2 Relationships Among Elasticities

This section demonstrates, at the individual level, how the revenue elasticity and the elasticity of taxable income combine to generate the elasticity of tax with respect to the marginal rate. For convenience, the distinction between gross income and taxable income is ignored, though this distinction is likely to be important for countries with extensive income tax deductions.[6] If there are endogenous, income-related deductions, the following analysis must be in terms of income after deductions have been made.

The literature on the tax revenue elasticity concentrates on the effects of changes in taxation resulting from exogenous changes in taxable income, with tax rates and thresholds held constant.[7] Furthermore, it is usual to assume that the exogenous change in income does not cause the individual to move into a higher tax bracket. Such a movement, where the tax function involves discrete changes in marginal rates, gives rise to a large jump in the elasticity, and this can - when carrying out appropriately tax-share weighted aggregation - distort the aggregate elasticity.

Suppose the multi-step tax function depends on a set of income threshold, and a corresponding set of marginal tax rates . Let the tax paid by an individual with income of y be denoted . The individual revenue elasticity, , is thus defined as:

and is given by the ratio of the marginal tax rate to the average tax rate faced by the individual. The following uses the general notation,, to denote a 'total' elasticity, and , to denote a partial elasticity. The revenue elasticity (where a tax threshold is not crossed) thus has the property that .

The above multi-step function can be written as:[8]

and so on. If y falls into the kth tax bracket, so that a_k < y < a_k+1, T(y) can be expressed for k≥2 as:

This can be rewritten as:

where:

and τ₀=0. Thus the tax function facing any individual taxpayer in the kth bracket is equivalent to a tax function with a single marginal tax rate, τ_k, applied to income measured in excess of a single threshold, . Therefore, unlike differs across individuals depending on the marginal income tax bracket into which they fall. For this structure, and supposing that the income thresholds remain fixed, the revenue elasticity is:

and the individual partial elasticity must exceed unity. Within each threshold (for which the marginal rate is fixed) the elasticity declines as income increases. As an individual crosses an income threshold, the revenue elasticity takes a discrete upward jump, before gradually declining again, as shown by the saw-tooth pattern in Figure 1.

Figure 1: Variation in the Revenue Elasticity with Income in a Multi-step Tax Function

For the multi-step function the partial individual elasticity, for j < k (that is, for changes in marginal tax rates below the tax bracket in which the individual falls) is given by:

which is simply the tax paid at the rate, τ_j , divided by total tax paid by the individual. Furthermore:

Hence .

Consider a change in the individual's tax liability resulting from an exogenous increase in one of the marginal tax rates (with other rates and the thresholds unchanged). This gives rise to a behavioural response, so that:

From (9), dividing by dτ and writing in elasticity form gives:

The first term may be said to reflect a pure 'tax rate' effect of a rate change, with unchanged incomes, while the second term reflects the combined 'tax base' effect, resulting from the incentive effects on taxable income, and the revenue consequences of that income change. When discussing the effect on total revenue of a change in the top income tax rate, Saez et al. (2009) refer to the tax rate effect as 'mechanical' and the second term as the 'behavioural' effect respectively. Thus, their 'behavioural effect' combines the revenue elasticity and elasticity of taxable income effects.[9] Tax paid by the individual increases, when the marginal rate increases, only if:

In any progressive income tax structure, the revenue elasticity, exceeds 1, unless (as shown in the following section) allowances vary sufficiently with income. Equation (11) shows that, for the individual's tax liability to increase when τ increases, the combination of (absolute) elasticities must lie to the south west of a rectangular hyperbola, as shown in Figure 2. The nature of the tax structure determines the position of the hyperbola (via the elasticity, and the revenue elasticity (which in turn depends on the individual's income).[10] It is of course expected that is non-zero only for a change in the marginal rate corresponding to the bracket in which the individual falls, otherwise a rate change affects only the average tax rate faced by the individual.

Figure 2: Revenue-Increasing Elasticity Combinations

The individual elasticity of taxable income, , measures the behavioural response of taxable income to a change in a marginal net-of-tax rate, 1 - τ, facing the individual. This can be applied to any particular tax rate (not simply the rate corresponding to the tax bracket in which the individual's income falls), and a subscript is omitted here for convenience. The elasticities and are related using . Hence the elasticity of revenue with respect to the marginal rate faced by an individual in the kth tax bracket is given by:

The first term, , is the partial or mechanical 'tax rate effect' of the rate change, while the second term combines the behavioural reduction in the tax base (via the individual elasticity of taxable income) with structural effect (via the individual revenue elasticity) to give the consequent effect on revenue.