Question

Wheat is produced according to the production function Q = 100 K^0.8 L^0.2

a. Beginning with a capital input of 4 and a labor input of 49, show that the marginal product of labor and the marginal product of capital are both decreasing.

b. Does this production function exhibit increasing, decreasing, or constant returns to scale?

please explain in 4 sentences thank you!

Answer 1

a) Marginal Product of Labour = MPL = dQ/dL = 20*K^0.8*L^-0.8

Marginal Product of Capital = MPK = 80*K^(-0.2)*L^0.2

As you can see clearly as labour and capital increases, the marginal product of labour and marginal product of capital respectively decreases (because L has negative power in MPL and K has negative power in MPK and therefore any increase in them will cause a decrease in MP)

b) According to the definition:

If both K and L increase by x times then if

1) Q increases by less than x times ---> Decreasing return to scale

2) Q increases by x times --> Constant return to scale

3) Q increases by more than x times --> Increasing return to scale

Increasing both K and L by x times,

Q* = 100*(x*K)^0.8*(x*L)^0.2 = x*(100*K^0.8*L^0.2) = x*Q

Since Q has also increased by x times therefore, there is constant returns to scale.