Incomplete markets and derivative assets

Abstract

We analyze derivative asset trading in an economy in which agents face both aggregate and uninsurable idiosyncratic risks. Insurance markets are incomplete for idiosyncratic risk and, possibly, for aggregate risk as well. However, agents can exchange insurance against aggregate risk through derivative assets such as options. We present a tractable framework, which allows us to characterize the extent of risk sharing in this environment. We show that incomplete insurance markets can explain some properties of the volume of traded derivative assets, which are difficult to explain in complete market economies.

Appendices

Appendix A: Proof of Proposition 1

We proceed in two steps: (i) We prove that our equilibrium exists in an economy without heterogeneity and with a zero mass tree, and (ii) we check that the result still holds by continuity.

First step: Equilibrium existence with zero volume and no heterogeneity.

We first assume $\alpha ^{1}=\alpha ^{2}$ and $V=0$. The asset price is given by:

$$\begin{aligned} P_{k}=\beta \left( 1+\alpha ^{1}\left( u^{\prime }(\delta )-1\right) \right) \sum _{j=1}^{n}\pi _{kj}\left[ P_{j}+y_{j}\right] . \end{aligned}$$

(25)

The first part of (11) stating that $\beta \left( 1+\alpha ^{1}\left( u^{\prime }(\delta )-1\right) \right) <1$ guarantees that the price $P_{k}$ is well defined in any state k. The condition of non-participation of unproductive agents is: $P_{k}u^{\prime }(\delta )> \beta (1+\rho (u^{\prime }(\delta )-1))\sum _{j=1}^{n}\pi _{kj}(P_{j}+y_{j})$. Indeed, the collateral constraint (6) implies that agents cannot short-sell some options to purchase other options: In the absence of assets, options cannot be traded. Using (25), the condition becomes: $\left( 1+\alpha ^{1}\left( u^{\prime }(\delta )-1\right) \right) u^{\prime }(\delta )>u^{\prime }(\delta )-1>1+\rho \left( u^ {\prime }(\delta )-1\right)$, which always holds whenever $u^{\prime }(\delta )>1$. No trade occurs.

Second step: Positive supply economy with one type of agent.

We assume $\alpha ^{2}=\alpha ^{1}$ but $V>0$. Every (identical) productive agent holds the same asset quantity $\frac{V}{\eta ^{1}+\eta ^{2}}$. Options are still not traded in this economy. The asset price $P_{k}$ in state $k=1,\ldots ,n$ verifies the following equation:

$$\begin{aligned} P_{k}=\beta \sum _{j=1}^{n}\pi _{kj}\left( 1+\alpha ^{1} \left( u^{\prime }(\delta +(P_{j}+y_{j})\frac{V}{\eta ^{1}+\eta ^{2}}) -1\right) \right) \left( P_{j}+y_{j}\right) . \end{aligned}$$

(26)

The condition for equilibrium existence is:

$$\begin{aligned} P_{k}u^{\prime }(\delta +(P_{k}+y_{k})\frac{V}{\eta ^{1}+\eta ^{2}}) >\beta \sum _{k^{\prime }=1}^{n}\pi _{kk^{\prime }}\left( 1+\rho \left( u^{\prime }(\delta )-1\right) \right) \left( P_{k^{\prime }} +y_{k^{\prime }}\right) . \end{aligned}$$

(27)

We can express (26) as $G\left( P,V\right) =0$, where $P=(P_{k})_{k=1,\ldots ,n}\in ({\mathbb {R}}^{+})^{n}$ and $G:({\mathbb {R}}^{+})^{n}\times {\mathbb {R}}^{+}\rightarrow ({\mathbb {R}}^{+})^{n}$ is continuous and differentiable in V. The Jacobian relative to P in $V=0$ is $G_{P}\left( P,0\right) =(1_{k=j}-\beta \pi _{kj}(1+\alpha (u^{\prime } \left( \delta \right) -1)))_{k,j=1,\ldots ,n}>0$.¹² This Jacobian matrix can be written as $I_{n}-\widetilde{G}$, where $I_{n}$ is the identity matrix and $\widetilde{G}$ a matrix whose norm is strictly smaller than one. The Jacobian $G_{P}$ is invertible, and the implicit function theorem implies that (26) defines P as a continuous function of V around $V=0$. As condition (27) holds for $V=0$, there exists a neighborhood $W_{1}\left( 0\right) \subset {\mathbb {R}}^{+}$ where (27) holds. We define $V^{*}\in W_{1}\left( 0\right) >0$ and $P^{*}$ ($Q^{*}$) the corresponding asset (option) price. The quantity of assets held by each agent is $x^{*}=\frac{V^{*}}{\eta ^{1}+\eta ^{2}}>0$, while no option is traded.

Third step: Positive supply economy with two types of agents.

In the general case, equilibrium quantities $(P_{k},Q_{k}^{1},\ldots ,Q_{k}^{l},x_{k}^{i},s_{k}^{1,i}, \ldots ,s_{k}^{l,i})_{k=1,\ldots ,n}^{i=1,2}$ are characterized by Eqs. (14)–(17) and must verify the following inequalities ($k,\hat{k}=1,\ldots ,n$; $h=1,\ldots ,l$; $i=1,2$):

$$\begin{aligned} P_{k}u^{\prime }\left( \delta +(P_{k}+y_{k})\, x_{\hat{k}}^{i}+ \sum _{l=1}^{H}\left( P_{k}-K_{\hat{k}}^{l}\right) ^{+}s_{\hat{k}}^{l,i}\right)&>\beta \sum _{j=1}^{n}\pi _{kj}\left( 1+\rho \left( u^{\prime }(\delta )-1\right) \right) \left( P_{j}+y_{j}\right) , \end{aligned}$$

(28)

$$\begin{aligned} Q_{k}^{h}u^{\prime }\left( \delta +(P_{k}+y_{k})\, x_{\hat{k}}^{i}+\sum _{l=1}^{H}\left( P_{k}-K_{\hat{k}}^{l}\right) ^{+}s_{\hat{k}}^ {l,i}\right)&>\beta \sum _{j=1}^{n}\pi _{kk^{\prime }}\left( 1+\rho \left( u^{\prime } (\delta )-1\right) \right) \left( P_{j}+y_{j}\right) . \end{aligned}$$

(29)

We proceed as above. We define $X=((y_{k})_{k},V,(\alpha _{k}^{1},\alpha _{k}^{2})_{k}) \in ({\mathbb {R}}^{+})^{n\times 1\times 2n}$ as the vector of parameters and $Z=((P_{k})_{k},(Q_{k}^{h})_{k}^{h},(x_{k}^{1})_{k},(x_{k}^{2})_{k}, (s_{k}^{1h})_{k}^{h},(x_{k}^{2h})_{k}^{h})\in ({\mathbb {R}}^{+})^{n\times nl\times 2n\times 2nl}$ as the vector of endogenous variables. We define the function F stacking pricing functions for both agent types and the market equilibrium equations (i.e., Eqs. (14)–(17)) such that for a given set of parameters X, an equilibrium Z is defined as a solution of $F\left( Z,X\right) =0$.

From the previous step, we know that there exists an equilibrium for $X^{*}=((y_{k})_{k},V^{*})$ in which the unproductive do not trade assets; this equilibrium is defined by $Z^{*}=(P^{*},Q^{*},$ $(x^{*},\ldots ,x^{*}),0_{2nl})$. We now show that the Jacobian $\Delta =\left( \frac{\partial F_{i}}{\partial z_{j}}\left( Z^{*},X^{*}\right) \right) _{i,j=1,\ldots ,3n(1+l)}$ is invertible.

