Managerial Economics

Managerial Economics Lecture Five: What is competition? The standard view (warts & all!)

Recap • Last week • Schumpeterian view of innovation process • Pricing freed from constraints of “supply/demand” by • Endogenous creation of new money to finance entrepreneur • Ability to undercut rivals/take over old markets with new technology • “creative destruction” • Logistic process of • product diffusion • Price setting/change (high for early adopters, falling to same input-output basis as for standard commodities) • This week: just what is competition? First, the standard view

Neoclassical models of strategic interaction • Standard (Marshallian) model critiqued in first 2 lectures has fundamental flaws • “Horizontal firm demand curve” can’t co-exist with downward sloping market demand • Neoclassical “profit-maximisation” formula ignores impact of other firms on profit in multi-firm industry • Corrected equilibrium profit maximisation formula predicts “competitive” industry will produce same amount as monopoly for same price • Assumption of upward sloping marginal cost doesn’t apply in real world • “Newer” Cournot-Nash-Game Theory “strategic interaction” model not so (obviously) flawed…

Cournot-Game Theoretic Competition Models • Game theory approach implicitly acknowledges firms produce more than profit-maximising amount • But then sees outcome as result of strategic interactions between firms • Key difference with “perfect competition”/Marshallian analysis • Marshallian presumes firms “atomistic” • Too small (blah blah blah) to worry about what competitors are doing; just tries to maximise profit w.r.t. market • Problem shown earlier: atomistic behavior means slope of individual firm demand curve equals slope of market demand curve • End result: competitive outcome same as monopoly

Cournot-Game Theoretic Competition Models • Game theory models don’t have this problem • Firms presumed to react to expected strategies of rivals • In Marshallian theory dqj/dqi=0: one firm doesn’t react to what other firms (might) do • In Game theory dqj/dqinot zero: one firm does react to what other firms (might) do • Curiousity point: model predates neoclassical economics! • French mathematician Augustin Cournot derived mathematical model of how two competing spas might price of access to spa in 1838! • Neoclassical school didn’t begin till 1870s • But one founder (Walras) corresponded with Cournot

Cournot-Game Theoretic Competition Models • Cournot concluded • Duopoly price lower than price if spas colluded • Price converges from monopoly level to perfect competition as number of competitors increases • Cournot’s analysis: firms reach “Cournot equilibrium” levels of output by independently pursuing “profit maximising” strategies… • Set “own-output” marginal revenue equal to marginal cost • NOT an interactive, strategic explanation—but reaches same result as later game theory approach of John von Neumann & “beautiful minds” John Nash • with a little problem we’ll discuss after trudging through the math…

Cournot on Monopoly & Oligopoly • Cournot’s analysis: assume linear demand & cost functions • (original paper actually assumed zero costs!) • Profit=Revenue minus Cost: • Maximum profit when gap between functions biggest • Biggest gap when differential w.r.t. Q is zero: Add these…

Cournot on Monopoly & Oligopoly • Monopoly profit-maximises by producing Q such that • Cournot applied same analysis to two firms in duopoly • Argued profit-maximise by equating marginal revenue and marginal cost w.r.t. own outputs (q1 & q2) • Each firm in a duopoly wants to maximise where Q=q1+q2 • “Therefore” each firm in a duopoly solves

Cournot on Monopoly & Oligopoly • Assume identical costs C(x)=c+dx • Working out quantity for firm 1: • Get symmetrical result for second firm:

Cournot on Monopoly & Oligopoly • To solve, substitute value for q2 into equation for q1: Cancel these… Multiply across Multiply all terms Subtract so Ditto for q2:

Cournot on Monopoly & Oligopoly • General rule is • The more firms the higher the output & lower the price • If n is the number of firms in the industry, then Output converges to perfect competition level as n • Perfect competition: price equals marginal cost: • But there’s a problem…

Cournot on Monopoly & Oligopoly • Equating MC and MR not profit maximising behavior! • Firm’s revenue a function not only of what it does, but what other firms do (especially since can’t control what they do!) • So need to find maximum gap not w.r.t own output, but w.r.t. industry output • This is total derivative: not • What happens when firm really does maximise profits by setting total derivative equal to zero? This is impact of 2nd firm on profit of 1st: ignored by Cournot This is MR-MC

Cournot on Monopoly & Oligopoly • The real profit maximisation point for firm 1 is where this equals zero: What is this bit? This we’ve already worked out This is zero This expands to For profit maximisation for firm 1,the sum of these must equal zero: This equals

Cournot on Monopoly & Oligopoly • Adding them together: Adding these three together Symmetrical result for q2: Substituting: Multiply by 3: Cancel terms Ditto for q2:

Cournot on Monopoly & Oligopoly • So total output is: • Oh dear… same as for monopoly! • General formula is: • Oh dear… same as for monopoly again! • Output independent of number of firms in industry… • So if firms are profit maximisers, competition makes no difference.

