Problem: Shutdown decision of the food manufacturer The production function of a typical food manufacturer has been estimated to be
q = K1/3 L1/2
where K denotes capital and L denotes labor. In the short run. the firm's capital is fixed at K = 1000. The prices of capital and labor are fixed at r = 1 and w = 2 in the short run and the long run.
a. Derive the short-run cost function of the firm.
b. Derive the marginal cost and average variable cost functions of the firm in the short run and represent them on a graph with price on the vertical axis.
c. Does the firm shut down at any positive output price in the short run? Explain your reasoning.
d. Derive the long-run cost function of the firm using the method of your choice.
e. Derive the marginal cost and average cost functions of the firm in the long run.
f. Using the graphing software available at https://www.flashandmath.com/mathlets/calc/fungraph/index.html, plot the long-run average cost curve and the long-run marginal cost curve of the firm.
g. Does the firm decide to shut down at any positive price in the long run? Explain your reasoning.