Accrual vs Realization in Capital Gains Taxation

Transcription

1 Accrual vs Realization in Capital Gains Taxation Giampaolo Arachi University of alento Massimo D Antoni University of iena Preliminary version: May, 06 Abstract Taxation of capital gains upon realization is known to bring about a distortion in portfolio choice lock-in effect). On the other hand, for given revenue, taxation upon accrual implies larger tax rates for consumption at later periods. Using a simple model of intertemporal choice with uncertainty, we assess and compare the effect of these two distortions. Our simulation shows that, for reasonable values of the parameters, either system can result to be less distortionary. Additionally, we find that the difference in efficiency is quite small. Keywords: capital gains taxation, taxation upon accrual, taxation upon realization

2 . Introduction The taxation of capital gains is one of the most critical area in modern tax system. As documented by a survey carried out by the OECD OECD, 006) most industrialized countries usually attempt to tax capital gains in a comprehensive way either under a countrys personal income tax or a separate capital gains tax. There are two main reasons behind this choice. The first one is related to the concern that an exemption of capital gains would produce relevant horizontal and vertical inequities between those earning primarily ordinary in particular wage) income and those making capital gains. The second one is the need to secure tax revenue. Where capital gains may be realized tax-free, taxpayers can be expected to take one or more steps to convert taxable income into exempt capital gains in order to avoid taxation. However the implementation of the tax on capital gains raises a number of difficulties. There are basically two different methods of collecting the tax: taxation of capital gains upon accrual or upon realization. Under an accrual system sometimes also referred to as yield-to-maturity approach the capital gain loss) is assessed and taxed with the same timing of ordinary income, usually on a yearly basis. Under a realization system the capital gain loss) is assessed and taxed when the asset is sold or disposed of. Economists have long been aware that taxing capital gains on a realisation basis entails a number of distortions in capital markets. First, the realization tax provides a timing option that allows the investor to realize capital losses immediately and defer capital gains in order to save taxes Constantinides, 98). The availability of the timing option distorts the optimal liquidation policy. econd, the opportunity to reduce the effective tax burden by deferring realization may lead investor to hold an appreciated asset even if an alternative investment provides a higher before-tax rate of return, resulting in a distorted allocation of capital and inefficient portfolio allocation Auerbach, 99). These two distortions are usually referred to as lock-in effect. Finally, if capital losses can be fully offset against other types of income, the overall tax burden can be further reduced down to nil) Constantinides, 98; tiglitz, 985) through the selective recognition of losses. The traditional response to these problems has been to limit loss offsets and impose restrictions on abuses. All of these remedies have ultimately been unsatisfactory and resulted in very complicated provisions in the tax code Alworth, Arachi and Hamaui, 00). These distortions will be absent under an accrual approach. For this reason the literature usually assume that accrual is more efficient than realization. However, the actual implementation of accrual taxation is hindered by a series of problems: a) it increases taxpayers compliance costs; b) marking to market is difficult for non-traded assets; c) in some extreme cases, unrealised gains may force liquidation at the time of tax payment Auerbach, 99; Alworth, 998; Alworth et al., 00). The pre-

