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?

Step by step solution

01

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

02

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.

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