ECO421: Signaling. Marcin P ski. March 26, 2018

Transcription

1 ECO421: Signaling Marcin P ski March 26, 2018

2 Plan Introduction Education Suits Signaling everywhere Conclusion

3 Introduction Signaling games We talked about communication cheap talk veriable talk Talk is cheap. Actions speak.

4 Introduction Signaling games Ann and Bob Ann observes the state of the world θ and chooses action a. Bob observes a (but not θ) and chooses action b. Players receive payos that depend on a, b, and θ. Special case: communication games with cheap talk, payos do not depend on a.

5 Introduction Signaling games Ann's choice of a may reect Ann's preferences. But, Ann may also have an interest in communicating her type to Bob Bob's action depend on his beliefs, Ann cares about Bob's action. Ann may want to use her action to signal her type to Bob.

6 Introduction Signaling games Signaling models in this class: Why do we go to the university? Why do consultants wear suits? Signaling everywhere.

7 Plan Introduction Education Suits Signaling everywhere Conclusion

8 Education Model Why do we go to the university? University is costly (tuition, lost opportunities, etc.) Is education worth it?

9 Education Model Classical theory of education: education builds human capital, human capital is valuable to the employers, higher education = higher wages.

10 Education Model Classical theory of education: education builds human capital, human capital is valuable to the employers, higher education = higher wages Signaling theory (M. Spence): education is intrinsically useless, it provides an evidence that a worker is smart and hard-working, employers do not care about the education, but they want to hire smart and hardworking workers.

11 Education Model M. Spence's model of education (1973). Two players: worker and employer. worker can be either high θ h or low quality θ l < θ h, θ = θ h θ l, workers knows her type, employer believes that π is the probability of θ h. Worker chooses education level e 0 education costs c (e, θ) = 1 θ e2, the cost is lower for high quality worker.

12 Education Model Employer observes the education, forms a belief that the worker is a high type and oers wage w: w = E p(e) θ = p (e) θ h + (1 p (e)) θ l beliefs p (e) In the original Spence's model, there are many employers who Bertrand compete for the worker and oer her the wage equal to their expected productivity. Here, we shortcut. We can assume that the employer maximizes: (w θ) 2. Optimal response is w = E p(e) θ.

13 Education Model Employer observes the education, forms a belief that the worker is a high type and oers wage w: w = E p(e) θ = p (e) θ h + (1 p (e)) θ l = θ l + p (e) θ.

14 Education Equilibrium worker's strategy e h, e l 0, employer beliefs p (e) (prob. that the worker is θ h ) after observing e, employer's strategy w (e).

15 Education Equilibrium Denition WPBE (e (.), w (.), p (.)): worker best responds to w (.), employer's strategy: for each e, the employer's strategy w (e) = θ l + p (e) θ, beliefs after after e {e (θ l ), e (θ h )}, p (e) = P (e, θ h), P (e)

16 Education Equilibrium Observation: In any wpbe, we can assume that for each o-path history e / {e (θ h ), e (θ l )}, Why? p (e) = 0, w (e) = θ l. take any equilibrium (e (.), w (.), p (.)), we can always modify the beliefs and wages so that the o-path beliefs put mass 1 on the low type, this lowers o-path wages, and strenghten incentives to stay on-path. The observation simplies looking for an equilibrium. But it is not necessary for an equilibrium: there are plenty of equilibria consistent with other o-path beliefs.

17 Education Equilibrium Two types of pure strategy equilibria: pooling equilibria e h = e l, separating equilibria e h e l.

18 Education Pooling equilibrium In a pooling equilibrium, both types of the worker choose the same eort e. Beliefs after e : Beliefs after e e, p (e ) = π and w (e ) = θ l + π θ = w, p (e) = 0 and w (e) = θ l.

19 Education Pooling equilibrium Worker type θ's payo from e w 1 θ e 2 = θ l + π θ 1 θ e 2. Worker's payo from deviation to e e w (e) 1 θ e2 = θ l 1 θ e2. the best deviation is e = 0, with payo θ l. Thus the worker best responds if θ l + π θ 1 θ e 2 θ l, or e θ (π θ).

20 Education Pooling equilibrium Example: e = 0 this equilibrium requires that the employer beliefs that any positive education means that the worker no smarter than the average, not very natural beliefs, but wpbe does not restrict o-path beliefs. There are many pooling equilibria.

