---
tags: micro, draft
---
# Adverse Selection

## Continuity of expected productivity

### Statement
In MWG P.441:
> For simplicity, from this point on, we assume that $F(\cdot)$ has an associated density function $f(\cdot)$, with $f(\theta)>0$ for all $\theta \in [\underline{\theta}, \bar{\theta}]$. This insures that the average productivity of those workers willing to accept employment, $E[\theta|r(\theta)\le w]$, varies continuously with the wage rate on the set $w \in [r(\underline{\theta}), \infty]$.

### Preliminary proof

Under the assumption,

$$E[\theta|r(\theta)\le w] = \frac{\int_{\underline{\theta}}^{a(w)} \theta f(\theta) d \theta}{\int_{\underline{\theta}}^{a(w)} f(\theta) d \theta}.$$, where $r(a(w))=w$.

If $r^{-1}$ exists and $a(w) = r^{-1}(w)$, then both the nominator and the denominator will be continuous functions of $w$ since the product of two continuous functions is continuous. Hence, the conditional expectation will be a continuous function of $w$. (Again, we acutually do not have the expectation when $a(w)=r(\underline{\theta})$.)

If $r$ is not continuous, we can construct a continuous function to replace $a(w)$. Define $h: [r(\underline{\theta}), \infty] \to [\underline{\theta}, \bar{\theta}]$ as

$$h(w) = \sup \{\theta: r(\theta) \le w\}$$

In other words, we use straight lines connecting all discontinuous points of $r$. A moment's thought shows that $h$ is a continuous function.

By the construction, if $r$ is continuous at $\theta$, then $h(r(\theta))=\theta$. Since this model assumes that $r$ is an increasing function, $r$ is discontinuous at most countable points. In other words, the set $\{\theta \in [\underline{\theta}, \bar{\theta}] : a(w) \neq h(w)\}$ has measure zero. The set $[\underline{\theta}, a(w)]$ could only different from the set $[\underline{\theta}, h(w)]$ at the end point. Therfore, the integral on two sets are identical as desired.

### Discussion

The generalization is unnecessary. We should just assume $r$ is a continuous function.

## Discrete model

### Motivation

1. The uncountable many types of setting is unrealistic.

2. The original model is unnecessarily complicated. Most students have not studied measure theory.

3. There are many meaningless results based on the original setting; we can construct a Cantor set on any continuous interval, but what is the meaning of uncountable many workers produce nothing in the economy?

So I try to propose a discrete model with the same problem.

### Setting

Consider an economy with $N$ type of workers, where $N \in \{2, 3, ...\}$. We use $I$ to represent the type of a worker, that is $I \in \{1, 2, ..., N\}$. We use the following notations in the model:

* $a(I): \{1, .., N\} \to \mathbb{R}_+$, the ability/productivity of type $I$ workers, which is an increasing function.

* $f(I): \{1, .., N\} \to \mathbb{N}_+$, the population of type $I$ workers.

* $r(I): \{1, .., N\} \to \mathbb{R}_+, r(I) \le a(I)$, the reservation wage of type $I$ workers.

There are some (finite) identical firms with CRTS technology in this economy. The production function is the summation of all workers' abilities. Assuming free entries and zero fixed cost, any firm should not have positive profit.

Each worker has the same linear utility. If they choose to work, they consume their wage. Otherwise, they will consume their reservation wage.

### When the ability is observable

Let $w(I)$ denote the wage a firm offered for type $I$ workers. To avoid negative profit and attract workers, we have

$$r(I) \le w(I) \le a(I).$$

Under the zero profit assumption, $w(I) = a(I)$ and everyone works.

### When the ability is unobservable

Because $a(I) \le r(I)$, at least the lowest ability workers will be employed, here we only consider the market equilibrium with the highest production.

Let $k$ denote the type of the highest ability work under wage $w$, we have the following inequality since $a(I)$ is an increasing function:

$$r(k) \le w \le \frac{\sum^k_{I=1} a(I) f(I)}{\sum^k_{I=1} f(I)}.$$

Under the zero profit assumption, $w = \frac{\sum^k_{I=1} a(I) f(I)}{\sum^k_{I=1} f(I)}$.

### Government intervention

The government may want to reallocate the resource. We assume the government can only tax, subsidize and nationalize the firm, but they cannot observe the ability and force people to work.

Most importantly, **the government cannot make lump-sum tax or lump-sum transfer**. The government can only tax and subsidize based on working status.

Assuming the government nationalizes firms and provides $w_e$ for employed workers and $w_u$ for unemployed workers. Both $w_e$ and $w_u$ could be negative as tax. The workers of type $I$ will work if and only if

$$w_u + r(I) \le w_e.$$

There is a $h \in \{\emptyset, 1, 2, ..., N\}$ as the highest type of employed workers.

We assume the government cannot issue bonds but can have a financial surplus. Hence, the budget constraint for the government is

$$ \sum^h_{I=1} w_e f(I) + \sum^N_{I=h+1} w_u f(I) \le \sum^h_{I=1} a(I) f(I).$$

Under competitive equilibrium, type $\{1, 2, ..., k\}$ workers work and the economy produce the GDP as:

$$GDP = \sum^k_{I=1} a(I) f(I) + \sum^N_{I=k+1} r(I) f(I).$$

The allocation under the same GDP could not be a Pareto improvement. Hence, the only chance is to encourage more people to work, i.e., $h >k$.

### Claim

Based on the result of competitive equilibrium with the highest production, the government could not make a Pareto improvement.

### Proof

Suppose the goverment can make a Pareto improvement with $w_e$ and $w_u$. That is, $h > k$.

For each type $I \in \{1,2, ..., h\}$ employed worker, we have

$$w_u + r(I) \le w_e.$$

However, if $w_u$ is negative, it is not a Pareto improvement for unemployed workers. Hence, the inequality becomes

$$r(I) \le w_e.$$

We can sum the inequalities and get:

$$\sum^h_{I=1}r(I) f(I) \le \sum^h_{I=1} w_e f(I).$$

Recall the government budget constraing,

$$ \sum^h_{I=1} w_e f(I) + \sum^N_{I=h+1} w_u f(I) \le \sum^h_{I=1} a(I) f(I).$$

Again, $w_u$ is non-negative,

$$ \sum^h_{I=1} w_e f(I) \le \sum^h_{I=1} a(I) f(I).$$

Combine the individual incentive constraint and the government budget constraing, we have

$$\sum^h_{I=1}r(I) f(I) \le \sum^h_{I=1} w_e f(I) \le \sum^h_{I=1} a(I) f(I).$$

However, the competive equilibrium wage $w$ is based on the highest type $k$ such that:

$$r(k) \le w \le \frac{\sum^k_{I=1} a(I) f(I)}{\sum^k_{I=1} f(I)}.$$

$$r(k)\sum^k_{I=1} f(I) \le \sum^k_{I=1} a(I) f(I)$$

By the monotone assumption

$$\sum^k_{I=1} r(I) f(I) \le r(k)\sum^k_{I=1} f(I) \le \sum^k_{I=1} a(I) f(I)$$

Hence, $h \le k$, which is a contradiction.

### Discussion

It seems that the key assumption is the inability of lump-sum tax?