Moreover, the foreign price elasticity with respect to exchange rates is proportional to domestic penetration of those markets (Dornbusch 1989). We use these relationships, assume that law of one price holds ex ante, and assume that the product of two trade share terms is approximately equal zero (cM = 0 and cM* = 0).11 Under these assumptions there is a very clean expression for the elasticity of L with respect to exchange rates itat on:
Equation (9) clearly shows the three channels through which optimal labor demand is exposed to exchange rate movements, and the key roles played by industry features and the producer’s external orientation. The three transmission channels are through industry import penetration (M), export orientation (c), and imported input use, a.
All else equal, from the derived elasticity of labor demand (under costless adjustment) to exchange rates we observe specific ways in which industry features magnify or reduce this elasticity: i) When the production technology is labor intensive (i.e. b is high), labor demand is less responsive to exchange rates; ii) Greater import penetration of domestic markets raises the sensitivity of labor demand to exchange rates; iii) Higher export orientation of an industry increases the sensitivity of its labor demand to exchange rates; and iv) Greater reliance on imported inputs into production (higher a) reduces labor demand following a stronger domestic currency (since domestic currency depreciation raises the cost of one of the factors of production).
Equation (9) also explicitly shows that the role of exchange rates in labor demand is strongest in industries that impart pricing power to firms. This occurs when к is high and when demand elasticities are low. Thus, all else equal, labor demand is most sensitive to exchange rates if foreign firms have pricing power in local markets. Product demand elasticities – in both domestic and foreign markets – also are important. The higher the price elasticity of demand facing producers, and the lower the implied price-over-cost markups in the industry, the more responsive will be labor demand to exchange rates.
Using equations (7) and (9), and log-linearizing, optimal labor demand in the absence of adjustment costs can be expressed in reduced form as:
where all variables other than c, M, and a are defined in logs.
We next use equations (6) and (10) to solve for optimal labor demand at any point in time. Recall that equation (6) shows that the reaction of employment today to an exchange rate shock depends not only on the current shock, but also on all future expected changes of the exchange rate through their effects on Zt+j. The actual structure of labor demand of any specific shock depends on whether the shock is permanent or transitory. A general form for optimal labor demand is given by a reduced form expression: