Chapter 73: Nonlinear Pricing

Overview

• What is linear pricing?

• What is nonlinear pricing?

• What is bundling, and how can it increase profitability?

• How can I find a profit-maximizing nonlinear pricing plan?

• What is linear pricing?

• In Chapter 71, “Pricing Products by Using Tie-Ins,” and Chapter 72, “Pricing Products by Using Subjectively Determined Demand,” I showed how to determine a profit-maximizing price for a product. We made the implicit assumption, however, that no matter how many units a customer purchases, the customer is charged the same amount per unit. This model is known as linear pricing because the cost of buying x units is a straight line function of x (cost of x units=(unit price)*x). We will see in this chapter that nonlinear pricing can often greatly increase a company’s profit.

• What is nonlinear pricing?

• A nonlinear pricing scheme simply means that the cost of buying x units is not a straight line function of x. We have all encountered nonlinear pricing strategies. Here are some examples:

  • Quantity discounts.   The first five units might cost $20 each and the remaining units $12 each. Quantity discounts are commonly used by companies selling software and computers. An example of the cost of purchasing x units is shown in the Nonlinear Pricing Examples worksheet in the file Nlp.xlsx, which is shown in Figure 73-1 on the next page. Notice that the graph has a slope of 20 for 5 or fewer units purchased and a slope of 12 for more than 5 units purchased.

    
    Figure 73-1: Cost of quantity discount plan

  • Two-part tariff.   When you join a country club, you usually pay a fixed fee for joining the club and then a fee for each round of golf you play. Suppose that your country club charges a membership fee of $500 per year and charges $20 per round of golf. This type of pricing strategy is called a two-part tariff. For this pricing policy, the cost of purchasing a given number of rounds of golf is shown in Figure 73-2. Again, look at the Nonlinear Pricing Examples worksheet in Nlp.xlsx. Note that the graph has a slope of 520 from 0 through 1 units purchased and a slope of 20 for more than 1 unit purchased. Because a straight line must always have the same slope, we can see that a two-part tariff is highly nonlinear.

    
    Figure 73-2: Cost of two-part tariff

• What is bundling, and how can it increase profitability?

• Price bundling involves offering a customer a set of products for a price less than the sum of the products’ individual prices. To analyze why bundling works, we need to understand how a rational consumer makes decisions. For each product combination available, a rational consumer looks at the value of what we are selling and subtracts the cost to purchase it. This yields the consumer surplus of the purchase. A rational consumer buys nothing if the consumer surplus of each available option is negative. Otherwise, the consumer purchases the product combination having the largest consumer surplus.

  So how can bundling increase our profitability? Suppose that we sell computers and printers and have two customers. The values each customer attaches to a computer and a printer are shown here:

  Open table as spreadsheet

  Customer	Computer value	Printer value
  1	$1,000	$500
  2	$500	$1,000

• We only offer the computer and printer for sale separately. By charging $1,000 for a printer and for a computer, we will sell one printer and one computer and receive $2,000 in revenue. Now suppose we offer the printer and computer in combination for $1,500. Each customer buys both the computer and the printer, and we receive $3,000 in revenue. By bundling the computer and printer, we can extract more of the consumer’s total valuation. Bundling works best if customer valuations for the bundled products are negatively correlated. In our example, the negative correlation between the values for the bundled products is a consequence of the fact that the customer who places a high value on a printer places a low value on a computer, and the customer who places a low value on a printer places a high value on a computer.

• We know that when you go to a theme park such as Disneyland, you don’t buy a ticket for each ride. You either buy a ticket to enter the theme park or you don’t go. This is an example of pure bundling because the consumer does not have the option of paying for a subset of the offered products. This approach reduces lines (imagine a line at every ride) and also results in more profit.

• To see why this bundling approach increases profitability, suppose there is only one customer and that the number of rides the customer wants to go on is governed by a demand curve that is calculated as (Number of rides)=20–2*(Price of ride). From the discussion of demand curves in Chapter 70, “Estimating a Demand Curve,” we know that the value the consumer gives to the ith ride is the price that makes demand equal to i–0.5. Thus, we know that i–0.5=20–2*(value of ride i) or, solving for the value of ride i, we find (value of ride i)=10.25–(i/2). The first ride is worth $9.75, the second ride is worth $9.25…the twentieth ride is worth $0.25.

• Assume we charge a constant price per ride and that it costs us $2 in variable costs per ride. We seek the profit-maximizing linear pricing scheme. In the OnePrice worksheet in the file Nlp.xlsx, shown in Figure 73-3, I show how to determine the profit-maximizing price per ride.

  
  Figure 73-3: Profit-maximizing linear pricing scheme

• I’ve associated the range names in C8:C10 with cells D8:D10 I enter a trial price in cell D8 and compute the number of ride tickets purchased in cell D9 with the formula 20–(2*D8). Then I compute our profit in cell D12 with the formula Demand*(price– unit_cost). I can now use the Microsoft Office Excel Solver feature to maximize the value in D12 (profit) by changing cell D8 (price). We find that a price of $6 results in eight ride tickets being purchased. We earn a maximum profit of $32.

