This is “Cournot Oligopoly”, section 17.1 from the book Beginning Economic Analysis (v. 1.0).

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- How do industries with only a few firms behave?
- How is their performance measured?

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

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

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:

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
$\sum _{i=1}^{n}\frac{p(Q)-{{c}^{\prime}}_{i}({q}_{i})}{p(Q)}{s}_{i}}=\frac{1}{\epsilon}{\displaystyle \sum _{i=1}^{n}{s}_{i}^{2}}=\frac{HHI}{\epsilon$
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/

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.

- Imperfect competition refers to the case of firms that individually have some price-setting ability or “market power” but are constrained by rivals.
- The Cournot oligopoly model is the most popular model of imperfect competition.
- In the Cournot model, firms choose quantities simultaneously and independently, and industry output determines price through demand. A Cournot equilibrium is a Nash equilibrium to the Cournot model.
- In a Cournot equilibrium, the price-cost margin of each firm is that firm’s market share divided by the elasticity of demand. Hence the share-weighted average price-cost margin is the sum of market squared market shares divided by the elasticity of demand.
- The Hirschman-Herfindahl Index (HHI) is the weighted average of the price-cost margins.
- In the Cournot model, larger firms deviate more from competitive behavior than do small firms.
- The HHI measures the industry deviation from perfect competition.
- The Cournot model generalizes the “inverse elasticity result” proved for monopoly. The HHI is one with monopoly.
- A large value for HHI means the industry “looks like monopoly.” In contrast, a small HHI looks like perfect competition, holding constant the elasticity of demand.
- With
*n*identical firms, a Cournot industry behaves like a monopoly facing a demand that is*n*times more elastic.