# Can we design a single simple taxation curve?

Greg Mankiw's blog links to an article on marginal tax rates for the poor. The summary is that you basically don't have any incentive to work more unless you can make more than $40k. This is about 40% of the population! In 2005 median income for families was about $44k according to Wikipedia's reading of the US Census data.

Here's the basic graph (take home value vs earned income):

Mankiw recently updated his post to include a link to this pdf from Larry Kotlikoff and David Rapson (now at UC Davis). See page 45 and onward to show the graphs that explain exactly why a family would have no financial incentive to get a better job or save more money.

The government loves to create programs that help the poor. Paradoxically to some, these programs can actually make people stay poor, hence the "trap" in "poverty trap."

Suppose we could eliminate all federal programs, and simply design a curve like the one above to be implemented by a single federal government program of providing cash to families based on their earned income. This is sometimes known as a "negative income tax". Could we eliminate this system by designing a curve like the one above but with better properties? To answer this briefly I played around with designing a simple curve, it took me about 2 or 3 hours to play with this in my favorite computer algebra system and then I gave up, because it's not my area of research and I had finals coming. But the shocking thing was that my first graph looked surprisingly like the one linked above.

First I assumed that the earned income of the US had a fixed lognormal distribution, with median = 1 (think of income as measured as a fraction of the median) and the standard deviation of the logarithm is log(2) meaning about 95% of people are between 1/4 and 4 times the median income. This probably isn't too bad an assumption and it makes doing the math much simpler.

Next I used the Graham Schmidt process to create the first few orthogonal polynomials which are orthogonal relative to the lognormally weighted 2 norm. In other words, these are the optimal building blocks for functions of income in my model.

Now I chose to try to design a tax policy based on a single curve that had the goal of "helping the poor" while also running the government. I made the following assumptions:

- The government needs to keep 25% of total GDP to run itself.
- Even if you have 0 net earned income you should get at least 1/4 of median income to keep you alive (inc(0) = 1/4).
- The government should take exactly 25% from the median earner, neither redistributing to them nor from them. (inc(1) = 1-1/4, the 25% is to run the government remember).
- As an incentive to work, for every extra dollar the 0 earned-income family makes, they should take home 2 dollars. (derivative of the curve = 2 at earned income = 0)
- The median earner should take home one extra dollar for every extra dollar they earn (derivative = 1 at earned income = 1)

With these conditions we can set the coefficients of a 5 term polynomial expansion in the orthogonal polynomials. This expansion focuses on the lower end of the curve, although there is a global consideration (restriction number 1 above) still, we shouldn't expect it to make sense for earned income much above say 1.5 or 2. Solving for those coefficients, and plotting the graph of income vs earned income, I got the following shockingly familiar looking graph.

See that long flat region between about 0.4 and 0.8. That's where even doubling your earned income doesn't get you anything more at the end of the day. Why not just work part time at a coffee shop with wages of 0.4 of median income (about $17.5k/yr?), sign up for tons of government programs, and take home around 0.6 of median income (about $26k dollars/yr?). If you work harder to get double that wage, you still don't take home anything more!

Compare this graph to the dark blue graph in this exerpted graph from the paper linked above:

Is the goal hopeless? Perhaps we need to reconsider how tax policy for the poor should work from a global perspective. Setting tax policy this way makes it clear what we're doing, whereas a plethora of individual programs can never show you the big picture. Let's get rid of poverty traps! Designing a good non-trap tax policy will require eliminating some of the desirable properties we've asked for above, and also modeling the effect of the tax policy on the earned income distribution. It's a hard problem, but qualitatively my result here tells me there's a fundamental flaw in some assumptions. I'm guessing it's the usual failure to acknowledge that TANSTAAFL.

The study of optimal income taxation curves is big in economics (look up "optimal taxation" or maybe "Mirrlees optimal taxation"). If you're curious, here are the big differences between your analysis and the approach that economists take.

1) Incentives. You assume there's a lognormal distribution of income. Economists assume there's some distribution of *potential* income (think: wages), but that the amount a person actually chooses to earn depends on the incentives given by the tax code. Implicitly you try to maintain "good incentives" by requiring a derivative of 2 at 0 and a derivative of 1 at 1, but economists work from a more micro-founded model.

2) Welfare. You find a polynomial featuring some desired properties 1-5. Assumption 1 & 2 are more or less fine, but instead of 3-5 economists find a tax function which maximizes a *welfare function* subject to *resource constraints*.

Welfare function: a family of potential earnings type X which actually earns Y many dollars has a welfare of W(X,Y). ("Earnings type" can be thought of as wage -- if your wage is $1,000 an hour, you can work very little and live very well. If your wage is $10 an hour, you could physically work 100 hours a week and pull in $50,000, but it's extremely unpleasant). The simplest formulation would be something like Log(Y) - (Y/X)^2: utility is logarithmic in income, and there's a quadratic cost of working Y/X hours in order to earn $Y. Population welfare would be the sum (integral) of every individual family's welfare.

Resource Constraints: the total amount of money earned must be equal to the total amount of money allocated. If you include a government financing constraint like the one you suggest, that means that .75 * Total Money Earned = Total After-Tax Income.

Maximizing Welfare subject to Resource Constraints now gives you an optimal control problem (ie, calculus of variations).

I see that you move towards this approach in your post of Dec 8. I just saw this link off of Andrew Gelman's blog.

The point of this article was to show what would happen if you naively tried to design a tax code. The fact that the naive ideas lead to the poverty trap was a little surprising at first but then it becomes informative about mechanisms for arriving at poverty trap style taxation.

Thanks for the references to the existing literature. Yes, in my second article I take a calculus of variations approach and it makes for a much more reasonable taxation structure. Glad to know that Economists are not unaware of the calculus of variations. If only politicians could be convinced to consider something like that.