• Now let’s pretend we’re like Disneyland and offer only a bundle of 20 rides to the customer. We set a price equal to the sum of the customer’s valuations for each ride ($9.75+$9.25+…$0.75+$0.25=$100.00). The customer values all 20 rides at $100.00, so the customer will buy a park entry ticket for $100.00. We earn a profit of $100.00– $2.00(20)=$60.00, which almost doubles our profit from linear pricing.

• How can I find a profit-maximizing nonlinear pricing plan?

• In this section, I’ll show how you can determine a profit-maximizing, two-part-tariff pricing plan for our amusement park example. We’ll proceed as follows:

  • Hypothesize trial values for the fixed fee and the price per ride.

  • Determine the value the customer associates with each ride (Value of ride i= 10.5–0.5i).

  • Determine the cumulative value associated with buying i rides.

  • Determine the price charged for i rides, Fixed fee + i*(price per ride).

  • Determine the consumer surplus for buying i rides, Value of i rides–price of i rides.

  • Determine the maximum consumer surplus.

  • Determine the number of units purchased. If the maximum consumer surplus is negative, no units are purchased. Otherwise, we’ll use the MATCH function to find the number of units yielding the maximum surplus.

  • Use a VLOOKUP function to look up our revenue corresponding to the number of units purchased.

  • Compute our profit as revenue–costs.

  • Use a two-way data table to determine a profit-maximizing fixed fee and price per ride.

• Our work is in the Two-Part Tariff worksheet in the file Nlp.xlsx, which is shown in Figure 73-4.

  
  Figure 73-4: Determination of optimal two-part tariff

• To begin, I named call F2.Fixed and call F3 LP. I entered trial values for the fixed fee and the price per ride in cells F2 and F3. Next, I determine the value the consumer places on each ride by copying from cell E6 to E7:E25 the formula 10.25–(D6/2). We find that the customer places a value of $9.75 on the first ride, $9.25 on the second ride, and so on.

• To compute the cumulative value of the first i rides, I copy from F6 to F7:F25 the formula SUM($E$6:E6). This formula adds up all values in column E that are in or above the current row. By copying from G6 to G7:G25 the formula fixed_fee+price_per_ride*D6, I compute the cost of i rides. For example, the cost of five rides is $68.50.

• Recall that the consumer surplus for i rides equals (Value of i rides)–(Cost of i rides). By copying from cell H6 to the range H7:H25 the formula F6–G6, I compute the consumer’s surplus for purchasing any number of rides. For example, the consumer surplus for purchasing five rides is –$24.75, which is the result of the large fixed fee.

• In cell H4, I compute the maximum consumer surplus with the formula MAX(H6:H25). Remember that if the maximum consumer surplus is negative, no units are purchased. Otherwise, the consumer will purchase the number of units yielding the maximum consumer surplus. Therefore, entering in cell I1 the formula IF(H4>=0,MATCH(H4,H6: H25,0),0) determines the number of units purchased (in our case, 15). Notice that the MATCH function finds the number of rows we need to move down in the range H6:H24 to find the first match to the maximum surplus.

• We now name the range D5:G25 as Lookup. We can then look up our total revenue in the fourth column of this range based on the number of units purchased (which is already computed in cell I1). Our total revenue is computed in cell I2 with the formula IF(I1=0,0, VLOOKUP(I1,lookup,4)). Notice that if no rides are purchased, we earn no revenue. We compute our total production cost for rides purchased in cell I3 with the formula I1*C3. In cell J6, I compute our profit as revenues less costs with the formula I2–I3.

• Now we can use a two-way data table to determine the profit-maximizing combination of fixed fee and price per ride. The data table is shown in Figure 73-5 on the next page. (Many rows and columns are hidden.) In setting up the data table, we vary the fixed fee between $10.00 and $60.00 (the values in the range K10:K60) and vary the price per ride between $0.50 and $5.00 (the values in L9:BE9). We recompute profit in cell K9 with the formula =J6.

  
  Figure 73-5: Two-way data table computes optimal two-part tariff

• We now select the table range (cells K9:BE60), and then on the Data tab of the Ribbon, in the Data Tools group, click What-If Analysis, and then select Data Table. Our column input cell is F2 (the fixed fee) and row input cell is F3 (the price per ride). After clicking OK in the Table dialog box, we’ve computed our profit for each fixed fee and price per ride combination represented in the data table.

• To highlight the profit-maximizing two-part tariff, I used conditional formatting, selecting the range L10:BE60. We choose Conditional Formatting from the Home tab of the Ribbon, click Top/Bottom Rules, and then click Top 10 Items. Then change the 10 in the dialog box to a 1, so only the largest profit is formatted. We find that a fixed fee of $56.00 and a price per ride of $2.50 earns us a profit of $63.50, which almost doubles the profit from linear pricing. A fixed fee of $59.00 and a price per ride of $2.30 also yields a profit of $63.50.

• Because a quantity discount plan involves selecting three variables (cutoff, high price, and low price), we cannot use a data table to determine a profit-maximizing, quantity-discount plan. You might think we could use a Solver model (with changing cells set to cutoff, high price, and low price) to determine a profit-maximizing, quantity-discount strategy. Unfortunately, Solver often has difficulty determining optimal solutions when the target cell is computed by using formulas containing IF statements. Therefore, Solver might fail to find a profit-maximizing, quantity-discount plan. For details about how to set up Solver models involving IF statements, read Chapter 15 of my book Introduction to Mathematical Programming (Duxbury Press, 2003).