On fairness of equilibria in economies with differential information

Abstract

The paper proposes a notion of fairness which overcomes the conflict arising between efficiency and the absence of envy in economies with uncertainty and asymmetrically informed agents. We do it in general economies which include, as particular cases, the main models of differential information economies, providing in this framework a natural competitive equilibrium notion which satisfies the fair criterion. The analysis is conducted allowing the presence of large traders, which may cause the lack of perfect competition.

Appendix

1.1 Fictitious economy

In order to prove the main results, following Wilson’s idea Wilson (1978) (see also de Clippel 2007 and Donnini et al. 2012), we consider a type-agent representation $\mathcal{E }^*$ of the original differential information economy $\mathcal{E }$ described before. It is a fictitious market for state-contingent claims that can be formalized as follows:

$$\begin{aligned} \mathcal{E }^* = \big \{(T^*, \mathcal{T }^*, \mu ^*); \, I\!\!R_{+}^{\ell \times |\varOmega |};\, \left( X_{(t, E)}, V_{(t,E)}, e_{(t,E)}\right) _{(t,E) \in T^*}\big \} \end{aligned}$$

where:

• 1. $T^*$ is the set of the type agents. In other words, $T^ *$ is the set of couple $(t,E),$ where $t$ is an agent and $E$ is an event of his information partition. $T^*$ coincides with the graph of the correspondence $\varPhi : T\rightarrow 2^\mathcal{F}$ defined by $\varPhi (t)=\mathcal{F }_ t$. $\mathcal{T }^*$ is the family of coalitions: a coalition $S^*$ is a measurable subset of $T^*$, i.e., $S^*\in {{\mathcal{T }}}\otimes {\mathcal{B }}(2^{\mathcal{F }})$, where ${\mathcal{B }}(2^{\mathcal{F }})$ denotes the Borel $\sigma$-algebra on the discrete topological space $2^{\mathcal{F }}$ and $\otimes$ denotes the product $\sigma$-algebra. Finally, the measure $\mu ^*$ on $\mathcal{T }^*$ is defined as the product measure of $\mu$ and the counting measure.

• 2. $X_{(t, E)}$ is the consumption set of the type agent $(t, E)$ defined as follows:

  $$\begin{aligned} X_{(t, E)}:=\bigcup _{\beta \in M_ t} \{\alpha : \varOmega \rightarrow I\!\!R_+^\ell :\,\, \alpha (\omega )= \beta (\omega )\chi _{E}(\omega )\,\, \mathrm{for\,\,all\,\, } \omega \in \varOmega \}. \end{aligned}$$

• 3. The couple $(V_{(t,E)}, e_{(t,E)})$ characterizes the type-agent $(t, E)$: given a type-agent $(t,E)$, his preference for a commodity $\alpha \in X_{(t, E)}$ is defined by

  $$\begin{aligned} V_{(t,E)}(\alpha )= \sum _{\omega \in {E}} u_t(\omega , \alpha (\omega ))\pi (\omega ), \end{aligned}$$

  while $e_{(t,E)}$ represents his initial endowment of physical resources defined as follows

  $$\begin{aligned} e_{(t, E)}(\omega ) = \left\{ \begin{array}{l@{\quad }l} e_t(\omega ) &{} \mathrm {if} \omega \in E \\ 0 &{} \mathrm {otherwise} . \end{array} \right. \end{aligned}$$

  Notice that since $e_t(\cdot )$ is $\mathcal G _t$-measurable for all $t \in T$, it follows that for all $(t, E) \in T^*$, $e_{(t, E)}\in X_{(t, E)}$.

Clearly, assuming that $\mathcal{F }_ t=\mathcal{G }_ t$ for each agent $t\in T$, then $X_{(t,E)}$ is made by all functions that are constant on $E$ and zero on $\varOmega \setminus E$. In the case the true state of nature is publicly announced, for each type-agent $(t, E)$ the consumption set is simply made by functions on $\varOmega$ that are null outside $E$. Notice also that if in the differential information economy $\mathcal{E }$ there is equal income, that is agents have the same initial endowment, the equal sharing of initial resources among the type-agents in the associated economy $\mathcal{E }^*$ may not hold.

