Price-Output Determination in Oligopolistic Market Structures

1. Price-Output Determination in Oligopolistic Market Structures We have good models of price-output determination for the structural cases of pure competition and pure monopoly. Oligopoly is more problematic, and a wide range of outcomes is possible.

2. Cournot Model1 • Illustrates the principle of mutual interdependence among sellers in tightly concentrated markets--even where such interdependence is unrecognized by sellers. • Illustrates that social welfare can be improved by the entry of new sellers--even if post-entry structure is oligopolistic. 1 Augustin Cournot. Research Into the Mathematical Principles of the Theory of Wealth, 1838

3. Assumptions • Two sellers • MC = $40 • Homogeneous product • Q is the “decision variable” • Maximizing behavior Let the inverse demand function be given by: P = 100 – Q [1] The revenue function (R) is given by: R = P • Q = (100 – Q)Q = 100Q – Q2 [2]

4. Thus the marginal revenue (MR) function is given by: MR = dR/dQ = 100 – 2Q [3] Let q1 denote the output of seller 1 and q2 is the output of seller 2. Now rewrite equation [1] P = 100 – q1 – q2 [4] The profit () functions of sellers 1 and 2 are given by: 1 = (100 – q1 – q2)q1 – 40q1 [5] 2 = (100 – q1 – q2)q2 – 40q2[6] Mutual interdependence is revealed by the profit equations. The profits of seller 1 depend on the output of seller 2—and vice versa

5. Monopoly case Let q2 = 0 units so that Q = q1—that is, seller 1 is a monopolist. Seller 1 should set its quantity supplied at the level corresponding to the equality of MR and MC. Let MR – MC = 0 100 – 2Q – 40 = 0 2Q = 60  Q = QM = 30 units Thus PM = 100 – QM = $70 Substituting into equation [5], we find that:  = $900

6. Finding equilibrium Question: Suppose that seller 1 expects that seller 2 will supply 10 units. How many units should seller 1 supply based on this expectation? By equation [4], we can say: P = 100 – q1 – 10 = 90 – q1 [7] The the revenue function of seller 1 is given by: R = P • q1 = (90 – q1)q1 = 90q1 – q12 [8] Thus: MR = dR/dq1 = 90 – 2q1 [9]

7. Subtracting MC from MR 90 – 2q1 – 40 = 0 [10] 2q1 = 50  q1 = 25 units [11] Thus the profit maximizing output for seller 1, given that q2 = 10 units, is 25 units. We repeat these calculations for every possible value of q2 and we find that the -maximizing output for seller 1 can be obtained from the following equation: q1 = 30 - .5q2[12]

8. Best reply function Equation [12] is a best reply function (BRF) for seller 1. It can be used to compute the -maximizing output for seller 1 for any output selected by seller 2. 60 30 - .5q2 Output of seller 2 30 10 25 30 0 15 Output of seller 1

9. In similar fashion, we derive a best reply function for seller 2. It is given by: q2 = 30 - .5q1 [13] q2 30 q2 = 30 - .5q1 0 60 q1

10. So we have a system with 2 equations and 2 unknowns (q1 and q2) : q1 = 30 – .5q2q2 = 30 – .5q1 The solutions are: q1 = 20 units q2 = 20 units q2 Equilibrium is established when both sellers are on their best reply function 60 Seller 1’s BRF 30 Equilibrium Seller 2’s BRF 20 0 20 30 60 q1

11. Cournot duopoly solution QCOURNOT = 40 Units (20 units each)PCOURNOT = $601 = 2 = $400 Note that: PCOMPETITIVE = $40QCOMPETITIVE = 60 Units Therefore PCOMPETITIVE < PCOURNOT < PMONOPOLY

12. Implications of the model The Cournot model predicts that, holding elasticity of demand constant, price-cost margins are inversely related to the number of sellers in the market This principle is expressed by the following equation [14] Where  is elasticity of demand and n is the number of sellers. So as n  , the price-margin approaches zero—as in the purely competitive case.