In the vicinity of the symmetric equilibrium, $\Delta$ has the following shape:

$$\begin{aligned} \Delta =\left[ \begin{array}{cccccc} I_{n}-A &{} 0{}_{n\times nl} &{} K_{a} &{} 0_{n\times n} &{} (K_{a1}\ldots K_{al}) &{} 0_{n\times nl}\\ I_{n}-A &{} 0{}_{n\times nl} &{} 0_{n\times n} &{} K_{a} &{} 0_{n\times nl} &{} (K_{a1}\ldots K_{al})\\ -\left( \begin{array}{c} B_{1}\\ \vdots \\ B_{l} \end{array}\right) &{} I_{nl} &{} \left( \begin{array}{c} K_{a1}\\ \vdots \\ K_{al} \end{array}\right) &{} 0_{nl\times n} &{} (K_{gh})_{g,h=1,\ldots ,l} &{} 0_{nl\times n}\\ -\left( \begin{array}{c} B_{1}\\ \vdots \\ B_{l} \end{array}\right) &{} I_{nl} &{} 0_{nl\times n} &{} \left( \begin{array}{c} K_{a1}\\ \vdots \\ K_{al} \end{array}\right) &{} 0_{nl\times n} &{} (K_{gh})_{g,h=1,\ldots ,l}\\ 0_{n\times n} &{} 0_{n\times nl} &{} E_{1} &{} E_{2} &{} 0_{n\times nl} &{} 0_{n\times nl}\\ 0_{nl\times n} &{} 0_{nl\times nl} &{} 0_{nl\times n} &{} 0_{nl\times n} &{} E_{1}\otimes I_{l} &{} E_{2}\otimes I_{l} \end{array}\right] , \end{aligned}$$

where:

• $I_{p}$ is the $p\times p$ identity matrix, $0_{n\times p}$ is the $n\times p$ null matrix, $1_{l\times 1}$ is a column vector of length l containing only 1, and $\otimes$ is the Kronecker product;

• A is an $n\times n$ matrix such that $A_{k,j}=\beta \pi _{k,j}(1+\alpha ^{1}(u^{\prime }(\delta +(P_{j}^{*} +y_{j})x^{*})-1+x^{*}u^{\prime \prime }(\delta +(P_{j}^{*} +y_{j})x^{*})(P_{j}^{*}+y_{j})))$;

• $B_{h}$ ($h=1,\ldots ,l$) is an $n\times n$ matrix such that $B_{h,kj}=\beta \pi _{k,j}(1+\alpha _{k}^{1}((u^{\prime }(\delta +(P_{j}^ {*}+y_{j})x^{*})-1)1_{P_{j}\ge K_{k}^{h}}+x^{*}u^{\prime \prime }(\delta +(P_{j}^{*}+y_{j})x^ {*})(P_{j}-K_{k}^{h})^{+}))$;

• $E_{i}$ ($i=1,2$) is an $n\times n$ diagonal matrix such that $E_{i,kk}=\eta _{k}^{i}$;

• $K_{a}$ is an $n\times n$ diagonal matrix such that $K_{a,kk}=-\beta \alpha _{k}^{1}\sum _{j=1}^{n}\pi _{k,j}u^ {\prime \prime }(\delta +(P_{j}^{*}+y_{j})x^{*})(P_{j}^ {*}+y_{j})^{2}$;

• $K_{ah}$ is an $n\times n$ diagonal matrix such that $K_{ah,kk}=-\beta \alpha _{k}^{1}\sum _{j=1}^{n}\pi _{k,j}u^{\prime \prime }(\delta +(P_{j}^{*}+y_{j})x^{*})(P_{j}^{*}+y_{j})(P_{j}-K_{k}^{h})^{+}$;

• $K_{gh}$ is an $n\times n$ diagonal matrix such that $K_{gh,kk}=-\beta \alpha _{k}^{1}\sum _{j=1}^{n}\pi _{k,j}u^{\prime \prime }(\delta +(P_{j}^{*}+y_{j})x^{*})(P_{j}^{*}-K_{k}^{g})^{+}(P_{j}^{*}-K_{k}^{h})^{+}$.

We now prove that $\Delta$ is invertible. Let $X=(X_{1},\ldots ,X_{6})\in \left( {\mathbb {R}}\right) ^{n+nl+2n+2nl}$. $X\in \ker \Delta$ implies the following set of equalities:¹³

$$\begin{aligned} {\left\{ \begin{array}{ll} 0_{n\times 1} &{}= (I_{n}-A)X_{1}+K_{a}X_{3}+(K_{a1}\ldots K_{al})X_{5}\\ 0_{n\times 1} &{}= (I_{n}-A)X_{1}+K_{a}X_{4}+(K_{a1}\ldots K_{al})X_{6}\\ 0_{nl\times 1} &{}= -(B_{1}\ldots B_{l})^{\top }X_{1}+X_{2}+(K_{a1}\ldots K_{al})^{\top }X_{3}+(K_{gh})X_{5}\\ 0_{nl\times 1} &{}= -(B_{1}\ldots B_{l})^{\top }X_{1}+X_{2}+(K_{a1}\ldots K_{al})^{\top }X_{4}+(K_{gh})X_{6}\\ 0_{n\times 1} &{}= E_{1}X_{3}+E_{2}X_{4}\\ 0_{nl\times 1} &{}= E_{1}\otimes I_{l}X_{5}+E_{2}\otimes I_{l}X_{6} \end{array}\right. }. \end{aligned}$$

(30)

Using the two first equations with the two last ones (together with the fact that any two diagonal matrices commute), we obtain $X_{1}=0$. By the same token, using the second and third equations with the two last ones, we obtain $X_{2}=0$. The system (30) simplifies to:

$$\begin{aligned} {\left\{ \begin{array}{ll} 0_{n\times 1}&{}= K_{a}X_{3}+(K_{a1}\ldots K_{al})X_{5}\\ 0_{n\times 1}&{}= K_{a}X_{4}+(K_{a1}\ldots K_{al})X_{6}\\ 0_{nl\times 1}&{}= (B_{1}\ldots B_{l})^{\top }X_{3}+(K_{gh})X_{5}\\ 0_{nl\times 1}&{}= (B_{1}\ldots B_{l})^{\top }X_{4}+(K_{gh})X_{6}\\ 0_{n\times 1}&{}= E_{1}X_{3}+E_{2}X_{4}\\ 0_{nl\times 1}&{}= E_{1}\otimes I_{l}X_{5}+E_{2}\otimes I_{l}X_{6} \end{array}\right. }. \end{aligned}$$

(31)

Since $K_{a}$ is invertible, we obtain $0_{nl\times 1}=\left( {\left( \begin{array}{c} K_{a1}\\ \vdots \\ K_{al} \end{array}\right) }(K_{a}^{-1}K_{a1}\ldots K_{a}^{-1}K_{al})-(K_{gh})\right) X_{5}$, which implies $X_{5}=0_{nl\times 1}$. To see this, we express $X_{5}=(X_{51},\ldots ,X_{5l})\in ({\mathbb {R}}^{n})^{l}$ and get that for any $g=1,\ldots ,l$, we have:

$$\begin{aligned} \sum _{h=1}^{l}\left( K_{ag}K_{ah}-K_{a}K_{gh}\right) X_{5h}=0. \end{aligned}$$

(32)

We introduce the bilinear form $(\cdot |\cdot )_{k}$: $\forall U,V\in ({\mathbb {R}}^{n})^{2}$, $(U|V)_{k}=-\beta \alpha ^{1}\sum _{j=1}^{n}\pi _{k,j}u_{k,j}^ {\prime \prime }U_{j}V_{j}$ and it is easy to check that it is an inner product. Multiplying (32) by $X_{5g}$ and summing for $g=1,\ldots ,l$, we obtain using $(\cdot |\cdot )_{k}$:

$$\begin{aligned} \left( \sum _{h=1}^{l}\varPi _{h}^{k}X_{5hk}\Bigg |\varPi _{a}\right) _{k}^{2}-(\varPi _{a}\Bigg |\varPi _{a}) _{k}\left( \sum _{h=1}^{l}\varPi _{h}^{k}X_{5hk}\Bigg |\sum _{h=1}^{l}\varPi _{h}^ {k}X_{5hk}\right) _{k}=0, \end{aligned}$$

where $\varPi _{a}=(P_{j}^{*}+y_{j})_{j=1,\ldots ,n}$ and $\varPi _{h}^{h}=(P_{j}^{*}-K_{k}^{h})_{j=1,\ldots ,n}^{+}$. The Cauchy–Schwarz inequality implies then that for all k, $\lambda _{k}\varPi _{a}=\sum _{h=1}^{l}\varPi _{h}^{k}X_{5hk}$, which means that $\lambda _{k}=0$ and $X_{5h}=0_{n\times 1}$, since by assumption option payoffs do not replicate asset payoffs. From the first equation in (31), we deduce $X_{3}=0_{n\times 1}$. By the same token, we obtain $X_{6}=0_{nl\times 1}$ and $X_{4}=0_{n\times 1}$.