An allocation in the fictitious economy is a function $\alpha : \varOmega \times T^* \rightarrow I\!\!R_{+}^{\ell }$ such that $\alpha (\omega , \cdot )$ is $\mu$-integrable for each $\omega$ and for each $(t,E)\in T^*,$ $\alpha _{(t,E)}:\varOmega \rightarrow I\!\!R_{+}^{\ell }$ belongs to $X_{(t, E)}$, where $\alpha _{(t,E)}$ represents the bundle that the type-agent $(t,E)$ receives under the allocation $\alpha$.

An allocation $\alpha$ is feasible for the coalition $S^*$ if

$$\begin{aligned} \int \limits _{S^*} \alpha _{(t,E)}(\omega )\, \mathrm{d}\mu ^* \le \int \limits _{S^*} e_{(t,E)}(\omega )\, \mathrm{d}\mu ^*\quad \mathrm{{for\,\,all\,\,}} \omega \in \varOmega , \end{aligned}$$

and it is feasible if it is feasible for the whole coalition of agents $T^*$. We shall rewrite, for reader convenience, the main equilibrium notions in the economy $\mathcal{E }^*$.

Definition 6.1

A feasible allocation $\alpha$ is said to be c-fair in the economy $\mathcal{E }^*$ if there do not exist an alternative allocation $\alpha ^\prime$, two coalitions $S^*_1$ and $S^*_2$, such that

• (1) $\mu ^*(S^*_1)>0,\,\, S_1^*\cap S_2^*=\emptyset ,$

• (2) $V_{(t, E)}(\alpha ^\prime _{(t, E)})> V_{(t, E)}(\alpha _{(t, E)}) \quad { {for\,\,almost \,\,all}}\,\, (t,E)\in S^*_1$

• (3) $\int _{ S^*_1} \left[ \alpha ^\prime _{(t,E)}(\omega )-e_{(t,E)}(\omega )\right] \,\mathrm{d}\mu ^* \!\le \! \int _{ S^*_2}\left[ \alpha _{(t,E)}(\omega )\!-\!e_{(t,E)}(\omega )\right] \, \mathrm{d}\mu ^* \quad {{for\,\,all}} \omega \in \varOmega .$

Definition 6.2

A feasible allocation $\alpha$ is a Walrasian (or Arrow-Debreu) allocation of the type-agent economy $\mathcal{E }^*$ if there exists a price $p:\varOmega \rightarrow I\!\!R^\ell _+$ such that for almost all $(t,E) \in T^*$, $\alpha _{(t,E)} \in \arg \max _{\alpha ^\prime \in B_{(t,E)}(p)} V_{(t,E)}(\alpha ^\prime ),\,\,{ where}$

$$\begin{aligned} B_{(t,E)}(p)=\left\{ \alpha ^\prime \in X_{(t,E)}\, |\, \sum _{ \omega \in \varOmega } p(\omega )\cdot \alpha ^\prime (\omega )\le \sum _{\omega \in \varOmega }p(\omega )\cdot e_{(t,E)}(\omega )\right\} . \end{aligned}$$

The pair $(\alpha , p)$ is said to be a Walrasian (or Arrow-Debreu) equilibrium.

The allocation $\alpha$ is said to be a quasi-Walrasian allocation of the type agent economy when $\alpha _{(t,E)} \in \arg \max _{\alpha ^\prime \in B_{(t,E)}(p)} V_{(t,E)}(\alpha ^\prime )$, for almost all $(t, E)$ such that $inf\, p\cdot X_{(t,E)}<inf\, p\cdot e_{(t,E)}$.

Remark 6.3

With standard arguments one can easily show that any Walrasian equilibrium allocation is c-fair in $\mathcal{E }^*$.