Cournot on Monopoly & Oligopoly • Cournot oligopoly analysis therefore flawed • But game theoretic version (which reaches same result) is not • Implicitly assumes strategic interactions force firms not to profit-maximise but to produce more than profit-maximising ouput… • Based on analogy to “police interrogation trick” • The “prisoners’ dilemma”…

Von Neumann: Games & Strategic Behavior • Two people have • Committed a crime together • Been captured • Suspected but only weak evidence • Police offer deal to each: • “rat on the other and we’ll let you go free” • “if you don’t, you’ll get a year anyway” • Prisoners face trade-off: • Keep quiet and go to gaol for a year • Rat on the other and go free • But accomplice gets 10 years • If both rat, the sentence is split 50:50

Von Neumann: Games & Strategic Behavior • Prisoners’ Dilemma trade-off table: • Dilemma: • If both keep mum, 1 year each • If one rats, gets off scot-free • If not and other does, could get ten years • Analysis: • Both have incentive to “rat”; both get 5 years

Von Neumann: Games & Strategic Behavior • Same analysis applied by analogy to duopolists: • Maximum equilibrium profit if both follow “Keen” profit-maximisation strategy (called “collusive” in economic literature, “cooperate” in game lit.) • If one “defects” to MC=MR strategy while other sticks with profit-maximisation, gets: • Large increase in sales ( 1/3 – 1/4 =1/12th increase) • Smaller fall in price • Reduces profit of other firm too • So both have incentive to “defect” • Profit-maximisation equilibrium “unstable” • Each gains if “defects” • Defect/Defect combination a “Nash equilibrium” • Higher output/Lower profit outcome of strategic interaction

Von Neumann: Games & Strategic Behavior • With any price/cost combination • Profit maximising strategy (MR-MC=[n-1]/n * [P-MC]) best for both • “Defect” (increase output to where MR=MC) better for one if other doesn’t change output • MC=MR strategy “best response” of initial non-defector to defection • E.g.: P(Q)=100- 1/100000000 Q • C(q) = 100000000 + 50 q

Von Neumann: Games & Strategic Behavior • Firstly, quantities: • Then the price

Von Neumann: Games & Strategic Behavior • Finally profits • Aggregate profits higher if both profit maximise • But both have incentive to increase output if other doesn’t do so • So both increase and get higher output, lower profit with some output sold at a loss…

Von Neumann: Games & Strategic Behavior • So game theory gives valid justification for Cournot outcome: convergence to PC as number of firms rises • Firm’s quantity not decided in isolation but with respect to expected strategies of competitors • Strategic interaction force non-profit-maximising outcome on each firm • Extrapolated to many firms, result gives convergence to perfect competition • Non-profit-maximising “strategy” of setting MR=MC converges to P=MC as number of firms rises • But there’s a problem (or four)…

Von Neumann: Games & Strategic Behavior • “Nash equilibrium” unstable • Both firms have incentive to reduce production • “Prisoners’ Dilemma” solution also breaks down • If firms don’t have complete information • If game played more than once; and • Experimental outcomes contradict theory • Artificial firms • Don’t follow Nash strategy in actual interactions • Get higher profits by deviating from it

Von Neumann: Games & Strategic Behavior • “Defect/Defect” strategy unstable • One firm can increase profits if reduces output to “Keen” level while other stays at “Cournot” level • Both firms have incentive to increase production • More on instability later…

Von Neumann: Games & Strategic Behavior • Information • In true Prisoners’ Dilemma: • Both suspects know they are guilty • Both know that the other one knows • Both know the consequences of ratting or not ratting • Assumption of “perfect information” (except for knowing whether accomplice will rat) justified • In true industrial competition • No firm knows any other firms costs • No firm knows any other firms sales • Assumption of perfect information unjustified • Experiments with artificial firms result in no equilibrium, or non-predicted equilibrium, once information reduced from “perfect”:

