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For Activities 1 through \(4,\) rewrite the statements into equivalent statements without using the term marginal. When labor is at 500 worker-hours, marginal product is -10 units per worker- hour.

Short Answer

Expert verified
An increase beyond 500 worker-hours decreases output by 10 units per worker-hour.

Step by step solution

01

Understand the Term 'Marginal Product'

The term 'marginal product' refers to the additional output produced by employing one more unit of input. In this context, it signifies the change in output resulting from an additional worker-hour.
02

Rephrase the Statement

To remove the term 'marginal', rephrase the statement in terms of change. The current statement says the marginal product at 500 worker-hours is -10.
03

Translate into Change in Output

Express the change without using 'marginal': 'When labor increases from 500 worker-hours, the output decreases by 10 units for every additional worker-hour.'

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Key Concepts

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

Marginal Product
The concept of 'marginal product' might initially sound complex, but it's manageable when broken down simply. Think about it as the result of adding one more unit of input—in this case, an hour of labor. If you bring in an extra worker-hour to your team, what’s the extra output you can expect? This effect is captured by the marginal product.

In mathematical terms, if you have a production function expressed as output (Q) depending on labor (L), the marginal product of labor (MPL) is the derivative of Q with respect to L. Thus, \[MPL = \frac{\partial Q}{\partial L}\]

A positive MPL implies more output with extra labor, while a negative MPL, as seen in the exercise, indicates a decrease. Understanding this helps businesses assess how efficiently they use labor and decide when it's wise to add more.
Worker-hours
Worker-hours are essentially the aggregate time contributed by the workforce in your production process. Imagine it like a giant clock tracking every work hour paid for, measuring productivity potential.

Whether it’s hiring more workers or increasing the working hours overtime, these decisions accumulate into total worker-hours, an essential input for calculating productivity and costs.
  • If you have ten workers each working eight hours a day, you total 80 worker-hours for that day.
  • Worker-hours give businesses clear data to understand labor input and plan resourcing needs.
Understanding worker-hours helps firms assess labor productivity and wage costs, impacting decision-making on employee management and overtime policies.
Change in Output
Change in output is the variation in production levels when factors like worker-hours shift. This concept reveals how output reacts to changes in labor input, critical to optimizing production strategies.

As per the solution, when additional worker-hours are employed, we express the change using output metrics rather than the abstract term 'marginal product.' Here’s how you express it:
  • "When labor increases from 500 worker-hours, output decreases by 10 units for every additional worker-hour."
  • This describes a relationship between labor input and production outcome, devoid of jargon.
Recognizing changes in output prepares businesses to adapt swiftly, ensuring that the input adjustments translate into desired productivity and efficiency outcomes.

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Most popular questions from this chapter

For Activities 5 through \(10,\) given the units of measure for production and the units of measure for cost or revenue a. Write the units of measure for the indicated marginal. b. Write a sentence interpreting the marginal as an increase. Revenue is given by \(R(q)\) million dollars when \(q\) billion units are sold; \(R^{\prime}(4)=2\).

A Cobb-Douglas function for the production of mattresses is $$ M=48.1 L^{0.6} K^{0.4} \text { mattresses } $$ where \(L\) is measured in thousands of worker hours and \(K\) is the capital investment in thousands of dollars. a. Write an equation showing labor as a function of capital. b. Write the related-rates equation for the equation in part \(a,\) using time as the independent variable and assuming that mattress production remains constant. c. If there are currently 8000 worker hours, and if the capital investment is \(\$ 47,000\) and is increasing by \(\$ 500\) per year, how quickly must the number of worker hours be changing for mattress production to remain constant?

Write the first and second derivatives of the function and use the second derivative to determine inputs at which inflection points might exist. \(j(x)=5 e^{-x}+\ln x\) with \(x>0\)

Write the indicated related-rates equation. \(v=\frac{1}{3} \pi r^{2} h ;\) relate \(\frac{d h}{d t}\) and \(\frac{d r}{d t},\) assuming that \(v\) is constant.

Garden Fencing \(A\) homeowner is building a fence around a rectangular garden. Three sides of the fence will be constructed with wire fencing. The fourth side is to be constructed with wooden privacy fencing. The wire fencing costs \(\$ 2\) per linear foot. The privacy fencing costs \(\$ 6\) per linear foot. The homeowner must stay within a \(\$ 320\) budget. a. Write a model for the area of the garden. b. Calculate the dimensions of the garden that will result in maximum area. What is the maximum possible area? c. How much of each type of fencing does the homeowner need to purchase?

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