Asymmetric information and mandatory social annuitization

Asymmetric information and mandatory social

annuitization

Yang Jiang

University of Groningen August 2013

Abstract: We study the role of mandatory social annuities in the society in the presence of asymmetric information in the annuity market: only the clients know their health status, but they are unwilling to reveal these information to the annuity firms or the government. Those with good health tend to buy more annuities while the relatively unhealthy abstain from the annuity market. Contrary to our intuition, the uniform mandatory social annuities prove to be welfare reducing, even though they offer a higher rate of return to the clients. This is because they crowd out private annuities and thus aggravate the adverse selection in the private annuity market. But if the government could collect more information and mitigate the damaging effect of adverse selection, will the mandatory social annuities be beneficial to the society? The answer is yes. In the paper we prove that a mandatory social annuity system with a progressive contribution rate (the relatively healthy contribute more into the annuity pool) enhances the social welfare. We also notice that its contribution is marginal: the whole society will be better off without any annuity market at all.

Keywords: Private annuity markets, mandatory social annuities, adverse selection, overlap-ping generation.

JEL Codes:D52, D91, E10, J10.

∗_{Faculty of Economics and Business, University of Groningen, P. O. Box 800, 9700 AV Groningen,}

1

Introduction

Although death is one of the certainties in life for everybody, the date at which it occurs is unknown to all of us. Faced with life-time uncertainty, rational agents must balance the risk of leaving unintended bequest against the risk of running out of resources in old age. As was shown in the classic analysis of Yaari (1965) and more recently by Davidoff et al. (2005), life annuities are very attractive insurance instruments in the presence of longevity risk. In-tuitively, annuities allow for risk sharing between long-lived and short-lived individuals (Heijdra and Reijnders, 2012). From a microeconomic perspective (i.e. in a partial equilib-rium setting), it is welfare improving to purchase life annuities, given that agents have no bequest motive and the payouts from the annuity exceeds that of conventional assets under market completeness (Davidoff et al., 2005).

However, empirical evidence has revealed that, in most countries voluntary annuitiza-tion is almost non-existent. The lack of annuitizaannuitiza-tion is especially surprising given the large welfare gains from annuities in life cycle models. Calibrated models suggest that typical 65-year-olds would be willing to pay one-fourth of their wealth for access to actuarially fair annuities (Mitchell et al., 1999). The drastic contrast between the theory and the real world has been dubbed the “annuity puzzle”.

Several explanations for the low participation in annuity markets have been given in the literature. First of all, individuals may have a bequest motive. They may wish to leave an inheritance to their offspring. Second, people may feel uncomfortable to bet on a long life. Annuities are paying off only if they live long enough, and most people would hate to die before having received at least their initial investment. Third, adverse selection plagues in annuity markets. Healthier people tend to buy more annuities from the annuity market. As a consequence, low-mortality (high-risk) individuals are overrepresented in the annuity mar-ket. Annuity firms will take this effect into account when pricing their products, otherwise they will incur a loss if they offer the rate of return based on average surviving probabilities. And this higher price will in turn aggravate the adverse selection problem, which lowers the participation rate in the annuity market.

im-proving. There are two key mechanisms that are ignored in the partial equilibrium analysis. First, in the absence of private annuities, there will be accidental bequests left by the short-lived agents, which are distributed to the surviving elders or to the younger generation. These accidental bequests may boost the consumption or investment opportunities of those agents who receive the intra-generational or inter-generational transfer. Second, the avail-ability of annuities affect the rate of return on an individual’s savings. Aggregate capital accumulation will depend on the annuity’s rate of return if it is available. In turn wages and the interest rate are determined by aggregate capital accumulation. In a general equilibrium framework, Heijdra et al. (forthcoming) find that there exists a phenomenon which they call the tragedy of annuitization: although full annuitization of assets is privately optimal, it may not be socially beneficial due to adverse general equilibrium repercussions. Intuitively, private annuities redistribute the assets to the surviving elderly for consumption whereas transferring the bequests to the young will boost their investment opportunities. In the end, the beneficial saving effects are dampened by the annuity market. Therefore, the welfare effects of the annuity market remains controversial in the general equilibrium.

To understand what makes the annuity market ineffective, we first need to tackle the adverse selection problem. To avoid the low rate of return in the private annuity market induced by the adverse selection, many policy makers introduce mandatory social annuities to the society, which is immune to adverse selection and forces everyone to participate. This makes it an attractive option for the government: it offers participants higher rate of return than the private annuity market, since the high-mortality (low risk) individuals are forced to join in the pool. However, Heijdra and Reijnders (2012) proved that in a closed econ-omy populated by overlapping generations of heterogeneous agents who are distinguished by their health status, the mandatory social annuities have a negative effect on steady-state capital intensity and welfare in a general equilibrium. The positive effect of a fair pooled rate of return on a fixed part of savings and a higher return on capital in equilibrium is out-weighed by the negative consequences of increased adverse selection in the private annuity market and a lower wage rate.

the macroeconomic equilibrium, and the wealth distribution. To this end, we build on the work of Heijdra and Reijnders (2012) and develop an analytical model of a closed economy populated by overlapping generations of heterogeneous agents. In contrast to the original model we acknowledge the existence of two (rather than one) sources of heterogeneity. In particular, individuals can be distinguished by their health status as well as their inherent ability. Following Heijdra and Reijnders (2012) we assume that an agent’s health type is pri-vate information so that there will be a pooling equilibrium in the pripri-vate annuity market. In addition we assume that inherent ability is positively correlated with an agent’s health status, although the correlation is less than perfect, i.e. there are healthy-unproductive and unhealthy-productive agents in the economy. Technically, we construct a bivariate uniform distribution of health status and inherent ability. The joint density can be derived using the Fairly-Morgenstern Family approach (Rice, 2007). Thus we could extract the information of agents’ health status by observing their inherent ability. However, an individual’s inherent ability is also private information that can not be directly observed. But the government may be able to indirectly observe an individual’s ability by looking at the wage rate he/she receives, which can be inferred from the income-tax system. With the information collected (although not perfect), the government may design a social security system that contain re-distributionary features (e.g. a progressive contribution rate). People with higher income will be asked to contribute more to the annuity pool. In the original work of Heijdra and Reijnders (2012) the positive effect of the social annuities (a fair pooled rate of return) is out-weighed by the negative consequences of increased adverse selection in the private annuity market. With an additional source of heterogeneity, which can be observed indirectly (there-fore more information could be extracted), we revisit this question and study whether this asymmetric information problem could be alleviated or solved.

2

Model with one source of heterogeneity (health status µ)

2.1 Bequests to the young

2.1.1 Consumers

In a two-period overlapping generation framework, we assume that consumers live for two periods, which are termed ‘youth’ (superscript y) and ‘old age’ (superscript o). Not every-body is lucky enough to live for two full periods, some people may pass away at the end of the first period because of poor health. Thus we differentiate the consumers by their health types, while healthy people have a higher probability to survive into the second period. The survival probability is denoted by µ. A type µ consumer is expected to live 1+µperiods. If the cohorts are large enough, exactly fraction µ of the young people will survive into the second period (law of large numbers). Note that the life uncertainty only occurs at the end of the first period, consumers are safe during the two periods (if they are alive).

