Production and Cost

    1. Consider the production function q = L1/4E1/4K1/4. a. Is this production function subject to diminishing marginal returns to labor? Show your logic and calculations. b. Showing your logic, find its returns-to-scale. 2. Consider the same production function, q = L1/4E1/4K1/4, used in (1) above. K is a fixed input (K = K). Let the input prices be represented by w, u, and r, respectively. a. Write the Lagrangean function to minimize variable cost, wL + uE, subject to the production constraint. The fixed value of K can simply be substituted into the production function: q = L1/4E1/4K1/4. Note that variable cost is the sum of the expenditures on variable inputs, wL + uE, while short-run total costs = variable cost + fixed cost = wL + uE + rK. b. Derive the first-order conditions. c. Solve for the expansion path. d. Solve for the short-run cost-minimizing demand function for labor and energy. Why are they short run? e. Solve for the variable cost function (conditional on K)