Optimum Taxation of Life Annuities - Semantic Scholar

Optimum Taxation of Life Annuities Johann K. Brunner and Susanne Pech * Revised Version April 2006

Abstract The market for private life annuities is characterised by adverse selection, that is, contracts offer lower than fair payoffs to individuals with low life expectancy. Moreover, longevity and income have been found to be positively correlated. We formulate an optimum income taxation model that incorporates these facts and discuss the conditions under which a linear tax on annuity payoffs, which raises more revenues from long-living individuals than from short-living, can serve as an instrument for redistribution. Further, we consider a nonlinear tax on annuity payoffs, and find that it can be employed to correct the distortion of the rate of return caused by asymmetric information. JEL Classification: H21, G22, D82. Keywords: Optimum taxation, life annuities, adverse selection.

* Address: Department of Economics, University of Linz, Altenberger Straße 69, A-4040 Linz, Austria. Phone: +43-732-2468-8248, -8593 FAX: -9821. E-mail: [email protected], [email protected].

I. Introduction As in many industrialised countries social security systems are under pressure because of an aging population, governments are trying to establish a so-called third pillar of old-age provision. The idea is that individuals should compensate a possible decline of the public pensions by increased private saving, that is, by shifting part of their income in the working period to the period of retirement. Among the various ways of how savings can be invested and how wealth, available for the financing of retirement consumption, can be accumulated, the purchase of private life annuities has the advantage of providing appropriate insurance against a long life. It allows the individual to avoid running out of assets before death as well as leaving unintended bequests. By transferring wealth from those, who die early, to those, who live long, annuities provide a higher rate of return than investments in the capital market (Yaari 1965). There is a tendency among governments to grant some way of preferential taxation to individuals who purchase private life annuities.1 The arguments for such a preferential treatment are rarely formulated explicitly; in the public discussion there seems to prevail a merit-good view, saying that individuals are myopic and therefore save too little in their active period of life.2 In particular, they might not have a sufficient perception of the likely reduction of the replacement rate offered by the public pension.3 In accordance with this view, several studies have investigated the incentive effect of taxes on the demand for private life annuities and on savings in general (for an overview see Bernheim 2002). Usually, economists have reservations against arguments based on irrationality of the individuals. If at all, these arguments might serve as a justification for public

1

2

3

For instance, in Austria the state subsidises the premium and, in addition, guarantees tax exemption of the payoffs. A similar regulation was introduced in Germany, the so-called Riester-Rente. In OECD countries the prevailing system seems to be that contributions to a pension fund are tax exempt, up to some limit, and pension payments are taxed, see, e.g., Whitehouse (1999) or OECD (1994). For the UK, Disney et al. (2001a,b) find some evidence, especially for low-wage earners, that the savings rate is too low in order to transfer sufficient income to the period of retirement. A similar argument rests on the suspicion that individuals might deliberately save too little, because they expect to receive some social assistance anyway.

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programs of limited size only, which means in the case of private pensions that preferential taxation for merit-goods reasons should be restricted to a minimum provision to cover basic needs. In principle, taxation of private pensions should follow the same rules that are guiding the design of the tax system as a whole. This leads one to the question of what is the appropriate tax treatment of annuities with regard to efficiency and equity. In particular, the distributive effect of the way of how annuities are taxed deserves a more thorough analysis than has been provided by the theoretical literature so far.4 To consider that together with the effect on economic efficiency in a coherent model is the subject of the present contribution. Obviously, besides providing insurance, the purchase of annuities represents an alternative to an investment on the capital market. It is therefore interesting to refer to results concerning the question of how such an investment should be taxed. On this, it seems fair to say that most theoretical studies have lead to the conclusion that capital income should be left tax-free, at least if some other instrument is applied for (redistributive) taxation in an optimal way and preferences are separable between leisure and consumption in different periods. Such a result can be derived in a static representative-consumer model of the Ramsey-type (see, e.g., Atkinson and Stiglitz 1980), but also rather generally in dynamic models with many households, which differ in their ability to earn income or in their capital endowment.5 The important question is then whether this result changes, if the insurance aspect of private pensions is taken into account. To answer this, one has to consider the functioning of the annuity market, in particular the problem of asymmetric information: insurance companies cannot distinguish between individuals with low and high life expectancy. This fact, together with the assumption of price competition among insurance companies, implies that firms are forced to offer the same rate of return to

4

5

Some studies concentrate on simulation results concerning distributional effects of different taxtreatments of annuities, see, e.g., Brown et al. (1999) and Burman et al. (2004). See Chamley (1986) and Judd (1985) for infinite-horizon models, and Ordover and Phelps (1979) for OLG-models. On the other hand, in both frameworks there are a few studies which establish the desirability of a tax on capital income, e.g. if credit constraints (Chamley 2001, Aiyagari 1995) or human capital accumulation (Jacobs and Bovenberg 2005) are incorporated.

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all customers, irrespective of their expected duration of life.6 This in turn induces an adverse-selection effect: the long-living individuals are overrepresented in annuity demand. This effect decreases the rate of return; it was estimated to make private pension more costly, to the extent of 7 - 15 percent.7 What we focus on in the present study are the distributive consequences of this phenomenon and whether it provides a specific rationale for taxation. As a first step of the analysis, we compare, by means of a simple example, the effects of two forms of tax exemption, namely either a wage tax, which leaves annuity payoffs tax-free, or a consumption tax, which leaves saving untaxed but taxes dissaving (i.e., annuity payoffs) fully. In particular, we show that, though both forms are equivalent as to consumption and welfare of the individuals, the latter extracts less revenue from the short-living individual than the former. This fact provides some intuition that a tax on annuity payoffs, without making purchasing tax exempt can indeed be a suitable instrument for a differentiated treatment of the individuals. For a more accurate analysis we have to allow for incentive effects on labour supply, where we take into account the empirical finding that income and life expectancy are positively correlated (see, among others, Attanasio and Hoynes 2000, Lillard and Panis 1998). We introduce this into the simplest possible model of optimum income taxation, consisting of two types of individuals, who live for two periods and differ in their wage rate and in their probability of survival to the second period. In this framework we consider two cases, that of a linear and of a nonlinear tax on annuity income. For a linear tax on annuity payoffs (in addition to the optimum nonlinear labour-income tax), we can show that its first-round effect (holding the rate of return constant) increases social welfare, given that demand for annuities does not decrease too much with leisure. In particular, the positive effect arises in case of 6

7

Price competition follows from the assumption that insurance firms cannot monitor whether costumers hold annuities also from other firms. In contrast, price and quantity competition, which was studied first by Rothschild and Stiglitz (1976) and Wilson (1977), requires that costumers can buy only one contract. Since this is regarded to be inapplicable for the annuity market, price competition is usually adopted for the analysis of the annuity market, see e.g. Pauly (1974), Abel (1986), Brugiavini (1993), Walliser (2000), Brunner and Pech (2005). Compared to a situation without adverse selection, i.e., where mortality of costumers is identical to that of the average population. See, e.g. Mitchell et al. (1999), Walliser (2000), Finkelstein and Poterba (2002).

