Customers who pay little or nothing and are subsidized by another set of customers are essential to a vast array of businesses. According to one estimate, this business model accounts for a majority of the revenues of 60 of the world’s 100 largest companies.

The rationale for this approach, of course, is that by charging one set of customers little or nothing, the business will attract the critical mass of them required to draw in large numbers of another set of customers, and the income generated by the latter will handsomely exceed the cost of acquiring and serving the former.

The high-stakes challenge is figuring out the true value of each “free” customer, which is crucial to answering questions that will decide your company’s future — for example, determining the optimal way for a company to grow. Just how much should a company spend at various points in time to acquire and retain free or heavily subsidized customers?

Traditional customer-valuation models are of no help: They focus exclusively on paying customers and ignore the nonpaying ones. Our model allows us to deduce the impact of each customer, whether paying or not.

DECIPHERING NETWORK EFFECTS

Essentially, the lifetime value of a free customer is his or her incremental effect on the net present value of cash flows from the population of fee customers. It depends on the degree to which a free customer attracts other fee and free customers and the ripple effects those customers have on still other customers. When these network effects are large, payments flow to the firm and the lifetime value of the customer increases.

Direct network effects can be positive or negative. For example, direct network effects may be negative in employment sites or malls. Employers may not want too much competition from other firms for good job candidates, or stores in a mall may not want a lot of direct competition for traffic.

Indirect network effects, between buyers and sellers, can be positive or negative as well. They’re positive in the video game industry, where a larger number of people who own a particular console will attract more developers to create new games for that console, increasing the variety of games available and attracting more users.

BUILDING A MODEL FOR AUCTIONS.COM

Our work for a major international online auction house in 2006 illustrates how to apply our model. Given the proprietary nature of the project, we can’t identify the firm, which we’ll call Auctions.com, and we have disguised its data here.

This auction house had been in operation for five years and had steadily increased the resources it was devoting to sellers — after all, they were the paying customers. Buyers paid nothing to the auction house for bidding or for winning an auction. Sellers paid both a per-item fee and a commission. However, the prices were based on a model intended to maximize short-term revenue.

Auctions.com’s managers wanted to take a more sophisticated approach to figuring out how much to spend on marketing to buyers and what prices to charge sellers. We helped the firm resolve those issues by taking it through the steps outlined next.

1. Auctions.com collected historical data on the numbers of its sellers and buyers, the growth rates of those groups, the prices it charged sellers and the marketing expenses it incurred to attract sellers and buyers.

2. Using those data, Auctions.com examined how the growth in the number of both sellers and buyers was affected by the firm’s marketing strategies — advertising, pricing and so on; direct network effects; and indirect network effects.

We devised two related equations that captured those relationships: one for the growth in the number of buyers and the other for the growth in the number of sellers. The two equations formed the core of our network model, which was used to determine the magnitude of the network and marketing effects. (For details on the specific equations in the model, see “The Value of a ‘Free’ Customer,” by Sunil Gupta, Carl F. Mela and Jose M. Vidal-Sanz, Harvard Business School working paper, 2008, http://www.hbs.edu/research/pdf/07-035.pdf.)

Auctions.com used the model to project the growth in sellers and buyers. The results suggested that the growth of both groups would be extremely rapid until the company’s 11th year but then would slow down, and that the firm would most likely reach a saturation point of about 10 million buyers and 2 million sellers in about 150 months, or 12 years to 13 years.

The model revealed that the direct and indirect network effects among both sets of customers were positive, but they were stronger among buyers than among sellers. Also, the effect of buyers on sellers was greater than the other way around. This sizable indirect effect of buyers made them especially valuable.

3. Auctions.com then assessed the monetary value of acquiring a new free customer at different points in time. Each additional buyer, the model showed, would have the ripple effect of bringing in more buyers and sellers. By multiplying the resulting growth in sellers by the fees they would pay, netting out marketing expense and discounting the remainder to present value, Auctions.com could assess the corresponding increase in firm profits added by each buyer.

They discovered that a bidder acquired early in the life of the company, when he or she was critical to starting the virtuous cycle of buyers attracting sellers and vice versa, was worth about $2,500. However, this value decreased over time. For example, a bidder acquired in month 50 was worth about $1,360, and one acquired later, in month 100, was worth just a couple hundred dollars. These estimates made the firm’s managers realize that buyers were worth much more than they’d anticipated, and helped them decide how much to spend to acquire buyers at different stages of the firm’s life.

4. Our network model simulated the effect of different price strategies on profits, and showed that a penetration pricing strategy — charging a low fee to its early paying customers and then raising rates over time — would increase profits the most. This is because a low price in the initial period attracts many more sellers, who in turn attract more buyers.

The analysis of the results also helped Auctions.com strengthen its marketing operations, cater more to buyers and make its case to investors. And the estimated lifetime value of the forecasted number of buyers and sellers is, in most instances, closely aligned with a company’s market value. This gave Auctions.com’s managers ammunition to address any of Wall Street’s concerns that its high-flying stock price was unwarranted.

Many of these changes were implemented only recently, but the early results are promising.

Understanding customer value in networked settings is a new and exciting frontier. While our model is not a panacea, it is an important step in understanding the value of free customers. If properly understood, the free customer can be a powerful weapon.

(Sunil Gupta is a professor of business administration at Harvard Business School and is a co-author of “Managing Customers as Investments.” Carl F. Mela is a professor of marketing at Duke University’s Fuqua School of Business in Durham, N.C.)