Unemployment cycles

Agents and Technology. Time is continuous, t ∈ R + . There is a measure one of risk neutral
workers in the economy. A worker is unemployed searching for a job 11 , or employed, in which case
he can choose to search actively or passively on-the-job. We assume that on-the-job search only takes
place in low productivity jobs. 12
The flow utility from being unemployed is b and the flow utility of
employment is equal to the wage, w. The search cost when unemployed or with passive on-the-job
search is normalized to zero and the search cost for active search when employed is k, so search costs
increase in search intensity. Workers maximize the value of employment: They search actively if the
gain from search exceeds the cost. Otherwise, they search passively at no cost (more below).
There is a large measure of potential jobs (firms). Firms can open a job paying a flow cost c.
If they stay inactive their payoff is zero.
Firms are risk neutral and maximize the discounted sum
of profits. Denote the measure of job openings by v. All jobs are ex ante identical, but ex post
heterogeneous in their job productivity y. We assume the technology is given by f(y) = py, where p
is aggregate and y ∈ {y, y} is match-specific productivity. 13
When a job is filled by an unemployed
worker, the productivity is y and when it is filled by a formerly employed worker the productivity is
y, with y ≤ y. 14 This captures in a stylized way the economy’s job ladder: Workers tend to be better
matched to the new job after they switch, which is reflected in the data by substantial wage gains after
EE transitions. We thus model the job ladder as improvements in the match-specific component of a
worker-firm pair.
Denote the measure of the unemployed by u; the measure of the employed in a low productivity
job by γ; the measure of the employed in high productivity jobs by ξ. Since the measure of workers is
equal to one, feasibility requires that u + γ + ξ = 1.
Market Frictions, Search and Wage Setting. Meetings between jobs and workers are stochastic,
and are modeled by means of a standard matching function m(v, s), where m is increasing, concave and
has constant returns to scale, and where v denotes the measure of vacancies and s the measure of job
searchers. Therefore the matching probability for a worker is m(θ), where θ = v s
, and that of a firm is
11 We do not include the possibility of endogenous search behavior by the unemployed (only on-the-job) for two reasons:
First, the empirical studies on how search intensity of the unemployed varies over the cycle are inconclusive. There
is evidence on both counter-cyclical search intensity (Mukoyama, Patterson, and Sahin (2014)) and pro-cyclical search
intensity (Schwartz (2014)). Second, in our model even if we introduced endogenous search intensity of the unemployed,
they would always (independent of the business cycle) choose a unique level of search intensity because, due to sequential
auction wage setting with constant value of unemployment, the gains for the unemployed from search are constant.
12 This assumption is for tractability; our results do not hinge on it. Those in high productivity jobs would have an
incentive to search merely to increase their wages. We can rationalize the no search behavior in high productivity jobs by
implicitly assuming that the wage increase does not compensate for the cost of on-the-job search.
13 Below we assume that not only match-specific productivity y but also the cost of on-the-job search k and unemployment
benefits b are proportional to p. This is consistent with the findings of Chodorow-Reich and Karabarbounis (2013) that
the value of unemployment is pro-cyclical.
14 If the surplus of a low type match is positive, it is optimal for the firm to accept this match even if that surplus is
lower than the surplus of a high type match. In our calibration below, the low type match surplus is positive. If the low
type match surplus were negative instead, our formulation implicitly assumes that firms commit to hire any worker type,
whether she is hired out of unemployment or from an on-the-job move.
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