Question

a) Show that the following Cobb-Douglas production function, f(K,L) = KαL1−α, has constant returns to scale.

(b) Derive the marginal products of labor and capital. Show that you the MPL is decreasing on L and that the MPK is decreasing in K.

Answer 1

A.

f (K, L) = K^α * L^(1-α)

When K and L are increased by X, then:

f(XK, XL) =  (XK)^α * (XL)^(1-α)

f(XK, XL) = X^(α + 1- α) * ( K^α * L^(1-α))

f(XK, XL) = X* f (K, L)

Above equation shows that increase in inputs shows the equal increase in output. So, it is a constant return to scale case.

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B.

f (K, L) = K^α * L^(1-α)

MPK = L^(1-α) * α *K^(α-1)

MPK = α * (L^(1-α) / K^(1-α))

MPL = K^α * (1-α)*L^1-α-1)

MPL = (1-α)*(K^α/L^α)

Above MPL and MPK shows that with each additional increase in unit of labor and capital, the marginal product decreases.