Ambiguous life expectancy and the demand for annuities

Abstract

In this paper, ambiguity aversion to uncertain survival probabilities is introduced in a static life-cycle model with a bequest motive to study the optimal demand for annuities. Provided that annuities’ return is sufficiently large, and notably when it is fair, positive annuitization is known to be the optimal strategy of ambiguity neutral individuals. Conversely, we show that the demand for annuities decreases with ambiguity aversion and that there exists a finite degree of aversion above which the demand is non-positive: the optimal strategy is then to either sell annuities short or to hold zero annuities if the former option is not available. To conclude, ambiguity aversion appears to be a relevant candidate for explaining the annuity puzzle.

Appendices

Appendix

Appendix A: Proof of Proposition 1.

As a preliminary, let us establish that $u^{\prime \prime }_{11}+R^2u^{\prime \prime }_{22}-2Ru^{\prime \prime }_{12}<0$, $u^{\prime \prime }_{12}-Ru^{\prime \prime }_{22} >0$ and $R u^{\prime \prime }_{12} - u^{\prime \prime }_{11}>0$.

First, as the Hessian of u is negative definiteness, it is straightforward that $u^{\prime \prime }_{11}+R^2u^{\prime \prime }_{22}-2Ru^{\prime \prime }_{12}<0$. Second, consider the static program $\max _{c,x} u[c,xR]$ subject to the constraint $\Omega = c+x$ where $\Omega$ is the life-cycle income of an agent. This program is equivalent to $\max _{c}u[c,(\Omega -c)R]$ or $\max _{x}u[\Omega -x,xR]$. Then, using the FOC of these programs and the implicit function theorem, there exist two functions A and B such that $c=A(\Omega )$ and $x=B(\Omega )$. It is straightforward that $A'(\Omega )$ has the sign of $u^{\prime \prime }_{12}-Ru^{\prime \prime }_{22}$ while $B'(\Omega )$ has the one of $R u^{\prime \prime }_{12} - u^{\prime \prime }_{11}$. By definition, c and x are normal goods if and only if $\partial c / \partial \Omega$ and $\partial x / \partial \Omega$ are positive. Then, Assumption 1 implies $u^{\prime \prime }_{12}-Ru^{\prime \prime }_{22} >0$ and $R u^{\prime \prime }_{12} - u^{\prime \prime }_{11}>0$.

A.1: Proof of statement (i).

Using (4), let us define $F(a,x)=-u^{\prime }_1[c (a,x),x R] +Ru^{\prime }_2[c(a,x),x R]=0$. Under Assumption 1, $F'_1(a,x)=(R_a-R)(Ru^{\prime \prime }_{12}-u^{\prime \prime }_{11})$ is non-negative whereas $F'_2(a,x)=u^{\prime \prime }_{11}+R^2u^{\prime \prime }_{22}-2Ru^{\prime \prime }_{12}$ is negative. Then, there exists a continuous and differentiable function f such that $x=f(a)$ and, under Assumption 1, $0\le f'(a)=-F'_1(a,x)/F'_2(a,x)\le R_a-R$. Replacing $x=f(a)$ and (4) in (3) allows us to define the solution $({\bar{a}},f({\bar{a}}))$ as a pair satisfying:

$$\begin{aligned} G(a,R_a,p)=p(R_{a}-R) u^{\prime }_2[{\widehat{c}}(a), f(a)R] -(1-p)R v^{\prime }[( w-a) R^{2}] =0. \end{aligned}$$

