d. Solve for the short-run cost-minimizing demands for labor and energy. Show all steps in your work.

e. Solve for the short-run variable cost function. Show all steps in the derivation.

f. Find the shadow price of capital (-∂VC/∂K).

4. What is the short-run total cost function?

a. Find short-run marginal cost.

b. Find short-run average total cost.

5. Using the short-run total cost function, derive the long-run cost function.

a. Set up the minimization problem and derive the first-order condition.

b. Interpret the first-order condition economically.

c. Solve the long-run demand for capital. Show all steps.

c. Solve for the long-run cost function. Show all steps.

Here's a problem focusing on deriving short-run and long-run cost functions from a production function. The production technology is given by q= K1/4 1/4 1/4 where output is given by q, capital by K, labor by L, and energy by E. The input prices are respectively denoted by r, w, and u. 1. Is this technology characterized by diminishing marginal returns to labor? Show your work and prove your point. 2. What degree of returns to scale does this technology exhibit? Prove it. Show all steps in your proof. 3. Capital is a fixed input in the short run: K= K. Find the short-run variable cost function. a. Set up the Lagrangean function for this cost-minimization problem. b. Derive the first-order conditions. c. Solve the first-order conditions for the expansion path.