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Chapter 9: Problem 3

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?

Short Answer

Expert verified
Answer: Sam can produce an additional 10 bar stools if he follows Cliff's plan.

Step by step solution

01

Equal input amounts and cost for Sam's initial plan

To find out how many hours of each input Sam needs to hire and how much will the renovation project cost, we first need to establish the constraint of producing 10 bar stools, i.e., $$10 = 0.1k^{0.2}l^{0.8}$$. Since Sam intends to hire equal amounts of both inputs, we have $$k = l$$. Now we can plug this into the equation: $$10 = 0.1k^{0.2}k^{0.8}$$. Solving for k, we get $$k = 55.36$$ hours. Therefore, Sam needs to hire 55.36 hours of both inputs, i.e., $$k=l=55.36$$. Since each input costs $$\$ 50$$ per hour, the total cost of the renovation project for Sam's initial plan is $$55.36 \times 50 \times 2 = \$5,536$$.
02

Optimal input levels when marginal productivities are equal

To find the optimal input levels, first, we need to find the marginal productivity of both inputs. The marginal productivity of labor (MP_l) and the marginal productivity of lathes (MP_k) can be obtained by taking the partial derivative with respect to l and k, respectively. $$MP_l = \frac{\partial q}{\partial l} = 0.8 \times 0.1 k^{0.2} l^{-0.2}$$ and $$MP_k = \frac{\partial q}{\partial k} = 0.2 \times 0.1 k^{-0.8} l^{0.8}$$. To set these marginal productivities equal, we need to find $$k$$ and $$l$$ such that $$MP_l = MP_k$$, i.e., $$0.8 \times 0.1 k^{0.2} l^{-0.2} = 0.2 \times 0.1 k^{-0.8} l^{0.8}$$. Upon simplifying and solving, we find that $$k = 4l$$. Now, we need to find the optimal values of $$k$$ and $$l$$ under the budget constraint. Since both inputs cost $$\$50$$ per hour, the total cost is $$50(k+l) = \$10,000$$, and we substitute the earlier found relationship $$k=4l$$ into this equation: $$5l = \$1,000$$. Thus, the optimal values are $$l = 200$$ hours and $$k = 800$$ hours. The total cost for the optimal input combination is $$50(200 + 800) = \$10,000$$, or equal to the allocated budget.
03

More bar stools for the same budget

To find out how many more bar stools Sam can produce if he follows Cliff's plan, we plug the optimal input amounts back into the production function: $$q = 0.1(800)^{0.2}(200)^{0.8} = 20$$ bar stools. Therefore, Sam can produce additional 10 bar stools (20 - initial 10) if he follows Cliff's plan.
04

Potential reasons for sticking to the original 10-bar stool plan

Carla may use the following arguments to convince Sam to stick to his original plan of producing only 10 bar stools: 1. Additional bar stools might lead to overcrowding and a noisy environment, making the bar less attractive to some patrons. 2. Extra seating capacity may not lead to a proportionate increase in revenues, especially if the bar already has enough seating to accommodate most patrons during peak times. 3. As Carla's concern suggests, more bar stools could mean more work for the employees without a proportional increase in their compensation, leading to potential dissatisfaction among the staff.

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Most popular questions from this chapter

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

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 \(\infty ?\)

Although much of our discussion of measuring the elasticity of substitution for various production functions has assumed constant returns to scale, often that assumption is not necessary. This problem illustrates some of these cases. a. In footnote 6 we pointed out that, in the constant returns-to-scale case, the elasticity of substitution for a two-input production function is given by $$\sigma=\frac{f_{k} f_{l}}{f \cdot f_{k l}}$$ Suppose now that we define the homothetic production function \(F\) as $$F(k, l)=[f(k, l)]^{\gamma}$$ where \(f(k, l)\) is a constant returns-to-scale production function and \(\gamma\) is a positive exponent. Show that the elasticity of substitution for this production function is the same as the elasticity of substitution for the function \(f\) b. Show how this result can be applied to both the Cobb-Douglas and CES production functions.

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 \((l) ?\)

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.
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