7 Marginal Welfare Costs

This section considers the efficiency costs of personal income taxation. Subsection 7.1 shows how marginal welfare costs can be obtained using the elasticity of taxable income concept. Subsection 7.2 reports empirical results. The welfare effects reported here do not allow for the possibility that some income may be shifted to other income sources which attract a lower tax marginal rate.

7.1 Revenue and Welfare Effects of the Top Marginal Rate

This subsection considers the efficiency effects of changes in the top marginal income tax rate in a multi-rate system. This has received most attention in the literature and is the policy change considered in Section 4. Suppose income above the threshold, z_τ , is taxed at the fixed rate, τ. If this is the top marginal rate, so that there are no higher income thresholds, the tax paid at that rate by an individual in the top bracket is given by τ ( z_i – z_τ ) and total revenue collected by the top marginal rate, R_Τ , is thus:

where is the arithmetic mean of those above the threshold, and N_Τ is the number of people whose taxable income is above the threshold. It is important to recognise that τ ( z_i – z_τ ) is not the total tax paid by person i, since the latter has to include tax paid at lower rates. However, changes in the top rate are of course expected to have no effect on revenue from the lower rates.

The effect on R_Τ , and thus on total revenue, of a change in the top rate is:

The first term is a pure 'tax rate' effect while the second term is a 'tax base' effect of the tax rate change. Using , and from the definition of η, , the revenue change becomes:[14]

Let denote the density function, the distribution function and the first moment distribution function of z. The Lorenz curve is, for example, the relationship between the proportion of people associated with a proportion of total income (cumulating from lowest to highest). In general it can be shown that:

where is the arithmetic mean of the complete distribution of z. Define:

so that (10) is more succinctly written as:

The tax rate, z*, which maximises revenue from the top marginal rate is thus a simple function ofα_Τ and the elasticity, η, whereby:

The various components of (11) can be obtained from information about the distribution of declared incomes.

With the crucial assumption of zero income effects, the marginal welfare cost, MWC, defined as the marginal excess burden divided by the change in tax revenue, can be shown to be:[15]

This expression is relevant only when the marginal tax rate is below the revenue-maximising rate in (14). The extension to cover all tax brackets is as follows. Suppose that in the kth tax bracket the marginal tax rate is τ_k above the income threshold a_k , for. For income in the kth bracket the tax paid at the rate τ_k is simply τ_k ( z – a_k ) , and for z > a_k+1 the tax paid at that rate is τ_k ( a_k+1 – a_k ) . Hence total tax paid at the kth rate is given by:

Creedy (2010) shows that the marginal welfare cost in the kth bracket becomes:

where:

The result for the top marginal rate is thus simply the special case where D_K = 0. If, as for example in New Zealand, there is no tax-free threshold in the income tax structure, so that a₁ = 0 it can be seen that