3 sumption of the superiority of accrual taxation and the acknowledgement of the practical difficulties in its implementation have prompted a long lived debate among economists as well as tax practitioners on viable alternative approaches. In particular, attention has been focused on imitating an accrual tax by retrospective taxation on a realization basis in order to circumvent the lock-in effect without running into the problems of liquidity and valuation. The basic idea has been first proposed by Vickrey 99): with full information regarding the path of asset prices and holdings, it is possible to adjust taxes paid on realisation and equalise them to those that would have accrued if portfolios had been marked to market. The adjustment is determined by calculating ex post accrued gains in each tax year between purchase and realisation, and capitalising the implicit or virtual tax payment or tax credit) for each year using the net-of-tax rate of return. everal alternatives, with varying informational requirements has been subsequently developed ahm, 009, see for a survey). However, despite the academic consensus, no country applies accrual taxation on a broad scale, only Italy made an attempt to introduce such a system in 998 but the reform has been repealed in the following years Alworth et al., 00). The gap between theory and policy practices has recently prompted a questioning of the superiority of accrual taxation, which is usually based on heuristic considerations rather than proper analysis in a formal model. ahm 008), using a simple general equilibrium model of an exchange economy with heterogeneous agents, shows that in the presence of accrued capital gains, asset prices are higher under a realization tax than under an accrual system. ince the realization system creates incentives to defer accrued gains to later periods, actual total demand for the asset and, thus, its price increase. Though a realization system distorts the individual liquidation decisions, the elimination of these distortions by an accrual system causes distributional effects, impeding a clearcut welfare improvement. While a realization system discriminates agents without accrued capital gains, it is in favour of individuals holding assets with such gains. As a result neither method of taxation Pareto dominates the other. The purpose of our paper is to shed light on the comparison of accrual and realization taxation by attempting a comparison of the distortions implied by the two system of taxation. We find that that, under some values of the relevant parameters, realization may be more efficient than accrual. The intuition behind the result is that while taxation upon realization entails an efficiency loss due to the lock in effect, it provides a lower distortion of saving choice with respect to taxation upon accrual. This point can be illustrated using an argument proposed by Banks and Diamond 00) to interpret the Chamley-Judd result of a zero optimal capital income tax in the long run. As the time horizon grows, a constant capital income tax rate levied upon accrual creates a growing tax wedge between current consumption and future consumption.

4 With interest rate r and no capital income taxes, a dollar today is worth + r) n after n years. If an investor is subject to an annual tax at rate t on capital income, then the investor can convert one unit of consumption today into only + t)r) n units after n years. Hence, the tax wedge + t)r) n / + r) n grows with n. This suggests that if people have a long enough horizon, capital income taxation with a rate that declines with the time lapse between saving and later consumption such as the effective tax rate of a tax levied at realization) may be more efficient than a tax with a constant rate such as the effective tax rate of a tax levied on accrual). Using a three periods model originally proposed by Auerbach 99) to simulate the welfare effect of transition to accrual taxation in the U economy, we show that, due the lower intertemporal distortion, a realization tax may bring about a higher utility than an accrual tax of equal yield. The model allows to jointly analyse the lock-in effect and the intertemporal distortion in a tractable way. We also show that even in the cases where accrual taxation dominatessuch as the ones identified by Auerbach 99)the welfare gain is very small.. The lock-in effect Capital gains taxation upon realization is known to produce a lock-in effect. This is easily illustrated by comparing the net of tax returns from holding an asset for two periods visa-vis the total return from two successive investments of one period. Indicating by r and r the returns in each period, with R = r + r + r r the total rate of return, we have respectively + r + r + r r ) t) = + R t) ) and + r t)) + r t)) = + R t) r r t t); ) where it is clear that the total net return is lower in the case of two successive investments. This difference is eliminated if taxation is upon accrual, so that the net return is as in ) in all cases, independently on the intertemporal profile of investments. Taxation upon realization introduces a distortion in the individual choices in that it makes it convenient to delay the liquidation of an asset and it discourages a portfolio adjustment when investment opportunities change, whenever the gain from a portfolio adjustment is lower than the benefit from delaying the tax. To illustrate, let B be the value of the asset held in portfolio, so that T B would be the tax due if it is liquidated, and let r its expected return. Let ˆr be the return from an This is true with a positive) capital gain. imilarly, it encourages early liquidation of the asset in case of a loss.

