Question

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

The marginal product of capital is 200 chips per hour.

Step-by-step solution

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.

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.

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.

Key concepts

Marginal Product of Labor

The marginal product of labor (MPL) refers to the additional output a company can produce by using one more unit of labor, typically measured in hours. It is an essential concept in economics because it helps firms determine how changes in labor input impact output levels. In simple terms, MPL shows how much more output a firm gets when it employs an extra worker, assuming all other factors remain constant.
For example, in the context of producing computer chips, if adding one hour of labor results in 50 extra chips, then the MPL is 50 chips per hour. This information is crucial for business decisions concerning the allocation of resources between labor and capital.
Understanding MPL helps firms optimize their production processes and ensure they operate efficiently. It is used to analyze labor productivity under various conditions, helping businesses to maximize profits.

Marginal Rate of Technical Substitution

The marginal rate of technical substitution (MRTS) is a crucial economic concept that explains how a firm can interchange labor and capital while maintaining the same level of output. This metric shows the trade-off between two inputs: labor and capital.
MRTS is defined as the ratio of the marginal product of labor (MPL) to the marginal product of capital (MPK), indicating how many units of capital can be reduced if one additional unit of labor is added, without affecting the output. For example, an MRTS of 1/4 suggests that one hour of capital can be replaced by four hours of labor with no change in production levels.
Understanding MRTS helps businesses determine the optimal combination of labor and capital, achieving cost efficiency while sustaining productivity. It guides decision-making on substituting labor for capital or vice versa, helping maintain output during variations in input availability.

Production Function

A production function illustrates the relationship between inputs used in production and the output produced. It is a fundamental concept in production theory, providing a mathematical description of the technological capabilities of a firm.
This function typically involves inputs like labor and capital and indicates how changes in these inputs affect output levels. In its simplest form, a production function can be expressed as an equation that shows output as a function of labor and capital: \[ Q = f(L, K) \] where \(Q\) is the quantity of output, \(L\) is the labor input, and \(K\) is the capital input.
Incorporating concepts like the marginal product of labor and capital, the production function becomes a powerful tool for understanding how different factors contribute to the production process. It helps businesses optimize resource allocation, plan future production, and effectively manage input constraints while maintaining desired output levels.