Triumph: Optimal Taxation

Mirrlees had been developing his understanding of economics and the mathematical modelling of it in the 1960s. What happened soon after his arrival at Oxford was that the problems he examined were at the epicentre of economic theory. He started to look in detail at the problem of income and commodity taxation leading to his development of the mathematical framework which was to have a much wider application across a significant range of economic models. The way Mirrlees formulated the issue was that optimal taxation was a problem in what would later be called asymmetric information: the tax had to be designed in a world where the underlying abilities (potential earnings) of individuals could not be observed. This meant that not only was there the standard trade-off between efficiency and (non-lump-sum) taxes but also an informational problem (how to tax people when you could not observe ability). The triumph was not simply to understand this problem, but to solve it. Depending on the social welfare function (the maximand), the aim of the tax regime is partly redistributive: you want to tax higher-ability households and transfer money to lower-ability households. However, you do not know who the higher-ability households are; all you can observe is their income and infer ability.

The key to the ‘Mirrleesian’ approach is the ‘sorting’ or ‘single-crossing’ condition, which says that the optimum consumption of households is a nondecreasing function of their ability (it is stated as condition B on page 182 of Mirrlees 1971). In his own words: ‘It is equivalent to assuming that (in the absence of taxation) the consumer’s demand for goods is an increasing function of the real wage rate (at any given non-wage income)’. This of course is very closely related to the Revelation principle. Mirrlees did not think of the model as an information revelation problem, but an optimal taxation problem can be thought of equivalently as one where the household faces a tax based on its ability and has to report its ability. The design of the optimal tax will then involve the household reporting truthfully its ability. In the fields of mechanism design and imperfect information, this became a key point. It is often referred to as the Mirrlees—Spence single-crossing condition; however, in Spence’s work there are only finitely many types (two), whereas Mirrlees covered the case of ability being a ‘continuous’ variable. It was working out how to deal with this problem that proved more useful than the application of optimal income taxation within which Mirrlees developed it. As he wrote a quarter of a century later:

In the income-tax problem, relatively simple conditions, easily checked for the particular model I was using, implied that the solution of the equations did give an optimum: the conditions were sufficient as well as necessary. When the computations were done, one knew one had the right answer, not just an answer that might be right (Mirrlees 1997: 1,317).

It is interesting to see where the solution came from. Mirrlees had been dealing for some years with the Ramsey growth model: this involves choosing consumption as a function of time. In the optimal income tax problem, you have to choose consumption as a function of income. It was the genius of Mirrlees to make the link that the same Pontryagin maximum principle applied to both, although in the 1971 paper he prefers to use the calculus of variations in his derivations—as had Ramsey—because it was more intuitive and familiar. The dynamic optimization framework Mirrlees had been working with on the optimal growth problem could thus be adapted and applied to the problem of income taxation. No one had ever done this before, and it provided the tools for use in a whole range of similar problems in different fields of economics over subsequent decades—in particular contract theory, the principal-agent problem and what has become known as Mechanism Design.

The work on taxation also extended to look at another of Ramsey’s interests: optimal indirect taxation. Ramsey had derived the famous result that the tax rate on a commodity should be equal to the inverse of the elasticity: it is better to tax goods that have inelastic demands since this will lead to less welfare loss. With Peter Diamond, Mirrlees considered an extension of Ramsey’s result. What happens if you have production? How should indirect taxes then be chosen? The result was the Diamond-Mirrlees Efficiency Theorem (Diamond and Mirrlees 1971a, b). In essence, the optimal indirect tax must involve productive efficiency (in the standard Pareto sense). Optimal indirect taxes cannot involve any distortion of the prices faced by producers (assuming no market imperfections such as externalities). Hence, indirect taxes should simply be on final consumption goods. Producer and intermediate goods prices should not be subject to taxation.

What does the optimal income tax schedule look like? This depends on the assumptions made about the distribution of abilities and the preferences of households. The original Mirrlees paper considered some explicit examples, and many more researchers went on to explore special cases. One interesting result that seems to be fairly common is that the marginal tax rate might be declining in income towards the top end, perhaps with a zero rate for the highest income. At the bottom income levels, the marginal tax rate could be low and even negative. To understand this, the special case of a degenerate distribution of types where everyone is the same can be used. Here, the incentive compatibility constraint falls away, and you can tax everyone the same. Efficiency will then dictate a universal lump-sum tax with zero marginal rate. If there is a little inequality, you want to introduce a small marginal tax but keep the average tax high enough to cover the required tax revenue. For the lowest-ability workers, the lump-sum element will result in low or possibly negative consumption that may need to be compensated with a lower, zero, or possibly negative income tax. This in turn means that the marginal tax rate will be increasing as you move to the middle-income earners. The zero top rate comes about because the real wage and hence marginal productivity of the top earner is the highest; hence, the disincentive effect of income taxation is the highest. The inverted U shape is quite common: the marginal tax is highest for middle (modal)-income earners. However, it is also possible to have the marginal rate declining everywhere, or indeed and approximately a ‘flat tax’.