If you collect carbon tax revenue, figure out what each household spent on carbon, and refund that amount to each household, then there is no incentive to reduce carbon. You're basically just making work for bureaucrats. This is why no one, as far as I know, is proposing such a scheme.
Rather, they are (or I am, at least) advocating a plan in which revenues are chopped into equal parts and redistributed, either to everyone or to lower income households. Then, you've successfully increased the relative price of carbon-intensive goods or services, and helped to offset the impact on household incomes with the refund. And household income will be spent on a less carbon-intensive basket of goods and services, based on the relative price change.
In fact, it seems to me that as long as you raise the relative price of carbon, it doesn't matter whether you rebate back to people on the basis of their carbon consumption, or simply use a flat rebate: carbon consumption goes down. That's because people still have to trade more of their rebate to purchase carbon-intensive goods. By making the terms of the tradeoff more favorable to low-carbon goods, you encourage people to substitute away from carbon.
The reason no to attempt to do this is that it would be administratively impossible. But I think this is an important point: in terms of carbon reduction, what's important is the price, not the distribution of the revenues from the tax (or the permit sales).
That's why I don't buy the argument that we need to auction permits in a cap-and-trade system in order to get the maximum carbon reduction. The auction changes where the revenues go, but it shouldn't fundamentally alter the amount of carbon emitted. A company deciding whether to buy a permit or reduce their emissions is not economically different from a company deciding whether to sell a permit or keep their emissions at the same level.
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