Question

The production function of good x is as follow: Q=5L⁰.²K⁰.⁸

A. Does this production function have increasing, constant or decreasing returns to scale?

B. Calculate the slope of the isoquant when the entrepreneur is producing efficiently with 75 (Explain your answer please.) laborers and 125 units of capital. (I.e.
C. Is diminishing return a characteristic of this production function? (Check the second derivative of each marginal product.)
D. What is the right or optimum bundle of inputs to produce 500 units of output using the slope relationship determined in b? (If it is, just write "no need to make a change.")
E. Derive the total cost function as a function of output. (Q)

Answer 1

The production function given has constant returns to scale due to the sum of the exponents of labor and capital being equal to 1. Diminishing returns can be checked by evaluating the second derivatives of the marginal products. The optimum bundle of inputs and total cost function require further computation based on the production function and input costs.

Step-by-step explanation:

A production function like Q=5L0.2K0.8A reflects the relationship between input factors - labor (L), capital (K), and technology (A) - and the quantity of output (Q) produced. To analyze returns to scale, we check the sum of the exponents of L and K. If the sum equals 1, it is constant returns to scale; if greater than 1, increasing returns to scale; and if less than 1, decreasing returns to scale. Here, 0.2 + 0.8 = 1, indicating constant returns to scale.

The slope of the isoquant, when producing efficiently, represents the rate at which labor can substitute for capital (the marginal rate of technical substitution). This involves deriving the marginal products, setting their ratio equal to the wage-rental ratio, and solving for the slope at the given quantities of labor and capital. To find it, we need to differentiate the production function with respect to L and K, find the marginal products, and then evaluate them at L=75 and K=125.

Regarding diminishing returns, we must check the second derivatives of the marginal products of labor and capital. If they are negative, it implies that the production function exhibits diminishing returns. However, we would need the actual calculations to confirm if diminishing returns are present within this function.

Finally, to determine the optimum bundle of inputs for a production level of 500 units using the slope relationship determined earlier, we would plug the value of Q into the production function along with the calculated slope and solve for L and K.

The total cost function for a production function of this kind requires knowing the cost of each input and the technology level. Assuming constant input prices and a given technology level, we express the total cost as a function of the quantity of output by substituting the optimal quantities of L and K back into their respective cost equations and then combining them.