Question

A firm applies a production technology to produce units of output (q) according to the production function q = 20 x L^.5 x K^.5 , where L is a variable input and K is a fixed input. Assume K = 100 in the short run.

solve for the short-run average product of labor (AP L) and the marginal product of labor (MP L) functions. If L = 100, what are the numerical values of AP L and MP L?

Answer 1

q = 20L^0.5K^0.5  

K = 100

q = 20L^0.5(100)^0.5  

= 20(10)L^0.5

q = 200L^0.5

AP_L = q/L  

= 200L^0.5/L  

= 200/L^0.5   

L = 100

AP_L = 200/(100)^0.5  

= 200/10  

= 20

q =   200L^0.5

dq/dL= MP_L = 200(0.5)L^{0.5 - 1}

= 100L^-0.5  

MP_L = 100/L^0.5  

L = 100

MP_L = 100/(100)^0.5

= 100/10

= 10