Saturday, January 24, 2009

The Tax Canon -- Mirrlees '71 (Part 2)

Well it turns out that my first guess about the bit of math that I was unsure of was pretty close to correct, and it wouldn't have mattered if I were way off anyway. Ah well. Anyhoo, I have now finished reading J. A. Mirrlees, An Exploration in the Theory of Optimum Income Taxation, 38 Rev. Econ. Stud. 175 (1971) (actually reading the tough math-y parts several times) and have a few observations to report.

It was actually alot of fun to read a real economics paper for the first time in about five years. I had almost forgotten about the incredible number of limiting assumptions that cause such papers to almost never end up being about what their titles imply. I actually began reading with the idea that this paper would tell me about how to design the income tax to be as efficient as possible; it technically did this insofar as one is willing to accept the umpteen limiting assumptions that mostly bear no relation to the real world. To be fair, this particular paper is quite brilliant and did make a real contribution to the economic study of optimal taxation theory. It is just not something that a policy-maker can grab an apply without taking a huge leap of faith.

The paper begins with a list of assumptions, including a complete disregard of intertemporal issues, uniformity of preferences and utility functions among taxpayers, the disallowance of migration, perfect information, and a single type of labor and a single consumer good in the economy.  You know, just like real life.

The model is set up such that there is a finite number of laborers/taxpayers in the economy each with a different amount of output that they can produce per unit of time. That is, worker 1 can produce one widget per hour, worker 2 can produce two widgets per hour, etc. up to some maximum. The utility of each laborer is a positive function of how much he consumes and a negative function of how much he works. That is, ideally every worker would want to consume infinite widgets and work zero hours. The job of the government is to set a tax rate for each type of worker (each level of per-hour productivity) so as to maximize aggregate utility given the amount of time that each worker will choose to work. Of course each worker must choose how much to work based on how much he wants to/gets to consume which is partially a function of the tax rate. It is this interplay between optimization problems that leads to quite a few pages of fancy math.

You may have noticed that the model makes taxation choices based on a taxpayer's productivity, not his income. Mirrlees attempts to square this with the real world when he says, "One might obtain information about a man's income-earning potential from his apparent I.Q., the number of his degrees, his address, age or colour; but the natural, and one would suppose the most reliable, indicator of his income-earning potential is his income." This may be so, but the assumption that productivity is indicated by income might be one of the larger assumptions that this model uses.

After a bit of brain-hurting math, Mirrlees demonstrates his results by running a number of numerical examples (each with different assumptions about preferences, utility functions, etc.) and spits out some results that are a bit surprising. Optimum marginal tax rates tend to be relatively unvaried among different income brackets (or productivity levels) which would indicate that we might want to consider a flat tax rate. The actual flat tax rate is sensitive to the distribution of skills (or productivity) among the population as well as the income/leisure preferences. These two points aren't particularly surprising, but what is is that of all the combinations of assumptions tested, the highest optimum marginal tax rate came out to be 60% with the great majority being less than half that.

Another important result is that the income tax is not shown to be a very good tool for the redistribution of wealth (well, really consumption and utility). It seems that it would be more efficient to tax at a (low) flat rate and then use some other mechanism to achieve redistributive ends. 

Well, again, at least insofar as our world correlates to Mirrlees' world of limiting assumption.

This was fun and I look forward to reading more about optimal tax theory, but next time I'll be going back to the legal tax canon and will thus be reading William D. Andrews, Personal Deductions in an Ideal Income Tax, 86 Harv. L. Rev. 309 (1972).

Oh, almost forgot. This time around the part that made me feel stupid was easily the amount of math that I have forgotten since finishing grad school only a few years ago. What I would have once flown through, my brain must now stare at and chug through at a snail's pace. This is something that I'm going to have to work on.

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