Question

An enterprising entrepreneur purchases two factories to produce widgets. Each factory produces identical products, and each has a production function given by \\[ q=\sqrt{k_{i} l_{i}}, \quad i=1,2 \\] The factories differ, however, in the amount of capital equipment each has, In particular, factory 1 has \(k_{1}=25,\) whereas factory 2 has \(k_{2}=100 .\) Rental rates for \(k\) and \(l\) are given by \(w=v=\$ 1\) a. If the entrepreneur wishes to minimize short-run total costs of widget production, how should output be allocated between the two factories? b. Given that output is optimally allocated between the two factories, calculate the short-run total, average, and marginal cost curves. What is the marginal cost of the 100 th widget? The 125 th widget? The 200 th widget? c. How should the entrepreneur allocate widget production between the two factories in the long run? Calculate the long-run total, average, and marginal cost curves for widget production. d. How would your answer to part (c) change if both factories exhibited diminishing returns to scale?

Short answer

In summary, the optimal allocation of output between the two factories in the short-run is given by \(q_1 = \frac{Q}{3}\) and \(q_2 = \frac{2Q}{3}\). The short-run and long-run total, average, and marginal cost curves for the constant returns to scale scenario are: \[C(Q) = \frac{Q^2}{75} + 125\] \[AC(Q) = \frac{Q}{75} + \frac{125}{Q}\] \[MC(Q) = \frac{2Q}{75}\] The marginal costs of the 100th, 125th, and 200th widgets are approximately $26.67, $16.67, and $26.67, respectively. In the long run, the entrepreneur can adjust both k and l, and the allocation remains the same as in the short run. If both factories exhibit diminishing returns to scale, the entrepreneur needs to reconsider the allocation between the two factories in the long run, and the total, average, and marginal cost curves will also change.

Step-by-step solution

We are given the production function \[q=\sqrt{k_i l_i}, \quad i=1,2\] The rental rates for k and l are given by \(w=v=\$1\). Then, the costs of inputs are equal, and we can write the cost function for each factory as: \[C_i = w\cdot l_i + v\cdot k_i\] We need to minimize costs, considering the production function constraints. To do this, we can substitute the production function into the cost function for each factory: \[C_i = w\left(\frac{q_i^2}{k_i}\right) + vk_i\] For factory 1, with \(k_1 = 25\), we have: \[C_1 = \frac{q_1^2}{25}+25\] Similarly, for factory 2, with \(k_2 = 100\), we have: \[C_2 = \frac{q_2^2}{100}+100\] The entrepreneur wants to minimize the total cost \(C = C_1 + C_2\) while producing a total output \(Q = q_1 + q_2\). To minimize the total cost function subject to the constraint \(Q = q_1 + q_2\), we can use the Lagrange multipliers method. This gives us the following problem: Minimize: \(\mathcal {L} = C_1+ C_2 - (\lambda)[Q-(q_1+q_2)]\) Which yields the following first-order conditions: 1. \(\frac{\partial \mathcal {L}}{\partial q_1}=2*q_1/25 - \lambda = 0\) 2. \(\frac{\partial \mathcal {L}}{\partial q_2}=2*q_2/100 - \lambda = 0\) Now solve the system of equations to find \(q_1\) and \(q_2\): Divide equation (1) by 2 and equation (2) by 4: \(\frac{q_1}{25} - \frac{\lambda}{2} = 0\) and \(\frac{q_2}{100} - \frac{\lambda}{4} = 0\) Solving for \(\lambda\), we get: \(\lambda = \frac{2q_1}{25} = \frac{q_2}{50}\). From the constraint, we also have \(q_1 = Q-q_2\). Thus, \[Q - q_2 = \frac{25q_2}{50}\] Now, we have found that the optimal allocation of output between the two factories will be: \(q_1 = \frac{Q}{3}\) and \(q_2 = \frac{2Q}{3}\) #b. Short-run Total, Average, and Marginal Costs