21 Education Separating equilibrium In a separating equilibrium, two eort levels e l e h. beliefs p (e l ) = 0, w (e l ) = θ l, p (e h ) = 1, w (e h ) = θ h, In equilibrium, the low type cannot get lower wage by choosing e = 0. 0 eort costs 0. Hence, it must be that e l = 0 < e h.

22 Education Separating equilibrium In equilibrium, none of the types should want to imitate each other: IC h : θ h 1 eh 2 θ θ l 1 el 2 h θ, h IC l : θ h 1 eh 2 θ θ l 1 el 2 l θ. l incentive compatibility! using el = 0, we can rewrite it as θ h θ eh 2 and e2 h θ l θ, or θl θ =: e eh e := θ h θ. the lower bound is for the low type not to choose the high type eort, the upper bound stops the high type from choosing low type eort.

23 Education Separating equilibrium w w 1 θ l e 2 = θ l θ h high type θ l low type w 1 θ h e 2 = θ l e e e e

24 Education Separating equilibrium O-path beliefs: if w (e) = θ l for e eh, e l, we ensure no protable o-path deviations. all such deviations lead to low wages Alternative o-path beliefs: we can also assume that p (e) = 1 and w (e) = θ h for each e eh any beliefs for e eh are good for equilibrium, check the incentives. Also, it is ok to assume (some) positive beliefs below e h, as long as the IC conditions are satised.

25 Education Separating equilibrium Lemma There exist a continuum of separating equilibria: e (θ l ) = 0, e (θ h ) = e [e, e], p (e) = 0 for each e < e and p (e) = 1 for each e = e, w (e) = θ l + ( θ) p (e).

26 Education Welfare So, a pooling equilibrium, and a continuum of separating equilibria. Which one is better?

27 Education Welfare The payos in pooling equilibrium: the same 0 eort, the same wage θ l + π ( θ) for each type, payos:w pool (θ) = θ l + π ( θ) for each type θ,

28 Education Welfare The payos in separating equilibrium with high eort e. Type θ l : wage θ l, zero eort, payo w sep (θ l ) = θ l < θ l + π ( θ) = w pool, the low type is always worse o. Type θ h : wage θ h, eort e, payo w sep (θ h ) = θ h 1 θ h (e ) 2, The most ecient separating equilibrium is when e is the smallest possible, or e = e = θ l ( θ). the high type type is better o if θ h 1 θ h e 2 w pool = θ l + π ( θ)

29 Education Welfare Summary: Separation is costly and wasteful. But it may raise welfare, because it improves match rate between employers and employees. Low types are worse o.

30 Education Welfare Is the signaling model of education correct? Alternative model: education creates human capital, more human capital creation for higher ability types. education is costly Very similar predictions: higher types get more education, more education means more wages.

31 Education Welfare Is the signaling model of education correct? Bryan Caplan (recent book) argues that 30% of the value of education is signaling,\ but he is rather alone among economists: intelligence is easy to test for, no need to prove it by 4 years of useless education, how about on-the-job signaling (by working for 4 years in dierent tasks)? most labor economists only work with human capital of higher education. Even if incorrect, nice cute model to understand signaling.

32 Education Intuitive criterion A theoretical point. Which of the separating equilibria is most natural? Depends on our theory of o-path beliefs.

33 Education Intuitive criterion Some deviations are very stupid for some types. We should not belief that certain deviation was done by a type θ if such a deviation would be very stupid for type θ. Seems intuitive.

34 Education Intuitive criterion The idea is as follows. The highest possible wage is θ h For any e > ẽ = θ l ( θ), the upper bound on the payos of lower type is w (e) 1 e 2 θ h 1 e 2. θ l θ l

35 Education Intuitive criterion The idea is as follows. The highest possible wage is θ h For any e > ẽ = θ l ( θ), the upper bound on the payos of lower type is w (e) 1 e 2 θ h 1 e 2 θ l θ l < θ h 1 ẽ 2 = 0 θ l the inequality comes from e > e, and the equality comes the denition of e as the eort for which the low type is indierent between signaling truthfully and imitation the high type.

36 Education Intuitive criterion We say that action e > e is equilibrium dominated for type θ. The intuitive criterion says that the beliefs after e should not assign positive value to any type θ for which e is equilibrium dominated. The intuitive criterion eliminates all equilibria e > e.