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weak separability between leisure and consumption in both periods, which is contrast to the result of a zero optimal tax on capital income mentioned above. In these models on capital income taxation, the rate of return – the interest rate – is usually taken as constant. However, when discussing old-age insurance it seems appropriate to consider the effect on adverse selection, that is, on the equilibrium rate of return as well. It turns out that if a linear tax on annuity payoffs increases the rate of return on annuities, then the effect on social welfare is positive; if the rate of return is decreased, then the welfare effect is ambiguous. Concerning the case of an optimum tax system, which is nonlinear not only with respect to wage income, but also with respect to annuity payouts, a remarkable, new feature turns out important: Such a fully nonlinear tax system has the advantage over the one with linear taxation of annuity payouts that it not only allows redistribution, but also a correction of the market failure arising from the adverse-selection problem. In particular, we find that the annuity payout of the long-living (and high-income) individual is reduced by the marginal tax rate to her individually fair payout. For the short-living individual the analogous effect increases her payout, however, a distortion may occur – familiar from optimum labour income taxation - and impede the (full) realisation of the first-best payout according to her low survival probability. This corrective role is characteristic for nonlinear annuity taxation, it differs markedly from the corresponding results for the optimum nonlinear tax on capital income. These typically show a zero marginal rate for the high-income individual and the same, but possibly distorted, for the low-income individual (see, e.g., Ordover and Phelps 1979, Brett 1998). The paper proceeds as follows: Section II provides an intuitive example, illustrating that taxation of annuity payoffs indeed extracts more tax revenue from the long-living individuals, compared to an equivalent wage tax. Section III contains the detailed analysis of the properties of a linear as well as of a nonlinear tax on annuity payoffs in a Mirrlees-type model of optimum income taxation. Section IV provides concluding remarks.

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II. Equivalent taxation As is well-known from straightforward textbook analysis (see, e.g., Stiglitz 2000), a proportional tax on wages (leaving capital income untaxed) is equivalent to a proportional consumption tax (leaving saving untaxed but burdening dissaving). We study some simple examples in order to derive an intuition of how this equivalence extends to the case of life annuities. Consider a group of N identical individuals who live for two periods. Each individual survives to the second period with probability π = 0.5. Let wage income in period 0 be w = 300. Suppose that after paying a wage tax with rate t = 0.2 each individual spends one third of net income wn = 240 on annuities, that is, a = 80. We assume, for simplicity and in order to concentrate on the insurance aspect of annuities, that the interest rate is zero. Given the fair payoff rate q = 1/π = 2 per unit of annuity, the individual receives 160 in period 1. Altogether we have w = 300, wn = 240, a = 80, c0 = 160, c1 = 160,

(1)

and total tax revenue is 60N. On the other hand, if expenses for annuities are deductible, but payoffs are fully taxed with rate t = 0.2, the individual is exactly in the same situation as before, if she chooses a higher annuity demand of a = 100; then we have: w = 300, wn = 260, a = 100, c0 = 160, c1 = 160.

(2)

Note that the government receives 40 as a wage tax from every individual and another 40 from the tax on annuity payouts, but only from the surviving individuals. Therefore, total tax revenue is 40N + 40N/2 = 60N, as before. Next assume that there are two groups L,H of individuals, where each group is of equal size N and characterised by a differing survival probability: πL = 1/3, πH = 2/3, otherwise they are identical. Insurance firms cannot distinguish between the groups, hence in equilibrium there exist only - so-called - pooling contracts with the same payoff rate for each individual. We assume for the moment that both groups buy the 5

same amount of annuities, then the payoff rate, which allows zero profits, is q = 1/((πL + πH)/2) = 2. In case of a wage tax with rate t = 0.2 the situation for each group is as described in (1) above, and (2) continues to reflect the effect of a consumption tax. However, there is one interesting aspect to observe: tax revenues TL from group L are 60N with the income tax, but only 40N + 40N/3 < 60N with the consumption tax. Still, consumption is the same with both kinds of taxes. This leads us to the question: Why is paying less taxes not to the advantage of group L? The answer is that with the consumption tax this group has to invest more into annuities, compared to the situation with the income tax (namely 100 instead of 80), which provide a lower than fair rate to them. If the tax payment of 60N were unchanged, this increased annuity demand ∆a = 20, given q < 1/πL, would reduce the short-living group’s lifetime income by the amount of ∆a(1 – qπL)N = 20(1 – 2/3)N. From these considerations, it is obvious that the smaller tax payment ∆TL = 20N/3 just compensates the short-living group for the disadvantage arising from the increased demand, given the lower than fair rate of return. Moreover, one could say that, via this increased annuity purchase, group L implicitly finances the additional tax amount, which in fact the group with the higher life expectancy (for which the situation is vice versa, as q > 1/πH) has to pay in case of consumption taxation. Altogether, both groups are as well off in either tax regime. We can generalise this example by assuming that the share αi of group i = L,H (with αL + αH = 1), and the first-period income wi need not be the same for each group i. In an annuity market, which is characterised by asymmetric information and price competition, a single rate of return, offered to both types of individuals, prevails in equilibrium (see the Introduction, in particular footnote 6). This so-called pooling rate of return is implicitly defined by the zero-profit condition that aggregate revenues must equal aggregate expected payoffs (the interest rate is still assumed to be zero), i.e.

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α La L + α Ha H − q( π L α La L + π Hα Ha H ) = 0 .8

(3)

Moreover, the individuals are assumed to have no bequest motive, which means that saving in riskless bonds is not an attractive strategy for them to provide for old-age. This follows from the result by Yaari (1965), mentioned in the Introduction, that annuities offer a higher return than bonds. Taking this into account, we determine the lifetime budget constraint of an individual i, in case of an income tax, by combining c i0 = w i (1 − t) − a i and c 1i = qa i and, in case of a consumption tax, by combining c i0 = (1 − t)(w i − a i ) and c 1i = q(1 − t)a i . In either tax regime one gets (by elimination of

ai) c i0 + c 1i q = w i (1 − t) .

From this it is obvious that an individual i chooses the same consumption path over her lifetime in either tax regime (for arbitrary preferences over c i0 and c 1i ), if the rate co of return is the same. In that case, a in i = a i (1 − t) , where the superscripts “in” and

“co” indicate the respective tax regime.9 As a result, total tax revenues from group i are Tiin = α i w i t with an income tax and

(

are Tico = α i tw i + t(qπ i − 1)a ico

)

with a consumption tax. One finds Tico ≶ Tiin

depending on πi ≶ 1/q. Thus, as above, group L pays less with a consumption tax than with an income tax in case of a pooling payout rate, while the opposite is true for group H. However, as with the former tax group L has to buy more insurance, which offers unfavourable conditions, it is equally well-off, though it pays less taxes, and vice versa for group H.10

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Note that in general demand ai depends on the rate of return q. One can show that at least one equilibrium rate of return exists, which is characterised by P’(q) ≤ 0, where P(q) denotes the LHS of (3). For a precise characterisation of the equilibrium rate of return, see Abel (1986). Note also that adding a bequest motive (and, thus, a motive for savings in bonds) would not change the result that in either tax regime the same lifetime budget constraint holds. Given a zero interest rate, the latter would read c i0 + c 1i / q + s i0 − s i0 / q + s 1i / q = w i (1 − t) , where sit , t = 0,1, denotes savings in bonds (or bequests) in each period. Hence, again we see that an individual i chooses the same consumption levels irrespective of the prevailing tax regime, for any given q. co in As above we find that the change in tax revenues T i − T i of each group i equals exactly the co in change of lifetime income α i (a i − a i )(qπ i − 1) for the respective group, which occurs due to their in co increased annuity demand a ico − a ini = ta ico (use the above result that a i = a i (1 − t) ) for in unchanged T i .

