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Chapter 9: Problem 2

Suppose the production function for widgets is given by $$q=k l-0.8 k^{2}-0.2 l^{2}$$ where \(q\) represents the annual quantity of widgets produced, \(k\) represents annual capital input, and \(l\) represents annual labor input. a. Suppose \(k=10 ;\) graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point? b. Again assuming that \(k=10\), graph the \(M P_{l}\) curve. At what level of labor input does \(M P_{l}=0\) ? c. Suppose capital inputs were increased to \(k=20 .\) How would your answers to parts (a) and (b) change? d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale?

Short Answer

Expert verified
And what is the average productivity of labor (APL) in that case? Is there a maximum APL? If so, what is it and how many widgets are produced at that point? b. What is the marginal productivity of labor (MPL) when k = 10? At what level of labor input does MPL equal 0? c. How would your answers change if k = 20? d. Does the widget production function show constant, increasing, or decreasing returns to scale?

Step by step solution

01

Substitute k=

Given that k=10, the production function can be simplified to: $$q=10l-0.8(10)^{2}-0.2l^{2}$$.
02

Calculate Total Productivity of Labor (TPL)

The given production function represents the total productivity of labor: $$q=10l-80-0.2l^{2}$$.
03

Calculate Average Productivity of Labor (APL)

Divide the TPL by l: $$\text{APL}=\frac{q}{l}=10-0.2l$$.
04

Calculate Maximum APL

To find the maximum APL, differentiate APL with respect to l and set the derivative equal to 0: $$\frac{d(\text{APL})}{dl}=-0.2=0$$. This yields no solution, as APL is constant with respect to labor input l.
05

Calculate q At Maximum APL

Since APL does not have a maximum value due to its linear relationship with labor, the number of widgets produced at the maximum APL cannot be determined. b. Graphing MP_l when k=10
06

Calculate Marginal Productivity of Labor (MP_l)

To determine MPL, differentiate TPL (q) with respect to l: $$\frac{dq}{dl}=10-0.4l$$.
07

Calculate MPL=0

Set MPL equal to 0 and solve for l: $$10-0.4l=0$$. Solving for l, we get: $$l=25$$. c. Analyzing the changes when k=20
08

Repeat Steps 1-5 For k=20

Following the same process as in parts a and b, we can find the new TPL, APL, Maximum APL, and MPL=0 for k=20. d. Determine returns to scale
09

Define Returns To Scale

Returns to scale refers to whether production increases, decreases, or stays the same proportionally as all input factors are increased.
10

Test For Returns To Scale

To test for returns to scale, multiply both k and l by a constant, t, and compare the resulting production function to the original: $$q(tK,tL)=tK(tL)-0.8(tK)^{2}-0.2(tL)^{2}$$ Simplify and compare to the original function: $$t^{3}(k l-0.8 k^{2}-0.2 l^{2})$$ Since the production increases as a cube of the scaling factor, the production function exhibits decreasing returns to scale.

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Key Concepts

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

Total Productivity of Labor
Total Productivity of Labor (TPL) refers to the overall output (or total quantity of goods produced) resulting from utilizing various inputs, in this case, labor. In the context of the given exercise, you can identify TPL as the function depending on labor (\(l\)) while holding capital constant, in this example at \(k=10\). The formula we derive for TPL when \(k=10\) becomes: - \(q=10l-80-0.2l^{2}\).This function captures how the total quantity of widgets produced changes as labor input varies.

Some important aspects to understand about TPL:
  • As labor input increases, total output initially increases too. However, due to the negative \(0.2l^{2}\) term, output eventually decreases, which indicates decreasing efficiency of each additional unit of labor.
  • This formula highlights that while more labor generally results in more production, the rate of increase in production diminishes over time.
  • By plotting the TPL curve, you can visualize how output changes at different labor input levels.
To craft an effective study of TPL, think of it as a way to see the big picture of labor's contribution to production.
Average Productivity of Labor
Average Productivity of Labor (APL) is a measure of output per unit of labor input. You calculate APL by dividing total output by total labor, giving an average view of how productive each unit of labor is:- \(\text{APL}=\frac{q}{l}=10-0.2l\).APL helps you assess efficiency and determine when labor is being used most effectively.

Key insights about APL:
  • APL shows how much each worker (or labor unit) contributes to production. In the formula given, APL decreases as more labor is added, indicating each worker adds less to total output.
  • By differentiating APL and setting the derivative to zero, you look for the point where APL reaches its peak, suggesting optimal labor allocation.
  • However, in this specific case, APL's derivative is a constant \(-0.2\); thus, no maximum point is found. In practical terms, this means APL decreases steadily as labor rises, without a turning point.
This concept highlights how changes in labor inputs impact average production efficiency, offering guidance on optimal labor usage for maximum efficiency.
Returns to Scale
Returns to Scale describes how a production process reacts when all input levels are proportionally increased. Put simply, it helps you understand how scaling up inputs affects output levels. For the widget production, increasing both labor and capital inputs leads us to consider the nature of returns to scale.

