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A firm has a production function given by f(x1; x2) = x₁^1/3x₂^1/3, where x1 and x2 are the two inputs used to produce output y. Let the price the firm pays per unit of input 1 be w1, and the price the ...rm pays per unit of input 2 be w2. The firm is planning to produce y units of output.

Long-run cost-minimization:

(a) Derive the firm's conditional input demand functions and its total cost function.

(b) Find the average total cost function and the marginal cost function. Does the average total cost remain constant, increase or decrease as output increases? What do we observe that? Justify your answer.

Short-run pro...t-maximization:

In the short-run, the firm uses ex2 units of input 2 and is unable to vary this quantity. In the short-run, the only thing that is left for the firm to choose is the amount of input 1. Let the price of the firm's output be p, and, once again, let the price the ...rm pays per unit of input 1 be w1, and the price the firm pays per unit of input 2 be w2.

(c) Derive the firm's short-run demand function for input 1 and its short-run supply function.

(d) Explain what happens to the short-run pro...t-maximizing production plan as the output price p decreases. Present graphically the short-run demand function for input 1 and show on the graph what happens to the quantity demanded when p decreases.

(e) Given than w1 = w2 = $1, p = $9 and x2 = 27, ...nd the ...rm's optimal level for input 1 and its optimal short-run output level. Calculate the ...rm's maximum short-run pro...t level.