In our model we assume that labour supply is endogenous, and let leisure enter the utility function (consumers derive utility not only from consumption but also from leisure). The time endowment of a consumer is normalized to 1, and leisure is the time left from work. The expected life-time utility of a consumer of health type µ born in period t is given by:

EΛt(µ) ≡U  C_ty(µ)εcy(1−Ly_t(µ))1−εcy  +µβU  Co_t+1(µ)εco(1−Lot+1(µ))1 −εco ₍₁₎

where C_ty(µ)and Ly_t(µ)denote consumption and labour supply during youth and C_to₊₁(µ) and Lo_t+1(µ)denote consumption and labour supply during old age. β is the parameter that describe the time preference(0< β<1), and U(·)is the felicity function:

U(x) ≡ x

1−1/σ₋₁

1−1/σ , σ>0 (2)

This felicity function features constant intertemporal substitution elasticity, σ. We assume that the agent does not have a bequest motive. The government then transfers their bequest to the younger generation (abbreviated TY).

The consumers’ periodic budget identities are given by:

C_ty(µ) +St(µ) =wtLy_t(µ) +Zt (3)

Where St(µ)is the savings of the consumer during youth, rt+1 is the rental rate of capital,

wtand wt+1 are wages during youth and old age, respectively. Zt is the lump-sum income

transfer from the government, which is the same for every young people. We also assume that workers are equally productive so that they earn the same wages.

Without the annuity market, consumers can only invest their savings in the capital mar-ket and earn a net interest rate of rt+1. For those who face a high risk of dying at the end of

the first period, they may have to leave their savings as unintended bequest. They are not allowed to die indebted, i.e. they can’t borrow from the capital market. Hence we impose the borrowing constraint St(µ) ≥0.

Now we solve the household’s optimization problem by using two-stage budgeting. Xy_t and Xo_t+1are defined as the full consumption in period t and t+1:

X_ty(µ) ≡C_ty(µ) +wt(1−Lyt(µ)) (5)

Xo_t+1(µ) ≡Cto+1(µ) +wt+1(1−Lot+1(µ)) (6)

In the first stage, the intratemporal maximization stage, the consumer chooses an optimal mix of consumption and leisure conditional upon the level of full consumption. They try to maximize life-time utility (1) subject to the budget constraint (5) and (6). Their optimal choice would be that the marginal rate of substitution between leisure and consumption equals the relative price of leisure and consumption:

(1−εc)/(1−Lyt(µ)) εc/Cyt(µ) =wt (7) (1−εc)/(1−Lot+1(µ)) εc/Co_t+1(µ) =wt+1 (8)

By substituting (7) and (8) into (5) and (6), we obtain expressions for consumption and leisure in terms of full consumption:

C_ty(µ) =εcyXy_t(µ) (9)

wt(1−Ly_t(µ)) = (1−εcy)Xy_t(µ) (10)

C_to+1(µ) =εcoXto+1(µ) (11)

wt+1(1−Lot+1(µ)) = (1−εco)Xto+1(µ) (12)

the lifetime utility function and get the following expression:

EΛt(µ) ≡U X_ty(µ)/qt

+µβU(X_to+1(µ)/qt+1) (13)

in which qtis the true price index at time t indicating the maximum attainable utility:

qt≡  1 εcy εcy wt 1−εcy 1−εcy , qt+1≡  1 εco εco wt 1−εco 1−εco (14)

Now the budget identities of the two periods can be rewritten as:

Xy_t(µ) +St(µ) =wt+Zt (15)

Xo_t+1(µ) =wt+1+ (1+rt+1)St(µ) (16) Maximize lifetime utility (13) subject to the budget identities (15) and (16), the full consump-tion Euler equaconsump-tion becomes:

U0 Xy_t(µ)/qt
βU0 Xo_t+1(µ)/qt+1
= (1+rt+1)µCq w1_t−εcy w1−εco t+1 with Cq≡ εεcoco(1−εco)1−εco εεcycy(1−εcy)1−εcy (17)

For unconstrained consumers, we can combine the two periodic budget identities to get the lifetime budget constraint:

Xy_t(µ) + X o t+1(µ) 1+rt+1 =wt+Zt+ wt+1 1+rt+1 (18)

Combining (17) and (18), we obtain the optimal paths of full consumption and saving:

Xy_t(µ) =Φ(µ, 1+rt+1, wt, wt+1)  wt+Zt+ wt+1 1+rt+1  (19) Xo_t+1(µ) 1+rt+1 = [1−_Φ(µ, 1+rt+1, wt, wt+1)]  wt+Zt+ wt+1 1+rt+1  (20) St(µ) = [1−Φ(µ, 1+rt+1, wt, wt+1)] [wt+Zt] −_Φ(µ, 1+rt+1, wt, wt+1) wt+1 1+rt+1 (21)

WhereΦ(µ, h, i, j)is the marginal propensity to consume out of lifetime wealth during youth:

Φ(µ, h, i, j) ≡ 1

1+ (µβ)σ_hσ−1_Cσ−1

q i(1−εcy)(σ−1)/j(1−εco)(σ−1)

(22)

Note that the consumer’s full consumption during youth equals her marginal propensity to

consume times her human wealth at birth. The marginal propensity to consume,Φ(µ, 1+rt+1, wt, wt+1),

Figure 1: Choices of a µ-type individual in the absence of annuities ! ! ! HBC X_to +1( )µ wt +1 w_t +Z_t Xt y ( )µ B A C EE(µ_{bc t,}) EE(µl) EE(µh)

more for old age. Equation (21) says if one should receive more income in youth and fewer income in old age, she would also increase her saving levels.

The optimal choice of a type µ consumer is shown in Figure 1. The line labeled HBC is the full consumption budget constraint as given in (18). A is the income endowment point, each individual can choose to consume at point A. However, they wish to consume such that: Xo_t+1(µ) X_ty(µ) = [βµ(1+rt+1)]σCqσ−1 w1_t−εcy w1−εco t+1 !σ−1 (23)

2.1.2 Demography

Let Ltdenote the population of the cohort born at time t. The distribution of consumers of

different health types within a cohort can be written as:

Lt(µ) ≡h(µ)Lt (24)

where h(µ)is the probability density function of health types.Rµh

µ_l h(µ)dµ=

Rµ_h

µ_l dH(µ) =1,

in which H(µ)is the cumulative distribution function.

Consumers of type µ (young and old) alive at time t is given by:

Pt(µ) ≡µLt−1(µ) +Lt(µ) (25)

If the population evolve according to Lt = (1+n)Lt−1(with n> −1), then we have:

Pt≡ Z µ_h µ_l Pt(µ)dµ= 1+n+ ¯µ 1+n Lt (26) where ¯µ≡ Rµ_h

µ_l µh(µ)dµ is the average survival rate of the cohort.

2.1.3 Government

After each period, a fraction of the younger generation could make it to the next period, while others passed away. Those who passed away left their savings as unintended bequests to the society. The government then collected the bequests and redistributed them equally among the up-coming generation. The lump-sum income transfer was made as soon as the new generation was born:

(1+rt)

Z _µ

h

µ_bc,t−1

(1−µ)Lt−1(µ)St−1(µ)dµ= LtZt (27)

The left-hand side is the total return from the capital market of the savings of those who passed away, while the right-hand side is the lump-sum income transfer made by the gov-ernment.

