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Chapter 6: Problem 13

Suppose that Manny, Jack, and Moe can hire workers for \(\$ 12\) per hour, or can rent capital for \(\$ 7\) per hour. a. Write an expression for Manny, Jack, and Moe's total cost as a function of how many workers they hire and how much capital they employ. b. Assume that Manny, Jack, and Moe wish to hold their total costs to exactly \(\$ 100\). Use your answer from (a) to find the equation for an isocost line corresponding to exactly \(\$ 100\) of costs. Rearrange your equation to isolate capital. c. Graph the equation for the isocost line, putting labor on the horizontal axis and capital on the vertical axis. d. What is the vertical intercept of the line you drew? The horizontal intercept? What does each represent? e. What is the slope of the line you drew? What does it represent? f. Suppose that bargaining with the local labor union raises wages. Manny, Jack, and Moe must now pay \(\$ 14\) per hour. What happens to the isocost line corresponding to \(\$ 100\) of expenditure? Explain. Show the new isocost line on your graph.

Short Answer

Expert verified
Total cost: \( TC = 12L + 7K \). Isocost line: \( K = \frac{100 - 12L}{7} \). Intercepts are \( L = 8.33, K = 14.29 \). Slope: \(-\frac{12}{7}\). Wage rise steepens the line.

Step by step solution

01

Define Cost Function

The total cost for hiring workers and renting capital can be defined as \( TC = wL + rK \), where \( w \) is the wage rate per hour, \( L \) is the number of worker hours hired, \( r \) is the rent for capital per hour, and \( K \) is the number of capital hours rented. Here, \( w = 12 \) and \( r = 7 \). Therefore, \( TC = 12L + 7K \).
02

Derive Isocost Line Expression

To find the isocost line for a total cost of \( \$ 100 \), set \( TC = 100 \). Thus, the equation becomes \( 12L + 7K = 100 \). Rearranging this to isolate \( K \), we get \( 7K = 100 - 12L \) and then \( K = \frac{100 - 12L}{7} \).
03

Draw Isocost Line

Graph the equation \( K = \frac{100 - 12L}{7} \) with labor \( L \) on the horizontal axis and capital \( K \) on the vertical axis.
04

Determine Intercepts

To find the vertical intercept, set \( L = 0 \), giving \( K = \frac{100}{7} \approx 14.29 \). To find the horizontal intercept, set \( K = 0 \), which results in \( L = \frac{100}{12} \approx 8.33 \). The vertical intercept represents the quantity of capital when no labor is used, while the horizontal intercept represents the quantity of labor when no capital is used.
05

Find Slope of Isocost Line

Re-write the isocost equation in the form \( K = mL + c \). The slope \( m \) is \( -\frac{12}{7} \) indicating the rate of substitution between labor and capital, i.e., how much capital must be reduced for each additional unit of labor to keep cost constant.
06

Adjust Isocost Line for Wage Increase

With the increased wage of \( \$ 14 \) per hour, the cost equation becomes \( 14L + 7K = 100 \). For the new isocost line, rearrange to \( K = \frac{100 - 14L}{7} \). Graph this new line, which will be steeper, reflecting a higher wage rate for labor, suggesting less labor can be hired for the same total cost.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cost Function
A cost function is essentially an equation that helps us understand the total cost incurred in production when using various inputs.
In the given exercise, Manny, Jack, and Moe need to account for two main inputs: labor and capital. Labor refers to the human resources needed, while capital refers to the machinery or tools required.
  • The cost function formula is represented as: \[TC = wL + rK \]where:
    • \(TC\) is the total cost,
    • \(w\) the wage rate or cost per labor hour,
    • \(L\) the number of labor hours employed,
    • \(r\) the rental rate of capital per hour, and
    • \(K\) the number of capital hours rented.
In the exercise, Manny, Jack, and Moe hire workers at \( \(12 \) per hour and rent capital at \( \)7 \) per hour. Plug these values into the formula, and you'll receive their total cost as: \[TC = 12L + 7K \] which tells us the cost of their production based on labor and capital adjustments.
Wage Rate
The wage rate is the price that Manny, Jack, and Moe must pay for each hour of labor.
Initially, the wage rate is set at \( \(12 \) per hour. However, changes in the wage rate can have significant repercussions on the isocost line.
When the exercise mentions that a local labor union causes the wage rate to rise to \( \)14 \) per hour, it implies a shift in how Manny, Jack, and Moe might optimize their resources. The higher wage rate increases the cost of labor, forcing adjustments.
  • The impact of such a wage increase means that for the same cost of \( $100 \), less labor can be employed compared to before, as each labor hour now costs more. This makes labor relatively more expensive than capital.
  • To reflect this change, the isocost line representing this new cost situation becomes steeper.
  • It's a clear example of how labor costs influence decisions on resource combinations to maintain desired cost levels.
Capital
In economic terms, capital refers to the assets used to produce goods and services. This could be equipment, machinery, or technology.
In the scenario of Manny, Jack, and Moe, capital is rented at a rate of \( $7 \) per hour.
  • Having a fixed rent per hour for capital makes it easier for businesses to plan and allocate resources efficiently.
  • Within the cost function \( TC = 12L + 7K \), the rent of capital is a crucial component. It balances the equation by compensating for labor costs, effectively determining how many hours of capital are necessary to maintain a specified budget.
When labor costs increase, it might become more favorable to substitute labor with capital. This means renting more capital hours while reducing labor hours to keep costs constant. This dynamic highlights the flexibility businesses need in responding to market or bargaining pressures.
Labor Union
A labor union is an organization that represents the collective interests of workers, facilitating bargaining for better wages, working conditions, and benefits.
In the exercise example, the local labor union causes the wage rate to increase from \( \(12 \) to \( \)14 \) per hour, showcasing how unions can influence economic conditions.
  • This action by the labor union impacts the labor market, affecting Manny, Jack, and Moe's decisions.
  • The bargaining power of labor unions can shift the balance of resource allocation by making labor relatively more expensive compared to capital.
  • This change necessitates a re-evaluation of how labor and capital are utilized to maintain operational costs at their desired level of \( $100 \).
It's a vivid representation of how external forces, like labor unions, shape economic strategies and business decisions. The presence and negotiation strength of a labor union could alter business landscapes significantly by influencing direct costs linked to human resources.

