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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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Most popular questions from this chapter

Power Goat Lawn Company uses two sizes of mowers to cut lawns. The smaller mowers have a 22 -inch deck. The larger ones combine two of the 22 -inch decks in a single mower. For each size of mower, Power Goat has a different production function, given by the rows of the following table. $$\begin{array}{lccc} \hline & \begin{array}{c} \text { Output per Hour } \\ \text { (square feet) } \end{array} & \begin{array}{c} \text { Capital Input } \\ \text { (# of 22" mowers) } \end{array} & \text { Labor Input } \\ \hline \text { Small mowers } & 5000 & 1 & 1 \\ \text { Large mowers } & 8000 & 2 & 1 \\ \hline \end{array}$$ a. Graph the \(q=40,000\) square feet isoquant for the first production function. How much \(k\) and \(l\) would be used if these factors were combined without waste? b. Answer part (a) for the second function. c. How much \(k\) and \(l\) would be used without waste if half of the 40,000 -square-foot lawn were cut by the method of the first production function and half by the method of the second? How much \(k\) and \(l\) would be used if one fourth of the lawn were cut by the first method and three fourths by the second? What does it mean to speak of fractions of \(k\) and \(l\) ? d. Based on your observations in part (c), draw a \(q=40,000\) isoquant for the combined production functions.

As we have seen in many places, the general Cobb-Douglas production function for two inputs is given by $$q=f(k, l)=A k^{\alpha} l^{\beta}$$ where \(0<\alpha<1\) and \(0<\beta<1 .\) For this production function: a. Show that \(f_{k}>0, f_{1}>0, f_{k k}<0, f_{l l}<0,\) and \(f_{k l}=f_{l k}>0\) b. Show that \(e_{q, k}=\alpha\) and \(e_{q, l}=\beta\) c. In footnote \(5,\) we defined the scale elasticity as$$e_{q, t}=\frac{\partial f(t k, t l)}{\partial t} \cdot \frac{t}{f(t k, t l)}$$ where the expression is to be evaluated at \(t=1 .\) Show that, for this Cobb-Douglas function, \(e_{q, t}=\alpha+\beta .\) Hence in this case the scale elasticity and the returns to scale of the production function agree (for more on this concept see Problem 9.9 ). d. Show that this function is quasi-concave. e. Show that the function is concave for \(\alpha+\beta \leq 1\) but not concave for \(\alpha+\beta>1\)

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 \(\alpha_{i} \geq 0 ?\)

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 \(\infty ?\)
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