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6 Nonlinear pricing is used in a situation in which the cost of purchasing q units is not equal to the unit cost c per unit times q. Bundling (discussed in Chapter 5, Price Bundling) is a special type of nonlinear pricing because the price of three items is not equal to the sum of the individual prices.
Other common examples of nonlinear pricing strategies include:
? Quantity discounts: If customers buy ?CUT units, they pay high price (HP) per unit, and if they buy >CUT units, they pay low price (LP) per unit. CUT is simply the cuto! point at which the charged price changes. For example, if customers buy ?1000 units, you charge $10 a unit, but if they buy more than 1000 units, you charge $8 per unit for all units bought. This form of nonlinear pricing is called the nonstandard quantity discount. Another type of quantity discount strategy is as follows: Charge HP for the fi rst CUT units bought, and charge LP for remaining units bought. For example, you charge $10 per unit for fi rst 1000 units and $8 per unit for remaining units bought. This form of nonlinear pricing is called the standard quantity discount. In both examples the value of CUT = 1000.
? Two-part tari! : The cost of buying q units is a fi xed charge K plus $c per unit purchased. For example, it may cost $500 to join a golf club and $30 per round of golf.
Many companies use quantity discounts and two-part tari! s. Microsoft does not charge twice as much for 200 units, for example, as for 100 units. Supermarkets charge less per ounce for a 2-pound jar of peanut butter as for a 1-pound jar of peanut butter. Golf courses often use a two-part tari! by charging an annual membership fee and a charge for each round of golf.
Just as in Chapter 5 you used the Evolutionary Solver to fi nd optimal price bun- dling strategies, you can use the Evolutionary Solver to fi nd the profi t maximizing parameters of a nonlinear pricing strategy. As in Chapter 5, you can assume the
Nonlinear Pricing
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consumer will choose an option giving her the maximum (non-negative) consumer surplus. You can see that nonlinear pricing can often signifi cantly increase your profi ts, seemingly creating profi ts out of nothing at all.
In this chapter you will fi rst learn how a consumers demand curve yields the consumers willingness to pay for each unit of a product. Using this information you will learn how to use the Evolutionary Solver to determine profi t or revenue maximizing nonlinear pricing strategies.
Demand Curves and Willingness to Pay A demand curve tells you for each possible price how many units a customer is willing to buy. A consumers willingness to pay curve is defi ned as the maximum amount a customer is willing to pay for each unit of the product. In this section you will learn how to extract the willingness to pay curve from a demand curve.
Suppose you want to sell a software program to a Fortune 500 company. Let q equal the number of copies of the program the company demands, and let p equal the price charged for the software. Suppose you estimated that the demand curve for software is given by q = 400 p. Clearly, your customer is willing to pay less for each additional unit of your software program. Locked inside this demand curve is information about how much the company is willing to pay for each unit of your program. This information is crucial to maximize profi tability of sales.
Now rewrite your demand curve as p = 400 q. Thus, when q = 1, p = $399, and so on. Now try to fi gure out the value your customer attaches to each of the fi rst two units of your program. Assuming that the customer is rational, the customer will buy a unit if and only if the value of the unit exceeds your price. At a price of $400, demand equals 0, so the fi rst unit cannot be worth $400. At a price of $399, however, demand equals 1 unit. Therefore, the fi rst unit must be worth somewhere between $399 and $400. Similarly, at a price of $399, the customer does not purchase the second unit. At a price of $398, however, the customer is purchasing two units, so the customer does purchase the second unit. Therefore, the customer values the second unit somewhere between $399 and $398. The customers willingness to pay for a unit of a product is often referred to as the units reservation price.
It can be shown that a good approximation to the value of the ith unit purchased by the customer is the price that makes demand equal to i 0.5. For example, by setting q equal to 0.5, you can fi nd that the value of the fi rst unit is 400 0.5 = $399.50. Similarly, by setting q = 1.5, you can fi nd that the value of the second unit is 400 1.5 = $398.50.
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Suppose the demand curve can be written as p = D(q). In your example this looks like D(q) = 400 q. The reader who knows integral calculus can exactly determine the value a consumer places on the fi rst n items by computing ?
n
0 D(q)dq. In your example
you would fi nd the value of the fi rst two units to be the following:
? 2
0 (400-q)dq = [400q .5q2]20 = 800 2 = 798
This agrees with your approximate method that yields a value of 399.5 + 398.5 = 798 for the fi rst two units.
Profi t Maximizing with Nonlinear Pricing Strategies Throughout this chapter assume a power company (Atlantis Power and Light, APL for short) wants to determine how to maximize the profit earned from a customer whose demand in kilowatt hours (kwh) for power is given by q = 20 2p. It costs $2 to produce a unit of power. Your analysis begins by assuming APL will use linear pricing; that is, charging the same price for each unit sold. You will fi nd that with linear pricing the maximum profi t that can be obtained is $32. Then you will fi nd the surprising result that proper use of quantity discounts or a two-part tari! doubles APLs profi t! The work for this chapter is in the fi le Powerblockprice.xls. To determine the profi t maximizing linear pricing rule, you simply want to maximize (20 2p) × (p 2). In the oneprice worksheet from the Powerblockprice.xls fi le, a price of $6 yields a maximum profi t of $32 (see Figure 6-1). The Solver model simply chooses a non-negative price (changing the cell that maximizes profi t [Cell D12]). Charging $6 per kwh yields a maximum profi t of $32.00.
Figure 6-1: Finding the profi t maximizing single price strategy
Optimizing the Standard Quantity Discount To determine a profit maximizing pricing strategy that uses the standard quantity discount, assume the quantity discount pricing policy is defined as
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follows: All units up to a value CUT are sold at a high price (HP). Recall that CUT is simply the cuto! point at which the per unit price is lowered. All other units sell at a lower price (LP). Assuming the customer chooses the number of kwh with the highest non-negative surplus, you can use the Evolutionary Solver to determine profi t maximizing values of CUT, HP, and LP. The work for this task is shown in sheet qd of file Powerblockprice.xls. Also, Figure 6-2 shows that the demand curve may be written as p = 10 (q/2), so the first unit is valued at 10 (.5/2) = $9.75, the second unit is valued at 10 (1.5/2) = $9.25, and so on.
Figure 6-2: Finding the profi t maximizing standard quantity discount strategy
To complete the determination of the profi t maximizing standard quantity dis- count strategy, complete the following steps:
1. Copy the formula =10-0.5*C6 from E6 to E7:E25 to determine the value of each unit.
2. In column F compute the cumulative value associated with buying 1, 2, 19 units. In F6 compute the value of the fi rst unit with formula =E6. Copy the formula =F6+E7 from F7 to F8:F25 to compute the cumulative value of buying 2, 3, 20 units.Complete questions 1 and 2 on page 132 using excel. Please see the attachments.
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