2 The Elasticity and Estimation

The central concept examined here is the elasticity, η, of declared income, z, with respect to the net-of-tax rate, 1 – τ, which is the net-of-tax income per marginal dollar of pre-tax income (Lindsey, 1987). It is defined as:

This elasticity captures all responses to tax rate changes, without attempting to model each form of response. A popular constant-elasticity reduced-form specification is the following:[2]

where z₀ denotes the individual's income in the absence of taxation (that is, when τ = 0). Importantly, this specification assumes that income effects of tax changes are assumed to be zero. The remainder of this section describes the alternative estimation methods used in this paper.

Let z_it and z_0it denote declared income of person i at time t and the income which would be declared in the absence of taxation. Furthermore, τ_it is the marginal tax rate facing individual i at time t, where T(z) is the tax function. Using the constant elasticity form given above:

where by assumption the elasticity η is the same for all individuals in the relevant population group considered.[3]One approach is to consider actual policy changes in the tax structure for which only a relatively small group of individuals are affected, using information about the distribution of taxable income before and after the policy change. For example, suppose there is a change in only the top marginal income tax rate, which has no effect on those subject to lower rates. Let P_t denote the share of income of the affected group at time t, and their average marginal tax rate is τ_Pt. Let t = 0 and t = 1 denote pre- and post-change periods. If the share of income in the relevant group would have remained constant in the absence of the policy change, an estimate can be obtained using:

This method requires only summary data relating to the (cross-sectional) taxable income distribution in two periods.

An alternative approach to a policy change involves using a difference-in-difference framework with panel data. Suppose that the treatment group, T, comprises the top P1 percentile of the income distribution and the control group, C, is made up of individuals in the next P2 percentile. Again, suppose tax policy changes from period 0 to period 1, and let E(.) denote the respective sample average. The difference between groups in the differences between average log-taxable income from one period to the next, denoted Δlog z, is given by:

The difference between groups in the differences between the logarithm of average net-of-tax rates from one period to the next, denoted Δlog(1 – τ), is given by:

The estimate of the elasticity of taxable income can be obtained using:

Again, this approach involves an assumption that without the policy change the incomes of the two groups would have grown at the same rate.[4] In addition, the elasticity of taxable income is assumed to be the same for both groups.

The difference-in-difference approach can also be applied in situations where there is no explicit policy change affecting the marginal tax rate faced by some individuals. In particular, fiscal drag gives rise to a general increase in average tax rates but it can also shift some individuals into the next bracket and thus subject them to a higher marginal rate. Such individuals are regarded as being in the 'treatment' group. Those in the lower section of an income range do not face a higher marginal rate, as they do not cross a threshold, and are regarded as being in the 'control' group. The expression for the elasticity can thus be used in this context. This method can be applied for each tax bracket, thereby allowing for variations in η with income (between brackets).[5]