2.1.4 Production

We assume that in this closed economy, a large number of perfect competitive firms produce homogeneous commodities with the production function:

where Yt is total output, Ωo is the given index of general factor productivity, Kt is the

ag-gregate capital stock, and Nt ≡ Lt ¯Lyt +1+1n¯Lot
is the effective working hours. ¯L y

t and ¯Lot

denote average working hours of young people and old people in period t:

¯Ly t ≡ Z _µ h µ_l h(µ)Ly_t(µ)dµ (29) ¯Lo t ≡ Z _µ h µ_l µh(µ)Lo_t(µ)dµ (30)

By defining yt ≡Yt/Ntand kt ≡Kt/Nt, we can write the intensive-form production function

as:

yt =Ωokεt (31)

Under the production constraint firms try to maximize their profit. Their factor demand equations can be written as:

rt+δ =εΩokεt−1 (32)

wt = (1−ε)Ωokεt (33)

where δ is the constant rate of depreciation of the capital stock (0<δ <1).

2.1.5 Fundamental mechanism

The basic model with lump-sum income transfer to young people is fully characterized by the following fundamental difference equation:

kt+1=φTY₁ [wt+Zt] −φTY₂ wt+1 1+rt+1

(34)

where φTY₁ and φTY₂ are given by:

φ₁TY≡ 1 (1+n)¯Ly_t+1+ ¯Lo_t+1 Z µ_h µ_bc,t [1−Φ(µ, 1+rt, wt−1, wt)]h(µ)dµ φ₂TY≡ 1 (1+n)¯Ly_t+1+ ¯Lo_t+1 Z _µ h µ_bc,t Φ(µ, 1+rt, wt−1, wt)h(µ)dµ

Equation (34) is obtained by imposing equilibrium in the saving market and using the de-mography setting described in section 2.2. (32), (33) and (34) constitute an implicit function determining kt , rt and wt given a predetermined k0. The cut-off value µ_bc,t is such that

Table 1: Stuctural parameters and functions

σ intertemporal substitution elasticity 0.7000

ε capital efficiency parameter 0.2750

n population growth factor 0.4889

δ capital depreciation factor 0.9158

ˆr interest rate factor 6.0400

Ω0 scale factor production function 2.4312

β time preference factor 0.4869

h(µ) PDF of the survival probability uniform on[µ_l, µ_h]

µ_l survival rate of the unhealthiest 0.0500

µ_h survival rate of the unhealthiest 0.9500

constant capital intensity ˆk and output per worker ˆy, so that the long-run economic growth is driven entirely by the population growth. This is a typical result of an exogenous growth model without the assumption of technology growth.

In this paper we simulate the models with realistic parameter values. We calibrate the model as follows. Firstly, we postulate plausible values for the intertemporal substitution elasticity (σ = 0.7), the efficiency parameter of capital (ε = 0.275), the annual capital de-preciation rate (δa = 0.06), the annual population growth rate (na = 0.01), and a realistic

annual steady-state interest rate (ˆra = 0.05). Together they ensure that the annual

steady-state capital-output ratio is ˆK/ ˆY =ε/(ˆra+ˆδa) =2.5. Secondly, we assume that each period

of the consumers lasts for 40 years. Then the values for n, δ and ˆr can be calculated1. Further-more, We assume that the distribution for the probability of surviving into the second period is uniform on the interval[0.05, 0.95]. A type-µ consumer has a life expectation of 40(1+µ) years. Thirdly we chooseΩ0such that ˆy = 1 in the initial steady state. This also pins down

the steady state value for ˆw and ˆk. Finally, we solve for the steady-state government transfer ˆ

Z, the discount factor β and the cut-off survival level for the borrowing constraint ˆµ_bc, such that a general equilibrium is obtained. The main structural parameters are summarized in Table 1. The key features of the steady-state equilibrium of the TY case are reported in col-umn (a) of Table 2. Government transfer to the young amount to ˆZ=0.0705, which is almost ten percent of wage income during youth. Consumption by the young and surviving old are

1_{1+n= (1+n}

Table 2: Allocation and welfare (a) TY (b) FI (c) AI (d) SA (e) SA θ=0.01 θ=0.02 ˆy 1.0000 1.0424 1.0022 0.9962 0.9910 ˆk 0.0395 0.0460 0.0398 0.0390 0.0383 %Q1 0.10 9.27 1.30 0.15 0.00 %Q2 14.43 20.70 17.47 12.65 7.48 %Q3 34.61 30.64 33.93 29.50 24.54 %Q4 50.85 39.39 47.30 43.18 38.54 %SAS 14.52 29.44 ˆr 6.0400 5.3180 6.0003 6.1105 6.2068 ˆra 0.0500 0.0472 0.0499 0.0503 0.0506 ˆ w 0.7250 0.7558 0.7266 0.7222 0.7185 ˆ Z 0.0705 H(ˆµ_bc) 0.2284 0.0000 0.1673 0.2210 0.2833 ˆ¯µp _{0.6834 0.7025 0.7239} ˆcy _{0.6084 0.5677 0.5517 0.5493 0.5471} %Q1 26.69 26.33 26.78 26.57 26.27 %Q2 25.71 25.36 25.57 25.65 25.75 %Q3 24.35 24.52 24.33 24.40 24.50 %Q4 23.25 23.79 23.32 23.39 23.48 ˆco _{0.3361 0.4101 0.3945 0.3922 0.3902} %Q1 4.81 8.69 4.34 4.72 5.33 %Q2 14.97 20.03 14.94 14.88 14.79 %Q3 30.60 30.63 30.72 30.60 30.40 %Q4 49.62 40.64 50.01 49.80 49.48 E ˆΛ(µ_l) −0.8996 −0.9779 −1.0276 −1.0437 −1.0598 E ˆΛ(µ₁) −1.0094 −1.0428 −1.1322 −1.1334 −1.1332 E ˆΛ(µ₂) −1.0836 −1.1109 −1.1847 −1.1870 −1.1875 E ˆΛ(µ₃) −1.1213 −1.1817 −1.2025 −1.2058 −1.2075 E ˆΛ(µ_h) −1.1377 −1.2551 −1.2002 −1.2044 −1.2072

Figure 2: Steady-state profiles

(a) Youth consumption (b) Old-age consumption ˆ Cy_{(µ) Cˆ}o_(µ) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 survival probability TY FI AI SA 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 survival probability TY FI AI SA

give by, respectively ˆcy _≡ hRµ_h µ_l εcyX y t(µ)dµ i /Nt and ˆco ≡ h Rµ_h µ_l εcoX o t(µ)dµ i /Nt. While the

youth consumption is quite even over quartiles, old-age consumption is very skewed with the healthiest quartile accounting for half of the consumption. More than twenty percent of the consumers face a binding borrowing constraint (H(ˆµ_bc) = 0.2284). The wealth distribu-tion (indicated by capital distribudistribu-tion) is very uneven, while the unhealthiest and healthiest quartile owing, respectively, 0.10% and 50.85% of the capital stock.

We illustrate the steady-state health profiles for youth and old-age consumption, saving, and expected utility in Figure 2. The dashed lines are for the TY case. The horizontal lines in (a) - (c) reflect the borrowing constraint of consumers with low surviving probability. For the unconstrained consumers, the youth consumption is increasing in health while the old-age consumption and saving are decreasing. The expected utility function is downward sloped, indicating an interesting result that unhealthy people live a happier life.

2.2 Private annuity markets

Now we introduce the private annuity market into this closed economy. Annuity is mainly used to insure against the uncertainty of life span. Its rate of return is generally higher than the market rate of interest to compensate for the risk of death. As soon as the annuitant passes away, the insurance company is held free of any obligations and the contract ends automatically.

Net savers in this closed economy would choose to fully annuitize, receiving a higher rate of return from their investment. As long as the annuity market exists and they don’t have a bequest motive, no bequests will be left behind. The savings left by those who pass away after ‘youth’ are in fact transferred to those who survive into ‘old age’, in the form of a premium in the return of annuities. Young people can no longer enjoy the benefit of income transfer. Wealth are flowing within the generation instead of across generations.