To determine returns to scale:
  • Multiply both capital \(k\) and labor \(l\) by a constant, \(t\), and observe the resulting changes in output.
  • The modified production function becomes \(q(tK,tL)=tK(tL)-0.8(tK)^{2}-0.2(tL)^{2}\).
  • Simplifying reveals that production increases by the cube of \(t\), or \(t^3\), indicating decreasing returns to scale.
When output increases by a proportion less than the increase in inputs (here, input scaling leads to a less than cubic output scaling), it signals decreasing returns to scale. This means gaining more output requires more proportional effort or input increase, highlighting limits to input-based growth strategies.

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Most popular questions from this chapter

A local measure of the returns to scale incorporated in a production function is given by the scale elasticity \(e_{q, t}=\partial f(t k, t l) / \partial t \cdot t / q\) evaluated at \(t=1\) a. Show that if the production function exhibits constant returns to scale, then \(e_{q, t}=1\) b. We can define the output elasticities of the inputs \(k\) and \(l\) as $$\begin{array}{l} e_{q, k}=\frac{\partial f(k, l)}{\partial k} \cdot \frac{k}{q} \\ e_{q, l}=\frac{\partial f(k, l)}{\partial l} \cdot \frac{l}{q} \end{array}$$ Show that \(e_{q, t}=e_{q, k}+e_{q, l}\) c. A function that exhibits variable scale elasticity is $$q=\left(1+k^{-1} l^{-1}\right)^{-1}$$ Show that, for this function, \(e_{q, t}>1\) for \(q<0.5\) and that \(e_{q, t}<1\) for \(q>0.5\) d. Explain your results from part (c) intuitively. Hint: Does \(q\) have an upper bound for this production function?

Sam Malone is considering renovating the bar stools at Cheers. The production function for new bar stools is given by $$q=0.1 k^{0.2} l^{0.8}$$ where \(q\) is the number of bar stools produced during the renovation week, \(k\) represents the number of hours of bar stool lathes used during the week, and \(l\) represents the number of worker hours employed during the period. Sam would like to provide 10 new bar stools, and he has allocated a budget of \(\$ 10,000\) for the project. a. Sam reasons that because bar stool lathes and skilled bar stool workers both cost the same amount ( \(\$ 50\) per hour), he might as well hire these two inputs in equal amounts. If Sam proceeds in this way, how much of each input will he hire and how much will the renovation project cost? b. Norm (who knows something about bar stools) argues that once again Sam has forgotten his microeconomics. He asserts that Sam should choose quantities of inputs so that their marginal (not average) productivities are equal. If Sam opts for this plan instead, how much of each input will he hire and how much will the renovation project cost? c. On hearing that Norm's plan will save money, Cliff argues that Sam should put the savings into more bar stools to provide seating for more of his USPS colleagues. How many more bar stools can Sam get for his budget if he follows Cliff's plan? d. Carla worries that Cliff's suggestion will just mean more work for her in delivering food to bar patrons. How might she convince Sam to stick to his original 10 -bar stool plan?

Consider a generalization of the production function in Example 9.3: $$q=\beta_{0}+\beta_{1} \sqrt{k l}+\beta_{2} k+\beta_{3} l$$ $$0 \leq \beta_{i} \leq 1, \quad i=0, \dots, 3$$ a. If this function is to exhibit constant returns to scale, what restrictions should be placed on the parameters \(\beta_{0}, \ldots, \beta_{3} ?\) b. Show that, in the constant returns-to-scale case, this function exhibits diminishing marginal productivities and that the marginal productivity functions are homogeneous of degree 0 c. Calculate \(\sigma\) in this case. Although \(\sigma\) is not in general constant, for what values of the \(\beta\) 's does \(\sigma=0,1\), or \(\infty ?\)

Suppose that the production of crayons \((q)\) is conducted at two locations and uses only labor as an input. The production function in location 1 is given by \(q_{1}=10 l_{1}^{0.5}\) and in location 2 by \(q_{2}=50 l_{2}^{0.5}\) a. If a single firm produces crayons in both locations, then it will obviously want to get as large an output as possible given the labor input it uses. How should it allocate labor between the locations to do so? Explain precisely the relationship between \(l_{1}\) and \(l_{2}\) b. Assuming that the firm operates in the efficient manner described in part (a), how does total output ( \(q\) ) depend on the total amount of labor hired \((l) ?\)

Suppose that a production function \(f\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) is homogeneous of degree \(k\). Euler's theorem shows that \(\sum_{i} x_{i} f_{i}=k f,\) and this fact can be used to show that the partial derivatives of \(f\) are homogeneous of degree \(k-1\) a. Prove that \(\sum_{i=1}^{n} \sum_{j=1}^{n} x_{i} x_{j} f_{i j}=k(k-1) f\) b. In the case of \(n=2\) and \(k=1\), what kind of restrictions does the result of part (a) impose on the second-order partial derivative \(f_{12} ?\) How do your conclusions change when \(k>1\) or \(k<1 ?\) c. How would the results of part (b) be generalized to a production function with any number of inputs? d. What are the implications of this problem for the parameters of the multivariable Cobb-Douglas production function \(f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\prod_{i=1}^{n} x_{i}^{\alpha_{i}}\) for \(\alpha_{i} \geq 0 ?\)
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