Asymmetric information - applications

Transcription

1 Asymmetric information - applications

2 Adverse selection and signaling How can we characterise market equilibria in settings of asymmetric information? Examples: 1. When a firm hires a worker, the firm may know less about the worker s innate ability than the worker herself; 2. In the used-car market, a prospective seller may have much better information about her car s quality than a prospective buyer. 1

3 3. When an individual buys health insurance, he may know more about his propensity to contract a serious disease than the insurance company does. In these cases, market equilibria may often fail to be Pareto optimal. Moreover, this problem may be further compounded by adverse selection. 2

4 Adverse selection arises when an informed individual s trading decisions depend on her privately-held information in a manner that adversely affects uninformed market participants. User-car example: individual more likely to sell her car when she knows it is not very good. 3

5 Akerlof s labour-market ( lemons ) model Many identical potential firms that can hire workers; Each produces identical output using a CRS technology; Labour is the only input; Firms are risk-neutral, seek to maximise expected profits and act as price takers; Price of output is 1 (in terms of numeraire) 4

6 Workers differ in their productivity, θ (number of units they can produce); [θ, θ] R - set of possible worker productivity levels, 0 θ θ < ; Proportion of workers with productivity of θ or less given by F (θ). We assume F(.) is non-degenrate; Total number (measure) of workers is N. 5

7 Workers seek to maximise amount they can earn from their labour (in terms of numeraire); A worker of type θ can earn r(θ) on her own (opportunity cost of working) she will accept employment at a form iff her wage is at least r(θ) What is the CE of this model when workers productivity levels are publicly observable? 6

8 There is a distinct equilibrium wage w (θ) for each type θ Given competitive, CRS nature of firms, w (θ) = θ for all θ. Set of workers accepting employment in a firm is {θ : r(θ) θ} This CE is Pareto optimal. 7

9 What is the CE when worker productivity levels are not observable by firms? Wage rate is now independent of worker type, so single wage rate w for all workers. Set of workers willing to accept employment at wage rate w is: Θ(w) = {θ : r(θ) w} Suppose firm believes that average productivity of workers who accept employment is µ. 8

10 What is demand for labour as a function of w? z(w) = 0, µ < w [0, ), µ = w, µ > w If worker types in set Θ are accepting employment offers in a CE, and if firms beliefs about productivity of potential employees correctly reflect the average productivity of workers hired in this equilibrium, then we must have µ = E[θ/θ Θ ] 9

11 Thus, demand for labour must equal supply in an equilibrium with a positive level of employment iff w = E[θ/θ Θ ] Definition: In a competitive labour market model with unobservable worker productivity levels, a CE is a wage rate w and a set Θ of worker types who accept employment such that Θ = {θ : r(θ) w } w = E[θ/θ Θ ] 10

12 Typically, a CE as defined above will not be Pareto optimal - i.e., there will be an inefficient allocation of workers between firms and home production. Consider the case where r(θ) = θ - every worker is equally productive at home. Suppose F (r) (0, 1) - there are some workers with θ > r and some with θ < r. Pareto optimal allocation will have those with θ r accepting employment at a firm and those with θ < r not doing so. 11

13 In a CE, set of workers willing to accept employment at a given wage Θ (w) is either [θ, θ] (if w r) or (if w < r). Thus E[θ/θ Θ(w)] = E[θ] for all w and so, equilibrium wage rate is w = E[θ]. If E[θ] r, all workers accept employment at a firm; if E[θ] < w, no one does. Which of these equilibria will arise depends on fraction of high and low productivity workers. 12

14 Signaling - Spence model Two types of workers with productivities θ H and θ L respectively, with θ H > θ L > 0 - private information; λ = P r(θ = θ H ) (0, 1) Before entering job-market, worker can get some education - amount of education a worker receives is observable. 13

15 Assume education has no effect on worker productivity! Cost of obtaining education level e for a type θ worker (monetary/psychic cost) given by twice continuously differentiable function c(e, θ) Assume c(0, θ) = 0; c e (e, θ) > 0; c ee (e, θ) > 0; c θ (e, θ) < 0 e > 0; c eθ (e, θ) < 0 - both cost and MC of education are lower for highability workers 14

16 Workers utility u(w, e/θ) = w c(e, θ) r(θ) - opportunity cost of working, or value of outside option. For simplicity, we assume r(θ H ) = r(θ L ) = 0 Implication: in the absence of ability to signal, unique equilibrium has all workers employed at firms at wage w = E[θ], and is Pareto efficient. Our analysis of signaling here therefore emphasises potential inefficiencies of signaling. 15

17 A set of strategies and a belied function µ(e) [0, 1] giving the firms common probability assessment that the worker is of highability after observing education level e is a weak PBE if: (i) The worker s strategy is optimal given the firms strategies; (ii) Belief function µ(e) is derived from the worker s strategy using Baye s rule, where possible; 16

18 (iii) Firms wage offers following each choice e constitute a NE of the simultaneous-move wage offer game in which the probability that the worker is of high-ability is µ(e). We begin our analysis at the end of the game. Suppose after seeing some education level e, firms attach probability of µ(e) that the worker is of type θ H. 17

19 Then, expected productivity of worker is µ(e)θ H + (1 µ(e)θ L In a simultaneous-move wage offer game, the firms pure strategy NE wage offers equal workers expected productivity. Thus, in any pure-strategy PBE, we must have both firms offering same wage which is exactly equal to expected productivity. 18

20 Knowing this, what is the worker s strategy - choice of education level contingent on her type? Workers preferences over (wage,education) pairs - single crossing property. Arises because worker s MRS between wages and education at any given (w,e) pair is ( dw de ) ū = c e (e, θ) which is decreasing in θ 19

21 w(e) - equilibrium wage offer that results for each education level. In any PBE, w(e) = µ(e)θ H + (1 µ(e)θ L for the equilibrium belief function µ(e), hence w(e) [θ L, θ H ] Separating equilibrium: Let e (θ) be worker s equilibrium education choice as a function of her type, and let w (e) be the firms equilibrium wage offer as a function of workers education level. 20

22 Lemma: In any separating PBE, w (e(θ H )) = θ H and w (e(θ L )) = θ L ; each worker type receives wage equal to her productivity level. Lemma: In any separating PBE, e (θ L ) = 0; a low-ability worker chooses to get no education. 21

23 Pooling equilibrium: In a pooling equilibrium, both types of workers choose same level of education, e (θ L ) = e (θ H ) = e. Since firms beliefs must be correctly derives from the equilibrium strategies and Baye s rule when possible, we must have w (e ) = λθ H + (1 λ)θ L = E[θ]. 22