Graphing Obama's Tax Plan

Char has a good point, I think, about a recent report on Obama's tax proposal by Alex Brill and Alan D. Viard, which was linked by Greg Mankiw. You can read about it in Char's posts, and in the comments thereto. One of the authors of the report, Alan Viard, replied to Char's original post. I added some comments too, less snarky than usual.

However, I still disagree on a technical point, which is the claim that you can increase marginal tax rates across the board and still lower the total tax.

Char writes:

I don't have the tax code in front of me, so I'll have to use a cartoon example (again). A firefighter with two kids earns $50,000 per year. Under the current tax code (let's say) after taking the personal exemption and so on, she pays taxes on $30,000. She's in the 20 percent tax bracket, so she pays $30k x .2 = $6000. She has two kids, so she gets a tax credit of $1000 x 2 = $2000, so her final tax burden is $4000.

Under Obama's plan (again, I'm making up numbers here just to demonstrate the point), say the marginal tax rate is raised to 25 percent while the child tax credit is doubled. She pays $30k x .25 = $7500 minus the tax credit of $4000 = $3500. She pays less in taxes despite the fact that she pays at a higher marginal tax rate.

Brill and Viard's point is that the higher marginal tax rate will reduce peoples' incentive to work and therefore will reduce growth. Debatable, but not an outlandish speculation. To avoid confusion, they should have made clear that low and middle income earners will at the same time be paying less under Obama's plan than they do now.

I understand his point, but since people who currently make no gross income pay no taxes, marginal rates can only decrease across the board while total taxes fall if there is to be negative taxation, i.e. if a person with two kids and no income gets a $2000 check in the mail. I assume marginal rates are non-negative under any plan..

Specifically, let T(y) be the average tax collected from everyone making y dollars gross income, and suppose m(y)>0 is the marginal rate, so that

T(y)= ∫₀^y m(z) z dz

and in particular T(0)=0. If a new marginal rate comes along, say with Obama's plan, and m₀(y)>m(y) for all y, then clearly T₀(y)>T(y) for all y.

I conclude that marginal rates must fall somewhere below $25k, the region not shown in the orginal graph of Brill and Viard. If the marginal tax rate falls for people earning less than $25k, then they will have more incentive to work. That's the basis of Brill and Viard's analysis after all. Perhaps it's better to provide such incentives to the poor than the rich? One could argue that incentives for the rich are more effective, because they will start new businesses which will help everyone. But then perhaps it would be better to fiddle with corporate income taxes and capital gains taxes instead.

Unfortunately, Alex Brill and Alan D. Viard don't provide a graph for the marginal rates under $25,000. Let's guess at what it might look like. In the first scenario, let's suppose Obama's plan lowers the marginal rate under $25k as shown, so that near zero income the marginal rate goes down by about 6% from a current level 10%. This seems to be in line with what the plan's goal is. I've guessed that the current rate decreases to 10% at zero income. Here is the graph:




It's easy to calculate the effective total tax by income, and the percent change in after-tax income, shown here:





You can see from the graph that with Obama's plan under scenario 1, people will be paying more taxes if they make $40k or more. That's not what the plan will do, according to the analysis in the report Alex Brill and Alan D. Viard cite. Let's guess again, making Obama's plan more progressive. Suppose the marginal tax rate under $25k is zero, under scenario 2. We get the following graphs:





Under this scenario, people making over about $67k will be paying more taxes. It doesn't seem to me that this is in agreement with the Obama plan either. My understanding is that the income level where people will be paying more tax is higher than that.

Something doesn't seem to be adding up right. It is unfortunate that Alex Brill and Alan D. Viard didn't provide all the data.

Update

Greg Mankiw posts a note from Alan Viard:

Given some confusion on the blogosphere, I want to reiterate that my and Alex' article does not find an increase in average tax rates, or in tax payments, at the income ranges shown in the chart. On the contrary, our article makes clear that Obama's proposed tax cuts would cause average tax rates and tax payments to decline throughout this income range...

Ok, this is not possible. Even if Obama's plan calls for 0 taxes below $25k and the current plan has a constant marginal rate of 21% from $0 through $40k, average tax rates go up at the $105,000 dollar point. This scenario clearly over-estimates of the tax break Obama will be giving to low-income people, so I don't think Mr. Viard's data and conclusion are in agreement. See the graphs below.