1. Nonlinear Pricing Models
Example 8.2
Nonlinear Pricing Models

2. Background Information
Suppose we sell Menthos candy. Most people value the first pack of Menthos they purchase more than the second pack. They also value the second pack more than the third pack, and so on.
How can we take advantage of this when pricing Menthos?
If we charge a single price for each pack of Menthos, then few people are going to buy more than one or two packs.
Alternatively, however, we can try the two-part tariff approach.

3. Background Information -- continued
Here we charge an “entry fee” to anyone who buys Menthos, plus a reduced price per pack purchased.
For example, a reasonable single price per pack is $1.10, then a reasonable two-part tariff might be an entry fee of $1.50 and a price of $.50 per pack.
This will give some customers an incentive to purchase many packs of Menthos. Because the total cost of purchasing n packs is no longer a linear function of n – it is now “piecewise linear” – we refer to the two-part tariff as a nonlinear pricing strategy.

4. Background Information -- continued
As usual with pricing models, the key input is customer sensitivity to price. Rather than having a single demand function, however, we now assume that each customer has his or her unique sensitivity to price.
To keep the example fairly small, we will assume that four typical customers from the four market segments for the product have been asked what they would pay for each successive pack of Menthos, the results are shown on the next slide.

6. Background Information -- continued
For example, customer 1 is willing to pay $1.24 for the first pack of Menthos, $1.03 for the second pack, and only $0.35 for the tenth pack.
These four customers are considered representative of the market.
If it costs $0.40 to produce a pack of Menthos, determine a profit-maximizing single price and a profit-maximizing two-part tariff.
Assume that the four market segments have 10,000, 5000, 7500, and 15,000 customers, respectively, and that the customers within a market segment all respond identically to price.

7. MENTHOS1.XLS
The completed single-price model appears on the next slide.
This file contains the spreadsheet model.

9. Developing the Single-Price Model
To develop this model, proceed as follows.
Inputs. Enter the inputs in the shaded ranges. Note that the large shaded range is just the price sensitivity table shown before.
Price. The only decision variable in this model is the single price charged for every pack of Menthos sold. Enter any value in this Price cell.
Total value table. The values in the shaded price sensitivity range are marginal values, the most each customer would pay for the next pack of Menthos. In the range H6:K15 calculate the total value of n packs to each customer (for n from 1 to 10).

10. Developing the Single-Price Model -- continued
First, enter the formula =B6 in cell H6 and copy it across row 6. Then enter the formula =H6+B7 in cell H77 and copy it to the range H7:K15.
Total cost column. Using the single-price scheme, each customer must pay np for n packs if the price is p. Calculate these amounts in the range E19:E28 by entering the formula =UnitPrice*D19 in cell E19 and copying it down.
Surplus table. This is the key to the model. We define the “surplus” for any customer from buying n packs as the total value of n packs minus the total cost of n packs, and we assume that the customer will buy the number of packs with the largest surplus.

11. Developing the Single-Price Model -- continued
This makes sense economically. If a customer places more value on n packs than it costs to buy n packs, then presumably the customer will consider purchasing n packs. But a customer will not purchase n packs if they cost more than the customer values them. To calculate these surpluses, enter the formula =H6-$E19 in cell H19 and copy it to the range H19:K28.
Maximum surplus. Calculate the maximum surplus for each customer by entering the formula =MAX(H19:H28) in cell B32 and copying it across row 32.

12. Developing the Single-Price Model -- continued
Packs purchased. For each customer we need to find the number of packs that corresponds to the maximum surplus. This can be done easily with Excel’s MATCH function. Specifically, enter the formula =IF(B32<0,0,MATCH(B32,H19:H28,0)) in cell B33 and copy it across row 33. This formula says that if the maximum surplus is negative, the customer will not purchase any packs at all. Otherwise, it matches the maximum surplus to the entries in the range H19:H28 and returns the index of the cell where the match occurs. In this example, the match for customer 1 occurs in the 4th cell of the range H19:H28, so that the MATCH function returns 4. Note that the last argument of the MATCH function is 0 if we want an exact match, as we do here.

13. Developing the Single-Price Model -- continued
Then calculate the total number of packs purchased by all customers with the formula =SUMPRODUCT(MktSize,Npurch) in the TotPurch cell.
Profit. Calculate the profit in the Profit cell with the formula =(UnitPrice-UnitCost)*TotPurch

14. Using the Evolutionary Solver
First, note that the standard Solver will have trouble with this model because of the IF and MATCH functions.
However, these present no difficulties to the Evolutionary Solver.
We set it up as shown on the next slide, using the same values for the various options as in the previous example.
Note that we use an upper limit of $1.50 for the unit price. This suffices because the most any customer will pay for any pack of Menthos is $1.49.

16. Using the Evolutionary Solver -- continued
Again, the Solver converges to the solution shown earlier quickly and then tries for a long time – unsuccessfully – to find a better solution.
We can be fairly certain that this solution is optimal, but this is not guaranteed. The single price of $0.80 produces a profit of $62,000.
It strikes the best balance for these four market segments. A lower price needlessly sacrifices revenue, whereas a higher price would cause at least one market segment to buy fewer packs.

17. MENTHOS2.XLS
The two-part tariff model is so similar that we made a copy of the Menthos1.xls spreadsheet file and then made the following modifications.
The completed single-price model appears on the next slide.
This file contains the spreadsheet model.

19. Developing the Two-Part Tariff Model
The steps that are the same as before are omitted.
Decision variables. Now there are two decision variables – the fixed entry fee and the variable cost per pack. Enter any values for these in the Fixed and Variable cells.
Total cost column. The total cost of purchasing n packs is now the fixed entry fee plus the variable cost times n. Calculate this in the range E19:E28 by entering the formula =Fixed+Variable*D19 in cell E19 and copying it to the rest of the range.

20. Developing the Two-Part Tariff Model -- continued
Revenues. Calculate the amount paid by the customers in row 34 by entering the formula =IF(B33>0,Fixed+Variable*B33,0) in cell B34 and copying it across. Note that the entry fee is evidently too high for customer 2, so she does not purchase any packs, and there is no corresponding revenue.
Profit. Calculate the profit in the Profit cell with the formula =SUMPRODUCT(Revenues, MktSize)-UnitCost*TotPurch

21. Using the Evolutionary Solver
The Evolutionary Solver setup is almost the same as before.
However, we now select both the Fixed and Variable cells as changing cells, and we put upper limits on each of them (We used $10 as an upper limit on Fixed and $1.50 for Variable, reasoning that these would almost certainly be large enough.
The solution shown earlier was found after a few seconds.

22. Using the Evolutionary Solver -- continued
It indicates that the company should charge all customers $3.39 plus $0.39 for each pack purchased.
This pricing scheme is too high for the second market segment, which doesn't buy any packs, but it entices segments 1, 3, and 4 to purchase many more packs than they purchased with the single price of $0.80.

23. Using the Evolutionary Solver -- continued
More important it yields a profit of $107,376, about 73% more than the profit from the single-price policy.
The moral is clear – clever pricing schemes can make companies significantly larger profits than the simple pricing schemes we are accustomed to.