Answer to Question #187418 in Microeconomics for que

Question

A competitive firm’s cost function is given as C = 100q − 4q 2 + 0.2q 3 + 450. When the market price is 75, what will the firm do in the short run? Draw a diagram that includes MC, AC, and AVC curves, and indicate the area for the profit in case the firm decides to produce.

Answer 1

The cost function will be $C=100q-4q^2+0.2q^3+450$

$MC=\frac{dc}{dq}=100-8q+0.6q^2$

$Price(P)=75$

Price maximization condition will be calculated as $P=MC$

$100-8q+0.6q^2=75$

$0.6q^2-8q+25=0$

Solving this equation we will get that q=8.33 or 5

Profit function will therefore be; $\pi=pq-c$

$\pi=75q-100q+4q^2-0.2q^3-450$

$\frac{d\pi}{dq}=8q-0.6q^2-25=0$

q=8.33 or 5.

$\frac{d^2\pi}{d\pi}=8-(0.6\times2)q$

$=8-1.2q$

At $q=8.33,\ \frac{d^2\pi}{dq}=8-(1.2\times8.33)$

$=-1.996$

At$\ q=5,\ \frac{d^2\pi}{2q^2}=8-(1.2\times5)$

$=2>0$

Therefore,

$\frac{d^2\pi}{dq^2}<0$ $for \ q\ =8.33$

hence profit maximization satisfied will produce 8.33 units.

Profits= $\pi=4q^2-0.2q^3-450-25q$

$=4(8.33)^2-0.2(0.33)^3-450-(25\times8.33)$

$=-496.3$

$AC=\frac{c}{q}=100-4q+0.2q^2+\frac{450}{q}$

At $q=8.33, \ AC=100-(4\times8.33)+0.2(8.33)^2+\frac{450}{8.33}$

$=134.579$

$VC=100q-4q^2+0.2q^3$

$AVC=\frac{VC}{q}=100-4q+0.2q^3$

At $q=8.33,\ AVC=100-(4\times8.33)+0.2(8.33)^2$

$=80.56$

The diagram below shows the profit area in case the firms decide to produce.