Adverse Selection

In reality, information of commodities is often asymmetrically held by market participants.

In the presence of asymmetric information, market equilibria often fail to be Pareto optimal.

Definition (Adverse Selection). 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.

Definition (Constrained Pareto Optimal Allocation). Allocations that cannot be Pareto improved upon by a central authority who, like market participants, cannot observe individuals’ privately held information.

Informational Asymmetries and Adverse Selection

Assumptions Many identical potential firms. Each produces the same output using an identical constant returns to scale technology in which labor is the only input. Risk neutral, seek to maximize their expected profits Act as price takers. For simplicity, we take the price of the firms' output to equal 1 (in units of a numeraire good). Workers differ in the number of units of output they produce if hired by a firm,which we denote by $\theta$. We let $[\underset{―}{\theta },\bar{\theta }]\subset \mathbb{R}$ denote the set of possible worker productivity levels, where $0\le \underset{―}{\theta }<\bar{\theta }<\mathrm{\infty }$. The proportion of workers with productivity of $\theta$ or less is given by the distribution function $F(\theta )$, and we assume that $F(\cdot )$ is nondegenerate, so that there are at least two types of workers. The total number of workers is $N$. Workers seek to maximize the amount that they earn from their labor (in units of the numeraire good). A worker can choose to work either at a firm or at home. Suppose that a worker of type $\theta$ can earn $r(\theta )$ on her own through home production. Assume that she accepts if she is indifferent.

Base: The competitive equilibrium with perfect information

Assumption Assume that workers' productivity levels are publicly observable.

Given the competitive, constant returns nature of the firms, in a competitive equilibrium we have

1. $w^{\ast }(\theta )=\theta$ for all $\theta$ (recall that the price of their output is 1);
2. The set of workers accepting employment in a firm is $\{\theta :r(\theta )\le \theta \}$.

As would be expected from the first fundamental welfare theorem, this competitive outcome is Pareto optimal.

Imperfect Information

Assumption Assume that workers' productivity levels are unobservable by the firms.

A single wage rate $w$ for all workers.

• Supply

  $$\mathrm{\Theta }(w)=\{\theta :r(\theta )\le w\}$$

• Demand

  $$z(w)=\{\begin{array}{ll}0& \text{if }\mu <w\\ [0,\mathrm{\infty }]& \text{if }\mu =w\\ \mathrm{\infty }& \text{if }\mu >w\end{array}$$

A firm believes that the average productivity of workers who accept employment is $\mu$.

• With rational expectation, we must have

  $$\mu =E[\theta |\theta \in {\mathrm{\Theta }}^{\ast }].$$

Definition (Competitive Equilibrium with Imperfect Information). In the competitive labor market model with unobservable worker productivity levels, a competitive equilibrium is a wage rate $w^{\ast }$ and a set ${\mathrm{\Theta }}^{\ast }$ of worker types who accept employment such that

$${\mathrm{\Theta }}^{\ast }=\{\theta :r(\theta )\le w^{\ast }\}$$

and

$$w^{\ast }=E[\theta |\theta \in {\mathrm{\Theta }}^{\ast }].$$

• When no workers are employed, we assume that $\mu =E[\theta ]$ and $w^{\ast }=E[\theta ]$ in such equilibrium.

A competitive equilibrium above will fail to be Pareto optimal.

Market Unraveling

Assumptions Assume that $r(\theta )\le \theta$ for all $\theta \in [\underset{―}{\theta },\bar{\theta }]$. Assume that $r(\cdot )$ is a strictly increasing function. The assumption that generates the adverse selection. $F(\theta )$ has an associated density function $f(\cdot )$, with $f(\theta )>0$ for all $\theta \in [\underset{―}{\theta },\bar{\theta }]$.

$$w^{\ast }=E[\theta |r(\theta )\le w^{\ast }]$$

Once the best workers are driven out of the market, the average productivity of the workforce falls, thereby further lowering the wage that firms are willing to pay. As a result, once the best workers are driven out of the market, the next-best may follow; the good may then be driven out by the mediocre.

Multiple Equilibria and Game Approach

The competitive equilibrium need not be unique.

It can arise because there is no restrictions on the slope of the function $E[\theta |r(\theta )\le w]$.

The equilibria can be Pareto ranked.

• The equilibrium with the highest wage Pareto dominates the others.
• The low-wage equilibria arise because of a coordination failure.

Assumption Assume that the structure of the market is common knowledge.

Two-stage game

1. Two firms simultaneously announce their wage offers.
2. Workers decide whether to work for a firm and, if so, which one.

