How re-usable shopping bags are like a regressive tax

A few weeks back, the New Zealand Herald reported:

As of July 1, retailers will no longer be able to sell or give away single-use plastic shopping bags.

Instead, people will have to purchase reusable bags that are more than 70 microns thick.

This follows Cabinet's decision last year to go ahead with a mandatory nationwide phase-out of single-use bags...

A regulatory impact assessment of the ban, written by the Ministry for the Environment, warned there was a risk it could impact on the poor the hardest.

"Requiring consumers to pay up-front for new multi-use shopping bags could disproportionately affect lower-income consumers," the assessment said.

A tax is described as regressive if lower income consumers (or households) pay a higher proportion of their income in the tax. In this case, assuming that higher-income and lower-income households use the same number of shopping bags (or at least, that shopping bag use increases with income, but at a diminishing rate, which seems reasonable [*]), then lower-income households will end up paying more of their income in shopping bag costs. Even though the bag cost is not strictly a tax, it is still a regressive cost imposed disproportionately (relative to income) on lower-income households by the government. The shopping bag cost is regressive.

MBIE proposed this solution, according to the article:

"This could be mitigated by retailers allowing consumers who have Gold Cards or Community Service Cards a discount or exceptions."

The ministry also proposed partnering with food banks and different donors to distribute free multi-use bags with food parcels.

That would ensure that most low-income households won't be paying a higher proportion of their income on shopping bags than high-income households. However, does that also open a possibility for black market shopping bag arbitrage?

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[*] Higher-income households might use more shopping bags because they regularly buy more shopping items, or because they are less careful with the bags they do have (because the cost of replacing the bags is less consequential to them). However, it seems unlikely that households with double the income would use twice as many shopping bags (they would probably use less than twice as many), hence the 'increase at a diminishing rate).