### Effective Marginal Tax Rates, and Capping Deductions

**Matthew Martin**3/08/2013 12:00:00 PM

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I was rummaging John Cochrane's blog and this post caught my eye. The thesis is that the effective marginal tax rate is what matters economically, and it often isn't equal to the marginal tax rate listed in the tax code. In particular, if a welfare benefit is phased out as income rises, then the rate at which it phases out is also part of your effective marginal tax rate. For example, suppose that, in your tax bracket, you pay 10% of each additional dollar to the government in taxes. But you are eligible for a welfare payment that phases out, so that increasing your income by one dollar reduces your welfare check by, say, 50 cents. What is your effective marginal tax rate? Answer: 60%. By raising your market income by one dollar, you pay ten cents to the government in taxes, and loose another fifty cents in welfare benefits, yielding an effective marginal tax rate of 60%.

This reasoning applies as much to actual welfare checks as it does to phase-outs of tax deductions. Which is why this passage caught my eye:

"In this weekend's Meet The Press interview, Gov. Romney said he wanted to "limit deductions and exemptions for people at the high end" only. Well, phasing out deductions is the same as a marginal tax rate. If earning an extra dollar lowers, say, the deductions you can take on your existing income by 50 cents, that counts as a 50% marginal tax rate every bit as much as if we just take the money."Ok, this is old-news Romney stuff. But Romney's back in the news again, so fair game. Here you have Romney touting his plan to spur GDP growth by lowering marginal tax rates on the wealthy. This spurs growth, so he says, because they have a bigger incentive to produce more. But wait, there's more! We are going to pay for this by phasing out deductions on the rich. We get the same revenue from the same people at lower marginal tax rates!

But...no. On the one hand we are lowering the statutory tax rate on the rich, while on the other hand we are matching this dollar-for-dollar by phasing out deductions on the rich. The effective marginal tax rate hasn't changed, see paragraph one. Romney's big idea was nothing more than the classic trick of trimming cloth off the bottom of the quilt to sew onto the top. And in the process, it makes the tax code even more complicated as we have to keep tables about which deductions phase out at what rate.

That said, optimal tax theory says that we should have welfare benefits for low income people that phase out as their incomes rise. If you follow this blog or my comments on other blogs a lot, you may have heard me assert that optimal tax theory says that we should have high effective marginal tax rates on the poor--this is exactly what I mean, they should receive a guaranteed basic welfare benefit, the size of which diminishes as their income rises. Beyond that, I'm generally opposed to any other tax deductions, except those specifically targeted at verifiable positive externalities.

But, we live in a world with lots of different kinds of welfare programs, benefits, deductions, exemptions, and tax rates. How do we guarantee that the effective marginal tax rate is never prohibitively high (whatever that threshold is)? It seems to me that one thing we would have to do is make sure that for the purposes of calculating gross income, all benefits from welfare and tax deductions are counted as income. Here is an example for illustration:

Suppose you are receiving two types of welfare benefits (say, the Earned Income Tax Credit and a health insurance subsidy) that phase out as your income grows. Suppose that if you increase your income by one additional dollar, one of these benefits is reduced by 50 cents, and the other is reduced by 60 cents. Now suppose that by "income" we are counting only market income and not welfare income. Then a one dollar raise actually decreases disposable income by 10 cents--your effective marginal tax rate is 110%. That's obviously problematic. Now suppose that instead of counting market income, we include all welfare benefits as income. Now if you increase market income by one dollar, the first benefit will perceive that your gross income rose by only 40 cents due to the reduction in the second benefit, while the second benefit will perceive that your income rose only 50 cents due to the reduction in the first benefit. Applying the marginal tax rates of 50% or 60% respectively implies that in sum, your disposable income rises by 20 cents. So instead of an effective marginal tax rate of 110%, this accounting system produces an effective rate of 80%.This system guarantees that the effective marginal tax rate is never greater than 100%. We can generalize this. Suppose [$]L[$] is the desired maximum effective marginal tax rate. We can insure this is the maximum by offering a universal deduction of a fraction [$]1-L[$] of off your gross income for the purposes of computing welfare benefits. Now suppose that there are [$]I[$] different types of welfare benefits, indexed by [$]i[$]. We denote the marginal tax rate (rate at which benefit phases out as income rises) associated with benefit [$]i[$] by [$]\tau_i[$]. Then your total effective marginal tax rate is given by [$$]\tau_{total}=1-\left(1-L\right)\prod_{i=1}^{I}\left(1-\tau_i\right)[$$]