A tale of two pricing systems for services

Abstract

Due to advances in technology and the rapid growth of online service offerings, various innovative web-based service models and delivery methods have appeared—including several free services. It is not always clear whether and how these emerging mechanisms for online service delivery will result in profitable businesses. In this paper, with an eye towards beginning to understand the issues involved, we present an analytical model of rational customer choice between available service plans. In particular, our model predicts how a monopoly service provider should devise its plans, if it understands such customer behavior. We then describe how this model would need to be extended in order to reflect increasingly inexpensive and even free service offerings.

Appendix 1

Proof of Proposition 3: With Plan 1, substituting $\hat{p}_{k} = c_{k}$ and $\hat{s}_{k} = 0$ into (3) yields a maximization problem that has first order conditions described by (9). So Plan 1 provides an (interior) solution for (8). Then $\hat{F} = f(\hat{q}_{1} , \ldots ,\hat{q}_{k} ) - \sum\nolimits_{k = 1}^{K} {c_{k} \hat{q}_{k} }$ also makes it so that (7) holds. Therefore, Plan 1 will also solve (6).

With Plan 2, if the marginal utility ${\frac{{\partial f(\hat{q}_{1} , \ldots \hat{q}_{k} , \ldots ,\hat{q}_{K} )}}{{\partial q_{k} }}}$ is positive, then $\hat{s}_{k} = \hat{q}_{k}$ and $\hat{p}_{k} = \infty$ will guarantee a corner solution with optimal quantity choices $\hat{q}_{k} ,\;k = 1, \ldots ,K,$ described by (8). Then $\hat{F} = f(\hat{q}_{1} , \ldots ,\hat{q}_{k} )$ also makes it so that (7) holds. Therefore, Plan 2 will also solve (6).

Proof of Proposition 4: With Plan 1, because ${\frac{{\partial f_{1} (\hat{q}_{11} , \ldots \hat{q}_{k1} , \ldots ,\hat{q}_{K1} )}}{{\partial q_{k1} }}} = c_{k} ,$ ${\frac{{\partial f_{2} (\hat{q}_{12} , \ldots \hat{q}_{k2} , \ldots ,\hat{q}_{K2} )}}{{\partial q_{k2} }}} = c_{k}$ and ${\frac{{\partial f_{1} (q_{1} , \ldots ,q_{k} \ldots ,q_{K} )}}{{\partial q_{k} }}} > {\frac{{\partial f_{2} (q_{1} , \ldots ,q_{k} \ldots ,q_{K} )}}{{\partial q_{k} }}},$ $\hat{s}_{k,1} = \hat{q}_{k1} > \hat{s}_{k,2} = \hat{q}_{k2} .$ And it is true that the lump-sum quantities in the two options are the optimal quantities that the customers should choose no matter who chooses which payment option. The reason is the really large additional unit prices. So if a customer in Segment 1 chooses option 2, he or she will choose $\hat{s}_{k,2} = \hat{q}_{k2} .$ And if a customer in Segment 2 chooses option 1, he or she will choose $\hat{s}_{k,1} = \hat{q}_{k1} .$ If the service provider wants to extract all the surplus from customers in Segment 1, i.e. M = 0, customers in Segment 1 would choose option 2 instead of option 1 because such customers can get a positive net utility from option 2, which is $M = f_{1} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} ) - f_{2} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} ).$ Thus, if the service provider wants customers in Segment 1 to choose option 2, the service provider needs to give up what such customers 1 can obtain from choosing option 2. That is why we have the M term in option 1. Segment 2 customers will not choose option 1 because they will obtain negative net utility with option 1).

With Plan 2, since option 2 is the same as in Plan 1, the service provider has to give up same amount in the fixed fee in option 1 to incent customer 1 to choose option 1, which is $M = f_{1} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} ) - f_{2} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} ).$ All the rest of the analysis in solution 1 applies to this solution.

Since the main concern is to incent customer 1 to choose option 1, it does not matter if fee-for-service or a fixed fee is emphasized in option 1. As long as option 2 is a fixed fee payment option, the service provider can differentiate the two segments using either Plan 1 or Plan 2.

Proof of Proposition 5A: When the service provider incent customer 1 to choose option 1 instead of option 2, it incurs an additional variable cost, $\sum\limits_{k = 1}^{K} {c_{k} \hat{q}_{k1} } - \sum\nolimits_{k = 1}^{K} {c_{k} \hat{q}_{k2} }$ because customer 1 chooses the higher quantity $\hat{q}_{k1}$. Thus, the service provider wants to differentiate the two customers only when the gain achieved from incenting customer 1 is no less than the cost of incenting, i.e. $f_{1} (\hat{q}_{11} , \ldots ,\hat{q}_{K1} ) - f_{1} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} ) \ge \sum\nolimits_{k = 1}^{K} {c_{k} \hat{q}_{k1} } - \sum\nolimits_{k = 1}^{K} {c_{k} \hat{q}_{k2} } .$ The condition can also be written as $f_{1} (\hat{q}_{11} , \ldots ,\hat{q}_{K1} ) - \sum\nolimits_{k = 1}^{K} {c_{k} \hat{q}_{k1} } \ge f_{1} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} ) - \sum\nolimits_{k = 1}^{K} {c_{k} \hat{q}_{k2} } .$ Then the left side is the gross market surplus that the service provider receives from customer 1 when the two customers are differentiated and the right side is the gross market surplus that the service provider receives when the two customers are not differentiated.

Proof of Proposition 5B: With Plans 1 and 2, customer 1 (who values the services more than customer 2) chooses quantity $\hat{q}_{k1} ;$ that is, more than what customer 2 chooses, $\hat{q}_{k2} .$ Customer 1 pays $f_{1} (\hat{q}_{11} , \ldots ,\hat{q}_{K1} ) - f_{1} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} ) + f_{2} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} )$ in both plans. Customer 2 pays $f_{2} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} )$ in both plans. Since $f_{1} (\hat{q}_{11} , \ldots ,\hat{q}_{K1} ) - f_{1} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} ) + f_{2} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} ) > f_{2} (\hat{q}_{12} , \ldots ,\hat{q}_{K2} ),$ customer 1 pays more than customer 2.