We conclude that $\ker \Delta =\{0\}$ and the Jacobian $\Delta$ in $(X^{*},Z^{*})$ is invertible. The implicit function theorem proves that there exists a continuously differentiable function $\widetilde{F}$ such that $Z=\widetilde{F}\left( X\right)$ for X close to $X^{*}$. In consequence, our equilibrium exists in the vicinity of $(X^{*},Z^{*})$.

Appendix B: Proof of Proposition 2

From Sect. 3.2, it is straightforward to deduce the price expressions (14) and (15). Market-clearing conditions (16) and (17) are easily deduced from equilibrium properties (only productive agents trade securities) and from the general market clearing conditions (9) and (10).

As explained in Sect. 3.2, two conditions have to hold for preventing unproductive agents to trade any securities. These conditions are $P_{t}u'(c_{t}^{i})>\beta E_{t}[u'(c_{t+1}^{i})(P_{t+1}+y_{t+1})]$ and $Q_{t}^{h}u'(c_{t}^{i})>\beta E_{t}[u'(c_{t+1}^{i})(P_{t+1}-K_{t}^{h})^{+}]$ for all unproductive agents. Plugging price expressions, we obtain the Eqs. (28) and (29) above.

Appendix C: Proof of Proposition 3

Since $\alpha ^{1}>\alpha ^{2}$, (19) implies that $u^{\prime }\left( \delta +(P_\mathrm{B}+y_\mathrm{B})\, x^{1}\right) <u^{\prime }\left( \delta +(P_\mathrm{B}+y_\mathrm{B})\, x^{2}\right)$ and therefore that $x^{1}>x^{2}$, since $u^{\prime }$ is decreasing. Moreover, $s^{1}<0$ if $\alpha ^{1}(u^{\prime }\left( \delta +(P_\mathrm{G}+y_\mathrm{G})\, x^{1}\right) -1)<\alpha ^{2}(u^{\prime }\left( \delta +(P_\mathrm{G}+y_\mathrm{G})\, x^{2}\right) -1)$ or if $\frac{u^{\prime }\left( \delta +(P_\mathrm{G}+y_\mathrm{G})\, x^{1}\right) -1}{u^{\prime }\left( \delta +(P_\mathrm{G}+y_\mathrm{G})\, x^{2}\right) -1}<\frac{u^{\prime }\left( \delta +(P_\mathrm{B}+y_\mathrm{B})\, x^{1}\right) -1}{u^{\prime }\left( \delta +(P_\mathrm{B}+y_\mathrm{B})\, x^{2}\right) -1}$. This holds if $\pi \mapsto \frac{u^{\prime }\left( \delta +\pi x^{1}\right) -1}{u^{\prime }\left( \delta +\pi x^{2}\right) -1}$ is decreasing, which is guaranteed by condition 18.

Appendix D: Proof of Proposition 4

We introduce as a benchmark the equilibrium without aggregate risk in which $y_\mathrm{G}=y_\mathrm{B}=y$. The asset (option) price is $P^{f}$ ($Q^{f}$). Options are not traded ($s^{f,i}=0$), and the asset holdings of type-i agents are denoted $x^{f,i}$, $i=1,2$:

$$\begin{aligned}&\frac{P^{f}}{P^{f}+y} =\beta \left( 1+\alpha ^{i}\left( u^{\prime } \left( \delta +(P^{f}+y)x^{f,i}\right) -1\right) \right) =\frac{Q^{f}}{P^{f}-K}, \end{aligned}$$

(33)

$$\begin{aligned}&\text {with: } \alpha ^{1}\left( u^{\prime }(\delta +(P^{f}+y)\, x^ {,f,1})-1\right) =\alpha ^{2}\left( u^{\prime }(\delta +(P^{f}+y)\, x^ {f,2})-1\right) ,\nonumber \\&\quad \eta ^{1}x^{f,1}+\eta ^{2}x^{f,2}=V \end{aligned}$$

(34)

Since $\alpha ^{1}>\alpha ^{2}$, we deduce that $x^{f,1}>x^{f,2}$; in the absence of aggregate risk, the agent facing the larger risk holds more assets. We also introduce the following constant $\kappa$:

$$\begin{aligned} \kappa =\frac{-\, x^{f,1}\,\frac{u^{\prime \prime }\left( \delta +(P^{f}+y)\, x^{f,1}\right) }{u^{\prime }\left( \delta +(P^{f}+y)\, x^{f,1}\right) -1}+x^{f,2}\,\frac{u^{\prime \prime }\left( \delta +(P^{f}+y)\, x^{f,2}\right) }{u^{\prime }\left( \delta +(P^{f}+y)\, x^{f,2}\right) -1}}{-\frac{1}{\eta ^{1}}\,\frac{u^{\prime \prime } \left( \delta +(P^{f}+y)\, x^{f,1}\right) }{u^{\prime }\left( \delta +(P^{f}+y)\, x^{f,1}\right) -1}-\frac{1}{\eta ^{2}}\,\frac{u^{\prime \prime } \left( \delta +(P^{f}+y)\, x^{f,2}\right) }{u^{\prime }\left( \delta +(P^{f}+y)\, x^{f,2}\right) -1}}>0. \end{aligned}$$

(35)

Condition 18 guarantees that $\kappa >0$. Before going further, we prove the following lemma:

Lemma 1

Let $\varPhi$ be a real continuously differentiable function of $y_\mathrm{G}$ and $y_\mathrm{B}$. We denote by V[y] (E[y]) the variance (mean) of the process y. A mean-preserving spread of y implies:

$$\begin{aligned} \left. \frac{\partial \varPhi }{\partial V[y]}\right| _{E[y]=\mathrm{const.}}=\frac{1}{2(y_\mathrm{G}-y_\mathrm{B})}\,\left[ \frac{2-\pi _\mathrm{GG}-\pi _{BB}}{1-\pi _{BB}}\frac{\partial \varPhi }{\partial y_\mathrm{G}}-\frac{2-\pi _\mathrm{GG}-\pi _{BB}}{1-\pi _\mathrm{GG}}\,\frac{\partial \varPhi }{\partial y_\mathrm{B}}\right] . \end{aligned}$$

(36)

Proof

Defining $q=\frac{1-\pi _{BB}}{2-\pi _\mathrm{GG}-\pi _{BB}}$, we have $y_\mathrm{G}=E[y]+(1-q)\sqrt{\frac{V[y]}{q(1-q)}}$ and $y_\mathrm{B}=E[y]-q\sqrt{\frac{V[y]}{q(1-q)}}$. Using basic differential calculus, it is straightforward to derive (36). $\square$

