Watson Strategy, Chap 10

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Watson Strategy, Chap 10

• Erik Gartzke
• Columbia University
• Department of Political Science
• W4209 Game Theory and Political Theory
• Spring 2002 Week 15
• Web www.columbia.edu/eg589

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Outline, Watson Chap 10

• Chap 10 Oligopoly, Tariffs, and Crime and
  Punishment
• Cournot Duopoly Model
• Bertrand Duopoly Model
• Tariff Setting by Two Countries
• A Model of Crime and Police

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Chap 10 Oligopoly, Tariffs, and Crime and
Punishment

• Cournot Duopoly Model
• 2 Firms produce identical good (red brick)
• Simultaneously and independently select how many
  bricks to produce (q1 for firm 1, q2 for 2)
• Price depends on quantity (p 1000 q1 q2)
• Production cost 100/1000 bricks
• Firms select quantity S1 0, ?) and S2 0,?)
• Firms payoff is profit (price x quantity cost)

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Cournot Duopoly Model

• Firm 1s payoff is
• u1(q1,q2) (1000 q1 q2)q1 100q1
• Firm 2s payoff is
• u2(q1,q2) (1000 q1 q2)q2 100q2
• Best response functions for Firms
• ?u1/?q1 1000 2q1 q2 100
• Setting this equal to zero and solving for q1, we
  get
• q1 450 q2/2 (Firm 1s best response
  function)
• Game is symmetric, so for Firm 2 BR2(q1) 450
  q1/2

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Cournot Duopoly Model

• Determine the quantities produced by each Firm
• BR1(q2) q1 450 (450 q1/2)/2
• q1 225 q1/4
• q1 300
• BR2(q1) q2 450 (450 q2/2)/2
• q2 225 q2/4
• q2 300
• Nash Eq. is inefficient
• Each firm better off producing at 225 (max
  payoff)
• Cannot credibly commit to this (PD-type problem)

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Bertrand Duopoly Model

• Cournot model has firms selecting quantity
• In reality, firms select both price and quantity
• Demand implies definite relationship between p
  and q
• Think of firms selecting p or q, then market set
  other
• Makes sense to think of firms selecting quantity
• Butstrategic differences between p and q
• Additional insight from model where firms select
  p
• Two firms simultaneously, independently select p
• Industry demand p 1000 q1 q2
• Can be written as Q 1000 p where Q q1 q2

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Bertrand Duopoly Model

• Facing p, consumers demand 1000 p 1000 units
  brick
• Specify the normal form of the game
• Firm that charges higher price sells no bricks
  (qi 0)
• Splits quantity if firms offer same price (1000
  p)/2
• Firm is payoff if pi lt pj
• ui(p1,p2) (1000 pi)pi 100(1000 pi)
• (1000 pi)(pi 100)
• Firm is payoff if pi gt pj is 0
• Firm is payoff if pi pj
• ui(p1,p2) (1000 pi)pi /2 100(1000
  pi)/2
• (1000 pi)(pi 100)/2

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Bertrand Duopoly Model

• Neither firm can price below 100 in equilibrium
• If so, at least one firm earns negative profit
• Firms can guarantee profit of 0 by pricing at 100
• pi gt pj ? 100 cannot be the case
• If pi gt 100, firm j raises profit charging pi ?
  pj gt 100
• pi pj gt 100 is not an equilibrium
• Firm i captures all of market by charging 100 ?
  pi lt pj
• Only possible equilibrium is pi pj 100
• Price equals marginal cost
• Equilibrium in Bertrand profit, Cournot profit
  gt 0
• Price setting markets more competitive than
  quantity setting

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Tariff Setting by Two Countries

• Governments influence trade by imposing barriers
  that restrict trade
• Tariff tax on import of foreign commodities
• Large states can be better off with small tariff
• Assuming other states do not raise tariffs
• Example EU tariff on bananas
• Tariff creates wedge between international
  price and domestic price for good
• If wedge is large enough, tariff revenue may be
  larger than welfare loss to own country from
  tariff

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Tariff Setting by Two Countries

• States are made worse off by uniformly high
  tariffs
• States benefit by finding way to enforce free
  trade
• Game
• Two countries (1 and 2)
• xi is tariff level of country i, for i 1,2 0 ?
  xi ? 100
• Payoffs ui(xi,xj) 2000 60xi xixj xi2
  90xj
• ?ui /?xi 60 xj 2xi , 60 xj 2xi 0
• xi 30 xj /2
• Substituting, xi 30 (30 xi)/2 60

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A Model of Crime and Police

• Optimal enforcement levels depend on
• How much it costs to catch and convict offenders
• The nature of punishments
• Response of offenders to changes in enforcement
• Game
• Two players criminal (C) and government (G)
• Gov. selects level of law enforcement x x ? 0
• Gov. payoff uG xc4 y2/x
• y2/x is negative effect of crime on society
• c4 is cost of law enforcement (positive constant)

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A Model of Crime and Police

• Criminals payoff uC y1/2/(1 xy)
• y1/2 is value of criminal activity to criminal
  (if not caught)
• 1/(1 xy) is the probability that criminal
  evades capture
• Solution
• ?uG/?x c4 y2/x2
• c4 y2/x2 0, c4 y2/x2, x y/c2
• ?uC/?y (xy1/2)/(1 xy)2 1/(2y1/2(1 xy))
• (xy1/2)/(1 xy)2 1/(2y1/2(1 xy)) 0
• (xy1/2)/(1 xy)2 1/(2y1/2(1 xy)), y 1/x
• Substituting
• x (1/x)/c2, x 1/c, y c