Question

For each production function below, find the marginal product of capital and labor, and the marginal rate of technical substitution. Show whether the production function exhibits CRS DRS, or IRS. For parts a and b, draw what the isoquant looks like for 10 units of output (a) f(K, L) 2K + 2L (b) f(K, L) 2K1/4L/4 (c) f(K, L) K1/2 L/2

Answer 1

Solution:

Marginal rate of technical substitution, MRTS = MPL/MPK

Marginal product of labor, MPL =

af(K, L)/AL

Marginal product of capital, MPK =

af(KL)/OK

To identify whether the production function exhibits decreasing, constant and increasing returns to scale, we proceed as follows:

f(tK, tL) = tⁿ*f(K, L);

where if n < 1, it is decreasing returns to scale (DRS)

if n = 1, it is exhibiting constant returns to scale (CRS)

if n > 1, it is exhibiting increasing returns to scale (IRS)

a) f(K, L) = 2K + 2L

MPL = 2

MPK = 2

So, MRTS = MPL/MPK

MRTS = 2/2 = 1

f(tK, tL) = 2tK + 2tL = t(2K + 2L)

f(tK, tL) = t*f(K, L)

Thus, this function exhibits constant returns to scale.

b) f(K, L) = 2K^1/4L^1/4

MPL = (1/4)*2K^1/4L^-3/4

MPK = (1/4)*2L^1/4K^-3/4

So, MRTS = MPL/MPK

MRTS = [(1/4)*2K^1/4L^-3/4]/[(1/4)*2L^1/4K^-3/4] = K/L

f(tK, tL) = 2*(tK)^1/4(tL)^1/4

f(tK, tL) = t^1/2*2K^1/4L^1/4 = t^1/2f(K, L)

Thus, this function exhibits decreasing returns to scale.

c) f(K, L) = K^1/2L^1/2

MPL = (1/2)*K^1/2L^-1/2

MPK = (1/2)*L^1/2K^-1/2

So, MRTS = MPL/MPK

MRTS = [(1/2)*K^1/2L^-1/2]/[(1/2)*L^1/2K^-1/2] = K/L

f(tK, tL) = (tK)^1/2(tL)^1/2

f(tK, tL) = t^1/2+1/2*K^1/2L^1/2 = t*f(K, L)

Thus, this function exhibits constant returns to scale.

Following is the required graph for (a) and (b) parts:

(K) * = 2A- 14 =10 (X=10 | 2g2L 507