Then, an optimal demand for annuities ${\bar{a}}$ is a real root of $G(a,R_a,p)=(R_a-R)p\varphi (a)-R(1-p)\psi (a)$ where $\varphi (a)=u^{\prime }_2[{\widehat{c}}(a), f(a)R]$ and $\psi (a)=v^{\prime }[( w-a) R^{2}]$. After computations, $\varphi '(a)=-R(R_a-R)(u^{\prime \prime \ 2}_{12} - u^{\prime \prime }_{11}u^{\prime \prime }_{22})/F'_2(a,x)$ and $\psi '(a)=-R^2v^{\prime \prime }[( w-a) R^{2}]$. Then, under Assumption 1, $\varphi$ is a decreasing function whereas $\psi$ is an increasing one, implying $G'_1(a,R_a,p)<0$. Moreover, $\varphi (-Rw/(R_a-R))=+\infty$, $0<\varphi (w)<+\infty$, $0<\psi (-Rw/(R_a-R))<+\infty$ and $\psi (w)=+\infty$. Then, as $G'_1(a,R_a,p)<0$, $G(-Rw/(R_a-R),R_a,p)>0$ and $G(w,R_a,p)<0$, there exists a unique optimal pair $({\bar{a}},{\bar{x}})$. This pair is such that ${\bar{a}}\in (-Rw/(R_a-R),w)$ and $0<{\bar{x}}=f({\bar{a}})<w$.

A.2: Proof of statement (ii).

Since $G({\bar{a}},R_a,p)=0$ and $G'_1(a,R_a,p)<0$, ${\bar{a}}$ is positive if and only if $G(0,R_a,p)$ is positive. Importantly, f(0) is independent of $R_a$ because (4) is independent of $R_a$ when $a=0$. Consequently, $\varphi (0)=u'_2[Rw-f(0),f(0)R]$ and $\psi (0)=v'[wR^2]$ are independent of $R_a$ and we have $G'_2(0,R_a,p)=p\varphi (0)>0$. Note that $G(0,R/p,p)=R(1-p)(u'_2[Rw-f(0),f(0)R]-v'[wR^2])$. As $f(0)<wR$ and $u'_2[c,y]\ge v'[y]$, we have $G(0,R/p,p)>0$. Since $G(0,R,p)<0$ and $G'_2(0,R_a,p)>0$, there exists a unique $\widehat{R}_a\in (R,R/p)$ such that $G(0,\widehat{R}_a,p)=0$. Consequently, ${\bar{a}}$ is positive if and only if $R_a$ is larger than $\widehat{R}_a$. $\square$

A.3: Proof of statement (iii).

According to Appendix A.2, ${\bar{a}}\ge 0$ if and only if $G(0,R_a,p)\ge 0$. As $G'_3(0,R_a,p)>0$, $G(0,R_a,0)=-Rv'[wR^2]<0$ and $G(0,R_a,1)=(R_a-R)u'_2[{\widehat{c}}(0),f(0)R]>0$, there exists a unique ${\widehat{p}}\in (0,1)$ such that ${\bar{a}}$ is positive if and only if p is larger than ${\widehat{p}}$. $\square$

Appendix B: Proof of Proposition 2.

As a preliminary, it would be useful to establish the following lemma:

Lemma 1

If (4) is satisfied (i.e., if ${\bar{x}}=f({\bar{a}})$) then the difference between u[.] the utility if alive and v[.] the utility if not is positive if and only if p is larger than a threshold $\check{p}\in (0,{\widehat{p}})$. Moreover, the demand for annuities ${\underline{a}}$ equalizing the utilities in the two states of nature is unique and negative.

Proof

Let $\tau (a)=u[{\widehat{c}}(a),f(a)R]$ be the utility if alive. As $0\le f'(a)\le R_a-R$ and $\tau '(a)=[(R_a-R)-f'(a)]u'_1[{\widehat{c}}(a),f(a)R]+Rf'(a)u'_2[{\widehat{c}}(a),f(a)R]$, the utility if alive increases with the demand for annuity while the utility if not, namely $v[(w-a)R^2]$, decreases with a. Hence, the difference between u[.] the utility if alive and v[.] the utility if not:

$$\begin{aligned} \xi (a)=u[{\widehat{c}}(a),f(a)R]-v[(w-a)R^2] \end{aligned}$$

increases with respect to a.

Let us define ${\underline{a}}$, the demand for annuities equalizing the utilities in the two states of nature (i.e., such that $\xi ({\underline{a}})=0$). Under Assumption 2 $\xi (-wR/(R_a-R))=u[0,0]-v[R^2R_aw/(R_a-R)]<0$ and $\xi (0)=u[{\widehat{c}}(0),f(0)R]-v[wR^2]>0$. As $\xi '(a)>0$, the demand for annuities ${\underline{a}}$ is unique and negative.

