Question

Firm A faces a total cost function defined by C = 8y2 + 10y + 8, where y is the number of the units produced.

(a) Find the average cost function (C/y) and denote it by AC.

(b) Find the marginal cost function (∂C/ ∂y ) and denote it by MC.

(c) Find the quantity y∗ that minimizes the average cost function

(d) Show that the marginal cost function intersects the average cost funtcion at the lowest point of the average cost function (i.e., show that AC and MC intersects at y∗).

Answer 1

C = 8y² + 10y + 8

a. Average Cost is the cost per output

AC = TC/y

AC = 8y + 10 + 8/y

b. Marginal Cost is the change in cost with respect to change in output

MC = first differentiation of TC with respect to output

MC = 16y + 10

c. Average Cost function is minimum when MC cuts it

So, at MC = AC, AC is minimum

8y + 10 + 8/y = 16y + 10

8 = 8y²

y = 1 unit

d. Lowest can be find out by double differentiating the AC function

d(AC)/dy = 8 - 8/y²

d²(AC)/dy² = 16/y³

Positive second serivative shows minimum AC