Answer to Question #183646 in Microeconomics for Trishool

Question

A competitive firm has the following average cost function: 𝐴𝐶 = 𝑦^2 − 8𝑦 + 30 + 5/𝑦. The corresponding marginal cost function is 𝑀𝐶 = 3𝑦 ^2 − 16𝑦 + 30. a) Derive the total cost function, then find the firm’s average variable cost, average fixed cost, and fixed cost. Is this firm in the short run or the long run? How do you know? b) At what quantity is marginal cost equal to average variable cost? At what quantity is average variable cost minimized? The firm will supply zero output if the price is less than what? c) What is the smallest positive amount that the firm will ever supply at any price? At what price would the firm supply exactly 6 units of output?  

Answer 1

a)

It is given that AC = y² – 8y + 30 + 5/y

AC means average cost. It can be derived by diving y by Total Cost TC.

Therefore, Tc can be obtained by multiplying AC by y.

 

TC = y³ – 8y² + 30y + 5

 

The total cost of a firm includes variable cost and fixed cost. From the derived TC function, one can see that fixed cost is 5 which does not vary with quantity of output produced.

Thus, FC = 5

AFC= 5/y

This implies, VC = y³ – 8y² + 30y

 Average variable cost (AVC) can be derived by dividing VC by y.

This implies, AVC = VC/y = y² – 8y + 30

 

The firm is in short run because it has fixed cost which is 5.

In the long run all the costs of the firm is variable with nothing being fixed or static.

b)

MC=AVC

$3y^2-16y+30=y^2-8y+30$

$3y^2-y^2-16y+8y+30-30=0$

$2y^2-8y=0$

$2y^2=8y$

$y=4$

Average variable cost will be equal to marginal cost when quantity produced is 4

Quantity at which AVC will be minimized

$AVC=y^2-8y+30$

solving the quadratic equation,

$\frac{8 ± \sqrt{64-120}}{2}$, assuming the imaginary part, y=4

therefore, average variable cost will be minimized when quantity is 4

Price at which firm will supply zero output

$AC=4^2-8(4)+30+\frac{5}{4}=15.25$

The firm will supply zero output when the price is less than 15.25 to avoid losses.

c)

the smallest positive amount the firm will supply at any price is 4 because production of this quantity has the minimum AVC

price at which the firm will supply exactly 6 units

$AC=y^2-8y+30+5/y$

substituting y with 6

$AC=6^2-8(6)+30+5/6 = 18.83$

The firm will supply exactly 6 units if price is 18.83