11. Search Matching

11.1. Introduction

The level of unemployment is measured as the number of people actively seeking work. (In practice, we focus on just “working-age population”.) In the data we often see non-trivial and quite persistent rates of unemployment.

The Walrasian general equilibrium approach to modeling labor markets is technically quite simple and convenient, as we saw in previous topics such as the real business cycle. Why don’t we just stop there? We don’t because, there are limitations to Walrasian equilibrium theories in terms of explaining unemployment. The neoclassical representative agent models of growth and business cycles (e.g. RBC) do not have unemployment by construction. All individuals work in those models, although they can vary the optimal level of employment measured at the “intensive margin” – i.e. they can vary the amount of hours work. The model is silent on the observed fact that individuals do fall out of employment altogether, or variations on aggregate employment on the “extensive margin”. Similarly, earlier unemployment models based on efficiency wages or bargaining theories adopt small departures from neoclassical theories by introducing information asymmetry or imperfect competition with sticky wages in the labor market. However, they are still silent on the “extensive margin” of unemployment.

Institutionally, frictions are more significant in labor markets than other markets. Thus an alternative means of building up a good story of labor market frictions should include the following ingredients:

  • Heterogenous goods (labor) with no single marketplace to clear excess demand or supply; and
  • Parties to a trade must search for someone to trade with.

Thus, such a model would involve transactions costs – it takes time for a person to look for a job and it takes time for a vacancy to find a worker. Search frictions can potentially rationalize significant transit times between unemployment and employment in a equilibrium where markets do not necessarily need to clear.

First, we study the one-sided, partial-equilibrium models of job search where the distribution of wage offers is assumed given: [mcc70]. From this simple model we can characterize the optimal reservation wage strategy quite easily. We will see that this is a simple benefit-cost analysis on the part of the worker. We also consider variations on the model such as allowing for quits, exogenous layoffs, and we also discuss the implications for aggregate unemployment dynamics.

Then we consider more recent general equilibrium treatment, or two-sided search, where workers and firms are looking for each other and must somehow match up through a matching process or function. This endogenizes the wage distribution and wage determination process. The early approach is due to the works of [mor82], [pis90] and [dia82]. For a recent survey of the state of the literature, see [rsw04]. In search models as theoretical applications, we will also see how our kung-fu grip on dynamic programming comes in handy.

11.2. The McCall one-sided search model

In this simple model a single worker who in unemployed is faced with a contemporaneous decision of accepting some unemployment benefit payout and thus remaining unemployed for one more period, or accept a stochastic wage offer and be employed indeterminately. This sort of model involves a decision process that is known as a “stopping time” problem which is also used in modeling call options in finance, in modeling an investors decision of when to sell an asset, or the very old “secretary problem” – i.e. one of an employer deciding when to hire a from sequence of secretarial candidates – originally due to [cay1875]. [1]

11.2.1. Defining the wage offer distribution

Let’s look at some basic properties of probability distributions used in this search theory first. We will soon be dealing with wage offer distributions with the properties that:

  • Wage offers are non-negative; and
  • It does not make sense to have infinitely positive wage.

To satisfy these properties we assume that there is zero probability on observing negative wage offers. There exists an upper bound on wage offers with probability one. So then we also have wage offers as nonnegative random variable that have first moments.

Let \(w\) be a continuous random variable with cumulative probability distribution function \(F(W)\) defined by

\[\Pr (w \leq W) =F(W).\]

ASSUMPTION 1.

The probability distribution function F satisfies

  1. \(F(0)=0.\)
  2. \(F( \infty ) =1.\)
  3. \(F\) is nondecreasing and continuous from the right.
  4. There exists \(B<+\infty\) such that \(F(B) =1\).

