MsC Microeconomics Lecture 8

1
MsC MicroeconomicsLecture 8

• Francesco Squintani
• University of Essex

2
Competitive Markets

• A competitive firm takes the market price of
  output as given and outside of its control.
• Let p be the market price.
• Then the demand curve faced by the firm is
• 0 if pgtp
• D(p) 0, infinity if pp
• infinity if pltp

3

• Because the firm takes price as given, the profit
  maximization problem is
• max pq c(q)
• q
• Hence, the firm produces the quantity q such
  that
• p c(q)
• price equals marginal cost,
• provided that c(q)lt0, costs are convex.

4

• This provides the firm supply function
• p c(q) if p gt AVC(q)
• 0 if p lt AVC(q)
• Note that the firm cannot recoup fixed costs, not
  even if it produces zero output. Hence it is
  willing to produce even if p lt AC.

c, p
MC
AC
AVC
q
5

• The market supply function is simply the sum of
  the individual supply functions Y(p) Si qi
  (p).
• The market demand function is the sum of
  individual demand functions X(p) Si xi (p).
• The market equilibrium is given by Si qi (p) Si
  xi (p).

p
Y(p)
X(p)
y
6
Competitive Markets - Welfare

• Say for simplicity that each consumer consumes
  only one unit of good.
• The welfare of a consumer j who buys the good at
  price p is uj p.
• All consumers j such that uj gt p buy the good.
• Hence the aggregate surplus of consumers is
  Sjuj gtp uj p.

p
Y(p)
The aggregate surplus of consumers is
depicted by the blue triangle, because consumer
js utility for the good coincides with her
willingness to pay.
X(p)
y
7

• Say for simplicity that each producer produces
  only one unit of good.
• The welfare of a producer i who sells the good at
  price p is p cj.
• All producers i such that cj lt p buy the good.
• Hence the aggregate surplus of producers is Sic
  i lt p p -c.

p
Y(p)
The aggregate surplus of producers is
depicted by the green triangle, because the
supply curve coincides with MC curve.
X(p)
y
8
Entry

• If firms make a profit in the short-run, new
  firms will enter the market, so that in the long
  run, profits are zero.
• Similarly, if firms make a loss in the short-run,
  some firms will exit the market, so that in the
  long run, profits are zero.
• In the long run, the supply curve is flat and
  coincides with the minimum of MC(q), i.e. the
  output q where AC(q)MC(q).

c, p
MC
AC
AVC
q
9
Monopoly

• A monopolist is the only supplier of a good or
  service.
• Because it does not face any competition, it is
  price-maker.
• Hence, given the inverse demand function P(q), it
  chooses output q so as to maximize p(q) P(q)q
  c(q).
• The first order conditions are
• p(q)p(q)q MR(q) MC(q) c(q).
• Rearranging,
• p(q)11/e(q) c(q),
• where e(q) is the elasticity of demand.

10

• Recalling that e(q) lt 0, we see that e(q) gt 1,
  or else MRlt0, and the monopolist would produce
  zero.
• Hence, the monopoly price p(qm) is larger than
  marginal cost, the price p(q) chosen by a
  competitive firm.

p
MC
p(qm)
p(q)
X(q)
MR
q
qm
q
11
Monopoly - Welfare

• The producer welfare is higher under monopoly
  than under competition. But the consumer welfare
  is smaller.
• There is an aggregate welfare loss, the
  red triangle.

p
MC
p(qm)
p(q)
X(q)
MR
q
qm
q
12
Price Discrimination

• The monopolist would like to price discriminate
  among its buyers, i.e. sell at a higher price to
  those who value the good most. This allows him to
  increase its profit.
• Examples
  of price discrimination
  include low prices for
  students and elderly.
• In the extreme case
• the monopolist
• captures all
• surplus.

p
MC
p(qm)
X(q)
MR
q
qm
13
Cournot's model of oligopoly

• A good is produced by n firms.
• Firm is cost of producing qi units is Ci(qi)
• (Ci is an increasing function).
• The firms' total output is Q q1 qn.
• The market price is P(Q)
• (P is the inverse demand function, it is
  decreasing when positive).
• Firm i's revenue is qi P(q1 qn).

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• Firm i's profit is revenue minus cost
• pi(q1 qn) qi P(q1 qn) - Ci (qi).
• Linear Costs and Demand
• Ci (qi) cqi, i1, , n.
• P (Q) a Q if a gt Q, P(Q) 0 if a lt Q.
• pi(q1 qn) qi a (q1 qn) - cqi.
• To find the optimal quantity, differentiate pi
  with respect to qi, set it equal to zero, and
  obtain
• dpi (q1 qn)/dqi a (q1 qn) qi
  c 0.
• Best Response functions
• bi (q-i) a (q1 qi-1 qi1 q n)
  c/2.

