Answer to Question #211570 in Microeconomics for Rida Awwal

Question

A profit maximizing firm in perfectly competitive market structure has average revenue of 10, average variable cost of 10-2q+q^2 and total fixed cost of 200

À)find profit maximizing output

B) Calculate the maximum profit

C) what price is needed for the firm to stay in the market

Answer 1

a)

Finding total variable cost:

It is average variable cost multiplied by units made. Using this:

$TVC = AVC×q\\= (10 −2q +q^2 )×q \\=10q −2q^2 +q^3$

Computing the marginal cost:

It is derivative of total variable cost with respect to output:

$MC =\frac{ dTVC}{dq}\\= 10×1 −2×2×q +3×q^2\\= 10−4q +3q^2$

The competitive firm will get the highest amount of profits when its price is same as marginal cost. Using this:

$Average \space revenue = Marginal \space cost\\10=10-4q+3q^2\\3q^2=4q\\3q=4\\q=1.33$

Hence, the maximum output is 1.33 units.

b)

Computing the maximum amount of profits:

It is the excess of total revenue over the business costs. Using this:

$Profits =TR −TC \\=Average\space revenue×q − (TVC + Fixed\space cost) \\=10q −(10q − 2q^2 +q^3+200)\\= 10×1.33 − (10×1.33 − 2×1.33^2 +1.33^3+200) −198.814$

So, the minimum amount of losses would be 198.814 dollars.

c)

The price at which the firm would stay in the market is the one equal to the minimum of average variable cost in the short-run . So, we will find the value of minimum average variable cost as follows:

Minimizing average variable cost

$AVC =10−2q +q^2 ,$ differentiate with respect to q:

$\frac{dAVC}{dq}=0 −2×1 +2×q\\=-2+2q$  putting it equal to zero:

$\frac{dAVC}{dq}=0 \\-2+2q=0\\q=1$

Putting it in the average variable function:

$AVC = 10−2q +q^2\\AVC = 10−2×1 +1^2\\AVC = 10 −2 +1\\AVC = 9$

Hence, price must be at least equal to 9 dollars for the firm to stay in the market.