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Finally, we observe that total revenue TLco + THco of both groups from the consumption tax equals total tax revenue TLin + THin = t(α L w L + α H w H ) from the income tax, if q is the pooling payoff rate determined by (3). Obviously, if q is below that rate, then a consumption tax raises less revenue than an income tax: the taxable base is reduced because of administrative costs and profits of the insurance companies, which are not accounted for in the present model.

III. An annuity tax in addition to an optimum nonlinear income tax From the example in Section II we learn that taxing payoffs from annuity contracts not only changes the time path of tax payments, but also the tax burden falling on a particular group, given that survival probabilities differ across groups and that there is a pooling rate of return. However, in this example, the two tax systems still were equivalent, leaving welfare of both groups unaltered, due to a compensating effect from increased annuity purchases. In the present section we keep the assumption of asymmetric information and price competition in the annuity market and turn to the question of what is the role of annuity taxation within an optimum-taxation framework. It is straightforward to see that in a Ramsey (1927) model with a representative individual (characterised by the wage rate and the life expectancy), the well-known results on indirect taxation, interpreted for the intertemporal consumption decision, apply (see, e. g., Atkinson and Stiglitz 1980, p. 442f). Therefore we analyse a model with differing individuals in the tradition of Mirrlees (1971), where a redistributive motive for taxation arises. That is, we consider an optimally designed tax on labour income, which redistributes from high- to low-income individuals, and ask whether, in addition, a tax on annuities increases social welfare by extending the scope for redistribution. The intuition is that, if income and life expectancy are positively associated, the latter tax, by shifting some burden from the short-living group to the long-living group, through a related mechanism as described above, also allows further redistribution along income. We consider an economy with two types of individuals, who differ in their ability and hence in their wage rate bi, i = L,H, with bL < bH, and in their probability of survival to 8

the second period. We again assume πL < πH and, thus, a positive correlation between the wage rate and life expectancy, which is plausible from empirical studies, as mentioned in the Introduction. Preferences are described by the utility function u(l i ,c i0 ,c 1i ; π i ) , where li denotes labour supply. The utility function is assumed to be

increasing in c i0 ,c 1i , decreasing in li, concave with respect to l i ,c i0 ,c 1i , and to depend positively on the survival probability, i.e. ∂u ∂πi > 0 . Note that in case of expected utility, which is typically assumed for the study of old-age provision under longevity risk (see e.g., Abel 1986, Brugiavini 1993, Walliser 2000), u(l i ,c i0 ,c 1i ; π i ) reads ˆ i ,c i0 ) + π i w(c ˆ 1i ) , u(l i ,c i0 ,c 1i ; π i ) = u(l

(4)

where uˆ is (strictly concave) per-period utility derived from labour and consumption ˆ describes utility of consumption in retirement. in period 0, while (strictly concave) w

As is usual in optimum-taxation theory in the tradition of Mirrlees, we assume that the authority does not know individual abilities, but only gross income. Thus, the tax system consists of a (nonlinear) tax on gross income and, in addition, of a tax on annuity payoffs. Concerning the latter, we consider two different models: In the first one, we ask whether the introduction of a linear tax improves welfare, while in the second we analyse an optimum tax system which is fully nonlinear with respect to income from both labour and annuities.

III.1 Linear taxation of annuity payoffs This case could be seen as a dual income tax system, where annuity payoffs are taxed separately from labour income, with a uniform rate τ. Let zi ≡ bili denote gross income, and xi net income. With the tax, consumption in the period of retirement is c 1i = q(1 − τ)a i and the budget constraint of an individual with given net income reads c i0 + c 1i (q(1 − τ)) ≤ x i . We use the notation Q ≡ 1 (q(1 − τ)) (this is the effective price of

retirement consumption), and define the indirect utility function, depending on zi and

{

}

xi, vi(xi,zi,Q) ≡ max u(z i b i ,c i0 ,c 1i ; π i ) c i0 + Qc 1i ≤ x i ,c i0 ,c 1i > 0 , for any price Q, influenced by the tax rate τ on annuity payoffs. This maximisation problem also gives us optimum consumption c 1i(x i ,z i ,Q) .

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As is usual in models of optimum nonlinear income taxation, we have to assume that preferences fulfill the single-crossing condition AM: − (∂v L / ∂z L ) /( ∂v L / ∂x L ) > −(∂v H / ∂z H ) /( ∂v H / ∂x H ) , for any x, z, and Q.11 Otherwise, redistribution via the tax on labour income could go from the less to the more able individual, because the latter would choose to work so little that her gross income is below that of the former. As already mentioned, the basic element of this Mirrlees-type model is asymmetric information concerning individual abilities, i.e., wage rates. In the model, this is accounted for in the usual way through the so-called "self-selection constraints": the government assigns two gross and net income positions to the individuals in such a way that each does not prefer the position assigned to the other: v L (xL ,zL ,Q) ≥ v L (xH ,zH ,Q) ,

(5)

v H (xH,zH,Q) ≥ v H (xL ,zL ,Q) .

(6)

The tax on wage income is implicitly defined as the difference between zi and xi. In a general formulation of the problem, the government takes into account that by imposing the taxes (i.e. by fixing zi and xi), it influences the demand for annuities and by this their rate of return before taxation. Then for any given gross and net income positions zL, zH, xL, xH and a given tax rate τ on annuities, equilibrium values of the rate of return q and the price Q are simultaneously determined by two equations, by the zero-profit condition (let αL = αH)12 [c 1L (x L ,z L ,Q) + c 1H (x H,z H,Q)] / q − [ π L c 1L (x L ,z L ,Q) + π Hc 1H (x H,z H,Q)] = 0

(3’)

and by Q = 1/(q(1−τ)). Let from now q(xL ,zL ,xH ,zH , τ) and Q(xL ,zL ,xH ,zH , τ) denote equilibrium values in this sense. Then v i (xi ,zi ,Q(xL ,zL ,xH ,zH , τ)) , which we abbreviate by v i (xi ,zi ,Q(⋅)) in the following, describes the welfare level of individual i associated 11

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The condition was called "agent monotonicity" by Seade 1982. It is a "single crossing condition", because it implies that indifference curves of the two individuals in (z,x)-space cross only once. See also Brunner 1989. Throughout Section III, for sake of simplicity we neglect - possibly differing - group shares αL, αH. Introducing them would not change the results.

