Question

The technical rate of substitution between factors \(x_{2}\) and \(x_{1}\) is \(-4 .\) If you desire to produce the same amount of output but cut your use of \(x_{1}\) by 3 units, how many more units of \(x_{2}\) will you need?

Short answer

You need to add 12 more units of \(x_2\).

Step-by-step solution

Identify Given Information

We are given that the technical rate of substitution (TRS) between factors \(x_2\) and \(x_1\) is \(-4\). It means that substituting 1 unit of \(x_1\) requires the addition of 4 units of \(x_2\) to maintain the same output level.

Understand the Concept of TRS

The technical rate of substitution is the ratio of the marginal products of the two inputs. In this case, it indicates how much of \(x_2\) is needed to replace a unit of \(x_1\) without altering the output. Thus, TRS \(-4\) means each unit cut from \(x_1\) requires adding 4 units of \(x_2\).

Determine Change in Inputs

Given that \(x_1\) is reduced by 3 units, according to the TRS, the reduction can be offset by increasing \(x_2\) by a certain amount. Since TRS = -4, for each unit decrease in \(x_1\), increase \(x_2\) by 4 units.

Calculate Additional Units of \(x_2\)

Using the TRS \(-4\), calculate the necessary increase in \(x_2\) when \(x_1\) is decreased by 3 units. \[ \text{Increase in } x_2 = 3 \times 4 = 12 \text{ units} \] Thus, to compensate for the reduction of 3 units in \(x_1\), you need to increase \(x_2\) by 12 units.

Key concepts

Understanding Marginal Products

The concept of marginal products is crucial for understanding how inputs contribute to output in production. Marginal product refers to the additional output generated by employing an extra unit of a particular input while keeping other inputs constant. For example, if adding one more worker increases the production of widgets by 10 units, the marginal product of labor is 10 widgets per worker.

In the context of the technical rate of substitution (TRS), marginal products play a vital role because TRS is the ratio of the marginal products of two inputs. If the marginal product of labor (input 1) is higher than the marginal product of raw materials (input 2), it indicates that workers contribute more to output than raw materials. Therefore, understanding marginal products helps firms optimize their resource allocation efficiently.

The Concept of Factor Substitution

Factor substitution occurs when a producer replaces one input with another while maintaining the same level of output. This is often driven by a change in relative input costs or availability. In our example, when the technical rate of substitution is \(-4\), it suggests that for every unit of input 1 reduced, four units of input 2 are needed to keep output constant.

This concept is crucial for businesses that might need to adjust their input mix depending on the prices or scarcity of inputs. For instance, if input 1 becomes expensive or scarce, a firm may opt to use more of input 2 to maintain its production levels. Understanding factor substitution allows businesses to adapt to changing economic environments efficiently by switching between inputs as necessary.

Exploring Input-Output Analysis

Input-output analysis is a method used to understand the relationship between different inputs and the resulting output in a production process. It helps in diagramming and analyzing how different sectors or inputs are interconnected within an economy or a production facility.

• Input-output models map out the flow of inputs (like labor and materials) into the production process and how they are transformed into outputs (finished goods or services).
• This analysis can identify bottlenecks or inefficiencies where adjustments in input use could enhance productivity.
• It is also used to anticipate the effects of changes in one part of the economy on others, serving as a vital tool for strategic planning.

Understanding input-output analysis allows firms and policymakers to predict how changes in input allocations affect overall production and to make informed decisions about resource management.