User talk:MrSchimpf

In macroeconomics, matching theory, also known as search and matching theory, is a mathematical framework describing the formation of mutually beneficial relationships over time. The key element that distinguishes matching theory from other approaches to macroeconomic modelling is the presence of a matching function.

Matching theory has been especially influential in labor economics, where it has been used to describe the formation of new jobs, as well as to describe other human relationships like marriage. Matching theory is closely related to an earlier framework called search theory which is somewhat more commonly applied in microeconomics.

One of the founders of matching theory is Dale T. Mortensen of Northwestern University. A textbook treatment of the matching approach to labor markets is Christopher A. Pissarides' book Equilibrium Unemployment Theory.

The matching function

A matching function is a mathematical relationship that describes the formation of new relationships (also called 'matches') from unmatched agents of the appropriate types. For example, in the context of job formation, matching functions are sometimes assumed to have the following 'Cobb-Douglas' form:

$m_{t}\;=\;M(u_{t},v_{t})\;=\;\mu u_{t}^{a}v_{t}^{b}$

where $\mu$, a, and b are positive constants. In this equation, $u_{t}$ represents the number of unemployed job seekers in the economy at a given time t, and $v_{t}$ is the number of vacant jobs firms are trying to fill. The number of new relationships (matches) created (per unit of time) is given by $m_{t}$.

A matching function is in general analogous to a production function. But whereas a production function usually represents the production of goods and services from inputs like labor and capital, a matching function represents the formation of new relationships from the pools of available unmatched individuals. Estimates of the matching function suggest that it has constant returns to scale, that is, $a+b\approx 1$.

If the fraction of jobs that separate (due to firing, quits, and so forth) from one period to the next is $\delta$, then the change in employment from one period to the next depends on the formation of new matches and the separation of old matches. A period may be treated as a week, a month, a quarter, or some other convenient period of time. (For simplicity, we are ignoring the entry of new workers into the labor force, as well as death or retirement of old workers, but these factors can easily be accounted for as well.) Suppose we write the number of workers employed in period t as $n_{t}=L_{t}-u_{t}$, where $L_{t}$ is the labor force in period t. Then given the matching function described above, the dynamics of employment over time would be given by

$n_{t+1}\;=\mu u_{t}^{a}v_{t}^{b}+(1-\delta )n_{t}$

Applications

Matching theory has been applied in many economic contexts, including:

• Formation of jobs, from unemployed workers and vacancies opened by firms
• Formation of marriages, from unmatched men and women
• Allocation of loans from banks to entrepreneurs
• Sales occurring when sellers and buyers meet

See also

• Search theory
• Labor economics
• Monetary economics