Question

5. (i) Do the following functions exhibit increasing, constant, or decreasing returns to scale? (ii) Do the following functions exhibit diminishing returns to labor? Capital? Show how you know. a. q = 2L^1.25 + 2K^1.25 b. q = (L + K)^0.7 c. q = 3LK^2 d. q = L^1/2 K^1/2

Answer 1

a. q=2L^1.25 + 2K^1.25

If L=1 &K=1, q=4

If L=2 &K=2, q= 2*(2)^1.25+ *2*(2)^1.25= 9.5

Doubling the input more than doubles the output.

Hence there is increasing returns to scale.

Differentiate the function with respect to K, then L.

MPk=2*1.25*K^0.25

As K rises, MPk rises. So increasing returns to capital.

MPL=2*1.25L^0.25

As L rises, MPL rises. So increasing returns to labor.

b. q=(L+K)^0.7

If L=1 &K=1, q= 1.62

If L=2 & K=2, q= (4)^0.7= 2.63

Decreasing returns to scale. q less than doubles

MPk= 0.7(L+K)^-0.3

Diminishing returns to capital

MPL=0.7(L+K)^-0.3

Diminishing returns to labor

c. q=3LK^2

When K=1 &L=1, q= 3

When K=2&L=2, q= 24

Increasing returns to scale. q more than doubles.

MPk= 6LK

Increasing returns to capital. When K rises, MPk rises.

MPL= 3K^2

Constant returns to labor. When L rises MPL remains constant.

d. q= L^1/2 K^1/2

When K=1 &L=1, q= 1

When K=2 &L=2, q= 2

Constant returns to scale. When input doubles output doubles.

MPk= 1/2L^1/2 K^-1/2

Decreasing returns to capital. When K rises MPk falls.

MPL=1/2L^-1/2K^1/2

Decreasing returns to labor. When L rises MPL falls.

If it helps kindly upvote

for doubts comment below