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with the tax policy, and c1i(xi ,zi ,Q(⋅)) is the corresponding optimum consumption in

the

retirement

period,

which

also

gives

us

annuity

demand

ai (x i,zi,Q(⋅)) = Qc1i (xi,zi,Q(⋅)) . Assuming a weighted utilitarian social objective (with weights ρL, ρH), the problem of the optimum nonlinear income tax reads, for any given tax τ on annuity payoffs: max ρL vL (xL ,zL ,Q(⋅)) + ρHvH (xH,zH,Q(⋅))

(7)

s. t. xL + xH ≤ zL + zH + σ[ πL c1L (xL ,zL ,Q(⋅)) + πHc1H (xH,zH,Q(⋅))] − G

(8)

vH (xH,zH,Q(⋅)) ≥ vH (xL ,zL ,Q(⋅))

(9)

xi ,zi ≥ 0 , i = L,H.

(10)

Equation (8) is the resource constraint of the economy with G as the required tax revenue and with σ ≡ τ /(1 − τ) . In principle, in addition to the self-selection constraint (9) for the high-wage (and long-living) individuals, the analogous restriction has to be included for the low-wage (and short-living) individuals. However, one can show that with sufficient importance of the latter in the objective function (7), the self-selection constraint for the low-wage individuals is not binding in the optimum, while (9) is. Note that we assume that individual H, when she considers mimicking, that is, when she thinks of opting for gross and net income assigned to the low-wage individual, takes the price Q(xL ,zL ,xH ,zH , τ) as fixed at its equilibrium level in case of non-

mimicking.13 This is in line with the usual assumption of price-taking consumers. The first-order conditions for an optimum solution of (7) - (10) with respect to xi,zi, i = L,H are given in Appendix A. We denote by S(τ) the optimum value of the objective function (7), for given τ. Inserting the zero-profit condition (3’) into (A.9) in Appendix A, the derivative of S with respect to τ, at τ = 0, can be written as, ∂S µ ∂q ∂q ∂q υ ∂vH ∂q = (aL + aH )( + aL + aH )+ (q − )(aH [L] − aL ). ∂τ τ=0 M ∂τ ∂xL ∂xL M ∂xL ∂τ

(11)

where aH [L] denotes annuity demand, which the high-wage individual would choose in 13

case

of

mimicking,

and

M

is

defined

in

Appendix

A

as

Otherwise it would read Q(xL ,zL ,xL ,zL , τ).

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M ≡ q + (∂q ∂xL )aL + (∂q ∂xH )aH . One observes that the first part on the RHS of (11) refers to that effect of an introduction of τ, which concerns the resource constraint (8) (multiplier µ), while the second part refers to the self-selection constraint (9) (multiplier υ).14 To facilitate interpretation, we first consider a situation where the equilibrium rate of return is taken as fixed, independent of xi and τ. In this case, using ∂q / ∂xi = ∂q / ∂τ = 0 , we have M = q and (11) becomes ∂v ∂S = υ H (aH [L] − aL ) , ∂τ τ=0 ∂xL

(12)

because the first term in square brackets in (11) vanishes: The effect on the resource constraint is zero. Intuitively, this follows from the fact that the amount of tax, collected through the introduction of τ, can be fully returned to the individuals via the reduction of the tax on labour income, that is, via an increase of xL and xH, thus welfare is not affected. Behind this mechanism is the well-known result that the deadweight loss of a distorting tax is of second order, thus the marginal deadweight loss of its introduction is zero. We conclude from (12), in view of υ > 0 and ∂v H / ∂x L > 0 , that a linear tax on annuity payoffs, in addition to the optimum nonlinear income tax, improves social welfare, if the more able individual, in case of mimicking the less able individual, has a larger annuity demand than the latter. The reason is that then the introduction of an annuity tax means that mimicking becomes less attractive, which gives slack to the selfselection constraint, and more redistribution via the income tax becomes possible. What can be said about this relation in our model? The difference in annuity demand is determined by two effects: the type-H individual has a higher life expectancy than the type-L individual, and she has to work less to earn the same gross income. One observes immediately that for arbitrary preferences no clear-cut statement on the overall effect can be given. However, for a specific, but plausible demand behaviour the following result applies: 14

The weights of the individuals do not appear in (11), because, when formulating (7) – (10), we have already assumed that with the optimum solution the self-selection constraint for individual H is binding. This assumption, which corresponds to the usual way the problem is formulated, in turn hinges on a sufficient importance of the disadvantaged individual L in the objective function.

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Result 1: Consider the case that the rate of return is independent of the taxes. Then a linear tax on annuity payoffs, in addition to the optimum nonlinear income tax, improves social welfare, if demand for annuities increases with the survival probability and does not increase with labour time.

To interpret this result, note first that the assumption of a larger annuity demand of individuals with high life expectancy is indeed reasonable. It holds unambiguously in case of expected utility as described in (4): For any given pair of net income x and working time l and for any given price Q, the long-living individual demands more annuities than the short-living. (This can be seen by implicit differentiation – with respect to π i – of the first-order condition for the maximisation of (4), subject to c i0 + Qc 1i ≤ x .) As to the second effect, a non-positive association of annuity demand

with labour time is clearly guaranteed, if preferences are weakly separable between labour (leisure) and consumption in both periods, i.e. if annuity demand is independent of labour time, for given net income.15 It is interesting to compare the above result with corresponding ones concerning the role of a tax on capital income. In these models the interest rate is usually assumed to be independent of taxation. As is well-known, given that preferences are weakly separable between labour (leisure) and consumption in different periods, no tax on capital income in addition to the optimum nonlinear income tax is desirable (see, e.g. Atkinson and Stiglitz 1980, Ordover and Phelps 1979). A tax on capital income is only desirable, if saving is positively associated with leisure (compare also Corlett and Hague 1953). In the present model, however, the desirability of the tax on annuity payoffs in fact results, as long as saving (that is, annuity demand) does not decrease too much with leisure, so that the increase in annuity demand of individual H due to her higher life expectancy dominates.

15

Note in passing that exactly the same formula (11) would arise in a first-best world, with individually fair rates of return qL = 1/πL, qH = 1/πH used instead of the zero-profit condition (3’) in (A.9) to derive (11). One can show that in the expected-utility case aH[L](qH) > aL(qL) holds, thus a result similar to the above applies.

