3. Worth 25 points total. Consider a firm that uses only labor to produce with f

Question

3. Worth 25 points total. Consider a firm that uses only labor to produce with function M ZL, where Y is output (a variable), Z is total factor productivity (a variable that the firm takes as given, i.e., it is an erogenous variable), L is labor input, and > 0 is a parameter. Let ue 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 firm's cost minimization problem can be formally stated as: min wLt Lt such that Derive mathematically the firm's cost minimizing level of labor

Explanation / Answer

Function that minimises cost can be solved using Lagrange method as below

O= wt×Lt-(Y_bar-Zt×Lt^alpha)

We have only variable here is labour hence differentiating this above function with respect to Labour L

dO/dL=wt-×Zt×alpha×Lt^(alpha-1)

dO/d= Y_-Zt×Lt^alpha=0

wt×Lt=×Zt×alpha×Lt^alpha

wt×Lt=×Y_

=wt×Lt/Y_

Now maximising profit function

=Zt×Lt^alpha-wt×Lt

d/dLt=Zt×Lt^(alpha-1)×alpha-wt

FoC equals to zero

wt×Lt=Y_

wt=Y_/Lt

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