Right-continuity of \(F\) on \([0,B]\) merely implies that we have well-defined probability measures. [2]

The expected value or mean of \(w\) is

\[\mathbb{E}\left( w\right) =\int_{0}^{B}wdF\left( w\right)\]

Note we can integrate by parts:

\[\begin{split}\begin{eqnarray*} \int_{0}^{B}wdF\left( w\right) &=&\left. wF\left( w\right) \right\vert _{0}^{B}-\int_{0}^{B}F\left( w\right) dw \\ &=&B-\int_{0}^{B}F\left( w\right) dw \end{eqnarray*}\end{split}\]

so we can also write

\[\mathbb{E}\left( w\right) =B-\int_{0}^{B}F\left( w\right) dw.\]

Consider event of independent drawing of \(w_{1}\) and \(w_{2}\) that are less than \(w\) from distribution \(F\): i.e. the event \(\left\{ \left( w_{1}<w\right) \cap \left( w_{2}<w\right) \right\} .\) Then

\[\begin{equation*} \Pr \left\{ \left( w_{1}<w\right) \cap \left( w_{2}<w\right) \right\} =\Pr \left\{ \max \left( w_{1},w_{2}\right) <w\right\} =F\left( w\right) ^{2} \end{equation*}\]

The random variable \(\max \left( w_{1},w_{2}\right)\) has mean

\[\begin{equation*} \mathbb{E}\left[ \max \left( w_{1},w_{2}\right) \right] =B-\int_{0}^{B}F\left( w\right) ^{2}dw. \end{equation*}\]

Similarly for \(n\) independent random variables drawn from \(F\),

\[\begin{equation*} \mathbb{E}\left[ \max \left( w_{1},w_{2},...,w_{n}\right) <w\right] =B-\int_{0}^{B}F\left( w\right) ^{n}dw. \end{equation*}\]

11.2.3. Detour on mean-preserving spreads

Following Rothschild and Stiglitz a useful way to compare riskiness of two distributions with the same mean is the concept of mean-preserving spreads. Consider a family of distributions that have the same mean. We can label distributions in this class by a parameter \(r\in R\), where \(R\) is some set of parameters. So the \(r^{th}\) distribution in our family of distributions with identical mean is denoted by \(\Pr \left( w\leq W\right) =F\left( W,r\right)\).

ASSUMPTION 2.

The class of probability distribution functions \(\{F\left( W,r\right)\}\) has the property that:

  1. \(F\left( W,r\right)\) is differentiable w.r.t. \(r\) for all \(W\in \left[0,B\right]\).
  2. \(F\left( 0,r\right) =0\) for all \(r\in R\).
  3. There is a single \(B<+\infty\) such that \(F\left( B,r\right) =1\) for all \(r\in R\).

So we focus on a class of probability distribution functions, \(R\), for nonnegative bounded random variables. Notice that

\[\begin{equation*} E\left( w\right) =B-\int_{0}^{B}F\left( w,r\right) dw \end{equation*}\]

Thus any two distributions with the same “area under the curve” \(F\left( w,r_{i}\right)\) for \(i=1,2\) have the same mean.

DEFINITION 1.

Two distributions indexed by \(r_{1},r_{2}\in R\) have identical means if

(3)\[\begin{equation} \int_{0}^{B}\left[ F\left( w,r_{2}\right) -F\left( w,r_{1}\right) \right] dw=0. \end{equation}\]

DEFINITION 2.

Two distributions indexed by \(r_{1},r_{2}\in R\) satisfy the single-crossing property if there exists \(\widehat{w}\) with \(0<\widehat{w}<B\) such that

(4)\[\begin{split}\begin{equation} F\left( w,r_{2}\right) -F\left( w,r_{1}\right) \left\{ \begin{array}{c} \leq 0\text{ for }w\geq \widehat{w} \\ \geq 0\text{ for }w\leq \widehat{w}% \end{array}% \right. \end{equation}\end{split}\]

We can think of distribution \(r_{2}\) as a redistribution of probability mass of distribution \(r_{1}\) toward the tails of the distribution while keeping the mean constant.

EXERCISE 1.

Illustrate two distributions that satisfy the single-crossing property (4).

