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Part 3: Game Theory INash Equilibrium: Applications
Oligopoly, Cournot Competition, Bertrand Competition,Free Riding Behavior, Tragedy of the Commons
June 2016
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Illustrating Nash Equilibrium
many models use notion of Nash equilibrium to study economic, political orbiological phenomena
often, these games involve continuous actions
examples:
firms choosing a business strategy in an imperfectly competitive market (price,output, investment in R&D)
candidates in an election choosing platforms (policies)
animals fighting over prey choosing time at which to retreat
bidders in auction choosing bid
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Oligopoly
a market or an industry is an oligopoly if it is dominated by a small numberof sellers (oligopolists) who each have a non-negligible effect on prices
oligopoly is a market form in between perfect competition and monopoly
various economic models study oligopoly:
Cournot model (quantities, homogeneous good)
Bertrand model (prices, homogeneous good)
price competition with differentiated products
Hotelling model of product differentiation
and many more...
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Cournot Competition
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The Cournot Model
two firms i = 1, 2 produce a homogeneous product
firm i’s output qi ≥ 0, constant marginal costs c
total industry output Q = q1 + q2inverse demand function p(Q) = a− b(q1 + q2)
firms simultaneously choose own output qi, taking the rival firm’s output qjas given → strategies are qi’s
firm i maximizes
πi = (a− bQ)qi − ciqi s.t. qj is given
FOC ∂πi
∂qi= 0 → best response functions qi = qbri (qj)
(q∗1 , q∗2) Nash equilibrium if q∗1 = qbr1 (q∗2) and q
∗2 = qbr2 (q∗1)
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Analysis
maximizing profit yields
maxq1
π1 = (a− b(q1 + q2))q1 − cq1 ⇒ qbr1 (q2) =
a− c2b−
1
2q2
maxq2
π2 = (a− b(q1 + q2))q2 − cq2 ⇒ qbr2 (q1) =
a− c2b−
1
2q1
solving for the NE q∗1 = qbr1 (q∗2) and q∗2 = qbr2 (q∗1) gives
q∗1 = q∗2 =a− c3b
and π∗1 = π∗2 =(a− c)2
9b
compare outcome to monopoly
q∗1 + q∗2 > qm =a− c2b
, p∗ < pm and π∗1 + π∗2 < πm =(a− c)2
4b
symmetric market with n firms:
q∗i =a− c
b(n+ 1), p∗ =
a+ nc
(n+ 1), and π∗i =
(a− c)2
b(n+ 1)2
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Graphic Illustration
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Graphic Illustration
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Graphic Illustration
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Graphic Illustration
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Graphic Illustration
firm 1’s isoprofit curve π1(q1, q2) = constant is tangent to q2 = q∗2 line
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Graphic Illustration
firm 1’s isoprofit curve π1(q1, q2) = constant is tangent to q2 = q∗2 line
firm 2’s isoprofit curve π2(q1, q2) = constant is tangent to q1 = q∗1 line
there are (q1, q2) combinations (the shaded area) that would make both firms better off → the NEis not efficient (Pareto optimal) from the firm’s point of view
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Summary
In a Cournot oligopoly
each firm’s profit is decreasing in the other firm’s quantityin producing output, firms impose negative externalities on rivals
⇒ in equilibrium, the firms produce too much output relative to joint profitmaximization (monopoly); they fail to maximize their joint profit
firms have an incentive to collude/cooperate
firms also have an incentive to cheat on any collusive/cooperative agreement(any agreement other than a NE is not self-enforcing)
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Bertrand Competition
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The Bertrand ModelNamed after Joseph Louis François Bertrand (1822-1900)
two firms i = 1, 2 produce a homogeneous product
firm i’s price pi ≥ 0, constant marginal cost ci
linear demand function
Di(pi) =
a− pi if pi < pj
(a− pi)/2 if pi = pj
0 if pi > pj
firms simultaneously choose prices pi, taking the rival firm’s price pj asgiven → strategies are pi’s
firm i maximizes πi = (pi − c)Di(pi) taking pj as given
(p∗1, p∗2) Nash equilibrium if p∗1 = pbr1 (p∗2) and p
∗2 = pbr2 (p∗1)
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Analysis
by pricing just below the rival firm, each firm can obtain the full marketdemand D(p) → strong incentive to ‘undercut’ one’s rival
claim: in the unique NE, p∗1 = p∗2 = c (marginal cost pricing)= same outcome as perfect competition!
