Question

A monopolistic company has to spend exactly $3000 as total cost for the production of a number of Basic Science and Mathematics textbooks for three schools. The Basic Science textbook sells at P1 = 455 – Q1 – Q2 and the Mathematics textbook at P2 = 910 – Q1 – 4Q2 where P1 and P2 denote the prices; Q1 and Q2 denote the number of Basic science and mathematics textbooks produced respectively. The joint cost of producing these textbooks is given as TC = 5Q1 + 10Q2

(i) Find the maximum profit the producer can make

(ii) Estimate the new profit if the company decides to reduce the total cost

by $50 (Assume that 2nd order conditions are satisfied)

Answer 1

Total cost for production of no. of basic science and mathematics books = $30

TC= 3000.........(i)

Basic Science price, $P_1$ = 455-$Q_1- Q_2$

Mathematics price, $P_2=910-Q_1-Q_2$

Where,

$Q_1$ = No. of basic Science books

$Q_2$ = No. of mathematics books

so, total revenue= Price * Quantity

$TR=TR_1+TR_2$

$TR=(455Q_1+910Q_2-2Q_1Q_2-Q_1^2-4Q_2^2)$

$TC=5Q+10Q_2=3000$ From(i)

$Q_1+2Q_2=600$

$Q_1=600-2Q_2 .................(iii)$

$TR= 455(600-2Q_2)+910Q_2-2Q_2(600-2Q_2)$

$TR=(1200Q_2-4Q_2-87000)$

$Profit=TR-TC$

$P=1200Q_2-4Q_2^2-87000-3000$

For maximum profit,

$\frac{dp}{dQ_2}=0 = 1200-8Q_2-0=0$

Hence maximum profit = 0

Also $\frac{d^2p}{dQ_2^2}= (0-8)=(-8)<0 : (Maximum\space\ Condition)$

If the total cost changes

TC=(3000-50)= 2950

Then from equation iii

$TC= 5Q_1+10Q_2=2950$

$TC= Q_1+Q_2=950$

$TC= Q_1=(950-2Q_2)$

So, $TR=455(590-2Q_2)+910Q_2-2Q_2(590-2Q_2)-(590-2Q_2)^2-4Q_2^2$

$TR=(1180Q_2-4Q_2^2-79650)$

So profit, p= $1180Q_2-4Q_2^2-82600$

for maximum profit: $\frac{dp}{dQ_2}(1180-8Q_2-0)=0= _2=(\frac{1180}{8})=147.5$

maximum profit= $(1180*147.5)-4(147.5)^2-82600= 4425$

Hence, maximum profit producer can make $4425