13

Next we turn to the general case, when the government takes into account that the taxes change the composition of annuity demand between the two types – i.e. the extent of adverse selection – and by this the equilibrium rate of return. Obviously, the above Result 1 still applies, if this change is sufficiently small, that is if ∂q / ∂τ and ∂q / ∂xL , ∂q / ∂xH in (11) are sufficiently close to zero. Further, one notes that the denominator M can regularly be taken as positive; this is guaranteed as long as an increase of xL or xH increases social welfare, in the absence of both the resource constraint and the self-selection constraint (see Appendix A). One observation in this general case is that the marginal effect of τ as far as the resource constraint is concerned, need not be zero. It depends, as the first part in (11) shows, on ∂q / ∂τ + (∂q / ∂xL )aL + (∂q / ∂xH )aH , which is the overall change of q (that is, how adverse selection is affected), caused by the introduction of τ and by the compensating increase of xL and xH to maintain a balanced public budget.16 Obviously, for general preferences, this effect can be positive or negative. However, in case of expected utility and separability, and with utility of per-period consumption characterised by constant relative risk-aversion, one can show (see Appendix B) that the overall effect is negative: (∂q / ∂xL )aL + (∂q / ∂xH )aH is negative and dominates ∂q / ∂τ , which can be negative or positive, depending on the degree of risk aversion. Further, as far as the self-selection constraint is concerned, ∂q / ∂τ , i.e. the marginal effect of τ on adverse selection, occurs in the general case (compare the second part of (11) with (12)). Still, q − ∂q / ∂τ can usually be taken as positive, which is equivalent to the statement that the introduction of the annuity tax indeed decreases the net rate of return, i.e. that ∂(q(1 − τ)) / ∂τ < 0 at τ = 0 . (In other words, q − ∂q / ∂τ is positive, if the second-round effect of τ does not lead to an increase of q which is large enough to outweigh the first-round effect. q − ∂q / ∂τ > 0 can explicitly be shown for expected utility, see Appendix B). As a consequence, the overall effect of τ on the selfselection constraint is still positive, given that aH [L] > aL , as in the specific case ∂q / ∂τ = 0 , discussed above.

16

Note that q(πLaL + πHaH) = aL + aH is the marginal increase of tax revenue due to the introduction of τ, thus aL + aH can be returned to the individuals to keep the public budget balanced.

14

Moreover, if the introduction of τ (and the compensating increase of xL and xH as above) increases adverse selection (that is, if ∂q / ∂τ + (∂q / ∂xL )aL + (∂q / ∂xH )aH < 0 , which implies (q − ∂q / ∂τ) M > 1), this has an effect similar to that of the introduction of τ itself, in reducing the attractiveness of mimicking further. This in turn allows even more redistribution and is positive for social welfare, given that aH [L] > aL . On the other

hand,

if

τ

decreases

adverse

selection

(that

is,

if

∂q / ∂τ + (∂q / ∂xL )aL + (∂q / ∂xH )aH > 0 , therefore (q − ∂q / ∂τ) M < 1 ), the effect on social welfare is reduced, but still positive. For expected utility we know already that the former case (q − ∂q / ∂τ) M > 1 occurs (see above). We conclude that there are two opposite effects related to the change of adverse selection due to the introduction of τ. If adverse selection decreases, this means a direct welfare gain, but favours mimicking of individual H and thus reduces the scope for redistribution to some extent. However, the overall welfare effect of τ is positive. If adverse selection increases, both effects are the other way around, and the total effect on social welfare is undetermined. Altogether, we have: Result 2: Assume that type H, when mimicking type L, has a larger annuity demand than the latter, i.e. aH[L] > aL. If the introduction of a linear tax on annuity payoffs increases (decreases) the rate of return q on annuities, then the effect on social welfare is positive (is undetermined). In case of a separable expected-utility function with constant relative risk aversion, the effect on social welfare is undetermined.

III.2 Nonlinear taxation of income from labour and annuities

A fully nonlinear system is based on the idea that in fact, by choosing the appropriate bundle of gross income and net income, an individual reveals her type in the first period of life (see, e. g., Brett 1998, Pirttila and Tuomala 2001 in the context of capital income taxation). This means that in the second period the tax on income from annuities can be imposed separately on each type. Following this reasoning, we consider a tax system, which consists of a (nonlinear) tax T(z), imposed on gross income, and two (nonlinear) taxes TL(qaL), TH(qaH), depending on annuity payoffs. Technically, this means that the self-selection constraint now has the form

15

v H (c H0 ,c1H,zH ) ≥ v H (cL0 ,c1L ,zL ) ,

where v i (c i0 ,c1i ,z) ≡ u(z / bi ,c i0 ,c1i , πi ) . That is, the government has to select two complete bundles of labour time (or gross income, equivalently) and consumption in both periods, such that the more able person does not prefer the bundle assigned to the less able. (Again we assume a-priori that the government wants to redistribute income from the former to the latter type.) As before, a single-crossing condition is required,

{

which

has

the

same

form

as

}

AM,

but

with

vi(x,z,Ti) ≡

max u(z b i ,c i0 ,c 1i ; π i ) c 0 + N i−1(c 1i ) q ≤ x, c i0 ,c 1i > 0 , where N i−1 is the inverse of the

net income function N i (qa i ) ≡ qa i − Ti (qa i ) . AM has to hold for appropriate Ti. With these preparations we can formulate the planner's problem as max ρL v L (cL0 ,c1L ,zL ) + ρHv H (cH0 ,c1H,zH ),

(13)

s. t. cL0 + c H0 + πLc1L + πHc1H ≤ zL + zH − G,

(14)

v H (c H0 ,c1H,zH ) ≥ v H (cL0 ,c1L ,zL ),

(15)

c i0 ,c1i ,zi ≥ 0 ,

(16)

c i0 ,c1i ,zi

i = L,H.

Note that in this maximisation problem the rate of return on annuities does not occur. As already mentioned, the government directly determines optimum second-best consumption and labour time bundles, subject to the overall resource constraint and to the self-selection constraint. A consequence of this formulation is that no effects on the rate of return caused by government policy have to be considered. However, these effects must be observed, when it comes to the implementation of the optimum second-best bundles through appropriate tax functions on labour income and on annuity payoffs. When constructing these functions, the government must be aware of their influence on annuity demand, and in particular, on adverse selection, which determines the rate of return. Put differently, the tax functions must be based on a rate of return, which indeed arises as a market equilibrium with zero profits.

16

The first-order conditions of the maximisation problem (13) – (16) can be found in Appendix C. The solution has the following properties: Proposition 1: For type H the optimum nonlinear tax on annuity payoffs exhibits a positive marginal tax rate equal to (q − 1/ πH ) q . For type L, the sign of the marginal tax rate is undetermined. Proof: Consider the optimisation problem of an individual i who determines labour

time and consumption, taking the tax function and the rate of return as given:

max u(zi / b,c i0 ,c1i ; πi ), s.t. c i0 = zi − T(zi ) − ai and c1i = qai − T(qa i i) From the corresponding first-order conditions, one derives an expression for the marginal tax rate Ti′ in terms of the marginal rate of substitution as Ti′ =

∂u / ∂c i0 ∂v i / ∂c i0 1 1 (q − ) = (q − ), q ∂u / ∂c1i q ∂v i / ∂c1i

i = L,H,

(17)

where the second equality is immediate from the definition of v i . On the other hand, from (C.4) and (C.5) in Appendix C we find that in the optimum ∂v H / ∂cH0 1 = 1 ∂v H / cH πH must hold, thus the first part of the Proposition is proved. For type-L individuals, (C.1) and (C.2) tell us that in the optimum  v H / ∂c L0 ρL ∂v L / ∂c L0 − υ∂ 1 = , 1 1  v H / ∂c L πL ρL ∂v L / ∂c L − υ∂

(18)

where ∂v H / ∂c Lt , t = 0,1, describes the marginal utility of individual H in case of mimicking, i.e. opting for the type-L bundle. In general, it cannot be concluded from (18), whether (∂v L / ∂c L0 ) /(∂v L / ∂c1L ) ≶ 1/ πL and therefore the marginal tax rate on annuity payoffs is undetermined.