Von Neumann: Games & Strategic Behavior • “Multiple equilibria are the norm in multi-period games of incomplete information, and Folk Theorems indicate that a small amount of incomplete information can produce almost any equilibrium payoffs when the discount rate is low and the horizon is long.” Richard Schmalensee, (1988), “Industrial Economics: An Overview”, Economic Journal, 98, p. 447 • And repeated games… • If prisoners know they will be made the same offer many times, sensible thing is not to rat ever… • If rat this time, one can “punish” other by ratting next time—so best strategy is to keep quiet • Ditto repeated “artificial economy” simulations: • “Firms” converge to “cooperate” strategy and make more profits as result

Von Neumann: Games & Strategic Behavior • Some economists try to sidestep problem by “backward iteration”: • “We know that if … the stage game is only played once … the potential for cooperation is quite slim: playing D is a strictly dominant strategy for each player… • An extension of this logic suggests that we should not expect to see cooperative behavior if the game is played twice, or 10 times, or even 100 times. To see this, apply backward induction: in the final game … playing D is a strictly dominant strategy for both players, so both players should expect the outcome (D, D) in the final game. We now look at the penultimate (second-to-last) game: knowing that (D, D) will be the outcome in the final game, playing D becomes a strictly dominant strategy in the penultimate game as well! We continue working backwards all the way to the beginning of the game, and conclude that the only rational prediction is for the players to play (D, D) in each of the stage games. So we cannot expect any cooperative behavior…” (Yoram Bauman 2004, Quantum Microeconomics, p. 98)

Von Neumann: Games & Strategic Behavior • But this depends on beliefs of other player’s behaviour: • “Consider the prisoners' dilemma… Players believe that the opponent cooperates in the first period. From period 2 until period m-3, the opponent cooperates if both players have always cooperated. Otherwise the opponent defects. If both players have cooperated in period m-3, then, in period m-2, the opponent cooperates or defects with equal probability. Otherwise the opponent defects. In the last two periods, the opponent defects. Both players' best response is to cooperate until period m-3 (included), and to defect in the last three periods.” (Alvaro Sandroni, 1998. “Does Rational Learning Lead to Nash Equilibrium in Finitely Repeated Games?” Journal of Economic Theory (78), p. 200)

Von Neumann: Games & Strategic Behavior • “Much interest in recent years has attached to repeated games … in which a relatively simple constituent or stage game (such as the one-period Cournot or Prisoners’ Dilemma games) is played repeatedly by a fixed set of players. Strategies of simply playing Nash equilibrium strategies of the constituent game in each period form a Nash equilibrium of the repeated game. But strategies in the repeated game may involve taking actions conditional on past history, and there are usually many other equilibria when the horizon is infinite. In fact, many of the main results in the supergame literature are variants on the so-called Folk Theorem, which says that virtually any set of payoffs can arise in a perfect equilibrium if the horizon is long enough and the discount rate is low enough.” Schmalensee p. 647.

Von Neumann: Games & Strategic Behavior • So game theory looks like a sound way to justify perfect competition… • Until you go beyond one-off game • “Perfect competition” only applies for “single shot” game • Recommended economic strategy is “always defect” • But “always defect” just one of many possible strategies in repeated games • Others include “tit for tat”: • cooperate on first play • cooperate if opponent cooperates • if opponent defects, defect yourself next time: “punish” non-cooperation • Tends to win game contests Reality check time!

Economic modelling • What is game theory supposed to do? • Make economic theory more realistic by considering interaction between agents • Marshallian theory assumes “atomism” • No reaction to what other firms might do • Game theory allows for reactions to what other firms might do • While still assuming rational behavior • Both Marshallian & game theory not supposed to be what firms actually do; • Instead, supposed to model “as if” behavior • “Whatever they say they’re doing, they must really be doing this if they are to be successful…” • Friedman’s argument:

Economic modelling • “Consider the problem of predicting the shots made by an expert billiard player… excellent predictions would be yielded by the hypothesis that the billiard player made his shots as if he knew the complicated mathematical formulas that would give the optimum directions of travel, …, could make lightning calculations from the formulas, and could then make the balls travel in the direction indicated by the formulas. Our confidence in this hypothesis is not based on the belief that billiard players, even expert ones, can or do go through the process described; it derives rather from the belief that, unless in some way or other they were capable of reaching essentially the same result, they would not in fact be expert billiard players.” (Friedman 1953 “The Methodology of Positive Economics”)

