3.1 Asymmetric information on survival probability

We consider a continuum of retirees who live for two periods at most: Period 1 with certainty and Period 2 with some probability ( $\theta$ ). The two periods correspond to, respectively, the early and advanced stages of retirement. Retirees have different probabilities of surviving to Period 2, represented by a cumulative distribution $F(\theta )$ where $\theta \in \lbrack \underline{\theta },\overline{\theta }]$ and $0\leq \underline{\theta }< \overline{\theta }\leq 1$ .Footnote ³ Generally, an individual knows her behavior and preference in health-related activities and has a better sense about her health and survival probability. In this model, we assume that $\theta$ is private information and is known by the individual in Period 1.

We assume that non-exclusive annuity contracts with linear pricing (as in Abel, Reference Abel1986; Hosseini, Reference Hosseini2015), rather than exclusive contracts with price convexity [as in Eckstein et al. (Reference Eckstein, Eichenbaum and Peled1985); Eichenbaum and Peled (Reference Eichenbaum and Peled1987)], are offered, because all observed PA contracts in various economies are similar to this type. Moreover, as pointed out by Abel (Reference Abel1986), it is hard to determine whether a buyer also holds annuities from other providers. This difficulty affects the effectiveness of exclusive contracts in the annuity market.Footnote ⁴ Motivated by observed practices, we also assume that quantity restrictions are imposed on the public annuities. As will be elaborated in Section 4.3, a distinguishing feature between public and private annuity provision in the presence of adverse selection is the effectiveness of imposing the quantity restrictions.

In the model, the lifetime utility of a retiree with survival probability $\theta$ is given by

(1) \begin{equation} U\left ( c_{1\theta },c_{2\theta };\,\ \theta \right ) =u\left ( c_{1\theta }\right ) +\frac{\theta }{1+\rho }u\left ( c_{2\theta }\right )\!, \end{equation}

where $c_{i\theta }$ is the level of consumption expenditure in Period $i$ ( $i=1,2$ ) and $\rho$ is the subjective discount rate. We assume that the utility function $U ( c_{1\theta },c_{2\theta };\,\ \theta )$ in (1) is homothetic, with the property that the marginal rate of substitution is a homogeneous function of degree 0:

(2) \begin{equation} \frac{\partial U\left ( tc_{1\theta },tc_{2\theta };\,\ \theta \right )/\partial c_{1\theta }}{\partial U\left ( tc_{1\theta },tc_{2\theta };\,\ \theta \right )/\partial c_{2\theta }}=\frac{\partial U\left ( c_{1\theta },c_{2\theta };\,\ \theta \right )/\partial c_{1\theta }}{\partial U\left ( c_{1\theta },c_{2\theta };\,\ \theta \right )/\partial c_{2\theta }} \end{equation}

for $t> 0$ .Footnote ⁵ Moreover, the standard assumptions that $u ( c )$ is strictly concave and $\lim _{c\rightarrow 0}u^{^{\prime }} ( c ) =\infty$ hold.

For simplicity, we do not consider bequest motive in this paper, because it is not likely to be a major factor in understanding the similarities and differences of voluntary versus mandatory PA plans. Assuming away bequest motive means that the annuity always dominates the risk-free bond as the financial tool to hedge longevity risk [e.g. Yaari (Reference Yaari1965), Davidoff et al. (Reference Davidoff, Brown and Diamond2005)]. As a result, retirees do not consider the purchase of risk-free bond in our model.

3.2 Before the introduction of the PA plan

We first consider the outcome before the introduction of the PA plan. A typical retiree makes the consumption and annuitization choices to maximize her expected lifetime utility, given by (1). The outcome under this environment will be useful in subsequent analysis, particularly the analysis (in Section 5) of the impact on different retirees when the government considers whether to adopt a particular PA plan or not.

In this paper, we consider an annuity contract with survival-contingent payments only. The private annuity contract operates as follows. If a retiree buys one dollar of annuity in Period 1, she will receive $\widehat{V}$ dollars in Period 2 if she is alive. With this annuity product, her budget constraints are given by:

(3) \begin{equation} \widehat{c}_{1\theta }=w-\widehat{\nu }_{\theta }, \end{equation}

and

(4) \begin{equation} \widehat{c}_{2\theta }=\widehat{V}\widehat{\nu }_{\theta }, \end{equation}

where $\widehat{\nu }_{\theta }$ is the annuitization amount of a retiree with parameter $\theta$ and $w$ is the retiree’s wealth. Since borrowing against annuity contracts is not allowed in most societies, the restriction of non-negative amounts of annuity purchase ( $\widehat{\nu }_{\theta }\geq 0$ ) is imposed. Note that we use the symbol $^{\wedge}$ above a variable to denote the corresponding variable in the economy before the introduction of the PA (i.e., with private annuity market only).

