Ambiguity sensitive preferences in Ellsberg frameworks

Abstract

We study the market implications of ambiguity sensitive preferences using the $\alpha$-maxmin expected utility ($\alpha$-MEU) model. In the standard Ellsberg framework, we prove that $\alpha$-MEU preferences are equivalent to either maxmin, maxmax or subjective expected utility (SEU). We show how ambiguity aversion impacts equilibrium asset prices, and revisit the laboratory experimental findings in Bossaerts et al. (Rev Financ Stud 23:1325–1359, 2010). Only when there are three or more ambiguous states, $\alpha$-MEU, maxmin, maxmax and SEU models induce different portfolio choices. We suggest criteria to discriminate among these models in laboratory experiments and show that ambiguity seeking agents may prevent the existence of market equilibrium. Our results indicate that ambiguity matters for portfolio choice and does not wash out in equilibrium.

Appendices

Proof of Proposition 2

In the following, we briefly summarize how the interaction among SEU and maxmin agents impacts the equilibrium asset prices. This will provide us the tools to prove Proposition 2.

Assume that $(p;w^1, \ldots , w^n)$ is an equilibrium with $p_\sigma >0$ for all $\sigma \in \{R,G,B\}$. Then, the equilibrium price p satisfies

$$\begin{aligned} \lambda _n p \in \partial U^n(w^n) \end{aligned}$$

(25)

for some $\lambda _n>0$; see (44). Here $\partial U^n(w)$ denotes the supergradient of the criterion $U^n$ of agent n at $w\in {\mathbb {R}}^3$.

The supergradient of a SEU agent with prior $\pi =(\pi _R, \pi _G, \pi _B)$ is simply the gradient

$$\begin{aligned} \partial U^n( w)= \left\{ \left( \pi _R u'(w_R),\pi _G u'(w_G) ,\pi _B u'(w_B) \right) \right\} . \end{aligned}$$

(26)

From (26) and the strict concavity of the utility function, it follows the well-known fact that the optimal portfolio $w=(w_R,w_G,w_B)$ of a SEU agent is always such that the optimal choices of state-dependent wealth are ranked opposite to the state-price/state-probability ratios, i.e.,

$$\begin{aligned} w_{\sigma }>w_{\nu } \Leftrightarrow \frac{p_{\sigma }}{\pi _{\sigma }}<\frac{p_{\nu } }{\pi _{\nu }}, \quad \sigma ,\nu \in \{R,B,G\}. \end{aligned}$$

(27)

The supergradient of an agent with maxmin (1-MEU) preferences represented as in (4) is

$$\begin{aligned} \partial U^m(w) = \left\{ \begin{array}{ll} \{(\pi _R u'(w_R), c u'(w_G), (1-\pi _R-c )u'(w_B) )\} &{} \text{ if } w_G> w_B \\ \{(\pi _R u'(w_R), d u'(w_G), (1-\pi _R-d )u'(w_B) )\} &{} \text{ if } w_G< w_B\\ \{(\pi _R u'(w_R), (\lambda c + (1-\lambda )d) u'(w_G), &{}\\ (1-\pi _R -(\lambda c + (1-\lambda )d))u'(w_B))\mid \lambda \in [0,1]\} &{} \text{ if } w_G= w_B. \end{array} \right. \nonumber \\ \end{aligned}$$

(28)

Using (25) and the shape of the supergradients we easily obtain the optimal portfolio choices that were already derived in Bossaerts et al. (2010).

In particular, from (28) and the strict concavity of u it follows that

$$\begin{aligned} \left\{ \begin{array}{lll} w_G>w_B &{} \hbox {if and only if} &{} \frac{p_G}{p_B}<\frac{c}{1-\pi _R-c}\\ w_G<w_B &{} \hbox {if and only if} &{} \frac{p_G}{p_B}>\frac{d}{1-\pi _R-d}\\ w_G=w_B &{} \hbox {if and only if} &{} \frac{p_G}{p_B}\in \left[ \frac{c}{1-\pi _R-c}, \frac{d}{1-\pi _R-d}\right] \end{array} \right. \end{aligned}$$

(29)

where $x/0:=\infty$. The larger the set of priors $\mathcal C$ in (5), the more likely a maxmin agent will take an unambiguous portfolio ($w_B=w_G$). In particular this will be always the case if ${\mathcal {C}}={\mathcal {C}}_{\max }:= \{(\pi _R, q, 1-q-\pi _R): q\in [0,1-\pi _R]\}$, because then the second, respectively, third coordinate of the supergradient in (28) will be 0 if either $w_G> w_B$ or $w_G< w_B$. Hence, $p_\sigma >0$ for all $\sigma \in \{R,G,B\}$ and (25) imply that in equilibrium this agent will only take an unambiguous portfolios w. If $c>0$ and/or $d<1-\pi _R$ in (5), then the multiple prior agent may also take an ambiguous portfolio in equilibrium. We observe that a maxmin agent holding an unambiguous optimal portfolio behaves as a SEU agent who is not differentiating between the ambiguous states G and B, but merges them to an unambiguous state $\{G,B\}$ with probability $(1-\pi _R)$. Indeed, from (28) and (25) it follows that

$$\begin{aligned} \frac{p_{\{G,B\}}}{p_R}= \frac{(1-\pi _R)u'(w_{\{G,B\}})}{\pi _R u'(w_R)}\left\{ \begin{array}{ll}<\frac{(1-\pi _R)}{\pi _R} &{} \text{ iff } w_{\{G,B\}}>w_R\\ >\frac{(1-\pi _R)}{\pi _R} &{} \text{ iff } w_{\{G,B\}}<w_R \end{array} \right. \end{aligned}$$

(30)

and thus

$$\begin{aligned} \frac{p_{\{G,B\}}}{(1-\pi _R)}< \frac{p_R}{\pi _R}\Leftrightarrow & {} w_{\{G,B\}}>w_R \end{aligned}$$

(31)

$$\begin{aligned} \frac{p_{\{G,B\}}}{(1-\pi _R)}> \frac{p_R}{\pi _R}\Leftrightarrow & {} w_{\{G,B\}}<w_R \quad (\hbox {compare this to } (27)), \end{aligned}$$

(32)

where $p_{\{G,B\}}:= p_G+p_B$ and $w_{\{G,B\}}:=w_G=w_B$.

Proof of Proposition 2

Case 1 Let $W_R>W_G>W_B$. Since the 1-MEU agents take an unambiguous portfolio, the optimal portfolio of some SEU agent must satisfy $y_G>y_B$ which according to (27) is equivalent to

$$\begin{aligned} \frac{p_B}{\pi _B}>\frac{p_G}{\pi _G}\end{aligned}$$

(33)

which only leaves the ranking of $p_R/\pi _R$ within (33) an open question. Suppose that the ranking of the ratios state-price/state-probability is as follows:

$$\begin{aligned} \frac{p_R}{\pi _R}\ge \frac{p_B}{\pi _B}>\frac{p_G}{\pi _G}. \end{aligned}$$

(34)

Then (27) implies that $y_G>y_B\ge y_R$ for any SEU agent, and rearranging (34) yields

$$\begin{aligned} \frac{p_G+p_B}{1-\pi _R}=\frac{p_G+p_B}{\pi _G+\pi _B} <\frac{p_R}{\pi _R}. \end{aligned}$$

Consequently, according to (30), we must have for each 1-MEU agent that $w_R<w_G=w_B$. But this contradicts the clearing of the market and $W_R>W_G>W_B$. If the ranking is (7), then we have $y_G>y_R>y_B$ for each SEU agent according to (27). Denote by $y^\Sigma =(y^\Sigma _R,y^\Sigma _G,y^\Sigma _B)$ the sum over all optimal portfolios of the SEU agents and similarly by $w^\Sigma =(w^\Sigma _R,w^\Sigma _G,w^\Sigma _B)$ the sum over all optimal portfolios of the 1-MEU agents. The market clearing condition says $W_\sigma = y^\Sigma _\sigma +w^\Sigma _\sigma$ for every $\sigma \in \{R,G,B\}$. Since $y^\Sigma _G>y^\Sigma _R$ we conclude that

$$\begin{aligned} w^\Sigma _R=W_R-y^\Sigma _R>W_G-y_G^\Sigma =w_G^\Sigma . \end{aligned}$$

Thus there must be at least one 1-MEU agent who’s portfolio $w=(w_R,w_G,w_B)$ satisfies $w_R>w_G=w_B$ which implies that $(p_G+p_B)/p_R>(1-\pi _R)/\pi _R$ due to (30). But then, again by (30), we must have $w_R>w_G=w_B$ for all 1-MEU agents. In case of (8) (27) and (30) imply the claimed ranking of payoffs in the portfolios y and w.

Case 2 Let $W_G>W_R>W_B$. As in case one we conclude that $y_G>y_B$. Assume that the ranking of the ratio state-price/state-probability is as follows:

$$\begin{aligned} \frac{p_R}{\pi _R}\ge \frac{p_B}{\pi _B}>\frac{p_G}{\pi _G}. \end{aligned}$$

(35)

Then as in case 1 it follows that $y_G>y_B\ge y_R$ and $w_R<w_G=w_B$ which together with the clearing of the market contradicts $W_R>W_B$. Similarly it follows that the ranking

$$\begin{aligned} \frac{p_B}{\pi _B}>\frac{p_G}{\pi _G}\ge \frac{p_R}{\pi _R} \end{aligned}$$

is not possible, since it would imply that $y_R\ge y_G>y_B$ and $w_R>w_G=w_B$ due to (30), again contradicting the assumed ranking of the aggregate wealth.

Case 3 Let $W_G> W_B>W_R$: Suppose that

$$\begin{aligned} \frac{p_B}{\pi _B}>\frac{p_G}{\pi _G}\ge \frac{p_R}{\pi _R} \end{aligned}$$

then, $y_R\ge y_G>y_B$, and in view of (30) we obtain $w_R>w_G=w_B$ for every 1-MEU agent which again contradicts the market clearing and the assumed ranking $W_G> W_B>W_R$. Again (27), (30), and the clearing of the market imply the claimed ranking of payoffs in the portfolios y, w for the remaining possible rankings.$\square$

Proof of Propositions 1 and 3

Proposition 1 is a special case of Proposition 3 since every set of priors in the standard Ellsberg framework is of the type ${\mathcal {C}}$ in (15), required in Proposition 3.

To prove Proposition 3 we observe that the maxmin utility with set of priors ${\mathcal {C}}$ can be written as

$$\begin{aligned}&u(w_\eta )+\sum _{\sigma \in S{\setminus } A}(u(w_\sigma )-u(w_\eta ))\pi _\sigma \nonumber \\&\quad +\sum _{\sigma \in A{\setminus }\{\eta \}}(u(w_\sigma )-u(w_\eta ))^+a_\sigma -(u(w_\sigma )-u(w_\eta ))^-b_\sigma , \end{aligned}$$

(36)

and the maxmax utility as

$$\begin{aligned} u(w_\eta )+\sum _{\sigma \in S{\setminus } A}(u(w_\sigma )-u(\eta ))\pi _\sigma +\sum _{\sigma \in A{\setminus }\{\eta \}}(u(w_\sigma )-u(w_\eta ))^+b_\sigma -(u(w_\sigma )-u(w_\eta ))^-a_\sigma . \end{aligned}$$

(37)

Consequently, the $\alpha$-MEU utility is

$$\begin{aligned} U(w)= & {} u(w_\eta )+ \sum _{\sigma \in S{\setminus } A}(u(w_\sigma )-u(w_\eta ))\pi _\sigma + \sum _{\sigma \in A{\setminus }\{\eta \}}(u(w_\sigma )\\&-\,u(w_\eta ))^+c_\sigma -(u(w_\sigma )-u(w_\eta ))^-d_\sigma . \end{aligned}$$

where $c_\sigma :=\alpha a_\sigma + (1-\alpha )b_\sigma$, and $d_\sigma :=\alpha b_\sigma + (1-\alpha )a_\sigma$, $\sigma \in A{\setminus }\{\eta \}$. If $\alpha >1/2$, then $c_\sigma <d_\sigma$; if $\alpha <1/2$, then $d_\sigma <c_\sigma$; and finally $c_\sigma =d_\sigma$ for $\alpha =1/2$. These facts, and comparing U for the different cases (i), (ii) and (iii) with (36) and (37) prove Proposition 3.

Lack of concavity of the $\alpha$-MEU utility

To see the lack of concavity of the $\alpha$-${\mathcal {C}}_{\max }$-MEU utility when $\alpha \ne 1$, consider portfolio $w^1$ such that $w^1_1=1, w^1_2=4 \hbox { and } w^1_j=2, \forall j= 2, \ldots , l$, and portfolio $w^2$ such that $w^2_1=1, w^2_2=2, w^2_3=6 \hbox { and } w^2_j=2, \forall j=4, \ldots , l$. Let $w^{\lambda }=(w^{\lambda }_1, \ldots , w^{\lambda }_l)$ be their convex combination, i.e., $w^{\lambda }_j=\lambda w^1_j +(1-\lambda ) w^2_j$, $j=1, \ldots , l$, $\lambda \in [0,1]$. Take for instance $\lambda ={1/2}$. Then $w^{\lambda }_1=1, w^{\lambda }_2=3, w^{\lambda }_3=4 \hbox { and } w^{\lambda }_j= 2, \forall j=4, \ldots , l$, and using (16)

$$\begin{aligned} \lambda U(w^1)+(1-\lambda )U(w^2)= & {} \alpha u(1) +(1-\alpha )\frac{1}{2} (u(4)+u(6))\\> & {} \alpha u(1)+(1-\alpha )u(4) =\alpha u(w^{\lambda \, A}_{\min })+(1-\alpha )u\left( w_{\max }^{\lambda \, A}\right) \\= & {} U(w^{\lambda })=U((\lambda w^1+ (1- \lambda )w^2)). \end{aligned}$$

Proof of Proposition 4

Proposition 4 follows from Lemmas 2–6 in the following.

Lemma 2

Suppose that the state-price vector $p=(p_\sigma )_{\sigma \in S}$ satisfies $p_\sigma >0$ for all $\sigma \in S$. Consider an $\alpha$-${\mathcal {C}}_{\max }$-MEU agent with $\alpha \in (0,1)$. Let $w=(w_\sigma )_{\sigma \in S}\in {\mathbb {R}}^n$ be an optimal portfolio for the $\alpha$-${\mathcal {C}}_{\max }$-MEU agent. Then, either w takes the same value on all ambiguous states, or there exist two disjoint subsets ${\overline{A}}$ and ${\underline{A}}$ of the set of ambiguous states A such that ${\overline{A}} \cup {\underline{A}} = A$ and two values $\overline{w},\underline{w}\in {\mathbb {R}}$ such that $w_\sigma = \overline{w}> \underline{w}=w_\eta$ for all $\sigma \in {\overline{A}}$ and all $\eta \in {\underline{A}}$.

Proof

Note that the only portfolio values on the ambiguous states on which the utility U in (16) depends are $w^A_{\max }$ and $w^A_{\min }$. We order the set of ambiguous states $A=\{\sigma _1,\ldots ,\sigma _l\}$ such that

$$\begin{aligned} w_{\sigma _1}\le w_{\sigma _2}\le \cdots \le w_{\sigma _l}. \end{aligned}$$

(38)

Let s be the number of strict inequalities in (38). Consider states $\nu _1,\ldots \nu _{s+1}\in A$ such that $w_{\nu _1}<w_{\nu _2}<\ldots <w_{\nu _{s+1}}$. Suppose there is a state $\eta \in A$ such that $w_\eta \ne w^A_{\max }$ and $w_\eta \ne w^A_{\min }$, namely suppose that $s\ge 2$. We now consider the function U in (16) as defined on ${\mathbb {R}}^{m+s+1}$, where we merge those ambiguous states in which w takes the same value. Let $\tilde{w}\in {\mathbb {R}}^{m+s+1}$ such that $\tilde{w}_R=w_R$ for all risky states $R\in S{\setminus } A$ and otherwise $\tilde{w}_{\sigma _i}=w_{\sigma _i}$ for $i=1,\ldots ,s+1$. Then, $\tilde{w}$ is a maximizer for the function U restricted to the open set $C:=\{x\in {\mathbb {R}}^{ m+s+1}\mid x_{\sigma _1}<x_{\sigma _2}<\ldots <x_{\sigma _{s+1}}\}$, which we call $U_C$, given the budget constraint $\tilde{p} \cdot \tilde{w}\le p\cdot e$. Here e is the initial portfolio and $\tilde{p}\in {\mathbb {R}}^{m+s+1}$ is obtained from p by summing up the prices of those states which are merged when forming $\tilde{w}$. As $U_C$ is concave, according to (44), a multiple of $\tilde{p}$ is in the supergradient of $U_C$ at $\tilde{w}$. However, this supergradient is equal to zero in any $x_{\sigma _i}$-direction, $i\in \{2,\ldots , s\}$, because only the largest value and the smallest value on the ambiguous states matter for U. This contradicts the assumption $p_{\sigma _i}>0$ for $i\in \{2,\ldots , s\}$. $\square$

Lemma 3

Assume Lemma 2. If $p_\sigma <p_\eta$ for $\sigma ,\eta \in A$, then the optimal portfolio w satisfies $w_\eta \le w_\sigma$.

Proof

Suppose that the optimal portfolio w is such that $w_\eta > w_\sigma$. Let $\tilde{w}$ given by $\tilde{w}_\nu =w_\nu$ for all $\nu \in S{\setminus } \{\sigma ,\eta \}$ and $\tilde{w}_{\sigma }=w_\eta$ and $\tilde{w}_{\eta }=w_{\sigma }$. Then $U(\tilde{w})=U(w)$, but $p\cdot \tilde{w} < p \cdot w$ because $p\cdot (w-\tilde{w})= (p_\eta -p_\sigma ) (w_\eta -w_\sigma )>0$. This contradicts the optimality of w, because increasing the wealth $\tilde{w}_{\sigma }$ one could achieve a strictly higher utility while still respecting the budget constraint. $\square$

Lemma 4

Assume Lemma 2. If the sets ${\overline{A}}$ and ${\underline{A}}$ associated to the optimal portfolio w are not empty, then ${\overline{A}}=\{{\overline{\sigma }}\}$ for a state ${\overline{\sigma }}\in I:=\{\sigma \in A\mid p_\sigma =\min _{\eta \in A} p_\eta \}$. Moreover, any portfolio which equals w on the risky states and assigns the weight $w^A_{\max }$ to a single state in I and $w^A_{\min }$ to all the other ambiguous states is optimal. Hence, there are |I| optimal portfolios.

Proof

By contradiction suppose that there are two different states $\sigma _1$ and $\sigma _2$ in ${\overline{A}}$, i.e., that the optimal portfolio w is such that $w_{\sigma _1}=w_{\sigma _2}=w^A_{\max }$, and without loss of generality we assume that $p_{\sigma _1}\le p_{\sigma _2}$. Consider $\tilde{w}$ given by $\tilde{w}_\eta =w_\eta$ for all $\eta \in S{\setminus } \{\sigma _1,\sigma _2\}$ and $\tilde{w}_{\sigma _1}=2w^A_{\max } -w^A_{\min }$ and $\tilde{w}_{\sigma _2}=w^A_{\min }$. Then $p\cdot \tilde{w} \le p\cdot w$, so $\tilde{w}$ satisfies the budget constraint, and $U(\tilde{w})>U(w)$ since $\tilde{w}^A_{\max }= \tilde{w}_{\sigma _1}>w^A_{\max }$ and $\tilde{w}^A_{\min }=w^A_{\min }$. This is a contradiction to optimality of w. Lemma 3 implies that ${\overline{\sigma }}\in I$.

The last statement of the lemma follows by observing that all these portfolios share the same price and utility. $\square$

Lemma 5

Assume Lemma 2 and let $\alpha <1$. Then w is unambiguous, i.e., $w_\sigma =w_\nu$ for all $\sigma ,\nu \in A$, if and only if (20) holds. In this case w is the only optimal portfolio. Condition (20) can only be satisfied if $\alpha \ge \frac{l-1}{l}$.

Proof

Suppose ${\overline{A}}=\{\sigma \}$ and thus ${\underline{A}}=A{\setminus }\{\sigma \}$. Then, the first-order conditions imply

$$\begin{aligned} \frac{p_R}{ \pi _R u'(w_R)}= \frac{p_\sigma }{(1-\alpha )(1-\sum _{R\in S{\setminus } A}\pi _R)u'(w^A_{\max })}= \frac{\sum _{\nu \in A{\setminus } \{\sigma \}}p_{\nu }}{\alpha (1-\sum _{R\in S{\setminus } A}\pi _R )u'(w^A_{\min })}\nonumber \\ \end{aligned}$$

(39)

where R denotes any risky state among the m ones. Thus,

$$\begin{aligned} \frac{p_\sigma }{\sum _{\nu \in A{\setminus } \{\sigma \}}p_{\nu }} =\frac{(1-\alpha ) u'(w^A_{\max })}{\alpha u'(w^A_{\min })} < \frac{1-\alpha }{\alpha } \end{aligned}$$

(40)

as $w^A_{\max }>w^A_{\min }$. Consequently, if there are no $\sigma \in A$ for which (40) is satisfied, i.e., if the condition (20) holds true, then w must be unambiguous. In order to prove necessity of (20), assume that (40) holds for some $\sigma \in A$. In the following we show that in this case the unambiguous portfolio cannot be optimal. To this end, suppose by contradiction that the unambiguous portfolio w is optimal and let $z:=w^A_{\max }=w^A_{\min }$. Then $\epsilon =0$ needs to maximize the function

$$\begin{aligned} F:{\mathbb {R}}\ni \epsilon \mapsto \alpha u(z-\epsilon ) + (1-\alpha )u\left( z +\delta (\epsilon )\right) \end{aligned}$$

over all $\epsilon \ge 0$, where $\delta (\epsilon ):=\epsilon \frac{\sum _{\sigma \in A{\setminus } \{\sigma \}}p_\nu }{p_\sigma }$ is chosen such that the portfolio which invests $z-\epsilon$ in the states $\nu \in {\underline{A}}$, and $z+\delta (\epsilon )$ in the state $\sigma$ satisfies the budget constraint (while the investment in the risky states is unaltered). F is a concave function and the first-order condition reads

$$\begin{aligned} \frac{u'(z+\delta (\epsilon ))}{u'(z-\epsilon )}= \frac{\alpha }{(1-\alpha )} \frac{p_\sigma }{\sum _{\sigma \in A{\setminus } \{\sigma \}}p_\nu }. \end{aligned}$$

By assumption, the right-hand side of the above equation is strictly smaller than 1. Hence, F attains its optimum for $\epsilon >0$, which contradicts the optimality at 0 over all $\epsilon \ge 0$. Finally, note that summing up (20) over all $\sigma \in A$ yields:

$$\begin{aligned} \alpha \sum _{\sigma \in A} p_\sigma \ge (1-\alpha )(l-1)\sum _{\nu \in A} p_\nu \quad \Leftrightarrow \quad \alpha \ge \frac{l-1}{l}. \end{aligned}$$

$\square$

Lemma 6

Assume Lemma 2. If $\alpha =1$, then w is unambiguous. If $\alpha =0$, then there is no optimal portfolio.

Proof

If $\alpha =1$, then (16) is a maxmin agent and also $\tilde{\pi }\in {\mathcal {C}}_{\max }$. Hence, Lemma 1 proves the claim. The optimization problem of a 0-MEU agent with the maximal set of priors ${\mathcal {C}}_{\max }$ is

$$\begin{aligned}&\sum _{R\in S{\setminus } A}\pi _R u(w_R) + (1-\sum _{R\in S{\setminus } A}\pi _R ) u(w^A_{\max }) \rightarrow \max \nonumber \\ \hbox { subject to }&p\cdot w \le p\cdot e \end{aligned}$$

(41)

where e denotes her initial endowment. Since the agent may go arbitrarily long in the ambiguous state $\sigma$ with $w_\sigma =w^A_{\max }$ and satisfy the budget constraint by going arbitrarily short in an other ambiguous state, the optimal value in (41) cannot be attained. $\square$

Proof of Lemma 1

Let w be an optimal portfolio of the maxmin agent and assume that $w_\sigma \ne w_\eta$ for $\sigma ,\eta \in A$. Consider the portfolio ${\hat{w}}$ given by ${\hat{w}}_R=w_R$ for any risky state $R\in S{\setminus } A$ and ${\hat{w}}_\sigma =z$ for any ambiguous state $\sigma \in A$ where

$$\begin{aligned} z:= \frac{\sum _{\sigma \in A}p_\sigma \, w_\sigma }{\sum _{\sigma \in A}p_\sigma }=\frac{1}{l}\sum _{\sigma \in A} w_\sigma . \end{aligned}$$

The portfolio ${\hat{w}}$ satisfies the budget constraint and

$$\begin{aligned} U({\hat{w}})= & {} \sum _{R\in S{\setminus } A}\pi _R \, u(w_R) + \left( 1-\sum _{R\in S{\setminus } A}\pi _R\right) \, u(z) \\> & {} \sum _{R\in S{\setminus } A}\pi _R \, u(w_R) + \frac{1}{l} \left( 1-\sum _{R\in S{\setminus } A}\pi _R\right) \, \sum _{\sigma \in A} u(w_\sigma )\ge U(w) \end{aligned}$$

where the strict inequality follows from the strict concavity of u and the last inequality is due to $\tilde{\pi }\in {\mathcal {C}}$. This contradicts the optimality of w.

Optimization in the partially concave case

Consider the optimization problem

$$\begin{aligned} \max _{x\in C} U(x) \quad \hbox {subject to}\quad px\le pe \end{aligned}$$

(42)

where $C\ne \emptyset$ is a convex subset of ${\mathbb {R}}^n$, $p,e\in {\mathbb {R}}^n$, and $U:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\cup \{-\infty \}$ is a concave function with ${\text {dom}}\,U=C$.

Lemma 7

If the optimal value in (42) is not $+\infty$ and if there exists at least one $\bar{x}\in {\text {ri}} C$ with $p\bar{x}\le pe$, then there is a multiplier $\lambda \ge 0$ such that the supremum of $h_\lambda (x)=U(x)-\lambda p(x-e)$, $x\in {\mathbb {R}}^n$, is finite and equal to the optimal value in (42). Moreover, suppose that $\lambda >0$ and that D is the set of points $x\in {\mathbb {R}}^n$ where h attains its maximum intersected with the set of points satisfying $px=pe$, then D is the set of all optimal solutions to (42).

Proof

see Theorem 28.1 and Corollary 28.2.2 in Rockafellar (1997). $\square$

Now suppose that agent n with choice criterium $U^n:{\mathbb {R}}^{|S|}\rightarrow {\mathbb {R}}$ maximizes her utility over all portfolios $w\in {\mathbb {R}}^{|S|}$ satisfying the budget constraint $pw\le pe^n$ for some $p\in {\mathbb {R}}^{|S|}$ with $p_i>0$ for all $i=1,\ldots ,|S|$. Furthermore, assume that an optimal portfolio ${\hat{w}}$ exists and that ${\hat{w}}\in C$ for a convex set $C\subset {\mathbb {R}}^{|S|}$ such that the restriction $U^n_C$ of $U^n$ to C is concave. Then, we may view $U^n_C$ as defined on all ${\mathbb {R}}^{|S|}$ by defining $U^n_C(x):=-\infty$ for $x\not \in C$, and we are thus in the setting of Lemma 7 where ${\hat{w}}$ is a solution to problem (42) with $U=U^n_C$. Hence, if there exists $x\in {\text {ri}} C$ with $px\le pe^n$, which is satisfied if for instance ${\hat{w}}\in {\text {ri}} C$, then there exists a multiplier $\lambda \ge 0$ such that

$$\begin{aligned} U_C^n({\hat{w}})=\sup _{x\in {\mathbb {R}}^n} h_\lambda (x) \end{aligned}$$

(43)

with $h_\lambda$ as in Lemma 7. If $C=C+{\mathbb {R}}_+\cdot (1,0,\ldots ,0)$ and given that the utility function u is strictly increasing we deduce that $\lambda >0$, since otherwise

$$\begin{aligned} h_\lambda ({\hat{w}}+ (1,0,\ldots ,0))=U^n_C({\hat{w}}+(1,0,\ldots ,0))>U^n_C({\hat{w}}). \end{aligned}$$

Moreover, any solution ${\hat{x}}$ to the right-hand side of (43) with $p{\hat{x}}=pe^n$ is a solution to the portfolio optimization problem, and in particular ${\hat{w}}$ is such a solution. Additionally, for any solution ${\hat{x}}$ to the right-hand side of (43) we have for all $y\in {\mathbb {R}}^{|S|}$ that

$$\begin{aligned} U_C^n(y)-\lambda p(y-e^n)\le U_C^n(\hat{x})-\lambda p({\hat{x}}-e^n) \end{aligned}$$

which shows that

$$\begin{aligned} \lambda p\in \partial U_C^n({\hat{x}})\end{aligned}$$

(44)

where $\partial U_C^n(w)$ denotes the supergradient of $U_C^n$ at w, i.e.,

$$\begin{aligned} \partial U_C^n(w):=\left\{ \nu \in \mathbb {R}^{|S|} \mid \forall y\in \mathbb {R}^{|S|}, \ U_C^n(y)\le U_C^n(w)+ \nu \cdot (y-w) \right\} . \end{aligned}$$