# 3.2 Calculating marginal welfare costs

## 3.2.1  Individuals

Consider a single individual, i, facing a linear income tax with marginal rate, t, supplying hi hours of work and paying Ri in tax. An increase in the tax rate imposes a welfare loss (equivalent variation) of EVi and an increase in tax of . The individual's marginal excess burden is thus:

The marginal welfare cost, the marginal excess burden per dollar of extra revenue from that individual, is:

The most common case is for to be positive.[20] However, it is possible that labour supply falls sufficiently for the tax paid by the individual to fall. The following simple elasticity relationship holds:

Where in general denotes the elasticity of a with respect to b. Hence an increase in the marginal tax rate generates an increase in revenue from person i, so long as the elasticity of hours worked with respect to the tax rate is greater than -1.

In cases where tax revenue falls, it makes no sense to refer to a ‘marginal excess burden' of taxation, despite the welfare loss. In practice, the tax and transfer system involves complex piecewise-linear budget constraints, depending for example on the extent of means-testing of benefits.[21] Effective marginal tax rates can decline (as certain benefits are exhausted) and increase (as the individual moves into a higher income tax bracket, or becomes eligible for a benefit that is subject to an abatement rate). It is possible for a small increase in a marginal rate to lead to a large reduction in labour supply and therefore for net tax payments to fall (that is, for tax net of benefit payments). This is more likely where the budget constraints for the particular group considered are dominated by non-convexities: this is relevant for lower-income groups who are recipients of means tested benefits. This applies in particular to sole parents: see the discussion in Section 4.1 below.

## 3.2.2  Aggregate measures

In aggregate, it may be expected that the New Zealand economy is on the 'right side' of the aggregate Laffer curve: that is, an increase in the marginal tax rate is expected to produce an increase, rather than a reduction, in revenue. Define as aggregate tax revenue, for a population of size, n. Then is expected to be positive even though some components may be negative. Similarly, let denote the aggregate welfare loss. Hence the aggregate marginal welfare cost is:

In computing these aggregates, there is no reason to omit those individuals for whom the marginal excess burden is not defined (that is, for whom the tax paid falls as a result of the marginal tax rate change). Omitting such people would understate the true aggregate welfare costs of the tax change.

In some contexts, it may be desired to report the average values for specified groups of individuals (those in well-defined demographic or income groups). In the case of the n individuals considered above, the population average, , would be obtained as:

Where m< n by excluding from the summation all those for whom is negative. Care is therefore needed in referring to aggregate or average values.

Even if all individuals in a particular demographic or income group were to have positive changes in tax payments when a marginal tax rate changes (that is, where m = n), the average marginal welfare cost of those in the group is not equal to the aggregate measure for those in the group. This is because the former involves an average of ratios (marginal excess burdens divided by marginal revenues). The latter effectively involves a ratio of averages, given that the above can be written as:

The above discussion is in terms of individuals only. In a microsimulation model with heterogeneous families, the welfare changes and revenue changes are all basically defined at the family level, given that for couples a single (joint) utility function is used. Simple aggregates of welfare changes and tax revenue changes therefore reflect total welfare losses and actual total revenue changes, so an aggregate MWC of taxation requires no adjustment for family size and composition.[22]

### Notes

• [20]In the expression for MWC there is clearly a possibility of a singularity where there is no change in tax paid. The same can apply when considering the aggregate MWC or groups of households. This situation arose in one case considered below, as noted in Table 4.
• [21]For details of the complex range of social benefits in New Zealand, many of which are subject to means-testing, see: https://www.workandincome.govt.nz/products/a-z-benefits/
• [22]However, sample weights provided with the HES are used to obtain aggregate results.
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