QED.

17

One observes that the marginal tax rate for group L will be negative, if the marginal rate of substitution between present and future consumption of individual L does not differ much from that of individual H in case of mimicking, because then (18) implies that (∂v L / ∂c L0 ) /(∂v L / ∂c1L ) is close to 1/πL, while the equilibrium rate of return q is smaller than 1/ πL . On the other hand, if individual H values future consumption more (in case of mimicking), because of her higher life expectancy, then we have (∂v L / ∂c L0 ) /(∂v L / ∂c1L ) > (∂v H / ∂cL0 ) /(∂v H / ∂c1L ) , which together with (18) implies that (∂v L / ∂c L0 ) /(∂v L / ∂c1L ) < 1/ πL .17 This in turn, used in (17), tells us that the marginal tax rate on annuity payoffs, for individual L may also be positive. It is important to notice the difference of the result in Proposition 1 to the previous ones: while the desirability of a linear tax on annuities essentially depends on the difference in annuity demand of high- and low-risk individuals, a further motive arises with a nonlinear tax: the correction of the rate of return in a pooling situation. This is more in line with the intuition developed in Section II: the loss (benefit) from pooling of the short-living (long-living) individual, compared to individually fair payoff rates, gives rise to a differentiated treatment by the tax system. It is obvious from the proof of Proposition 1 that if q was equal to the individually fair rates qL, qH, resp., then the marginal tax rate for individual H is zero, while a distortion - familiar from other Mirrlees-type models - arises for individual L. The correction of this market failure arising from asymmetric information is specific for annuity taxation; in models investigating the optimum nonlinear capital income tax, the marginal tax rate for the high-wage individual is zero, while for the low-wage individual it is distorted, except when preferences are weakly separable between consumption and leisure (Ordover and Phelps 1979). As already mentioned, in the present model the payoff rate, fixed on the annuity market, does not directly enter the planner's optimisation problem, which aims at determining optimum second-best bundles. The marginal tax rate follows from a comparison with the rate of return offered by the market. Such a correction of the rate 17

This

follows

immediately

by

rewriting

( ∂v H / ∂c1L ) ( ∂v L / ∂c1L ) > ( ∂v H / ∂c L0 ) ( ∂v L / ∂c L0 )

( ∂v L / ∂c L0 ) /( ∂v L / ∂c1L ) > ( ∂v H / ∂c L0 ) /( ∂v H / ∂c1L )

as

and

to

by

transforming

equation

(18)

(∂v L / ∂c ) ( ρL − υ (∂v H / ∂c ) (∂v L / ∂c ) )  (∂v L / ∂c ) ( ρL − υ (∂v H / ∂c ) (∂v L / ∂c ) )  = 1 πL . 0 L

0 L

0 L

1 L

1 L

1 L

18

of return through the tax system becomes possible, if we follow the idea that in the retirement period the authority can indeed identify individuals by their types, because these are revealed when gross (and net) income is reported by the end of the working period. Yet, annuity demand is expressed during this period already, when neither tax authority nor insurance firms can distinguish the types, therefore the latter are unable to offer individually fair rates. Finally, it should be mentioned that the usual properties concerning the optimum tax rates on labour income can be derived: It is zero for the more able individual but positive for the less able.

IV. Concluding comment

Private life annuities are becoming a more wide-spread instrument for old-age provision, as public pension systems are expected to provide less support in the future. However, it is well-known that the annuity market is affected by an adverseselection problem, which is a typical obstacle to many insurance markets. As a consequence, it provides only less than fair contracts for individuals with low life expectancy. In addition, empirical studies have found that life expectancy and income are positively correlated. Therefore, it appears a natural question to ask whether these facts should have an influence on the tax system, in particular on the balancing of efficiency and equity considerations. Theoretical studies on capital income taxation have shown that in a variety of models such a tax cannot fulfil any further redistributive task, given an optimally designed tax on labour income. Intuitively, one might be willing to accept a similar statement for the taxation of annuity payoffs. On the other hand, intuition also shows that such a tax falls on long-living individuals to a larger extent than on short-living and has, thus, a different effect compared to a tax on income from labour or capital. To clarify this question, we formulated a Mirrlees-type model of optimum taxation, where we made allowance for the stylised fact that life expectancy and income of an individual are positively correlated.

19

In this framework we were able to find a clear cut result in favour of a linear tax on annuity payoffs only if the rate of return on annuities is taken as constant. Then it can be used for redistribution (above the scope possible with optimal income taxation alone), if annuity demand increases with life expectancy, which is quite plausible. Generally, if also the influence of the government’s tax policy on the equilibrium rate of return is taken into account, the effect of a linear tax on social welfare is found to consist of two opposite components: On the one hand, as expected, there is a direct welfare gain, if the introduction of the linear annuity tax alleviates adverse selection in the annuity market and hence increases the rate of return on annuities. On the other hand, however, this increase in the rate of return reduces the scope for redistribution (and hence social welfare) to some extent through its aggravating effect on the selfselection constraint, but the overall welfare effect is positive. In case that adverse selection is exacerbated by the linear annuity tax, the two effects are vice-versa and the overall welfare effect is undetermined. Further we investigated an optimum tax system, which is nonlinear not only with respect to wage income but also with respect to annuity income. We found that such a tax system does not only represent an instrument for redistribution, but can also be directly employed to correct the distortion of the rate of return caused by asymmetric information in the annuity market. The normative framework of our analysis is that of the theory of optimum income taxation, where it is typically assumed that society favours redistribution from the high-ability to the low-ability individuals. In our model the high-ability types are also those with the high life expectancy, which allows us to investigate how this characteristic can be used – via annuity taxation – to increase redistribution above the extent possible with labour-income taxation alone. Obviously, a basic normative view underlying this analysis is that longevity itself does not change society’s preference for redistribution, or that society even favours redistribution from longliving to short-living individuals as well. Only then longevity represents an adequate instrument which a social planner, who strives for more redistribution, but is restricted by asymmetric information, can use to get closer to the desired distribution.