to see this, look at firm 1’s best response to p2
if p2 ≤ c, firm 1 can set p1 = c (makes no profit anyway)
if c < p2 ≤ pm, firm 1 should set p1 = p2 − ε
if p2 > pm, firm 1 should set p1 = pm
analogous for firm 2
⇒ best responses intersect only at p1 = p2 = c
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Graphic Illustration
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Summary
In a Bertrand oligopoly:
each firm’s profit is increasing in the other firm’s pricein raising their price, firms impose positive externalities on rivals
⇒ in equilibrium, the firms set the price too low relative to joint profitmaximization
Similarities to Cournot:
price below monopoly price
firms fail to maximize their joint profit → incentive to collude/cooperate
still: incentive to cheat on any collusive/cooperative agreement(any agreement other than a NE is not self-enforcing)
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Summary
Differences to Cournot:
Bertrand predicts two firms is enough to generate marginal cost pricing
in practice: differentiated products → p > MC
if capacity and output can be easily changed, Bertrand fits situation better;otherwise Cournot
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The Free Rider Problem
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Games of Collective Action
games of collective action = situations where the benefit to group dependson the actions (efforts) of all members
in collective action games, individuals have a tendency to free ride on thecontribution of others; they contribute nothing/too little themselves but stillreap the benefit
primary example: contributing to a public good
in general, the free-rider problem means that
there are too few ‘volunteers’
there is too little group effort
⇒ benefit from collective action too low → Pareto inefficient outcomes (marketfailure)
the problem gets worse as the group becomes larger
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Ex: Private Provision of a Public Good
two roomates, Harry (H) and Sally (S)
public good = cleanness of apartment G, utilities are
uS = 40√G+money uH = 20
√G+money
a cleaning lady will take G hours to produce G and costs $ 10/hour →C(G) = 10G and MC(G) = 10
Pareto optimal amount is at MBS +MBB =MC,
201√G
+ 101√G
= 10 ⇒ Geff = 9
if i = H,S contributes $xi, total cleaning budget is xH + xS and cleanness isG = 1
10 (xH + xL)
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Contributing to a Public Good (contd.)
how much will each person contribute?
answer depends on how much each expects the roommate to contribute
→ strategic game
→ solve using NE concept
if S expects H to contribute xH , her optimal contribution solves
maxxS
uS = 40
√1
10(xS + xH)− xS s.t. xH given, xS ≥ 0
FOC gives
xbrS (xH) = 40− xH for xH ≤ 40 (otherwise xbrS (xH) = 0)
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Contributing to a Public Good (contd.)
if H expects S to contribute xS , his optimal contribution solves
maxxH
uH = 20
√1
10(xS + xH)− xH s.t. xH given, xH ≥ 0
FOC gives
xbrH (xS) = 10− xS for xS ≤ 10 (otherwise xbrH (xS) = 0)
(x∗S , x∗H) Nash equilibrium if x∗S = xbrS (x∗H) and x∗H = xbrH (x∗S)
NE is at x∗S = 40 and x∗H = 0 ⇒ G∗ = 4 < 9 = Geff
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Graphic Illustration
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Graphic Illustration
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Graphic Illustration
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Graphic Illustration
Sally’s indifference curve uS(xS , xH) = constant is tangent to xH = x∗H line
Harry’s indifference curve uH(xS , xH) = constant is not tangent to xS = x∗S line
there are (xS , xH) combinations that would make both better off→ the NE is not efficient (Pareto optimal) from collective point of view
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Contributing to a Public Good (contd.)
In private provision of public goods:
each person’s utility is increasing in the other person’s contribution→ positive externality (not internalized)
each person’s optimal contribution is decreasing in the other person’scontribution→ free riding behavior
⇒ in equilibrium, people contribute too little relative to joint surplus (welfare)maximization
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Contributing to a Public Good (contd.)
Other instances where similar problem occur:
tragedy of the commons = overuse of common resource
public infrastructure (roads, parks)
natural resources (oceans, air, water)
other games of collective action need not share same problems(e.g. adopting a common standard)
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Solving Collective Action Problems
Free riding and other problems in collective action games can often be mitigatedor solved by:
detection and punishment/rewards
sanctions, customs, and social norms(in repeated interaction)
government provision
government regulation (taxes, subsidies)
Coasian bargaining
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