Intermediate Microeconomics

IMPERFECT COMPETITION BEN VAN KAMMEN, PHD PURDUE UNIVERSITY Outline of objectives 1. Solve a mathematical model of duopoly that results in behaving like competitive firms. 2. Solve a mathematical model of duopoly that results in behaving like a monopoly. 3. Solve a mathematical model of duopoly using Nash Equilibrium or “simultaneous best response.” 4. Solve multiple functions simultaneously and apply simple differential calculus to solve models of duopoly. Imperfect competition Firms compete but do not necessarily erode all profits. Oligopoly: A market with few firms but more than one. Duopoly: A market with two firms. Cartel: Several firms collectively acting like a monopolist by charging a common price and dividing the monopoly profit among them. Bertrand competition Profit for each firm: = , [ ] and = ( , )[ ]. 𝐷𝐷 𝐷𝐷 𝐾𝐾 𝐷𝐷 Π 𝑄𝑄 𝑃𝑃 𝑃𝑃 𝑃𝑃 − 𝑐𝑐 th This model Πis 𝐾𝐾the Bertrand𝑄𝑄 𝑃𝑃𝐷𝐷 𝑃𝑃𝐾𝐾 model𝑃𝑃𝐾𝐾 −—𝑐𝑐after 19 Century economist, Joseph Louis Francois Bertrand. To find the equilibrium, however, we’re not going to use calculus—just reason. Equilibrium in the Bertrand model > , = , 𝑃𝑃𝐾𝐾 <𝑃𝑃𝐷𝐷 . 𝐾𝐾 𝐷𝐷 Same price both firms𝑃𝑃 means:𝑃𝑃 1 𝑐𝑐 ≤ 𝑃𝑃𝐾𝐾 𝑃𝑃𝐷𝐷 , . 2 𝑄𝑄 𝑃𝑃𝐷𝐷 𝑃𝑃𝐾𝐾 𝑃𝑃𝐾𝐾 − 𝑐𝑐 Bertrand equilibrium Price exactly equal to marginal cost. Both firms zero profit—the same as perfect competition. and are both zero and = = in Bertrand equilibrium. 𝛱𝛱𝐷𝐷 𝛱𝛱𝐾𝐾 𝑃𝑃𝐷𝐷 𝑃𝑃𝐾𝐾 𝑐𝑐 The competitive outcome with only 2 firms. Monopoly outcome with 2 firms Multiple firms act like a monopoly: dividing the monopoly quantity between/among the members of the cartel. Unstable. ◦ Incentive to “cheat,” produce “too much” output. Outline of objectives 1. Solve a mathematical model of duopoly that results in behaving like competitive firms. 2. Solve a mathematical model of duopoly that results in behaving like a monopoly. 3. Solve a mathematical model of duopoly using Nash Equilibrium or “simultaneous best response.” 4. Solve multiple functions simultaneously and apply simple differential calculus to solve models of duopoly. Cournot competition 2 firms produce the same good. Market Demand for their good has negative slope. Each firm chooses a quantity to maximize its own profits. ◦ Each firm takes the behavior of the other firm(s) as given. The market price is determined by the sum of the two firms’ quantities, i.e., ( , ).

𝑃𝑃 𝑞𝑞1 𝑞𝑞2 Example 2 firms, “Johnsonville” and “Klements.” Demand for bratwurst: = 200 with inverse demand: = 20 0. . 𝑄𝑄 − 10𝑃𝑃 The market quantity (Q) is: + . 𝑃𝑃 − 1𝑄𝑄 Marginal cost is constant, i.e., = . 𝑄𝑄 ≡ 𝑞𝑞𝐽𝐽 𝑞𝑞𝐾𝐾 𝑇𝑇𝑇𝑇 2𝑞𝑞 Profit functions in Cournot competition = , 2 and = , 2 . Π𝐽𝐽 𝑃𝑃 𝑞𝑞𝐽𝐽 𝑞𝑞𝐾𝐾 𝑞𝑞𝐽𝐽 − 𝑞𝑞𝐽𝐽 Π𝐾𝐾 𝑃𝑃 𝑞𝑞𝐽𝐽 𝑞𝑞𝐾𝐾 𝑞𝑞𝐾𝐾 − 𝑞𝑞𝐾𝐾 Market demand Solving the Cournot model

