Chapter summary

As the discussion of Chapter 3 showed, the taxation of income is a major source of government revenue. It is also a major source of contention. One perspective sees the income tax as a disincentive to effort and enterprise implying the rate of tax should be kept as low as possible. A competing perspective is that income taxation is well-suited to the task of redistribution, with equity requiring that high earners pay proportionately more tax on their incomes than low earners. The determination of the optimal structure of income taxation involves the resolution of these
contrasting views. The chapter begins by conducting an analysis of the interaction between income taxation and labour supply. A number of theoretical results are derived and these are related to the empirical evidence. This evidence makes clear how different are the responses of male and female labour supply to taxation. A model that permits the efficiency and equity aspects of taxation to be incorporated into the design of the optimal tax is then described. A series of results describing the optimal tax are then derived using this model and these are interpreted in terms of practical policy recommendations. The chapter is completed by reviewing calculations of the optimal tax rates that emerge from the model.

15.1 Taxation and labour supply

The effect of income taxation upon labour supply can be investigated using the standard model of consumer choice. The analysis will begin with the general question of labour supply and then move on to a series of specific analyses concerning the effect of variations in the tax system. The major
insight this gives will be to highlight the importance of competing income and substitution effects

As is standard, it is assumed that the consumer has a given set of preferences over allocations of consumption and leisure. The consumer also has a fixed stock of time available which can be divided between labour supply and leisure, all of which can be expressed as a utility function. . Labour is assumed to be unpleasant for the consumer so utility is reduced as more labour is supplied, implying that U = U (x, T- l).

The labour supply decision can be depicted on a graph of pre-tax income against consumption. The optimal choice occurs at the tangency between the highest attainable indifference curve and the budget constraint. An increase in the wage rate causes a change in the indifference curve. The direction of the substitution effect can always be signed since it is given by a move around the indifference curve. In contrast, the income effect cannot be signed: it may be positive or negative. The net effect is to raise pre-tax income but the effect on labour supply is ambiguous.

A common feature of the income tax in many countries is that there is a threshold level of income below which income is untaxed. The economic importance of this threshold is that it puts a kink into the budget constraint. However, it can be expected that a number of consumers will cluster or
‘bunch’ at the kink point b. The observation that consumers will bunch at a kink point is a common feature and reflects the fact that an extra unit of labour will receive net pay [1−t]w whereas the previous unit received w. It is therefore helpful to distinguish between interior solutions and corner solutions

The final issue that is worth investigating in this framework is that of participation in the labour force. The basic assumption so far has been that the worker can continuously vary the number of working hours in order to arrive at the most preferred outcome. In practice it is often the case that
hours are either fixed or else there is a minimum that must be undertaken with the possibility of more. Either case leads to a discontinuity in the budget constraint at the point of minimum hours. The choice for the consumer is then between either undertaking no work or working at least
the minimum. This is the participation decision whether or not to join the workforce. If a change in the tax rate changes the participation decision there will be a discrete change in working hours.

• The modelling of the choice of labour supply
• Income and substitution effects
• Tax thresholds and corner solutions

15.2 Empirical evidence

The theoretical analysis has identified the three major issues in the study of labour supply. These are the potential conflict between income and substitution effects which prevent clear-cut results, kinks in the budget constraint which make behaviour insensitive to taxes, and the participation
decision which can be very sensitive to taxation.

It still remains a fact that males continue to remain the dominant income earner in most families. This leaves the married female as typically a secondary income earner and for them there is often no necessity to work. From this position, it is the participation decision that is paramount. In contrast, most males consider work to be a necessity so the participation decision is an irrelevance. It can therefore be expected that the labour supply of males and females will show different degrees of sensitivity to taxation.

[Author is not really up to date with contemporary conditions]

Surveys on labour supply have normally arrived at the conclusion that changes in the tax rate have little effect on the labour supply decision. If this were correct, the labour supply function would be approximately vertical. In terms of the theoretical analysis the survey results point to an income effect that almost entirely offsets the substitution effect. However, the discussion has already suggested that different groups in the population may have different reactions to changes in the tax system.

• Empirical evidence is needed to evaluate labour supply responses
• The elasticity of labour supply varies between groups
• Elasticity is low for men, high for lone mothers

15.3 A model of income taxation

The optimal income tax trades off efficiency and equity to maximise welfare. A model that can provide an interesting analysis of this question
must have the following attributes:

1. There must be an unequal distribution of income in order for there to be equity motivations for taxation.
2. The income tax must affect the labour supply decisions of the consumers so that it has efficiency effects.
3. There must be no prior restrictions placed upon the optimal tax function.

The government is subject to two constraints when it chooses the tax function. The first constraint is that the income tax must achieve the government’s revenue requirement. The second constraint is that the tax function must be incentive compatible. To understand this it is helpful to view the government as assigning to each consumer a pre-tax income/consumption pair. Incentive compatibility requires that each consumer must find it in their own interest to choose the pair that the government intends for them rather than a pair assigned to a different consumer.

In the absence of taxation, income would be equal to consumption and this is depicted by the 45 o line. Where the consumption function lies above the 45 o line, the tax payment is negative. It is positive when the consumption function is below the line. The gradient of the consumption function is equal to 1 minus the marginal rate of tax.

Combining preferences with the consumption function it is now possible to display the utility maximisation decision of a consumer. Each consumer makes the choice of income (which is equivalent to choosing labour supply) and consumption demand to maximise their utility subject to satisfying the
consumption function.

To derive results from the model requires that one further assumption be placed upon preferences. This involves relating the gradient of the indifference curves through a given consumption-income point for consumers of different abilities. The required assumption is termed agent monotonicity.

