Firm Dynamics, On-the-Job Search, and Labor Market Fluctuations

2.4. Labor demand and turnover

We can now proceed to consider optimal labor demand and turnover decisions. Recall that the latter is a key challenge that arises from the interaction of firm dynamics and on-the-job search, as labor demand decisions and turnover rates are intertwined in this environment. In this subsection, we provide a solution in which the joint determination of labor demand and turnover takes a surprisingly simple and tractable form.
We begin by returning to the firm’s problem in (7). Optimal hires and separations satisfy (see, for example, Harrison and Taksar, 1983)11
(−c+Πn)dH∗=0,and,ΠndS∗=0.
(12)
The marginal value of labor Πn is set equal to the marginal hiring cost c in the event of hiring, dH∗>0⁠, and to zero in the event of firing, dS∗>0⁠. It follows that the maximized value of the firm satisfies
rΠ=xnα−wn−δnΠn+μxΠx+12σ2x2Πxx.
(13)
The proximate effects of on-the-job search on the firm are thus distilled in the turnover costs δnΠn⁠. Intuitively, each of the firm’s n employees quits at rate δ⁠, and is valued on the margin by the firm at Πn⁠. The magnitude of these turnover costs, and the firm’s response to them, will play a central role in the model.
Equation (12) provides conditions on the marginal value of labor to the firm Πn⁠. For brevity, in what follows we shall denote the latter by J≡Πn⁠. Differentiating the firm value in (13) implies that
rJ=xαnα−1−∂(wn)∂n−∂(δnJ)∂n+μxJx+12σ2x2Jxx,
(14)
The marginal value of labor to the firm is determined by the flow marginal product xαnα−1 net of the marginal cost of labor ∂(wn)/∂n and the marginal turnover costs ∂(δnJ)/∂n⁠, together with the capital gains associated with shocks to the firm’s idiosyncratic productivity.
Solution approach. As stated, the worker surplus in (11), the firm’s value in (13) and the firm’s marginal value in (14) require two idiosyncratic state variables: the firm’s employment n⁠, and productivity x⁠. We now show how the structure of the problem admits a normalization that allows one to distill these into a single idiosyncratic state, namely the firm’s flow marginal product, which we shall hereafter denote m≡xαnα−1⁠. Thus, we propose and verify a solution in which the marginal product m is a sufficient statistic for worker and firm behavior. We gather this together in the following definition.