Linear and Non-linear Income Taxation:

Static models are of two types: linear and non-linear (general) income taxes. Linear income tax models have two parameters, a lump-sum tax (or a lump-sum grant to each individual) and a fixed marginal tax rate. The marginal tax rate creates disincentive for the supply of labour and thereby distorts the decision of the workers.

It is thus associated with an efficiency cost. Let Y indicate income as earned by the individual, R is the value of total tax revenue paid by the individual, S is the income subsidy given by the government. Total taxes paid by an individual is therefore,

R = -S + t.Y

All individuals receive an income transfer from the government and all of them face the same marginal tax rate. It is clear, in view of the budget constraint of the government, that the rate of taxation determines the amount of income transfer which could be financed. It brings into focus the trade-off between inefficiency caused by the tax rate and the redistributional goals the  government intends to achieve.

The government can choose both the parameters and can raise revenue and redistribute revenue across income groups. The  optimal choice of parameters depends on (1) how much revenue does the government need to collect, (2) society's preference for redistribution as reflected in the social welfare function, (3) responsiveness of the labour supply decisions to post-tax wage, and (4) the distribution of the pre-tax wage in the economy, or, to put it differently the inequality in the pre-tax income distribution.

However, the tax schedule need not be linear. In practice, we obtain non-linear tax schedules, which is progressive. The rationale behind designing a progressive tax schedule is provided by the concept of vertical equity as discussed in Unit 11. Mirrlees (1971) initiated an interesting debate how progressive an income tax structure should be. In his framework, the government seeks to  maximise a utilitarian social welfare function and has to determine an income tax schedule subject to collection of a given amount of revenue. In the non-linear income tax systems, marginal tax rate changes with the change in income level. The goal of the  government remains one of raising revenue with minimum distortions in an equitable manner. The distortions are created as in the linear tax model by the non-zero marginal tax rate on labour supply decisions.

Suppose we would like to determine the general optimal income tax structure that maximises a social welfare function. We are, in effect, looking for a relationship between the tax rate and earned income that could be progressive as well as regressive, at least  in principle. Let us assume a smooth income tax schedule as follows, t = f(Y) The solution to the problem tells us how to set the  tax rate t for all levels of income. Finding an optimal income tax schedule entails a compromise between progressive taxation to  achieve social justice and at the same time, progressivity causing inefficiency. The efficiency question is addressed if we know response of the taxpayers towards work and leisure. If the government thinks that inefficiency caused by high tax rate need not be a matter of concern and empirical estimates show that work and leisure substitution is low and people pay in accordance with their ability, a highly progressive tax structure may be opted for. If the government instead emphasises more on the extent of 'leaks' causing inefficiencies and the  excess burden caused by progressive taxation and less on equity, the decision makers would advocate a moderate tax schedule or, low progressivity.

It may be mentioned that ideally government could impose tax on the innate ability of the taxpayer, which would have been  non-distortionary. But the government cannot observe ability and hence government has to tax income, which is taken to be a proxy for ability. In presence of heterogeneous taxpayers, one has to take note of income elasticity of commodities and  importance assigned in terms of social weight to redistribution of welfare for designing an optimal tax system. The important results which follow from above formulation are as follows:

The marginal tax rate should lie between zero and one.

The marginal tax rate for the person with the highest income should be zero.

If the person with the lowest income (or hourly wage rate) is working at the optimum, then the marginal tax rate should be zero.

What follows with general tax schedules is that marginal rates do not rise uniformly throughout the range of income. The actual tax schedule however imposes tax burden, which rises with the level of income.

In order to understand the intuition behind the second result, let us suppose that there are two tax schedules. In the first one,  the tax rate on the highest income earner is positive and in the second one, the marginal tax rate at the top of the income ladder is reduced to zero. The taxpayers faced with the second tax schedule, may decide to work more in response to an increase in post tax income. So the effect of setting a tax rate at zero for the highest income earner results in a Pareto optimal situation as welfare of the highest income individual increases and the associated gains in social welfare and even the government collects for  revenue.

What follows is that optimum tax schedule must have a schedule of rising marginal rates near the bottom and a segment of  declining marginal rates near the top. This contradicts most of the characteristics of an actual income tax schedule. The logic that an optimal schedule should necessarily have zero tax rate at the top applies only to the individual at the top of the income bracket and does not throw any light on how marginal income tax rate should be just below the top income level. From a practical point of view, it is virtually impossible to ascertain the top of the income level in the income distribution.

For lowest income individual, the question of redistribution would not arise as there is no taxpayer with lower income to gain from redistribution. So there is no rationale for imposing inefficiency causing tax on the lowest income individual in absence of any gain from redistribution.