The marginal cost for the firm of an additional unit of labor has three components: 1) the additional wage that has to be paid; 2) the costs incurred in adjusting the level of input use by that additional unit; and 3) the present value of the change in additional costs of changing the optimal labor amount in the future by the firm. The resulting first-order condition has the form:
Equation (5) is a second order difference equation in units of labor. As a step toward solving this equation, it is convenient at this point to define a new variable, L , as that level of input use which would be the optimal amount chosen by the firm in the absence of adjustment costs, i.e. at b=0. Nickell (1986) shows that, under reasonable assumptions, the stable root of the resulting fundamental equation implies a partial adjustment path of optimal employment:
where g denotes expected real wage growth rate (assumed to be a constant), and m denotes the stable root of the fundamental equation for employment. m is increasing in b and decreasing in the wage sensitivity of marginal revenue product.
Equation (6) shows that the target level of current employment is simply a convex combination of last period’s employment and a weighted sum of all future values of L. The weights on future values of L decline geometrically. The “speed of adjustment” of labor demand to the levels that would exist in the absence of costly adjustment is given by 1-m. This weighting structure is intuitive: changes in employment will be slower in industries with large adjustment costs and faster in those industries with more wage sensitive marginal revenue product.
At this point, we need to elaborate on the solution for L at any date t. The solution to the first-order conditions of the producer problem, after invoking Euler’s theorem, shows that optimal labor demand by a firm in the absence of adjustment costs is:
Equation (7) shows that optimal labor demand in the absence of adjustment costs depends on the structure of and importance to the firm of both domestic and foreign demand, and on the substitutability between productive factors measured alongside their costs.
Our goal is to use the optimal labor demand defined by equations (6) and (7) to derive a workable and intuitive relationship between exchange rates — a variable assumed exogenous to the firm — and labor demand. Recall from our framework in equation (1) that there are three potential sources of shocks that the firm faces, through aggregate domestic demand, through foreign demand, and through the exchange rate. Our emphasis here is on the effect of changes in
where c represents the share of export sales in revenues, c‘ = P *‘q *’/(p‘q‘ + P * q *). hp,e and hp*e are domestic and foreign price elasticities with respect to exchange rates. These price elasticities are best understood in the context of theories of exchange-rate pass through. For monopolistically competitive markets, the domestic price elasticity with respect to exchange rates is proportional to import penetration of domestic markets, i.e. hP,e x kM. online payday loan direct lenders