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

April 2009

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