The annuity buyer’s optimization problem, after knowing her private information $\theta$ , is to choose $\widehat{\nu }_{\theta }$ to maximize

(5) \begin{equation} U\left ( \widehat{c}_{1\theta },\widehat{c}_{2\theta };\,\ \theta \right ) =u\left ( \widehat{c}_{1\theta }\right ) +\frac{\theta }{1+\rho }u\left ( \widehat{c}_{2\theta }\right ) =u\left ( w-\widehat{\nu }_{\theta }\right ) +\frac{\theta }{1+\rho }u\left ( \widehat{V}\widehat{\nu }_{\theta }\right ). \end{equation}

It is easy to show that the optimal choice of $\widehat{\nu }_{\theta }$ is an interior solution, which is characterized by:Footnote ⁶

(6) \begin{equation} u^{\prime }\left ( w-\widehat{\nu }_{\theta }^{\ast }\right ) =\frac{\theta }{1+\rho }\widehat{V}u^{\prime }\left ( \widehat{V}\widehat{\nu }_{\theta }^{\ast }\right ), \end{equation}

where the symbol ^* associated with a variable denotes the optimal choice of that variable.

The left-hand side (LHS) in the first-order condition (6) is the marginal cost of annuitizing one more dollar, because the buyer has to reduce one unit of consumption in Period 1. In terms of utility, the marginal cost is $u^{\prime } ( \widehat{c}_{1\theta }^{\ast } ) =u^{\prime } ( w-\widehat{\nu }_{\theta }^{\ast } )$ . If the buyer is alive in Period 2, she receives $\widehat{V}$ extra units of annuity income. Thus, the increase in her utility level is given by $\widehat{V}u^{\prime } ( \widehat{c}_{2\theta }^{\ast } ) =\widehat{V}u^{\prime } ( \widehat{V}\widehat{\nu }_{\theta }^{\ast } )$ . After discounting the expected value of this future utility benefit back to Period 1, we have the right-hand side (RHS) term of (6). At the optimal choice, the marginal benefit and marginal cost are equal according to (6).

Differentiating (6) totally, we obtain

(7) \begin{equation} \frac{\partial \widehat{\nu }_{\theta }^{\ast }}{\partial \theta }=\frac{-\widehat{V}u^{\prime }\left ( \widehat{c}_{2\theta }^{\ast }\right ) }{\left ( 1+\rho \right ) u^{\prime \prime }\left ( \widehat{c}_{1\theta }^{\ast }\right ) +\theta \widehat{V}^{2}u^{\prime \prime }\left ( \widehat{c}_{2\theta }^{\ast }\right ) }> 0, \end{equation}

which suggests that the optimal annuity choice $\widehat{v}_{\theta }^{\ast }$ is increasing in the survival probability $\theta$ . The relationship is shown in Panel A of Figure 1.

Figure 1. Annuity choices under various plans.

Based on the buyers’ choices described above, annuity providers’ revenue in Period 1 is given by $\int _{\underline{\theta }}^{\overline{\theta }}\widehat{\nu }_{\theta }^{\ast }dF(\theta )$ , and their expected payment in Period 2 is $\int _{\underline{\theta }}^{\overline{\theta }}\theta \widehat{V}^{\ast }\widehat{\nu }_{\theta }^{\ast }dF(\theta )$ . Following many researchers [such as Abel (Reference Abel1986), Hosseini (Reference Hosseini2015)], we assume the zero-profit condition for the provision of annuities. This assumption offers the advantage that the correlation between survival probability and annuity purchase, which is the source of adverse selection, is reflected in the equilibrium annuity payout level. Under the zero-profit condition for private annuity market, the equilibrium value of the payout term $\widehat{V}$ , denoted by $\widehat{V}^{\ast }$ , is determined according to

(8) \begin{equation} \widehat{V}^{\ast }=\frac{\left ( 1+r\right ) \int _{\underline{\theta }}^{\overline{\theta }}\widehat{\nu }_{\theta }^{\ast }dF(\theta )}{\int _{\underline{\theta }}^{\overline{\theta }}\theta \widehat{\nu }_{\theta }^{\ast }dF(\theta )}, \end{equation}

after using the discount factor $1/(1+r)$ to adjust for the revenue and payment terms in the two different periods, where $r$ is the interest rate.Footnote ⁷ According to Lau et al. (Reference Lau, Ying and Zhang2022), the equilibrium payout $\widehat{V}^{\ast }$ is unique for all time-separable utility functions (1) if the distribution of retirees’ survival probabilities satisfies a rather weak condition. For example, a uniform distribution or a truncated normal distribution ( $\underline{\theta }\leq \theta \leq \overline{\theta }$ ) with variance larger than some threshold level satisfies this sufficient condition.

Comparing the payout $\widehat{V}^{\ast }$ with the actuarially fair payout $\frac{1+r}{E\left ( \theta \right ) }$ based on average survival probability of the population, it is straightforward to conclude that

(9) \begin{equation} \widehat{V}^{\ast }< \frac{1+r}{E\left ( \theta \right ) }, \end{equation}

because $\widehat{\nu }_{\theta }^{\ast }$ and $\theta$ are positively correlated according to (7).

The value of payout $\widehat{V}^{\ast }$ is below the actuarially fair level, indicating some efficiency loss due to adverse selection, which is often cited as an important reason calling for government intervention.