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• Solving for the Nash equilibrium.
• Because the system of best-response functions is
  linear and symmetric, equalize qi across i
  1,,n
• qi bi (qi) a (n-1) qi c/2.
• The Nash equilibrium quantity is
• qi a c/(n1).
• The Nash equilibrium price is
• pi (q) a Q a na c/(n1)
• a nc/(n1).
• The Nash equilibrium profits are
• pi (q) qia Q cqi
• a c 2/(n-1)2

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q2

• With n 2,
• b1(q2) a q2 c/2.
• b2(q1) a q1 c/2.

b1(q2)
(q1, q2) qi a c/3, i1,2.
a c/3
b2(q1)
q1
a c/3
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• Comparison with Perfect Competition
• In a perfectly competitive market,
• price equals marginal cost pi c,
• profits are zero pi 0.
• In the Nash Equilibrium of Cournot model,
• price is pi (qi) c a c /(n1) gt c,
• profits are pi (qi) a c 2/(n-1)2 gt 0.
• In the limit for large n,
• price pi (qi) c a c /(n1) converges
  to c
• profits pi (qi) a c 2/(n-1)2 go to zero.

18
Comparison with Collusion

• Suppose the firms collude and act like a
  monopolist maximizing the sum of profits
• p(Q) Qa Q cQ.
• Differentiating the optimal quantity, we have
• Qm a c/2.
• In the Nash Equilibrium of Cournot model,
• Q na c/(n1).
• The firms would like to collude, but cannot
  commit
• (as in the Prisoners Dilemma).

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• In Cournot competition, firms cannot collude.
• But perfectly competition is not achieved either.

P
Cournot competition
P(Qm)
P(Q)
Q
Qm
Q
20
Bertrand Competition

• Unlike Cournot competition, firms compete in
  prices.
• The demand function is denoted by D, if the good
  is available at the price p then the total amount
  demanded is D(p).
• The firm setting the lowest price sells to all
  the market.

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Linear Costs and Demand

• Ci (qi) cqi, i1, , n.
• D (p) a p if a gt p, D (p) 0 if a lt p.
• Let pj min p1, , pn.
• Profit is
• pi(p1, , pn) (pi c)(a - pi) if pi lt pj,
• pi(p) (pi c)(a - pi)/m if pi pj,
• pi(p) 0 if pi gt pj.

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Best-Response Correspondence

• If pj lt c, then pi(p) lt 0 for pi lt pj,
• pi(p) 0 for pi gt pj bi(p) pi pi gt pj.

pi
pj
pi
c
pm
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• If pj c, then pi(p) lt 0 for pi lt pj,
• pi(p) 0 for pi gt pj bi(p) pi pi gt pj.
• If pj gt pm, then bi(p) pm.

pi
pi
pj
c
c
pm
pm
pj
pi
pi
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• If c lt pj lt pm then pi(p) increases in pj, but
  discontinuously drops at pi pj. So, bi(p) f.
• The best response correspondence is empty.

pi
pj
pi
c
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• In sum, the best-response correspondence is
• bi(p) pi pi gt pj, if pj lt c,
• bi(p) pi pi gt pj, if pj c,
• bi(p) f, if c lt pj lt pm,
• bi(p) pm, if pj gt pm.
• The Nash equilibrium is pi c, for all i
  1,,n.
• Intuitively, selling at any price pi lt c yields
  negative profit. If the lowest industry price
  were pj gt c, then firm i sells to the whole
  industry at any price pi with c lt pi lt pj. In
  equilibrium, pi c, for all i.

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Stackelbergs model of duopoly

• There are two firms in the market.
• Firm i s cost of producing qi units of the good
  is Ci (qi) the price at which output is sold
  when the total output is Q is P(Q).
• Each firms strategic variable is output, as in
  Cournots model, but the firms make their
  decisions sequentially, rather than
  simultaneously one firm chooses its output, then
  the other firm does so, knowing the output chosen
  by the first firm.

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• Extensive form of Stackelberg Game
• Players The two firms.
• Terminal histories The set of all sequences (q1,
  q2) of outputs for the firms (where each qi, the
  output of firm i, is a nonnegative number).
• Player function P(Ø) 1 and P(q1) 2 for all
  q1.
• Preferences The payoff of firm i to the terminal
  history (q1, q2) is its profit qi P(q1 q2) -
  Ci(qi), for i 1, 2.

