1 Introduction

Governments are often faced with the possibility that a future contingency may arise involving higher public expenditure, although much uncertainty is usually involved. For example there may be a risk of an earthquake, though the timing and cost implications are subject to considerable uncertainty. A different context is that of population ageing, which is being experienced by most developed economies. While the time profile of the population age distribution in a country can be predicted with a reasonable amount of confidence,[1] it is very difficult to know how markets will respond since a wide range of behavioural and general-equilibrium adjustments to factor and goods prices is involved. Productivity and labour force participation changes are just two examples of important factors which influence future tax-financed public expenditure. Faced with such uncertainty, despite the more confident expection of population ageing, it is by no means clear whether governments should take immediate action or wait to see what actually happens, and thereby collect useful information. Intuitively, decision makers are more likely to take immediate and larger action, the higher the perceived probability of the contingency arising, the larger the potential cost, and the higher their degree of risk aversion. But such intuition is rather vague. The aim of this paper is therefore to clarify the nature of the various relationships, and the orders of magnitude involved, in the context of very simple models where uncertainty is involved.

In particular, this paper investigates the tax policy choices facing a government in a multi-period framework in which a future contingency may or may not arise. A single disinterested decision-maker or judge is considered to select an optimal tax policy by maximising a social welfare, or evaluation, function expressed in terms of net incomes in each period. The context is one in which it may be necessary to incur a higher expenditure in future, and this may be financed using some form of tax smoothing, by increasing the present tax rate. The incentive for tax smoothing arises from the fact that the excess burden arising from taxation increases disporportionately with the tax rate, and from the assumption that the judge is averse to risk. [2]

The question considered here is whether, and to what extent, a decision-maker should act by immediately raising the tax rate above current requirements, even though some or all of the resulting extra revenue that is accumulated may not be needed. If the extra revenue is not in fact needed, the future tax rate can be reduced accordingly. However, the higher net income and lower excess burden may not compensate for the initial tax increase. A further possibility is that it may also be possible to reduce other tax-financed expenditure to help finance the contingency, either in the initial or future periods. The present paper focusses on the essential nature of the trade-offs involved. The approach is to consider an independent judge making the decision on the basis of the expected value of a welfare function, expressed in terms of aggregate amounts of net income and the excess burden of taxation, in a range of situations. [3] The welfare function makes explicit the value judgements of the judge. The optimal degree of tax smoothing where the outcome is uncertain can be compared with the optimal policy where future costs are known with certainty.

The analyses are exercises in welfare economics: they consider optimal policies which involve answers to the question of, ‘what if a decision maker has a particular type of objective function?'. Thus, it is clear that the results cannot provide direct policy advice. However, it is useful to consider the elements of the decision-making problem, suggesting the type of information needed, and the nature of the possible trade-offs involved. The analysis does indeed highlight the fact that the nature and extent of uncertainty is crucial in considering practical policy issues and therefore cannot be ignored.

The framework may in some ways be compared with the consideration of investment in a multi-period project where the future returns are not known with certainty, there is a non-recoverable sunk cost of investing in the first period and there exists the option of waiting until later periods before making the investment (and thereby waiting for some of the uncertainty to be resolved). [4] The sunk cost consists of any fixed costs which cannot be recovered (such as the difference between the cost of investing in specific equipment and its resale value) and the foregone value of waiting and obtaining more information, referred to as the option value. [5] The concept of an option value provides valuable insights: this is discussed further in Appendix A.

In the present context an increase in the tax rate in the first period, in order to accumulate a fund that can be used in the event of a future (possible) expenditure requirement, involves a sunk cost and possibly a positive option value. [6] This sunk cost arises partly from the nature of the excess burden of taxation which increases more than proportionately with the tax rate. A subsequent reduction in the tax rate, if the extra revenue is not needed, allows the later tax rate to be reduced below the value needed for the other (fully anticipated) tax-financed expenditure. But this cannot fully recover the extra excess burden from the initial tax increase. However, the present context does not require, as in the standard investment framework, a given lumpy amount of investment (such as the construction of a factory) in the period in which it is decided to invest. Here it is possible in the first period to commit to a policy which involves only a small increase in the tax rate, leaving the additional (uncertain) revenue to be obtained by a higher tax rate increase in future, if this turns out to be necessary.

Section 2 begins by examining a two-period model in which there is a probability that an event, which has a given known cost, will occur. In section 3, the model is extended to allow for some uncertainty over the size of the cost, if the event occurs. Section 4 introduces the ability to reduce expenditure on other items in the second period if the need arises. [7] Despite the fact that the models are highly simplified, closed-form analytical solutions cannot be obtained. Thus, in each case numerical methods are used to provide illustrations. Conclusions are in Section 5.