Question

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 in order 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

Allocate labor such that \(l_1 = 25l_2\). Total output relation: \( q = 10 \sqrt{2l} \).

Step-by-step solution

Setup for Labor Allocation

The firm's goal is to maximize output by optimally distributing the labor between the two locations. Given the production functions, the marginal product of labor in each location must be equal to ensure efficiency. The marginal product is found by differentiating the production functions: \( MP_{l_1} = \frac{5}{\sqrt{l_1}} \) and \( MP_{l_2} = \frac{25}{\sqrt{l_2}} \).

Solve for Efficient Labor Use

To maximize output, equate the marginal products: \( \frac{5}{\sqrt{l_1}} = \frac{25}{\sqrt{l_2}} \). Solving for \(l_1\) in terms of \(l_2\), we get: \( \sqrt{l_1} = 5\sqrt{l_2} \), hence, \( l_1 = 25l_2 \). This equation shows the relationship: the labor in location 1 should be 25 times the labor in location 2.

Express Total Labor

The total labor \( l \) is the sum of the labor used at each location: \( l = l_1 + l_2 \). Using the relationship \( l_1 = 25l_2 \), substitute into the total labor expression to get: \( l = 25l_2 + l_2 = 26l_2 \).

Derive Total Output Function

To express total output \( q \) in terms of total labor \( l \), first find \( l_2 = \frac{l}{26} \) from the total labor expression. Then, \( l_1 = 25 \times \frac{l}{26} \). Substitute these into the production functions to find total output: \( q_1 = 10 \left( 25 \times \frac{l}{26} \right)^{0.5} \) and \( q_2 = 50 \left( \frac{l}{26} \right)^{0.5} \).

Combine Outputs to Find Total Output

Set \( q = q_1 + q_2 \). Substitute the expressions from the previous step: \( q = 10 \times 5 \times \sqrt{\frac{l}{26}} + 50 \times \sqrt{\frac{l}{26}} \). Simplify to get: \( q = 100 \sqrt{\frac{l}{26}} \), or simplified further: \( q = 10 \sqrt{2l} \).

Key concepts

Marginal Product of Labor

In production economics, the concept of Marginal Product of Labor (MPL) is crucial for determining how adding more labor affects output. Marginal Product of Labor represents the change in output when an additional unit of labor is employed. Mathematically, it can be obtained by differentiating the production function with respect to labor. This gives firms insights into how efficiently labor is being used and guides them in labor allocation decisions.

For the given production functions of crayons, the Marginal Product of Labor at location 1 is calculated as the derivative of the production function with respect to labor, denoted as \( MP_{l_1} = \frac{5}{\sqrt{l_1}} \). At location 2, the MPL is given by \( MP_{l_2} = \frac{25}{\sqrt{l_2}} \).

• These expressions tell us how much additional output is gained from each additional unit of labor at each location.
• By comparing these marginal products, the firm can determine how best to allocate labor between the two locations to maximize production.
• The goal is to have the MPL equal across different locations, indicating that labor is being employed efficiently.

Labor Allocation

Labor allocation is the process of distributing labor resources across different production sites or tasks to ensure efficiency and maximize output. In the context of our crayon production example, efficiently allocating labor means achieving equal marginal products of labor at both locations.

To achieve this, the equation \( \frac{5}{\sqrt{l_1}} = \frac{25}{\sqrt{l_2}} \) is used, which equates the marginal products of labor at both sites. When this equation is solved, it results in the relationship \( l_1 = 25l_2 \).

• This means that the labor at location 1 should be 25 times what it is at location 2.
• Understanding and applying this relationship ensures that the firm utilizes its labor efficiently and produces the maximum possible output with the resources available.
• This illustrates the critical principle of equating marginal products for optimal resource allocation.

Total Output Function

The Total Output Function illustrates how the total production varies with the quantity of input used. In the exercise, the total output is expressed as a function of the total labor employed across two locations.

Starting with the combined labor equation \( l = 26l_2 \), we derive expressions for each location's labor, \( l_1 = 25\times \frac{l}{26} \) and \( l_2 = \frac{l}{26} \). These are substituted back into the respective production functions to determine each site's output.

• The total output \( q \) is calculated by adding the outputs from both locations: \( q_1 + q_2 \).
• Simplifying this expression yields \( q = 10\sqrt{2l} \), representing the relationship between total labor hired \( l \) and total output \( q \).
• This function shows that as more labor is applied, output increases according to the square root of twice the labor applied, reflecting the inherent efficiency in labor allocation and production expansion.