A firm has the following production function q 5LK05 2L2K

A firm has the following production function: q = 5LK^0.5 + 2L²K - L³K, with capital fixed at K = 9

Show that the firm\'s elasticity of output with respect to labour in the short run is a function of marginal product of labour and average product of labour.

Calculate the short-run elasticity of output with respect to labour.

Solution

Labor elasticity of output measures the percentage change in output divided by percantage change in labor input.

Output elasticity e=dQ/dL*Q/L

Given `q = 5LK^0.5 + 2L²K - L³K

we know, that Marginal product of labor MP_L=dQ/dL

Avg. product of labor MP_L=Q/L

so, by substituting these in the elasticity formula, we get

e=Marginal product of labor/Average product of labor

b. Given K is fixed at K=9

q = 5LK^0.5 + 2L²K - L³K

substituting K=9 in this equation we get

q = 5L9^0.5 + 2L²9 - L³9

q= 15L+18L² - 9L³

MP_L=dq/dL and AP_L=q/L

dQ/dL=15+36L-27L²

q/L=15+18L-9L²

^e=MP_L/AP_L

₌15+36L-27L^2/15+18L-9L²

^{for L=1}

24/24

e=1