3.3 Introducing the PA plan

In the presence of market imperfection such as adverse selection, the government may choose to intervene in the annuity market in different ways, such as taxation, regulation, or annuity provision. Drawing on the PA practices in various economies [see, e.g. Section 2 and Table 1 of Lau and Zhang (Reference Lau and Zhang2023)], we consider the case that the government (directly or indirectly through a statutory body) steps in the market as the PA provider.

We assume that the PA contract provides survival-contingent payments only. If a retiree buys one dollar of the PA in Period 1, she will receive $G$ dollars in Period 2 if she is alive. Therefore, the retiree’s budget constraints, when both private and public annuities are available, are given by

(10) \begin{equation} c_{1\theta }=w-\gamma _{\theta }-\nu _{\theta }, \end{equation}

and

(11) \begin{equation} c_{2\theta }=G\gamma _{\theta }+V\nu _{\theta }, \end{equation}

where $\gamma _{\theta }$ ( $\gamma _{\theta }\geq 0$ ) is her purchase amount of PA and $\nu _{\theta }$ ( $\nu _{\theta }\geq 0$ ) is her purchase amount of private annuity. The other variables are the same as before, except that the symbol $^{\wedge}$ is absent after the PA plan is introduced.

Combining (1), (10), and (11), the problem becomes

\begin{equation*} \max _{\gamma _{\theta },\nu _{\theta }}U\left ( c_{1\theta },c_{2\theta };\,\ \theta \right ) =u\left ( w-\gamma _{\theta }-\nu _{\theta }\right ) +\frac {\theta }{1+\rho }u\left ( G\gamma _{\theta }+V\nu _{\theta }\right ). \end{equation*}

Solving the above problem leads to the following two conditions:

(12) \begin{equation} \frac{\partial U}{\partial \gamma _{\theta }}=-u^{\prime }\left ( w-\gamma _{\theta }-\nu _{\theta }\right ) +\frac{\theta }{1+\rho }Gu^{\prime }\left ( G\gamma _{\theta }+V\nu _{\theta }\right ), \end{equation}

and

(13) \begin{equation} \frac{\partial U}{\partial \nu _{\theta }}=-u^{\prime }\left ( w-\gamma _{\theta }-\nu _{\theta }\right ) +\frac{\theta }{1+\rho }Vu^{\prime }\left ( G\gamma _{\theta }+V\nu _{\theta }\right ), \end{equation}

where $u^{\prime } ( w-\gamma _{\theta }-\nu _{\theta } )$ in either (12) or (13) is the marginal cost of buying one more unit of a public or private annuity. On the other hand, the marginal benefit of buying a private annuity is different from that of buying the PA, with $\frac{\theta }{1+\rho }Gu^{\prime }\left ( G\gamma _{\theta }+V\nu _{\theta }\right )$ in (12) being the marginal benefit of buying the PA, and $\frac{\theta }{1+\rho }Vu^{\prime }\left ( G\gamma _{\theta }+V\nu _{\theta }\right )$ in (13) being the marginal benefit of buying the private annuity.

3.4 Pure mandatory PA plan

Before analyzing two PA plans which are empirically relevant, we first consider a benchmark case of a pure mandatory public annuity plan (pure MPA plan) in which all retirees are required to purchase the same level of PA. The results of this case, together with those in the opposite extreme case of a pure VPA plan,Footnote ⁸ are helpful to understand the similarities and differences of the two PA plans to be considered in the next section.

The pure MPA plan is represented by

(14) \begin{equation} \gamma _{\theta }=m. \end{equation}

For this plan, the behavior of PA purchase is straightforward, as all retirees are required to buy the same amount ( $m$ ) of public annuities. On the other hand, the behavior in the private annuity market is more interesting, particularly when $m$ is set at a not-too-high level such that retirees with high value of $\theta$ still have residual annuitization demand after purchasing the mandated amount of public annuities. In this case, retirees with good health satisfy their residual demands by purchasing annuities from the private sector. The optimal choice of private annuity ( $\nu _{\theta }^{\ast }$ ), which is obtained by combining (14) and $\left. \frac{\partial U}{\partial \nu _{\theta }}\right \vert _{\nu _{\theta }=\nu _{\theta }^{\ast }}=0$ in (13), is given by

(15) \begin{equation} u^{\prime }\left ( w-m-\nu _{\theta }^{\ast }\right ) =\frac{\theta }{1+\rho }Vu^{\prime }\left ( Gm+V\nu _{\theta }^{\ast }\right ). \end{equation}

Using similar procedure as in (7), we can show that

(16) \begin{equation} \frac{\partial \nu _{\theta }^{\ast }}{\partial \theta }> 0. \end{equation}

Since $\nu _{\theta }^{\ast }$ is always increasing for those who buy private annuity, we can determine the threshold level of survival probability $\theta _{mb}$ by substituting $\nu _{\theta }^{\ast }=0$ in (15) to obtain

(17) \begin{equation} \theta _{mb}=\frac{\left ( 1+\rho \right ) u^{\prime }\left ( w-m\right ) }{V^{\ast }u^{\prime }\left ( G^{\ast }m\right ) }, \end{equation}

where $V^{\ast }$ is the equilibrium value of payout of the private annuity, which is determined according to the zero-profit condition as

