Question

You are given this estimate of a Cobb-Douglas production function: Q = 10K0.6L0.8 A. Calculate the output elasticities of capital and labor. (Note: As shown on p. 300, for the Cobb-Douglas production function Q = 10KaLb the output elasticity of capital is EK = (%ΔQ/%ΔK) = a and the output elasticity of labor is EL = (%ΔQ/%ΔL) =

b. B. Using what you found in Part (A), by how much will output increase if the firm increases capital by 10 percent, while holding labor constant? By how much would output increase if, instead, the firm increases labor by 10 percent, while holding capital constant?

C. By how much will output increase if the firm increases both the quantity of capital and the quantity of labor by 10 percent?

Answer 1

a) Generally the output elasticities of capital and labor are the indexes of capital and labor respectively which implies 0.6 and 0.8 in this case. Find them formally as

EL = (dQ/dL)*(L/Q) = (10*0.8K^0.6L^-0.2) * (L/10*K^0.6L^0.8)

= 0.8*L/L

= 0.8

EK =  (dQ/dK)*(K/Q) = (10*0.6K^-0.4L^0.8) * (K/10*K^0.6L^0.8)

= 0.6K/K

= 0.6

This implies EK = (%ΔQ/%ΔK) = 0.6 and the output elasticity of labor is EL = (%ΔQ/%ΔL) = 0.8

b) Now we have %ΔQ/%ΔK = 0.6 or %ΔQ/10 = 0.6 or %ΔQ = 6%. Similarly %ΔQ/%ΔL = 0.8 or %ΔQ/10% = 0.8 or %ΔQ = 8%

Output rises by 6% when the firm increases capital by 10 percent, while holding labor constant and it increases by 8% when the firm increases labor by 10 percent, while holding capital constant

c) If the firm increases both the quantity of capital and the quantity of labor by 10 percent, then output will increase by 8% + 6% = 14%.