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Chapter 6: Problem 10

In Example \(6.4,\) wheat is produced according to the production function \\[ q=100\left(K^{0.8} L^{0.2}\right) \\] a. Beginning with a capital input of 4 and a labor input of \(49,\) show that the marginal product of labor and the marginal product of capital are both decreasing. b. Does this production function exhibit increasing, decreasing, or constant returns to scale?

Short Answer

Expert verified
a. Given the function and computation, it's found that both MPL and MPK are decreasing for increasing inputs (L and K). b. The sum of exponents of inputs is 1 which means that the production function exhibits constant returns to scale.

Step by step solution

01

Compute Marginal Products

Margin products are computed by taking partial derivatives of the production function. The marginal product of labor (MPL) is the derivative of \(q\) with respect to \(L\), and the marginal product of capital (MPK) is the derivative with respect to \(K\). Therefore: \(MPL = \frac{dq}{dL} = 20(K^{0.8}L^{-0.8})\) and \(MPK = \frac{dq}{dK} = 80(K^{-0.2}L^{0.2})\)
02

Evaluate the Marginal Products

Evaluate MPL and MPK at the values provided (K=4, L=49): \(MPL = 20(4^{0.8}49^{-0.8})\) and \(MPK = 80(4^{-0.2}49^{0.2})\). To show these are decreasing, evaluate the marginal products at slightly higher values of inputs, for instance K=5, L=50 and establish that the values reduce as the inputs increase.
03

Determine Returns to Scale

Returns to scale is determined by summing up the exponent values of the inputs in the production function, here they are 0.8 and 0.2. If the sum is equal to 1, it exhibits constant returns to scale. If it is more than 1, increasing returns to scale and if it's less than 1, it indicates decreasing returns to scale.

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Most popular questions from this chapter

Do the following functions exhibit increasing, constant, or decreasing returns to scale? What happens to the marginal product of each individual factor as that factor is increased and the other factor held constant? a. \(q=3 L+2 K\) b. \(q=(2 L+2 K)^{1 / 2}\) c. \(q=3 L K^{2}\) \(\mathbf{d} . q=L^{1 / 2} K^{1 / 2}\) e. \(q=4 L^{1 / 2}+4 K\)

The production function for the personal computers of DISK, Inc., is given by \\[ q=10 K^{0.5} L^{0.5} \\] where \(q\) is the number of computers produced per day, \(K\) is hours of machine time, and \(L\) is hours of labor input. DISK's competitor, FLOPPY, Inc., is using the production function \\[ q=10 K^{0.6} L^{0.4} \\] a. If both companies use the same amounts of capital and labor, which will generate more output? b. Assume that capital is limited to 9 machine hours, but labor is unlimited in supply. In which company is the marginal product of labor greater? Explain.

The menu at Joe's coffee shop consists of a variety of coffee drinks, pastries, and sandwiches. The marginal product of an additional worker can be defined as the number of customers that can be served by that worker in a given time period. Joe has been employing one worker, but is considering hiring a second and a third. Explain why the marginal product of the second and third workers might be higher than the first. Why might you expect the marginal product of additional workers to diminish eventually?

Suppose a chair manufacturer is producing in the short run (with its existing plant and equipment). The manufacturer has observed the following levels of production corresponding to different numbers of workers: $$\begin{array}{|cc|} \hline \text { NUMBER OF WORKERS } & \text { NUMBER OF CHAIRS } \\ \hline 1 & 10 \\ \hline 2 & 18 \\ \hline 3 & 24 \\ \hline 4 & 28 \\ \hline 5 & 30 \\ \hline 6 & 28 \\ \hline 7 & 25 \\ \hline \end{array}$$ a. Calculate the marginal and average product of labor for this production function. b. Does this production function exhibit diminishing returns to labor? Explain. c. Explain intuitively what might cause the marginal product of labor to become negative.

A political campaign manager must decide whether to emphasize television advertisements or letters to potential voters in a reelection campaign. Describe the production function for campaign votes. How might information about this function (such as the shape of the isoquants) help the campaign manager to plan strategy?
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