Wednesday, January 02, 2013

Amusement Park Rides

The Alzheimer's problem of the day.


Col. Tom Barker is about to open his newest amusement park,
Elvis World. Elvis World features a number of exciting attractions: you
can ride the rapids in the Blue Suede Chutes, climb the Jailhouse Rock
and eat dinner in the Heartburn Hotel. Col. Tom  argues that Elvis World
will attract 1,000 people per day, and each person will take x = 50 50p
rides, where p is the price of a ride. Everyone who visits Elvis World is
pretty much the same and negative rides are not allowed. The marginal
cost of a ride is essentially zero.

(a) What is each person's inverse demand function for rides? 
 (b) If Col. Tom sets the price to maximize profit, how many rides will be
taken per day by a typical visitor? 
 (c) What will the price of a ride be? 
 (d) What will Col. Tom's profits be per person? 
 (e) What is the Pareto efficient price of a ride? 
 (f) If Col. Tom charged the Pareto efficient price for a ride, how many
rides would be purchased? 
 (g) How much consumers' surplus would be generated at this price and
quantity? 
 (h) If Col. Tom decided to use a two-part tariff, he would set an admission
fee of ____  and charge a price per ride of ____?

a.  The inverse demand function puts P, price, on the left side of the equals sign.  Thus, P = 1 - x/50
b.  Find the profit function as total revenue minus total cost then find profit max at the point where marginal revenue equals marginal cost.  TR = (1-x/50)x; 1x-x^2/50; 1-2X/50 = 0; 50-2X=0; X = 25
C.  1 - X/50 = 1-25/50 = 1-.50 OR .50.
d.  Each person will ride 25 rides at 50 cents so profit will be 12.5.
e.  The Pareto efficient price will be where no one can be made better off.  This happens at a price of Zero.
f.  He would sell 50 rides per customer.
g.  take one-half base times height. (.50 x 50)/2 =12.5
h.  Colonel Tom would have to capture all of the consumers surplus. So he would charge $12.5 to enter and charge $0 per ride.

If you find errors, please comment.

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