Abstract
We saw in the previous chapter that there are two types of oligopoly models, those that assume cooperative behavior and those that assume noncooperative behavior. In Chaps. 10 and 11, we develop the classic models of oligopoly where firms behave noncooperatively. These models represent the most abstract material that is found in the book. Here you will see how some of the great figures in history have thought about the oligopoly problem.
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Notes
- 1.
This assumes an auctioneer who quotes a market price that just clears the market, which is p *.
- 2.
In fact, firm i’s profit equation would be identical to that of a monopolist if q j = 0.
- 3.
This produces a maximum because the profit equation for each firm is concave. That is, the second-order condition of profit maximization holds, because the second derivative of the profit equation for each firm is −2b < 0. For further discussion of second-order conditions, see the Mathematics and Econometrics Appendix at the end of the book.
- 4.
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 .
- 5.
For a discussion of comparative static analysis, see the Mathematics and Econometrics Appendix.
- 6.
This symmetry condition is sometimes called a level playing field assumption or an exchangeability assumption (Athey and Schmutzler 2001).
- 7.
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.
- 8.
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.
- 9.
As we demonstrate in Appendix 10.A, the equilibrium is stable because BR1 is steeper than BR2.
- 10.
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.
- 11.
In addition, this firm typically takes a leadership role in choosing output or price, an issue we take up in the next chapter.
- 12.
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).
- 13.
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 is
$$ \frac{{\partial {\pi_i}}}{{\partial {q_i}}} = p + \frac{{\partial p}}{{\partial {q_i}}}{q_i} - {\hbox{M}}{{\hbox{C}}_i} = 0, $$where MC i is firm i’s marginal cost. Given symmetry, q i = Q/n, where Q is industry output. Thus,
$$ \frac{{\partial {\pi_i}}}{{\partial {q_i}}} = p + \frac{{\partial p}}{{\partial {q_i}}}\frac{Q}{n} - {\hbox{M}}{{\hbox{C}}_i} = 0. $$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.
- 14.
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.
- 15.
The proof assumes that prices are infinitely divisible.
- 16.
This also assumes that c 2 is less than firm 1’s simple monopoly price (p m).
- 17.
Notice that the second-order conditions of profit maximization hold, because the second derivative of the profit equation for each firm is −2 < 0.
- 18.
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.
- 19.
Detailed derivations can be found in Shy (1995, 162–163).
- 20.
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.
- 21.
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).
- 22.
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.
- 23.
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.
- 24.
In this model, the number of consumers (N) is normalized to 1 for simplicity.
- 25.
Later we will see that another constraint will be important, that is φ H > 2φ L > 0.
- 26.
The second-order conditions of profit maximization hold, because the second derivative of the profit equation for each firm is −2/z < 0.
- 27.
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.
- 28.
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.
- 29.
With this assumption, p i can be thought of as the difference between the price and marginal cost.
- 30.
Notice that the second-order condition holds for each firm. That is ∂2 π 1/∂q 21 = –2(1–d 2) < 0, and ∂2 π 2/∂p 22 = –2.
- 31.
- 32.
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.
- 33.
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.
- 34.
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.
- 35.
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.
- 36.
This concept is discussed more fully in the Mathematics and Econometrics Appendix.
References
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Appendices
Appendix A: Stability of the Cournot and Bertrand Models
Here, we are interested in the stability of the Nash equilibrium (NE) in a Cournot, Bertrand, or Cournot–Bertrand model. According to Mas-Colell (1995: 414), an equilibrium in a static model is stable when the “adjustment process in which the firms take turns myopically playing a best response to each others’ current strategies converges to the Nash equilibrium from any strategy pair in a neighborhood of the equilibrium.” For a stable NE, the best-reply functions must meet certain regularity conditions.
First, we consider the Cournot model developed in Sect. 10.2.1. The Cournot equilibrium is stable when BR1 is steeper than BR2, as in Fig. 10.18. To see this, assume that firm 1 chooses a disequilibrium level of output, q 1′. Firm 2’s best reply to q 1′ is q 2″. When firm 2 chooses q 2″, firm 1’s best response is q 1′′′. Thus, the adjustment process moves from point A, to B, to C in the graph, a process that continues until the NE is reached. At equilibrium, Firm 1’s best reply to q *2 is q *1 , and firm 2’s best reply to q *1 is q *2 (i.e., they are a mutual best reply). The equilibrium is unstable when BR1 is flatter than BR2, as illustrated in Fig. 10.19. In this case, when starting at q 1′ the adjustment process moves away from the NE.
A stable Cournot model
An unstable Cournot model
We now investigate stability of the Cournot equilibrium more generally. In Appendix 10.B, we prove that the slopes of the best-reply functions are ∂q BR1 /∂q 2 = –π 12/π 11 for firm 1 and ∂q BR2 /∂q 1 = –π 21/π 22 for firm 2. In the graph with q 2 on the vertical axis, the slope of firm 1’s best reply is −π 11/π 12. Thus, stability of the equilibrium in the Cournot model requires that |–π 11/π 12| > | – π 21/π 22|. Because π ii < 0 and π ij < 0, we can rewrite the stability condition as π 11 π 22 – π 12 π 21 > 0. In the example from Sect. 10.2.1, π 11 = π 22 = –2 and π 12 = π 21 = –d. Thus, the slope of firm 1’s best reply is −2/d, the slope of firm 2’s best reply is −d/2, and the stability condition is π 11 π 22 – π 12 π 21 = 4 – d 2 > 0. Thus, the equilibrium is stable when d < 2.
Next, we consider the differentiated Bertrand model developed in Sect. 10.2.2. The Bertrand equilibrium is stable when BR1 is steeper than BR2, as in Fig. 10.20. If we begin at a disequilibrium point, p 1′, firm 2’s best reply is p 2″. When firm 2 chooses p 2″, firm 1’s best response is p 1′′′, etc. Thus, the adjustment process moves from point A, to B, to C and converges to the NE. This equilibrium is unstable, however, when BR1 is flatter than BR2, as in Fig. 10.21. In this case, the adjustment process moves away from the NE.
A stable Bertrand model
An unstable Bertrand model
In the example from Sect. 10.2.2, π 11 = π 22 = –2β and π 12 = π 21 = δ. Thus, the slope of firm 1’s best reply is 2β/δ, the slope of firm 2’s best reply is δ/2β, and the stability condition is π 11 π 22 – π 12 π 21 = 4β 2 – δ 2 > 0. Therefore, Bertrand equilibrium is stable when β > δ/2.
Analysis of stability conditions for the Cournot–Bertrand model can be found in V. Tremblay et al. (forthcoming-a).
Appendix B: Strategic Substitutes and Complements and the Slope of the Best-Reply Functions
As discussed in the text, the two strategic variables of firms i and j, s i and s j , are strategic complements when π ij > 0 and are strategic substitutes when π ij < 0. The proof follows from the first- and second-order conditions of profit maximization and the application of the implicit-function theorem, which is discussed in the Mathematics and Econometrics Appendix at the end of the book. Recall that firm i’s best-reply function is derived by solving the firm’s first-order condition for s i , s i BR, which is the optimal value of s i given s j . Even though we are using a general function, embedded in the first-order condition is s i BR. Thus, we can use the implicit-function theorem to obtain the slope of firm i’s best-reply function:
where π ii ≡ ∂2 π i /∂s i 2, which is negative from our concavity assumption (ensuring that the second-order condition of profit maximization is met). Thus, the sign of ∂s i BR/∂s j equals the sign of π ij . To summarize:
-
When π ij < 0, the best-reply functions have a negative slope and s i and s j are strategic substitutes, as in the Cournot model.
-
When π ij > 0, the best-reply functions have a positive slope and s i and s j are strategic complements, as in the differentiated Bertrand model.
In the mixed Cournot and Bertrand model developed in Sect. 10.3, π 12 = d > 0 and π 21 = –d < 0. This verifies that firm 1’s best-reply function has a positive slope, and firm 2’s best-reply function has a negative slope (Fig. 10.15). It also implies that q 1 and p 2 are strategic complements for the Cournot-type firm and are strategic substitutes for the Bertrand-type firm.
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Tremblay, V.J., Tremblay, C.H. (2012). Quantity and Price Competition in Static Oligopoly Models. In: New Perspectives on Industrial Organization. Springer Texts in Business and Economics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3241-8_10
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