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 Model 1 1 Augustin Cournot. Research Into the Mathematical Principles of the Theory of Wealth, 1838 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.

3. Assumptions 1.Two sellers 2.MC = $40 3.Homogeneous product 4.Q is the “decision variable” 5.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 – Q 2 [2]

4. Thus the marginal revenue (MR) function is given by: MR = dR/dQ = 100 – 2Q [3] Let q 1 denote the output of seller 1 and q 2 is the output of seller 2. Now rewrite equation [1] P = 100 – q 1 – q 2 [4] The profit (  ) functions of sellers 1 and 2 are given by:  1 = (100 – q 1 – q 2 )q 1 – 40q 1 [5]  2 = (100 – q 1 – q 2 )q 2 – 40q 2 [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 q 2 = 0 units so that Q = q 1 —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 = Q M = 30 units Thus P M = 100 – Q M = $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 – q 1 – 10 = 90 – q 1 [7] The the revenue function of seller 1 is given by: R = P q 1 = (90 – q 1 )q 1 = 90q 1 – q 1 2 [8] Thus: MR = dR/dq 1 = 90 – 2q 1 [9]

7. Subtracting MC from MR 90 – 2q 1 – 40 = 0 [10] 2q 1 = 50  q 1 = 25 units [11] Thus the profit maximizing output for seller 1, given that q 2 = 10 units, is 25 units. We repeat these calculations for every possible value of q 2 and we find that the  -maximizing output for seller 1 can be obtained from the following equation: q 1 = 30 -.5q 2 [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. Output of seller 1 Output of seller 2 30 -.5q 2 60 30 0 10 15 25

9. In similar fashion, we derive a best reply function for seller 2. It is given by: q 2 = 30 -.5q 1 [13] q1q1 q2q2 0 30 60 q 2 = 30 -.5q 1

10. So we have a system with 2 equations and 2 unknowns (q 1 and q 2 ) : q 1 = 30 –.5q 2 q 2 = 30 –.5q 1 The solutions are: q 1 = 20 units q 2 = 20 units q2q2 q1q1 0 20 Equilibrium Seller 1’s BRF Seller 2’s BRF 60 30 60 Equilibrium is established when both sellers are on their best reply function

11. Cournot duopoly solution Q COURNOT = 40 Units (20 units each) P COURNOT = $60  1 =  2 = $400 Note that: P COMPETITIVE = $40 Q COMPETITIVE = 60 Units Therefore P COMPETITIVE < P COURNOT < P MONOPOLY

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.