Question

Suppose you are the manager of a theatre company. You have identified two groups of customers. Group 1 has a demand given by Q 1 = 100 - P and Group 2 has a demand given by Q 2 = 120 -3P. You are currently charging the same price - 40 euros - to both groups. To maximize revenue, you should charge a price of

(a) 12.50 euros to group 1 and 20 euros to group 2

(b) 50 euros to group 1 and 20 euros to group 2

(c) 30 euros to group 1 and 20 euros to group 2

(d) 25 euros to group 1 and 60 euros to group 2

Answer 1

The demand for group 1 is

Q1 = 100 - P1

or, P1 = 100 - Q1

Revenue from group 1 is

R1 = P1.Q1 = (100 - Q1).Q1

or, R1 = 100.Q1 - Q1^{​​​​​2}​​​​.........(1)

Now, demand for grouo 2 is

Q2 = 120 - 3.P2

or, P2 = 40 - (1/3).Q2

Revenue from group 2 is

R2 = P2.Q2 = (40 - Q2/3).Q2

or, R2 = 40.Q2 - (1/3).Q2^{​​​​​2}​​​​..........(2)

Hence, total revenue is

R = R1 + R2

or, R = 100.Q1 - Q1^{​​​​​2} + 40.Q2 - (1/3).Q2^{​​​​​2}

If, revenue is maximized then,

dR/dQ1 = 0

or, 100 - 2.Q1 = 0

or, Q1* = 50

Also, dR/dQ2 = 0

or, 40 - (2/3).Q2 = 0

or, Q2* = 60

Putting Q1* and Q2* in the corresponding demand curves we get,

P1* = 100 - Q1* = 100 - 50

or, P1* = 50 euros

And, P2* = 40 - (1/3).Q2* = 40 - (1/3)×60

or, P2* = 20 euros

Hence, the company should charge 50 euros to group 1 and 20 euros to group 2.

We eliminate options (a), (c) and (d) as these fo not match our result.

Answer is option (b).

Hope the solution is clear to you my friend.