4 For each of the following production functions determine t

4. For each of the following production functions, determine the returns to scale. Check to see whether diminishing returns for labor and capital exist for each function (quantity, L K capital) labor, a)

Solution

Powers of L and K should be considered for finding returns to scale.

Qa)

Q = 3L + 2K

The power of L is 1 and the power of K is 1 too. Since there is “+” sign in between, these two powers should be multiplied; (1 × 1 =) 1; as it is equal to 1, the production function has constant returns to scale.

Qb)

Q = (2L + 2K) ^ (1/2)

Q = 2L^0.5 + 2K^0.5

The power of L is 0.5 and the power of K is 0.5 too. Since there is “+” sign in between, these two powers should be multiplied; (0.5 × 0.5 =) 0.25; as it is smaller than 1, the production function has decreasing returns to scale.

Qc)

Q = 3(L × K^2)

The power of L is 1 and the power of K is 2. Since there is “×” sign in between, these two powers should be added; (1 + 2 =) 3; as it is higher than 1, the production function has increasing returns to scale.

Qd)

Q = L^(1/2) × K^(1/2)

The power of L is 0.5 and the power of K is 0.5 too. Since there is “×” sign in between, these two powers should be added; (0.5 + 0.5 =) 1; as it is equal to 1, the production function has constant returns to scale.

Qe)

Q = 4L^(1/2) + 4K

The power of L is 0.5 and the power of K is 1. Since there is “+” sign in between, these two powers should be multiplied; (0.5 × 1 =) 0.5; as it is smaller than 1, the production function has decreasing returns to scale.