As $\xi ({\underline{a}})=0$ and $\xi '(a)>0$, $\xi ({\bar{a}})>0$ if and only if ${\bar{a}}>{\underline{a}}$, i.e., if and only if $G({\underline{a}},R_a,p)>0$. As $G'_3({\underline{a}},R_a,p)>0$, $G({\underline{a}},R_a,0)=-Rv'[(w-{\underline{a}})R^2]<0$ and $G({\underline{a}},R_a,1)=(R_a-R)u'_2[{\widehat{c}}({\underline{a}}),f({\underline{a}})R]>0$, there exists a unique $\check{p}\in (0,1)$ such that $\xi ({\bar{a}})$ is positive if and only if p is larger than $\check{p}$. As $G({\underline{a}},R_a,\check{p})=G({\bar{a}},R_a,{\widehat{p}})=0$, ${\underline{a}}<0$, $G'_1(a,R_a,p)<0$ and $G'_3({\underline{a}},R_a,p)>0$, we have $\check{p}<{\widehat{p}}$. $\square$

We now consider the problem (${\mathcal {P}}_\phi$) and denote by $(a^{\star },x^{\star })$ its solution which is the solution of (10) and (11).

B.1: Proof of statement (i).

We first show that $a^\star$ is lower than ${\bar{a}}$. Let us define the random variable $\widehat{\mathcal {U}}(a,{\widetilde{p}})={\mathcal {U}}(a,f(a),{\widetilde{p}})$ where the application $x=f(a)$ is derived from (11). Then, $\widehat{\mathcal {U}}(a,{\widetilde{p}})$ represents the expected utility when the budget constraint (1) and the consumption-bequest optimal allocation (11) are satisfied. It is a function of the random variable ${\widetilde{p}}$. Moreover, let $\widehat{\mathcal {U}}(a,p_\varepsilon )$ denote the expected utility associated to $p_\varepsilon$, a realization of ${\widetilde{p}}$. Then, the system of equations (10) and (11) rewrites as a single equation in a:

$$\begin{aligned} \eta _{_\phi }(a)= & {} E(\phi ^{\prime }(\widehat{\mathcal {U}}(a,{\widetilde{p}}))\{{\widetilde{p}}(R_{a}-R) u_1^{\prime }[{\widehat{c}}(a),f(a)R] \\&-\,( 1-{\widetilde{p}}) R^{2}v^{\prime }[(w-a) R^{2}]\}) =0. \end{aligned}$$

We are going to prove that $\eta _{_\phi }({\underline{a}})>0$, $\eta _{_\phi }({\bar{a}})<0$ and $\eta '_{_\phi }(a)<0$ for $a\in [{\underline{a}},{\bar{a}}]$.

As $\widehat{\mathcal {U}}({\underline{a}},{\widetilde{p}})$ is independent of ${\widetilde{p}}$, $\eta _{_\phi }({\underline{a}})$ has the sign of:

$$\begin{aligned} E\left( {\widetilde{p}}(R_{a}-R) u_1^{\prime }[{\widehat{c}}({\underline{a}}),f({\underline{a}})R] -( 1-{\widetilde{p}}) R^{2}v^{\prime }[(w-{\underline{a}}) R^{2}]\right) , \end{aligned}$$

i.e., since $E{\widetilde{p}}=p$ and (11), the one of $G({\underline{a}},R_a,p)$. Under Assumption 3 and according to Lemma 1 we have $G({\underline{a}},R_a,p)>0$ and, consequently, $\eta _{_\phi }({\underline{a}})>0$. Since $G({\bar{a}},R_a,p)=0$, using (11) we have $\eta _{_\phi }({\bar{a}})=E(\phi ^{\prime }(\widehat{\mathcal {U}}({\bar{a}},{\widetilde{p}}))RG({\bar{a}},R_a,{\widetilde{p}}))=Cov(\phi '(\widehat{\mathcal {U}}({\bar{a}},{\widetilde{p}})),$ $RG({\bar{a}},R_a,{\widetilde{p}}))<0$. Since $\phi$ is concave and $G'_1(a,R_a,{\widetilde{p}})<0$, $\eta '_{_\phi }(a)<0$. Consequently, $\eta _{_\phi }$ has a unique real root that shall be denoted $a^{\star }$ and that belongs to $({\underline{a}},{\bar{a}})$.

