Managerial Economics

1
Managerial Economics

• Lecture Five
• What is competition?
• The standard view (warts all!)

2
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

3
Neoclassical models of strategic interaction

• Standard (Marshallian) model critiqued in first 2
  lectures has fundamental flaws
• Horizontal firm demand curve cant 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
  doesnt apply in real world
• Newer Cournot-Nash-Game Theory strategic
  interaction model not so (obviously) flawed

4
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/Marshall
  ian 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

5
Cournot-Game Theoretic Competition Models

• Game theory models dont have this problem
• Firms presumed to react to expected strategies of
  rivals
• In Marshallian theory dqj/dqi0 one firm doesnt
  react to what other firms (might) do
• In Game theory dqj/dqi not 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 didnt begin till 1870s
• But one founder (Walras) corresponded with Cournot

6
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
• Cournots 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 explanationbut
  reaches same result as later game theory approach
  of John von Neumann beautiful minds John Nash
• with a little problem well discuss after
  trudging through the math

7
Cournot on Monopoly Oligopoly

• Cournots analysis assume linear demand cost
  functions
• (original paper actually assumed zero costs!)

• ProfitRevenue minus Cost

• Maximum profit when gap between functions biggest
• Biggest gap when differential w.r.t. Q is zero

Add these
8
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 Qq1q2

• Therefore each firm in a duopoly solves

9
Cournot on Monopoly Oligopoly

• Assume identical costs C(x)cdx
• Working out quantity for firm 1

• Get symmetrical result for second firm

10
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
11
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 theres a problem

12
Cournot on Monopoly Oligopoly

• Equating MC and MR not profit maximising
  behavior!
• Firms revenue a function not only of what it
  does, but what other firms do (especially since
  cant 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
13
Cournot on Monopoly Oligopoly

• The real profit maximisation point for firm 1 is
  where this equals zero

What is this bit?
This weve already worked out
This is zero
This expands to
For profit maximisation for firm 1,the sum of
these must equal zero
This equals
14
Cournot on Monopoly Oligopoly

• Adding them together

Adding these three together
Symmetrical result for q2
Substituting
Multiply by 3
Cancel terms
Ditto for q2
15
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.

16
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

17
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 well let you go free
• if you dont, youll 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 5050

18
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

19
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 MCMR 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

20
Von Neumann Games Strategic Behavior

• With any price/cost combination
• Profit maximising strategy (MR-MCn-1/n
  P-MC) best for both
• Defect (increase output to where MRMC) better
  for one if other doesnt change output
• MCMR strategy best response of initial
  non-defector to defection

• E.g. P(Q)100- 1/100000000 Q
• C(q) 100000000 50 q

21
Von Neumann Games Strategic Behavior

• Firstly, quantities

• Then the price

22
Von Neumann Games Strategic Behavior

• Finally profits

• Aggregate profits higher if both profit maximise
• But both have incentive to increase output if
  other doesnt do so
• So both increase and get higher output, lower
  profit with some output sold at a loss

23
Von Neumann Games Strategic Behavior

• So game theory gives valid justification for
  Cournot outcome convergence to PC as number of
  firms rises
• Firms 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 MRMC
  converges to PMC as number of firms rises
• But theres a problem (or four)

24
Von Neumann Games Strategic Behavior

• Nash equilibrium unstable
• Both firms have incentive to reduce production
• Prisoners Dilemma solution also breaks down
• If firms dont have complete information
• If game played more than once and
• Experimental outcomes contradict theory
• Artificial firms
• Dont follow Nash strategy in actual interactions
• Get higher profits by deviating from it

25
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

26
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

27
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 timeso best strategy is to keep
  quiet
• Ditto repeated artificial economy simulations
• Firms converge to cooperate strategy and make
  more profits as result

28
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)

29
Von Neumann Games Strategic Behavior

• But this depends on beliefs of other players
  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)

30
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.

31
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!
32
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 theyre doing, they must
  really be doing this if they are to be
  successful
• Friedmans argument

33
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)

34
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)

35
Multi-agent modelling

• Friedmans purpose in The Methodology of
  Positive Economics to stop economists looking
  at empirical research (Blinders 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

36
Multi-agent modelling

• Rather than predicting what profit-maximising
  firms will do, lets 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!

37
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

38
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 MRMC equals (n-1)/n times P-MC

39
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

40
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?

41
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

42
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

43
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
44
Multi-agent modelling

• Result for 400 firms

• Converges towards Keen rather than Cournot
• BUT doesnt quite reach it apparent complex
  interaction effects between firms

45
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

46
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

47
Multi-agent modelling

• Outputs from 3 sample firms

• Many varying individual behaviors

48
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