(18) \begin{equation} V^{\ast }=\frac{\left ( 1+r\right ) \int _{\underline{\theta }}^{\overline{\theta }}\nu _{\theta }^{\ast }dF(\theta )}{\int _{\underline{\theta }}^{\overline{\theta }}\theta \nu _{\theta }^{\ast }dF(\theta )}=\frac{\left ( 1+r\right ) \int _{\theta _{mb}}^{\overline{\theta }}\nu _{\theta }^{\ast }dF(\theta )}{\int _{\theta _{mb}}^{\overline{\theta }}\theta \nu _{\theta }^{\ast }dF(\theta )}. \end{equation}

It is straightforward to see that the equilibrium value of PA payout ( $G^{\ast }$ ) for the pure MPA plan under the zero-profit condition is given byFootnote ⁹

(19) \begin{equation} G^{\ast }=\frac{\left ( 1+r\right ) \int _{\underline{\theta }}^{\overline{\theta }}\gamma _{\theta }^{\ast }dF(\theta )}{\int _{\underline{\theta }}^{\overline{\theta }}\theta \gamma _{\theta }^{\ast }dF(\theta )}=\frac{\left ( 1+r\right ) \int _{\underline{\theta }}^{\overline{\theta }}mdF(\theta )}{\int _{\underline{\theta }}^{\overline{\theta }}\theta mdF(\theta )}=\frac{1+r}{E\left ( \theta \right ) }, \end{equation}

the actuarially fair level.

Buyers’ choices after the introduction of the pure MPA plan are summarized as follows. Buyers with $\theta \leq \theta _{mb}$ purchase the PA ( $\gamma _{\theta }^{\ast }=m$ ) only, where $\theta _{mb}$ is defined in (17). On the other hand, buyers with $\theta > \theta _{mb}$ purchase both the PA (at the mandated amount, $\gamma _{\theta }^{\ast }=m$ ) and the private annuity ( $\nu _{\theta }^{\ast }$ ), where $\nu _{\theta }^{\ast }$ is determined according to (15) and is increasing in $\theta$ .

Buyers’ annuitization choices under the pure MPA plan are shown in Panel B of Figure 1.

4.1 VPA plan with ceiling

The Hong Kong Mortgage Corporation (HKMC) Annuity Plan was first launched in July 2018. The plan is a voluntary one. Individuals over a certain age (60 currently) are allowed to purchase the public annuities, with the amount of purchase ranging from HKD50,000 (roughly USD6,400) to a level set by the PA provider from time to time. Currently, the maximum amount of purchase is HKD5 million (roughly USD641,000), which was set at June 2022.Footnote ¹⁰

Motivated by the observed practices in Hong Kong, we consider the VPA plan with ceiling, which is specified as:Footnote ¹¹

(20) \begin{equation} 0\leq \gamma _{\theta }\leq m, \end{equation}

where

(21) \begin{equation} m\leq \overline{m}, \end{equation}

with $\overline{m}$ being determined according toFootnote ¹²

(22) \begin{equation} u^{\prime }\left ( w-\frac{G^{\ast }-V^{\ast }}{\widehat{V}^{\ast }-V^{\ast }}\overline{m}\right ) =\frac{\overline{\theta }}{1+\rho }V^{\ast }u^{\prime }\left ( \widehat{V}^{\ast }\frac{G^{\ast }-V^{\ast }}{\widehat{V}^{\ast }-V^{\ast }}\overline{m}\right ), \end{equation}

and $V^{\ast }$ is determined according to (18) and $G^{\ast }$ is the equilibrium payout of the VPAc plan, given by (26) below. Since setting a very high level of $m$ may not be financially prudent and may lead to a complete or substantial crowding out of the private annuity market, it is not likely that many governments want to do so. As will be shown in subsequent analysis, imposing the not-too-high ceiling condition (21) ensures that the private annuity market is not substantially crowded out.

When the payout of the PA is higher than that of the private annuity (to be shown in Proposition 1) and when the optimal amount of PA purchased by individual $\theta$ , denoted by $\gamma _{\theta }^{\ast }$ , is less than the ceiling level $m$ , the optimal choice $\gamma _{\theta }^{\ast }$ is defined by

(23) \begin{equation} u^{\prime }\left ( w-\gamma _{\theta }^{\ast }\right ) =\frac{\theta }{1+\rho }Gu^{\prime }\left ( G\gamma _{\theta }^{\ast }\right ), \end{equation}

which is obtained by combining $\nu _{\theta }=0$ and $\left. \frac{\partial U_{\theta }}{\partial \gamma _{\theta }}\right | _{\gamma _{\theta }=\gamma _{\theta }^{\ast }}=0$ in (12). Following similar procedure as in (7), it is straightforward to show that

(24) \begin{equation} \frac{\partial \gamma _{\theta }^{\ast }}{\partial \theta }=\frac{-Gu^{\prime }\left ( c_{2\theta }^{\ast }\right ) }{\left ( 1+\rho \right ) u^{\prime \prime }\left ( c_{1\theta }^{\ast }\right ) +\theta G^{2}u^{\prime \prime }\left ( c_{2\theta }^{\ast }\right ) }> 0. \end{equation}

