Marketing Analytics: Data-Driven Techniques with Microsoft Excel (2014)

Part II. Pricing

Chapter 7. Price Skimming and Sales

To solve many interesting pricing problems, you can use the assumption that consumers choose the option that gives them the maximum consumer surplus (if it is non-negative) as discussed in Chapter 5, “Price Bundling,” and Chapter 6, “Nonlinear Pricing.” This chapter uses this assumption and the Evolutionary Solver to explain two pricing strategies that are observed often:

·        Why do prices of high tech products usually drop over time?

·        Why do many stores have sales or price promotions?

Dropping Prices Over Time

The prices of most high-tech products as well as other products such as new fashion styles tend to drop over time. You might be too young to remember that when VCRs were first introduced, they sold for more than $1,000. Then the price of VCRs quickly dropped. There are a few reasons for this behavior in prices, and this section examines three of them.

Learning Curve

The most common reason that prices of products—high-tech items in particular—drop over time is due to a learning or experience curve. As first observed by T.P. Wright in 1936 during his study of the costs of producing airplanes, it is often the case that the unit cost to produce a product follows alearning curve. Suppose y = unit cost to produce the xth unit of a product. In many situations y=ax-b where a>0 and b>0. If the unit cost of a product follows this equation, the unit cost of production follows a learning curve. If it can easily be shown (see Exercise 3) that if a product's cost follows a learning curve, then whenever cumulative product doubles, unit cost drops by the same percentage (1 – 2-b, see Exercise 1). In most observed cases, costs drop between 10 percent and 30 percent when cumulative production doubles. If unit costs follow a learning curve, costs drop as more units are sold. Passing this drop in costs on to the consumer results in prices dropping over time. The learning curve also gives an incentive to drop prices and increase capacity so your company sells more, increases your cost advantage over competitors, and perhaps puts them out of business. This strategy was popularized in the 1970s by Bruce Henderson of the Boston Consulting Group. Texas Instruments followed this strategy with pocket calculators during the 1980s.


Aside from a learning curve causing these types of reductions, prices also drop over time as competitors enter the market; this increases supply that puts a downward pressure on prices.

Price Skimming

A third reason why prices of products drop over time is price skimming. When a new product comes out, everyone in the market places a different value on the product. If a company starts with a low price, it foregoes the opportunity to have the high-valuation customers pay their perceived product value. As time passes, the high-valuation customers leave the market, and the company must lower the price to sell to the remaining lower valuation customers.

The following example shows how price skimming works. Suppose there are 100 people who might be interested in buying a product. One person values the product at $1; one person values the product at $2; one person values the product at $100. At the beginning of each of the next 10 years, you set a price for the product. Anyone who values the product at an amount at least what you charge buys the product. The work in the Skim.xls file shows the pricing strategy that maximizes your total revenue over the next 10 years. The following steps walk you through the maximization.

1. Enter trial prices for each year in C5:C14.

2. Copy the formula =C5-1 from D5 to D6:D14 to keep track of the highest price valuation person that is left after a given year. For example, after Year 1 people valuing the product at $91 or less are left.

3. Copy the formula =D4-D5 from E5 to E6:E14 to track the unit sale for each year. For example, in Year 1 all people valuing the product at $92 or more (9 people) buy the product.

4. Copy the formula =E5*C5 from F5 to F6:F14 to compute each year's revenue.

5. In cell F15 compute the total revenue with the formula =SUM(F5:F14).

6. Use the Solver to find revenue-maximizing prices. The Solver window is shown in Figure 7.1. Choose each year's price to be an integer between $1 and $100 with a goal to maximize revenue. Figure 7.2 shows the sequence of prices shown in C5:C14.

Figure 7-1: Solver window for price skimming model


Figure 7-2: Price skimming model


You should discover that prices decline as you “skim” the high-valuation customers out of the market early. You could model prior purchasers coming into market after, say, three years, if the product wears out. You just need to track the status of the market (that is, how many people with each valuation are currently in the market) at each point in time and then the Solver adjusts the skimming strategy.

Companies often have other reasons for engaging in price skimming. For instance, a firm may engage in price skimming because it wants its products to be perceived as high quality (e.g. Apple's iPhone and iPad devices). Alternatively, the firm may engage in price skimming to artificially manipulate demand and product interest (e.g. Nintendo's Wii console.)

There are, however, downsides to a price skimming strategy:

·        The early high prices lead to high profit margins that may encourage competitors to enter the market.

·        Early high prices make it difficult to take advantage of the learning curve.

·        Early high prices reduce the speed at which the product diffuses through the market. This is because with fewer earlier purchasers there are fewer adopters to spread the word about the product. The modeling of the diffusion of a product is discussed in Chapter 27, “The Bass Diffusion Model.”

Why Have Sales?

The main idea behind retailers having sales of certain products is that different people in the market place different values on the same products. For a durable good such as an electric razor, at different points in time, there will be a different mix of people wanting to buy. When people with low product valuations predominate in the market, you should charge a low price. When people with high product valuations predominate in the market, you should charge a higher price. The following example develops a simple spreadsheet model which illustrates this idea.

