# Cournot’s Duopoly, IIM Nagpur and IMT Nagpur

For a long time Nagpur lacked a reputed management institute. Nagpur University had its department of business management, but it was stuck with outdated syllabus. In spite of outdated syllabus, students did quite well in their professional career. Unfortunately, Nagpur University could not take advantage of talented faculty and alumni base to upgrade itself.

Later IMT Gaziabad started its Nagpur Campus and Nagpur got a reputed management institute, now Chief Minister has announced that Nagpur will now have an Indian Institute of Management- IIM. IIM Nagpur will be mentored by IIM Ahmedabad.

There will be competition between IMT and IIM to attract brightest talent from Nagpur. Each of them will charge certain fees, now question is how many seats should each institute have to get maximum profit?

In Game theory, there is concept of Cournot’s Duopoly which helps in solving this problem. It is named after French philosopher and mathematician Antoine Augustin Cournot.

Everything starts with market demand; the market demand curve in turn is function of price and quantity. Let us assume that market curve is defined by equation

P= 120-0.5 Q

Where P is market price and Q is market quantity demanded. Here Q is sum of number of students who will either join IMT or IIM i.e. Q= q1 +q2, where q1 is number of seats IMT should have and q2 is number of seats IIM should have.

With this equation now becomes P= 120-0.5(q1 + q2).

There is certain cost which institute incurs on each student i.e. marginal cost, let us assume that for IMT it is MC1 and for IIM it is MC2. Let MC1 be 10 lakhs and MC2 be 11 lakhs.

To make profit the institutes should earn revenue to cover up the cost, for profit to be maximum marginal cost should be equal to marginal revenue. We have assumed marginal cost, now we need to calculate marginal revenue.

Let us first take case of IMT, for calculating marginal revenue say MR1, we need to first calculate total revenue.

Total revenue for IMT = Price x Quantity

Total revenue for IMT= TR1= P x q1= (120-0.5 q1-0.5 q2) x q1= 120q1-.05q1 q1-0.5 q1q2

To find marginal revenue we need to apply differential calculus to the equation i.e. dTR1/dq1, which will give us

MR1= dTR1/dq1= 120- q1-0.5 q2

For maximum profit MC1= MR1 i.e. 120-q1-0.5q2= 10 i.e. q1= 110- 0.5 q2

Similarly we can find Marginal revenue for IIM i.e MR2

Now MR2= dTR2/dq2= 120-0.5q1- q2, here profit will be maximum when q2= 109-0.5 q1

Solving two equations q1=110-0.5q2 and q2=109-0.5q1 we get q1= 74 and q2= 72.

So for given fee and market demand, both IMT and IIM should have 74 and 72 seats respectively to  make maximum profit.