A Contribution to the Theory of Taxation

TILE problem I propose to tackle is this: a given revenue is to be raised by proportionate taxes on some or all uses of income, the taxes on different uses being possibly at different rates; how should these rates be adjusted in order that the decrement of utility may be a minimum? I propose to neglect altogether questions of distribution and considerations arising from the differences in the marginal utility of money to different people; and I shall deal only with a purely competitive system with no foreign trade. Further I shall suppose that, in Professor Pigou's terminology, private and social net products are always equal or have been made so by State interference not included in the taxation we are considering. I thus exclude the case discussed in Marshall's Principles in which a bounty on increasing-return commodities is advisable. Nevertheless we shall find that the obvious solution that there should be no differentiation is entirely erroneous. The effect of taxation is to transfer income in the first place from individuals to the State and then, in part, back again to rentiers and pensioners. These transfers will slightly alter the demand schedules in a way depending on the incidence of the taxes and the manner of their expenditure. I neglect these alterations; 1 and I also suppose that " a given revenue " means a given money revenue, " money " being so adjusted that its marginal utility is constant. This problem was suggested to me by Professor Pigou, to whom I am also indebted for help and encouragement in its solution. In the first part I deal with the perfectly general utility function and establish a result which is valid for a sufficiently small revenue, and takes a peculiarly simple form if we can treat the revenue as an infinitesimal. I prove, in fact, that in raising an infinitesimal revenue by proportionate taxes on given commodities the taxes should be such as to diminish in the same proportion the production of each commodity taxed. In the second part I assume that the utility function is quadratic, which means roughly that the supply and demand