Answer to Question #219724 in Microeconomics for Ermi

Question

If the inverse demand curve of profit maximizing monopolist is given as P =1200 − 2Q , and cost function as

C = Q3 − 61.25Q2+1528.5Q + 2000, find equilibrium output level, monopolist price, and profit.

Answer 1

Meaning

$p =1200Q − 2Q ,$

$c = Q^3 − 61.25Q^2+1528.5Q + 2000$

Now the inverse demand function is one where price P has to be expressed as a function of quantity q

Now, we know that total revenue equals price times quantity, so that

$R(Q)= P(Q)×Q$

$P(Q)×Q=1200Q-2Q^2$

Dividing through by Q We get

$\frac{Q}{Q}P[Q]=\frac{1200Q}{Q}-\frac{2Q^2}{Q},$

$P[Q]=1200-2Q$

Which is the required inverse demand function

Let denote the profit function by. $π(Q)$

Now,

$π(Q)=R(Q)-TC(Q)$

$π(Q)=(1200Q-2Q^2)-(Q^3 − 61.25Q^2+1528.5Q + 2000)$

$π(Q)=1200Q-2Q^2-Q^3 + 61.25Q^2-1528.5Q - 2000$

$π(Q)=-Q^3 + 59.25Q^2-328.5Q + 2000$

Which is the required profit function

Now finding profit maximizing output we need to find $Q^*$ such that

$π(Q^*)=0....................(i)$

$π(Q^*)<0................. .(ii)$

Now

$π(Q)=-Q^3 + 59.25Q^2-328.5Q + 2000$

Differentiating both sides with respect to Q;

$π'(Q)=-3Q^2 + 118.5Q-328.5$

Differentiating both sides again with respect to Q;

$π''(Q)=-6Q + 118.5$

Now let the first satisfy condition (i)

$π(Q^*)=0....................(i)$

$-3(Q^*)^2s + 118.5Q^*-328.5=0$

$(Q^*)^2s - 39.5Q^*+109.5=0$

$2(Q^*)^2s -79Q^*+219=0$

$2(Q^*)^2s -6Q^*- 73Q^*+219=0$

$2Q^*(Q^*-3)-73(Q^*-3)=0$

$(2Q^*-73)(Q^*-3)=0$

Q= 3 or 36.5