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, Ω^m, into the four distinct regions and .

Clearly, a merchant will lie in if and only if max or in other words, if and only if:^[32]

Next, among those merchants not in , a given merchant will give preference to accepting the cards of platform i over those of platform j if and only if .^[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 To do this observe that a merchant not in will choose to sign up to both platforms if and only if ; 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 in -space, with slope . Similarly Inequality (A4) will hold if and only if

for which the case of equality again corresponds to a line through the point in -space, this time with slope . Inequalities (A1), (A2), (A5) and (A6) then immediately yield the breakdown of Ω^m 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, Ω^c, into the five distinct regions and .

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

Next, among those consumers not in (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 . 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 and 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 into two parts, separated by the line corresponding to equality in (A8) – which passes through the point and has slope . Inequality (A9) represents a different subdivision of Ω^c into two parts, this time separated by the line (Line 3 in Figures 2 and 3), which passes through the origin in -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, , and those who will prefer to use card j, . Starting with , by definition a consumer in will lie in this subset if and only if and . Hence, by Equations (16) to (18), such a consumer will lie in if and only if and the following two inequalities hold:

and

The first of these inequalities readily reduces to the condition that

where 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 will therefore lie in if and only if Inequalities (A9), (A12) and (A13) all hold. Moreover, it is readily checked that, in the event that , none of these three inequalities is redundant in delineating the region in -space (see Figure 2). By contrast, in the event that , the latter constraint, Inequality (A13), does become redundant in specifying , as illustrated in Figure 3.

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

and

Footnotes

Here, and in what follows, we assume without loss of generality that and , 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 and has slope . [34]