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• A strategy of firm 1 is simply an output choice
  q1.
• Firm 2 moves after every history in which firm 1
  chooses an output. A strategy of firm 2 is a
  function that associates an output q2 to each
  possible output q1 of firm 1.
• We use backward induction to find its subgame
  perfect equilibria.
• For any q1 output of firm 1, we find the output
  b2 (q1) of firm 2 that maximize its profit q2
  P(q1 q2) C2 (q2).
• In any subgame perfect equilibrium, firm 2s
  strategy is b2.
• We find the output q1 of firm 1 that maximize its
  profit, given the strategy b2 (q1) of firm 2.
• Firm 1s output in a subgame perfect equilibrium
  is the value q1 that maximizes q1P(q1 b2(q1)) -
  C1 (q1).

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• Suppose that Ci(qi) cqi for i 1, 2, and P(Q)
  a - Q if Q a, P(Q) 0 if Q gt a, where c gt 0
  and c lt a.
• We know that firm 2s best response to output q1
  of firm 1 is b2(q1) (a - c - q1)/2 if q1 a -
  c, b2(q1) 0 if q1 gt a - c.
• In a subgame perfect equilibrium of Stackelbergs
  game firm 2s strategy is this function b2 and
  firm 1s strategy q1 maximizes q1 (a - c - (q1
  (a - c - q1)/2))
• q1 (a - c - q1)/2.
• The FOC yields q1 (a - c)/2.
• The unique subgame perfect equilibrium is (q1,
  b2).

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• The SPE outcome of the Stackelberg game is
• q1S (a - c)/2, q2S (a - c - q1 )/2 (a -
  c)/4.
• Firm 1s profit is q1S(P(q1Sq2S) - c) (a -
  c)2/8, firm 2s profit is q1S(P(q1Sq2S) - c)
  (a - c)2/16.
• Recall that in the unique Nash equilibrium of
  Cournots (simultaneous-move) game, q1C q2C
  (a - c)/3, so that each firms profit is (a -
  c)2/9.
• First-Mover Advantage Firm 1 produces more
  output and obtains more profit in the subgame
  perfect equilibrium of the sequential game where
  it moves first than it does in the Nash
  equilibrium of Cournots game, and firm 2
  produces less output and obtains less profit.

31
Repeated Oligopoly

• Suppose that n firms repeatedly play a Bertrand
  oligopoly game.
• If the discount factor is not too small, they can
  collude on the monopoly price and split the
  profits equally.
• Suppose that there are two firms, for simplicity.
• Consider the following Grim Trigger strategy,
  which prescribes that the player initially
  collude, and continues to do so if all players
  collude at all previous times.
• pi (a1, . . . , aT) pm if at (pm, pm) for
  all t 1, . . . , T.
• si (a1, . . . , aT) c otherwise.

32
Grim Trigger SPE

• Suppose that both players adopt the grim trigger
  strategy.
• There are two sets of histories. Those for which
  grim trigger strategy prescribes that the players
  play (pm, pm) and those for which the grim
  trigger strategy prescribes that they do not play
  (pm, pm) .
• In the first set of histories, if player i plays
  grim trigger, then the outcome is (pm, pm) in
  every period with payoffs P(qm)qm c(qm)/2,
  whose discounted average is P(qm)qm c(qm)/2.
• If i deviates only once, she plays p lt pm. Then
  she reverts to the grim trigger strategy, that
  prescribes to play c at all subsequent periods.

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• The opponent, playing grim trigger strategy,
  plays c forever as a consequence of is one-shot
  deviation.
• Let q P-1(p). The OSD yields the stream of
  payoffs
• (P(q)q c(q), 0, 0, . . .) with discounted
  average
• (1 - d) P(q)q c(q).
• Thus player i cannot increase her payoff by
  deviating if and only if
• P(qm)qm c(qm)/2 supqltq m P(q)q
  c(q)(1 - d),
• or d 1/2.
• In the second set of histories, if player i plays
  grim trigger, then the outcome is (c, c) in every
  period with payoffs (0, 0, . . .), whose
  discounted average is 0.
• If I deviates only once, she plays C. Then she
  reverts to the grim trigger strategy, that
  prescribes to play D at all subsequent periods.