Importantly, the derivative of function $\widehat{\mathcal {U}}(a,p_\varepsilon )$ with respect to a realization $p_\varepsilon$ is:

$$\begin{aligned} \widehat{\mathcal {U}}_2^{\prime }(a,p_\varepsilon )=u[{\widehat{c}}(a),f(a)R] -v[(w -a) R^{2}]\equiv \xi (a). \end{aligned}$$

Then, $\xi (a)=\widehat{\mathcal {U}}_2^{\prime }(a,p_\varepsilon )$ is independent of any realizations of ${\widetilde{p}}$. Since $\xi ^{\prime }(a) >0$, $\xi ({\underline{a}})=0$ and $a^\star >{\underline{a}}$, we have $\xi (a^{\star })\ge 0$.

We now show that $a^\star$ decreases with ambiguity aversion. The strategy is to consider two independent DMs. The problem of the first DM is given by (${\mathcal {P}}_\phi$) and her optimal demand for annuities is denoted $a^{\star }$. The second DM faces the similar problem $({\mathcal {P}}_\psi )$ with $\psi \equiv T\circ \phi$ and where T is an increasing and concave function. Her optimal demand, denoted $a^{\star \star }$, is a solution of:

$$\begin{aligned} \eta _{_{T\circ \phi }}(a)= & {} E(\psi ^{\prime } (\widehat{\mathcal {U}}(a,{\widetilde{p}}))\{{\widetilde{p}}(R_{a}-R) u_1^{\prime }[{\widehat{c}}(a),f(a)R]\\&-\,( 1-{\widetilde{p}}) R^{2}v^{\prime }[(w-a) R^{2}]\}) =0, \end{aligned}$$

where $\psi ^{\prime }(\widehat{\mathcal {U}}(a,{\widetilde{p}}))=T^{\prime }(\phi (\widehat{\mathcal {U}}(a,{\widetilde{p}})))\phi ^{\prime }(\widehat{\mathcal {U}}(a,{\widetilde{p}}))$. Note that $\eta _{_{T\circ \phi }}(a)$ is a decreasing function of a.

From Step 1 to Step 4 it is supposed that ${\widetilde{p}}$ has only two realizations, $p_{1}$ and $p_{2}$, satisfying $1\ge p_{1}>p_{2}\ge 0$, and whose occurrence probabilities are, respectively, q and $1-q$. The result obtained is generalized in Step 5 for any distribution of ${\widetilde{p}}$.

$\underline{Step\; 1 -- \widehat{\mathcal {U}}(a^{\star },p_{1})\ge \widehat{\mathcal {U}}(a^{\star },p_{2}) \;and \; \widehat{\mathcal {U}}(a^{\star \star },p_{1}) \ge \widehat{\mathcal {U}}(a^{\star \star },p_{2}).}$

Let $a^\sharp$ an optimum (i.e., $a^\sharp =a^\star$ or $a^{\star \star }$). Then, we have $\widehat{\mathcal {U}}(a^\sharp ,p_1)-\widehat{\mathcal {U}}(a^\sharp ,p_2)=(p_1-p_2)\xi (a^\sharp )$. As we have proved that $\xi (a^\sharp )\ge 0$, we have $\widehat{\mathcal {U}}(a^\sharp ,p_1)\ge \widehat{\mathcal {U}}(a^\sharp ,p_2)$ for $a^\sharp =a^\star$ and $a^\sharp =a^{\star \star }$.

