1OligopolyECON 370: Microeconomic TheorySummer 2004 – Rice UniversityStanley GilbertEcon 370 - Oligopoly 2Oligopoly: Introduction• Alternative Models of Imperfect Competition– Monopoly and monopolistic competition– Duopoly - two firms in industry– Oligopoly - a few (> 2) firms in industry• Essential Features– Nature of interaction between firms (beyond those captured in price) is essence of theories– No single “grand theory”Econ 370 - Oligopoly 3Oligopoly: Analysis• Simplest Model of Oligopoly: Duopoly– Assume only two firms (to limit interactions)– Assume homogeneous output• No product differentiation• Single market price• No competition in quality– Equilibrium: Solve for output, price of each firmEcon 370 - Oligopoly 4Oligopoly Models• We use Game Theory to model strategic behavior– Strategic Behavior takes into account how others will react to one’s actions• Non-cooperative simultaneous games– Simultaneously choose quantities (or prices) • Non-cooperative sequential games– Quantity (or price) leader (dominant/barometric firm)– Quantity (or price) follower• Cooperative games– Collusion -- jointly set quantities (or prices)2Econ 370 - Oligopoly 5Quantity Competition: Introduction• Assume firms choose output and allow prices to adjust to clear markets• Each firm chooses output to max profits, given output level of competitor• “Firms compete in outputs”• Firm 1: y1units; Firm 2: y2units– total quantity supplied is y1+ y2– market price will be p(y1+ y2)– total cost functions are c1(y1) and c2(y2)Econ 370 - Oligopoly 6Quantity Competition: Profits• Firm 1 maximizes profit, given y2• Firm 1 profit function:• π1(y1; y2) = p(y1+ y2) y1–c1(y1)• Firm 1 “Reaction Function”– What output y1maximizes firm 1 profit?–Given y2(expected or observed)– Solve for reaction function y1=f(y2)Econ 370 - Oligopoly 7Quantity Competition: Example• Let market inverse demand function be– p(yT) = 60 – yT– yT= y1+ y2• Let firms’ (different) total cost functions be– c1(y1) = y12– c2(y2) = 15y2+ y22Econ 370 - Oligopoly 8Example: Firm 1• Firm 1 profit function is– π1(y1; y2) = (60 – y1– y2)y1– y12• So, given y2, solve for firm 1 profit-maximizing y102yy2y60y12111=−−−=∂∂π2211y4115)(yRy −==• Firm 1’s reaction function (best response) is3Econ 370 - Oligopoly 9Graph: Firm 1Firm 1’s “Reaction Curve” R1(y2))460(4115)(122211yyoryyRy−=−==y2y16015Econ 370 - Oligopoly 10Example: Firm 2• Similarly, given y1, Firm 2’s profit function is– π2( y2; y1) = (60 – y1– y2)y2–15y2– y22• To get Firm 2’s profit-maximizing output021526022122=−−−−=∂∂yyyyπ445)(1122yyRy−==• Firm 1’s reaction function (best response) isEcon 370 - Oligopoly 11Graph: Firm 2y2y1Firm 2’s “Reaction Curve” R2(y1)445)(1122yyRy−==45/445Econ 370 - Oligopoly 12Equilibrium• Equilibrium is a Cournot-Nash equilibrium• Each firm’s output level is best response to other firm’s output level • Stable: neither firm wants to change output• Thus, (y1*, y2*) such that – y1*= R1(y2*) and– y2*= R2(y1*)• Essentially solving a pair of simultaneous equations4Econ 370 - Oligopoly 13Equilibrium*2*21*14115)( yyRy −==445)(*1*12*2yyRy−==Substitute for y2*to get13y4y454115y*1*1*1=⇒⎟⎟⎠⎞⎜⎜⎝⎛−−=841345*2=−=yCournot-Nash equilibrium is (y1*, y2*) = (13, 8)Econ 370 - Oligopoly 14Graph: Equilibriumy2y1601545/445EquilibriumEcon 370 - Oligopoly 15Cournot v Monopoly•Price – Less than monopoly – Greater than perfect competition• Quantity– Greater than monopoly– Less than perfect competition• Total profit– Less than monopoly – Greater than perfect competitionEcon 370 - Oligopoly 16Price Competition: Bertrand Games• Alternative strategic behavior• Firms compete using only price (not quantity)• Bertrand games– Simultaneous game– Firms use price as strategic variable• Get results dramatically different from quantity competition5Econ 370 - Oligopoly 17Bertrand Games: Introduction• Example of Bertrand game– Each firm’s MC = c, constant– All firms simultaneously set their prices• Nash Equilibrium: All firms set p = c– All firms have same p, or high p loses all sales– Any p > c, slight price reduction yields big profit– Any p < c, lose moneyEcon 370 - Oligopoly 18Sequential Games• Sequential games• One firm (larger firm) moves first• Then “follower firms” react• Both consider reactions of other• Can compete in – Quantity—von Stackelberg Model– Price—Price leadership modelsEcon 370 - Oligopoly 19The von Stackelberg Model• Outputs are strategic variables•Firm 1—leader firm—chooses y1first•Firm 2—follower—then reacts• Leader anticipates reaction of follower (doesn’t assume y2constant as in C-N)• Issues– What are prices, outputs, profits– Is there a “first mover” advantage?Econ 370 - Oligopoly 20The von Stackelberg Model• Follower firm will choose y2to maximize profit, given leader firm y1(C-N assumption)• Thus, follower reaction function: y2= R2(y1)• Leader firm (1) anticipates follower firm’s (2) reaction function, so chooses y1to max profit– π1s(y1) = p[y1+ R2(y1)]y1– c1(y1)6Econ 370 - Oligopoly 21Von Stackelberg Game: Profits• Note: leader firm makes a profit at least as large as Cournot-Nash profit– Can always choose y1= C-N output– Follower will respond with y2= C-N output– So, can at least achieve C-N profit• Return to duopoly example w / different MC’s– Leader firm 1 has lower costs c1(y1) = y12– Follower firm 2 has higher costs c2(y2) = 15y2+ y22Econ 370 - Oligopoly 22Von Stackelberg Game: Example• Same characteristics as before• Market inverse demand function is – p = 60 – yT• The firms’ cost functions are – c1(y1) = y12and c2(y2) = 15y2+ y22• Firm 2 is follower, with reaction function445)(1122yyRy−==Econ 370 - Oligopoly 23Leader’s profit function isFor a profit-maximum, first order condition is9.13yy274195s11=⇒=Von Stackelberg Game: Example2111144560 yyyy −⎟⎠⎞⎜⎝⎛−−−=()[]2111211160)( yyyRyys−−−=π211474195yy −=Econ 370 - Oligopoly 247.8413.945)(yRys12s2=−==Von Stackelberg Game: Example• Recall C-N outputs are (y1*, y2*) = (13,8)• So leader produces more than C-N output, follower produces less than its C-N output• First mover advantage to leader (but modest because leader also