Chapter 7 of Erik Haugom’s Essentials of Pricing Analytics (Price Optimization) covers the following topics:
- Basic price optimization.
- Price optimization with capacity constraints.
- Optimal price differentiation with capacity constraints.
- Optimal time-based price differentiation.
- Elasticity and optimization.
- Pricing with competition.
As always, at the end of the chapters are Exercises, and for the purpose of this blog post I will post my solution to Exercise 07.
1 Case Study
A newly established hotel wants to make it simple for its customers by offering only two rates: one weekend rate and a midweek rate. The variable costs associated with making each room ready for new guests are estimated at 10 USD during midweek days (Monday–Thursday) and 18 USD for weekend days (Friday–Sunday). The midweek days and weekend days have the following price–response functions (PRF):
Midweek PRF:
\[ d(p) = 1200 - 8p\]
Weekend PRF:
\[ d(p) = 1200 - 12p\]
The hotel has a capacity of 500 rooms in both the midweek period and the weekend period. What is the profit-maximizing single price? What are the profit-maximizing variable prices? What is the percentage increase in total profit from introducing variable pricing?
2 Solution
Profit Model of Midweek Customers:
\[ \pi_{midweek} = d(p_{midweek}) \times (p_{midweek} - vc_{midweek}) \]
Profit Model of Weekend Customers:
\[ \pi_{weekend} = d(p_{weekend}) \times (p_{weekend} - vc_{weekend}) \]
Both models are subject to the same constraint:
\[ d(p) \leq 500 \]
The results are below:
| PriceMid | PriceWeek | DemandMid | DemandWeek | ProfitMid | ProfitWeek | ProfitTotal | |
|---|---|---|---|---|---|---|---|
| SamePrice | 87.5688 | 87.5688 | 499.45 | 149.175 | 38741.7 | 10377.9 | 49119.6 |
| VarPrice | 87.5688 | 58.9795 | 499.45 | 492.246 | 38741.7 | 20172 | 58913.7 |
| RelDiff | 0 | -0.326478 | 0 | 2.2998 | 0 | 0.943746 | 0.199393 |
| AnalyticalSolution | 87.5 | 58.3333 | 500 | 500 | 38750 | 20166.7 | 58916.7 |
So, if we charge 87.5688 to both customer groups, we can expect profit of 49.199,60 USD. Unfortunately, that price will mean that most of the hotel’s rooms will be empty on the weekend.
On the other hand, if we charge 87.5688 to midweek customers and 58.9795 to weekend customers, we can expect profit bump of almost 20%.
The SamePrice and VarPrice scenarios were found on brute force, but it was not necessary. This problem has an analytical solution, through which we charge 87.50 USD to Midweek Customers and 58.3333 to Weekend Customers.
As the saying goes, picture is worth a thousand words:
Due to the capacity constraint the hotel has, only the first quadrant of the graph has feasible solutions. Why? Well, if the \(p_{midweek} \le 87.5\), \(d(p_{midweek}) \gt 500\) and we will have unsatisfied customers (and probably loss of goodwill). On the other hand, if \(p_{week} \le 58.33\), \(d(p_{week}) \gt 500\), and we have the same problem on weekends.
This problem, though solved, poses another challenge, which is presented in the next exercise:
It turns out that introducing lower prices for the weekend induces some customers with great flexibility (such as retirees) to move some of their demand from the higher-priced midweek days to the lower-priced weekend days. An analysis shows that ten customers are likely to shift from midweek to weekend for every dollar in price difference between the two periods. What are the optimal variable prices now?
The technical details are left to the reader of this blog, but I’ve got the following solutions:
- PriceMid: 87.57 USD
- PriceWeek: 73.84 USD
- DemandMid: 499.45 rooms
- DemandWeek: 313.96 rooms
- ProfitDS: 53,287.91 USD
Basically, we should increase the PriceWeek of VarPrice scenario by 25.20%. Demand of Weekend Customers will drop, but the price offset of PriceWeek should be more than enough to keep our profits on high levels (53.2k vs. 58.9k that VarPrice gave us). Updated plot looks like this:
