A competitive firm has estimated its average variable cost function as: AVC = 20

Question

A competitive firm has estimated its average variable cost function as:


AVC = 20 - 0.04Q + 0.00005Q2


Its total fixed cost is $500.


a) Calculate the short-run marginal cost function associated with the average variable cost function.




b) Calculate the level of output at which average variable cost reaches its minimum and what value that is.




c) Calculate the level of output this firm should produce in order to maximize profits, and what level of profit (loss) this firm would incur.



d) Suppose the forecasted price of its product is $10, calculate the level of output and profit (loss) this firm would incur.

Explanation / Answer

A) AVC = VC/Q, so if we multiply AVC by Q we can find the variable costs VC = (20 - 0.04Q + 0.00005Q^2) * Q = 20Q - 0.04Q^2 + 0.00005Q^3 The marginal cost is the first derivative of the variable costs f'(vc) = MC = 20 - 0.08Q + 0.00015Q^2 B) When AVC is at its minimum it is equal to the marginal cost so 20 - 0.04Q + 0.00005Q^2 = 20 - 0.08Q + 0.00015Q^2 0.04Q - 0.0001Q^2 = 0, Q = 400 min variable cost = 20 - 0.04*400 + 0.00005*400^2 = 20 - 16 + 8 = $12 C) Max profit occurs when MR = MC but since I don't have a price level or a demand curve I can't calculate a value for you. D) 10 = MC 10 = 20 - 0.08Q + 0.00015Q^2 0 = 10 - 0.08Q + 0.00015Q^2, Q = 333.3 or 333 units. profit(loss) = 10* 333 - [500 + (20Q - 0.04Q^2 + 0.00005Q^3)] = 10*333 - (500 + 20*333 - 0.04*333^2 + 0.0005*333^3 = 10*333 - (500 + 6660 - 4435 + 1846) = 3333 - 4571 = - 1238, since this loss is bigger than the fixed cost it would be better for the firm to shut down.

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