Properties (3) and (4) imply

(5)\[\begin{equation} \int_{0}^{y}\left[ F\left( w,r_{2}\right) -F\left( w,r_{1}\right) \right] dw\geq 0, \qquad 0\leq y\leq B. \end{equation}\]

We can define the concept of mean-preserving spread using (3) and (5) – i.e. distribution \(r_{2}\) is said to be obtained from a distribution \(r_{1}\) by a mean-preserving increase in spread if the two distributions satisfy (3) and (5).

Note that (3) and (5) provide an ordering of distributions according to riskiness that is transitive. But (3) and (4) do not.

For very very very very small changes in \(r\), we can use the differential version of (3) and (5) to rank distributions with identical mean, according to their riskiness.

Invoking Assumption 2 we have

DEFINITION 3.

An increase in \(r\) represents a mean-preserving increase in risk if

(6)\[\begin{equation} \int_{0}^{B}\frac{\partial F\left( w,r\right) }{\partial r}dw=0 \end{equation}\]

and

(7)\[\begin{equation} \int_{0}^{y}\frac{\partial F\left( w,r\right) }{\partial r}dw\geq 0, \qquad 0\leq y\leq B. \end{equation}\]

11.2.4. Effect of mean preserving spreads in the McCall model

Now we’re back in the saddle again. For another way to see how \(\overline{w}\) is determined we go back to the equation

\[\begin{equation*} \overline{w}-c=\frac{\beta }{1-\beta }\int_{\overline{w}}^{B}\left( w^{\prime }-\overline{w}\right) dF\left( w^{\prime }\right) . \end{equation*}\]

which equates the cost to the benefit of searching. Now re-write as

\[\begin{split}\begin{multline*} \overline{w}-c=\frac{\beta }{1-\beta }\int_{\overline{w}}^{B}\left( w^{\prime }-\overline{w}\right) dF\left( w^{\prime }\right) +\frac{\beta }{% 1-\beta }\int_{0}^{\overline{w}}\left( w^{\prime }-\overline{w}\right) dF\left( w^{\prime }\right) \\ -\frac{\beta }{1-\beta }\int_{0}^{\overline{w}}\left( w^{\prime }-\overline{w% }\right) dF\left( w^{\prime }\right) . \end{multline*}\end{split}\]

This gives us

\[\begin{split}\begin{eqnarray*} \overline{w}-c &=&\frac{\beta }{1-\beta }\int_{0}^{B}\left( w^{\prime }-% \overline{w}\right) dF\left( w^{\prime }\right) -\frac{\beta }{1-\beta }% \int_{0}^{\overline{w}}\left( w^{\prime }-\overline{w}\right) dF\left( w^{\prime }\right) \\ &=&\frac{\beta }{1-\beta }\left[ E\left( w\right) -\overline{w}\right] -% \frac{\beta }{1-\beta }\int_{0}^{\overline{w}}\left( w^{\prime }-\overline{w}% \right) dF\left( w^{\prime }\right) \end{eqnarray*}\end{split}\]

or

\[\begin{equation*} \overline{w}-\left( 1-\beta \right) c=\beta E\left( w\right) -\beta \int_{0}^{\overline{w}}\left( w^{\prime }-\overline{w}\right) dF\left( w^{\prime }\right) \end{equation*}\]

Exercise 2.

Show that

\[\begin{equation*} \int_{0}^{\overline{w}}\left( w^{\prime }-\overline{w}\right) dF\left( w^{\prime }\right) =-\int_{0}^{\overline{w}}F\left( w^{\prime }\right) dw^{\prime }. \end{equation*}\]

Hint: Let \(u=\left( w^{\prime }-\overline{w}\right)\) and \(v=F\left( w^{\prime }\right)\). Use integration by parts formula, \(\int udv=uv-\int vdu\).

