A Simple Screening Example

Micro EC487_Advanced_Microeconomics

Background

See Asymmetric Information and Mechanism Design .

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How should a seller sell to a buyer with an unknown marginal willingness to pay?

Suppose the buyer has the preferences

$$\theta q-p$$

where $\theta >0$ is the buyer’s marginal value, $q⩾0$ is the quantity purchased, and $p$ is the total price paid. The type $\theta$ is the buyer’s private information.

The seller has costs given by the increasing convex function $c(q)$ , satisfying $c^{\prime }(0)=0$ , $c^{\prime \prime }⩾0$ and $\begin{array}{r}{\lim }_{q\to \infty }{c}^{\prime }(q)=\infty \end{array}$ .

Benchmark: If the seller knew the value of $\theta$ , then he would set exactly $p=\theta q$ and choose a certain $q$ (or $p$ ) to maximize $p-c(q)$ .

But how if seller doesn’t know the value of $\theta$ ? We need to involve expectation. And that’s where we introduce virtual value.

A Binary case

先考虑一个最简单的情形：假设 agent 的 type 只可能有 ${\theta }_L$ 和 ${\theta }_H$ .成为 ${\theta }_H$ 的可能性是 ${\alpha }_H$。

对于卖家来说，它想要完成的是：

$$\underset{q_H,q_L,p_H,p_L}{\max }{\alpha }_L(p_L-c(q_L))+{\alpha }_H(p_H-c(q_H))$$

subject to

$$\begin{array}{rl}{\theta }_L{q}_L-p_L& \ge {\theta }_L{q}_H-p_H(I{C}_L)\\ {\theta }_H{q}_H-p_H& \ge {\theta }_H{q}_L-p_L(I{C}_H)\\ {\theta }_L{q}_L-p_L& \ge 0(I{R}_L)\\ {\theta }_H{q}_H-p_H& \ge 0(I{R}_H)\end{array}$$

$1-F(\theta )$ measures the amount people who are above the $\theta$ . And $f(\theta )$ measures the weight (how many “me” here in the market). This is an inevitable distortion compared with the First-best objective.