Question

Consider a firm with a short run Total Cost (TC) given by TC=2000 + 1000Q - 40Q² + Q³ . What is the firms marginal cost? What is firm's shut down price?

Answer 1

Answer

Marginal cost (MC) is the change in total cost(TC) for an extra unit increase in the output (Q).

So, MC = TC / Q , where '' stands for change

Here,

TC = 2000 + 1000Q - 40Q² + Q³

Differentiating the above equation with respect to Q,

d(TC)

MC = --------- = 1000 - 80Q + 3Q²

dQ

marginal cost = 1000 - 80Q + 3Q²

In short-run , the shut down price is the price that covers at least the average variable cost (AVC) of production , i.e.

Price (P) must be equal to the AVC or greater than AVC.

P > = AVC

AVC is the total variable cost (VC) per unit of output.

AVC = VC / Q

In the given cost equation, TC= 2000 + 1000Q - 40Q² + Q³ ,

Fixed cost (FC) = 2000

Variable Cost (VC), that starts when the production starts and thus depends on the variable input = 1000Q - 40Q² + Q³ ,

AVC = (1000Q - 40Q² + Q³ ) / Q

AVC = 1000 - 40Q + Q²

Now, a perfectly competitive firm in the short-run maximizes profit at the quantity, where P = MC

Again for the shut-down condition, P at least should be equal to AVC

Now to stay in the market, price should be equal to the MC, which should be equal to the AVC.

P = MC = AVC

Or, MC = AVC

Now, putting the values of MC and AVC, we get,

1000 - 80Q + 3Q² = 1000 - 40Q + Q²

Or, - 80Q + 3Q² = - 40Q + Q²

Or, - 80Q + 40 Q = Q² - 3Q²

Or, -40Q = - 2 Q²

Or, 2Q² = 40Q

Or, 2Q = 40

Q = 20

Now, putting the value of Q in MC equation, MC = 1000 - 80Q + 3Q² , we get,

MC =1000 - 80 * 20 + 3 * 20 *20

Now, as P= MC (profit maximization),

P = 1000 - 80 * 20 + 3 * 20 *20

Or, P = 1000 - 1600 + 1200

Or, P = 2200 - 1600

Or, P = 600

Price (P) = 600 money unit.

So, the shut down price of this firm is 600 money unit.

__________________________________________________________________