2.2.1 Full Information equilibrium

receive their actuarially fair rate of return and achieve perfect insurance against life span uncertainty.

Consumers of health type µ receive a rate of return r_tp₊₁(µ)for their annuity holdings Ap_t(µ). For each health type of consumers there is a separate annuity market, and the clear-ing condition for this market is given by:

(1+rt+1)Lt(µ)A_tp(µ) =µ(1+rp_t+1(µ))Lt(µ)Ap_t(µ), µ∈ [µ_l, µ_h]. (35) Equation (35) says that the gross return of the savings of type µ clients in period t from the capital market are redistributed to those surviving clients in the form of insurance claims in period t+1. It follows that:

1+r_tp₊₁(µ) = 1+rt+1

µ , µ∈ [µl, µh]. (36)

We can see that the return of annuities decreases as the survival probability increases. That is to say the annuity firms pay a higher rate of return to those who have a lower survival probability. Since µ is between 0 and 1, the return in the annuity market is higher than in the capital market, therefore agents would invest all their savings in the annuity market, so their lifetime budget constraint is given by:

Xy_t(µ) + X o t+1(µ) 1+r_tp₊₁(µ) = wt+ wt+1 1+r_tp₊₁(µ) (37)

We can see that the lump-sum transfer disappears from the budget constraint and the re-turn becomes type dependent. The full consumption Euler equation for full information equilibrium is: U0 Xy_t(µ)/qt
βU0 Xo_t+1(µ)/qt+1
= (1+r p t+1(µ))µCq w1−εcy t w1−εco t+1 = (1+rt+1)Cq w1−εcy t w1−εco t+1 (38)

The marginal rate of substitution between current and future consumption depends on the gross interest rate and the wage rates. The survival rate µ is absent because each client is perfectly insured against the lifetime uncertainty.

optimal plans for full consumption and the demand for annuities: Xy_t(µ) =Φ  µ,1+rt+1 µ , wt, wt+1   wt+ µwt+1 1+rt+1  (39) µXo_t+1(µ) 1+rt+1 =  1−Φ  µ,1+rt+1 µ , wt wt+1   wt+ µwt+1 1+rt+1  (40) Ap_t(µ) =  1−_Φ  µ,1+rt+1 µ , wt, wt+1  wt −Φ  µ,1+rt+1 µ , wt, wt+1  µwt+1 1+rt+1 (41)

whereΦ(µ, h, i, j)is defined in (22) above. The optimal choice of a type µ consumer is shown in figure 2. The line labeled HBC(µ_h) and HBC(µ_l) are household budget constraint for the healthiest and unhealthiest type of consumers, respectively. The annuity firms pay a higher rate of return to their unhealthy clients, so their budget lines are steeper than the healthy. Every budget line passes through the initial endowment point A. In the full information equilibrium, every consumer is a net saver. They would like to consume such that:

Xo t+1(µ) X_ty(µ) = [β(1+rt+1)]σCqσ−1 w1_t−εcy w1−εco t+1 !σ−1 (42)

This Euler equation is represented by EE, a straight line from the origin in the diagram. The healthiest type of consumers choose the consumption bundle B while the unhealthi-est choose to consume at point C. The optimal consumption choices of the population are located along the line segment BC.

In Figure 2 the steady-state health profiles for youth and old-age consumption, annuity demand, and expected utility are depicted in solid lines for the FI case. Youth and old-age consumption are decreasing in health while the demand for annuities is increasing. The expected utility is a downward-sloped straight line, indicating that unhealthier people live a happier life.

2.2.2 Asymmetric Information equilibrium

Figure 3: Choices of a µ-type individual with fully-insured annuities wt wt +1 Xt o +1( )µ Xt y ( )µ

A

B

C

status, since unhealthy clients get paid a higher rate of return. As a consequence, the firm will not trust the claim of their clients.

Under these circumstances the Asymmetric Information (AI) equilibrium in the annuity market is a pooling equilibrium. Annuity firms will offer a single annuity rate ¯r_tp₊₁ to all their clients. In the pooled equilibrium, all agents with µ ∈ hµ_bc,t, µ_h

i

(net savers) purchase annuities. The clearing condition for the annuity market is given by:

(1+rt+1) Z _µ h µ_bc,t Lt(µ)A_tp(µ)dµ= (1+¯r_tp₊₁) Z _µ h µ_bc,t µLt(µ)A_tp(µ)dµ (43) As in the full information equilibrium, the gross return of the savings of all clients in period t from the capital market are redistributed to surviving clients in the form of insurance claims in period t+1. It follows that:

1+¯r_tp₊₁ = 1+rt+1

¯µ_tp (44)

where ¯µp_t denotes the asset-weighted average survival rate of annuity purchasers:

The lifetime budget constraint for annuitants is given by: Xy_t(µ) + X o t+1(µ) 1+¯r_tp₊₁ =wt+ wt+1 1+¯r_tp₊₁ (46)

where ¯r_tp₊₁ is defined in (42) above. This budget constraint is very similar with the one ap-peared in the full information equilibrium. the only difference is that each annuitant (al-though with different health profiles) face the same rate of return from the annuity market.

The full consumption Euler equation for asymmetric equilibrium is: U0 Xy_t(µ)/qt
βU0 Xo_t+1(µ)/qt+1
= µ(1+¯r p t+1)Cq w1_t−εcy w1−εco t+1 = µ ¯µ_tp(1+rt+1)Cq w1_t−εcy w1−εco t+1 (47)

this means that the marginal substitution between current and future consumption depends on expected gross return on pooled annuities. The survival rate µ is present because clients are not perfectly insured against lifetime uncertainty.

Combining the lifetime budget constraint (46) and the Euler equation (47), we obtain the optimal plans for full consumption and the demand for annuities:

Xy_t(µ) =Φ µ,1+rt+1 ¯µp_t , wt, wt+1 ! " wt+ ¯µp_twt+1 1+rt+1 # (48) ¯µ_tpX_to₊₁(µ) 1+rt+1 = " 1−_Φ µ,1+rt+1 ¯µ_tp , wt, wt+1 !# " wt+ ¯µ_tpwt+1 1+rt+1 # (49) A_tp(µ) = " 1−Φ µ,1+rt+1 ¯µ_tp , wt, wt+1 !# wt (50) −Φ µ,1 +rt+1 ¯µ_tp , wt, wt+1 ! ¯µ_tpwt+1 1+rt+1 (51)

The case of Asymmetric Information is illustrated in Figure 4. The line labeled HBC(AI) is the household budget constraint faced by all consumers as given in (46). We can see from the diagram that it’s a little bit steeper than the budget line in TY case. That is because people now enjoy a higher rate of return (1+¯rp_t+1 = (1+rt+1)/ ¯µtp > 1+rt+1, ¯µtp ∈ (0, 1)). F is

Figure 4: Choices of a µ-type individual with a single rate of return annuities ! ! ! A B C HBC(TY) HBC(AI) ! ! D E F G ! ! X_to +1( )µ X_ty ( )µ wt +1 wt +Zt wt EE(µl) EE l l p (µ µ, ) EE(µbc t, ) EE bc t l p (µ ,,µ ) EE(µ_h) EE h l p (µ µ, )

Compare the two cases in figure 3, we find that for relative healthy consumers, their consumption choice sets are enlarged while the unhealthy suffered the loss. This is because under the single rate of return system, the relative healthy annuitants receive a better than actuarially fair rate of return on their annuity holdings whereas the unhealthy receive a less than fair return. Intuitively, this is because the redistribution scheme favors the healthy, since the bequests are transferred implicitly to those surviving consumers instead of redistributed equally to the younger generation.

