Question 1 - Suppose a firm's production process is characterized by the production function: Q = 0.4L^0.5K^0.5. Let the price of labor be P_L = $8 per unit and the price of capital be P_K = $16 per unit. The budget available for production (i.e., the amount of production costs that can be covered) is given by C = $1200. Use the Lagrangian method to identify the combination of K and L that maximizes output, given the input prices and budget. Specifically:

A. Specify the Lagrangian function that will be maximized.

B. Derive the 3 first order necessary conditions for a maximum.

C. Using the FOC's derived in 'B', calculate the optimal combination of K and L that should be employed. In addition, calculate the maximum amount of output that can be produced. Show all steps for credit.

Question 2 - Suppose that a firm's total cost (TC) function is given by: TC = 100 + 4Q - 3Q² + 2Q³ where Q denotes units of output measured in thousands.

A. Derive/identify the firm's TFC, TVC, AVC, ATC and MC functions from the TC function.

B. Calculate the level of output that i) minimizes AVC and ii) minimizes MC. Show all steps for credit.