= 20 0.1 + 2 and

Π𝐽𝐽 = 20− 0.1𝑞𝑞𝐽𝐽 +𝑞𝑞𝐾𝐾 𝑞𝑞𝐽𝐽 − 2𝑞𝑞𝐽𝐽 . Π𝐾𝐾 − 𝑞𝑞𝐽𝐽 𝑞𝑞𝐾𝐾 𝑞𝑞𝐾𝐾 − 𝑞𝑞𝐾𝐾 Cournot profit maximization

= 20 0.2 0.1 2 𝜕𝜕Π𝐽𝐽 − 𝑞𝑞𝐽𝐽 − 𝑞𝑞𝐾𝐾 − 𝜕𝜕𝑞𝑞𝐽𝐽 = 20 0.2 0.1 2 𝜕𝜕Π𝐾𝐾 − 𝑞𝑞𝐾𝐾 − 𝑞𝑞𝐽𝐽 − 𝜕𝜕𝑞𝑞𝐾𝐾 = 0 0.2 = 18 0.1 𝜕𝜕Π𝐽𝐽 → 𝑞𝑞𝐽𝐽 − 𝑞𝑞𝐾𝐾 𝜕𝜕𝑞𝑞𝐽𝐽 = 0 0.2 = 18 0.1 𝜕𝜕Π𝐾𝐾 → 𝑞𝑞𝐾𝐾 − 𝑞𝑞𝐽𝐽 𝜕𝜕𝑞𝑞𝐾𝐾 Cournot firms’ reaction functions Solving for the q’s: = 90 0.5 = 90 0.5 𝑞𝑞𝐽𝐽 − 𝑞𝑞𝐾𝐾 Reaction functions. 𝑞𝑞𝐾𝐾 − 𝑞𝑞𝐽𝐽 Solving for equilibrium = 90 0.5 = 90 0.5 90 0.5 = 90 45 + 0.25 0.75 = 45 . . . 𝑞𝑞𝐽𝐽 − 𝑞𝑞𝐾𝐾 − − 𝑞𝑞𝐽𝐽 = 60. 𝑞𝑞𝐽𝐽 − 𝑞𝑞𝐽𝐽 → 𝑞𝑞𝐽𝐽 𝑞𝑞𝐽𝐽 Solving for equilibrium, cont’d = 90 0.5 = 90 0.5 60 = 60. = = 60 Equilibrium𝑞𝑞𝐾𝐾 − is: 𝑞𝑞𝐽𝐽 − , the intersection→ 𝑞𝑞𝐾𝐾 of the reaction functions. 𝑞𝑞𝐾𝐾 𝑞𝑞𝐿𝐿 Reaction functions graphed

Guess where equilibrium occurs . . . Cournot example (summary)

Structure Market Q Market P Firm Profit Cartel 90 $11 $405 (Monopoly) Cournot 120 $8 $360

Perfect 180 $2 $0 Comp’n. The cartel profit is ½ the monopoly profit of [(11-2)*90]. Comparing Cournot competition to being in a cartel Being in a cartel: more profit than competing. ◦ If you expect the other firm to produce the quantity chosen by the cartel, what should you do? Betray the cartel! ◦ Both firms have the incentive to do this if they expect their competitor to stick to the agreement. The cartel game A game in which both “players” decide “cartel” or “Cournot.” Set of strategies: q = {45, 60}, ◦ 45 loyalty to the cartel and ◦ 60 Cournot. Combination of strategies determines the market price and each firm’s profit. Cartel game payoff matrix Klements Klements Cartel Cournot

Johnsonville Cartel $405, $405 $337.50, $450

Johnsonville Cournot $450, $337.50 $360, $360

Each cell is structured as follows: {Johnsonville payoff, Klements payoff}. Cartel game dominant strategy

Klements Klements Cartel Cournot

Johnsonville Cartel $405, $405 $337.50, $450

Johnsonville Cournot $450, $337.50 $360, $360

The underlined payoffs indicate each firm’s best action, given what the other player does. Note that the lower right cell has both payoffs underlined. Nash Equilibrium Nash Equilibrium: a simultaneous best response by all players. ◦ Named after economist and mathematician, John Nash— the author of the proof of its existence. Summary & conclusion In single period “games” it is likely that cartels will dissolve as a result of the firms’ individual profit motives. Under the right conditions, however, a repeated cartel game can result in a sustained cartel. Imperfect competition models are part of the most active areas of research in Economics.