The short-run total cost function for a firm producing skateboards is TC= 18q^2-

Question

The short-run total cost function for a firm producing skateboards is TC= 18q^2-9q+72 where q is the number of skateboards per week. (1) What is the fixed cost? What is the variable cost? (2) Please calculate the average cost function for skateboards. (3) Please calculate the marginal cost function for skateboards. (4) Please show that when average cost reaches a minimum, marginal cost intersects average cost. Also, at what level of skateboard output does average cost reach a minimum? What is the average cost at this level of output?

Explanation / Answer

1. Here fixed cost is 72. Which is fixed because it is not deependent on variable of production.

Variable cost = (18q2 - 9q).

2. Average cost = Total cost / q = 18q + (72 / q) - 9

3. Marginal cost is calculated by differentiating total cost function with respect to quantity-

MC = 36q - 9.

4. AC = 18q + (72 / q) - 9

Differentiating it-

AC ' = 18 - 72 / (q2) = 0

q = 2.

Second order conditon,

AC " = 144 / q3 >0.

So, q = 2 and as AC " > 0 that means the AC function is in minimum level.

So, at q = 2 level average cost reaches its minimum.

At q = 2 ,

AC = 18 * 2 - 9 = 27.

That is at q = 2, average cost is 27.

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