3 Aggregate Revenue

For tax policy purposes attention is often devoted to aggregate revenue and its variation as component tax rates are changed. This section therefore examines aggregation over individuals. Emphasis is on the effect on total income tax revenue of a change in a single tax rate, and the effect of a simultaneous similar change in all rates. As above, attention is restricted to the case of the multi-rate tax function.[11] First, components of total revenue are examined in subsection 3.1. Aggregate elasticities are derived in subsection 3.2. The potential orders of magnitude involved are then examined in subsection 3.3.

3.1 Components of Total Revenue

When dealing with population aggregates it is necessary to distinguish various tax and revenue terms, for both clarity and succinctness. In the previous section, the tax liability facing an individual with an income of y was denoted by T(y). In the multi-tax form, if income is in the kth tax bracket a distinction can be made between and the tax paid by the individual at the kth marginal rate, thereby ignoring tax paid on income falling into lower thresholds.

For aggregate revenue amounts defined over populations, or population sub-groups, R is used. Thus, in this section R represents aggregate revenue, while R_k refers to the aggregate revenue obtained from all individuals whose incomes fall in the kth tax bracket: that is, R_k is the aggregate over individuals in the kth bracket of values. Let R_(k) denote the aggregate amount raised only at the rate k from individuals who fall into the kth bracket: that is, R_(k) is the sum over individuals in the kth bracket of values. Furthermore, R_(k) refers to the aggregate revenue obtained at the kth rate from individuals whose incomes fall into higher tax brackets: that is, the number of all individuals in higher tax brackets multiplied by .

In the multi-step tax function with K brackets, suppose P_k people are in the kth bracket, for, and the arithmetic mean income in each bracket is . Then aggregate revenue is:

Let denote the number of people above the kth tax bracket. For the top marginal rate, clearly . Thus aggregate revenue can be written more succinctly as:

Thus, using :

In this expression, , although their sums over k = 1,...,K are equal.

3.2 Changes in Aggregate Revenue

Consider the response of aggregate revenue to a change in the kth marginal tax rate. This has two basic components. First, there is the direct effect of the change in the kth tax rate on tax from that bracket alone. From previous sections, in addition to the partial elasticity this is made up of the behavioural effect of the tax rate change on the incomes of those in the kth bracket, along with the revenue elasticity effect (which is not a reflection of behaviour but depends on the tax structure). Second, there is an indirect effect on individuals in higher tax brackets, as a result of the term . Assume first that there are no behavioural responses. It can be seen that, letting

For this ‘no behavioural response' case, these elasticities sum to unity, so that the elasticity of total revenue with respect to an equal proportional change in all rates is unity. Any behavioural response clearly reduces the elasticity below 1, as shown below.

In the case where there are behavioural effects of marginal rate changes, it is convenient to assume that all those in a given bracket have the same elasticity, . In this case, it can be shown that an appropriate adjustment to the average income level within the tax bracket gives:

The expression in (17), while quite straightforward, does not bring out the separate elements influencing the elasticity in a transparent way. First, rewrite this as:

Then multiplying and dividing by :[12]

From equation (6), is the revenue elasticity at arithmetic mean income in the kth bracket. This expression therefore shows how the elasticity, on the left hand side of (19), depends on the elasticity of taxable income of those in the kth tax bracket, along with the revenue elasticity at , and various tax-share terms. Furthermore, it can be shown that is positive if:

For the top bracket, the final term within square brackets in equation (20) is zero and the elasticity is positive if:

and although the first term in brackets exceeds 1 as long as the top tax rate is less than 0.5, the second term in brackets is likely to be well below 1. Hence the elasticity of taxable income must be relatively low for a tax rate increase to increase aggregate revenue. If all marginal tax rates change, but income thresholds remain fixed, the elasticity of aggregate revenue with respect to a simultaneous equal proportional change in all tax rates is the sum of the separate elasticities over all K.