So then we have

\[\begin{split}\begin{eqnarray*} \overline{w}-c &=&\beta \left[ E\left( w\right) -c\right] -\beta \int_{0}^{% \overline{w}}\left( w^{\prime }-\overline{w}\right) dF\left( w^{\prime }\right) \\ &=&\beta \left[ E\left( w\right) -c\right] +\beta \int_{0}^{\overline{w}% }F\left( w^{\prime }\right) dw^{\prime }. \end{eqnarray*}\end{split}\]

Let \(g\left( w\right) =\int_{0}^{w}F\left( w^{\prime }\right) dw^{\prime }\). Note:

  • \(g\left( 0\right) =0\),
  • \(g\left( w\right) \geq 0\),
  • \(g^{\prime }\left( w\right) =F\left( w\right) >0\),
  • \(g^{\prime \prime }\left( w\right) =F^{\prime }\left( w\right) >0\), for \(w>0\).

Then we can write

\[\begin{split}\begin{eqnarray*} \overline{w}-c &=&\beta \left[ E\left( w\right) -c\right] +\beta \int_{0}^{% \overline{w}}F\left( w^{\prime }\right) dw^{\prime } \\ &\equiv &\beta \left[ E\left( w\right) -c\right] +\beta g\left( \overline{w}% \right) . \end{eqnarray*}\end{split}\]

[We can show this in a diagram...]

Consider all else given. Now compare two wage offer distributions \(F\left( w^{\prime },r_{1}\right)\) and \(F\left( w^{\prime },r_{2}\right)\) where (6) and (7) are satisfied. Thus a mean-preserving increase in spread, with \(r_{2}>r_{1}\), makes the curve for the RHS steeper. A mean-preserving increase in spread makes \(g\left(\overline{w}\right)\) larger, and hence, \(\overline{w}\) larger.

The model implies:

  • Increase in unemployment benefit, \(c\), increases \(\overline{w}\) since it lowers to cost of searching one more time.
  • A mean-preserving increase in riskiness of wage offers increases \(\overline{w}\).
    • Greater probability of high wage offers increases value of (being unemployed and) searching.
    • Higher probability of drawing bad offers not so detrimental since option to work at bad wage offers will not be exercised anyway.
    • an option holder (here, option to accept \(w\)) receives payoffs from the tail of the distribution.

11.2.5. Allowing quits

Read this from Section 6.3.2 in [ls04]. Dooyit! It turns out that if we relax the no-quit assumption – i.e. give a worker option to quit her job and search again, she would never exercise that option after being unemployed for one period. Prove that this is true as an exercise.

11.2.6. Waiting times

We can also infer the duration of unemployment in the model. Since we assumed independent sequential draws of wage offers this is easy to calculate. Let \(N\) be time until a successful offer is drawn. Then

\[\begin{equation*} \Pr \left( N=1\right) =1-\int_{0}^{\overline{w}}dF\left( w^{\prime }\right) \equiv 1-\lambda. \end{equation*}%\]
\[\begin{equation*} \Pr \left( N=2\right) =\left( 1-\lambda \right) \lambda \end{equation*}\]

and more generally

\[\begin{equation*} \Pr \left( N=j\right) =\left( 1-\lambda \right) \lambda ^{j-1} \end{equation*}\]

So waiting time is a random variable that is geometrically distributed. The mean waiting time is then \(\left( 1-\lambda \right) ^{-1}\).

11.2.7. Layoffs

Consider the same setup as before. Now assume after the first period on the job, there is a probability \(\alpha \in \left(0,1\right)\) of being laid off. For simplicity, assume \(\alpha\) independent of tenure. When employed, the worker receives \(w\) per period until fired. Consider a worker facing offer \(w\). Now let \(\widehat{v}\left( w\right)\) be this (previously unemployed) worker’s expected present value of income modified for probability of layoffs.

