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Chapter 9: Problem 4

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 \((l) ?\)

Short Answer

Expert verified
Answer: The firm should allocate labor in such a way that for every unit of labor employed at location 2, they should employ 5 units of labor at location 1. The total output as a function of the total labor hired is given by \(q(l) = 10 (5 \frac{l}{6})^{0.5} + 50 (\frac{l}{6})^{0.5}\).

Step by step solution

01

Calculate the Marginal Product of Labor at Each Location

First, we'll calculate the marginal product of labor (MPL) for both locations. The MPL is the additional amount of output produced when an extra unit of labor is employed. To find the MPL for each location, we'll differentiate the production functions with respect to the labor input: For location 1: MPL\(_{1} = \frac{d(q_{1})}{d(l_{1})} = \frac{d(10 l_{1}^{0.5})}{d(l_{1})} = 5 l_{1}^{-0.5}\) For location 2: MPL\(_{2} = \frac{d(q_{2})}{d(l_{2})} = \frac{d(50 l_{2}^{0.5})}{d(l_{2})} = 25 l_{2}^{-0.5}\)
02

Equate Marginal Products to Find the Relationship Between \(l_{1}\) and \(l_{2}\)

To maximize output, the firm should allocate labor between the two locations in such a way that the marginal products are equal. This ensures that the last unit of labor employed in each location produces the same amount of output. Equating the marginal products: \(5 l_{1}^{-0.5} = 25 l_{2}^{-0.5}\) To express the relationship between \(l_{1}\) and \(l_{2}\), we will isolate one of the variables: \(l_{1} = 5 l_{2}\) This relationship shows that the firm should allocate 5 times more labor to location 1 than location 2 to maximize output.
03

Express Total Output as a Function of Total Labor Hired

Now, we need to express the total output (q) as a function of the total amount of labor hired (l). To do this, first, combine the production function of both locations: \(q = q_{1} + q_{2} = 10 l_{1}^{0.5} + 50 l_{2}^{0.5}\) Next, we can use the relationship between \(l_{1}\) and \(l_{2}\) from step 2: \(l_{1} = 5 l_{2}\) Now, express \(l_{1}\) as a function of \(l_{2}\): \(l_{1}^{0.5} = (5 l_{2})^{0.5}\) Substitute this back into the total output function: \(q = 10 (5 l_{2})^{0.5} + 50 l_{2}^{0.5}\) Finally, notice that total labor hired is given by \(l = l_{1} + l_{2}\). To express total output (q) as a function of total labor hired (l), use the relationship we found earlier between \(l_{1}\) and \(l_{2}\) and the fact that \(l_{1} = 5 l_{2}\): \(l = 5 l_{2} + l_{2}\). Therefore, \(l_{2} = \frac{l}{6}\). Substitute this back into the total output function: \(q = 10 (5 \frac{l}{6})^{0.5} + 50 (\frac{l}{6})^{0.5}\) Now we have the total output as a function of the total amount of labor hired: \(q(l) = 10 (5 \frac{l}{6})^{0.5} + 50 (\frac{l}{6})^{0.5}\)

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Key Concepts

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

Production Function
The production function is a mathematical representation of the relationship between the inputs used in production and the resulting output. In our crayon manufacturing example, we have two distinct production functions for two locations, expressed as location 1: \(q_{1} = 10 l_{1}^{0.5}\) and location 2: \(q_{2} = 50 l_{2}^{0.5}\). These functions indicate that the output of crayons at each location is dependent on the amount of labor allocated, with labor being the only input in this case.

To comprehend the production function further, let's break down the elements:\(q_{1}\) and \(q_{2}\) represent the quantity of crayons produced in location 1 and location 2, respectively. The input, labor, is shown by \(l_{1}\) and \(l_{2}\), and is raised to the power of 0.5, signifying a square root relationship between labor input and output. This power indicates diminishing marginal returns, a common concept in economics where each additional unit of input produces less additional output. The production function's coefficients (10 for location 1 and 50 for location 2) help to quantify the efficiency of each location in transforming labor into crayons.
Labor Allocation Optimization
Optimizing labor allocation is crucial for firms aiming to maximize their output. In our scenario, the company needs to distribute labor between two locations to optimize production of crayons. The solution entails equating the Marginal Product of Labor (MPL) for each location, which represents the incremental output achieved by employing an additional unit of labor. Mathematically, it is the derivative of the production function regarding labor.

The step by step solution provided uses calculus to find the MPL for both locations, leading to the relationship \(5 l_{1}^{-0.5} = 25 l_{2}^{-0.5}\). By isolating the variables, we can reveal that \(l_{1} = 5 l_{2}\), suggesting that for every unit of labor assigned to location 2, five units should be assigned to location 1 to achieve optimal output. This strategic resource allocation is vital for businesses that want to utilize their labor force effectively and is an exemplary application of marginal analysis in the decision-making process.
Total Output as a Function of Labor
Understanding how total output varies with the amount of labor hired is fundamental in managerial decision-making. With our firm's production situation, total output \(q\) is the sum of outputs from both locations. To derive a single equation for total output as a function of total labor hired \(l\), we combined the individual production functions and utilized the established relationship between \(l_{1}\) and \(l_{2}\).

After some algebraic manipulation, as shown in the solution, we arrived at the total output function \(q(l) = 10 (5 \frac{l}{6})^{0.5} + 50 (\frac{l}{6})^{0.5}\). This function allows the firm to determine the expected total output of crayons for any given level of total labor hired. It consolidates the production capabilities of both locations into a single perspective, empowering the firm to forecast production and make informed labor hiring decisions according to their operational requirements and market demands.

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Most popular questions from this chapter

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 \(\infty ?\)

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?

Show that Euler's theorem implies that, for a constant returns-to-scale production function \([q=f(k, l)]\) $$q=f_{k} \cdot k+f_{l} \cdot l$$ Use this result to show that, for such a production function, if \(M P_{l}>A P_{l}\) then \(M P_{k}\) must be negative. What does this imply about where production must take place? Can a firm ever produce at a point where \(A P_{l}\) is increasing?

Suppose we are given the constant returns-to-scale CES production function \\[ q=\left(k^{\rho}+l^{\rho}\right)^{1 / \rho} \\] a. Show that \(M P_{k}=(q / k)^{1-\rho}\) and \(M P_{l}=(q / l)^{1-\rho}\) b. Show that \(R T S=(k / l)^{1-\rho} ;\) use this to show that \\[ \sigma=1 /(1-\rho) \\] c. Determine the output elasticities for \(k\) and \(l ;\) and show that their sum equals 1 d. Prove that \\[ \frac{q}{l}=\left(\frac{\partial q}{\partial l}\right)^{\sigma} \\] and hence that \\[ \ln \left(\frac{q}{l}\right)=\sigma \ln \left(\frac{\partial q}{\partial l}\right) \\] Note: The latter equality is useful in empirical work because we may approximate \(\partial q / \partial l\) by the competitively determined wage rate. Hence \(\sigma\) can be estimated from a regression of \(\ln (q / I)\) on \(\ln w\).

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 \(\alpha+\beta>1\)
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