37 Plan Introduction Education Suits Signaling everywhere Conclusion

38 Suits Why do we wear suits to the job interview? Or why do we wear suits at all? They are uncomfortable.

39 Suits Example A consulting rm is interviewing job market candidates. There are two types θ = l, h and the prior probability of the high type is π (0, 1). The type of the candidate is unobservable, but the rm can see whether the candidate wears a suit. The cost of the suit is the same for both types c > 0. The rm decides whether to employ the candidate. If the candidate θ is hired, the payos are u θ for the candidate and v θ for the rm. We assume that v h > 0 > v l, πv h + (1 π) v l < 0, u h > u l.

40 Suits Assumptions v h > 0 > v l, the rm would like to hire the high type, but not the low type, πv h + (1 π) v l < 0 u h > u l, nobody would be hired without the interview. the high type has a higher payo from being hired, perhaps because he expects better career than the low type. in the Spence model, the employment payo was the same for both types. cost of the suit is the same for everybody: model works if the cost is lower for the high type, in the Spence model, the cost was lower for the high type.

41 Suits Types of equilibria pooling equilibrium nobody wears suits: always exists, everybody wears suits: never exists, separating equilibrium high types only wear suit, exists if u h > c > u l, low types only wear suit: do not exist. partially separating equilibrium mixed strategies.

42 Suits Pooling equilibrium There is an equilibrium in which nobody wears a suit. Indeed, take beliefs p 0 = π, p 1 = π. The suit is treated as uninformative. Because average candidate is not worthy, he is not hired. Because wearing suit does not increase chances of being hired, nobody wears the suit. Pooling equilibrium. If c > u h, this is the only equilibrium.

43 Suits Equilibrium with suits - full separation If u h > c > u l, then there is a fully separating equilibrium. If c < u l, u h, then, no fully separating equilibrium.

44 Suits Strategies and beliefs From now on, assume c < u h, u l. Notation: mixed actions α θ - probability that candidate type θ wears the suit. βi - probability that candidate is hired if he wears the suit (i = 1) or no (i = 0), beliefs p i - probability that suited (i = 1) or unsuited (i = 0) candidate is type h.

45 Suits Strategies and beliefs We are looking for equilibrium, where 1 > πα (h) + (1 π) α (l) > 0, or suits are worn, but not always. all other cases already considered: nobody ever wears suits (equilibrium) everybody wears suits (no equilibrium) no o-path beliefs. On-path beliefs: Bayes formula: p 1 = p 0 = πα (h) πα (h) + (1 π) α (l), and π (1 α (h)) π (1 α (h)) + (1 π) (1 α (l)).

46 Suits Employer's decision Employer with beliefs p i : payos from hiring p i v h + (1 p i ) v l, not hiring: 0, Dene the threshold belief as p = v l v h + ( v l ), vl < 0 = p > 0, πvh + (1 π) v l < 0 = p > π, the latter means that the employer needs to have a better information than the average to hire the candidate.

47 Suits Employer's decision Employer with beliefs p i : best response hire (β i = 1) if p i > p, indierent (β i [0, 1]) if p i > p, hire (β i = 0) if p i > p.

48 Suits Worker's decision Worker type θ wants to wear the suit if β 1 u θ c > β 0 u θ. Dene t θ = c u θ. notice that t h < t l. Worker's best response wear suit (α θ = 1) if β 1 β 0 > t θ, indierent (α θ [0, 1]) if β 1 β 0 = t θ, not wear the suit (α θ = 0) if β 1 β 0 < t θ.

49 Suits Worker's decision Monotonicity: because t h < t l, there are 5 possibilities: 1. α h = α l = 1 2. α h = 1,α l (0, 1), 3. α h = 1, α l = 0, 4. α h (0, 1), α l = 0 5. α h = 0 = α l. notice that both types of the worker cannot beindierent because it would require t h = β 1 β 0 = t l > t h -> contradiction.

50 Suits Worker's decision Monotonicity: because t h < t l, there are 5 possibilities: α h = α l = 1 everybody wears suits never exists α h = 1,α l (0, 1), α h = 1, α l = 0 full separation never exists α h (0, 1), α l = 0 α h = α l = 0, nobody wears suits already considered.