5 alternative asset, with ˆr > r. The portfolio will not be modified if B T ) + ˆr t)) < B + r t) T ) ) i.e. if ˆr T ) < r. Note that, taxing capital gains upon realization, it would be possible to eliminate the lock-in effect simply by applying a tax rate increasing with the time horizon. This is the logic followed in the Italian reform of 997, where an equalized equalizzatore ) was introduced. Indicating with t the tax rate applied to an investment of one period and t the rate for an investment of two periods, it is possible to calculate the tax rate upon realization equivalent to a tax upon accrual by equation ) and ); we have the following condition: t t = + r r R t ) > 4) where it is clear that the time profile of tax rates depend on the rates of return in the two periods. For example, if the return is constant and equal to 4% in both periods and t =,5%, the equalized rate in second period will be t =,7%. The lock-in effect has been given much emphasis in the literature on capital gains taxation; however, this is only one of the possible distortion induced by capital income taxation. This means that there is no warrancy that its elimination brings about an increase in overall efficiency. In the next paragraphs we will consider the cost of the lock-in effect, the distortion induced in the incentives to adjust the portfolio to new opportunities, jointly with the overall distortion in savings decision induced by capital income taxation. To this purpose, the first step will be to consider the optimal structure of capital taxation from the point of view of intertemporal allocation of consumption in a simple three period model. Next, we will introduce the lock-in effect by assuming that in the second period it will be possible to buy a new asset which is not initially available.. Optimal tax rates in a three period model We will assume that a representative individual allocates a given amount of wealth across three periods in order to finance consumption, with no bequest. He uses his asset to finance consumption in the first period and saves the rest in an asset whose return is treated as a capital gain. Because capital gains taxation affects relative price of consumption in different period, its effect are similar to that of a tax on consumption in the second and third period. If we allow different tax rate for shorter or longer term savings, we can represent the problem 4

6 of finding the optimal capital income tax as equivalent to finding an optimal consumption taxes m k with k =, on consumption C and C. The intertemporal budget constraint is M = C + + m )C + + m )C + r + R 5) where + R = + r ) + r ) e M is income, which we assume as given and is received in the first period. The optimal tax rates depend on the distortion induced, which on turn depends on how substitutable is consumption at different times. A particularly convenient functional form for intertemporal utility is the CE Constant Elasticity of ubstitution) UC,C,C ) = ) C β +C β +C β β 6) with β > 0, where /β is the elasticity of substitution, measuring how much the individual is willing to reallocate consumption in response to variations in relative prices of consumption at different periods. As it is shown in the Appendix, the optimal tax implies uniform taxes on different commodities in this case, on consumption at different periods) or m = m. As a function of m, the optimal tax rate on capital income in the first period is obtained by solving: + r t ) = + m = t = + r m. 7) + r r + m On the other hand, m = m = m implies that the relative price of consumption at period and is not modified when the tax is in place. With taxation upon accrual this could be obtained only having t = 0 zero tax in second period). With taxation upon realization, the optimal tax rate t as a function of t is obtained by solving or: + R t ) + r t ) = + r 8) t = r + r r r + r + r r t < t 9) for example, with r = r = 4%, we would have approximately t =.5 t ). Therefore, a capital gains tax with rate uniform across periods is equivalent to a tax upon realization with rate decreasing with time horizon of the investment). This functional form is general in that it allows to consider different degrees of intertemporal substitutability of consumption; this is controlled by the parameter β. It should be reminded, however, that the CE is an additively separable function, and this property has well known consequences on the optimal structure of the tax. This is consistent with standard results in the theory of optimal taxation with separable utility functions. 5