Proposition 13.B.1. Let $W^{\ast }$ denote the set of competitive equilibrium wages for the adverse selection labor market model, and let $w^{\ast }=max\{w:w\in W^{\ast }\}$

1. If $w^{\ast }>r(\underset{―}{\theta })$ and $\mathrm{\exists }\epsilon >0$ such that $E[\theta |r(\theta )\le w^{\prime }]>w^{\prime }$ for all $w^{\prime }\in (w^{\ast }-\epsilon ,w^{\ast })$, then there is a unique pure strategy SPNE of the two-stage game-theoretic model.
   • In this SPNE, employed workers receive a wage of $w^{\ast }$, and workers with types in the set $\mathrm{\Theta }(w^{\ast })=\{\theta :r(\theta )\le w^{\ast }\}$ accept employment in firms.
2. If $w^{\ast }=r(\underset{―}{\theta })$, then there are multiple pure strategy SPNEs. However, in every pure strategy SPNE each agent's payoff exactly equals her payoff in the highest-wage competitive equilibrium.

The game-theoretic model tells us that if sophisticated firms have the ability to make wage offers, then we break the coordination problem.

Market Intervention

Definition (Constrained/Second-Best Pareto Optimum). An allocation that cannot be Pareto improved by an authority who is unable to observe agents’ private information is known as a constrained (or second-best) Pareto optimum.

A central authority who is unable to observe worker types can always implement the best (highest-wage) competitive equilibrium outcome.

• She need only set $w_e=w\ast$, the highest competitive equilibrium wage, and $w_u=0$.

Proposition 13.B.2. In the adverse selection labor market model where $r(\cdot )$ is strictly increasing with $r(\theta )\le \theta$ for all $\theta \in [\underset{―}{\theta },\bar{\theta }]$ and $F(\cdot )$ has an associated density $f(\cdot )$ with $f(\theta )>0$ for all $\theta \in [\underset{―}{\theta },\bar{\theta }]$, the highest-wage competitive equilibrium is a constrained Pareto optimum. Proof

• In more general situation, this may fail. Details...

Signaling

Definition (Signaling). Individuals on the more informed side of the market (workers) chose their level of education in an attempt to signal information about their abilities to uninformed parties (the firms).

Assumptions Assume $2$ types of workers, ${\theta }_H$ and ${\theta }_L$. A worker can get some oberservable education. Education does nothing for a worker's productivity. The cost $c(e,\theta )$ $c(0,\theta )=0$ $c_e(e,\theta )>0$ $c_{ee}(e,\theta )>0$ $c_{\theta }(e,\theta )<0$ $c_{e\theta }(e,\theta )<0$ $u(w,e|\theta )$ denote the utility. $u(w,e|\theta )=w-c(e,\theta )$

Conclusion

1. The otherwise useless education may serve as a signal of unobservable worker productivity.

2. The welfare effects of signaling activities are ambiguous.

Inefficiency of Signaling

Assumptions $r({\theta }_H)=r({\theta }_L)=0$.

The unique equilibrium has all workers employed by firms at a wage of $w^{\ast }=E[\theta ]$ and is Pareto efficient.

The equilibrium concept is of a weak perfect Bayesian equilibrium, but with an added condition.

Assumptions Assume the firms' beliefs have the property: for each possible choice of $e$, there exists a number $\mu (e)\in [0,1]$ firm 1’s belief that the worker is of type ${\theta }_H$ after seeing her choose $e$ is $\mu (e)$. After the worker has chosen $e$, firm 2’s belief that the worker is of type ${\theta }_H$ and that firm 1 has chosen wage offer $w$ is precisely $\mu (e){\sigma }_1^{\ast }(w|e)$, where ${\sigma }_1^{\ast }(w|e)$ is firm 1’s equilibrium probability of choosing wage offer $w$ after observing education level $e$. Firm's belief is common knowledge.

Definition (Perfect Bayesian Equilibrium). A set of strategies and a belief function $\mu (e)\in [0,1]$ giving the firms’ common probability assessment that the worker is of high ability after observing education level $e$ is a PBE if

1. The worker’s strategy is optimal given the firm’s strategies.
2. The belief function $\mu (e)$ is derived from the worker’s strategy using Bayes’ rule where possible.
3. The firms’ wage offers following each choice $e$ constitute a Nash equilibrium of the simultaneous-move wage offer game in which the probability that the worker is of high ability is $\mu (e)$.

Analysis

Firms

• $E(\theta |e)=\mu (e){\theta }_H+(1-\mu (e)){\theta }_L$
• $w=\mu (e){\theta }_H+(1-\mu (e)){\theta }_L$

Worker

• $u(e|\theta )=w-c(e,\theta )$

Definition (Single-Crossing Property). The indifference curves cross only once and that, where they do, the indifference curve of the high-ability worker has a smaller slope.

Equilibria

Separating Equilibria

Let $e^{\ast }(\theta )$ be the worker’s equilibrium education choice as a function of her type, and let $w^{\ast }(e)$ be the firms’ equilibrium wage offer as a function of the worker’s education level.

