Job Market Signaling (Spence model)

1
Job Market Signaling (Spence model)

• Perfectly competitive firms are bidding for
  services of workers. Competition bids up the wage
  rate to the level of the expected productivity of
  the worker, so firms make 0 profits.
• There are two types of workers low productivity
  (y1 1) and high productivity (y2 2).
  Productivity is private information of each
  worker.
• Before entering the job market, each worker
  decides how much education (e) to get.
• Education does not affect productivity or utility
  of the worker.
• Education is costly ct e/ yt

2
Job Market Signaling

• After observing the level of e, firms make
  individual wage offers, which can depend on the
  level of education.
• The goal of the firm is to maximize expected
  profit.
• Workers maximize expected wage.
• This game has many equilibria (pooling and
  separating)
• The following is a separating PBE in this game
• Type 1 worker chooses e 0, type 2 worker
  chooses e 1
• Firms set wages w(elt1) 1, w(egt1) 2
• Firms believe that if egt1 then type 2 and if
  elt1 then type 1

3
Limit pricing (Milgrom-Roberts model)

• Limit pricing is a situation, where na incumbent
  monopolist charges a below-cost price to deter
  entry of a new firm (or limit the scale or
  scope of entry)
• The classic rationale for limit pricing, that the
  entering firm will get scared of fierce
  competition, does not survivie the game-theoretic
  logic
• Milgrom and Roberts (1982) showed that what looks
  like limit pricing could be an equilibrium in a
  signaling game

4
Limit pricing the game

• Nature decides the incumbent's type. His marginal
  cost is low with probability x or high with prob.
  (1 x)
• The incumbent sets the price, observed by the
  potential entrant. The entrant may update her
  beliefs about the incumbent based on that price
• Entrant decides whether or not to enter. If no
  entry incumbent enjoys unthreatened monopoly
  position. If entry occurs firms receive duopoly
  profits

5
Limit pricing - notation

• Let i1 denote the incumbent and i2 the entrant.
• Mit(p) - monopoly profit of firm i if her type is
  t and charges price p
• Mit max Mit(p) - monopoly profit of firm i if
  her type is t and charges the price pmt (the
  optimal monopoly price for that type)
• Dit - duopoly profit of firm i if incumbent's
  type is t
• Assume that entrant wants to enter iff type is
  high, i.e. D2H gt 0 gt D2L (example Bertrand
  competition with cH gt cE gt cL)

6
Separating equilibria

• Let us look at conditions for and properties of
  separating equilibria in this game
• In a separating equilibrium, the two types set
  different prices pL ? pH
• And entry will occur only if the entrant observes
  pH. It follows that pH pmH

7
Separating equil. cont.

• We have the following constraints on pL
• ICH M1H ?D1H ? M1H(pL) ?M1H or M1H
  M1H(pL) ? ?(M1H D1H)i.e. that the high-cost
  type will not rather mimic the low-cost type
• IRL M1L(pL) ?M1L ? M1L ?D1L or M1L
  M1L(pL) ? ?(M1L D1L)i.e. that the low-cost
  type would not rather deviate and charge pLm
• In equilibrium, the entrant stays out if sees pL
  (satisfying the above), enters if sees any price
  other than pL (in particular pmH) (beliefs?)

8
Separating equil. cont.

• It is not easy to prove that separating
  equilibria exist, but indeed they do, with pL lt
  pLm (but not necessarily below cost)
• the incumbent of type L has to give up some
  profit in order to discourage entry, the price is
  below monopoly price
• social welfare is higher than under perfect
  information entry occurs only when it is
  efficient and type L charges a below-monopoly
  price in period 1 (limit pricing of this type is
  good!)

9
Pooling equilibria

• In a pooling equilibrium both types charge the
  same price pP
• If (1 x)D2H xD2L gt 0 then the entrant wants
  to enter if sees pP, but then at least one of
  the types has an incentive to charge a different
  price (i.e. Pooling equilibrium must involve
  entry deterrence)
• We must therefore have (1 x)D2H xD2L lt 0

10
Pooling equil. cont.

• We have the following constraints on pP
• IRL M1L(pP) ?M1L ? M1L ?D1L or M1L
  M1L(pP) ? ?(M1L D1L)i.e. that the low-cost
  type would not rather deviate and charge pLm
• IRH M1H(pP) ?M1H ? M1H ?D1H or M1H
  M1H(pP) ? ?(M1H D1H) i.e. that the low-cost
  type would not rather deviate and charge pHm
• In equilibrium, the entrant stays out if sees pP
  (satisfying the above), enters if sees any price
  other than pP (beliefs?)

11
Pooling equil. cont.

• It can be shown that there are many prices that
  satisfy the above conditions, but most
  importantly, pLm satisfies them (intuitive)
• The low-cost type does not have to worry about
  entry, so she chooses her monopoly price. The H
  type has to give up some profit to discourage
  entry, and charges pLm which is below her
  monopoly price
• There is no entry in equilibrium no matter what
  the costs are (inefficient)
• Social welfare effect is ambiguous entry never
  occurs, but price sometimes lower

12
One more thing

• Notice that in a separating equilibrium the L
  type is engaged in limit pricing, while in a
  pooling equilibrium - the H-type is engaged in
  limit pricing!
• In both cases it is used as a deterrent by the
  type that is most endangered by entry.

13
Cooperative game theory

• We do not model how the agents come to an
  agreement, we just characterize the players in
  terms of their bargaining power and then look for
  solutions that satisfy certain desirable
  mathematical conditions (axioms)
• Axiomatic bargaining the bargaining power is
  defined by the threat point
• Coalitional Games slightly more complex,
  bargaining power depends on threat points of
  every coalition. Coalition formation may be but
  does not have to be explicitly modeled or
  assumed.

14
Coalitional games

• Coalitional game with transferable payoffs the
  threat point of any coalition can be represented
  by a single number (the value of the coalition).
  We therefore assume that what a coalition can
  achieve can be turned into some transferable good
  (money?) and distributed among the coalition
  member in an arbitrary manner. The existence of
  the value of a coalition is assumed, the usual
  interpretation is that what coalition can achieve
  is independent of what happens outside of the
  coalition (if it isnt, then non-cooperative GT
  is more appropriate).
• Formally, the CG with transferable payoffs
  consists of
• N - the finite set of players
• v(S) the (value) function that assigns a real
  number to every nonempty set S ? N
• Cohesiveness Assumption v(N) the sum of
  values of any partition of N

15
The Core

• The Core is the equivalent of NE for coalitional
  games
• Let
• The Core of a coalitional game is the set of
  payoff profiles x (N payoff vectors) for which
  x(S) v(S) for every S ? N
• So x is in the core if no coalition can obtain a
  total payoff that exceeds the sum of its members
  payoffs in x
• The core is a prediction of the level of
  transferable utility (money) that each player
  will end up with. Problem (just as with Nash
  equilibrium) may not be unique, may not exist.