Question

An enterprising entrepreneur purchases two factories to produce widgets. Each factory produces identical products, and each has a production function given by $$q_{i}=\sqrt{k_{i} l_{i}} \quad i=1,2$$ 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 the short run, the optimal allocation of output between the two factories is producing half the output in factory 1 and the other half in factory 2. The short-run total cost curve is \(TC(q) = 125 + \frac{4q^2}{225}\), the average cost curve is \(AC(q) = \frac{125}{q} + \frac{4q}{225}\), and the marginal cost curve is \(MC(q) = \frac{8q}{225}\).

Step-by-step solution

a. Optimal Allocation of Output in the Short Run

The production function for each factory is given by \(q_{i}=\sqrt{k_{i} l_{i}}\) (i=1,2) and the cost of capital (k) and labor (l) are both equal to 1. To minimize the short-run total cost, we need to minimize the cost of producing widgets for a given level of output. Let \(q = q_1 + q_2\) be the total output. We will derive the cost function for each factory and then find the optimal allocation of output between the factories. First, we solve for labor (l) in the production function: \(l_{i}=\frac{q_{i}^{2}}{k_{i}}\) Since the rental rates for both capital and labor are equal to 1, the cost function for each factory is given by: \(C_{i} = k_{i} + l_{i} = k_{i} + \frac{q_{i}^{2}}{k_{i}}\) For factory 1, \(k_1 = 25\). Thus, the cost function is: \(C_1 = 25 + \frac{q_1^2}{25}\) For factory 2, \(k_2 = 100\). Thus, the cost function is: \(C_2 = 100 + \frac{q_2^2}{100}\) Now, we set the marginal costs equal to each other and solve the equation: \(\frac{dC_1}{dq_1} = \frac{dC_2}{dq_2}\) Calculating the derivatives, we get: \(\frac{2q_1}{25} = \frac{2q_2}{100}\) Solving the equation, we find that \(q_1 = \frac{1}{2}q_2\). Hence, the optimal allocation in the short run is to produce half the output in factory 1 and the other half in factory 2.

b. Short-Run Total, Average, and Marginal Cost Curves

Using the optimal allocation found in part a, we can now compute the short-run total, average, and marginal cost curves for the system. We know that \(q_1 = \frac{1}{2}q_2\), and the total output q is given by \(q = q_1 + q_2\). Thus, we have \(q = \frac{3}{2}q_1\). The short-run total cost is: \(TC = C_1 + C_2 = 25 + \frac{q_1^2}{25} + 100 + \frac{q_2^2}{100} = 25 + \frac{q_1^2}{25} + 100 + \frac{4q_1^2}{100}\) Using the substitution \(q = \frac{3}{2}q_1\), we can express the total cost as a function of total output q: \(TC(q) = 25 + \frac{(\frac{2}{3}q)^2}{25} + 100 + \frac{4(\frac{2}{3}q)^2}{100} = 125 + \frac{4q^2}{225}\) The short-run average cost is: \(AC(q) = \frac{TC(q)}{q} = \frac{125}{q} + \frac{4q}{225}\) The short-run marginal cost is the derivative of the total cost with respect to q: \(MC(q) = \frac{dTC(q)}{dq} = \frac{8q}{225}\) To find the marginal cost of the 100th, 125th, and 200th widgets, we plug these values of q into the marginal cost formula: \(MC(100) = \frac{8(100)}{225}\) \(MC(125) = \frac{8(125)}{225}\) \(MC(200) = \frac{8(200)}{225}\)

c. Long-Run Optimal Allocation and Cost Curves

In the long run, the entrepreneur can freely allocate capital to both factories in order to minimize total cost. Since both factories have the same production function, the entrepreneur should allocate capital to both factories equally. This means producing the same amount of output in each factory: \(q_1 = q_2\). The total, average, and marginal cost curves for widget production will also be equal for both factories. The long-run total, average, and marginal cost curves can be calculated by: 1. Allocating capital equally: \(k_1 = k_2 = \frac{k}{2}\), where k is the total capital in the system. 2. Deriving the long-run production function given the equal allocation of capital: \(Q = 2\sqrt{\frac{k}{2} L}\), where \(Q\) is the total output and \(L\) is the total labor. 3. Computing the long-run total cost: \(TC = k + L\). 4. Calculating the long-run average cost: \(AC = \frac{TC}{q}\). 5. Finding the long-run marginal cost: \(MC = \frac{dTC}{dq}\).

d. Impact of Diminishing Returns to Scale

If both factories exhibit diminishing returns to scale, then doubling the inputs (capital and labor) will result in less than double the output. Consequently, the long-run average and marginal cost curves will be upward sloping as the level of output increases. The entrepreneur would allocate production towards the factory that has a lower marginal cost until the marginal costs of both factories are equal. In this case, the optimal allocation of output across both factories would no longer involve producing the same output in both factories. Instead, a more intricate optimal production allocation will emerge depending on the scale parameter of each factory's production function. In the presence of diminishing returns to scale, the long-run average and marginal cost curves are expected to rise as output increases, and the optimal allocation between the two factories would be more complex and sensitive to the specific scale parameter values.