$\underline{Step \, 2 -- \widehat{\mathcal {U}}(a^{\star },p_{1})-\widehat{\mathcal {U}}(a^{\star \star },p_{1}) \, and \,\widehat{\mathcal {U}}(a^{\star },p_{2}) -\widehat{\mathcal {U}}(a^{\star \star },p_{2}) \; have \; opposite} \underline{signs.}$

Proceed by contradiction. Suppose first they are both positive. This implies that $E(\psi (\widehat{\mathcal {U}}(a^{\star },{\widetilde{p}}))) >E(\psi (\widehat{\mathcal {U}}(a^{\star \star },{\widetilde{p}})) )$, which is not possible since $E(\psi (\widehat{\mathcal {U}}(a^{\star \star },{\widetilde{p}})))$ is a maximum. Similarly, they cannot be both negative because this would imply that $E(\phi (\widehat{\mathcal {U}}(a^{\star }, {\widetilde{p}})))$ is not a maximum.

$\underline{Step \; 3 -- \widehat{\mathcal {U}}(a^{\star },p_{1})>\widehat{\mathcal {U}}(a^{\star \star },p_{1}) \;and \;\widehat{\mathcal {U}}(a^{\star },p_{2})<\widehat{\mathcal {U}}(a^{\star \star },p_{2}).}$

As a preliminary, the definition of a maximum yields $E(\phi (\widehat{\mathcal {U}}(a^{\star },{\widetilde{p}})))>E(\phi (\widehat{\mathcal {U}}(a^{\star \star },{\widetilde{p}})))$, which rewrites:

$$\begin{aligned} q[\phi (\widehat{\mathcal {U}}(a^{\star \star },p_{1})) -\phi (\widehat{\mathcal {U}}(a^{\star },p_{1}))]<(1-q) [\phi (\widehat{\mathcal {U}}(a^{\star },p_{2}))-\phi (\widehat{\mathcal {U}}(a^{\star \star },p_{2}))]. \end{aligned}$$

(13)

Equivalently, $E(\psi (\widehat{\mathcal {U}}(a^{\star \star },{\widetilde{p}})))>E(\psi (\widehat{\mathcal {U}}(a^{\star },{\widetilde{p}}) ))$ rewrites:

$$\begin{aligned} q[\psi (\widehat{\mathcal {U}}(a^{\star \star },p_{1}))-\psi (\widehat{\mathcal {U}}(a^{\star },p_{1}))] >(1-q)[\psi (\widehat{\mathcal {U}}(a^{\star },p_{2}))-\psi (\widehat{\mathcal {U}}(a^{\star \star },p_{2}))]. \end{aligned}$$

(14)

Now proceed by contradiction by supposing that $\widehat{\mathcal {U}}(a^{\star },p_{1})<\widehat{\mathcal {U}}(a^{\star \star },p_{1})$. Using Steps 1 and 2, this implies that $\widehat{\mathcal {U}}(a^{\star \star },p_{2})<\widehat{\mathcal {U}}(a^{\star },p_{2})<\widehat{\mathcal {U}}(a^{\star },p_{1}) <\widehat{\mathcal {U}}(a^{\star \star },p_{1})$. Since $\phi$ and $\psi$ are both increasing, dividing (14) by (13) yields the following inequalities:

$$\begin{aligned} \frac{\psi ( \widehat{\mathcal {U}}(a^{\star \star },p_{1}) ) -\psi ( \widehat{\mathcal {U}}(a^{\star },p_{1}) ) }{\phi ( \widehat{\mathcal {U}}(a^{\star \star },p_{1}) ) -\phi ( \widehat{\mathcal {U}}(a^{\star },p_{1}) ) }> \frac{\psi ( \widehat{\mathcal {U}}(a^{\star },p_{2}) ) -\psi ( \widehat{\mathcal {U}}(a^{\star \star },p_{2}) ) }{\phi ( \widehat{\mathcal {U}}(a^{\star },p_{2}) ) -\phi ( \widehat{\mathcal {U}}(a^{\star \star },p_{2}) ) }>0. \end{aligned}$$

(15)

