Do the poor pay for card rewards of the rich?

Card payment systems are sometimes accused of taking from the poor and giving to the rich. The argument is as follows: High card fees are leading to higher retail prices for both, card users and cash users. However, high income card holders are receiving rewards when purchasing by card. The result may be a net transfer of, mostly low-income, cash users to, mostly high-income, card users. In this article a model with monopolist product differentiation is used to show that rich card holders may actually be paying for their card rewards themselves. In this case, there is perverse distribution effect.