First, use the optimal output allocation to find the total cost function: \[C(Q) = C_1+ C_2 = \frac{(\frac{Q}{3})^2}{25}+25+\frac{(\frac{2Q}{3})^2}{100}+100\] \[C(Q) = \frac{Q^2}{75}+125\] Now calculate the average cost and marginal cost functions: \[AC(Q) = \frac{C(Q)}{Q}=\frac{Q}{75}+\frac{125}{Q}\] \[MC(Q) = \frac{dC(Q)}{dQ}=\frac{2Q}{75}\] Next, we need to find the marginal costs of the 100th, 125th, and 200th widgets: \[MC(100) = \frac{2*100}{75} = \frac{80}{3}\approx26.67\] \[MC(125) = \frac{2*125}{75}=\frac{50}{3}\approx16.67\] \[MC(200) = \frac{2*200}{75}= \frac{80}{3} \approx 26.67 \] #C. Long-run Allocation and Costs

In the long run, the entrepreneur can adjust both k and l to minimize costs. Such optimization will result in equal marginal costs in both plants, as they produce identical products. In this scenario, the long-run allocation will be the same as in the short run, \(q_1 = \frac{Q}{3}\) and \(q_2 = \frac{2Q}{3}\). The long-run total, average, and marginal cost curves remain the same as those calculated for the short run: \[C(Q) = \frac{Q^2}{75}+125\] \[AC(Q) =\frac{Q}{75}+\frac{125}{Q}\] \[MC(Q) =\frac{2Q}{75}\] #d. Diminishing Returns to Scale

If both factories exhibited diminishing returns to scale, then the production function would change, meaning an increase in k and l would lead to a less-than-proportional increase in q. In this case, the entrepreneur needs to reconsider the allocation between the two factories in the long run to achieve cost minimization, and it might not be optimal to allocate the output in the same ratio as found in the short run. The new optimal allocation would depend on the specific production functions exhibiting diminishing returns to scale. However, the calculated total, average, and marginal cost curves would be different from those obtained for constant returns to scale. They would still be derived similarly by using the respective (new) production functions, but the actual numbers will change.

Key concepts

Cost Minimization

In production economics, cost minimization involves choosing the optimal combination of inputs to produce a given level of output at the lowest possible cost. The entrepreneurial challenge in the exercise is to minimize costs while producing widgets across two factories with different capital levels. Each factory has its own production function, yet they face the same rental costs for labor and capital. To achieve cost minimization, the entrepreneur allocates output such that the marginal cost in one factory equals the marginal cost in the other. This is achieved by using methods like the Lagrange multipliers to find the point where costs are minimized given the constraints of the production function and the total desired output. Thus, understanding cost minimization is crucial for efficient resource allocation and profitability.

Short-run vs Long-run Costs

The distinction between short-run and long-run costs is essential in production planning. In the short run, some inputs, such as capital, are fixed while others like labor can be varied. In the given exercise, factories operate with fixed capital but can vary labor. Therefore, the short-run total cost equation considers these fixed inputs and only varies with the number of widgets produced. In contrast, in the long run, all inputs can be adjusted. This flexibility allows the entrepreneur to optimize production across both factories, rebalancing capital and labor to minimize costs further. In both scenarios, cost curves—total, average, and marginal—provide insight into cost behavior and guide decision-making. While short-run costs depend on fixed constraints, long-run costs focus on efficiency and full flexibility.

Lagrange Multipliers Method

The Lagrange multipliers method is a mathematical approach used for finding the maximum or minimum of a function subject to constraints. In the exercise, it's employed to allocate output between two factories while minimizing costs. By setting up a Lagrangian function that incorporates the total cost function and the constraint (total production needed), we derive equations that lead to optimal output allocation. The method reveals how small changes in constraints affect the optimal solution, marked by a multiplier—lambda (\( \lambda \)). Here, it ensures that production costs are minimized by allocating production output such that the marginal cost for each widget is equal across both factories. Understanding this method is integral for economists seeking to optimize under constraints, as it uncovers the interplay between different economic variables.

Diminishing Returns to Scale

Diminishing returns to scale refer to a situation in production where increasing all inputs results in a less-than-proportional increase in output. This is based on the assumption that further input additions become less productive over time. In the context of the exercise, if the factories experience diminishing returns to scale, it implies the entrepreneurial strategy must adapt. The cost curves calculated in prior sections assume constant returns to scale. However, under diminishing returns, the output allocation may need recalibration to prevent inefficiency. This means re-evaluating how resources are distributed among factories to ensure output increases do not lead to disproportionately high costs. Recognizing diminishing returns helps businesses avoid the inefficiencies of overproduction and may prompt investments in innovation or technology to boost productivity.