Question

An amusement park has estimated the following demand equation for the average park guest

Q=16?2P

Where Q represents the number of rides per guest and P the price per ride. The total cost of providing rides to a guest is

TC=2+0.5Q

If a one-price policy is used, how much should it charge per ride if the park wishes to maximize its profit?

What is the park's profit for each guest?

If a two-part tariff policy is used, what admission fee should the park charge to maximize its profit?

What is the park's profit for each guest?

Answer 1

Part-a)

The demand equation for average park guest is given as: Q=16-2P where Q represents the number of rides per guest and P denotes the price per ride.

We can rewrite the demand equation as following:-

Q=16-2P

Q-16=-2P

8-Q/2=P

Total Revenue(TR)=PQ=(Q/2+8)Q=8Q-Q^2/2

Marginal Revenue(MR)=dTR/dQ=8-Q

Total Cost(TC)=2+0.5Q

Marginal Cost(MC)=dTC/dQ=0.5

Based on the profit maximizing condition of the monopoly,we can state:-

MR=MC

8-Q=0.5

-Q=-8+0.5

-Q=-7.5

Q=7.5

Now,plugging the value of Q obtained above into the demand equation,we get:-

Q=16-2P

7.5=16-2P

7.5-16=-2P

-8.5=-2P

4.25=P

Thus,the price charged per ride by the amusement park is 4.25

Marginal Profit(MP) or profit per guest=P-MC

Now plugging the value of P into the MP function,we obtain:-

P-MC

=4.25-0.5

=3.75

Hence,the profit per guest earned by the park would be 3.75

Part-2)

Now,if we equate the price P and marginal cost MC for the park,we get:-

P=MC

8-Q/2=0.5

-Q/2=-8+0.5

-Q/2=-7.5

-Q=-15

Q=15

Now,considering Q=0 in the demand equation,we obtain:-

Q=16-2P

0=16-2P

-16=-2P

8=P

Therefore,Consumer Surplus(CS)=0.5*(8-0.5)*(15-0)=0.5*7.5*15=56.25

Thus,the park will charge an access or membership fee of 56.25 to maximize its profit under the two part tariff system.

Now,MP or profit per guest=(56.25+MR)-MC

Using the MR and MC equations into the MP function,we obtain:-

(56.25+MR)-MC

=(56.25+8-Q)-0.5

=64.25-Q-0.5

=63.75-Q

Now,plugging the value of Q obtained in part-2) into the MP function,we derive:-

63.75-Q

=63.75-15

=48.75

Under two part tariff system the amusement park will earn a profit of 48.75 per guest.