Suppose that a firm's fixed proportion production function is given by \\[ q=\min (5 K, 10 \mathrm{L}) \\] and that the rental rates for capital and labor are given by \(v=1, w-3\) a. Calculate the firm's long-run total, average, and marginal cost curves. b. Suppose that Xis fixed at 10 in the short run. Calculate the firm's short- run total, aver age, and marginal cost curves, What is the marginal cost of the 10 th unit? The 50 th unit? The 100 th unit?

Short Answer

Expert verified
Answer: The long-run total cost function is \(TC(K) = 2.5K\), the average cost function is \(AC(q) = \frac{1.25}{L}\), and the marginal cost function is \(MC(q) = 1.25\). Question: What are the short-run total, average, and marginal cost curves for the given firm when capital is fixed at 10? Answer: The short-run total cost function is \(STC(L) = 25 + 3L\), the average cost function is \(SAC(q) = \frac{2.5 + 0.6L}{L}\), and the marginal cost function is \(MC(q) = 3\).

Step by step solution

01

a. Long-Run Total, Average, and Marginal Cost Curves

1. Find the total cost function in the long run: Since the production function is given by \(q = \min(5K, 10L)\) and the rental rates for capital and labor are \(v = 1\) and \(w = 3\), we can find the total cost as follows: \\[ TC(K, L) = vK + wL \\] 2. Solve for L in terms of K using the production function: Since the production function has fixed proportions, the amount of capital and labor needed to produce a certain level of output is fixed. We can write: \\[ 5K = 10L \\] \\[ L = 0.5K \\] 3. Substitute L in terms of K in the total cost function: Substitute for L gives: \\[ TC(K) = K + 3(0.5K) = 2.5K \\] 4. Calculate the average cost function: The average cost function is given by: \\[ AC(q) = \frac{TC(K)}{q} \\] Substitute the total cost function to get: \\[ AC(q) = \frac{2.5K}{\frac{10L}{5}} = \frac{1.25}{L} \\] 5. Calculate the marginal cost function: Since the production function is in fixed proportions, the marginal cost is constant. The constant marginal cost can be found by taking the derivative of the total cost function with respect to q: \\[ MC(q) = \frac{d(TC(K))}{dq} = \frac{d(2.5K)}{dq} = \frac{d(10L)}{dq} = \frac{d(5K)}{dq} = 1.25 \\] So, the long-run total cost function is \(TC(K) = 2.5K\), the average cost function is \(AC(q) = \frac{1.25}{L}\), and the marginal cost function is \(MC(q) = 1.25\).
02

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

1. Find the short-run total cost function: Given that capital is fixed at 10 in the short-run, substitute this value into the total cost function we derived above: \\[ STC(L) = 2.5(10) + 3L = 25 + 3L \\] 2. Calculate the short-run average cost function: The short-run average cost function is given by: \\[ SAC(q) = \frac{STC(L)}{q} \\] Substitute the short-run total cost function to get: \\[ SAC(q) = \frac{25 + 3L}{\frac{10L}{5}} = \frac{2.5 + 0.6L}{L} \\] 3. Calculate the short-run marginal cost function: The short-run marginal cost function can be found by taking the derivative of the short-run total cost function with respect to L: \\[ SMC(q) = \frac{d(STC(L))}{dq} = \frac{d(25 + 3L)}{dq} = 3 \\] 4. Calculate the marginal cost of the 10th, 50th, and 100th units: Since the short-run marginal cost is constant, the marginal cost of the 10th, 50th, and 100th units will all be 3. So, the short-run total cost function is \(STC(L) = 25 + 3L\), the average cost function is \(SAC(q) = \frac{2.5 + 0.6L}{L}\), and the marginal cost function is \(MC(q) = 3\).

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Most popular questions from this chapter

Suppose the total cost function for a firm is given by \\[ T C=(.5 v+W v w+.5 w) q \\] a. Use Shephard's lemma to compute the constant output demand function for each in put, ifand \(L\) b. Use the results from part (a) to compute the underlying production function for \(q\) c. You can check the result by using results from Extension \(\mathrm{E} 12.2\) to show that the CES cost function with \(a-(3-.5\) generates this total cost function.

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