10. Quantity and Price Competition in Static Oligopoly Models

This assumes an auctioneer who quotes a market price that just clears the market, which is p ^*.

In fact, firm i’s profit equation would be identical to that of a monopolist if q _j  = 0.

We derive firm i’s marginal revenue as follows. Firm i’s total revenue function is TR _i  = aq _i –bq _i ²–bq _i q _j . We obtain the partial derivative of TR _i by taking its derivative and holding rival output (q _j ) fixed. Thus, ∂π _i /∂q _i  = a–2bq _i –bq _j .

For a discussion of comparative static analysis, see the Mathematics and Econometrics Appendix.

This symmetry condition is sometimes called a level playing field assumption or an exchangeability assumption (Athey and Schmutzler 2001).

We solve for q ₂ because q ₂ will be on the vertical axis and q ₁ will be on the horizontal axis in our figures.

That is, the q ₂ intercept is (a–c)/b for BR₁ and (a–c)/(2b) for BR₂. The slope is –2 for BR₁ and –½ for BR₂.

As we demonstrate in Appendix 10.A, the equilibrium is stable because BR₁ is steeper than BR₂.

Recall from Chap.​ 4 that two players have reached a NE when firm i’s best reply to s _j ^* is s _i ^*, for all i = 1 or 2 and j ≠ i. In other words, firm i chooses s _i ^* based on the belief that firm j chooses s _j ^*. The NE is reached when this belief is correct for both firms. In the Cournot model, this means that (1) when firm 2 chooses q ₂ ^* , firm 1’s best reply is q ₁ ^* and (2) when firm 1 chooses q ₁ ^* , firm 2’s best reply is q ₂ ^* . Thus, the q ₁ ^* –q ₂ ^* pair is a mutual best reply and neither firm has an incentive to change its level of output.

In addition, this firm typically takes a leadership role in choosing output or price, an issue we take up in the next chapter.

This is true only in equilibrium. We can set Q _−i  = (n–1)q _i in the first-order condition because optimal output levels are embedded in it. In other words, it is true that q ₁ ^*  = q ₂ ^*  = q ₃ ^*  = … = q _n ^*, but it need not be true that q ₁ = q ₂ = q ₃ = … = q _n . Thus, we can make this substitution in the first-order condition but not in the profit equation, (10.21).

We can see this more generally from firm i’s first-order condition of profit maximization. Assume that the firm’s profit equals π _i  = p(Q)q _i –TC(q _i ), where p(Q)q _i is total revenue and TC(q _i ) is total cost. The first-order condition iswhere MC _i is firm i’s marginal cost. Given symmetry, q _i  = Q/n, where Q is industry output. Thus,

Notice that if n = 1, this is the first-order condition of a monopolist [see Chap.​ 6, Eq. (6.7)]. Furthermore, as n approaches infinity, Q/n approaches 0 and price approaches marginal cost, the perfectly competitive outcome.

That is, dQ/dp = –1/b, while the slope of the inverse demand function (dp/dQ) is −b. In addition, the price intercept equals a.

The proof assumes that prices are infinitely divisible.

This also assumes that c ₂ is less than firm 1’s simple monopoly price (p _m).

Notice that the second-order conditions of profit maximization hold, because the second derivative of the profit equation for each firm is −2 < 0.

The effects of a change in marginal cost and a change in the demand intercept are the same as in the case with homogeneous goods.

Detailed derivations can be found in Shy (1995, 162–163).

Notice that the second-order conditions of profit maximization hold, because the second derivative of the profit equation for each firm is −2\( \beta \ <\ 0 \), as β > 0.

For BR₁, the slope is 2β/δ and the p ₂ intercept is −(α + βc)/δ. For BR₂, the slope is δ/2β and the p ₂ intercept is (α + βc)/2β. For the equilibrium to be stable, an issue that we discuss in the Appendix 10.A, BR₁ must be steeper than BR₂ (i.e., β > δ/2).

To simplify the analysis, we also assume that the market is covered (i.e., no consumer refrains from purchase) and that consumers have unit demands (i.e., each consumer buys just one unit of brand 1 from store 1 or one unit of brand 2 from store 2). To review these concepts, see Chap.​ 7.

Notice that the second-order conditions of profit maximization hold, because the second derivative of the profit equation for each firm is −N/t < 0.

Later we will see that another constraint will be important, that is φ _H > 2φ _L > 0.

For BR₁, the slope is 2 and the p ₂ intercept is −(c + zφ _H). For BR₂, the slope is 1/2 and the p ₂ intercept is (c–zφ _L)/2.

Historically, the market for personal computers provides another example of Cournot–Bertrand type behavior. That is, Dell set price and built computers to order, while IBM shipped completed computers to dealers who let price adjust to clear the market. Cournot–Bertrand behavior can also be found in the aerospace connector industry where leading distributors compete in price and smaller distributors compete in output.

With this assumption, p _i can be thought of as the difference between the price and marginal cost.

Notice that the second-order condition holds for each firm. That is ∂² π ₁/∂q ₁ ²  = –2(1–d ²) < 0, and ∂² π ₂/∂p ₂ ²  = –2.

This is similar to the outcome of a “contestable market”, as discussed in Chap.​ 5. For further discussion, see C. Tremblay and V. Tremblay (2011a) and C. Tremblay, M. Tremblay, and V. Tremblay (2011).

For BR₁, the slope is 2(1–d ²)/d and the p ₂ intercept is (ad–a)/d. For BR₂, the slope is –d/2 and the p ₂ intercept is a/2.

Kreps and Scheinkman actually proposed a two-stage game, where each firm makes its decision on the sticky (long run) variable in the first stage and the flexible (short-run) variable adjusts to equilibrium in the second stage. This leads to the same result, however: (1) When output is sticky, firms compete in output, and price adjusts to meet demand, as in Cournot; (2) When price is sticky, firms compete in price, and output adjusts to meet demand, as in Bertrand.

The slopes of firm and market demand functions converge as the degree of product differentiation diminishes, and the slopes are the same when products are homogeneous.

Although there are exceptions when demand and cost functions are nonlinear, Amir and Grilo (1999) call this the “typical geometry” for the Cournot and Bertrand models. Throughout the book, we assume this typical geometry.

This concept is discussed more fully in the Mathematics and Econometrics Appendix.