publick economic

publick The exercise

No loss offsets

So far it has been assumed that losses may be set fully against tax; this may not however be the case. To see whether the results depend crucially on the assumption. We now consider the extreme case of no loss offsets: where the return to the risky asset equais  if  and x otherwise. Initially, we can contrast the tax and no-tax situations. Later we compare taxes with different degrees of loss offset.

To see the effect of the tax (without loss offset), let us take the special two-state case with the return in the second state negative. A dollar invested in the risky asset now yields an extra  in state (1) and  in state (2), so that the downward slope of the budget line is

This is steeper; i.e., the absence of teh loss offset provision makes the risky asset less attractive at the margin, relative to the no-tax situation. It does not however follow that this kind of income tax will necessarily discourage risk-taking, since there is in addition the wealth effect. This is illustrated by Fig. 4-6 for the case where r=0. The no-tax budget line is ST, the with-tax line is ST’. The move from p to p’ may be decomposed into a wealth effect PQ and the movement round the indifference curve QP’. The net result may be seen from the fact that the vertikal distance from S to P’ in Fig. 4-6 is equal to  ; in other words, it is an index of the amount invested in the risky asset. Depending of the wealth elasticity of demand, this may decrease or increase, the latter being shown in Fig. 4-6. Even therefore without loss offsets it is possible that social risk-taking may be increased by income taxation.

On the other hand, with sufficiently large tax rates, the demand for risky asset is reduced where there are no loss offsets. To see this, all we have to observe is taht for tax rates near 100 per cent, almost the entire portfolio is allocated to the safe asset. This follows because, as the tax rate approaches 100 per cent, the maximum return on the risky asset approaches zero and the expected return becomes negative. Since the indifference curves are convex, the demand curves for the different assets are continuous functions of the tax rate.

We turn now to the comparison of taxes with and without loos affsets. It seems likely taht a reduction in the extent to which losses can be set against tax will reduce risk-taking. The effect does however depend on the basis for comparison. To examine this, let us assume that α of losses can be offset . Then, if the tax rate is held constant, so that expected revenue falls (and expected utility rises) as α is increased, we can show that  denote the affter-tax return to the risky asset with loss offsets provosion, α, and r^ the after-tax return to the safe asset, so taht terminal wealth is,................................................................................

Now on the right-hand side c^x^/ is strictly positive where x < 0 and zero otherwise. It follows that the second term is evaluated only at positive values (where  then x^<r^ and U”<0), and that the whole right-hand side is positive. Since the coefficient of  is positive, it follows that α is increased.

                This conclusion does not however necessarily hold where the tax without loss offsets has a lower tax rate. Suppose, for example, that we compare taxes with equal expected utility. Since the removal of the loss offset provosion unambiguously makes individuals worse off in all states in which losses are incrurred, it is clear that to compensate the tax rate must be lower. Thus, for any particular portfolio allocation, the pattern of returns is such that there is a larger probability of very small and very large incomes. The new distribution. To analyse the consequences for portfolio allocation, we use general result of Diamond and Stiglitz (1974). To do this, we note that, since U is by assumption a strictly increasing function of A, and hence of x, we may therefore invert this relationship and write the first-order condition (4-17) as,......................................

                The basic result which can now be used is that an increase in the dispersion of a random variable, keeping the mean constant, increase the expected value of a convex function and decreases that of a concave function. Applying this to left-hand side of the first order condition (4-18), the removal of loss offsets leads to an increase in the dispersion of U, and hence to a fall in the expected value where the function is concave. It is left as an exercise to the reader to explore the relationship between this condition and the properties of absolute and relative risk aversion (and whether they are increasing or decreasing). It can be shown that it is by no means guaranteed that social risk-taking is lower as a result of the (expected utility-preserving) reduction in the loss offset provision (see Diamond and Stiglitz, 1974).

                The empirical relevance of the no-loss offset case depends of the form of a country’s income tax law, and on the range of an individual’s economic activities. Where losses may be set against other forms of income, where losses may be carried forward, and where capital losses may be set against investment income, the full loss offset case may be a reasonable approximation. On the other hand, there are typica;;y restrictions on the transfer of losses (e.g., capital losses not being eligible for relief on income tax) or where no carry-over is allowed. The no-loss offset case may therefore be more applicable.