The first result follows as a direct consequence of agent monotonicity: high-ability consumers will never earn less income than low ability. Generally, they will earn more. This result arises because at the point where the indifference curve of the low-ability consumer is tangential to the
consumption function, that of the high ability is flatter and so cannot be at a tangency. The solution for the high ability must then be further to the right.

The second result relates to the maximum tax rate that will be charged. If the consumption function slopes downward then the shape of the indifference curves ensures that no consumer will choose to locate on the downward sloping section. This part of the consumption function is therefore redundant and could be replaced by the flat dashed section without altering any of the consumers’ choices

A negative tax rate like this represents a marginal subsidy to the tax payer from the tax system. That is, the after-tax wage for additional work will be greater than the pre-tax wage ; a negative marginal rate can never be optimal.

There has been considerable debate about this result due partly to its contrast with what is observed. There are several points that can be made in this respect. The result is valid only for the highest-ability consumer and it makes no prediction about the tax rate that will be faced by even
the second-highest ability. Therefore it does not demonstrate that those close to the top of the ability range should face a tax rate of zero or even close to zero. For them the tax rate may have to be significantly different to zero. If this is the case, observed tax systems may only be ‘wrong’ at the very top which will not result in too great a divergence from optimality. The result also relies on the fact that the highest ability person can be identified and the tax system adjusted around their needs. Putting this into practice is clearly an impossible tax. In summary, the result is important in that it questions preconceptions about the structure of taxes but it has
limited immediate policy relevance.

• Income is observed, skill is not
• Agent monotonicity implies income increases with skill
• Optimal marginal tax rate is between 0 and 1

15.4 Numerical results

The standard analysis of optimal income taxation has been introduced above and a number of results have been derived that provide some characterisation of the shape of the tax schedule. It has been seen that the marginal rate is between zero and one but as yet no idea has been developed, except for the endpoints, of how close it should be to either. Similarly, although equity considerations are expected to raise the marginal rate, this has not been demonstrated formally nor has consideration been given to how efficiency criteria, particularly the effect of taxation upon labour supply, affects the choice of tax schedule. Due to the analytical complexity of the model, these questions are best addressed via numerical analysis.

The form of this social welfare function permits variations in the degree of concern for equity by changes in e . Higher values of e represent greater concern for equity, with e= 0 representing the utilitarian case. The individual utility function has the constant elasticity of substitution form
and the skill distribution is log-normal.

The first fact to be noticed from these results is that the average rate of tax for low-ability consumers is negative. These consumers are receiving an income supplement from the government. This is in the nature of a negative income tax where income below a chosen cut-off is supplemented by the
government through the tax system. The average rate of tax then increases with ability. The maximum average rate of tax is actually quite small.

These results provide an interesting picture of the optimal income tax function. They suggest that it should subsidise low-skill consumers through a negative income tax but should still face them with a high marginal rate of tax. The maximum marginal rate of tax should not be at the top of the
skill distribution but should occur much lower. Generally the marginal rate should be fairly constant. These are not results that would have been discovered without the use of this model.

• Simulations provide new insights
• Average rate of tax starts negative then rises
• The marginal rate of tax rises and then falls.

READING: Hindriks and Myles, Intermediate Public Economics, MIT, 2004

Chapter 16: Income Taxation

In this chapter we will only consider welfaristic equity criterion (like the utilitarian and rawlsian social welfare functions). So inso far as the social objective is entirely based on individual welfare, we are not assessing the tax structure on the basis of its capacity of either redressing inequality, or eliminating poverty. We do not either consider egalitarian social objective like equal sacrifice or equality of opportunities. There is indeed a interesting literature on fair income tax examining the distribution of taxes that imposes the same loss of utility to everyone, either in absolute or relative terms. It is related to the ability to pay principle according to which $1 tax is less painful for a rich than for a poor
(due to the decreasing marginal value of income). This equal sacrifice approach predicts that the resulting tax struture must be progressive (in the sense that everyone sacrifices equally if they pay increasing percent of their income in tax as their income rises) p378

Although the estimates vary widely within the groups, indicating some im- precision in the estimates, some general conclusions can still be drawn. Firstly, the elasticity of labor supply is not uniform across the population of workers. It clearly varies between the three groups identified in this discussion and probably varies within these groups. Despite this, it is still clearly apparent that the labor supply elasticity for married men is small with estimates grouped around zero. Such a finding has immediate implications for the efficiency consequences of a tax rate increase. In contrast, the elasticity of women is higher and reflects the participation effect and the greater flexibility they have in the choice of hours. p385

A tax system is progressive if the marginal rate of tax increases with income. Since it has been shown that the marginal rate should be zero at the top of the income distribution, the optimal tax system cannot be a progressive one. There has been considerable debate about this result due partly to its contrast with what is observed. There are several points that can be made in this respect. The result is valid only for the highest ability consumer and it makes no prediction about the tax rate that will be faced by even the second-highest ability. Therefore it does not demonstrate that those close to the top of the ability range should face a tax rate of zero of even close to 0. For them the tax rate may have to be significantly different to 0. If this is the case, observed tax systems may only be “wrong” at the very top which will not result in too great a divergence from optimality. The result also relies on the fact that the highest ability person can be identified and the tax system adjusted around their needs. Putting this into practice is clearly an impossible tax. In summary, the result is important in that it questions preconceptions about the structure of taxes but it has limited immediate policy relevance. p393

These results provide an interesting picture of the optimal income tax function. They suggest that it should subsidize low skill households through a negative income tax but should still face them with a high marginal rate of tax. The maximum marginal rate of tax should not be at the top of the skill distribution but should occur much lower. Generally the marginal rate should be fairly constant. These are not results that would have been discovered without the use of this model. p399h