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.

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.

Answer 1

Solution:

We have the following inverse demand curves for the two visitors, A and B:

PA = 5 - 2*QA so, the demand curve for A is: QA = 2.5 - 0.5*PA

PB = 2.5 - 0.5*QB so, the demand curve for B is: QB = 5 - 2*PB

So, the market demand curve becomes: Q = QA + QB

Q = (2.5 - 0.5*P) + (5 - 2*P)

Q = 7.5 - 2.5*P

Or P = 3 - 0.4*Q (so, the market inverse demand curve derived above is wrong)

Under two-part tariff, same access fee is charged from both, and additionally per unit price is charged (or in this case, per ride price). The common access fee equals the (lower) consumer surplus among the two consumers (lower, because otherwise one consumer shall not participate as access fee will exceed the consumer surplus, and monopolist will never charge below the marginal cost)

With Price, P = MC,

QA = 2.5 - 0.5*1 = 2

So, for consumer A, consumer surplus = (1/2)*(vertical intercept - MC)*Qa

= (1/2)*(5 - 1)*2 = $4

And QB = 5 - 2*1 = 3

Consumer surplus for consumer B = (1/2)*(vertical intercept - MC)*QB

= (1/2)*(2.5 - 1)*3 = $2.25

So, the access fee will equal the consumer surplus of consumer B. Now, optimizing for the maximum profit by finding the equilibrium price.

Profit, M = 2*(consumer surplus of consumer B) + (P - MC)*(QA + QB)

Consumer surplus is multiplied by 2 as that is the access fee charged from both consumers. Rest part: P is the per ride price charged and subtracting marginal cost for the profit.

So,

M = 2((1/2)*(2.5 - P)*(5 - 2P)) + (P - 1)*(7.5 - 2.5*P)

M = 12.5 - 5P - 5P + 2P² + 7.5P - 2.5P² - 7.5 + 2.5P²

M = 4.5P² + 5

Now, optimizing using the first order condition: = 0

= 2*4.5P = 9P

Using the FOC, then 9P = 0, so P = 0

So, equilibrium price, P = 0, and equilibrium Q = 7.5 - 2.5*0 = 7.5

This seems a weird answer (as then access fee = 6.25, which exceeds the consumer surplus of both consumers and so none of them would participate in the market)

Another option is to charge the access fee equaling consumer surplus of consumer A, but then consumer B will never purchase, and maximum profit = access fee from A = consumer surplus of A = 4

Profit can still be increased, next option is to charge access fee equaling the consumer surplus of consumer B, so access fee = $2.25, meaning from two consumers (as none leaves the market in this case), excess or entry fee = 2*2.25 = $4.5. Equilibrium price charged equals marginal cost = $1.

So, profit = $4.5

equilibrium quantity, Q = 7.5 - 2.5*1 = 5 (as already seen, at P = MC, consumer A takes 2 rides, and consumer B takes 3 rides)