Specifically, the Bellman equation is now [3]

\[\begin{split}\begin{multline*} \widehat{v}\left( w\right) =\max \left\{ w+\beta \left( 1-\alpha \right) \widehat{v}\left( w\right) +\beta \alpha \left( c+\beta \int \widehat{v}% \left( w^{\prime }\right) dF\left( w^{\prime }\right) \right) ,\right. \\ \left. c+\beta \int \widehat{v}\left( w^{\prime }\right) dF\left( w^{\prime }\right) \right\} . \end{multline*}\end{split}\]

Guess \(\widehat{v}\left( w\right)\) is a nondecreasing function in \(w\). So solution of the reservation wage form:

\[\begin{split}\begin{equation*} \widehat{v}\left( w\right) =\left\{ \begin{array}{ll} \frac{w+\beta \alpha \left( c+\beta \int \widehat{v}\left( w^{\prime }\right) dF\left( w^{\prime }\right) \right) }{1-\beta \left( 1-\alpha \right) }\text{,} & \text{if }w\geq \overline{w} \\ c+\beta \int \widehat{v}\left( w^{\prime }\right) dF\left( w^{\prime }\right) , & \text{if }w\leq \overline{w}% \end{array}% \right. \end{equation*}\end{split}\]

where \(\overline{w}\) solves

\[\begin{equation*} \frac{w+\beta \alpha \left( c+\beta \int \widehat{v}\left( w^{\prime }\right) dF\left( w^{\prime }\right) \right) }{1-\beta \left( 1-\alpha \right) }=c+\beta \int \widehat{v}\left( w^{\prime }\right) dF\left( w^{\prime }\right) . \end{equation*}\]

Rearranging,

\[\begin{equation} \frac{\overline{w}}{1-\beta }=c+\beta \int \widehat{v}\left( w^{\prime }\right) dF\left( w^{\prime }\right) \tag{*} \end{equation}\]

Contrast this with the reservation wage in the model previously where there is no risk of layoffs, \(\alpha =0\):

(8)\[\begin{equation} \frac{\overline{w}}{1-\beta }=c+\beta \int_{0}^{B}v\left( w^{\prime }\right) dF\left( w^{\prime }\right) \end{equation}\]

The difference now is that \(\widehat{v}\left( w^{\prime }\right) \neq v\left( w^{\prime }\right)\). The two value function are not the same! In particular, \(\widehat{v}\left( w^{\prime }\right) <v\left( w^{\prime }\right)\). And, \(\widehat{v}\left( w^{\prime }\right) <v\left( w^{\prime }\right)\) implies that reservation wage \(\overline{w}\) is strictly lower in the economy with firing. Unemployed workers optimally search less because expected payoffs (continuation value) is lower in the presence of a probablity of being fired. That is, there is less incentive to hold out or to keep searching for high wage offers when a job obtained is expected to last for a short time.

11.2.8. Lake Model

What are the implications of the model on the aggregate unemployment rate? Now suppose our economy has a continuum of ex ante identical workers who face the McCall job-search problem previously.

Each worker may be drawing different \(w \text{'s}\) so that ex post they are heterogenous. The agents move recurrently between unemployment and employment.

From previous calculations, we know the mean duration of each employment spell is \(\alpha ^{-1}\) and mean duration of unemployment is \(\left( 1-\lambda \right) ^{-1}\).

So the average unemployment rate is governed by the difference equation

\[\begin{equation*} U_{t+1}=\alpha \left( 1-U_{t}\right) +\lambda U_{t} \end{equation*}\]

where

  • \(\lambda =\int_{0}^{\overline{w}}dF\left( w^{\prime }\right) =F\left(\overline{w}\right)\) is probability of rejecting an offer.
  • \(\alpha\) is “hazard rate of leaving employment”.
  • \(1-F\left( \overline{w}\right)\) is “hazard rate of leaving unemployment”.

Since \(\lambda <1\), the difference equation is stable and thus, \(U_{t}\rightarrow U\).

A stationary unemployment rate is one where \(U_{t+1}=U_{t}=U\). Solving for this we get

\[\begin{equation*} U=\frac{\left[ 1-F\left( \overline{w}\right) \right] ^{-1}}{\left[ 1-F\left( \overline{w}\right) \right] ^{-1}+\alpha ^{-1}}. \end{equation*}\]

Thus we can deduce the long-run determinants of unemployment in this model. Specifically, \(U\) depends on average duration of unemployment as a fraction of sum of average duration of unemployment and average duration of employment.