1.1 Quantities

Deriving (19) and (20) relative to $y_{l}$ in the vicinity of the riskless equilibrium yields:

$$\begin{aligned}&\alpha ^{1}u^{\prime \prime }\left( \delta +(P^{f}+y)\, x^{f,1}\right) \left( x^{f,1}\left( \frac{\partial P_\mathrm{B}}{\partial y_{l}}+1_{l=B})+(P^{f}+y\right) \frac{\partial x^{1}}{\partial y_{l}}\right) \\&\quad =\alpha ^{2}u^{\prime \prime }\left( \delta +(P^{f}+y)\, x^{f,2}\right) \left( x^{f,2}\left( \frac{\partial P_\mathrm{B}}{\partial y_{l}}+1_{l=B})+(P^{f}+y\right) \frac{\partial x^{2}}{\partial y_{l}}\right) ,\\&\alpha ^{1}u^{\prime \prime }\left( \delta +(P^{f}+y)\, x^{f,1}\right) \left( x^{f,1}\left( \frac{\partial P_\mathrm{G}}{\partial y_{l}}+1_{l=G})+(P^{f}+y\right) \frac{\partial x^{1}}{\partial y_{l}}+(P^{f}-K)\frac{\partial s^{1}}{\partial y_{l}}\right) \\&\quad =\alpha ^{2}u^{\prime \prime }\left( \delta +(P^{f}+y)\, x^{f,2}\right) \left( x^{f,2}\left( \frac{\partial P_\mathrm{G}}{\partial y_{l}}+1_{l=G})+(P^{f}+y\right) \frac{\partial x^{2}}{\partial y_{l}}+(P^{f}-K)\frac{\partial s^{2}}{\partial y_{l}}\right) . \end{aligned}$$

Note that the derivation of (23) and (24) wrt to $y_{l}$, $l=B,G$ implies that $\frac{\partial x^{1}}{\partial y_{l}}=-\frac{\partial x^{2}}{\partial y_{l}}$ and $\frac{\partial s^{1}}{\partial y_{l}}=-\frac{\partial s^{2}}{\partial y_{l}}$. Dividing both equations by $\alpha ^{1}\left( u^{\prime }\left( \delta +(P^{f}+y)\, x^{,f,1}\right) -1\right) =\alpha ^{2}\left( u^{\prime } \left( \delta +(P^{f}+y)\, x^{f,2}\right) -1\right)$ and after some algebra, we deduce using the definition (35) of $\kappa$ that ($i=1,2$):

$$\begin{aligned}&\eta ^{i}\,\left( P^{f}+y\right) \frac{\partial x^{i}}{\partial y_{l}} =(-1)^{i}\kappa \left( \frac{\partial P_\mathrm{B}}{\partial y_{l}}+1_{l=B}\right) , \end{aligned}$$

(37)

$$\begin{aligned}&\eta ^{1}\,\left( P^{f}+y\right) \frac{\partial x^{i}}{\partial y_{l}}+\eta ^{i}\left( P^{f}-K\right) \frac{\partial s^{i}}{\partial y_{l}} = (-1)^{i}\kappa \left( \frac{\partial P_\mathrm{G}}{\partial y_{l}}+1_{l=G}\right) . \end{aligned}$$

(38)

1.2 Prices

Differentiating (14) with respect to $y_{l}$ in the vicinity of the riskless equilibrium yields:

$$\begin{aligned} \frac{\partial P_{k}}{\partial y_{l}} =&\beta \,\widehat{M}\,\sum _{j=B,G}\pi _{k,j}\left( \frac{\partial P_{j}}{\partial y_{l}}+1_{j=l}\right) ,\nonumber \\ \text{ with: } \widehat{M} =&1+\alpha ^{1}(u^{\prime } (\delta +(P^{f}+y)x^{f,1})-1)\nonumber \\&\times \left( 1-\frac{V}{\eta ^{1}\eta ^{2}}\frac{u^{\prime \prime } \left( \delta +(P^{f}+y)x^{f,1}\right) }{u^{\prime }\left( \delta +(P^{f}+y)x^{f,1}\right) -1}\frac{u^{\prime \prime }\left( \delta +(P^{f}+y)x^{f,2}\right) }{u^{\prime }\left( \delta +(P^{f}+y)x^{f,2} \right) -1}\right) , \end{aligned}$$

(39)

Denoting $\widetilde{M}=\frac{\beta \widehat{M}}{(1-\beta \widehat{M}) (1-(\pi _\mathrm{GG}+\pi _{BB}-1)\beta \widehat{M})}$, we obtain that:

$$\begin{aligned} \left[ \begin{array}{c} \frac{\partial P_\mathrm{G}}{\partial y_{l}}\\ \frac{\partial P_\mathrm{B}}{\partial y_{l}} \end{array}\right] =\widetilde{M}&\left[ \begin{array}{cc} \left( \pi _\mathrm{GG}-\beta \widehat{M}(\pi _\mathrm{GG}+\pi _{BB}-1)\right) 1_{l=G} +\left( 1-\pi _\mathrm{GG}\right) 1_{l=B}\\ \left( 1-\pi _{BB}\right) 1_{l=G}+\left( \pi _{BB}-\beta \widehat{M} (\pi _\mathrm{GG}+\pi _{BB}-1)\right) 1_{l=B} \end{array}\right] >0. \end{aligned}$$

(40)

Differentiating Eq. (22) with respect to $y_{l}$ finally yields:

$$\begin{aligned} \frac{\partial Q_{k}}{\partial y_{l}}&=\beta \widehat{M}\left( \frac{\partial P_\mathrm{G}}{\partial y_{l}}+1_{l=G}\right) . \end{aligned}$$

(41)

1.3 Back to Proposition 4

Using (37) and (38) with Lemma 1, we deduce the impact of a mean-preserving spread of dividends on security quantities (recall that V[y] (E[y]) is the variance (mean) of the dividend process):

$$\begin{aligned} \eta ^{1}\left( P^{f}+y\right) \left. \frac{\partial x^{1}}{\partial V[y]}\right| _{E[y]\text { cst}}&=\kappa \,\frac{2-\pi _\mathrm{GG}-\pi _{BB}}{2(y_\mathrm{G}-y_\mathrm{B})(1-\pi _\mathrm{GG})}\, \frac{1}{1-(\pi _\mathrm{GG}+\pi _{BB}-1)\beta \widehat{M}}>0,\\ \eta ^{1}\left( P^{f}-K\right) \left. \frac{\partial s^{1}}{\partial V[y]}\right| _{E[y]\text { cst}}&=-\eta ^{1}(P^{f}-K)\left. \frac{\partial s^{1}}{\partial V[y]}\right| _{E[y]\text { cst}}\frac{2-\pi _{BB}-\pi _\mathrm{GG}}{1-\pi _{BB}}<0. \end{aligned}$$

The derivatives of the asset price in (21) relative to V[y] can be expressed as ($l=B,G$):

$$\begin{aligned} \frac{2(y_\mathrm{G}-y_\mathrm{B})}{2-\pi _\mathrm{GG}-\pi _{BB}}\left. \frac{\partial P_{l}}{\partial V[y]}\right| _{E[y]\text { cst}} =(1_{l=G}-1_{l=B})\frac{1}{1-\pi _{ll}}\frac{(\pi _\mathrm{GG} +\pi _{BB}-1)\beta \widehat{M}}{1-(\pi _\mathrm{GG}+\pi _{BB}-1)\beta \widehat{M}} \end{aligned}$$

We finally deduce $\left. \frac{\partial Q_\mathrm{G}}{\partial V[y]}\right| _{E[y]}>\left. \frac{\partial Q_\mathrm{B}}{\partial V[y]}\right| _{E[y]}$ using (41).