7 It should be clear that, if implemented through taxes upon realization with decreasing with respect to the time horizon) rates, such a solution would exacerbate the lock-in problem discussed above. Indeed, with respect to a uniform tax rate it would increase difference between the tax on a series of short term investments and the tax on a single longer term investment of equal total time horizon. 4. Uncertainty and the lock-in effect In order to take into account both saving and portfolio choice decisions and therefore the lock-in effect we adapt the analytical framework suggested by Auerbach 99); namely, we assume that in the second period, after having invested part of his income, the individual has the possibility to modify his portfolio in order to choose the desired combination of return and risk. To this purpose, the individual must sell part of his portfolio and invest in a previously unavailable) asset with different risk/return properties. The lock-in effect manifests itself as a disincentive to adjust the portfolio in the second period. To take into account that the adjustment can go either way i.e. towards more or less risk) we will consider two different cases. In the first case, the individual has access to a riskless activity in the first period, while in the second period a new risky investment opportunity is made available. In the second case, the individual invests in a risky asset in the first period although the risk manifests only in the second period), and in the second period he has the opportunity to invest in a riskless asset. Hence, the adjustment is in the direction of a higher exposure to risk in the first case, and to a lower exposure to risk in the second case. 4 To simplify, it is assumed that there are two possible occurrences of the return of the risky asset, a high return g and a low return g, occurring with equal probability. We redefine the utility function to allow for choice under uncertainty by considering consumption in the third period contingent on two different states of the world. Namely, C and C represent respectively consumption conditional on states or, where the two states are defined with reference to the return from the risky asset. The utility function is now UC,C,C,C ) = ) ) C β +C β + C γ β β + C γ γ with β > 0 and γ > 0; /β represents as before) the intertemporal elasticity of substitution, while /γ is the elasticity of substitution between states, which is related to risk aversion: γ is the coefficient of relative risk aversion, hence a lower γ should be interpreted as a higher propensity to risk. This functional form allows us to consider independently 4 Auerbach 99) considers only our second case, in which the adjustment is aimed at reducing the riskiness of the portfolio. 6 0)

8 the effect of these two elasticities, which affect differently the distortions coming from the tax. 5 The indirect utility function is see Appendix for a formal derivation): V q,m) = M + q b + Q b ) b = + q b + a b) a M q a + q a ) a b ) b ) with a = /γ and b = /β the elasticities of substitution. Tax revenue is G = q p )C + q p )C + q p )C ) where C = q b M + q b + Q b C i = Q b M q a + q b + Q b q a + q a i =, ) with Q = a a q a + q a ) a. This general formula can be used to evaluate the effect of different hypotheses with regard to investment opportunities and taxation, which are reflected in consumer prices q, q e q. We derive such prices under different hypotheses about the tax regime, considering first the case in which the asset available in the first period is safe, while the risky asset becomes available only in the second period. We write the budget constraints for the first two periods in this way: M = C + B 4) + r) = C + B 5) where B is the amount invested in the asset either safe or risky) available in the first period; of this amount, will be disinvested in the second period in order to finance consumption C and the new investment opportunity B, while B will be kept in the initial asset until the third period. For notational simplicity, we indicate by r, R and g i the net of tax returns on different assets, i.e. r = r t), R = R t) and g i = g i t). Consumption in the third period depends on the tax regime. In the case of taxation upon accrual we have C i = B ) + r) + B + g i ) i =,. 6) 5 Note that, while we allow β to be different from γ, a representation in terms of expected utility requires that β = γ. 7

9 so that, substituting from 4) and 5), we obtain the intertemporal budget constraint: M = C + C + r + g r C + r g ) C. 7) + R g g g g Consumer prices under accrual are q = / + r) and q = + r) r g q = g g + r) g r. 8) g g Consider instead the case of taxation upon realization. We have C i = B ) + R) + B + g i ) i =, 9) where R = R t), so that the intertemporal budget constraint is M = C + C + r + [ + r) + g ) + R)]C + [ + R) + r) + g )]C. 0) + R) + r) g g ) Therefore, consumer prices for period are now q = g g ) + r + g ) + R q = + g g g ) + R ) + r ) The effect of taxation on prices can be better appreciated by comparing 8) and ) with the expression of consumer prices with no taxes. In this case t = 0) we have p = / + r) and p = + r) r g p = g g + r) g r. ) g g Another way to look at the distortion introduced by taxation is through its effect on the relative price between consumption in the two states of the world in the third period. With taxation upon accrual: while in the case of realization, for t > 0 we have q q = r g g r = p p ) q = + R) + g ) + r) q + g ) + r) + R) = r g + rr g ) g r + r g r) > r g g r. 4) In other words, taxation at realization discourages a portfolio adjustment towards more risk taking, reducing variance in third period consumption; it encourages consumption in state in which the return is lower. 8

