Answer to Question #276713 in Microeconomics for cyah

Question

Determine whether the following production functions exhibit constant, increasing, or decreasing returns to scale. Provide relevant workings to support your answers.

i. Q = 0.25X + 5Y + 30Z

ii. Q = 4L²Â + 6LK + 3K²

iii. Q = 2L^0.2K^0.6

Describe the condition of finding the optimum level of single input and multiple inputs. Illustrate and discuss the concept of finding the optimum level of multiple inputs. 

Answer 1

if all inputs increase by b

i)

Q = 0.25X + 5Y + 30Z

0.25bX + 5bY + 30bZ = b(0.25X + 5Y + 30Z) = bQ

Therefore, Q rises by b, hence it is constant and return to scale.

ii)

Q = 4L²Â + 6LK + 3K²

4(bL)² 6b² LK + 3(bK)^{2 = b2}( 4L² +6LK + 3K²) = Qb²

Therefore, Q rises by more than b, hence it is increasing returns to scale.

ii)

Q = 2L^0.2K^0.6

⁼ 2 (bL)^0.2 (bK)^0.6 = 2b^0.8 L^0.2 K^{0.6 =} b^0.8Q

Therefore, Q rises by less than b, hence it is decreasing returns to scale.

single input

Employ the input by the amount such that input's marginal revenue product is equal to the price of input with regard to the labor input L which is paid at wage rate W.

Optimal quantity L is where the Marginal product of L = (W/P0 = real wages'

When more than one input is involved, such as labor and capital, optimal amounts of capital and labor are employed according to (Marginal product of capital/price of capital) = (marginal product of labor/ price of labor