Buyers’ choices under the VPAc plan are summarized as follows. When this plan is introduced, (a) buyers with $\theta < \theta _{om}$ purchase the PA only, with the PA purchase ( $\gamma _{\theta }^{\ast }$ ) being determined according to (23) and increasing in $\theta$ , where

(25) \begin{equation} \theta _{om}=\frac{\left ( 1+\rho \right ) u^{\prime }\left ( w-m\right ) }{G^{\ast }u^{\prime }\left ( G^{\ast }m\right ) }; \end{equation}

(b) buyers with $\theta _{om}\leq \theta \leq \theta _{mb}$ purchase the PA only, and purchase the maximum amount of PA ( $\gamma _{\theta }^{\ast }=m$ ), where $\theta _{mb}$ is defined in (17); and (c) buyers with $\theta > \theta _{mb}$ purchase both the PA (at the maximum amount, $\gamma _{\theta }^{\ast }=m$ ) and the private annuity ( $\nu _{\theta }^{\ast }$ ), with $\nu _{\theta }^{\ast }$ being determined according to (15) and increasing in $\theta$ . (The proof of the above results, which is straightforward but tedious, is given in the Online Appendix.)

Consistent with the buyers’ behavior, the equilibrium value of PA payout of the VPAc plan is given byFootnote ¹³

(26) \begin{equation} G^{\ast }=\frac{\left ( 1+r\right ) \int _{\underline{\theta }}^{\overline{\theta }}\gamma _{\theta }^{\ast }dF(\theta )}{\int _{\underline{\theta }}^{\overline{\theta }}\theta \gamma _{\theta }^{\ast }dF(\theta )}=\frac{\left ( 1+r\right ) \left [ \int _{\underline{\theta }}^{\theta _{om}}\gamma _{\theta }^{\ast }dF(\theta )+\int _{\theta _{om}}^{\overline{\theta }}mdF(\theta )\right ] }{\int _{\underline{\theta }}^{\theta _{om}}\theta \gamma _{\theta }^{\ast }dF(\theta )+\int _{\theta _{om}}^{\overline{\theta }}\theta mdF(\theta )}. \end{equation}

Buyers’ annuitization choices under the VPAc plan are shown in Panel C of Figure 1.

4.2 MPA plan with flexibility

The Central Provident Fund (CPF) Lifelong Income for the Elderly (LIFE) program in Singapore was introduced in 2009 and has become mandatory since 2013. Participants are required to set aside the Full Retirement Sum (FRS) to buy the lifelong annuity provided by the CPF Board. The FRS level is adjusted periodically and the current level is, for example, 192,000 Singaporean dollars (about USD142,000) for those aged 55 in 2022. In 2016, the CPF Board introduced the Retirement Sum Topping-Up Scheme. Participants can use this scheme to top-up their Retirement Sum up to the Enhanced Retirement Sum (ERS), which is 1.5 times of the FRS.Footnote ¹⁴ We observe that a floor (i.e. the FRS) and a ceiling (i.e. the ERS) are imposed on the amount of PA purchase in the CPF LIFE plan. Some flexibility between these two levels is given to the participants in this MPA program.

Motivated by the observed practices in Singapore,Footnote ¹⁵ the MPA plan with flexibility is specified as:

(27) \begin{equation} f\leq \gamma _{\theta }\leq m, \end{equation}

where parameters $m$ and $f$ satisfy (21) andFootnote ¹⁶

(28) \begin{equation} \widehat{\nu }_{\underline{\theta }}^{\ast }\leq f< m. \end{equation}

If condition (28) does not hold, the lowest mandated level ( $f$ ) of the MPAf plan is even lower than the purchase level of private annuity of the least healthy retiree before the PA plan is introduced. We impose condition (28) to eliminate this uninteresting case.Footnote ¹⁷

Following similar procedure as above, buyer’s behavior under this PA plan is given as follows. After the PA plan is introduced, (a) buyers with $\theta \leq \theta _{fo}$ only purchase the minimum required level ( $\gamma _{\theta }^{\ast }=f$ ) of PA, where

(29) \begin{equation} \theta _{fo}=\frac{\left ( 1+\rho \right ) u^{\prime }\left ( w-f\right ) }{G^{\ast }u^{\prime }\left ( G^{\ast }f\right ) }; \end{equation}

(b) buyers with $\theta _{fo}< \theta < \theta _{om}$ purchase the PA only, with the PA purchase ( $\gamma _{\theta }^{\ast }$ ) being determined according to (23) and increasing in $\theta$ , where $\theta _{om}$ is defined in (25); (c) buyers with $\theta _{om}\leq \theta \leq \theta _{mb}$ purchase the maximum amount of PA ( $\gamma _{\theta }^{\ast }=m$ ) only, where $\theta _{mb}$ is defined in (17); and (d) buyers with $\theta > \theta _{mb}$ purchase both the PA (at the maximum amount, $\gamma _{\theta }^{\ast }=m$ ) and the private annuity ( $\nu _{\theta }^{\ast }$ ), with $\nu _{\theta }^{\ast }$ being determined according to (15) and increasing in $\theta$ . Moreover, the equilibrium PA payout of the MPAf plan is given by