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• The opponent, playing grim trigger strategy,
  plays c forever as a consequence of is one-shot
  deviation.
• The OSD yields the stream of payoffs (0,0,) with
  discounted average 0
• Player i cannot increase her payoff by deviating
  1 d.
• We conclude that if d ½ then the strategy pair
  in which each players strategy is the grim
  trigger strategy is a Subgame-Perfect equilibrium
  of the infinitely repeated Bertrand Game.

35
Public Goods

• A good is called excludable if people can be
  excluded from consuming it.
• A good is called nonrival if one persons
  consumption does not reduce the amount available
  to others.
• Goods that are nonrival and not excludable are
  called public goods.
• Goods that are nonrival but excludable are called
  club goods.

36
Public Goods Discrete Case

• Consider a market with two agents and two goods.
• The good xi is private, we call it income.
• Agent i has endowment wi of the private good and
  chooses the contribution gi towards the public
  good.
• So, the consumption of private good is xi wi -
  gi
• The public good production is
• G 0 if g1g2 lt c
• G 1 if g1g2 gt c.
• The utility of agent i is ui (G,xi).

37
Public Goods Discrete Case

• The provision of the public good with
  contributions g1, g2 is a Pareto improvement if
  for i1,2,
• ui(1, wi gi) gt ui(0, wi).
• Let the reservation price ri solve
• ui(1, wi ri) ui(0, wi).
• The provision of the public good is a Pareto
  improvement if and only if r1 r2 gt c.

38
Public Goods Discrete Case

• The private provision of a discrete public good
  is similar to a Prisoners Dilemma.
• Suppose that c/2 lt r1 lt c, c/2 lt r2 lt c.

Agent 2 Contribute Not
Agent 1 Contribute
Not
u2(1, w1-c), u2(1, w2)
u1(1, w1 c/2), u2(1, w2 c/2)
u1(1, w1), u2(1, w2 c)
u1(0, w1), u2(0, w2)
39
Public Goods Continuous Case

• Suppose that the public good is provided in
  continuous amounts, so that G f (g1, g2).
• The utility of agent i is ui(f(g1g2), wi gi).
• Efficiency requires
• max a1u1(f(g1g2), w1 g1) a2u2(f(g1g2), w2
  g2)
• First-Order conditions are, for i 1,2,
• a1(du1/dG) f a2(du2/dG) f ai(dui/dxi)
  0
• It follows that a1 du1/dx2 a2 du1/dx1, and
  rearranging
• (du1/dG) f (du2/dG) f
• (du1/dx1) (du1/dx1)

1 or MRS1 MRS2 1
40
Public Goods Private Provision

• Suppose that agent j contributes gj, agent choose
  gi
• maxg ui(f(g1g2), wi gi) s.t.
  gi gt 0.
• Kuhn-Tucker conditions are
• dL/dgi (dui/dG) f (dui/dxi) lt 0, gi gt 0,
  (dL/dgi)gi 0.
• Let the function bi(wi) be the solution of dL/dgi
  0, it follows that agent is
  reaction function is
• gi maxbi(wi gj )
  gj , 0.
• These two equations determine the Nash
  Equilibrium g
• When utility is quasilinear in income, ui vi
  (f(g1g2)) wi gi, bi(wi) gi constant in
  wi, so gi maxgj gj , 0.

41
Lindahl Equilibrium

• Suppose that we offer each consumer i the right
  to buy any quantity of G at price pj.
• Agent i choose gi
• maxG ui(G, xi) s.t. xi pi G
  wi.
• First-Order Conditions are
• dui/dG
• dui/dxi
• Hence, letting G and x be the Pareto efficient
  allocations,
• dui (G, xi)/dG
• dui (G,
  xi)/dxi
• delivers the efficient allocations. Each agent i
  pays the tax pi G.

pi
choosing pi
42
Groves-Clarke Mechanism

• Consider the discrete case again, G 0,1.
• Let si be the share of cost paid by agent i.
• Let vi ri sic, be the net value of agent i
  for the public good.
• Each agent is asked to report her net value, we
  denote the report by bi.
• The public good is provided if and only if Si bi
  gt 0.
• If the good is provided, each agent i receives a
  (possibly negative) side-payment equal to the sum
  Sji bj of other agents bids.

