^n\frac{p(Q)-{c^{\prime }}_i(q_i)}{p(Q)}{s}_i}=\frac{1}{\epsilon }{\displaystyle \sum _{i=1}^n{s}_i^2}=\frac{HHI}{\epsilon }$ where $HHI={\displaystyle \sum _{i=1}^n{s}_i^2}$ is the Hirschman-Herfindahl Index (HHI)The weighted average of the price-cost margins of all firms in the market..The HHI is named for Albert Hirschman (1915– ), who invented it in 1945, and Orris Herfindahl (1918–1972), who invented it independently in 1950. The HHI has the property that if the firms are identical, so that s_i = 1/n for all i, then the HHI is also 1/n. For this reason, antitrust economists will sometimes use 1/HHI as a proxy for the number of firms, and describe an industry with “2 ½ firms,” meaning an HHI of 0.4.To make matters more confusing, antitrust economists tend to state the HHI using shares in percent, so that the HHI is on a 0 to 10,000 scale.

We can draw several inferences from these equations. First, larger firms, those with larger market shares, have a larger deviation from competitive behavior (price equal to marginal cost). Small firms are approximately competitive (price nearly equals marginal cost), while large firms reduce output to keep the price higher, and the amount of the reduction, in price-cost terms, is proportional to market share. Second, the HHI reflects the deviation from perfect competition on average; that is, it gives the average proportion by which price equal to marginal cost is violated. Third, the equation generalizes the “inverse elasticity result” proved for monopoly, which showed that the price-cost margin was the inverse of the elasticity of demand. The generalization states that the weighted average of the price-cost margins is the HHI over the elasticity of demand.

Because the price-cost margin reflects the deviation from competition, the HHI provides a measure of how large a deviation from competition is present in an industry. A large HHI means the industry “looks like monopoly.” In contrast, a small HHI looks like perfect competition, holding constant the elasticity of demand.

The case of a symmetric (identical cost functions) industry is especially enlightening. In this case, the equation for the first-order condition can be rewritten as $0=p(Q)+p^{\prime }(Q)\raisebox{1ex}{$Q$}\!\left/ \!\raisebox{-1ex}{$n$}\right.-c^{\prime }\left(\raisebox{1ex}{$Q$}\!\left/ \!\raisebox{-1ex}{$n$}\right.\right)$ or $p(Q)=\frac{\epsilon n}{\epsilon n-1}{c}^{\prime }\left(\raisebox{1ex}{$Q$}\!\left/ \!\raisebox{-1ex}{$n$}\right.\right)\text{.}$

Thus, in the symmetric model, competition leads to pricing as if demand was more elastic, and indeed is a substitute for elasticity as a determinant of price.