Assume that all customers value an electric razor at $30, $40, or $50. Currently, there are 210 customers in the market, and an equal number value the razor at $30, $40, or $50. Each year 20 new customers with each valuation enter the market. A razor is equally likely to last for one or two years. The work in the sales worksheet of the Sales.xls workbook (see Figure 7.3) determines a pricing policy that maximizes your profit over the next 20 years. These steps are outlined in the following list:

1. Enter a code and the valuation associated with each code in D4:E6. The code for each year is the changing cell and determines the price charged that year. For instance, a code of 2 for one year means a price of $40 will be charged.

2. Enter a trial set of codes in B8:B27.

3. Compute the price charged each year in the cell range C8:C27. The Year 1 price is computed in C8 with the formula =VLOOKUP(B8,lookup2).

4. Copy this formula to the cell range C8:C27 to compute the price charged each year.

5. Enter the number of people valuing a razor at $30, $40, and $50 (classified as buyers and nonbuyers) in Year 1 in D8:I8. Half of all new people are classified as buyers and half as nonbuyers. Also 50 percent of holdovers are buyers and 50 percent are nonbuyers.

6. The key to the model is to accurately track each period for the number of buyers and nonbuyers of each valuation. The key relationships include the following two equations:

1 c07-math-001

2 c07-math-002

7. Copy the formula =E8+(D8-J8)+0.5*J8+0.5*New from D9 to D10:D28 to use Equation 1 to compute the number of high-valuation buyers in periods 2–20.

8. Copy the formula 0.5*New+0.5*J8 from E9 to E10:E28 to use Equation 2 to compute the number of high-valuation nonbuyers in periods 2–20.

9. In J8:L8 determine the number of people of each type (High, Medium, and Low valuation) who purchase during Year 1. Essentially, all members of High, Medium, or Low in the market purchase if the price does not exceed their valuation. Thus in K8 the formula =IF(C8<=$E$5,F8,0)determines the number of Medium valuation people purchasing during Year 1. The Solver window for the sales example is shown in Figure 7.4.

Figure 7-3: Why Have Sales? model


Figure 7-4: Solver window for Why Have Sales?


In the Solver model the target cell is meant to maximize the total profit (cell M4). The changing cells are the codes (B8:B27). Constrain the codes to be integers between 1 and 3. You use codes as changing cells rather than prices because prices need to be constrained as integers between 30 and 50. This would cause the Solver to consider silly options such as charging $38. During most years a price of $30 is optimal, but during some years a price of $40 is optimal. The price of $40 is optimal only during years in which the number of $40 customers in the market exceeds the number of $30 customers in the market. The $50 people get off easy! They never pay what the product is worth. Basically, the razors are on sale 50 percent of the time. (Ten of the 20 years' price is $30!)

It is interesting to see how changing the parameters of this example can change the optimal pricing policy. Suppose the high-valuation customers value the product at $100. Then the Solver finds the optimal solution, as shown in Figure 7.5 (see the high price worksheet in the Sales.xls file). You cycle between an expensive price and a fire sale!

Figure 7-5: Sales solution when high valuation = $100


Of course stores have sales for other reasons as well:

·        Drugstores and supermarkets often put soda on sale as a loss leader to draw people into the store in the hope that the customer will buy other products on their current trip to the store or return to the store in the future.

·        Stores often have sales to clear out excess inventory to make room for products that have better sales potential. This is particularly true when a new version of a product comes out (like a new PC or phone.)

·        Stores often have a sale on a new product to get customers to sample the product in the hopes of maximizing long-term profits from sales on the product. The importance of long-term profits in marketing analytics will be discussed in Chapters 19-22, which cover the concept of lifetime customer value.


In this chapter you learned the following:

·        Prices of high-tech products often drop over time because when the product first comes out, companies want to charge a high price to customers who value the product highly. After these customers buy the product, the price must be lowered to appeal to customers who have not yet purchased the product.

·        Because customers have heterogeneous valuations for a product, if you do not have sales, you never can sell your product to customers with low valuations for the product.


1. Joseph A. Bank often has a deal in which men can buy one suit and get two free. The file Banks.xlsx contains the valuations of 50 representative customers for one, two, or three suits. Suppose it costs Joseph A. Banks $150 to ready a suit for sale. What strategy maximizes Joseph A. Banks profit: charging a single price for each suit, charging a single price for two suits, or charging a single price for three suits?

2. The file Coupons.xlsx gives the value that a set of customers associates with a Lean Cuisine entree. The file also gives the “cost” each person associates with clipping and redeeming a coupon. Assume it costs Lean Cuisine $1.50 to produce a dinner and 10 cents to redeem a coupon. Also the supermarket sells the entrée for twice the price it pays for the entrée. Without coupons what price should Lean Cuisine charge the store? How can Lean Cuisine use coupons to increase its profit?

3. Show that if the unit cost to produce the xth unit of a product is given by ax-b, then doubling the cumulative cost to produce a unit always drops by the same fraction.

4. If unit cost drops by 20 percent, when cumulative production doubles, then costs follow an 80 percent learning curve. What value of b corresponds to an 80 percent learning curve?

5. (Requires calculus). Suppose the first computer of a new model produced by Lenovo costs $800 to produce. Suppose previous models follow an 85 percent learning curve and it has produced 4,000 computers. Estimate the cost of producing the next 1,000 computers.