Appendix E: Proof of Proposition 5

We consider the evolution of prices and quantities around the symmetric equilibrium $\alpha ^{1}=\alpha ^{2}=\alpha$, where the asset prices $P_{k}^{s}$ and option prices $Q_{k}^{s}$ are:

$$\begin{aligned}&P_{k}^{s}=\beta \sum _{j=B,G}\pi _{k,j}\left( 1 +\kappa ^{s}_{j}\right) (P_{j}^{s}+y_{j}),\\&Q_{k}^{s}=\beta \pi _{k,G}\left( 1+\kappa ^{s}_\mathrm{G}\right) (P_\mathrm{G}^{s}-K), \end{aligned}$$

where $\kappa ^{s}_{j}=\alpha \left( u^{\prime }\left( \delta +(P_{j}^{s}+y_{j})\,\frac{V}{\eta ^{1}+\eta ^{2}}\right) -1\right)$

1.1 Quantities

Differentiating (19) and (20) wrt $\alpha ^{i}$ ($\alpha ^{i}=1,2$) yields (close to the symmetric equilibrium):

$$\begin{aligned}&\eta ^{1}(P_\mathrm{B}+y_\mathrm{B})\,\frac{\partial x^{1}}{\partial \alpha ^{i}} =(-1)^{i}\frac{\eta ^{1}\,\eta ^{2}}{\eta ^{1} +\eta ^{2}}\frac{u^{\prime }\left( \delta +(P_\mathrm{B}^{s}+y_\mathrm{B})\, \frac{V}{\eta ^{1}+\eta ^{2}}\right) -1}{\alpha \, u^{\prime \prime }\left( \delta +(P_\mathrm{B}^{s}+y_\mathrm{B})\,\frac{V}{\eta ^{1}+\eta ^{2}}\right) },\\&\eta ^{1}(P_\mathrm{G}+y_\mathrm{G})\frac{\partial x^{1}}{\partial \alpha ^{i}} +\eta ^{1}(P_\mathrm{G}-K)\frac{\partial s^{1}}{\partial \alpha ^{i}} =(-1)^{i}\frac{\eta ^{1}\,\eta ^{2}}{\eta ^{1}+\eta ^{2}} \frac{u^{\prime }\left( \delta +(P_\mathrm{G}^{s} +y_\mathrm{G})\,\frac{V}{\eta ^{1}+\eta ^{2}}\right) -1}{\alpha \, u^{\prime \prime }\left( \delta +(P_\mathrm{G}^{s}+y_\mathrm{G}) \,\frac{V}{\eta ^{1}+\eta ^{2}}\right) }. \end{aligned}$$

We deduce that $\frac{\partial x^{1}}{\partial \alpha ^{1}}>0$ and $\frac{\partial s^{1}}{\partial \alpha ^{1}}<0$ whenever condition (18) holds.

1.2 Prices

We differentiate the expressions of both asset and option prices with respect to $\alpha ^{i}$ ($i=1,2$) in the neighborhood of the symmetric equilibrium:

$$\begin{aligned} \frac{\partial P_{k}}{\partial \alpha ^{i}}&=\beta \,\frac{\eta ^{i}}{\eta ^{1}+\eta ^{2}}\, \sum _{j=B,G}\pi _{k,j}\Delta _{j}+\beta \sum _{j=B,G}\pi _{k,j}\, M_{j}\,\frac{\partial P_{j}}{\partial \alpha ^{i}},\\ \text {with:}\ M_{j}&=1+\kappa _{j}^{s}\left( 1+\frac{(P_{j}^{s} +y_{j})V}{\eta ^{1}+\eta ^{2}}\frac{u^{\prime \prime } (\delta +(P_{j}^{s}+y_{j})\frac{V}{\eta ^{1} +\eta ^{2}})}{u^{\prime }(\delta +\frac{(P_{j}^{s} +y_{j})V}{\eta ^{1}+\eta ^{2}})-1}\right) \\ \Delta _{j}&=\left( u^{\prime }\left( \delta +(P_{j}^{s} +y_{j})\,\frac{V}{\eta ^{1}+\eta ^{2}}\right) -1\right) (P_{j}^{s}+y_{j}). \end{aligned}$$

Using matrix notation, we obtain after denoting $\widetilde{M}_{GB}=1-\beta \pi _\mathrm{GG}M_\mathrm{G}-\beta \pi _{BB}M_\mathrm{B} +\beta ^{2}(\pi _\mathrm{GG}+\pi _{BB}-1)M_\mathrm{G}M_\mathrm{B}$:

$$\begin{aligned}&\widetilde{M}_{GB}\left[ \begin{array}{c} \frac{\partial P_\mathrm{G}}{\partial \alpha ^{i}}\\ \frac{\partial P_\mathrm{B}}{\partial \alpha ^{i}} \end{array}\right] \\&\quad =\beta \frac{\eta ^{i}}{\eta ^{1}+\eta ^{2}}\left[ \begin{array}{c} (\pi _\mathrm{GG}-\beta M_\mathrm{B}(\pi _\mathrm{GG}+\pi _{BB}-1))\Delta _\mathrm{G}+(1-\pi _\mathrm{GG})\Delta _\mathrm{B}\\ (\pi _{BB}-\beta M_\mathrm{G}(\pi _\mathrm{GG}+\pi _{BB}-1))\Delta _\mathrm{B}+(1-\pi _{BB})\Delta _\mathrm{G} \end{array}\right] >0. \end{aligned}$$

Analogously for the option price, we have:

$$\begin{aligned} \frac{\partial Q_{k}}{\partial \alpha ^{i}}&=\beta \pi _{k,G}\widehat{M}_\mathrm{G}\frac{\partial P_\mathrm{G}}{\partial \alpha ^{i}}+\beta \frac{\eta ^{i}}{\eta ^{1} +\eta ^{2}}\pi _{k,G}\left( u^{\prime }(\delta +(P_\mathrm{G}^{s}+y_\mathrm{G})\frac{V}{\eta ^{1}+ \eta ^{2}})-1\right) (P_\mathrm{G}^{s}-K).\\ \text{ with: } \widehat{M}_\mathrm{G}&=1+\kappa _\mathrm{G}^{s} \left( 1+\frac{(P_\mathrm{G}^{s}-K)\frac{V}{\eta ^{1}+\eta ^{2}} u^{\prime \prime }(\delta +(P_\mathrm{G}^{s}+y_\mathrm{G})\frac{V}{\eta ^{1} +\eta ^{2}})}{u^{\prime }(\delta +(P_\mathrm{G}^{s}+y_\mathrm{G})\,\frac{V}{\eta ^{1}+\eta ^{2}})-1}\right) . \end{aligned}$$

We easily deduce that $\frac{\partial Q_\mathrm{G}}{\partial \alpha ^{i}}>\frac{\partial Q_\mathrm{B}}{\partial \alpha ^{i}}>0$, which proves the last result in Proposition 4.

Appendix F: Extension to heterogeneous suboptimal production levels

In this section, we show that allowing agents to be endowed with heterogeneous levels of suboptimal production may make our results consistent with those of Franke et al. (1998). We assume that type-1 and type-2 agents have access to different suboptimal production levels denoted, respectively, $\delta _{1}$ and $\delta _{2}$. The rest of the setup is unchanged. As long as $\delta _{1}$ and $\delta _{2}$ are not too different from each other, we can still prove the existence of an equilibrium as in Proposition 1.

In this setup, we can prove the following lemma:

Lemma 2

(Portfolio holdings with heterogeneous $\delta$) Assume that type-1 and type-2 agents are endowed with different suboptimal production levels denoted, respectively, $\delta ^{1}$ and $\delta ^{2}$, as described above. If $\delta ^{1}>\delta ^{2}$, there exist probabilities $\alpha ^{1}>\alpha ^{2}$ such that:

• type-2 agents hold a greater quantity of assets than type-1 agents, i.e., $x^{2}>x^{1}>0$;

• type-2 agents sell options to type-1: $s^{2}<0<s^{1}$.

