Question

1. Let us assume that there are two visitors, A and B, in an amusement park. The demand curve for the visitors facing the amusement park are as follows.

P_A= 5 – 2Q_A

P_B= 2.5 – 0.5Q_B

Marginal cost (MC) to serve each visitor is equal to $1.

a. If the amusement park decides to set the price using two-part tariff, given the demand curve

P=6 – 2.5Q and MC =$1, how much is the equilibrium P and Q

b. Calculate the maximum upfront fee the park could charge each visitor

Answer 1

a.

Since there is two-part pricing, the equilibrium would be as below:

P = MC

6 – 2.5Q = 1

2.5Q = 6 – 1

Q = 5/2.5 = 2

By putting this value in the demand function,

P = 6 – 2.5Q

   = 6 – 2.5 × 2

   = 6 – 5

   = 1

Answer: P = $1; Q = 2 units

b.

Now it is out of individual comparison.

PA = 5 – 2QA

TR = PA × Q = 5Q – 2QA^2

MR = Derivative of TR with respect to QA

       = 5 – 4QA

Equilibrium is,

MR = MC

5 – 4QA = 1

4QA = 4

QA = Q = 1

By putting this value in the price function

P = 5 – 2QA

   = 5 – 2 × 1

   = 5 – 2

   = $3

Again,

PB = 2.5 – 0.5QB

TR = PA × Q = 2.5Q – 0.5QB^2

MR = Derivative of TR with respect to QB

       = 2.5 – 1QB

Equilibrium is,

MR = MC

2.5 – 1QB = 1

QB = Q = 1.5

By putting this value in the price function

P = 2.5 – 0.5QB

   = 2.5 – 0.5 × 1.5

   = 2.5 – 0.75

   = $1.75

The lower price between $3 and $1.75 should be considered because of keeping both the customers.

Answer: The maximum fee is $1.75.