(30) \begin{equation} G^{\ast }=\frac{\left ( 1+r\right ) \int _{\underline{\theta }}^{\overline{\theta }}\gamma _{\theta }^{\ast }dF(\theta )}{\int _{\underline{\theta }}^{\overline{\theta }}\theta \gamma _{\theta }^{\ast }dF(\theta )}=\frac{\left ( 1+r\right ) \left [ \int _{\underline{\theta }}^{\theta _{fo}}fdF(\theta )+\int _{\theta _{fo}}^{\theta _{om}}\gamma _{\theta }^{\ast }dF(\theta )+\int _{\theta _{om}}^{\overline{\theta }}mdF(\theta )\right ] }{\int _{\underline{\theta }}^{\theta _{fo}}\theta fdF(\theta )+\int _{\theta _{fo}}^{\theta _{om}}\theta \gamma _{\theta }^{\ast }dF(\theta )+\int _{\theta _{om}}^{\overline{\theta }}\theta mdF(\theta )}. \end{equation}

Buyers’ annuitization choices under the MPAf plan are shown in Panel D of Figure 1.

4.3 Severity reduction and private market distortion effects in a two-tier annuity market

In Sections 4.1 and 4.2, we obtained the results about annuity buyers’ choices conditional on a higher payout for the PA ( $G^{\ast }> V^{\ast }$ ). The following proposition, which compares each of the equilibrium payouts of the public and private annuities with the payout level of the private annuity market before the PA plan is introduced, confirms that $G^{\ast }> V^{\ast }$ is an endogenous outcome under either the VPAc or MPAf plan.Footnote ¹⁸

Proposition 1. Consider the introduction of either the VPAc plan or MPAf plan. Compared with the equilibrium payout of the private annuity ( $\widehat{V}^{\ast }$ ) before the introduction of the PA plan,

(a) the equilibrium payout of the PA is higher:

(31) \begin{equation} G^{\ast }> \widehat{V}^{\ast }; \end{equation}

and (b) the equilibrium payout of the private annuity is lower:

(32) \begin{equation} V^{\ast }< \widehat{V}^{\ast }. \end{equation}

Combining parts (a) and (b) of Proposition 1, together with the positive correlation of the annuity purchase amount and $\theta$ according to (24) for those buyers whose purchase amounts are not constrained by the quantity restrictions of the PA plan, we obtainFootnote ¹⁹

(33) \begin{equation} V^{\ast }< G^{\ast }< \frac{1+r}{E\left ( \theta \right ) }. \end{equation}

An implication of (33) is the outcome of a two-tier annuity market. Less healthy people only purchase the public annuities, but healthier people purchase both public and private annuities. Moreover, we observe from the above analysis that there are two key effects when the PA plan is introduced.

First, the design of the PA plan matters. The imposed quantity restrictions (for both the ceiling and floor) on the PA purchase lead to a reduction in the severity of adverse selection, which originally arises from the positive correlation of the risk type (based on survival probability) and the PA purchase amount. Specifically, the ceiling restriction of both PA plans reduces the positive correlation of these two variables at the upper end by restricting the purchase amount of healthier retirees to the maximum level ( $m$ ), and the floor restriction of the MPAf plan reduces the positive correlation at the lower end by requiring the less healthy retirees to purchase the minimum mandated level ( $f$ ). We call this effect the severity reduction effect.

The restrictions imposed by the PA plans may not be implemented in the private annuity market, and even if some of them are implemented, the effects are different. Obviously, private annuity companies cannot make retirees’ purchase of private annuity products mandatory. On the other hand, imposing the maximum purchase level is possible for private annuity companies, but this policy is not effective in reducing the severity of adverse selection of the competitive private market. In a competitive market where different companies offer similar annuity products, if a company sets a purchase ceiling, a retiree with good health can buy the maximum amount from this company and then go to another one to satisfy her unfulfilled demand. It is not likely (and also not legal in many countries) that different companies share their customer lists and restrict the customers from going to other companies to buy annuities.Footnote ²⁰ Since the overall severity of adverse selection is determined by the total amount of private annuity purchased by all retirees, the effect of imposing purchase restriction on the severity of adverse selection in the PA sector cannot be replicated in the private market.

Second, the PA plan affects indirectly the private annuity market. Since the public and private annuity contracts provide the same financial function to the retirees, and the payout of the PA is higher, the retirees satisfy their annuity demand by first purchasing the more attractive PA. Because of the restrictions imposed by either VPAc or MPAf plan, the demands of some retirees may not be completely satisfied. As shown in (18), the presence of the PA plan causes less healthy retirees to drop out completely from the private annuity market. The equilibrium price of private annuities will be higher (and annuity payout lower) since the low-risk buyers do not participate in the private annuity market. At the same time, healthier retirees’ purchases of the private annuity are also distorted by the PA plan, since their budget constraints are changed after purchasing $m$ units of PA at a cheaper price.Footnote ²¹ When the utility function is homothetic, combining these two factors leads to (32), as shown in Appendix A. We label this effect the private market distortion effect.