Denote $y_{1}=\phi ( \widehat{\mathcal {U}}(a^{\star \star },p_{2}) ) ,$ $y_{2}=\phi ( \widehat{\mathcal {U}}(a^{\star },p_{2}) ) ,$ $y_{3}=\phi ( \widehat{\mathcal {U}}(a^{\star },p_{1}) ) ,$ and $y_{4}=\phi ( \widehat{\mathcal {U}}(a^{\star \star },p_{1}) )$. Hence, $y_{1}<y_{2}<y_{3}<y_{4}$ and (15) rewrites as follows:

$$\begin{aligned} \frac{T( y_{4}) -T( y_{3}) }{y_{4}-y_{3}}>\frac{ T( y_{2}) -T( y_{1}) }{y_{2}-y_{1}}. \end{aligned}$$

(16)

This latter inequality is true if and only if T is convex. Since T is concave, conclude that $\widehat{\mathcal {U}}(a^{\star },p_{1}) >\widehat{\mathcal {U}}(a^{\star \star },p_{1})$ and, using Step 2, that $\widehat{\mathcal {U}}(a^{\star },p_{2}) <\widehat{\mathcal {U}}(a^{\star \star },p_{2})$.

$\underline{Step \; 4 -- a^{\star }>a^{\star \star }\; for\; a\; binary\; distribution.}$

For an optimum a, we have $\widehat{\mathcal {U}}(a,p_{1}) -\widehat{\mathcal {U}}(a,p_{2})=(p_1-p_2)\xi (a)$ where $\xi (a)>0$ and $\xi '(a)>0$. Then, $\widehat{\mathcal {U}}(a,p_{1}) -\widehat{\mathcal {U}}(a,p_{2})$ is an increasing function of a. According to Steps 1 and 2, $\widehat{\mathcal {U}}(a^{\star },p_{1}) -\widehat{\mathcal {U}}(a^{\star },p_{2}) >\widehat{\mathcal {U}}(a^{\star \star },p_{1}) -\widehat{\mathcal {U}}(a^{\star \star },p_{2}) \ge 0$. Consequently, $a^{\star }>a^{\star \star }$.

$\underline{Step \; 5 -- a^{\star }>a^{\star \star } \; for \;any\; distribution\; of \; {\widetilde{p}}.}$

We have previously shown that $\eta _{_\phi }(a)$ and $\eta _{_{T\circ \phi }}(a)$ are decreasing functions of a. Consequently, $(a^{\star }>a^{\star \star })$ if and only if $(E(g({\widetilde{p}} )) =0\Rightarrow E(h({\widetilde{p}})) <0)$ where $g({\widetilde{p}}) =\phi ^{\prime }(\widehat{\mathcal {U}}(a^{\star },{\widetilde{p}} ))\{{\widetilde{p}}(R_{a}-R) u_1^{\prime }[{\widehat{c}}(a^\star ),f(a^\star )R] -( 1-{\widetilde{p}}) R^{2}v^{\prime }[(w-a^\star ) R^{2}]\}$ and $h({\widetilde{p}}) =T^{\prime }(\phi (\widehat{\mathcal {U}}(a^{\star },$ ${\widetilde{p}}) ) )\phi ^{\prime }( \widehat{\mathcal {U}}(a^{\star }, {\widetilde{p}}) )\{{\widetilde{p}}(R_{a}-R) u_1^{\prime }[{\widehat{c}}(a^\star ),f(a^\star )R] -(1-{\widetilde{p}}) R^{2}v^{\prime }[(w-a^\star ) R^{2}]\}$.

The diffidence theorem then applies [see Lemma 1 page 84 in Gollier (2001)] and, thus, the result $a^{\star }>a^{\star \star }$ holds for any distribution of ${\widetilde{p}}$. Then, the demand for annuities decreases with ambiguity aversion.

B.2: Proof of statement (ii).