Lemma 13.C.1. In any separating perfect Bayesian equilibrium, $w^{\ast }(e^{\ast }({\theta }_H))={\theta }_H$ and $w^{\ast }(e^{\ast }({\theta }_L)={\theta }_L$; that is, each worker type receives a wage equal to her productivity level.

Lemma 13.C.2. In any separating perfect Bayesian equilibrium, $e^{\ast }({\theta }_L)=0$; that is, a low-ability worker chooses to get no education.

Belief

• $\mu (0)=0$, $\mu (\tilde{e})=1$.

The fundamental reason that education can serve as a signal here is that the marginal cost of education depends on a worker’s type.

DANGER

Any education level between $\tilde{e}$ and $e_1$ can be the equilibrium education level of the high-ability workers.

• These various separating equilibria can be Pareto ranked.

Welfare Comparison

It is of interest to compare welfare in these equilibria with that arising when worker types are unobservable but no opportunity for signaling is available.

• Firms earn expected profits of zero in both situation.
• Low-ability workers are strictly worse off when signaling is possible.
• High-ability workers may be better or worse off when signaling is possible.

Pooling Equilibria

Let $e^{\ast }({\theta }_L)=e^{\ast }({\theta }_H)=e^{\ast }$.

Belief

• $\mu (e^{\ast })=\lambda$

Firms

• $w^{\ast }(e^{\ast })=\lambda {\theta }_H+(1-\lambda ){\theta }_L=E(\theta )$

Any education level between $0$ and the level $e^{\prime }$ can be sustained.

• Given the wage schedule, we draw the indifference curves and move them upward.

Welfare Comparison

A pooling equilibrium in which both types of worker get no education Pareto dominates any pooling equilibrium with a positive education level.

Thus, pooling equilibria are weakly Pareto dominated by the no-signaling outcome.

Pareto Improvement

In the presence of signaling, a central authority who cannot observe worker types may be able to achieve a Pareto improvement relative to the market outcome.

The best separating equilibrium can be Pareto dominated by the outcome that arises when signaling is impossible.

• A Pareto improvement can be achieved simply by banning the signaling activity.

It may be possible to achieve a Pareto improvement even when the no-signaling outcome does not Pareto dominate the best separating equilibrium.

• The central authority introduces cross-subsidization, where high-ability workers are paid less than their productivity level while low-ability workers are paid more than theirs, an outcome that cannot occur in a separating signaling equilibrium.

Screening

Definition (Screening). The uninformed parties take steps to try to distinguish, or screen, the various types of individuals on the other side of the market.

Assumptions Suppose that jobs may differ in the “task level” required of the worker. Suppose that higher task levels add nothing to the output of the worker; rather, their only effect is to lower the utility of the worker. Assume that the utility of a type $\theta$ worker who receives wage $w$ and faces task level $t>0$ is $$u(w,t|\theta )=w-c(t,\theta )$$ $c(t,\theta )=0$, $c_t(t,\theta )>0$, $c_{tt}(t,\theta )>0$, $c_{\theta }(t,\theta )<0$ for all $t>0$, and $c_{t\theta }(t,\theta )<0$.

The model.

1. A worker come to find a job.
2. Two firms simultaneously announce sets of offered contracts.
   • A contract is a pair $(w,t)$.
3. Workers choose whether to accept a contract and which one.
   • Assume that if a worker is indifferent between two contracts, she always chooses the one with the lower task level and that she accepts employment if she is indifferent about doing so.
   • If a worker’s most preferred contract is offered by both firms, she accepts each firm’s offer with probability $1/2$.

Base: Perfect Information

Proposition 13.D.1. In any SPNE of the screening game with observable worker types, a type ${\theta }_i$ worker accepts contract $(w_i^{\ast },t_i^{\ast })=({\theta }_i,0)$, and firms earn zero profits.

Imperfect Information

To determine the equilibrium outcome with unobservable worker types, it is useful to begin by drawing three break-even lines: the zero-profit lines for productivity levels ${\theta }_L$, $E[\theta ]$, and ${\theta }_H$, respectively.

Proposition 13.D.2. In any subgame perfect Nash equilibrium of the screening game, low-ability workers accept contract $({\theta }_L,0)$, and high-ability workers accept contract $({\theta }_H,{\hat{t}}_H)$, where ${\hat{t}}_H$ satisfies ${\theta }_H-c({\hat{t}}_H,{\theta }_L)={\theta }_L-c(0,{\theta }_L)$.

The equilibrium may not exist.

More generally, an equilibrium exists only if there is no such profitable deviation.

Welfare Properties of Screening Equilibria

One difference from the signaling model, however, is that in cases where an equilibrium exists, screening must make the high-ability workers better off.

Indeed, when an equilibrium does exist, it is a constrained Pareto optimal outcome.

• if no firm has a deviation that can attract both types of workers and yield it a positive profit, then a central authority who is unable to observe worker types cannot achieve a Pareto improvement either.