Finally, the dash-dotted lines in Figure 2 shows the steady-state health profiles for youth and old-age consumption, annuity demand and expected utility. Youth consumption and expected utility is decreasing in health while the old-age consumption and demand for an-nuities are increasing. The graph looks like a downward shift of the graph in the TY case.

2.2.3 Transitional Dynamics

Figure 5: Capital intensity relative to TY kt/ˆkTY 0 1 2 3 4 5 6 7 8 0.95 1 1.05 1.1 1.15 1.2 1.25 post−shock time FI AI SA

The results are given in Figures 5 and 6 for both the full information (FI) equilibrium and the asymmetric information (AI) equilibrium in the private annuity market. The paths for the capital intensity are scaled by the steady-state equilibrium value in the no annuities case (TY). This means that a value in excess of unity indicates that the capital intensity increases relative to its benchmark scenario, and vice versa. For lifetime utility we report the difference with the steady-state values obtained in the TY case, i.e. ∆EΛt(µ_j) ≡ EΛt(µ_j) −E ˆΛTY(µ_j). Table 2 gives their unscaled steady state values.

Figure 6: Expected lifetime utility relative to TY

(a) Type µ_l (b) Type µ₁

information, and the unhealthy agents suffer from incomplete information in this case. At the time the annuity market is opened up there are still accidental bequests in the economy which have been left by the previous generation. Hence, in the first period of the new regime young agents benefit from the higher return on their savings through full an-nuitization while also receiving the intergenerational transfer. Their expected utility level increases relative to the benchmark. However, from period 2 (i.e. post-shock time 1) on-wards the transfers are abolished and all health types have a lower utility level under either equilibrium in the private annuity market than in the absence of annuities. This is an ex-ample of the so-called ‘tragedy of annuitization’, as described in Heijdra et al. (2010). Even though it is individually rational to fully annuitize (as the annuity rate of return exceeds the return on capital), this is not optimal from a social point of view. If all agents invests their financial wealth in the annuity market, then the resulting long-run equilibrium leaves future newborns worse off compared to the case where annuities are absent and accidental bequests are redistributed to the young. In other words, the introduction of an annuity mar-ket is welfare improving when considered in isolation, but not when the general equilibrium repercussions are taken into account.

2.3 Social Annuities

In order to reduce the negative effects of adverse selection and improve the gross rate of return and participation rate in the annuity market, some decision-makers introduce the mandatory annuity system to the society. This can be seen as a fully funded pension sys-tem. Young people save a fixed fraction of their income As_t into the system and receive

(1+¯rs

t+1)Ast if they survive into the old age. The government pooled all their savings and

invested them in the capital market, and then distributed the return equally among the sur-vivors. The net return of social annuities is implicitly determined by:

(1+rt+1)LtAts = (1+¯rst+1)Ast

Z µ_h

µ_l

µLt(µ)dµ (52)

Solving for the implied rate of return yields:

1+¯r_ts+1 =

1+rt+1

¯µ (53)

where ¯µ≡Rµ_h

µ_l µh(µ)dµ is the average survival rate. Apart from the mandatory social

for the private annuity market is given by: (1+rt+1) Z µ_h µSA_bc,t Lt(µ)A_tp(µ)dµ= (1+¯r_tp,SA₊₁ ) Z µ_h µSA_bc,t µLt(µ)Ap_t(µ)dµ (54) It follows that: 1+¯r_tp,SA₊₁ = 1+rt+1 ¯µp,SA_t (55)

The social annuity system is immune to adverse selection, So ¯µ < ¯µ_tp,SA. That is to say the average survival rate of the population is lower than the asset-weighted survival rate of annuity purchasers, since people with higher survival rate are willing to purchase more annuities. Thus the return of the social annuity is higher than the private annuity market, i.e. ¯rs_t+1 > ¯r_tp,SA₊₁ . Intuitively, it seems a wise decision for the government to introduce the social annuity system as it pays a higher rate of return.

Consumers’ periodic budget identities can be written as:

Xy_t(µ) +A_tp(µ) +As_t =wt (56)

Xo_t+1(µ) =wt+1+ (1+¯rp,SAt+1 )A p

t(µ) + (1+¯rst+1)Ast (57)

By combining the two equations, their lifetime budget constraint is given by:

Xy_t(µ) + Xo_t+1(µ) 1+¯r_tp,SA₊₁ =wt+ wt+1 1+¯r_tp,SA₊₁ + ¯µp,SA_t − ¯µ ¯µ A s t (58)

Compared to the AI case, the human wealth increases after the government introduces the social annuity system, which is immune to the adverse selection and gives a higher rate of return. The full consumption Euler equation is given by:

U0 Xy_t(µ)/qt
βU0 Xo_t+1(µ)/qt+1
= µ(1+¯r p,SA t+1 )Cq w1_t−εcy w1−εco t+1 = µ ¯µ_tp,SA (1+rt+1)Cq w1_t−εcy w1−εco t+1 (59)

choices of full consumptions and the total demand for annuities: X_ty(µ) =Φ µ,1+rt+1 ¯µ_tp,SA , wt, wt+1 ! " wt+ ¯µp,sA_t wt+1 1+rt+1 + ¯µ p,SA t − ¯µ ¯µ A s t # (60) ¯µ_tp,SAXo t+1(µ) 1+rt+1 = " 1−Φ µ,1+rt+1 ¯µp,SA_t , wt, wt+1 !# " wt+ ¯µp,SA_t wt+1 1+rt+1 + ¯µ p,SA t − ¯µ ¯µ A s t # (61) A_tp(µ) +As_t = " 1−Φ µ,1+rt+1 ¯µp,SA_t , wt, wt+1 !# wt −Φ µ,1+rt+1 ¯µ_tp,SA , wt, wt+1 ! " ¯µ_tp,SAwt+1 1+rt+1 + ¯µ p,SA t − ¯µ ¯µ A s t # (62)

Compared to the model without social annuities (the AI case), there are some changes in the equilibrium. Firstly, from (60) and (61) we find that the human wealth is augmented by a term that reflect the relative return of social annuities. As mentioned before, consumers’ investment in social annuities receive a higher rate of return. Secondly, part of the private annuity holdings has been crowded out by social annuity holdings. For those with very few demand for annuities, their holdings are completely replaced. Thirdly, because of the crowding-out effect, relatively unhealthy types of consumers turn to social annuities. Only those with high survival rate are willing to invest in the private annuities. In other words, the introduction of social annuities aggravate the adverse selection in private annuity market. This leads to a higher marginal health type ( ¯µ_tp,SA > ¯µP_t) and a lower rate of return in private market (¯r_tp,SA₊₁ < ¯r_tp₊₁).

We illustrate the two cases in figure 4. In the presence of social annuities, the initial endowment point shifts from point A to point D. This is exactly the effect of mandatory so-cial annuities. Compared to the AI case, the household budget line flattens since the rate of return in private annuity market drops. For the same reason the line representing Euler equation rotates clockwise. We also see from the diagram that, in the Social Annuity equilib-rium, the budget constraints are loosened for the healthier part of the population, while the borrowing constraint becomes binding for more unhealthier consumers.

ex-Figure 7: Choices of a µ-type individual with mandatory social annuities Xto+1( )

µ

µ

•

•

•

•

•

•

µ µ

µ µ

B

A

E

D

F

C

HBC(AI)

HBC(SA)

(

1−

θ

)

w

w

pected utility profile also shows that unhealthier people are negatively affected by the social security system since they are more constrained in their youth consumption.