20

That indeed redistribution from the high- to the low-life expectancy persons (neglecting income for the moment) is desirable can be justified by the Rawlsian social welfare concept (which aims at maximising the utility of the worst-off group), as long as – other things equal – utility increases with survival probability, which is the case with expected utility. On the other hand, one could argue that – other things equal – a higher life expectancy means a higher marginal utility of income, which is again a property of expected utility. Then the other prevalent social objective, the (unweighted) utilitarian social welfare function, asks for less redistribution from the high-ability to the low-ability types, given the positive correlation between life expectancy and ability. Thus, it is a difficult, but important normative question in which way, if at all, life expectancy itself should be an argument for a differentiated treatment of individuals. Economic analysis like ours cannot answer that question, but can point out the appropriateness of instruments to achieve the goals set by society’s objectives.

21

Appendix A

The Lagrangian to the maximization problem (7) - (10) reads L = ρL vL (xL ,zL ,Q(⋅)) + ρHvH (xH,zH,Q(⋅)) −

(

)

− µ xL + xH − zL − zH − σ[πLc1L (xL ,zL ,Q(⋅)) + πHc1H (xH,zH,Q(⋅))] + G + + υ ( vH (xH,zH,Q(⋅)) − vH (xL ,zL ,Q(⋅)) ) ,

which

gives

us

the

first-order

conditions

(we

use

the

abbreviation

vH [L] ≡ vH (xL ,zL ,Q(⋅)) ) ∂v L ∂v L ∂Q ∂v ∂Q ∂c 1 ∂c 1 ∂Q + − µ + µσ[ πL ( L + L ] + ρH H )+ ∂x L ∂Q ∂x L ∂Q ∂x L ∂x L ∂Q ∂x L ∂c 1 ∂Q ∂v ∂Q ∂v H [L] ∂v H [L] ∂Q +πH H − − ] + υ[ H ] = 0, ∂Q ∂x L ∂Q ∂x L ∂x L ∂Q ∂x L

(A.1)

∂v L ∂v L ∂Q ∂v ∂Q ∂c 1 ∂c 1 ∂Q + − µ + µσ[πL ( L + L ] + ρH H )+ ∂z L ∂Q ∂z L ∂Q ∂z L ∂z L ∂Q ∂z L ∂c 1 ∂Q ∂v ∂Q ∂v H [L] ∂v H [L] ∂Q +πH H − − ] + υ[ H ] = 0, ∂Q ∂z L ∂Q ∂z L ∂z L ∂Q ∂z L

(A.2)

∂v L ∂Q ∂v ∂v ∂Q ∂c 1 ∂Q + ρH [ H + H + ] − µ + µσ[πL L ∂Q ∂x H ∂x H ∂Q ∂x H ∂Q ∂x H ∂c 1 ∂c 1 ∂Q ∂v ∂v ∂Q ∂v H [L] ∂Q +πH ( H + H − )] + υ[ H + H ] = 0, ∂x H ∂Q ∂x H ∂x H ∂Q ∂x H ∂Q ∂x H

(A.3)

∂v L ∂Q ∂v ∂v ∂Q ∂c 1 ∂Q + ρH [ H + H + ] − µ + µσ[πL L ∂Q ∂z H ∂z H ∂Q ∂z H ∂Q ∂z H ∂c 1 ∂c 1 ∂Q ∂v ∂v ∂Q ∂v H [L] ∂Q +πH ( H + H − )] + υ[ H + H ] = 0, ∂z H ∂Q ∂z H ∂z H ∂Q ∂z H ∂Q ∂z H

(A.4)

ρL [

ρL [

ρL

ρL

where ∂Q ∂x i , ∂Q ∂z i , i = L,H, describes the effect of a marginal increase of type i’s net, gross income, resp., on the equilibrium price Q(⋅), as explained in the text. Using the Envelope Theorem we get

22

∂v ∂v ∂Q ∂S ∂σ = (ρ L L + ρ H H ) + µ ( π Lc 1L + π Hc 1H ) + ∂τ ∂Q ∂Q ∂τ ∂τ ∂c 1 ∂c 1 ∂Q ∂v ∂v [L] ∂Q , + µσ( π L L + π H H ) + υ( H − H ) ∂Q ∂Q ∂τ ∂Q ∂Q ∂τ

(A.5)

where ∂Q ∂τ describes the effect of a marginal increase of the annuity tax on the 2

equilibrium price Q(⋅). For τ = 0, ∂Q / ∂τ = (q − ∂q / ∂τ) q where ∂q / ∂τ is determined by implicit differentiation of the zero-profit condition (3’). By Roy's Lemma we have ∂v i / ∂Q = −c1i (∂v i / ∂xi ) . Using this, (A.5) reads at τ = σ = 0 ∂v ∂v ∂S 1 ∂q = −(ρL L c1L + ρH H c1H ) 2 (q − ) + µ( πLc1L + πHc1H ) + ∂τ τ=0 ∂xL ∂xH ∂τ q ∂v ∂v [L] 1 ∂q + υ( − H c1H + H c1H [L]) 2 (q − ). ∂xH ∂xL ∂τ q

(A.6)

where c1H [L] ≡ c1H (xL ,zL ,Q(⋅)) denotes retirement consumption, which individual H would choose, if endowed with gross and net income of individual L. Using again Roy’s Lemma and solving the two equations (A.1) and (A.3) simultaneously for ρL ∂vL ∂xL and ρH ∂vH ∂xH at τ = σ = 0 yields ρL

ρH

∂vL ∂xL ∂vH ∂xH

=

∂v [L] q ∂Q ∂Q ∂Q ∂Q {µ [1 + ( − )c1H ] + υ H [1 − c1H [L] − c1H )]} , M ∂xL ∂xH ∂xL ∂xL ∂xH

(A.7)

=

∂v [L] ∂Q 1 ∂v q ∂Q ∂Q (cL − c1H [L] )} − υ H , {µ [1 + ( − )c1L ] + υ H M ∂xH ∂xL ∂xL ∂xH ∂xH

(A.8)

τ=0

τ=0

2

where M ≡ [1 − (∂Q ∂xL )c1L − (∂Q ∂xH )c1H ]q . By use of ∂Q / ∂xi τ=0 = −(∂q / ∂xi ) q c1i

τ=0

and

= ai q , we obtain M ≡ q + (∂q ∂xL )aL + (∂q ∂xH )aH . Substituting (A.7) and (A.8)

into (A.6) and appropriate grouping yields ∂S µ 1 ∂q 1 ∂q = [c1L ( πLM − (q − )) + c1H ( πHM − (q − ))] + ∂τ τ=0 M q ∂τ q ∂τ υ ∂vH 1 ∂q + (q − )[c1H [L] − c1L ]. M ∂xL q ∂τ

(A.9)

Positivity of M follows, if we assume that the partial derivatives of the objective function (7) with respect to xL and xH are positive (which is quite plausible, because it

23

means, in other words, that the resource constraint is indeed binding: welfare could be increased, if more net income could be distributed to the individuals.). To see the positivity of M, we apply Roy’s Lemma to transform the partial derivatives, i.e. the first two terms in (A.1) and (A.3). That they are assumed to be positive reads then ρL

∂v L ∂v ∂Q ∂Q [1 − c 1L ] − ρHc 1H H > 0, ∂x L ∂x L ∂x H ∂x L

−ρLc 1L

∂v L ∂Q ∂v ∂Q + ρH H [1 − c 1H ] > 0. ∂x L ∂x H ∂x H ∂x H

(A.10) (A.11)

We know that ∂Q ∂xH > 0 (given the usual assumption that annuities are a normal good, therefore increasing type-H income increases adverse selection and, thus the effective price of annuities). Moreover, ∂Q ∂xL < 0 implies that 1 − c 1L ∂Q ∂x L > 0 . Multiplying (A.11) by 1 − c 1L ∂Q ∂x L and (A.10) by c1L ∂Q ∂xH and adding up does not change the sign and gives ρH

∂v H ∂Q ∂Q [1 − c 1L − c 1H ]>0, ∂x H ∂x L ∂x H

The expression in square brackets equals M/q. As ρH and ∂v H ∂x H are positive, M must be positive as well.

