# RDP 2009-03: Competition Between Payment Systems: Results Appendix C: Generalised Versions of Our PTP Model

In this appendix we briefly describe the main changes to our PTP model which would result from allowing for the two possible generalisations canvassed in Section 4.2, namely: having the merchant rather than consumer choose the payment instrument at the moment of sale; and introducing a parameter κ representing disutility to consumers from holding the cards of more than one platform. These generalisations give rise to three variants of the basic PTP model – corresponding to incorporating either one or both of the possible changes.

In each case we focus on the intuition underlying the changes which would result to the geometric frameworks determining the card holding and acceptance behaviour of consumers and merchants. Technical details of the derivations of these frameworks are provided in Gardner and Stone (2009b).

## C.1 The PTP Model without κ but with Merchant Choice

Consider first the case of introducing merchant choice into the basic PTP model but with κ = 0. This affects the geometric frameworks described in Section 4.1 in a straightforward way, since the only asymmetry between merchants and consumers in both this and our main PTP model relates to who holds the choice of payment instrument at the moment of sale. Hence, granting this power to merchants rather than consumers simply switches the roles of these two groups: the card holding decisions of consumers are now described by a geometric framework exactly akin to that shown in Panel 1 of Figure 1 (with m-superscripts suitably replaced by c- or c,*-superscripts); and the corresponding card acceptance decisions of merchants are likewise described by a geometric framework exactly akin to that shown in Panel 2 of Figure 1 (with c- or c,*-superscripts replaced by m-superscripts).

## C.2 The PTP Model with κ > 0 but Retaining Consumer Choice

In the case where we allow for κ > 0, but with consumer choice restored, the situation for merchants is simple. For them, the only difference resulting from the introduction of κ, relative to the basic PTP model, is indirect, through its effect on the consumer market fractions . Hence, the geometric framework describing merchants' card acceptance decisions is exactly as in Figure 1 for the basic PTP model, with the slopes of Lines 1 and 2 again given by and respectively.[31]

As for the consumer side, here the introduction of κ > 0 does have a direct effect on which cards consumers choose to hold and use, as shown in Panel 2 of Figure C1 (for the case where ). Intuitively, the effect of κ is to diminish the incentive for consumers to hold both platforms' cards, pushing the boundaries of the region up and to the right compared with the situation in the basic PTP model.

Specifically, the lower boundary of is now given by the line

reflecting the trade-off now facing those consumers who would choose to hold card i rather than card j if they could only hold one, and who would prefer to use card i over card j if they held both. These consumers will now opt to hold both platforms' cards rather than just card i if, and only if, their aggregate benefit from also holding card j, , exceeds the disutility, κ, now entailed by holding multiple cards.

Similarly, the left-hand boundary of is now correspondingly given by the line

Finally, the remaining boundary of is now given by the line joining the points and , where the quantities and are defined by

and

This line (Line 5) has slope . It represents the boundary along which consumers who would choose to hold card j rather than card i if they could only hold one, but who would prefer to use card i over card j if they held both, will be indifferent between holding both platforms' cards or just that of platform j (that is, will have in utility terms).

## C.3 The PTP Model with κ > 0 and Merchant Choice

Turning finally to the case where we allow for κ > 0, but switch to merchant choice of the payment instrument, here the situation for merchants is exactly akin to that for consumers in our basic PTP model. In particular, the introduction of κ now has no effect at all on merchants' card acceptance (and selection) decisions, even indirectly. Hence, as shown in Panel 1 of Figure C2, the geometric framework describing these decisions here is just the identical twin of that for consumers' card holding and use decisions in Figure 1.

As for the consumer side, here (Figure C2) the effect of the presence of κ is once again to push the boundaries of the region up and to the right compared with the situation in the basic PTP model (Figure 1); while the inability to choose the payment instrument also now creates an incentive for ‘steering’ behaviour on the part of some consumers. The upshot is the geometric framework shown in Panel 2 of Figure C2, where Lines 3 and 4 are not horizontal or vertical (respectively) because of this ‘steering’ incentive, and where the quantities and are given by: [32]

and

## Footnotes

Of course, for any given {,,,}, the actual slopes of these lines will be different from the basic PTP model because of the different values taken in each case by the relevant consumer market fractions in the face of such fees. [31]

See Gardner and Stone (2009b) for details. [32]