10 Existing asset is risky, alternative asset is safe. Consider now the alternative case in which the individual has access to the risky asset in the first period while the alternative safe asset B is available only in the second period recall we have assumed for simplicity that, although the asset is purchased in the first period, the capital gain is uncertain only in the second period). With taxation upon accrual we have C i = B ) + r) + g i ) + B + r); 5) we see that in this case the intertemporal budget constraint is the same as in 0) and consumer prices are as in 8). With taxation upon realization, consumption in the third period is: C i = B ) + R i ) + B + r) i =, 6) where R i = R i t) and R i = + r) + g i ) = g i + r + g i r is the return of the risky asset over the two periods in state i. It follows that the intertemporal budget constraint is now M = C + C + r + {[ + ] [ } R + R R R + r) C + ]C + r). 7) Consumer prices for consumption in the third period are now: q = so that the relative prices are [ + ] R R R + r) q = [ ] + R R R + r), 8) q = + r) + R ) q + R ) + r) = r g + r r g ) g r + rg r) < r g g r = p. 9) p The lock-in effect is equivalent, in this case, to an incentive to consume more in the high return state, i.e. to bear more risk. 5. imulations We can summarise the analysis above by saying that, when the savings opportunities are modified, taxation upon realization introduces a distortion as it makes portfolio adjustment more costly. uch distortion is absent with taxation upon accrual. This is how the well known lock-in effect of taxation upon realization manifests itself in our framework. However, in the light of what we said in section we have to consider that the distortion 9

11 in intertemporal allocation of consumption is larger with taxation at accrual, due to the fact that the tax wedge increases with the time horizon of the investment. In this section we will rely on numerical simulations using the model we have developed to show that there may be cases in which the latter effect intertemporal distortion) dominates the former distortion from lock-in). To this purpose, we calculate the tax rate which is necessary to raise a given revenue G under the two regimes of taxation upon accrual and taxation upon realization, and compare the implied utility levels each other and with the utility we would have should we raise the same revenue using an ideal lump sum tax on first period wealth. This gives us a direct measure of the deadweight loss of different tax schemes we will normalize the utility by setting at the level reached with the lump sum tax; because the indirect utility is proportional to income, they can be interpreted as the income equivalent loss from distortions). We repeat the simulation for different values of the elasticites. As we said above, the chosen functional form allows to assume different values of the elasticity of substitution among periods b) and between states a, related to risk aversion). The other parameters to be specified are rates of return. Considering periods of 5 years, we assume a rate of return of, corresponding.8% compounded rate per year, fo the safe asset, while we will consider different combination of return and risk for the risky asset. In all simulations, we assume that revenue G = 0 and we normalize utility in the case of lump sum taxation to. The simulation will show the value of consumptions and of B, B and. We show in the tables the ratio d = B / + r ) B + B / + r ) representing the portfolio allocation at the end of period but discounted at period in order to make variables comparable); namely, it is the ratio between the amount invested in the alternative asset B / + r ) and total wealth at the end of the second period. The higher is d, the larger we expect the lock-in effect. We will consider both the case in which the alternative asset is risky, and the case in which the alternative is safe and the risky asset is the one in portfolio. 0) Returns on risky asset g = and g = 0. With g = and g = 0 we have, for the risky asset, with an average overall return of 50% over a 5 year period, corresponding an average yearly compounded return of.7%. Results of our simulation are reproduced in Table, where LUMP, ACCR and REAL indicate respectively lump sum taxation, accrual and realization. The shaded areas of the table help identify the cases in which realization 0