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Most popular questions from this chapter

Mad Max's Road Warriors fix potholes in interstate highways. Max's road crews fill potholes using workers and shovels in 1 to 1 correspondence. A worker with 1 shovel can fill 10 potholes in a day. A worker with 2 shovels can still only fill 10 potholes, as can 2 workers with 1 shovel. a. Draw the production isoquant corresponding to filling 30 potholes. b. Assume that production displays constant returns to scale and draw a few more isoquants. c. If shovels rent for \(\$ 5\) and workers must be paid \(\$ 25\), draw several isocost lines. d. If Mad Max has received a state contract to fill 30 potholes, what is the minimum cost at which it can fulfill the contract? e. If the cost of renting a shovel suddenly rises from \(\$ 5\) to \(\$ 6,\) what will happen to the composition of inputs that Mad Max uses to fill potholes? Why?

Nobody fixes more fender benders than Crazy Bob! At his auto body shop, the production function showing the number of cars repaired each year is \(Q=100 \sqrt{K L}\), where \(K\) is the number of arc welding machines available and \(L\) is the number of employees. Currently, \(K\) is fixed at \(9 .\) a. Write an equation for Crazy Bob's short-run production function, showing output as a function of labor only. b. Calculate the total number of cars Bob can repair each year for \(L=1,2,3,4,\) and 5 . c. Calculate the marginal product of labor for each of Bob's first five workers. Does the \(M P_{L}\) diminish? d. Calculate the average product of labor for Bob's first five workers. Is the \(M P_{L}\) greater than, equal to, or less than the \(A P_{L}\) at each level of employment? Why?

Miguel and Jake run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's long-run production function is \(Q=4 K^{075} L^{025},\) where \(Q\) is the number of reams produced, \(K\) is the quantity of capital rented, and \(L\) is the quantity of labor hired. For this production function, \(M P_{L}\) \(=K^{075} / L^{075}\) and \(M P_{K}=3 L^{025} / K^{025} .\) The weekly cost function for the paper plant is \(C=10 K+2 L,\) where \(C\) is the total weekly cost. a. What ratio of capital to labor minimizes Miguel and Jake's total costs? b. How much capital and labor will Miguel and Jake need to rent and hire in order to produce 1,000 reams of paper each week? c. How much will hiring these inputs cost them?

Determine whether each of the production functions below displays constant, increasing, or decreasing returns to scale: a. \(Q=10 K^{075} L^{025}\) b. \(Q=\left(K^{075} L^{025}\right)^{2}\) c. \(Q=K^{075} L^{075}\) d. \(Q=K^{025} L^{025}\) e. \(Q=K+L+K L\) f. \(Q=2 K^{2}+3 L^{2}\) g. \(Q=K L\) h. \(Q=\min (3 K, 2 L)\)

Suppose that a firm's production function is given by \(Q=K^{033} L^{067},\) where \(M P_{K}=0.33 K^{-067} L^{067}\) and \(M P_{L}=\) \(0.67 K^{033} L^{-033}\) a. As \(L\) increases, what happens to the marginal product of labor? b. As \(K\) increases, what happens to the marginal product of labor? c. Why would the \(M P_{L}\) change as \(K\) changes? d. What happens to the marginal product of capital as \(K\) increases? As \(L\) increases?
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