According to Lemma 2, despite the fact that type-1 agents face a higher probability of losing their production opportunity, they hold riskier portfolios: They hold less stocks and sell call options (and thus insurance) to type-2 agents.¹⁴

Proof

It is rather straightforward to show pricing Eqs. (14) and (15) generalize to ($k=B,G$ and $i=1,2$):

$$\begin{aligned} P{}_{k} =&\beta \pi _{k,G}\left( 1+\alpha ^{i}\left( u^{\prime } \left( \delta ^{i}+(P_\mathrm{G}+y_\mathrm{G})x^{i} +(P_\mathrm{G}-K)s^{i}\right) -1\right) \right) (P_\mathrm{G}+y_\mathrm{G})\nonumber \\&+\beta \pi _{k,B}\left( 1+\alpha ^{i}\left( u^{\prime } \left( \delta ^{i}+(P_\mathrm{B}+y_\mathrm{B})x^{i}\right) -1\right) \right) (P_\mathrm{B}+y_\mathrm{B}), \end{aligned}$$

(42)

$$\begin{aligned} Q{}_{k} =&\beta \pi _{k,G}\left( 1+\alpha ^{i}\left( u^{\prime } \left( \delta ^{i}+(P_\mathrm{G}+y_\mathrm{G})x^{i}+(P_\mathrm{G}-K)s^{i}\right) -1\right) \right) (P_\mathrm{G}-K). \end{aligned}$$

(43)

Market-clearing conditions (23) and (24) still hold. Equations (42) and (43) imply that equations (19) and (20) characterizing the participation of both agent types to both markets become:

$$\begin{aligned}&\alpha ^{1}\left( u^{\prime }(\delta ^{1}+(P_\mathrm{B}+y_\mathrm{B})x^{1}) -1\right) =\alpha ^{2}\left( u^{\prime }(\delta ^{2}+(P_\mathrm{B} +y_\mathrm{B})x^{2})-1\right) , \end{aligned}$$

(44)

$$\begin{aligned}&\alpha ^{1}\left( u^{\prime }(\delta ^{1}+(P_\mathrm{G}+y_\mathrm{G})x^{1} +(P_\mathrm{G}-K)s^{1})-1\right) \nonumber \\&\quad =\alpha ^{2}\left( u^{\prime } (\delta ^{2}+(P_\mathrm{G}+y_\mathrm{G})x^{2}+(P_\mathrm{G}-K)s^{2})-1\right) . \end{aligned}$$

(45)

Let us first assume that $\alpha ^{1}=\alpha ^{2}$. Equations (44) and (45) imply:

$$\begin{aligned} x^{2}-x^{1}&=\frac{\delta ^{1}-\delta ^{2}}{P_\mathrm{B}+y_\mathrm{B}}>0,\\ s^{1}-s^{2}&=\frac{(P_\mathrm{G}+y_\mathrm{G})-(P_\mathrm{B}+y_\mathrm{B})}{P_\mathrm{B}+y_\mathrm{B}}(\delta ^{1}-\delta ^{2})>0. \end{aligned}$$

If $\alpha ^{1}=\alpha ^{2}$, type-2 agents hold more stock than type-1 and sell their calls. By continuity of (44) and (45) in $\alpha ^{1}$ and $\alpha ^{2}$, we can find two values $\alpha ^{1}>\alpha ^{2}$ such that $x^{2}-x^{1}>0$ and $s^{1}-s^{2}>0$, which concludes the proof.

Appendix G: Extension to a three-state economy with riskless bonds

In this section, we provide a detailed presentation of an extension of our setup to a three-state economy with riskless bonds. More formally, we consider an economy similar to the one described in Sect. 2 of the paper. We introduce a riskless bond that pays off one unit of consumption in every state and whose price is denoted $R_{t}$ at date t. The size of the risky tree is now denoted $V_{X}>0$, while we assume that the net supply of bonds is denoted $V_\mathrm{B}>0$. We denote $b_{t}^{i}$ the bond holdings of agent i at date t. The option is still in zero net supply. We further assume that agents cannot short-sell the bond. The program of a type-i agent can be expressed as follows:

$$\begin{aligned}&\max _{(c_{t}^{i},e_{t}^{i},x_{t}^{i},b_{t}^{i},(s_{t}^{i,h})_{h}) _{t\ge 0}}E_{0}\left[ \sum _{t=0}^{\infty }\beta ^{t} \left( u(c_{t}^{i})-e_{t}^{i}\right) \right] \end{aligned}$$

(46)

$$\begin{aligned}&\text {s.t.} c_{t}^{i}+P_{t}\, x_{t}^{i}+R_{t}b_{t}^{i}+\sum _{h=1}^{H}Q_{t}^{h}s_{t}^{h,i} =\xi _{t}^{i}e_{t}^{i}+(1-\xi _{t}^{i})\delta \end{aligned}$$

(47)

$$\begin{aligned}&\quad +(P_{t}+y_{t})\, x_{t-1}^{i}+b_{t-1}^{i}+\sum _{h=1} ^{H}(P_{t}-K_{t-1}^{h})^{+}s_{t-1}^{h,i} \end{aligned}$$

(48)

$$\begin{aligned}&c_{t}^{i}\ge 0 \text{ and } e_{t}^{i}\ge 0 \end{aligned}$$

(49)

$$\begin{aligned}&P_{t}x_{t}^{i}+R_{t}b_{t}^{i}+\sum _{h=1}^{H}Q_{t}^{h}s_{t}^{h,i}\ge 0, \end{aligned}$$

(50)

$$\begin{aligned}&x_{t}^{i}\ge 0,\ b_{t}^{i}\ge 0, \end{aligned}$$

(51)

$$\begin{aligned}&\sum _{h=1}^{l}Q_{t}^{h}(s_{t}^{h,i,j})^{+}\le P_{t}\, x_{t}^{i,j}+R_{t}b_{t}^{i}, \end{aligned}$$

(52)

$$\begin{aligned}&\lim _{t\rightarrow \infty }\beta ^{t}E_{0}\left[ u^{\prime }(c_{t} ^{i})\, x_{t}^{i}\right] =\lim _{t\rightarrow \infty }\beta ^{t}E_{0} \left[ u^{\prime }(c_{t}^{i})\, b_{t}^{i}\right] =\lim _{t\rightarrow \infty }\beta ^{t}E_{0}\left[ u^{\prime } (c_{t}^{i})\, s_{t}^{h,i}\right] , \end{aligned}$$

(53)

$$\begin{aligned}&\{x_{-1}^{i},b_{-1}^{i},s_{-1}^{1,i},\ldots ,s_{-1}^{H,i}, \xi _{0}^{i},y_{0}\} \text{ are } \text{ given. } \end{aligned}$$

(54)

To express market-clearing conditions, we need to adapt the definition of the distribution $\varLambda _{t}^{i}$ of type-i agents that is now a function of all security holdings, including bonds, and labor status using the probability measure $\varLambda _{t}^{i}:{\mathcal {B}}({\mathbb {R}}) ^{2+H}\times {\mathcal {B}}(E^{t})\rightarrow [0,1]$. This probability measure can be interpreted as follows: $\varLambda _{t}^{i}(X,B,S^{1},\ldots ,S^{H},I)$ (with $(X,B,S^{1},\ldots ,S^{H},I)\in {\mathcal {B}} ({\mathbb {R}})^{2+H}\times {\mathcal {B}}(E^{t})$) is the measure of agents of type i, with stock holdings $x\in X$, bond holdings $b\in B$, option positions $s^{h}\in S^{h}$ ($h=1,\ldots ,H$) and with an individual history $\xi \in I$. Using these probability measures, market-clearing conditions become:

$$\begin{aligned} \sum _{i=1,2}\int _{{\mathbb {R}}^{2+H}\times E^{t}}x\varLambda _{t}^{i}(\mathrm{d}x, \mathrm{d}b,\mathrm{d}s^{1},\ldots ,\mathrm{d}s^{H},\mathrm{d}\xi )&=V_{X}, \end{aligned}$$

(55)

$$\begin{aligned} \sum _{i=1,2}\int _{{\mathbb {R}}^{2+H}\times E^{t}}b\varLambda _{t}^{i}(\mathrm{d}x,\mathrm{d}b,\mathrm{d}s^{1},\ldots ,\mathrm{d}s^{H},\mathrm{d}\xi )&=V_\mathrm{B}, \end{aligned}$$

(56)

$$\begin{aligned} \sum _{i=1,2}\int _{{\mathbb {R}}^{2+H}\times E^{t}}s^{h}\varLambda _{t}^{i}(\mathrm{d}x,\mathrm{d}b,\mathrm{d}s^{1},\ldots ,\mathrm{d}s^{H},\mathrm{d}\xi )&=0\ (h=1,\ldots ,H). \end{aligned}$$

(57)

The equilibrium is defined very similarly to Definition 1:

Definition 2

(Sequential competitive equilibrium in an economy with bonds) A sequential competitive equilibrium is a collection of consumption and effort levels $(c_{t}^{i},e_{t}^{i})_{t\ge 0}$, of stock demands $(x_{t}^{i})_{t\ge 0}$, of bond demands $(b_{t}^{i})_{t\ge 0}$, of derivative demands $(s_{t}^{1,i},\ldots ,s_{t}^{H,i})_{t\ge 0}$ for $i=1,2$ and of security prices $(P_{t},R_{t},Q_{t})_{t\ge 0}$ such that for an initial distribution of security holdings and of idiosyncratic and aggregate shocks $\{(x_{-1}^{i},b_{-1}^{i},s_{-1}^{1,i}\ldots ,s_{-1}^{H,i}, \xi _{0}^{i})_{i=1,2},y_{0}\}$, we have:

• 1. Individual strategies solve the optimization program (46) when prices are given;

• 2. Security prices adjust such that security markets clear at all dates and equations (55)–(57) hold;

• 3. The evolution of the probability measures $\varLambda _{t}^{1}$ and $\varLambda _{t}^{2}$ is consistent with individual choices.

Provided that security volumes $V_{X}$ and $V_\mathrm{B}$ are not too large, that heterogeneity remains limited and that condition (11) still holds, we can prove the existence of a limited heterogeneity equilibrium characterized by the set of quantities and prices $(x_{k}^{i},b_{k}^{i},s_{k}^{i},P_{k},$ $R_{k},Q_{k})_{k=B,G}^{i=1,2}$ solving the following equations ($k=1,\ldots ,n$, $h=1,\ldots ,H$ and $i=1,2$):¹⁵

$$\begin{aligned} P_{k}&=\beta \sum _{j=1}^{n}\pi _{k,j}(1+\alpha ^{i} (u^{\prime }(\delta +(P_{j}+y_{j})\, x_{k}^{i}+b_{k}^{i}+ \sum _{h=1}^{H}(P_{j}-K_{k}^{h})^{+}s_{k}^{h,i})-1))(P_{j}+y_{j})\\ R_{k}&=\beta \sum _{j=1}^{n}\pi _{k,j}(1+\alpha ^{i} (u^{\prime }(\delta +(P_{j}+y_{j})\, x_{k}^{i}+b_{k}^{i} +\sum _{h=1}^{H}(P_{j}-K_{k}^{h})^{+}s_{k}^{h,i})-1))\\ Q_{k}^{h}&=\beta \sum _{j=1}^{n}\pi _{k,j}(1+\alpha ^{i} (u^{\prime }(\delta +(P_{j}+y_{j})\, x_{k}^{i}+b_{k}^{i}+ \sum _{l=1}^{H}(P_{j}-K_{k}^{l})^{+}s_{k}^{l,i})-1))(P_{j} -K_{k}^{h})^{+}\\ V_{X}&=\eta ^{1}\, x_{k}^{1}+\eta ^{2}\, x_{k}^{2},\\ V_\mathrm{B}&=\eta ^{1}\, b_{k}^{1}+\eta ^{2}\, b_{k}^{2},\\ 0&=\eta ^{1}\, s_{k}^{h,1}+\eta ^{2}\, s_{k}^{h,2}. \end{aligned}$$

We further simplify the model along the lines of Sect. 4:¹⁶

• 1. There are three states of the world $n=3$ that we denote G, M and L (from good too bad) and that one single call is traded ($H=1$).

• 2. Aggregate states are persistent, i.e., $\pi _\mathrm{GG}+\pi _{MM}+\pi _{BB}>1$.

• 3. The utility function u is such that $\lim _{c\rightarrow \infty }u^{\prime }(c)=0$ and:

  $$\begin{aligned} X\mapsto -X\frac{u^{\prime \prime }(\delta +X)}{u^{\prime }(\delta +X)-1} {\,\,is\,\, increasing\,\, for\,\,} X\in [0,u^{\prime -1}(1) -\delta ). \end{aligned}$$

• 4. The strike K of the option is such that the option exactly pays off in the good state G of the world.

There are three non-redundant securities and three states of the world. Every agent type therefore holds a security portfolio, which is independent of the state of the world. The simplified equilibrium is then characterized by 15 variables $\{ x^{1},x^{2},b^{1},b^{2},s^{1},s^{2},P_\mathrm{G},P_\mathrm{M},P_\mathrm{B},$ $R_\mathrm{G},R_\mathrm{M},R_\mathrm{B},Q_\mathrm{G},Q_\mathrm{M},Q_\mathrm{B}\}$ together with the following eight equations:

$$\begin{aligned}&\alpha ^{1} \left( u^{\prime }(\delta +(P_\mathrm{B}+y_\mathrm{B})x^{1}+b^{1}) -1\right) =\alpha ^{2}\left( u^{\prime }(\delta +(P_\mathrm{B}+y_\mathrm{B})x^{2} +b^{2})-1\right) \end{aligned}$$

(58)

$$\begin{aligned}&\alpha ^{1} \left( u^{\prime }(\delta +(P_\mathrm{M}+y_\mathrm{M})x^{1} +b^{1})-1\right) =\alpha ^{2}\left( u^{\prime }(\delta +(P_\mathrm{M} +y_\mathrm{M})x^{2}+b^{2})-1\right) \end{aligned}$$

(59)

$$\begin{aligned}&\alpha ^{1} \left( u^{\prime }(\delta +(P_\mathrm{G}+y_\mathrm{G})x^{1} +b^{1}+(P_\mathrm{G}-K)s^{1})-1\right) \nonumber \\&\quad =\alpha ^{2}\left( u^{\prime }(\delta +(P_\mathrm{G}+y_\mathrm{G}) x^{2}+b^{2}+(P_\mathrm{G}-K)s^{2})-1\right) \end{aligned}$$

(60)

$$\begin{aligned}&P{}_{k} =\beta \sum _{j=B,M,G}\pi _{k,j}\left( 1+\alpha ^{1}\left( u ^{\prime }\left( \delta +(P_{j}+y_{j})x^{1} +b^{1}+1_{j=G}(P_\mathrm{G}-K)s^{1}\right) -1\right) \right) (P_{j}+y_{j}) \end{aligned}$$

(61)

$$\begin{aligned}&R{}_{k} =\beta \sum _{j=B,M,G}\pi _{k,j}\left( 1+\alpha ^{1} \left( u^{\prime }\left( \delta +(P_{j}+y_{j}) x^{1}+b^{1}+1_{j=G}(P_\mathrm{G}-K)s^{1}\right) -1\right) \right) \end{aligned}$$

(62)

$$\begin{aligned}&Q{}_{k} =\beta \pi _{k,G}\left( 1+\alpha ^{1}\left( u^{\prime }\left( \delta +(P_\mathrm{G}+y_\mathrm{G})x^{1}+b^{1}+(P_\mathrm{G}-K)s^{1}\right) -1\right) \right) (P_\mathrm{G}-K), \end{aligned}$$

(63)

$$\begin{aligned}&V_{X} =\eta ^{1}\, x^{1}+\eta ^{2}\, x^{2} \end{aligned}$$

(64)

$$\begin{aligned}&V_\mathrm{B} =\eta ^{1}\, b^{1}+\eta ^{2}\, b^{2} \end{aligned}$$

(65)

$$\begin{aligned}&0 =\eta ^{1}\, s^{1}+\eta ^{2}\, s^{2} \end{aligned}$$

(66)

We can now state a proposition similar to Proposition 3 about portfolio compositions.

Proposition 6

(Agents’ portfolios in a three-state economy) Type-1 agents, facing a greater risk of becoming unproductive, choose to hold a less risky portfolio than type-2 agents. More precisely:

• type-1 agents hold more bonds than type-2 agents, i.e., $b^{1}>b^{2}>0$;

• type-1 agents always sell call options to hedge their stock holdings.

Moreover, if we further assume $x\mapsto -\frac{u^{\prime \prime }(x)}{u^{\prime }(x)}$ to be weakly decreasing, we obtain that in the neighborhood of $V_{X}=0$, we have $x^{1}\ge x^{2}$.

High-risk agents use securities to hold less risky portfolios than low-risk agents. Indeed, high-risk agents hold more bonds, which provide insurance in the bad state of the world and they sell calls, which allow smoothing out payoffs of stock holdings. The position in stocks of high-risk agents can be smaller or greater than the one of low-risk agents. The intuition is the following. Bonds are mainly purchased to provide insurance in the bad state, but they also payoff in the medium state. If bonds provide insufficient insurance in medium state, high-risk agents will need to further purchase stocks and their holdings will be larger than the ones of low-risk agents. Conversely, if bonds provide “too much” insurance in medium state, type-1 agents will hold less stocks than type-2. We prove that this latter case never holds when few stocks are available and when agents have DARA utility.

Portfolio compositions in the extended economy with three states and riskless bonds are therefore very consistent with our findings of Proposition 3: High-risk agents hold less risky portfolios than low-risk agents.

Proof

We denote $u_{j,k}^{\prime }=u^{\prime }(\delta +(P_{j}+y_{j})x^{k}+b^{k})$ and $u_{j,k}^{\prime \prime }=u^{\prime \prime }(\delta +(P_{j}+y_{j})x^{k}+b^{k})$ for $k=1,2$ and $j=B,M,G$. We consider the two Eqs. (58) and (58) together with (64) and (65) in $s^{1}$and $x^{1}$. Values $P_\mathrm{M}+y_\mathrm{M}$ and $P_\mathrm{B}+y_\mathrm{B}$ are considered as distinct fixed parameters. Parameters $\alpha ^{1}$ and $\alpha ^{2}$ can be varied. Remark that for $\alpha ^{1}=\alpha ^{2}$, we have $x^{1}=x^{2}$ and $b^{1}=b^{2}$. Computing the derivative of (58) and (58) with respect to $\alpha ^{1}$ yields

$$\begin{aligned} (P_{j}+y_{j})\frac{\partial x^{1}}{\partial \alpha ^{1}}+\frac{\partial b^{1}}{\partial \alpha ^{1}}=-\frac{u_{j,1} ^{\prime }-1}{\alpha ^{1}u_{j,1}^{\prime \prime } +\alpha ^{2}u_{j,2}^{\prime \prime }}>0 \end{aligned}$$

and

$$\begin{aligned} \left( \frac{1}{P_\mathrm{B}+y_\mathrm{B}}-\frac{1}{P_\mathrm{M}+y_\mathrm{M}}\right) \alpha ^{1}\frac{\partial b^{1}}{\partial \alpha ^{1}}&=\frac{1}{\frac{-(P_\mathrm{B}+y_\mathrm{B}) u_\mathrm{B,1}^{\prime \prime }}{u_\mathrm{B,1}^{\prime }-1}+\frac{- (P_\mathrm{B}+y_\mathrm{B})u_\mathrm{B,2}^{\prime \prime }}{u_\mathrm{B,2}^{\prime }-1}}\\&\quad -\frac{1}{\frac{-(P_\mathrm{M}+y_\mathrm{M})u_\mathrm{M,1}^{\prime \prime }}{u_\mathrm{M,1}^{\prime }-1}+\frac{-(P_\mathrm{M}+y_\mathrm{M})u_\mathrm{M,2} ^{\prime \prime }}{u_\mathrm{M,2}^{\prime }-1}}. \end{aligned}$$

Let us assume $(P_\mathrm{M}+y_\mathrm{M})>(P_\mathrm{B}+y_\mathrm{B})$ (note that if we make the reverse assumption, the result still holds; both inequalities below will be reversed). We have using Assumption (18) about utility shape:

$$\begin{aligned} \left( \frac{1}{P_\mathrm{B}+y_\mathrm{B}}-\frac{1}{P_\mathrm{M}+y_\mathrm{M}}\right) \alpha ^{1}\frac{\partial b^{1}}{\partial \alpha ^{1}}=-\frac{u_\mathrm{M,1}^{\prime }-1}{\alpha ^{1}u_\mathrm{M,1}^{\prime \prime }+\alpha ^{2}u_\mathrm{M,2} ^{\prime \prime }}+-\frac{u_\mathrm{B,1}^{\prime }-1}{\alpha ^{1}u_\mathrm{B,1}^{\prime \prime }+\alpha ^{2}u_\mathrm{B,2}^{\prime \prime }} \end{aligned}$$

which implies

$$\begin{aligned} \left( \frac{1}{P_\mathrm{B}+y_\mathrm{B}}-\frac{1}{P_\mathrm{M}+y_\mathrm{M}}\right) \alpha ^{1}\frac{\partial b^{1}}{\partial \alpha ^{1}}>0. \end{aligned}$$

Therefore $(P_\mathrm{M}+y_\mathrm{M})>(P_\mathrm{B}+y_\mathrm{B})$ implies $\frac{\partial b^{1}}{\partial \alpha ^{1}}>0$. Market clearing implies then that $b^{1}>b^{2}$, which proves the first part of the proof.

Regarding $s^{1}$, we define the following function of $s^{1}$:

$$\begin{aligned} \psi (s^{1})&=\alpha ^{1}\left( u^{\prime }(\delta +(P_\mathrm{G}+y_\mathrm{G})x^{1}+b^{1}+(P_\mathrm{G}-K)s^{1})-1\right) \\&-\alpha ^{2}\left( u^{\prime }(\delta +(P_\mathrm{G}+y_\mathrm{G})x^{2} +b^{2}-(P_\mathrm{G}-K)\frac{\eta ^{1}}{\eta ^{2}}s^{1})-1\right) , \end{aligned}$$

which is a strictly decreasing function of $s^{1}$. To prove that $s^{1}<0$, it is sufficient to prove that $\psi (0)<0$. To do so, let us consider $\tilde{\psi }_{0}:\pi \mapsto \alpha ^{1}\left( u^{\prime }(\delta +\pi x^{1}+b^{1})-1\right) -\alpha ^{2}\left( u^{\prime }(\delta +\pi x^{2}+b^{2})-1\right)$. The function $\tilde{\psi }_{0}$ admits at most two zeros, which are $P_\mathrm{B}+y_\mathrm{B}$ and $P_\mathrm{M}+y_\mathrm{M}$. Therefore, since $P_\mathrm{G}+y_\mathrm{G}>P_\mathrm{B}+y_\mathrm{B},P_\mathrm{M}+y_\mathrm{M}$ (since the option only pays off in the best state by construction), $\tilde{\psi }_{0}(P_\mathrm{M}+y_\mathrm{M})$ has the sign as $\alpha _{2}-\alpha _{1}<0$. We deduce that $\tilde{\psi }_{0}(P_\mathrm{M}+y_\mathrm{M})=\psi (0)<0$, which implies $s^{1}<0$. This proves the second part of the proof.

If $|V_{X}|\ll 1$, we obtain developing (58) and (58) at the first order in $x^{1}$ and $x^{2}$:

$$\begin{aligned} -x^{1}\frac{u^{\prime \prime }(\delta +b^{1})}{u^{\prime }(\delta +b^{1})} =-x^{2}\frac{u^{\prime \prime }(\delta +b^{2})}{u^{\prime }(\delta +b^{2})}. \end{aligned}$$

Since $b^{1}>b^{2}$ and if we further assume $x\mapsto -\frac{u^{\prime \prime }(x)}{u^{\prime }(x)}$ to be decreasing, we obtain $0<-\frac{u^{\prime \prime }(\delta +b^{1})}{u^{\prime }(\delta +b^{1})}\le -\frac{u^{\prime \prime }(\delta +b^{2})}{u^{\prime }(\delta +b^{2})}$ and $x^{1}\ge x^{2}$, which terminates the proof.