43
Groves-Clarke Mechanism

• We now show that the profile b v is weakly
  dominant.
• Agent i payoff is
• ui(bi, b-i) vi Sji bj if bi
  Sji bj gt 0,
• ui(bi, b-i) 0 if bi Sji bj lt 0.
• Suppose that vi Sji bj gt 0.
• ui(bi, b-i) vi Sji bj gt0 if bi vi,
• ui(bi, b-i) vi Sji bj gt0 if bi gt - Sji bj
• ui(bi, b-i) 0 if bi lt - Sji bj

44
Groves-Clarke Mechanism

• Suppose that vi Sji bj lt 0.
• ui(bi, b-i) 0 if bi vi,
• ui(bi, b-i) vi Sji bj lt 0 if bi gt - Sji bj
  
• ui(bi, b-i) 0 if bi lt - Sji bj
• In general, any mechanism with side transfer of
  the form
• ui(bi, b-i) h(b-i) Sji bj if bi gt - Sji
  bj
• ui(bi, b-i) h(b-i) if bi lt - Sji bj
• will yield truthful reporting as a weakly
  dominant strategy.

45
Groves-Clarke Mechanism continuous case

• Suppose that the good is continuous.
• If G units of good are provided, and utilities
  are quasilinear
• vi (G) ui (G) siG.
• Each agent reports a function bi(.) of the public
  good provided.
• The government chooses G argmaxG Si bi(G).
• Each agent receives a transfer Sji bj(G).
• Again, we show that reporting the truthful
  function vi(.) is a weakly dominant strategy.

46
Groves-Clarke Mechanism continuous case

• Take any profile of opponents reports b i.
• Agent i wants to maximize her net utility
• vi(G) Sji bj(G)
• The government maximizes
• bi(G) Sji bj(G).
• Hence, by truthfully reporting, agent i makes
  sure that the government maximizes her own net
  utility.

47
Externalities

• A choice by an agent that effects the utility of
  other agents is called an externality.
• Among the example of negative externalities
  pollution.
• Among the example of positive externalities
  education.
• When there are externalities, the First Welfare
  Theorem does not hold, because there are external
  effects that are not priced.

48
Externalities - An Example

• Suppose there are two firms.
• Firm 1 produces output x at cost c(x), sold in a
  competitive market.
• Output x imposes a negative externality e(x) on
  firm 2.
• We assume that c(x) and e(x) are increasing and
  convex.
• The profits are p1 maxx px c(x), p2
  -e(x).
• The market solution satisfies pc(x).
• But the socially efficient solution maximizes p1
  p2,
• Hence it satisfies pc(x) e(x).
• Firm 1 should internalize the cost e(x) borne on
  firm 2.

49
Externalities Pigouvian Taxes

• Pigouvian taxes are a means to solve the
  externality problem.
• The firms producing the externalities are taxed
  so that they internalize the externalities
  produced.
• In the example, suppose that the Government
  imposes a tax t on the good x.
• Firm 1 now maximizes (p-t)x c(x),
• and chooses x so that p-t c(x).
• Evidently, choosing t e(x) solves the
  externality problem.

50
Externalities Missing Markets

• Suppose that the Government establishes a
  competitive market for pollution.
• Firm 1 can decide how much pollution it wants to
  sell to firm 2, and firm 2 can decide how much
  pollution it wants to buy.
• Let r be the price of a unit of pollution.
• Firm 1 maximizes px1 rx1 c(x1).
• Firm 2 maximizes rx2 e(x2).
• The first-order conditions are pr c(x1), -r
  e(x2).
• Equating demand and supply, x1x2x, we recover
  the optimality condition p c(x) e(x).

51
Compensation Mechanism

• The missing markets solution is often impractical
  because of high administrative costs.
• The Pigouvian taxes are also difficult to
  implement because the Government is unlikely to
  know the cost function of externality producing
  firms.
• The government may try to elicit Pigouvian taxes
  from the firms.
• First, each firm i 1,2 is asked to report a
  Pigouvian tax rate ti.
• Second, firm 1 chooses x, pays tax t2x and firm 2
  receives transfer t2x. If t1 t2, each firm pays
  (t1 t2)2 to the Government.

52
Compensation Mechanism

• We show that reporting t1 e(x) and t2
  e(x) is the unique Subgame Perfect Equilibrium.
• By backward induction, in stage 2, firm 1s
  chooses x so as to
• maxx px c(x) t2x (t1 t2)2
• Hence, firm 1 chooses x so that p c(x) t2
• At the first stage, the best response of firm 1
  is t1(t2) t2, because t1 has no influence on
  the x chosen in second stage.
• The best response of firm 2 at stage 1
  incorporates the effect of t2 on the second
  period choice of x
• max t1x(t2) e(x(t2)) (t2 t1)2,
• yields (t1 e(x))x(t2) (t2 t1) 0
• Solving, we recover the efficiency condition p
  c(x) e(x).