>> Credit card transaction fees are regulated at a much lower level in Europe.

> I don't understand why we don't do that here.

It really comes down to allowing markets to operate efficiently. Such regulation, in order to be work as intended, creates a price ceiling for the credit card market. This ceiling produces a deadweight loss (in this case, you would likely see it as either a reduction in the aggregate quantity of credit extended to consumers or the perks associated with using the card like fraud protection, ability to file charge backs, and customer support). On the whole (producer surplus + consumer surplus), you will observe such a loss.

Now what makes the government more capable of determining appropriate prices than market participants themselves? Allowing a competitive market to determine prices for credit services will drive the cost of the services to an efficient level. Given that governments are tasked with many pressing issues, it's reasonable to believe that they can only afford to put so many resources into assessing price levels for each regulated good or service. However, each market participant is able to focus solely on determining a price for the product offered. Even if the government can determine the optimal price level for the market, there is necessarily a lag time for the regulation to be passed and implemented, reducing the ability of the market to respond to systemic shocks and changing conditions.

Along different lines, if price regulations on credit card fees are acceptable, should the government install price ceilings on housing (causing reduced incentive to build more) and other markets? Should the price of all meals at restaurants be capped at $10 so more people can dine out? In the latter example it's easier to imagine the ramifications of a price ceiling, but the same market forces are at work.

> People like the rewards but that only works because the cost of those rewards is subsided by a price increase for everyone, even the people paying cash.

That's not true in specific, observable cases, and I suspect it's not true in general either.

To provide concrete examples, many gas stations and vending machines offer discounts for paying with cash instead of card, specifically to avoid the impact of fees on their prices. Additionally, vendors can choose to set minimum thresholds at which they accept credit cards, in order to mitigate the impact on their pricing and profitability.

In general, I assume the rewards are not truly subsidized by merchant fees. From my recollection of previous reading on the subject (I'd appreciate more accurate numbers if someone can provide them), merchants pay 3% to accept the credit card. ~2.5% of the fee is consumed by the cost of fraud. The remainder is split between the payment processor (e.g. Stripe), the issuing bank, and the credit card company. Clearly in a world of 2% rewards cards, a 3% fee does not allow for fraud protection, servicing expenses, and rewards.

My guess is that rewards programs only become feasible when you consider the interest charged on unpaid balances. Somewhere on the order of 50% - 70% of consumers carry debt on their credit cards. At 6% - 30% APR, this expense can reasonably cover the perks conferred by a credit card. If all consumers suddenly stopped carrying debt on credit cards, I suspect rewards programs would disappear rapidly.