12 dominates accrual in terms of welfare. We see that taxation upon accrual is not always better than taxation upon realization when the existing asset is safe so that the adjustment is towards a riskier portfolio). The latter gives higher utility when a is high and b is low. This is not surprising, as a high a reflects a high intertemporal substitution and a low b reflects a low propensity to risk bearing. However, we note that all differences in utility are very small, in the order of magnitude of 0.0 i.e. 0.% of tax revenue). Table shows the case in which the alternative asset is safe, so that the reallocation in the second period is towards a reduction in risk exposure. We note that in this case accrual always dominates realization. Why do we have a different outcome in this case? The reason is that when the asset in portfolio is risky, an increase in risk aversion has two effects. The first is to make the distortion associated to portfolio risk adjustment less costly, as in the previous case. The latter is to induce the individual to change the portfolio to reduce as much as possible exposure to risk. In the extreme, we have that the individual disinvests almost all his asset to purchase the safe alternative. This implies that the difference between realization and accrual is minimized, as a vey small part negligible in the limit) of the portfolio is kept for the two periods. Returns on risky asset g = 5 and g = 0. We repeat the simulation assuming the variance of the return from th risky asset is higher: we now assume that the average cumulated return for the risky asset is 50%, corresponding to an average yearly return of 5.%. The results, shown in tables and 4, confirm the previous conclusions, and reinforce our conclusion, as the values of elasticities where realization dominates accrual is larger in this case. This was expected, given that a higher variance enhances the role played by attitude towards risk, which makes a portfolio reallocation less attractive and reduces the role played by the lock-in effect. 6. Conclusion In this paper we have compared taxation upon realization and taxation at accrual of capital gains. If on the one hand taxation upon realization introduces a distortion in portfolio allocation the well known lock in effect on the other hand, viz-a-viz taxation upon accrual, it reduces the distortion in intertemporal allocation brought about by capital taxation. To show how the two effects compare, we have developed a formal model of intertemporal and portfolio choice and used it to make some numeric simulation.

13 b = 0.0 V C C C B B b = 0. V C C C B B b = 0.6 V C C C B B b = 0.9 V C C C B B G = 0 r = g = g = 0 M = a = 0.0 a = 0. a = 0.4 a = 0.6 LUMP ACCR REAL LUMP ACCR REAL LUMP ACCR REAL LUMP ACCR REAL Table. Existing asset is safe, alternative asset is risky

14 b = 0.0 V C C C B B b = 0. V C C C B B b = 0.6 V C C C B B b = 0.9 V C C C B B G = 0 r = g = g = 0 M = a = 0.0 a = 0. a = 0.4 a = 0.6 LUMP ACCR REAL LUMP ACCR REAL LUMP ACCR REAL LUMP ACCR REAL Table. Existing asset is risky, alternative asset is safe

15 b = 0.0 V C C C B B b = 0. V C C C B B b = 0.6 V C C C B B b = 0.9 V C C C B B G = 0 r = g = 5 g = 0 M = a = 0.0 a = 0. a = 0.4 a = 0.6 LUMP ACCR REAL LUMP ACCR REAL LUMP ACCR REAL LUMP ACCR REAL Table. Existing asset is safe, alternative asset is risky 4

16 b = 0.0 V C C C B B b = 0. V C C C B B b = 0.6 V C C C B B b = 0.9 V C C C B B G = 0 r = g = 5 g = 0 M = a = 0.0 a = 0. a = 0.4 a = 0.6 LUMP ACCR REAL LUMP ACCR REAL LUMP ACCR REAL LUMP ACCR REAL Table 4. Existing asset is risky, alternative asset is safe 5

