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The Built-in Flexibility of Income and Consumption Taxes in New Zealand - WP 03/05

2  Individual revenue elasticities

This section examines tax revenue elasticities for individuals. The variation in individual elasticities with income provides a useful independent indication of the local progressivity of the tax structure, and of course the individual elasticities provide the basic components on which aggregate values are based. The appropriate formulae are given in subsection 1, and empirical estimates for the 2001 tax structure in New Zealand are reported in subsection 2.

2.1  Elasticity formulae

Suppose denotes the income tax paid by individual with a nominal income of Changes in nominal income with respect to nominal tax allowances affect built-in flexibility.[2] The revenue elasticity of the income tax with respect to a change in individual income, is defined as:


where is the marginal tax rate and is the average tax rate faced by . Here, the first subscript of the revenue elasticity, refers to the type of tax revenue considered (so that in the present case of the individual elasticity, the subscript is dropped for convenience) and the second subscript refers to the income (or tax base) that is considered to change. In a progressive tax structure, for all so that This elasticity is also a local measure of progressivity: it is the concept of liability progression defined by Musgrave and Thin (1948).

Consider an individual with gross income of and facing a multi-step income tax function, such that if the tax paid is ; iftax paid is ; if tax paid is and so on. Hence if falls into the th tax bracket, so that and income tax can be expressed for as:


where Hence the tax paid under a multi-step function is equivalent to that paid with a single-step tax structure having a marginal rate, imposed on the individual’s income in excess of an effective threshold of

Creedy and Gemmell (2003) show that, for this tax function, the individual elasticity, is:


This indicates the potentially important role of the elasticity of effective allowances, . The individual revenue elasticity must exceed unity if . A positive value of can be expected where allowances are income-related, for example when there is tax relief for mortgage interest payments or pension contributions.[3]

To derive the individual revenue elasticity for consumption taxes, define as individual ’s net income, so that:


Suppose a proportion, , of is consumed, so that total consumption expenditure, is In general, can vary with and hence with , and can exceed unity over some ranges of as discussed below.

If the tax-exclusive ad valorem indirect tax rate imposed on the th good (for is the equivalent tax-inclusive rate is . Define as person ’s budget share of the th good. The consumption tax paid by person on good can be written as:


It is required to obtain the consumption tax revenue elasticity for each good, that is, . Writing as expenditure on the th good, first note that:

(6)    fig implies: fig1

fig3 implies: fig4

fig5 implies fig6

Differentiating (5) with respect to income, and using the relationships in (6), it can be shown that:


where is the elasticity of disposable income, , with respect to . The second term in (7) can be expressed in terms of , the total expenditure elasticity of demand for the th good by person , whereby:


The last term in (7), , is determined by the progressivity of the income tax, such that:


In fact, is the familiar measure of residual progression. Combining (7), (8) and (9) it follows that:


Equation (10) demonstrates that the consumption tax revenue elasticity for good can be decomposed into three terms, reflecting the total expenditure elasticity for good , the way in which the proportion of disposable income consumed by changes with income, and the degree of residual progression determined by individual ’s marginal and average income tax rates.

The consumption tax revenue elasticity for all goods combined, for person , can be obtained directly from the expression for the consumption tax paid on all goods, . Aggregating (5) over goods gives:


Differentiation of (11) then reveals that is given by:


This result shows that, compared with the revenue elasticity for a single good in (10), the tax-share weighted expenditure elasticity appears in (12). To calculate the weighted elasticity, it is necessary to distinguish only between goods facing different ad valorum tax rates.

The elasticity of the consumption proportion with respect to income in (10), , also varies with incomes if saving rates vary across disposable income levels. While a non-proportional relationship is generally accepted for cross-sectional income differences and, to a lesser extent, for time-series changes over the short-term, changes in the consumption proportion over the long-run are probably best regarded as proportional.

Creedy and Gemmell (2003) allow for the possibility of a non-proportional relationship by using the specification:[4]


For this case, it can be shown that in (12) is equal to . Hence for a proportional consumption function (including zero savings, where ), . The elasticity therefore depends on the three terms as follows:


The first two bracketed terms of equation (14) are less than or equal to unity, but the third component, shown in curly brackets, may exceed unity for some income levels and tax structures. However, tends towards unity as income increases.[5]


  • [2]The exception is where the tax function is homogeneous of degree one in income and allowances, and both are indexed similarly.
  • [3]For example, Creedy and Gemmell (2003) found that takes values around 0.4 for the UK, but varies significantly over time in response to changes in the tax deductability of various income-related reliefs suchas those for families,pensions and mortgages. The value of is of course unlikely to exceed unity.
  • [4]The over-spending at very low income levels can be viewed in terms of the existence of transfer payments and consumption out of savings.
  • [5]This is because all expenditure elasticities converge towards unity, although the convergence may not be monotonic, along with the first two terms in (14). Creedy and Gemmell (2003a) illustrate this decomposition for the UK.
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