RDP 2009-02: Competition Between Payment Systems Appendix A: Derivation of the Geometric Frameworks in Section 3

Appendix A: Derivation of the Geometric Frameworks in Section 3

In this Appendix we formally derive the geometric frameworks described in Sections 3.1 and 3.2 for understanding the card holding, use and acceptance decisions of consumers and merchants in our model.

A.1 The Framework for Merchants' Card Choices

We begin by deriving the subdivision – shown in Figure 1 – of the population of all merchants, Ωm, into the four distinct regions Inline Equation and Inline Equation.

Clearly, a merchant will lie in Inline Equation if and only if maxInline Equation or in other words, if and only if:[32]

Next, among those merchants not in Inline Equation, a given merchant will give preference to accepting the cards of platform i over those of platform j if and only if Inline Equation.[33] Hence, by Equations (22) and (23), they will prefer to sign up to platform i over platform j if and only if

while they will prefer to sign up to platform j over platform i if and only if the reverse inequality holds.

Lastly, we wish to identify those merchants that will choose to accept the cards of both platforms; that is, which lie in Inline Equation To do this observe that a merchant not in Inline Equation will choose to sign up to both platforms if and only if Inline Equation; or in other words, by Equations (22) to (24), if and only if

and

Yet then, on re-arranging, Inequality (A3) will hold if and only if

with the case of equality here corresponding to a line through the point Inline Equation in Inline Equation-space, with slope Inline Equation. Similarly Inequality (A4) will hold if and only if

for which the case of equality again corresponds to a line through the point Inline Equation in Inline Equation-space, this time with slope Inline Equation. Inequalities (A1), (A2), (A5) and (A6) then immediately yield the breakdown of Ωm shown in Figure 1.

A.2 The Framework for Consumers' Card Choices

We now derive the corresponding but more complex subdivision – shown in Figures 2 and 3 – of the population of all consumers, Ωc, into the five distinct regions Inline Equation and Inline Equation.

To begin with, by Equations (15) to (17) a consumer will clearly lie in Inline Equation if and only if maxInline Equation; or in other words, if and only if

Next, among those consumers not in Inline Equation (that is, who will hold at least one card), a given consumer will prefer to hold platform i's card over platform j's if and only if Inline Equation. Hence, by Equations (16) and (17), they will prefer to sign up to platform i over platform j if and only if

while they will prefer to sign up to platform j over platform i if and only if the reverse inequality holds.

Note that the issue of which platform a consumer would prefer to subscribe to is distinct from the question of which platform's card they would prefer to use, at the moment of sale, if they held both. This latter preference will depend only on the relative magnitudes of Inline Equation and Inline Equation with a given consumer preferring to use platform i's card over that of platform j if and only if

Inequality (A8) may be thought of as dividing the region Inline Equation into two parts, separated by the line corresponding to equality in (A8) – which passes through the point Inline Equation and has slope Inline Equation. Inequality (A9) represents a different subdivision of Ωc into two parts, this time separated by the line Inline Equation (Line 3 in Figures 2 and 3), which passes through the origin in Inline Equation-space and has slope 1.

Finally, we want to identify those consumers who will choose to hold the cards of both platforms – further subdivided into those who will prefer to use card i, Inline Equation, and those who will prefer to use card j, Inline Equation. Starting with Inline Equation, by definition a consumer in Inline Equation will lie in this subset if and only if Inline Equation and Inline Equation. Hence, by Equations (16) to (18), such a consumer will lie in Inline Equation if and only if Inline Equation and the following two inequalities hold:

and

The first of these inequalities readily reduces to the condition that

where Inline Equation is as defined in Section 2.4; while the second, after some simplification, may be re-expressed as the requirement that:[34]

A consumer not in Inline Equation will therefore lie in Inline Equation if and only if Inequalities (A9), (A12) and (A13) all hold. Moreover, it is readily checked that, in the event that Inline Equation, none of these three inequalities is redundant in delineating the region Inline Equation in Inline Equation-space (see Figure 2). By contrast, in the event that Inline Equation, the latter constraint, Inequality (A13), does become redundant in specifying Inline Equation, as illustrated in Figure 3.

As for Inline Equation a consumer not in Inline Equation will, analogously, lie in this subset if and only if

and

Footnotes

Here, and in what follows, we assume without loss of generality that Inline Equation and Inline Equation, since otherwise one or other platform would be attracting no consumers, and so making no profit. [32]

Here, the terminology ‘preferred’ platform means that platform whose cards the merchant would choose to accept if it could only sign up to one platform, not both. [33]

Note that the line corresponding to equality in Inequality (A13) passes through the point Inline Equation and has slope Inline Equation. [34]