I point out that such claims can be misleading, even on their own terms. Excerpt:
Suppose gas is originally $3 per gallon. Our poor household in a typical week buys 10 gallons. In contrast, the rich household in a typical week buys 20 gallons, because they drive more and because they drive an SUV.
Now imagine the government imposes a draconian $4 per gallon carbon tax that more than doubles the price of a gallon of gasoline to $7. In response, the poor household practically stops driving; people take the bus whenever they can. They end up buying only 2 gallon of gas per week. So they implicitly pay 2 x $4 = $8 total in carbon taxes per week.
The rich household however is better able to shoulder the blow. They reduce their gasoline consumption to 15 gallons per week. At this level of consumption, they implicitly pay 15 x $4 = $60 in carbon taxes per week.
Now if the government took these tax receipts and then issued an even rebate, the $8 + $60 = $68 total would get split into payments of $34 each. So Schultz and Halstead would conclude that the poor household “gained” $34 – $8 = $26 per week in net dividends from the scheme, while the rich household lost $60 – $34 = $26 per week in net payments. (This makes sense: If these are the only two households, the gain to one must be the loss to the other.)
But would we conclude that the poor household in our scenario is better off? Not at all! Yes, the household would have an extra $26 per week in money, but now (by assumption) gasoline would cost $7 per gallon. This artificially high price led the household to “choose” much lower gasoline consumption, but this was a coerced choice. The extra $26 per week in monetary terms would not mean the same as it would have in the original scenario, when gasoline was cheaper.