Suppose a firm uses two good in the production process, x 1 and x 2 . The price

Question

Suppose a firm uses two good in the production process, x1 and x2. The price for a unit of x1 costs the firm g1 and the price for a unit of x2 costs the firm g2. The firms production function is defined as f(x1,x2)=x11/3x21/3.   MP1= (x21/3) / (3x12/3), Mp2= (x11/3) /(3x12/3). The firm sells its output in a perfectly competitive market at a price of 'p' per unit.

a. Write down the profit-maximization problem facing the firm.

b. Suppose that x2 is fixed at 8 units.

i. Is the firm in the short-run, long-run, or not enough information?

ii. What condition (equation) must hold for the firm to maximize profits?

iii. Explain the economics of the equation in part ii as if you were talking to Martha who lives in the Oxford county Nursing Home.

iv. Solve for the amount of x1, as a function of p,g1, and g2, that maximizes profit for the firm.

v. Suppose that g1 = 1, g2 = 1, and p = 6, solve for the optimal amount of x1.

vi. At the optimal amount of x1 (in part v), calculate output, total revenue, variable cost, fixed cost, and profit.

vii. What is the own price elasticity of demand for the firm?

Explanation / Answer

a. i. f(x1,x2)=1/9x1.x2

MP1=1/3.x2/x1 ; MP2=1/3.x1/x2

Profit-maximization facing problem:

z=1/9x1x2-g1x1-g2x2

b. i. As x2 is fixed it is short-run production function.

ii. FOC requires dz/dx1=0

or, 1/9.x2-g1=0

or,1/9.8=g1

or, 8/9=g1

iii. Short run optimal price of x1 is 8/9

iv.

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