Economic modelling • “It is only a short step from these examples to the economic hypothesis that under a wide range of circumstances individual firms behave as if they were seeking rationally to maximize ... "profits" ... as if, that is, they knew the relevant cost and demand functions, calculated marginal cost and marginal revenue from all actions open to them, and pushed each line of action to the point at which the relevant marginal cost and marginal revenue were equal. Now, of course, businessmen do not actually and literally solve the system of simultaneous equations ... The billiard player, if asked how he decides where to hit the ball, may say that he "just figures it out" … the businessman may well say that he prices at average cost, with of course some minor deviations when the market makes it necessary. The one statement is about as helpful as the other, and neither is a relevant test of the associated hypothesis.” (Friedman 1953)

Multi-agent modelling • Friedman’s purpose in “The Methodology of Positive Economics”: to stop economists looking at empirical research (Blinder’s predecessors) • Economists prefer to try to prove what firms do rather than find out… • But fail… proofs either wrong (Marshallian) or inconclusive, not robust as realism increases (game theory) • Computers offer new approach between proof and questionnaires: simulate & see what happens • Model large populations using computer simulations of agents • Carefully design behavior; see what well-defined agents do

Multi-agent modelling • Rather than predicting what profit-maximising firms will do, let’s “find out” with simulation • Define profit maximisers in terms of behavior rather than calculus • “instrumental profit maximisers” • Try something (e.g., increase output) • If profit increases, do same again • If profit falls, reduce output • Model single market, demand curve • No assumptions about knowing/applying calculus, etc. • Just computer programming: • Give computer precise instructions • See what happens!

Multi-agent modelling • Basic idea: • Define demand curve & cost functions • Create random list of initial outputs for n firms • Work out initial price given sum of outputs • Create random list of variations in output for n firms • FOR a number of iterations • Add variation to output of each firm • Work out new price level • FOR each firm • Work out whether profit has risen or fallen • IF rose, keep going the same way; ELSE • IF fell, reverse direction • See whether output converges to Cournot or “Keen” prediction • Trying it out with sample demand curve & ten firms…

Multi-agent modelling • P(Q)=100- 1/100000000 Q • C(q) = 100000000 + 50 q • Cournot/Game theory prediction: firms equate MR & MC • “Keen” profit maximisation prediction: firms produce where MR–MC equals (n-1)/n times P-MC

Multi-agent modelling • Start with randomly allocated list of outputs by ten firms • Random initial quantity between Cournot & Keen predictions for each firm • Initial outputs • Next step: work out initial profits:

Multi-agent modelling • Work out vector of changes in output (much smaller amounts than the initial output so that firms won't end up “producing” negative amounts) • 6 firms reduce output and 4 increase: • Aggregate output drops a bit: • Next what are new profit levels?

Multi-agent modelling • Curiousity point: some firms lose profit by reducing output; others increase! • Firms 0-4 decreased output & saw profit fall • Firm 6 decreased output & saw profit rise • Cause: elasticity interactions between size of aggregate price change & size of individual output change

Multi-agent modelling • Firms 0-4 saw profit fall, so they will alter the direction of their output changes • Firms 4-9 increased profit, so they continue in the same direction • If profit rose, this function returns 1; if it fell -1: • This is multiplied by the dq amounts and added to second period output to work out third period…

Multi-agent modelling • The entire program: Random number generator Random initial outputs Random change amounts For r iterations Calculate market price Change outputs Calculate new market price For each firm Change direction if profit has fallen

Multi-agent modelling • Result for 400 firms: • Converges towards Keen rather than Cournot… • BUT doesn’t quite reach it; apparent complex interaction effects between firms…

Multi-agent modelling • But not ones predicted by game theory • Reason: local instability of Cournot equilibrium, stability of Keen • If duopoly begins at Cournot or Keen level & firms change output seeking higher profit, 1st mover will gain at expense of 2nd • 2nd will change behavior, affecting 1st mover • Sustainable change only if both increase profit • Local region around equilibria determines what happens next • Region around Cournot unstable: both firms gain if both reduce output • Region around Keen stable: no sustainable pattern of output change possible:

Multi-agent modelling • Cournot near-equilibrium dynamics • “True” means profit rise from change in output • Only sustainable pattern where both firms have “True” result • Sustainable change from Cournot, not from Keen… • Keen near-equilibrium dynamics

Multi-agent modelling • Outputs from 3 sample firms • Many varying individual behaviors…

Multi-agent modelling • Multi-agent results show • Generally Keen formula better predictor of behavior than Cournot • Increasing number of firms reduces likelihood of Cournot outcome • Model still makes false assumption of rising marginal cost • But shows new analytic approach: • Accurately define conditions of economy • Simulate behavior to see whether theories describe reality well • Next week: sophisticated multi-agent model of innovation & competition

Managerial Economics