This one comes from Chapter 3 of Erik Haugom’s Essentials of Pricing Analytics:
1 Case study
An amusement park will introduce a new pricing scheme before the next season. It wants to set an entrance fee and a fee for each ride in the park. A market survey shows that the price–response function for the entrance part can be described by the following function:
\[ d(E) = 5,000 - 100E \]
where E is the entrance fee. The price–response function for each ride can be described by:
\[ d(p) = 5,000 - 500p \]
where p is the price per ride. The variable unit costs are estimated at 2.00 USD per entrance and 1.50 USD per ride. What is the optimal (profit-maximizing) two-part tariff in this case? You can solve the problem any way you want, but the objective function that shall be maximized is:
\[ \underset{E, p}{maxZ} = E \times d(E) + (p - c_{v}) \times d(p) - c(f) \]
2 Solution
Unfortunately, the author did not define fixed costs, but let’s set them to 5,000.
Problem will be solved through Python, using following functions:
def demand_entrance(E):
return 5000 - 100 * E
def demand_rides(p):
return 5000 - 500 * p
def vc_entrance():
return 2.00
def vc_ride():
return 1.50
def profit_entrance(E):
return (E - vc_entrance()) * demand_entrance(E)
def profit_ride(p):
return (p - vc_ride()) * demand_rides(p)
def profit_total(E, p, fc=5000):
total = profit_entrance(E) + profit_ride(p)
return total - fcAfter some trials and errors, the solutions can be plotted in 3D space:
So, the pricing levels for the entry to the amusement park should be 25.97 USD, while fee per ride should be 5.73 USD. This should bring us profit of 61.6k USD.
I don’t prefer 3D plots: searching for Z variable (profit) given X (Entrance Fee) and Y (Ride Fee) variable takes some time. Two-dimensional plot seems to solve that problem:
On this plot there is a black line which represents all combinations of entrance fee and ride fee where the total profit is zero. Do note that line gets thicker when Ride Fee is greater than 16.87 since for the given level of Ride Fee there exists two Entrance Fee levels that give profit of zero.