2.3.1 Transitional Dynamics

The transitional paths of the capital intensity and lifetime utility for the SA case are given in figure 5 and 6. We depict the case that the mandatory social security contribution is two percent of youth wage (θ=0.02). The transitional paths of SA case is quite similar with that of the AI case. The capital intensity increases in the second period (i.e. post-shock time 1) and then declines sharply to below the benchmark scenario. The lifetime utility increases in the first period (because they still receive the accidental bequests), but then declines for all health types. Form figure 6 we can see that in the long run the lifetime utility in the SA case is even lower than in the AI case.

par-tial equilibrium effect. Because the social annuities offer a higher rate of return than private annuities, there is a decrease in the demand for private annuities. And the unhealthy agents tend to purchase less annuity products than healthy agents, so they reduce their private an-nuity demand share more than the healthy agents do. This aggravates the degree of adverse selection in the private annuity market, which leads to a decrease in the rate of return on private annuities. Second, there is also a general equilibrium effect which partly offsets the decrease. In the steady state, the capital intensity decreases, which leads to a rise in its rental rate. However, the net effect on the rate of return of private annuities is still negative.

Overall, due to a lower return on private annuities and a lower wage rate, the negative effects of introducing social annuities outweigh the positive effect of offering a fairer rate of return on a fixed part of savings. This is evident in figure 6: we can see welfare drops in all scenarios in the long run (compared with the AI case).

3

Model with two sources of heterogeneity (health status µ and

innate ability η)

In the previous section we showed that the mandatory social annuities have a negative effect on steady-state capital intensity and welfare in a general equilibrium, due to the aggravating adverse selection. In this section we aim to solve this problem by adding another source of heterogeneity: agents’ innate ability, which is denoted by η. By indirectly observing the agents’ innate ability, we could extract the information of their health status, assuming that these two sources are to a certain extent correlated. With more information collected, we reexamine the net effect of social annuities and try to find out if the asymmetric information problem could be alleviated or solved.

3.1 Bequests to the young

3.1.1 Consumers

period t is given by: EΛt(η, µ) ≡U  C_ty(η, µ)εcy(1−Ly_t(η, µ))1−εcy  +µβU  Co_t+1(η, µ)εco(1−Lot+1(η, µ))1 −εco ₍₆₃₎

where U(·)is the felicity function:

U(x) ≡ x

1−1/σ₋₁

1−1/σ , σ>0 (64)

In this case we assume that agents do not have bequest motive, and the bequests are au-tomatically transfered to the younger generation by the government. Therefore, the con-sumers’s periodic budget identities are given by:

C_ty(η, µ) +St(η, µ) =wt(η)Ly_t(η, µ) +Zt (65)

Co_t+1(η, µ) =wt+1(η)Lo_t+1(η, µ) + (1+rt+1)St(η, µ) (66) we can see that consumers’ consumption and saving not only depend on their health type µ, but also on their innate ability η. One thing worth mentioning is that their wage rates depend solely on their innate ability, which we will prove in the following Section 3.1.4 (Production). Now we solve the household’s optimization problem by using two-stage budgeting. Xy_t and Xo

t+1are defined as the the full consumption in period t and t+1:

X_ty(η, µ) ≡Cy_t(η, µ) +wt(η)(1−Ly_t(η, µ)) (67) Xo_t+1(η, µ) ≡Co_t+1(η, µ) +wt+1(η)(1−Lo_t+1(η, µ)) (68) In the first stage, the intratemporal maximization stage, the consumer chooses an optimal mix of consumption and leisure conditional upon the level of full consumption. They try to maximize life-time utility (63) subject to the budget constraint (67) and (68). Their optimal choice would be that the marginal rate of substitution between leisure and consumption equals the relative price of leisure and consumption:

By substituting (69) and (70) into (67) and (68), we obtain expressions for consumption and leisure in terms of full consumption:

Cy_t(η, µ) =εcyXty(η, µ) (71)

wt(η)(1−Ly_t(η, µ)) = (1−εcy)Xyt(η, µ) (72)

C_to₊₁(η, µ) =εcoXot+1(η, µ) (73)

wt+1(η)(1−Lo_t+1(η, µ)) = (1−εco)Xto+1(η, µ) (74)

In the second stage, the intertemporal stage, the consumer optimally smoothes full consump-tion over lifetime. We substitute consumpconsump-tion and leisure in terms of full consumpconsump-tion into the lifetime utility function and get the following expression:

EΛt(η, µ) ≡U V(C_ty(η, µ), 1−Ly_t(η, µ))
+µβU(V(C_to₊₁(η, µ), 1−Lo_t+1(η, µ))) (75)

in which qtis the true price index at time t indicating the maximum attainable utility:

qt≡  1 εcy εcy_{ w} t(η) 1−εcy 1−εcy , qt+1≡  1 εco εco_{ w} t+1(η) 1−εco 1−εco (76)

Now the budget identities of the two periods can be rewritten as:

Xy_t(η, µ) +St(η, µ) =wt(η) +Zt (77)

X_to+1(η, µ) =wt+1(η) + (1+rt+1)St(η, µ) (78) Maximize lifetime utility (75) subject to the budget identities (77) and (78), the full consump-tion Euler equaconsump-tion becomes:

U0 Xy_t(η, µ)/qt
βU0 X_to₊₁(η, µ)/qt+1
= (1+rt+1)µCq wt(η)1−εcy wt+1(η)1−εco (79) with Cq≡ εεcoco(1−εco)1−εco εεcycy(1−εcy)1−εcy (80)

For unconstrained consumers, we can combine the two periodic budget identities to get the lifetime budget constraint:

Figure 8: Choices of a µ-type individual in the absence of annuities X_to +1( )µ

X

( )

µ

µ

µ

EE

(

µ η

( ))

EE

(

µ η

( ))

•

•

•

•

•

•

A

_{C F}

E

B

D

η

η

Combining (79) and (81), we obtain the optimal paths of full consumption and saving:

Xy_t(η, µ) =Φ(µ, 1+rt+1, wt(η), wt+1(η))  wt(η) +Zt+ wt+1(η) 1+rt+1  (82) Xo_t+1(η, µ) 1+rt+1 = [1−Φ(µ, 1+rt+1, wt(η), wt+1(η))]  wt(η) +Zt+ wt+1(η) 1+rt+1  (83) St(η, µ) = [1−Φ(µ, 1+rt+1, wt(η), wt+1(η))] [wt(η) +Zt] −Φ(µ, 1+rt+1, wt(η), wt+1(η))wt+1(η) 1+rt+1 (84)

WhereΦ(µ, h, i, j)is the marginal propensity to consume out of lifetime wealth during youth:

Φ(µ, h, i, j) ≡ 1

1+ (βµ)σ_hσ−1_Cσ−1

q i(1−εcy)(σ−1)/j(1−εco)(σ−1)

(85)

Note that the consumer’s full consumption during youth equals her marginal propensity to consume times her human wealth at birth. The marginal propensity to consume,Φ(µ, h, i, j), is decreasing in µ. That means healthy people tend to consume less during youth and save more for old age. Equation (84) says if one should receive more income in youth and fewer income in old age, she would also increase her saving levels.

