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Price as a function of Quantity

A park has a capacity to hold 60,000 people on any particular day.

Using third degree price discrimination - demand for 2 groups (boys & girls)
Q /6000 + P = 8 Q / 6000 + P = 6
b b g g

The hint provided is: Marginal Revenue in the two markets must be equal

How do I figure out what to charge for admission to the boys and girls respectively?

Solution Preview

Let's first set each demand as Price as a function of Quantity:

For boys (let's call Qb and Pb to their quantity and price)
Qb/6000 + Pb = 8
Pb = 8 - Qb/6000

For girls:
Qg/6000 + Pg = 6
Pg = 6 - Qg/6000

Total profits from the sales of tickets are thus:
Pb*Qb + Pg*Qg = (8 - Qb/6000)*Qb + (6 - Qg/6000)*Qg =
= 8Qb - (Qb^2)/6000 + 6Qg - (Qg^2)/6000
[the ^ symbol means "to the power of"]

The problem we have to solve is:

Max 8Qb - (Qb^2)/6000 + 6Qg - (Qg^2)/6000
Qb, Qg

subject to
Qb + Qg = 60,000

That is: choose how many tickets you want to sell to ...

Solution Summary

Price as a function of Quantity is utilized in this case are determined. The marginal revenues in the two markets which must be equal are discussed.