Question

The demand and total cost functions of a good are respectively  and

a) Find expressions for TR, (profit) , MR, and MC in terms of Q.

b) Solve the equation

and hence determine the value of Q which maximizes profit.

c) Verify that, at the point of maximum profit, MR=MC.

Answer 1

a)

Since the two equations for TR and TC are missing for this equation, I will use the following equations to explain this answer.

$TR=-2Q^2+20Q$

$TC=Q^3-8Q^2+20Q+10$

$MR=-4Q+20$

$MC=3Q^2-16Q+20$

We need to set $MR=MC$

$3Q^2-16Q+20=-4Q+20$

$3Q^2-12Q=0.$

b)

Solving the equation.

$3Q^2-12Q=0$

$3Q(Q-4)=0$

But we know that $Q>0$ for $MC=MR$

Therefore,

$Q-4=0$

$Q=4$ .

c)

When the value of the final unit of product (marginal revenue) matches the cost of manufacturing the last unit of product, a manager optimizes profit (marginal cost). As a result, the company will not produce that unit.

Profit is maximized where MR=MC in a perfectly competitive market.

Maximum profit will be expressed as $P=MC$ or $P=MR$

$MC=$ $-4(4)+20$

$=4$

Maximum profit will be obtained at 4 units.