η(η₁ < η< η₂) laborers is parallel to HBC(η₁)and HBC(η₂)and lies in between. A and D are the income endowment points for type η₁ and η₂laborers. Each individual can choose to consume at their initial endowment point (A–D), but they wish to consume such that:

Xo_t+1(µ) X_ty(µ) = [βµ(1+rt+1)]σCqσ−1  wt(η)1−εcy wt+1(η)1−εco σ−1 (86)

This is the Euler equation for the felicity defined in (64). It is a straight line from the origin. The higher the survival probability µ, the steeper the slope of the line. EE(µ_h)denote Eu-ler equation for the healthiest type of consumers. They consume at segment (B–E), which requires a lot of saving during youth. For the unhealthiest type of consumers their Euler equation EE(µ_l)passes through point C and F. However, they can not possibly consume at segment (C–F) as it violates the borrowing constraint, St(η, µ) >0. The best option for them is to consume at their initial income endowment, segment (A–D). There is a marginal health type µ_bc,t(η₁)and µ_bc,t(η₂)such that EE(µ_bc,t(η₁))passes through point A and EE(µ_bc,t(η₂)) passes through point D. Any relatively unhealthy consumers (A–C–F–D) face borrowing constraint and have to consume their initial income endowment bundles (A–D) while rela-tively healthy consumers (A–B–E–D) are not constrained and choose freely between (A–D) and (B–E).

3.1.2 Demography

Let Ltdenote the population of the cohort born at time t. The distribution of consumers of

different health types and innate abilities within a cohort can be written as:

Lt(η, µ) ≡h(η, µ)Lt (87)

We construct a bivariate uniform distribution for η and µ, where h(η, µ)is the probability density function of health types and innate abilities:

h(η, µ) = 1 µ_H−µ_L

1

η₂−η₁[1+ρηµ(µ− ¯µ)(η− ¯η)] (88) In equation (87) ¯η and ¯µ denote the mean of η and µ, respectively. σ2_η and σ2_µ are the vari-ances of η and µ, cov_√(η,µ)

σ2ησ2µ

= ρ_ηµσησµis the correlation coefficient of η and µ in the bivariate

distribution, and we haveRη2 η₁

Rµ_h

Consumers of type µ (young and old) alive at time t is given by:

Pt(µ) ≡µLt−1(µ) +Lt(µ) (89)

If the population evolve according to Lt = (1+n)Lt−1(with n> −1), then we have:

Pt≡ Z _µ h µ_l Pt(µ)dµ= 1+n+ ¯µ 1+n Lt (90) where ¯µ≡ Rµ_h

µ_l µh(µ)dµ is the average survival rate of the cohort.

3.1.3 Government

The government then collected the bequests from those who passed away and redistributed them equally among the younger generation:

(1+rt) Z η₂ η₁ Z µ_h µ_bc,t−1(η) (1−µ)Lt−1(η, µ)St−1(η, µ)dµdη= LtZt (91)

The left-hand side is the total return of the savings of those who passed away, while the right-hand side is the lump-sum income transfer made by the government.

3.1.4 Production

In this close economy, a large number of perfect competitive firms produce homogeneous commodities with the production function:

Yt =ΩoKεtNt1−ε, 0<ε<1 (92)

where Nt ≡ Lt ¯Lyt +1+1n¯Lot
is the total effective working hours (workers’ innate ability

times the working hours) of the young and the old. ¯Ly_t and ¯Lo_t are defined as: ¯Ly t ≡ Z _η 2 η₁ Z _µ h µ_l ηh(η, µ)Ly_t(η, µ)dµdη (93) ¯Lo t ≡ Z η₂ η₁ Z µ_h µ_l ηµh(η, µ)Lo_t(η, µ)dµdη (94)

By defining yt ≡Yt/Ntand kt ≡Kt/Nt, we can write the intensive-form production function

as:

Under the production constraint firms try to maximize their profit. Their factor demand equations can be written as:

rt+δ =εΩokεt−1 (96)

wt = (1−ε)Ωokεt (97)

where δ is the constant rate of depreciation of the capital stock (0<δ <1). Note that Ntcan also be written as:

Nt ≡ Z η₂ η₁ η Z µ_h µ_l  Lt(η, µ)Ly_t(η, µ) +µLt−1(η, µ)Lo_t(η, µ) dµ  dη = Z η₂ η₁ ηEt(η)dη (98)

where Et(η)stands for the working hours of η type laborers. Therefore, the profit of those perfect competitive firms can be rewritten as:

Πt≡ΩoKεtNt1−ε− (rt+δ)Kt−

Z _η

2

η₁

wt(η)Et(η)dη (99)

It’s easy to obtain the first order conditions: ∂Πt ∂Kt =ΩoεK_tε−1N_t1−ε− (rt+δ) =0 (100) ∂Πt ∂Et(η) =Ωo(1−ε)K ε tNt−εη−wt(η) =0 (101) It follows that: wt(η) =ηwt (102) 3.1.5 Fundamental mechanism

The basic model with lump-sum income transfer to young people is fully characterized by the following fundamental difference equation:

kt+1=ψTY₁ wt+ψ₂TYZt−ψTY₃ wt+1 1+rt+1

(103)

(96), (97) and (103) constitute an implicit function determining kt, rt and wt given a

prede-termined k0. The cut-off value µ_bc,t(η)is such that St(µ_bc,t(η)) =0. Ztis in turn determined

by rt , wt , kt and the constraint of µbc,t(η). In the steady state we have a constant capital

intensity ˆk and output per worker ˆy, so that the long-run economic growth is driven entirely by the population growth.

we simulate the models and keep the parameter values unchanged. The calibration pro-cedure is almost the same as the last section. The key features of the steady-state equilib-rium of the TY case are reported in column (a) of Table 3. Government transfer to the young amount to ˆZ =0.1488, which is almost twenty percent of wage income during youth. Con-sumption by the young and surviving old are give by ˆcy _≡hRη₂

η₁ Rµ_h µ_l Lt(η, µ)C y t(η, µ)dµdη i /Nt and ˆco ≡hRη₂ η₁ Rµ_h µ_l µLt−1(η, µ)C o t(η, µ)dµdη i

/Nt, respectively. While the youth consumption

is quite even over quartiles, old-age consumption is very skewed with the healthiest quartile accounting for half of the consumption. The wealth distribution (indicated by capital distri-bution) is very uneven, while the unhealthiest and healthiest quartile owing, respectively, 0.08% and 51.54% of the capital stock

We illustrate the steady-state health profiles for youth and old-age consumption, saving, and expected utility in Figure 9. The dashed lines are for the TY case. The horizontal lines in (a) - (c) reflect the borrowing constraint of consumers with low surviving probability. For the unconstrained consumers, the youth consumption is increasing in health types µ while the old-age consumption and saving are decreasing. Interestingly, the expected utility function is U sloped, indicating that the most unhealthy and healthy people live the happiest life.

