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Chapter 6: Problem 7

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?

Short Answer

Expert verified
The marginal product of capital is 200 chips per hour.

Step by step solution

01

Understand the concept of Marginal Rate of Technical Substitution

The Marginal Rate of Technical Substitution (MRTS) is given as the ratio of the marginal product of labor to the marginal product of capital. It tells us about the trade-offs between labor and capital, which in this case is \(1 / 4\). This means that the output will remain constant even if 1 hour of capital is replaced by 4 hours of labor in the production process.
02

Using the relation between MRTS, MPL and MPK

We know that the formula that relates MRTS, MPL and MPK is MRTS = MPL / MPK. We can rearrange this formula to solve for the unknown, which is the marginal product of capital: MPK = MPL / MRTS.
03

Calculate the marginal product of capital

Substitute the given values MPL = 50 and MRTS = \(1 / 4\) into the formula to find MPK. Hence MPK = 50 / \(1 / 4\) = 200 chips per hour. Therefore the marginal product of capital in the production of computer chips is 200 chips per hour.

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Most popular questions from this chapter

Suppose life expectancy in years \((L)\) is a function of two inputs, health expenditures \((H)\) and nutrition expenditures \((N)\) in hundreds of dollars per year. The production function is \(L=\mathrm{c} H^{0.8} N^{0.2}\) a. Beginning with a health input of \(\$ 400\) per year \((H=4)\) and a nutrition input of \(\$ 4900\) per year \((N=49),\) show that the marginal product of health expenditures and the marginal product of nutrition expenditures are both decreasing. b. Does this production function exhibit increasing, decreasing, or constant returns to scale? c. Suppose that in a country suffering from famine, \(N\) is fixed at 2 and that \(c=20 .\) Plot the production function for life expectancy as a function of health expenditures, with \(L\) on the vertical axis and \(H\) on the horizontal axis. d. Now suppose another nation provides food aid to the country suffering from famine so that \(N\) increases to \(4 .\) Plot the new production function. e. Now suppose that \(N=4\) and \(H=2 .\) You run a charity that can provide either food aid or health aid to this country. Which would provide a greater benefit: increasing \(H\) by 1 or \(N\) by \(1 ?\)

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?

For each of the following examples, draw a representative isoquant. What can you say about the marginal rate of technical substitution in each case? a. A firm can hire only full-time employees to produce its output, or it can hire some combination of fulltime and part-time employees. For each full-time worker let go, the firm must hire an increasing number of temporary employees to maintain the same level of output. b. \(A\) firm finds that it can always trade two units of labor for one unit of capital and still keep output constant. c. \(A\) firm requires exactly two full-time workers to operate each piece of machinery in the factory.

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?

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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