4.4 Comparing PA plans with different degrees of choice restriction

Based on Proposition 1, we trace the effects of the PA plans on the two equilibrium annuity payouts ( $G^{\ast }$ and $V^{\ast }$ ) to the severity reduction and private market distortion effects. The above analysis focuses on the similarities of the VPAc and MPAf plans and shows that both effects are present in each plan. We now examine their differences by analyzing the relative importance of the two effects in the two plans.

In general, different economies adopting different PA plans are also likely to select different ceiling ( $m$ ) and floor ( $f$ ) levels. It is usually difficult to obtain unambiguous results when comparing different PA plans with changes in both $m$ and $f$ parameters. Instead, we conduct a comparative static exercise based on the following distinction when comparing parameters $m$ and $f$ of various PA plans: there is no floor ( $f=0$ ) for the VPAc plan based on (20), there is a floor ( $0\leq \widehat{\nu }_{\underline{\theta }}^{\ast }< f< m$ ) for the MPAf plan based on (28), and the floor is the same as the ceiling ( $f=m$ ) for the pure MPA plan based on (14).

We now compare the equilibrium payouts of various PA plans with the same level of $m$ but different values of $f$ . The results are summarized in the following proposition.

Proposition 2. Comparing a VPAc plan, a MPAf plan, and a pure MPA plan such that each plan has the same level of $m$ , the equilibrium payouts of the private and public annuities in the three plans are ranked as follows:

(34) \begin{equation} V_{MPA}^{\ast }< V_{MPAf}^{\ast }< V_{VPAc}^{\ast }< \widehat{V}^{\ast }< G_{VPAc}^{\ast }< G_{MPAf}^{\ast }\,< G_{MPA}^{\ast }. \end{equation}

The proof of Proposition 2 is very similar to that of Proposition 1. It is available in the Online Appendix.

Among different PA plans with the same ceiling level ( $m$ ), the floor parameter ( $f$ ) of a particular PA plan can be interpreted as representing its degree of choice restriction. Comparing with the pure MPA plan with the same ceiling level, the MPAf plan has less choice restriction ( $f< m$ ). As a result, the severity reduction effect is not so strong and the private market distortion effect is also weaker, leading to the results that $G_{MPAf}^{\ast }\,$ is lower (than $G_{MPA}^{\ast }$ ) in the PA sector but $V_{MPAf}^{\ast }\,$ is higher (than $V_{MPA}^{\ast }$ ) in the private annuity market. On the other hand, we observe that while both the VPAc and MPAf plans with the same ceiling level have elements of flexibility and restrictiveness, the relative emphasis is different. Comparing with the VPAc plan, the MPAf plan has a higher degree of choice restriction ( $f> 0$ ). As a result, $G_{MPAf}^{\ast }\,$ is higher (than $G_{VPAc}^{\ast }$ ) in the PA sector but $V_{MPAf}^{\ast }\,$ is lower (than $V_{VPAc}^{\ast }$ ) in the private annuity market.

More generally, the intuition of Proposition 2 can be understood as follows. When the degree of choice restriction of the PA plan is increased by raising the floor parameter $f$ from 0 (VPAc plan) to an intermediate value $0\leq \widehat{\nu }_{\underline{\theta }}^{\ast }< f< m$ (MPAf plan) and then to $m$ (pure MPA plan), poor health buyers are required to purchase more units of the PA. As a result, the severity reduction effect in the PA sector becomes stronger (i.e. $G_{MPA}^{\ast }$ is the highest). On the other hand, fewer buyers (only those who are very healthy) participate in the private annuity market, resulting in a stronger private market distortion effect (i.e. $V_{MPA}^{\ast }$ is lowest).

To summarize, the tradeoff between the severity reduction and private market distortion effects is present in each of the three PA plans: VPAc, MPAf, and pure MPA plans. The PA plan with a lower degree of choice restriction (such as the VPAc plan) has a smaller gain in the severity reduction effect but also a smaller loss in the private market distortion effect. On the other hand, the PA plan with a higher degree of choice restriction has a larger gain in the severity reduction effect but also a larger loss in the private market distortion effect.

5.1 Effects under a VPA with ceiling plan

We consider the VPAc plan with the not-too-high ceiling condition (21). We first obtain a useful result in Lemma 2. The proof is given in the Online Appendix.

Lemma 2. If ( 21 ) holds, then

(38) \begin{equation} U_{\overline{\theta }}^{\ast }-\widehat{U}_{\overline{\theta }}^{\ast }< 0. \end{equation}

The intuition of Lemma 2 is as follows. When the VPAc plan is introduced, the retirees benefit from the higher level of PA payout $G^{\ast }$ (the severity reduction effect), but they can at most enjoy this benefit up to $m$ units of PA purchase. On the other hand, retirees who are very healthy and have residual demand for the private annuity suffer from the lower level of private annuity payout $V^{\ast }$ (the private market distortion effect). Lemma 2 shows that when $m$ is set according to condition (21), the loss from the private market distortion effect for the healthiest retirees (with $\theta =\overline{\theta }$ ) dominates the benefit from the severity reduction effect. As a result, (38) holds and the healthiest annuity buyers are adversely affected by the introduction of the VPAc plan.