3.2 Private annuity markets

3.2.1 Full Information equilibrium

In this case we assume that private annuity market exists and the health status of each indi-vidual is public information. Consumers of health type µ receive a rate of return r_tp₊₁(µ)for their annuity holdings Ap_t(µ). For each health type of consumers there is a separate annuity market, and the clearing condition for this market is given by:

Table 3: Allocation and welfare (a) TY (b) FI (c) AI (d) SA (e) SA θ=0.01 θ=0.02 ˆy 1.0000 1.0203 0.9729 1.0145 1.0210 ˆk 0.0395 0.0425 0.0358 0.0417 0.0426 %Q1 0.08 8.99 1.29 1.02 0.17 %Q2 13.90 20.37 17.19 14.65 10.77 %Q3 34.48 30.64 33.78 29.16 25.02 %Q4 51.54 40.00 47.74 41.40 37.04 %SAS 13.76 27.01 ˆr 6.0400 5.6813 6.5618 5.7808 5.6695 ˆra 0.0500 0.0486 0.0519 0.490 0.0486 ˆ w 0.7250 0.7379 0.7054 0.7355 0.7402 ˆ Z 0.1488 H(ˆµ_bc) 0.1662 0.1699 0.2162 ˆ¯µp _{0.6853 0.5136 0.5133} ˆcy _{0.6396 0.5579 0.5200 0.5542 0.5556} %Q1 26.17 25.77 26.20 26.11 25.95 %Q2 25.57 25.20 25.39 25.37 25.43 %Q3 24.54 24.72 24.54 24.57 24.62 %Q4 23.73 24.31 23.87 23.95 23.99 ˆco _{0.3405 0.4027 0.3780 0.4577 0.4562} %Q1 4.73 8.45 4.19 4.19 4.51 %Q2 14.71 19.74 14.66 14.62 14.58 %Q3 30.49 30.62 30.60 30.58 30.48 %Q4 50.07 41.19 50.55 50.60 50.43 E ˆΛ(µ_l) −0.3335 −0.4589 −0.5072 −0.4798 −0.4838 E ˆΛ(µ₁) −0.3816 −0.4739 −0.5536 −0.5081 −0.5023 E ˆΛ(µ₂) −0.4029 −0.4912 −0.5582 −0.4978 −0.4927 E ˆΛ(µ₃) −0.3952 −0.5106 −0.5353 −0.4614 −0.4568 E ˆΛ(µ_h) −0.3707 −0.5318 −0.4962 −0.4098 −0.4057

Figure 9: Steady-state profiles

(a) Youth consumption (b) Old-age consumption ˆ Cy_{(µ) Cˆ}o_(µ) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.65 0.7 0.75 0.8 0.85 0.9 0.95 survival probability TY FI AI SA 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 survival probability TY FI AI SA

Figure 10: Choices of a µ-type individual with fully-insured annuities X_to +1( )

µ

X

( )

µ

EE

Equation (104) says that the gross return of the savings of type µ clients in period t from the capital market are redistributed to those surviving clients in the form of insurance claims in period t+1. It follows that:

1+r_tp₊₁(µ) = 1+rt+1

µ , µ∈ [µl, µh]. (105)

Annuity firms pay a lower return to people who have a higher probability to survive (the higher the µ, the lower the return 1+r_tp₊₁(µ)). Since µ < 1, the return from the annuity market is higher than the capital market (1+rt+1

µ >1+rt+1), and consumers would invert all

their savings into the annuity market. Their lifetime budget constraint is given by :

Xy_t(η, µ) + Xo_t+1(η, µ) 1+r_tp₊₁(µ) =wt(η) + wt+1(η) 1+r_tp₊₁(µ) (106)

The full consumption Euler equation is: U0 Xy_t(η, µ)/qt
βU0 Xo_t+1(η, µ)/qt+1
= (1+r p t+1(µ))µCq wt(η)1−εcy wt+1(η)1−εco = (1+rt+1)Cq wt(η)1−εcy wt+1(η)1−εco (107)

obtain the optimal plans for full consumption and the demand for annuities: Xy_t(η, µ) =Φ  µ,1+rt+1 µ , wt(η), wt+1(η)   wt(η) + µwt+1(η) 1+rt+1  (108) µXo_t+1(η, µ) 1+rt+1 =  1−Φ  µ,1+rt+1 µ , wt(η), wt+1(η)   wt(η) + µwt+1(η) 1+rt+1  (109) Ap_t(η, µ) =  1−_Φ  µ,1+rt+1 µ , wt (η), wt+1(η)  wt(η) −Φ  µ,1+rt+1 µ , wt(η), wt+1(η)  µwt+1(η) 1+rt+1 (110)

whereΦ(µ, h, i, j)is defined in (85) above. The optimal choice of a type µ consumer is shown in figure 10. The lines labeled H(l, 1)and H(h, 1)are household budget constraint for the healthiest and unhealthiest type of consumers with innate ability η₁, and lines H(l, 2)and H(h, 2)are the budget constraint for healthiest and unhealthiest consumers with innate abil-ity η₂, respectively. The annuity firms pay a higher rate of return to their unhealthy clients, so their budget lines are steeper than the healthy. Every budget line for type η₁ consumers passes through the initial endowment point A, and for type η₂consumers it passes through the point D. In the full information equilibrium, every consumer is a net saver. They would like to consume such that:

Xo t+1(η, µ) X_ty(η, µ) = [β(1+rt+1)]σCσq−1  wt(η)1−εcy wt+1(η)1−εco σ−1 (111)

This Euler equation is represented by EE, a straight line from the origin in the diagram. For type η₁consumers, the healthiest choose the consumption bundle B while the unhealthiest choose to consume at point C. For type η₂consumers the optimal consumption choices for the healthiest and unhealthiest are points E and F. The optimal consumption choices of the population are located along the line segment (B–E–C–F).

In Figure 9 the steady-state health profiles for youth and old-age consumption, annuity demand, and expected utility are depicted in solid lines for the FI case. Youth and old-age consumption are decreasing in health while the demand for annuities is increasing. The expected utility is a downward-sloped straight line, indicating that unhealthier people live a happier life.

3.2.2 Asymmetric Information equilibrium

to all their clients. In the pooled equilibrium, all agents with µ ∈ hµ_bc,t, µ_h i

(net savers) purchase annuities. The clearing condition for the annuity market is given by:

(1+rt+1) Z _η 2 η₁ Z _µ h µ_bc,t Lt(η, µ)A_tp(η, µ)dµdη= (1+¯r_tp₊₁) Z _η 2 η₁ Z _µ h µ_bc,t µLt(η, µ)A_tp(η, µ)dµdη The gross returns of the savings of all clients in period t are redistributed to the surviving clients in the form of insurance claims in period t+1. It follows that:

1+¯r_tp₊₁ = 1+rt+1

¯µ_tp (112)

where ¯µp_t denotes the asset-weighted average survival rate of annuity purchasers: ¯µp_t ≡ Z _µ h µ_bc,t µϕ_t(µ)dµ, ϕ_t(µ) ≡ Rη₂ η₁ A p t(η, µ)h(η, µ)dη Rη₂ η₁ Rµ_h µ_bc,t A p t(η, µ)h(η, µ)dµdη , (113)

The lifetime budget constraint for annuitants is given by:

Xy_t(η, µ) + Xo t+1(η, µ) 1+¯r_tp₊₁ = wt(η) + wt+1(η) 1+¯r_tp₊₁ (114)

where ¯rp_t+1 is defined in (112) above. This budget constraint is very similar with the one appeared in the full information equilibrium, the only difference is that each annuitant (al-though with different health profiles) face the same rate of return from the annuity market.

The full consumption Euler equation for asymmetric equilibrium is: U0 Xy_t(η, µ)/qt
βU0 Xo_t+1(η, µ)/qt+1
=µ(1+¯r p t+1)Cq w1_t−εcy w1−εco t+1 = µ ¯µ_tp(1+rt+1)Cq wt(η)1−εcy wt+1(η)1−εco (115) this means that the marginal substitution between current and future consumption depends on expected gross return on pooled annuities. The survival rate µ is present because clients are not perfectly insured against lifetime uncertainty.

Combining the lifetime budget constraint (114) and the Euler equation (115), we obtain the optimal plans for full consumption and the demand for annuities: