1. Explain why the marginal cost curve must cross the average cost curve at the minimum of the average cost.
Consider the following production function: f(L,K) = (L^1/2+ K^1/2)^2 . The short run value of capital is K ¯ = 4, and the input prices are w = 2 for labor and r = 1 for capital.
2. Derive the short run average cost, marginal cost, and average variable cost functions. (Hint: Before you start, what is the output if L = 0?)
3. Check that the short run average cost reaches its minimum for q = 9. What is the minimum value of the short run average cost? (Hint: To check the minimum of the SRAC, show that the derivative of the SRAC function is equal to zero at q = 9.)
4. Verify analytically that at q = 9, the SRMC and the SRAC intersect. (Hint: Simply show that the two functions take the same value at q = 9.)
5. At what point do the short run MC and the short run AVC curve intersect? Comment.