Derive the marginal and average cost functions calculate

Question

QUESTION 1: Suppose that you are given the following cost function

C(w, r, Q) = 2w^1/2r^1/2Q^3/2

where w is the wage rate for labor, r is the rental rate of capital and q is the output level.

(a) Derive the marginal and average cost functions.

(b) Calculate the supply function for the firm in the short run.

(c) Does output increase or decrease as input and output price change (Assume this is a competitive firm with market output price p).

(d) Does the production function that gives rise to this cost function have increasing, decreasing or constant returns to scale?

QUESTION 2: A firm has a production function Q = f(x₁, x₂) = X₁^1/2 + X₂^1/2

(a) Find the input requirement functions that minimize the cost of producing a level of output Q given input prices w₁ and w₂.

(b) Find the cost function C(w1, w2, Q).

(c) Find the firm's optimal choice of Q(w1, w2, p) assuming that it faces an output price p.

(d) Find the firm's unconditional (marshallian) demand for inputs X₁^*(w1, w2, p) and X₂^*(w1, w2, p).

QUESTION 3: Respond each part of this question with a graph and a brief explanation. Let Q* be the profit maximizing quantity of output in a competitive firm, which corresponds to the Q* where P=MC.

(a) Would a lump sum profit tax affect Q*?

(b) Would a proportional tax on profits affect Q*?

(c) Would a unit-specific tax t assessed on each unit of output affect Q*?

(d) Would a tax on labor input affect Q*?