17 The result of the simulation can be summarized as follows: contrary to what seems to be the common view, taxation upon accrual is not always more efficient; depending on the parameters of the model, the inefficiency from the intertemporal distortion can be larger than the inefficiency from the lock-in effect; because the two effects work in opposite directions, the difference in terms of inefficiency are however quite small: the magnitude is never larger and very often much smaller, than % of the revenue from the capital gains tax. [TO BE COMPLETED] References Alworth, J., 998. Taxation and integrated financial markets: the challenges of derivatives and other financial innvoations. International Tax and Public Finance, pp Alworth, J., Arachi, G., Hamaui, R., 00. Whats come to perfection perishes: adjusting capital gains taxation in Italy. National Tax Journal, vol. 56, pp Auerbach, A. J., 99. Retrospective capital gains taxation. American Economic Review, vol. 8, pp Auerbach, A. J., 99. On the design and reform of capital-gains taxation. American Economic Review Papers and Proceedings), vol. 8, pp Banks, J., Diamond, P., 00. The base for direct taxation. In: for Fiscal tudies, I. ed.), Dimensions of tax design: the Mirrlees Review, Oxford University Press, Oxford, ch. 6, pp Constantinides, G. M., 98. Capital market equilibrium with personal tax. Econometrica, vol. 5, pp ahm, M., 008. Methods of capital gains taxation and the impact on asset prices and welfare. National Tax Journal, vol. 6, pp ahm, M., 009. Imitating accrual taxation on a realization basis. Journal of Economic urveys, vol., pp tiglitz, J. E., 985. The general theory of tax avoidance. National Tax Journal, vol. 8, pp Vickrey, W., 99. Averaging of income for income tax purposes. Journal of Political Economy, vol. 47, pp

18 APPENDIX Optimal tax rate in a three period model Maximizing 6), with the budget constraint M = C + q C + q C C is the numeraire, hence q = ) we obtain the demand function k =,,) so that the indirect utility function is V q,m) = C k q,m) = q b k M k q b k [ i q b β) i i q b i ] β M b = /β ) M = [ i q b i ] b. ) The elasticity of substitution among periods is represented by b = /β 0. The compensated demand is: ) /b C k q,u) = q b k + q b + q b u ) Let p k = +r) k with k =, be production prices with no taxes, while consumption prices are q k = p k +m k ) with m k the ad valorem tax rate). Given revenue G and setting q = p = no taxes on consumption in period ), we maximize the indirect utility under the revenue constraint, or maxv,q,q,m) s.t. q p )C + q p )C = G 4) q,q The first order optimality condition for q the condition for q is analogous) is q p ) C + q p ) C = µ ) C 5) q q λ where µ = V M + q p ) C M + q p ) C M Therefore, considering that: 6) C = C q b) + q b ) the first order conditions become: q p q q p q + q b + q b )q C q b + q b + q b q p q q b + q b + q b q p q q = C a)q b + q b + q b 7) )q + q b + q b + q b = µ ) b λ + q b + q b + q b = b µ ) λ 8) 9) 7

19 hence the conclusion that: q p q = q p q 40) in the case under exam in which q k p k )/q k = m k / + m k ), we have m = m = m. Indirect utility with uncertainty in the third period We solve the individual optimization problem in two stages, considering first the choice of C and C given income allocated to consumption in the third period M ; it is We define: M = M C q C = q C + q C. 4) ψq,q,m ) max C γ C,C + ) C γ γ s.t. M = q C + q C. 4) The conditional demand in each state is i =,) C i = M q a i q a + q a ) a = γ where a is the elasticity of substitution between states. By substituting, we have: 4) The optimization problem is: max C,C,M U C,C, ψq,q,m ) = M a a q a + q a ) a M a a q a + q a ) a ). 44) s.t. C + q C + M = M. 45) ubstituting C = M /Q, where Q = a a q a + q a ) a, this is equivalent to: max C,C,C UC,C,C ) s.t. C + q C + Q C = M 46) hence the demand function for C and C can be easily obtained from 4) using ) and M = Q C. 8