MACROECONO 180.302 -Consider a firm that uses only labor to produce

Consider a firm that uses only labor to produce with function Yt = ZtL t , where Yt is output (a variable), Zt is total factor productivity (a variable that the firm takes as given, i.e., it is an exogenous variable), Lt is labor input, and > 0 is a parameter. Let wt denote the wage rate (which the firm takes as given) and normalize the price of output to 1. Suppose that the firm wants to choose a cost-minimizing level of labor input in order to produce the constant level of output Y > 0. As such, the firms cost minimization problem can be formally stated as: min Lt wtLt such that ZtL t = Y . Derive mathematically the firms cost minimizing level of labor.