According to (9) the following maxmin problem, denoted, is obtained when the absolute ambiguity aversion, given by $-\phi ^{\prime \prime }/\phi ^{\prime }$, is infinite: (${\mathcal {P}}_{\infty }$)

First observe that the optimum $(a^\sharp ,x^\sharp )$ is such that $x^\sharp =f(a^\sharp )$. Indeed, for any optimal value $p_\varepsilon ^\sharp$, $(a^\sharp ,x^\sharp )$ is determined by maximizing ${\mathcal {U}}(a,x,p_\varepsilon ^\sharp )$ subject to (1) and (2). The FOC (4) of the expected utility problem (${\mathcal {P}}_{0}$) (where $p=p_\varepsilon ^\sharp$) is independent of $p_\varepsilon ^\sharp$ and implies that $x^\sharp =f(a^\sharp )$.

Consequently, there are only three candidates for the optimum $a^\sharp$ according to the sign of $\xi (a^\sharp )$.

The first candidate corresponds to the case where $\xi (a^\sharp )=0$. Then it corresponds to the (unique) pair (${\underline{a}},f({\underline{a}})$) exhibited in Lemma 1. The maxmin utility attainable is then $v[(w-{\underline{a}})R^2]$.

The second candidate $\breve{a}$ corresponds to the case where $\xi (a^\sharp )<0$. As $\widehat{\mathcal {U}}(\breve{a},p_\varepsilon )=p_\varepsilon \xi (\breve{a}) +v[(w-\breve{a})R^2]<v[(w-\breve{a})R^2]$, the maxmin utility attainable is lower than $v[(w-\breve{a})R^2]$. Importantly, $\breve{a}$ is obviously the solution of the expected utility problem (${\mathcal {P}}_{0}$) where $p=p_{M}$ is the upper bound of Supp$({\widetilde{p}})$, the support of ${\widetilde{p}}.$ Then, according to Assumption 3 and Lemma 1, we have ${\underline{a}}<\breve{a}$ and, consequently, $v[(w-\breve{a})R^2]< v[(w-{\underline{a}})R^2]$. Since the maxmin utility attainable with $\breve{a}$ is strictly lower than the one attainable with ${\underline{a}}$, we have necessarily $\xi (a^\sharp )\ge 0$ at the optimum.

The third candidate ${\check{a}}$ corresponds to the case where $\xi (a^\sharp )>0$. Importantly, ${\check{a}}$ is the solution of the expected utility problem (${\mathcal {P}}_{0}$) where $p=p_{m}$ is the lower bound of Supp$({\widetilde{p}})$. Then, according to Assumption 3 and Lemma 1, we have ${\underline{a}}<{\check{a}}$. We now show that this third candidate cannot solve the problem (${\mathcal {P}}_{\infty }$) because if ${\check{a}}$ solves the problem (${\mathcal {P}}_{\infty }$) we have necessarily ${\underline{a}}\ge {\check{a}}$. Indeed, assume that ${\check{a}}$ solves the problem (${\mathcal {P}}_{\infty }$). Then we have $\widehat{\mathcal {U}}({\underline{a}},p_{m})\le \widehat{\mathcal {U}}({\check{a}},p_{m})$. As $0=\xi ({\underline{a}})=\widehat{\mathcal {U}}^{\prime }_2({\underline{a}},p_\varepsilon )<\xi ({\check{a}})=\widehat{\mathcal {U}}^{\prime }_2({\check{a}},p_\varepsilon )$, we obtain $\widehat{\mathcal {U}}({\underline{a}},1)\le \widehat{\mathcal {U}}({\check{a}},1)$, i.e., $u[{\widehat{c}}({\underline{a}}),f({\underline{a}})R]\le u[{\widehat{c}}({\check{a}}),f({\check{a}})R]$. As, $u[{\widehat{c}}(a),f(a)R]$ is increasing function of a, we have ${\underline{a}}\ge {\check{a}}$.

Consequently, we have $\xi (a^\sharp )=0$ at the optimum and the optimal pair $(a^\sharp ,x^\sharp )$ which solves the problem (${\mathcal {P}}_{\infty }$) is the pair $({\underline{a}},f({\underline{a}}))$ exhibited Lemma 1. Then, according to Lemma 1, the demand for annuities is always negative when ambiguity aversion is infinite. $\square$