A Reconsideration of Some Aspects of the Utility Sacrifice Rules of Taxation

This paper derives the type of tax system (progressive, proportional or regressive) required by different marginal income utility functions to satisfy the equal absolute and equal proportional sacrifice of utility among taxpayers. The context is that in which the rules were originally formulated, namely, the old welfare economics of measurable and comparable utility. It is found that equal absolute sacrifice calls for progression only if marginal income utility decreases at a faster rate than income increases (inelastic schedule). All other marginal utility curves require regressive taxation, save the unit elastic Bernoulli curves which requires proportional taxation. Equal proportional sacrifice indicates proportional taxation if the ratio of the area formed by the product of income and marginal utility to the total area under the marginal curve up to that income level is the same for all income levels. If this ratio decreases, progressive taxation is required; if it increases, regressive taxation is needed. With equal absolute sacrifice, the degree of progression is shown to be independent of the amount of utility sacrificed, whereas with equal proportional sacrifice the greater the proportion of utility to be sacrificed the more progressive (or less regressive) must be the tax structure. Finally, Musgrave's assertion that a unit elastic marginal income utility curve calls for proportional taxation under either rule is shown to be incorrect.