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Chapter 6: Problem 9

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.

Short Answer

Expert verified
To find out which company will generate more output, the same amounts of capital and labour need to be substituted into each company's production function. The company with the higher result produces more. MPL can be calculated by taking the derivative of the production function with respect to labour. The company with the higher MPL, given a certain amount of capital, has a greater additional output for each additional unit of labour.

Step by step solution

01

Compare Outputs

Given that both companies use the same amounts of capital and labour, simply substitute the same values of \(K\) and \(L\) in both equations and compare the outputs. The company with the higher output would generate more.
02

Derive Marginal Product of Labor

The marginal product of labour (MPL) measures how much additional output is produced when one more unit of labour is utilized, holding all other inputs constant. It can be found by taking the derivative of the production function with respect to labour (L). Perform this operation for both the companies.
03

Compare Marginal Products

Given that capital is limited to 9 machine hours, substitute this value in the derived expressions from step 2. This will give the MPL when \(K\) equals 9. Compare these values. The company with the higher value has a greater MPL.

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Most popular questions from this chapter

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?

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.

The marginal product of labor in the production of computer chips is 50 chips per hour. The marginal rate of technical substitution of hours of labor for hours of machine capital is \(1 / 4 .\) What is the marginal product of capital?

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?

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?
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