Question

5. Consider the following cost function: c(q; F) = F + 10q + q2 , where2

F > 0 represents the fixed cost: F = c(0; F).

1. (a) Compute the marginal cost function, MC(q) = c0(q; F).

2. (b) Show that the marginal cost function MC(q) is increasing.

3. (c) Recall the average cost function, AC(q; F) = c(q;F) . Find qˉ(F),q

   the value of q (given F) at which AC(q; F) = MC(q).

Answer 1

5. The cost function is given as $\dpi{80} C = F + 10q +q^2$ .

(a) The marginal cost will be $\dpi{80} MC = \frac{\partial C}{\partial q}$ or $\dpi{80} MC = \frac{\partial }{\partial q}(F + 10q +q^2)$ or $\dpi{80} MC = 10 + 2q$ .

(b) A function is increasing if it has a positive slope. We have $\dpi{80} \frac{\partial }{\partial q}(MC) = 2 > 0$ , and hence, MC is increasing for all q>0.

(c) The average cost function will be $\dpi{80} AC = \frac{C}{q}$ or $\dpi{80} AC = \frac{F + 10q +q^2}{q}$ or $\dpi{80} AC = \frac{F}{q} + 10 +q$ .

We have $\dpi{80} AC = MC$ as $\dpi{80} \frac{F}{q} + 10 +q = 10 + 2q$ or $\dpi{80} \frac{F}{q} = q$ or $\dpi{80} q^2 = F$ or $\dpi{80} q = \sqrt{F}$ .

Hence, at production point $\dpi{80} q = \sqrt{F}$ , the marginal cost is equal to the average cost.

Answer 2

4. Consider the following cost function: C(Q) = 100 + 10Q +Q: a. What are the formulas for the fixed cost, variable cost, average cost, and marginal cost? b. Compute total fixed cost, total variable cost, average fixed cost, average variable cost, average total cost, and marginal cost when level of output (Q) is 10 units. c. At what output level of output average cost reaches its minimum? Compute minimum level of average cost.