# A Model of Part-time Employment

To study the implications of firm-level changes in demand in the aggregate labor market over the business cycle, I build a model of part-time employment. I incorporate a part-time margin in a competitive search environment with large firms and costly recruitment. The model builds on the framework of Kaas and Kircher (2015), with the extension of the firm’s problem to account for a part-time employment decision. Firm size is determined by decreasing returns to scale production in labor, while firm growth is sub ject to search frictions and convex vacancy posting costs.^{[1]} Alternative frameworks incorporating firm size and search include the random search models of Elsby and Michaels (2013) and Moscarini and Postel-Vinay (2013) and the directed search model in Schaal (2012). These environments, however, all incorporate linear vacancy posting costs. The inclusion of convex recruitment costs produces rich firm dynamics over the life-cycle of the firm, rather than instantaneous firm growth as would occur under linear vacancy costs.

The model is in discrete time with a continuum of risk-neutral workers of mass one and an endogenous mass of firms. Workers and firms discount future payoffs at a rate *<* 1. The timing of the model is as follows: first, aggregate productivity is revealed, new firms pay an entry cost, and idiosyncratic productivity for firms is revealed. Next, firms choose the fraction of their labor force to employ on a part-time basis, a. Firms then produce using their stock of employed labor, post contracts for new hires and vacancy postings, and choose the separation rate for workers. Firms exogenously exit at the end of recruitment with probability *Ỗ.* Workers consume their wages, and unemployed or part-time employed workers consume leisure. Lastly, unemployed workers and vacancies are matched, and separations occur, changing the stocks of unemployment and employment for the next period.

**Firms**

Firms operate a decreasing returns to scale production technology. A firm’s output in one period is xzF(L) when utilizing a mass L *>* 0 of labor in production x G X is the firm’s level of idiosyncratic productivity, and z G Z is aggregate productivity: X and Z are finite state spaces. Because workers are identical, and hours and bodies are assumed to be perfect substitutes, a firm produces the same output from utilizing one unit of labor in production, regardless of this unit coming from two part-time workers (working half-time) or one full-time worker. I assume that the firm only has the ability to employ any individual worker either at full-time or at half-time. The firm’s part-time utilization is decided by choosing what fraction *a G* [0,1] of its labor force to employ at half-time (henceforth part-time), while the fraction 1 — *a* of its labor force is employed full-time and supplies one unit of labor per worker.

Firms choose separation and the fraction of their labor force to employ parttime as a function of contingent histories of x and z. A firm of age j at time t experiencing history (x^{j}, *z** ^{t})* is subject to an exogenous exit probability

*s,*and chooses a separation probability s(x

^{j}, z

^{t}) G [s

**0**, 1], and a fraction of its labor force to employ part-time,

*a(x*

^{j}, z*[0,1]. The probabilities*

^{t}) G*Ỏ >*0 and s

_{0}> 0 reflect the possibility of exogenous firm death and worker separation, respectively.

Firms choose history-contingent recruitment policies to hire workers. A firm with workforce L and productivity (x, z) that posts V vacancies pays recruitment costs C (V, L). I adopt a constant-returns specification for the firm’s vacancy cost function C(V, L), as in Merz and Yashiv (2007):

With this specification, the average cost per vacancy for a firm depends on the firm’s vacancy rate ( V /L ). The flow cost per vacancy is equal for all firms recruiting the same percentage of their current labor force.

**Matching and Contracts**

The labor market features competitive search, under which firms compete for workers by posting long-term contracts. Unemployed workers direct their search to those postings that yield the highest expected utility, taking the fact that better contracts will have a lower probability of matching into account. With a standard matching function, at each contract type there is a corresponding queue length of A unemployed per vacancy, yielding a matching probability for the firm’s vacancy of m. If a firm is to fill a vacancy with probability m, it must offer a contract that attracts A( m) workers. With the usual assumptions on the matching function, the function A(m) is the inverse of the matching function, and satisfies A(0) = 0, A'(0) *>* 1, and A'(1) = ^. The worker’s probability of matching is thus m/A(m).

Firms offer contracts to each cohort of workers recruited in a given period. These contracts specify a sequence of policies fwhich apply to each worker in this cohort for every contingent history so long as the match exists. If the state of a contract offered by an age j firm in a particular history is defined as T = {j, (x^{j}, z^{t})}, then a contract is denoted as:

The retention probability *ộ**(**t*) is the firm’s exit probability *Ỗ* multiplied by the probability of match separation *s*(*t*), so that the probability of separation for a worker is 1 — *ộ**(**t*). *a(T*) specifies the probability of a worker being placed on part-time work. The wage is a function of full-time or part-time employment: *w*f(*T*) or *w*p(*T*), respec- tively.^{[1]} Due to risk-neutrality, workers simply value the expected present value of being employed in a contract, *W*(c(*T*)).