Note that:

  • Increase in \(\overline{w}\) raises mean duration of unemployment \(\left[1-F\left( \overline{w}\right) \right] ^{-1}\) and thus steady-state \(U\).
  • Increase in job separation rate \(\alpha\) increases \(U\).

Exercise 3.

What happens to \(U\) when we have a more generous unemployment benefits scheme, \(c\)?

11.3. A simple two-sided search model

XXXXX [To be added ]

11.4. Appendix A

11.5. Well-defined probability measures

We adapt some results and definitions from [dud02].

DEFINITION 4.

Let \(X\) be a set and \(\mathcal{S}\) a collection of subsets of \(X\). A function \(\mu\) is said to be countably additive if whenever

  • \(A_{i}, A_{j} \in \mathcal{S}\) and \(A_{i} \cap A_{j} = \emptyset\), \(i,j = 1,2,...\), and
  • \(A := \cup_{i=1}^{\infty}A_{i} \in \mathcal{S}\),

we have

\[\mu(A) = \sum_{i=1}^{\infty}\mu(A_{i}).\]

In measure theory we can show that the length \(\mu = b-a\) of any interval \((a,b] \subset \mathbb{R}\) is countably additive. What if we replace this measure by \(F(b)-F(a)\) for some suitable function \(F\)? In general we have the following definition.

DEFINITION 5.

A function \(F\) from \(\mathbb{R}\) into itself is called nondecreasing if \(F(x) \leq F(y)\) whenever \(x \leq y\). Then for any \(x\), the limit \(F(x^{+}) := \lim_{y \downarrow x}F(y)\) exists. \(F\) is said to be continuous from the right if and only if \(F(x^{+}) = F(x)\) for all \(x\).

Let \(\mathcal{S} := \{(a,b] : - \infty < a\leq b < +\infty \}\). The function \(\mu := \mu_{F}\) is defined on \(\mathcal{S}\) by \(\mu((a,b]):= F(b)-F(a) \in [0,1]\) and if \(a=b\), such that \((a,b] = \emptyset\), then \(\mu(\emptyset) = 0\). Then \(\mu(A) \in [0,1]\) for each \(A \in \mathcal{S}\).

THEOREM 1.

If \(F\) is nondecreasing and continuous from the right, then on \(\mathcal{S}\), \(\mu\) is countably additive.

proof: See [dud02], pp.87-88.

DEFINITION 6.

A countably additive function \(\mu\) from a \(\sigma\)-algebra \(\mathcal{S}\) of subsets of \(X\) into \([0,1]\) is called a probability measure. Then \((X,\mathcal{S},\mu)\) is a measurable space.

11.6. Search Matching References

[cay1875]Cayley, A. (1875): “Mathematical Questions with Their Solutions,” Educational Times, 23(18), 1875.
[dia82]Diamond, P. (1982): “Wage Determination and Efficiency in Search Equilibrium,” Review of Economics Studies, 49, 217–227.
[dud02](1, 2) Dudley, R. M. (2002): Real Analysis and Probability. Cambridge University Press, New York, NY.
[ls04]Ljungqvist, L., and T. J. Sargent (2004): Recursive Macroeconomic Theory. MIT Press, Cambridge, MA.
[mcc70](1, 2) McCall, J. (1970): “Economics of Information and Job Search,” Quarterly Journal of Economics, 84, 113–126.
[mor82]Mortensen, D. T. (1982): “The Matching Process as a Noncooperative Bargain- ing Game,” in The Economics of Information and Uncertainty, ed. by J. McCall, pp. 233–258, Chicago, IL. University of Chicago Press.
[pis90]Pissarides, C. (2000): Equilibrium Unemployment Theory. MIT Press.
[put05]Puterman, M. L. (2005): Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, Hoboken, NJ.
[rsw04]Rogerson, R., R. Shimer, and R. Wright (2004): “Search-Theoretic Models of the Labor Market – A Survey,” NBER Working Paper.

Footnotes

[1]For more discussion on these examples, see section 3.5 of [put05].
[2]See Appendix Well-defined probability measures
[3]Note if \(\alpha =0\), back to same Bellman equation as before.