Appendix B

In this Appendix we restrict our attention to preferences, which are separable between labour and lifetime consumption and are of expected-utility type, with a utility function of per-period consumption showing a constant coefficient of relative risk aversion R > 0. Hence the utility function of an individual i = L,H reads  i bi) + u(z i b i ,c i0 ,c 1i ; π i ) = u(z

(c i0 ) 1−R − 1 (c 1 ) 1−R − 1 + πi i . 1− R 1− R

(B.1)

We show that with (B.1) at τ = 0 (compare the terms in (11)) the following inequalities hold: ∂q ∂τ

0, if R

1,

aL ∂q ∂aL + aH∂q ∂aH < 0 , for any R > 0,

24

∂q ∂τ + aL ∂q ∂aL + aH∂q ∂aH < 0 , for any R > 0, q − ∂q ∂τ > 0 , for any R > 0.

As a preparation, we derive, by maximisation of (B.1) subject to the budget constraints c i0 + a i ≤ x i , c1i ≤ q(1− τ)ai in both periods, an explicit formula for annuity demand ai = πi1 R xi K i ,

where K i ≡ ( q(1 − τ) )

1−1 R

(B.2) + π i 1 R , Ki > 0. The corresponding derivatives are given by

∂ai πi1 R xiq1−1 R (1− 1 R) , = ∂τ (1− τ)1 R K i2

(B.3)

∂ai πi1 R = , ∂xi Ki

(B.4)

∂ai π 1 R xi (1− τ)1−1 R (1− 1 R) . =− i ∂q q1 RK i2

(B.5)

It is appropriate to rewrite the LHS of the zero-profit condition (3’) as P ≡ 1− q

π La L + π Ha H , aL + aH

(B.6)

where aL, aH are described in (B.2). Implicit differentiation of the zero-profit condition gives us the two relations ∂q ∂P ∂τ =− , ∂τ ∂P ∂q

(B.7)

a ∂P ∂xL + aH∂P ∂xH ∂q ∂q aL + aH = − L , ∂xL ∂xH ∂P ∂q

(B.8)

with ∂P ∂q < 0 (see footnote 8). The numerators of (B.7) and (B.8) are obtained by differentiating (B.6) with respect to τ and xL, xH, resp., and appropriate grouping as ∂a ∂τ ∂a H ∂τ ∂P = N( L − ), ∂τ aL aH aL

∂a ∂a ∂P ∂P + aH = N( L − H ) , ∂x L ∂x H ∂x L ∂x H

(B.9) (B.10)

25

where N ≡ ( π H − π L )qa L a H (a L + a H ) 2 , with N > 0, as πH > πL. Further, by use of (B.2), (B.3), (B.4) and the definition of Ki we obtain for the expressions in the brackets ∂a L ∂τ ∂a H ∂τ (1 − 1 R)q 1−1 R − =− (π L 1 R − π H 1 R ) , 1R aL aH (1 − τ) K LK H

(B.11)

∂a L ∂a H ( q(1 − τ) ) − = ∂x L ∂x H K LK H

(B.12)

1−1 R

(π L 1 R − π H 1 R ) .

Inserting (B.11) in (B.9), we find ∂P ∂τ and ∂P ∂q < 0 that ∂q ∂τ

0, if R

0, if R

1, which implies by use of (B.7)

1. Analogously we find by combining (B.8),

(B.10) and (B.12) that (∂q ∂xL )aL + (∂q ∂xH )aH < 0 for any R > 0. Using (B.7) – (B.12) gives us for τ = 0 ∂q ∂q ∂q + aL + aH ∂τ ∂x L ∂x H

τ=0

1 Nq 1−1 R =− (π L 1 R − π H 1 R ) , ∂P ∂q RK LK H

(B.13)

which is negative, as ∂P ∂q < 0 , N > 0, R > 0 and πH > πL. Finally, we use (B.7) and write q − ∂q ∂τ as q−

∂q 1 ∂P ∂P = (q + ). ∂τ ∂P ∂q ∂q ∂τ

(B.14)

Differentiating (B.6) with respect to q and substituting the result, together with (B.9), into (B.14) yields q−

π a +π a ∂q 1 = {− q L L H H + ∂τ ∂P ∂q aL + aH ∂a ∂q ∂aH ∂q ∂a ∂τ ∂aH ∂τ + N[q( L − − )+( L )]}. aL aH aL aH

(B.15)

Using (B.2) and (B.5) and appropriate grouping yields

∂a ∂q ∂a H ∂q (1 − 1 R) ( q(1 − τ) ) q( L − )= aL aH K LK H

1−1 R

(π L 1 R − π H 1 R ) ,

(B.16)

26

which at τ = 0 is equal to −[(∂a L ∂τ) a L − (∂a H ∂τ) a H ] (see (B.11)). Hence, the last term in the squared brackets on the RHS of (B.15) is zero and (B.15) reduces to (q −

∂q q πLaL + πHaH ) =− , ∂τ τ=0 ∂P ∂q aL + aH

(B.17)

which is positive, as ∂P ∂q < 0 .

Appendix C

We write the Lagrangian function for (13) - (16) as

(

)

L = ρL v L (cL0 ,c1L ,zL ) + ρHv H (cH0 ,c1H,zH ) − µ cL0 + cH0 + πL c1L + πHc1H − zL − zH + G +

(

+ υ v H (cH0 ,c1H,zH ) − v H (cL0 ,c1L ,zL )

)

and get the first-order conditions: ρL

∂v L ∂v − µ − υ H0 = 0, 0 ∂c L ∂c L

(C.1)

ρL

∂v L ∂v  L − υ H1 = 0, − µπ 1 ∂c L ∂c L

(C.2)

ρL

∂v L ∂v H   + µ − υ = 0, ∂zL0 ∂zL

(C.3)

ρH

∂v H ∂v − µ + υ H0 = 0, 0 ∂c H ∂c H

(C.4)

ρH

∂v H ∂v H   − µπ + υ = 0, H ∂c1H ∂c1H

(C.5)

ρH

∂v H ∂v + µ + υ H = 0. ∂zH ∂zH

(C.6)

27

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