Question

1) A firm operating in a purely competitive environment is faced with a market price of $250. The firm’s total cost function (short run) is as follows:

TC = 6000 + 400Q – 20Q² + Q³

1. Should the firm produce at this price in the short run?
2. If the market price is $300, what will total profits (losses) be if the firm produce 10 units of output? Should the firm produce at this price?
3. If the market price is greater than $300, should the firm produce in the short run?

___________________________________________________

2) Consider the following short-run production function (where X = variable input, Q = output)

Q= l0X - 0.5X²

Suppose that output can be sold for $10 per unit. Also assume that the firm can obtain as much of the variable input (X) as it needs at $20 per unit.

1. Determine the marginal revenue product function and the marginal factor cost function.
2. Determine the optimal value of X, given that the objective is to maximise profits.

Answer 1

1)
a) Shut down criteria is P>=AVC
AVC = TC/Q = 6000/Q + 400– 20Q+ Q2
250 =  6000/Q + 400– 20Q+ Q2
6000/Q + 150– 20Q+ Q2 = 0

As there are no positive roots, the firm shall not produce at this price

Price = 300
Revenue = 300*10 = 3000
TC = 6000 + 400*10 – 20*100 + 1000 = 9000
Losses = 6000
AVC = 9000/10 = 900
Since P<AVC, the firm shall not produce

Production criteria:
P >= 6000/Q + 400 – 20Q+ Q2
For a price = 1400>300, firm can produce in the short run. Therefore, if the market price is higher than 300, firm can produce in the short run.

2)
a) Revenue (R) = 10Q
Revenue Product = dR/dX = 100-10X

b) For profit max, Revenue Product = 20
100-10X = 20
X = 8