2 A Two-Period Model
Suppose tax revenue is used to finance government expenditure in each of two periods. The term 'two-period model' need not be taken literally, since the periods can consist of multiple sub-periods, in which case care is needed in specifying interest and growth rates. The required revenues of b1 and b2 per period are obtained from a proportional income tax at the rate, τ1 = b1/y1 and τ2 = b2/y2, where yi is income in period i. In the present model, labour supply effects of income taxation are ignored, so that the yi are considered to be exogenous and known with certainty. [8] In a more complex model, it would be desirable to allow the tax rate to affect the growth rate of income, thereby contributing an additional component of sunk cost. There is exogenous real income growth at the rate gy, so τ1 = τ2 only if desired expenditure growth, gb, is the same rate as income growth.
2.1 A Possible Future Cost
There is a probability of p that an event could take place in the second period that involves an additional cost of C. This section assumes that C (along with p) is known, but in the following section there is a distribution of cost values. Two basic options are possible. The first option is simply to wait to see if the uncertain event arises and then, in the second period, take take the necessary tax and expenditure decision. If the decision is to wait, the expenditure (if the need arises) must be met from additional taxation in the second period, assuming (in the present section) that reducing b2 is not viable. The second option is to take action in the first period by contributing to a fund obtained from additional tax in the first period. But of course it would never be desirable to save enough to cover the full contingency, even if there were no uncertainty, since tax smoothing involves tax rates in both periods being higher than in the absence of any chance of the costly 'event'. Tax smoothing under certainty within this simple framework is examined in Appendix B. Hence, the fund may be designed to meet only a proportion, γ, of the possible cost, the remaining cost being met from additional taxation in the second period if the event actually occurs. [9]
The approach taken here is to suppose that the choice of policy is taken by an independent and thus disinterested judge who is considered to maximise the expected value of a social welfare function expressed in terms of net income and the excess burden of taxation in each period. Of course there is no 'correct' form of this function: it represents the value judgements of the decision maker and so a wide range of assumptions could be modelled. The present form has been chosen simply to emphasise the potentially important roles of net income, the excess burden and risk aversion. [10] Disposable income in period i is yi (1 - τi), where τi is the appropriate overall proportional tax rate. The excess burden is assumed to be proportional to the square of the tax rate; hence it can be expressed as βτi2. The approximation of the excess burden in terms of τi2 is of course standard. [11] These components are assumed to be additive.
Suppose the judge displays constant relative risk aversion of ε. Welfare in each period is therefore:
(1)
The social welfare function is the expected value of the present value of welfare in each period, using a discount rate of r.
To consider the effects of varying ε, consider a simple example of a single-period context in which two values of an uncertain outcome, x1 and x2 can arise with probabilities p and 1 - p respectively, where welfare under each outcome is now W (xi) for i = 1,2. The expected value of W (x) is thus:
(2)
The value of x, say xε, which - if obtained with certainty - gives rise to the same value as the uncertain prospect is given by:
(3)
Table 1 illustrates the effects of varying both p and ε for two different combinations of x1 and x2. For example, a person with this kind of welfare function and having ε = 0.2 would regard the certain sum of 468 as equivalent to the uncertain prospect of receiving 200 with probability 0.1, and 500 with probability 0.9. As the probability of the low-value outcome, of 200, increases to 0.8, the certainty equivalent falls to 255; with high risk aversion of ε = 2.0, the certainty equivalent falls to 435 and 227 for p = 0.1 and p = 0.8 respectively. As ε increases, the certainty equivalent approaches the minimum alternative value of x quite rapidly.
| x1=200; x2=500 | x1 = 10; x2=500 | |||||
|---|---|---|---|---|---|---|
| p = 0.1 | p = 0.5 | p = 0.8 | p = 0.1 | p = 0.5 | p = 0.8 | |
| ε = 0.2 | 468 | 343 | 255 | 441 | 222 | 82 |
| ε = 0.8 | 460 | 322 | 244 | 378 | 103 | 29 |
| ε = 1.5 | 447 | 300 | 233 | 194 | 31 | 15 |
| ε = 2.0 | 435 | 286 | 227 | 85 | 20 | 12 |
The form of the welfare function in (1) also has implications for trade-offs between periods in the multi-period framework. For example, in the case where there is no uncertainty, a welfare function of the form x1-ε/(1-ε) implies a type of aversion to variability in terms of a preference for a steady 'consumption' stream. This differs from time preference which of course involves only a preference for present over future consumption. [12]
The following two subsections describe the form of the welfare function under each policy. The third subsection presents numerical examples. In the fourth subsection the role of inequality aversion is examined in more detail.
Notes
- [8]Other individual behaviour is ignored, such as saving, which may be affected by uncertainty and views about how the government responds to it.
- [9]Constraints on flexibility of government policy, such as the ability to change tax rates and expenditure, are ignored here. Such constraints are examined by Auerbach and Hassett (2001), who suggested that, faced with uncertainty, the inability to have frequent changes suggests early action but on the other hand inaction may be chosen because of the inability to reverse any adverse effects on particular groups. He concluded that, 'the optimal policy response over time might best be characterized by great caution in general, but punctuated by occasional periods of apparent irresponsibility'.
- [10]As the values of bi are fixed, it is not necessary here to allow for any benefits arising from this form of tax-financed expenditure. This is discussed further in section 4.
- [11]However, the approximation strictly applies to small tax rates, or small increases in rates: see Creedy (2004). The term, β, involves income and the (compensated) labour supply elasticity, but these need not be considered explicitly here as, to keep the model as simple as possible, income is assumed to be exogenous.
- [12]The inter-temporal elasticity of substitution is 1/ε. Furthermore the consumption discount rate is, from the standard Ramsey equation, δ + εg, where g is the growth rate and δ is the pure time preference rate.