                A natural question to ask is why tax authorities so commonly impose limitations on the extent to which losses are to  be offset. After all, it is exactly in those situations where individuals incur losses that risk-sharing with the goverment ought to be important. The answer is that it is extremely difficult for the goverment to distinguish, in many cases, between production and comsuption activities. An individual coul raise horses because he enjoys raising horses, or he could raise horse as a meaningful economic activity, i.e., for profit.  The goverment would not like to subsidize the former, but might not want to discriminate against the lateer as an econimic activity. They only way it can distinguish is to require the individual who claims that he is raising the horses for profit to make a profit. If he return out to be unsuccessful, then he is classified as having embarked on the activity for enjoyment, even if that were not his motive (and even if he hates horses).

Limited deductibility of interest

                If interest expenses are not deductible, there is a kink in the budget constraint at the point where all of the individual’s wealth is invested in the risky asset, i.e., below and to the right of the point T in Fig. 4-1. The effect depends on the balance of the substitution effect, which discourages risk-taking by individuals who  previously borrowed, and the wealth effect.

                Exercise 4-4 examine the effect of not allowing interest deductibility (but with full loss offsets) and how it depends on the properties of the wealth-portfolio locus.

                Again, we can enquire into the reason for this limitation which some (but not all) countries impose ; again, we find the answer in limitations on the government’s ability to identify the objective of borrowing. For example, a parent could, in principle, give a dollar to his child, and have the child lend the dollar back to parent ; the parent then could pay an arbitrary amount of interest to the child. This is a mechanism by which income from the parent could be transferred to the child ; so long as the two have different marginal rates, it is desirable for them to do this. In fact, of course, restrictions are imposed on the rates of interest that could “quality” ; but there is clearly room for considerable discretion in transferring income from one taxpayer to another.

                Similarly, the ability to deduct interest enable individuals to take advantage of special provisions of the tax code. For instance, in the united states interest on municipal bonds is tax-exempt. Consider an individual with a 70 per cent marginal tax rate. Assume the borrowing rate is 10 per cent, but the interest rate on municipal bonds is at 7 per cent. He borrows $100. He pays @10 in interest every year, which is tax-deductible, so his “net cost” is $3. He receives $7 in tax-free interest. Thus, for a zero investment, he receives annually $4. Obviously, if he could do this he would demand bonds up to the point where his marginal tax rate falls to 30 per cent. In fact, there are restrictions in that one cannot borrow to buy tax-free bonds, but since funds are fungible, he may be able to borrow to current consumption, and use what he would have spent on current consumption to purchase bonds.

Exemption of capital gains

                Most countries provide special treatment of capital gains. Levying lower rate of tax than on other forms of investment income. In this section we examine the extreme case where there is no tax on capital gains : the extension to the case of partial exemption should be apparent to the reader. The implication depend on the characterization of the two assets. For purpose of illustration we assume that the safe asset yields a return solely in the form of taxable interest, and that the return to the risky asset is entirely capital gains (again the extension to the partial case is immediate). Although this is a caricature, it allows us to examine the frequently made assertion that the special provisions  for capital gains encourage investment in risky assets.

                Terminal wealth now becomes, with a tax at rate  on the safe asset only

                Again, the slope of the budget constraint is altered by the tax, but in contrast to the no-loss offset case it now slopes less steeply. Moreover, it continues to pass through T (since at this point individual is neither holding the safe asset not borrowing)-see Fig. 4-7. Suppose now that we consider the indifference curve passing through the new equilibrium P’. Q is the point on this curve with the same slope as the original budget line, and it is clear that we can again distinguish a “wealth” effect  (P to Q) and subtitution effect (Q to P’). The latter is in the direction of increased risk-taking. The former depends on the wealth elasticity of demand for the risky asset, but where this is greater than it operates to reduce risk-taking. This is illustrated by Fig. 4-7. From (4-19) we can see that tax, for a given value of a, reduces the wealth in each state nature by the same absolute amount. The locus of points of constant values of a has therefore a slope of 45--see the dashed kine through P in Fig. 4-7. If the wealth elasticity of demand for the risky asset is positive, then the slope of the wealth-portfolio locus is less than 45 (refer back to Fig. 4-2), and Q involves a lower value of α, and hance lower social risk-taking, than at P. As shown, the net effect is for risk-taking to increase (P’ is below the dashed line). However, it is quite possible for the tax on the safe asset to reduce risk-taking, counter-intuitive though that may seen. As in other situations. The substituation effect operating in the expected direction may be more than offset by the wealth effect.

                The justification sometimes given for the exemtion of capital gains, or for their being taxed at a lower rate, is that this provision encourages risk-taking. In this section we have seen that this is not necessarily the case : the outcome depends on the properties of the asset demand function.