The separation of efficiency and redistribution in the second theorem of welfare economics rests on the assumption that lump-sum transfers are feasible. As soon as the bases for taxation can be affected by agents’ behavior, dead-weight losses are created. Then raising money for redistributive purposes destroys efficiency. More redistribution requires more inefficiency. A trade-off appears between redistribution and efficiency. When labor income is taxed, the leisure-consumption choices are distorted and the incentives for work are decreased. Sidgwick (1883) in his Method of Ethics was apparently the first writer to recognize the incentive problems of redistribution policies:
It is conceivable that a greater equality in the distribution of products would lead ultimately to a reduction in the total amount to be distributed in consequence of a general preference of leisure to the results of labor.
—Sidgwick (1883, chap. 7, sec. 2)
The informational difficulty associated with income taxation is that the supply of labor is not observable and therefore not controllable, hence the distortion. However, if the wage was observable, as well as income, the supply of labor would be easily recovered. The next stage in the modelling of the problem was to assume that the wage of an agent equates his innate ability (equal itself to his marginal productivity), which is private information of the agents.21 Income, the observable variable, is the product of a moral hazard variable—the supply of labor—and of an adverse selection variable—ability.
A major step was achieved by Vickrey, who was senior economist of the tax research division of the United States Treasury Department and a tax expert for the governor of Puerto Rico. As early as 1945, he used the insights of Von Neumann and Morgenstern to model the optimal income tax problem as a principal-agent problem where the principal is the tax authority and the agents are the taxpayers. In his 1945 article, Vickrey defined the objective function of the government:
If utility is defined as that quantity the mathematical expression of which is maximized by an individual making choices involving risk, then to maximize the aggregate of such utility over the population is equivalent to choosing that distribution of income which such an individual would select were he asked which of various variants of the economy he would become a member of, assuming that once he selects a given economy with a given distribution of income he has an equal chance of landing in the shoes of each member of it.
—Vickrey (1945, p. 329)
Equipped with this utilitarian social welfare criterion (with, in passing, the Harsanyi  interpretation of expected utility as a justice criterion), Vickrey formulated the fundamental problem of optimal income taxation:
It is generally considered that if individual incomes were made substan- tially independent of individual effort, production would suffer and there would be less to divide among the population. Accordingly some degree of inequality is needed in order to provide the required incentives and stimuli to efficient cooperation of individuals in the production process.
—Vickrey (1945, p. 330)
The question of the ideal distribution of income, and hence of the proper progression of the tax system, becomes a matter of compromise between equality and incentives.
—Vickrey (1945, p. 330)
He then proceeded to a formalization of the problem that is still the current one. The utility function of any individual is made a function of his consumption and of his productive effort. There is a relationship between the amount of output and the amount of effort and unknown productive characteristics of the individual. This leads to an alternative form of the utility function that depends on con- sumption, output, and the individual’s characteristics. Taxation creates a relation- ship between output and consumption. Adjusting his effort or output optimally, the individual obtains his supply of effort characterized by a first-order condition, which is the first-order condition of incentive compatibility for an adverse selection problem. Vickrey stated the government’s optimization problem, which is to max- imize the sum of individuals’ utilities under the incentive compatibility conditions and the budget equation of the government. Recognizing a calculus of variation problem, he wrote the Euler equation and gave up:
Thus even in this simplified form the problem resists any facile solution.
—Vickrey (1945, p. 332)
The Pontryagin principle was still years away, and it would be twenty-six years before Mirrlees’s (1971) neat formulation and solution of the problem.23
Note that the problem analyzed here is not in the strict sense a delegation problem as we defined it earlier. The principal is actually delegated by the taxpay- ers the task of redistributing income, i.e., the choice of a particular public good. The principal observes neither the effort level of a given agent nor his productive characteristics. However, by observing output, which is a function of both, it can reduce the problem to a one-dimensional adverse selection problem. The principal is not facing a single agent over the characteristics of which he has an asymmetry of information, but a continuum of them for which he knows only the distribu- tion of characteristics. Nevertheless, the problem is mathematically identical to a delegation problem with a budget balance equation instead of a participation constraint.
Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.