51 Suits Equilibrium with suits Let us consider α h (0, 1), α l = 0. Bayes formula -> p 1 = 1, p 0 (0, 1) suit is only worn by the high type, Because 1 > p, we have β 1 = 1. Because α h (0, 1), we have β 0 = β 1 t h = 1 c u h (0, 1). Becuase β 0 (0, 1), we have p 0 = p.

52 Suits Equilibrium with suits But then Contradiction. π < p =p 0 π (1 α h ) = π (1 α h ) + (1 π) (1 α l ) π (1 α h ) = π (1 α h ) + (1 π) =π 1 α h 1 πα h < π. Hence, no equilibrium so that α h (0, 1), α l = 0.

53 Suits Equilibrium with suits Let us consider α h = 1, α l (0, 1). Bayes formula -> p 0 = 0, p 1 (0, 1) suit is never worn by the low type, Because 0 < p, we have β 0 = 0. Because α l (0, 1), we have β 1 = β 0 + t l = c u l (0, 1). Becuase β 1 (0, 1), we have p 1 = p.

54 Suits Equilibrium with suits Hence, p =p 1 πα h = πα h + (1 π) α l π =, π + (1 π) α l which can be used to nd the strategy of the low type α l = π ( ) 1 1 π p 1.

55 Suits Equilibrium with suits Theorem If c < u l, u h, there is an equilibrium in which α h = 1, α l (0, 1) and the unsuited candidate is never hired.

56 Signaling everywhere Suit is a signal that the worker is of the high type. The high type is willing to pay the signal because it increases her chances of getting a job. The signal is more valuable for the high type, because she has a higher utility from being hired. But some low types also wear the suit. If not, the suited candidates would be always hired, which would make the low type want to wear the suit. In equilibrium, the unsuited type is never hired and the suited type is hired with probability that makes the low type indierent between wearing the suit or not. the high type strictly prefers the suit.

57 Plan Introduction Education Suits Signaling everywhere Conclusion

58 Signaling everywhere Features of signaling models Ann wants to convince Bob that she is the high type. She uses a costly signal: either the cost of signal is lower for high types (Spence), or the benets from convincing Bob are higher for high types (Suits).

59 Signaling everywhere Conspicuous consumption. Handicap principle. Contracts.

60 Suits Conspicuous consumption Conspicuous consumption consumption that is not done for its utility value, but for show, T. Veblen (1899) Example: luxury watches Signaling explanation it is very expensive and wasteful, but I can aord it because my income ability is very high.

61 Signaling everywhere Other examples: Diamond ring for engagement. Large wedding. Charity. Purdah.

62 Signaling everywhere Handicap principle Some animal behavior or features seems clearly useless or harmful peacock tail Gazelle stotting larks songs Theory of Honest Signals (Amotz Zahavi): a certain debilitating feature can represent a proof of otherwise very t genes, if I survived with a long tail, I must be amazing otherwise mates who select for this feature can have healthier progeny -> evolution.

63 Signaling everywhere Handicap principle Other examples begging for food (see also Osborne, Ex 335.2) autumn colors, Red frogs

64 Signaling everywhere Contract negotiations Often contract negotiations are dicult because of the uncertainty of the partner's quality adverse selection (market for lemons) Possible solutions: trade terms that are bad on some dimensions to credibly distinguish themselves from bad types lifetime warranty! we accept all returns, no questions asked we will match the lowest price

65 Plan Introduction Education Suits Signaling everywhere Conclusion

66 Conclusions What did we learn - concepts Signaling games features, how to identify signaling in real world. Pooling and separating equilibria. Intuitive criterion.

67 Conclusions What did we learn - skills Find equilibria of signaling games, separating and pooling, mixed and pure strategy. Find the best payo equilibria.

68 Conclusions Further reading Spence's model of education Michael Spence (1973). "Job Market Signaling". Quarterly Journal of Economics. 87 (3): Andrew Weiss (1995). "Human Capital vs. Signaling Explanations of Wages". The Journal of Economic Perspectives. 9 (4): Handicap principle Honest Signaling Theory: A Basic Introduction. By Carl T. Bergstrom Conspicuous consumption Theory of the Leisure Class by Thorstein Veblen A signaling explanation for charity A Glazer, KA Konrad - The American Economic Review, 1996