# An Introduction to Search Theory¶

Stigler, George (1961) “The Economics of Information” *Journal of Political Economy* 69:3 213-225.

Original Question:

Static formulation misses the *recursive* nature of the problem

- It lacks the time element essential to most search
- Why decide in advance how many times to search?
- What if I find the lowest price on my first drawing?

## Sequential Search¶

McCall, John J. (1970) “Economics of Information and Job Search” *Quarterly Journal of Economics* 84:1 113-126.

Sequential setup

If I draw a price one at a time, with each draw costing \(c\), when should I stop?

Solution will be an optimal stopping time

## Labor Market Search¶

Each period, an unemployed individual draws a single offer, \(w\), from a known distribution, \(F(w)\).

- \(w\) is a non-negative random variable
- \(F(0) = 0\) and \(F(\infty) = 1\)

The individual’s choice is to either

- accept the offer, work forever at the accepted wage, or
- reject the offer, receive unemployment insurance, \(c\), this period and draw a new offer \(w'\) next period.

There is no recall of past offers

The agent evaluates income streams according to

where

Let \(v(w) = E \sum_{t=0}^{\infty} \beta^t y_t\) for a currently unemployed worker who has a wage offer, \(w\). So the worker’s Bellman equation can be written

The optimal policy is a *reservation wage* strategy.

- If \(w < \bar{w}\), reject offer
- If \(w \ge \bar{w}\), accept offer

The solution to the function equation is

We can illustrate this graphically

What can we say about the reservation wage, \(\bar{w}\)?

From the first part of this equation we know

Since \(\int_0^{\infty} F(w') dw' = 1\), we can re-write the left-hand side

If we add \(\bar{w} \int_{\bar{w}}^{\infty} F(w') dw'\) to both sides and do some more algebra we get

Interpretation

- LHS – cost of searching one more time when an offer \(\bar{w}\) is in hand
- RHS – expected benefit of searching one more time (expected present value associates with drawing \(w' > \bar{w}\)).

We could show (but we won’t here), that \(\bar{w}\) is increasing in \(c\).

The important policy questions are:

1. Does increasing unemployment insurance lead to longer durations of unemployment? The McCall search model suggests that the answer is yes.

2. Are longer unemployment durations a bad thing? The answer to this question is harder answer. On the one hand, we would prefer have people working rather than collecting unemployment. But on the other hand, more search may lead to higher wages.

To read more on this topic, check out the following Washington Post article and NBER Digest article.