Question

Suppose there are three industrial sectors in the economy, defense production, home construction and transportation, and only two categories of workers, machinists and electricians. Assume that it takes a long time and much investment in retraining programs before one category of workers can do the work of the other category. Defense production needs 30,000 machinists and 50,000 electricians for \(\$ 1\) billion output, home construction needs 20,000 machinists and 70,000 electricians per \(\$ 1\) billion of output, and transport needs 60,000 machinists and 30,000 electricians per \(\$ 1\) billion of output. If the defense production is cut by \(\$ 40\) billion, because of a reduced involvement in war, by how much should home construction and transport be increased to insure as low as possible unemployment of either type of worker.

Short answer

To minimize unemployment among machinists and electricians when defense production is cut by \$40 billion dollars, we should increase home construction spending by approximately \$2.93 billion dollars, and transportation spending by approximately \$12.22 billion dollars.

Step-by-step solution

Formulate the problem in terms of linear equations

Let the variables be as follows: - \(x_1\): decrease in defense production spending in billions of dollars - \(x_2\): increase in home construction spending in billions of dollars - \(x_3\): increase in transportation spending in billions of dollars We can then write two linear equations based on the number of machinists and electricians needed in each sector: - Machinists: \(30,000x_1 = 20,000x_2 + 60,000x_3\) - Electricians: \(50,000x_1 = 70,000x_2 + 30,000x_3\) We are given that the \(x_1 = \$40\) billion. Plugging this into our equations, we get: - Machinists: \(30,000(40) = 20,000x_2 + 60,000x_3\) - Electricians: \(50,000(40) = 70,000x_2 + 30,000x_3\)

Simplify the linear equations

To make the equations easier to work with, simplify by dividing both sides of each equation by 10,000. We get: - Machinists: \(120 = 2x_2 + 6x_3\) - Electricians: \(200 = 7x_2 + 3x_3\)

Solve the system of linear equations

To solve the system of linear equations, we can use a method such as substitution or elimination. Here, we'll use the substitution method. Solving the first equation for \(x_2\), we get: \(x_2 = \frac{120 - 6x_3}{2}\) Now, substitute this expression for \(x_2\) into the electricians equation: \(200 = 7\left(\frac{120 - 6x_3}{2}\right) + 3x_3\) Multiply both sides by 2 to eliminate the fraction: \(400 = 7(120 - 6x_3) + 6x_3\) Expand and simplify the equation: \(400 = 840 - 42x_3 + 6x_3\) Combine like terms: \(-440 = -36x_3\) Now, divide by -36: \(x_3 = \frac{440}{36} \approx 12.22\) Now that we have \(x_3\), substitute it back into our expression for \(x_2\): \(x_2 = \frac{120 - 6(12.22)}{2}\) \(x_2 \approx 2.93\) To minimize unemployment among machinists and electricians when defense production is cut by \(40\) billion dollars, we should increase home construction spending by approximately \(2.93\) billion dollars, and transportation spending by approximately \(12.22\) billion dollars.

Key concepts

Linear Equations

Linear equations are mathematical expressions that present a relationship between two variables in a linear form. This relationship is usually characterized by a straight line when graphed on a coordinate plane. In the context of the given problem, the utilization of linear equations is crucial to determine the most efficient way to allocate the workforce among different industrial sectors.

To begin with, constructing linear equations requires an understanding of the resources needed for each sector's output. As the exercise indicates, defense production, home construction, and transportation sectors have specific demands for machinists and electricians per billion dollars of output. The linear equations formulated from this information reflect how changes in each sector's spending will impact the employment of both categories of workers.

When faced with such economic allocation issues, linear equations provide a straightforward method for exploring possible outcomes. They allow us to model our constraints and objectives into equations that can be manipulated algebraically. By representing the decrease in defense production and the potential increase in home construction and transportation spending as variables, the problem is transformed into a set of solvable linear equations.

Workforce Allocation

Workforce allocation involves distributing human resources across different divisions or sectors within an economy or organization to optimize productivity and efficiency. It's a strategic decision-making process that considers workforce skills, industry needs, demand fluctuations, and other factors. The goal is to minimize unemployment and underemployment while maximizing output and worker satisfaction.

In economic terms, workforce allocation becomes quite complex, particularly when diverse skill sets are involved and cannot be easily transferred between sectors due to constraints like retraining costs and time. This is exactly the situation outlined in our textbook exercise, where machinists and electricians cannot quickly shift roles. When allocation necessitates keeping unemployment low, careful planning is vital.

From the exercise, it's clear that each \(1\) billion dollar decrease in defense production necessitates a proportional increase in spending on home construction and transportation to absorb the displaced workforce. Strategic reallocation of resources, as guided by the solution of linear equations, can therefore help in maintaining full employment levels.

Sectoral Resource Distribution

Sectoral resource distribution is an economic strategy that ensures resources like labor and capital are appropriately disbursed across various sectors within an economy. This equilibrium aids in achieving macroeconomic goals such as growth, efficiency, and full employment.

Our exercise illustrates a practical application of resource distribution. A key element is grasping the economic repercussions of reallocating funds from one sector to another, especially when the sectors involved have different labor requirements. By reducing defense spending and redistributing it into home construction and transportation, we encounter the challenge of how to mitigate the impact on the workforce.

Effective sectoral resource distribution requires analysis of the labor market and industry capabilities. In the real world, this approach is often accompanied by government policies and economic planning to accommodate shifts in demand or supply, technological changes, or international market forces. An understanding of linear equations enhances our ability to predict the outcomes of such redistributions and to craft solutions that strive for an optimal balance.