# Search theory

In microeconomics, search theory studies buyers or sellers who cannot instantly find a trading partner, and must therefore search for a partner prior to transacting.

Search theory has been influential in many areas of economics. It has been applied in labor economics to analyze frictional unemployment resulting from job hunting by workers. In consumer theory, it has been applied to analyze purchasing decisions. From a worker's perspective, an acceptable job would be one that pays a high wage, one that offers desirable benefits, and/or one that offers pleasant and safe working conditions. From a consumer's perspective, a product worth purchasing would have sufficiently high quality, and be offered at a sufficiently low price. In both cases, whether a given job or product is acceptable depends on the searcher's beliefs about the alternatives available in the market.

More precisely, search theory studies an individual's optimal strategy when choosing from a series of potential opportunities of random quality, under the assumption that delaying choice is costly. Search models illustrate how best to balance the cost of delay against the value of the option to try again. Mathematically, search models are optimal stopping problems.

Macroeconomists have extended search theory by studying general equilibrium models in which one or more types of searchers interact. These macroeconomic theories have been called 'matching theory', or 'search and matching theory'.

## Search from a known distribution

George J. Stigler proposed thinking of searching for bargains or jobs as an economically important problem.[1][2] John J. McCall proposed a dynamic model of job search, based on the mathematical method of optimal stopping, on which much later work has been based.[3][4][5] McCall's paper studied the problem of which job offers an unemployed worker should accept, and which reject, when the distribution of alternatives is known and constant, and the value of money is constant.[6] Holding fixed job characteristics, he characterized the job search decision in terms of the reservation wage, that is, the lowest wage the worker is willing to accept. The worker's optimal strategy is simply to reject any wage offer lower than the reservation wage, and accept any wage offer higher than the reservation wage.

The reservation wage may change over time if some of the conditions assumed by McCall are not met. For example, a worker who fails to find a job might lose skills or face stigma, in which case the distribution of potential offers that worker might receive will get worse, the longer he or she is unemployed. In this case, the worker's optimal reservation wage will decline over time. Likewise, if the worker is risk averse, the reservation wage will decline over time if the worker gradually runs out of money while searching.[7] The reservation wage would also differ for two jobs of different characteristics; that is, there will be a compensating differential between different types of jobs.

An interesting observation about McCall's model is that greater variance of offers may make the searcher better off, and prolong optimal search, even if he or she is risk averse. This is because when there is more variation in wage offers (holding fixed the mean), the searcher may want to wait longer (that is, set a higher reservation wage) in hopes of receiving an exceptionally high wage offer. The possibility of receiving some exceptionally low offers has less impact on the reservation wage, since bad offers can be turned down.

While McCall framed his theory in terms of the wage search decision of an unemployed worker, similar insights are applicable to a consumer's search for a low price. In that context, the highest price a consumer is willing to pay for a particular good is called the reservation price.

## Search from known distributions and heterogeneous costs

Opportunities might provide payoffs from different distributions. Costs of sampling may vary from an opportunity to another. As a result, some opportunities appear more profitable to sample than others. These problems are referred to as Pandora box problems introduced by Martin Weitzman.[8] Boxes have different opening costs. Pandora opens boxes, but will only enjoy the best opportunity. With ${\displaystyle x_{i}}$ the payoff she discovered from the box ${\displaystyle i}$, ${\displaystyle c_{i}}$ the cost she has paid to open it and ${\displaystyle S}$ the set of boxes she has opened, Pandora receives

${\displaystyle \max _{i\in S}x_{i}-\sum _{i\in S}c_{i}}$

It can be proven Pandora associates to each box a reservation value. Her optimal strategy is to open the boxes by decreasing order of reservation value until the opened box that maximizes her payoff exceed highest reservation value of the remaining boxes. This strategy is referred as the Pandora's rule.

In fact, the Pandora's rule remains the optimal sampling strategy for complex payoff functions. Wojciech Olszewski and Richard Weber[9] show that Pandora's rule is optimal if she maximizes

${\displaystyle u\left(x_{1},...,x_{S}\right)-\sum _{i}^{S}c_{i}}$

for ${\displaystyle u}$ continuous, non-negative, non-decreasing, symmetric and submodular.

## Endogenizing the price distribution

Studying optimal search from a given distribution of prices led economists to ask why the same good should ever be sold, in equilibrium, at more than one price. After all, this is by definition a violation of the law of one price. However, when buyers do not have perfect information about where to find the lowest price (that is, whenever search is necessary), not all sellers may wish to offer the same price, because there is a trade-off between the frequency and the profitability of their sales. That is, firms may be indifferent between posting a high price (thus selling infrequently, only to those consumers with the highest reservation prices) and a low price (at which they will sell more often, because it will fall below the reservation price of more consumers).[10][11]

## Search from an unknown distribution

When the searcher does not even know the distribution of offers, then there is an additional motive for search: by searching longer, more is learned about the range of offers available. Search from one or more unknown distributions is called a multi-armed bandit problem. The name comes from the slang term 'one-armed bandit' for a casino slot machine, and refers to the case in which the only way to learn about the distribution of rewards from a given slot machine is by actually playing that machine. Optimal search strategies for an unknown distribution have been analyzed using allocation indices such as the Gittins index.

## Matching theory

More recently, job search, and other types of search, have been incorporated into macroeconomic models, using a framework called 'matching theory'. Peter A. Diamond, Dale Mortensen, and Christopher A. Pissarides won the 2010 Nobel prize in economics for their work on matching theory.[12]

In models of matching in the labor market, two types of search interact. That is, the rate at which new jobs are formed is assumed to depend both on workers' search decisions, and on firms' decisions to open job vacancies. While some matching models include a distribution of different wages,[13] others are simplified by ignoring wage differences, and just imply that workers pass through an unemployment spell of random length before beginning work.[14]

## References

1. ^ Stigler, George J. (1961). "The economics of information". Journal of Political Economy. 69 (3): 213–225. doi:10.1086/258464. JSTOR 1829263.
2. ^ Stigler, George J. (1962). "Information in the labor market". Journal of Political Economy. 70 (5): 94–105. doi:10.1086/258727. JSTOR 1829106.
3. ^ Mortensen, D. (1986). "Job search and labor market analysis". In Ashenfelter, O.; Card, D. The Handbook of Labor Economics. 2. Amsterdam: North-Holland. ISBN 0-444-87857-2.
4. ^ Lucas, R.; Stokey, N. (1989). Recursive Methods in Economic Dynamics. Cambridge: Harvard University Press. pp. 304–315. ISBN 0-674-75096-9.
5. ^ Adda, J.; Cooper, R. (2003). Dynamic Economics: Quantitative Methods and Applications. Cambridge: MIT Press. p. 257. ISBN 0-262-01201-4.
6. ^ McCall, John J. (1970). "Economics of information and job search". Quarterly Journal of Economics. 84 (1): 113–126. doi:10.2307/1879403.
7. ^ Danforth, John P. (1979). "On the role of consumption and decreasing absolute risk aversion in the theory of job search". In Lippman, S. A.; McCall, J. J. Studies in the Economics of Search. New York: North-Holland. ISBN 0-444-85222-0.
8. ^ Weitzman, Martin L. (1979). "Optimal Search for the Best Alternative". Econometrica. 47 (3): 641–654. doi:10.2307/1910412. JSTOR 1910412.
9. ^ Olszewski, Wojciech; Weber, Richard (2015-12-01). "A more general Pandora rule?". Journal of Economic Theory. 160 (Supplement C): 429–437. doi:10.1016/j.jet.2015.10.009.
10. ^ Butters, G. R. (1977). "Equilibrium distributions of sales and advertising prices". Review of Economic Studies. 44: 465–491. doi:10.2307/2296902.
11. ^ Burdett, Kenneth; Judd, Kenneth (1983). "Equilibrium price dispersion". Econometrica. 51 (4): 955–969. JSTOR 1912045.
12. ^ The Prize in Economic Sciences 2010
13. ^ Mortensen, Dale; Pissarides, Christopher (1994). "Job creation and job destruction in the theory of unemployment". Review of Economic Studies. 61 (3): 397–415. doi:10.2307/2297896.
14. ^ Pissarides, Christopher (2000). Equilibrium Unemployment Theory (2nd ed.). MIT Press. ISBN 0-262-16187-7.