With the help of Lemmas 1 and 2, we obtain the systematically different effects of introducing the VPAc plan on two categories of buyers, based on their survival probabilities ( $\theta$ ).Footnote ²³ The result is summarized in the following proposition.

The policy implications of Proposition 3, together with those of Proposition 4 regarding the MPAf plan in Section 5.2, will be discussed in Section 5.3.

5.2 Effects under a MPA with flexibility plan

We now consider the MPAf plan. The following lemma is useful for analyzing the utility effects under this plan. The proof is given in the Online Appendix.

Lemma 3. If

(39) \begin{equation} f> \underline{f}, \end{equation}

where $\underline{f}$ is the larger root to

(40) \begin{equation} U_{\underline{\theta }}\left ( \underline{f}\right ) =u\left ( w-\underline{f}\right ) +\frac{\underline{\theta }}{1+\rho }u\left ( G^{\ast }\underline{f}\right ) =\widehat{U}_{\underline{\theta }}^{\ast }, \end{equation}

with $G^{\ast }$ given by ( 30 ) and $\widehat{U}_{\underline{\theta }}^{\ast }=u\left ( w-\widehat{v}_{\underline{\theta }}^{\ast }\right ) +\frac{\underline{\theta }}{1+\rho }u\left ( \widehat{V}^{\ast }\widehat{v}_{\underline{\theta }}^{\ast }\right )$ , then

(41) \begin{equation} U_{\underline{\theta }}^{\ast }-\widehat{U}_{\underline{\theta }}^{\ast }< 0. \end{equation}

If

(42) \begin{equation} \widehat{\nu }_{\underline{\theta }}^{\ast }\leq f\leq \underline{f}, \end{equation}

then ( 41 ) does not hold.

Under the MPAf plan, retirees with poor health (such as those with $\theta < \theta _{fo}$ ) benefit from the higher PA payout ( $G^{\ast }$ ) but are adversely affected by the restriction to buy the mandated level ( $f$ ) of PA purchase. Lemma 3 shows that if the floor is set at a not-too-low level (i.e. (39) holds), then for retirees with the lowest survival probability ( $\underline{\theta }$ ), the mandated floor level deviates quite substantially from their annuity demand (as measured by $\widehat{\nu }_{\underline{\theta }}^{\ast }$ before the PA plan is introduced). As a result, (41) holds and annuity buyers with the lowest survival probability are adversely adversely by the introduction of the MPAf plan, because the loss from the restriction to buy the mandated PA level dominates the benefit from the higher PA payout.

We now focus on the more interesting case that the not-too-low floor condition (39) holds.Footnote ²⁴ The following proposition summarizes the results that introducing the MPAf plan has systematically different effects on three categories of buyers, based on their survival probabilities ( $\theta$ ).

The proof of Proposition 4 is given in the Online Appendix. Most parts of the proof are similar to those of Proposition 3.

Propositions 3 and 4 appear to be quite different, with one threshold for the VPAc plan in Proposition 3 but two thresholds for the MPAf plan in Proposition 4 when (39) holds.Footnote ²⁵ There is actually a similarity between Propositions 3 and 4 in that the utility difference ( $U_{\theta }^{\ast }-\widehat{U}_{\theta }^{\ast }$ ) for either the VPAc or MPAf plan is first increasing in $\theta$ and then decreasing in $\theta$ after reaching a maximum. As a result, there is a single peak in the $U_{\theta }^{\ast }-\widehat{U}_{\theta }^{\ast }$ function for either PA plan. The above framework also helps understand the main factor leading to Proposition 4. When $f$ is set according to condition (39), the floor has a strong effect in restricting the PA purchases by retirees of poor health. Thus, the distortion caused by the mandated floor level dominates the severity reduction effect, leading to (41) for the least healthy buyers. Together with the single peak property of the $U_{\theta }^{\ast }-\widehat{U}_{\theta }^{\ast }$ function, it can be shown (in the Online Appendix) that there are three categories of buyers.Footnote ²⁶

5.3 Which PA plan to adopt: Voluntary or mandatory?

Our analysis provides some guidance regarding which PA plan the government wants to adopt. Whatever the MPAf or VPAc plan is adopted, the average health group benefits but the good health group is adversely affected.Footnote ²⁷ The intuition is that the annuity buyers with average health purchase some, but not excessive, amount of annuity before the PA plan is introduced. Thus, when the PA plan is introduced, they benefit from the higher PA payout but their optimal choices of annuity quantity are not severely distorted by the restriction of either plan. On the other hand, the negative private market distortion effect is more important to the good health group, leading to a drop in utility level. For the poor health group, the change in utility level depends on which PA plan is adopted. In particular, retirees with poor health are adversely affected in the MPAf plan with condition (39) because the minimum mandated level of annuity purchase differs substantially from their desired levels in the absence of the plan, but benefit from the VPAc plan because there is no floor restriction that would otherwise distort the amount of their PA purchase.

While the government’s ultimate decision to adopt a particular PA plan depends on many economic and political factors, our analysis based on a simple two-period model brings out some important factors (the severity reduction and private market distortion effects) clearly and shows how the two existing PA plans have systematically different utility effects on the retirees with different health characteristics.Footnote ²⁸