We are going to show that any competitive market equilibrium of the economy $\mathcal{E }$ corresponds to a Walrasian equilibrium of the associated type economy $\mathcal{E }^*$. To this end, we construct a natural isomorphism between $\mathcal{E }$ and its type-agent representation $\mathcal{E }^*.$

Given an allocation $a$ of $\mathcal{E },$ its type-agent representation is the allocation $\alpha$ such that for each $(t,E)$ in $T^*$

$$\begin{aligned} \alpha _{(t, E)}(\omega ) = a_ t(\omega )\chi _{E}(\omega ). \end{aligned}$$

Notice that since $a_t(\cdot )$ is $\mathcal G _t$-measurable, then $\alpha _{(t, E)} \in X_{(t, E)}$.

Given an allocation $\alpha$ of $\mathcal{E }^*$, its associated allocation $a$ in the original economy $\mathcal{E }$ is such that for each $t$ in $T$ and each $\omega$ in $\varOmega$

$$\begin{aligned} a_t(\omega )=\alpha _{(t, {F}_t(\omega ))}(\omega ). \end{aligned}$$

Since $\alpha$ is an allocation of $\mathcal{E }^*$, by definition $\alpha _{(t, E)}\in X_{(t,E)}$ for all $(t, E)\in T^*$. Thus, $\alpha _{(t, {F}_t(\omega ))}(\omega )=\alpha _{(t, {F}_t(\omega ))}(\omega ^\prime )$ for all $\omega ^\prime \in {G}_t(\omega )$. This implies that $a_t(\cdot )$ is $\mathcal G _t$-measurable.

Therefore, $a_t(\cdot )$ is $\mathcal G _t$-measurable if and only if $\alpha _{(t, E)}\in X_{(t, E)}$ for all $(t, E) \in T^*$.

Remark 6.4

By adapting similar arguments used in de Clippel (2007) and Donnini et al. (2012), it is easy to show that there exists a one to one correspondence between a differential information economy $\mathcal{E }$ and the associated economy $\mathcal{E }^*$ in terms of competitive equilibrium and c-fair allocations. Precisely, if $(a,p)$ is a competitive market equilibrium for $\mathcal{E }$, then the associated allocation $\alpha$ is such that the pair $(\alpha , p)$ is an Arrow-Debreu equilibrium for $\mathcal{E }^*$. Conversely, if $(\alpha ,p)$ is an Arrow-Debreu equilibrium for $\mathcal{E }^*$, then the associated allocation $a$ is such that the pair $(a,p)$ is a competitive market equilibrium for $\mathcal{E }$. Analogously, to any c-type fair allocation $a$ in $\mathcal{E }$ corresponds a c-fair allocation $\alpha$ in $\mathcal{E }^*$ and vice versa.

From the above remark we can deduce Proposition 5.3 as follows.

1.2 Proof of Proposition 5.3

Let $a$ be a competitive market equilibrium allocation. We need to show that it is c-fair. To this end, consider the associated allocation $\alpha$ in the type-agent economy $\mathcal{E }^*$, which is a Walrasian equilibrium allocation (see Remark 6.4). From Remark 6.3 it follows that $\alpha$ is c-fair for $\mathcal{E }^*$, and hence by coming back to the original differential information economy $\mathcal{E }$ we get that $a$ is c-type fair (see Remark 6.4).$\square$

1.3 Proof of Theorem 4.2

Our next goal is to prove the existence of a competitive market equilibrium. We will proceed by following the steps below:

• 1. Starting from the mixed differential information economy $\mathcal{E }$, we construct the associated mixed type-agent economy $\mathcal{E }^*$ as before. For each atom $t\in T_1$, the type agent $A^*=(t, E)$ for any $E$ is an atom of $\mathcal{E }^*$. Thus, the set of type agents $T^*$ can be decomposed into the disjoint union of the nonatomic sector $T^*_0$ and the atomic part $T^*_1$, where $T^*_ 1$ is the disjoint union of at most countable many atoms $A^*_ i$. Precisely,

  $$\begin{aligned} T^*_0&= \{ (t, E)\in T_0 \times \mathcal F \,:\,\, E \in \mathcal{F }_t\}\quad \mathrm{and}\\ T^*_1&= \{ (A, E)\in T_1 \times \mathcal F \,:\,\, E \in \mathcal{F }_A\}\quad \mathrm{that\,\,is\,\,}T^*_1= T^*\setminus T^*_0. \end{aligned}$$

  We have already observed that there is a one to one correspondence between $\mathcal{E }$ and $\mathcal{E }^*$ in terms of competitive equilibrium (Remark 6.4; see also de Clippel 2007 and Donnini et al. 2012), that is if $(a,p)$ is a competitive market equilibrium for $\mathcal{E }$, then the associated allocation $\alpha$ is such that the pair $(\alpha , p)$ is an Arrow-Debreu equilibrium for $\mathcal{E }^*$. Conversely, if $(\alpha ,p)$ is an Arrow-Debreu equilibrium for $\mathcal{E }^*$, then the associated allocation $a$ is such that the pair $(a,p)$ is a competitive market equilibrium for $\mathcal{E }$.

• 2. It has been proved in Greenberg and Shitovitz (1986) that $\mathcal{E }^*$ can be identified with the atomless economy $\tilde{\mathcal{E }}^*$ in which the set of agents $\tilde{T}^*$ is the union of $T^*_0$ and the intervals $\tilde{A}^*_ i$ corresponding to atoms $A^*_ i$, such that $\mu ^*(A^*_i)=\tilde{\mu }^*(\tilde{A}^*_i)$. Agents of $\tilde{A}^*_ i$ have the same characteristics (i.e., the same utility function, consumption set and initial endowment) of atom $A^*_ i$ (see also De Simone and Graziano 2003 for economies with infinitely many commodities and Pesce 2010 for differential information economies). It is easy to show that given a quasi equilibrium $(\tilde{\alpha }, p)$ of the atomless economy $\tilde{\mathcal{E }}^*$, the pair $(\alpha ,p)$ where $\alpha _t=\tilde{\alpha }_t$ for all $t \in T^*_0$ and $\alpha (A^*_ i)=\int _{\tilde{A}^*_ i}\tilde{\alpha }_t \mathrm{d}\tilde{\mu }^*$ is a quasi equilibrium of $\mathcal{E }^*$.

• 3. Consider the atomless economy $\tilde{\mathcal{E }}^*$ and notice each consumption set $X_{(t,E)}$ and each utility function $V_{(t,E)}$ satisfy standard requirements ensuring the existence of a quasi equilibrium $(\tilde{\alpha }, p)$ of $\tilde{\mathcal{E }}^*$ (see Hildenbrand 1974), that is for almost all $(t,E) \in \tilde{T}^*$,

  • (1) $\tilde{\alpha }_{(t,E)} \in B_{(t,E)}(p)$ where

    $$\begin{aligned} B_{(t,E)}(p)=\left\{ \alpha ^\prime \in X_{(t,E)}\, |\, \sum _{ \omega \in \varOmega } p(\omega )\cdot \alpha ^\prime (\omega )\le \sum _{\omega \in \varOmega }p(\omega )\cdot e_{(t,E)}(\omega )\right\} . \end{aligned}$$

  • (2) $\tilde{\alpha }_{(t,E)} \in \arg \max _{\alpha ^\prime \in B_{(t,E)}(p)} V_{(t,E)}(\alpha ^\prime ),\,\,\mathrm{whenever}\,\, p\cdot e_{(t,E)}\not = 0$.

  This pair $(\tilde{\alpha }, p)$ corresponds to a quasi equilibrium $(\alpha ,p)$ of the mixed economy $\mathcal{E }^*$.

• 4. Prove that $(\alpha ,p)$ is actually an Arrow-Debreu equilibrium of $\mathcal{E }^*$; use Remark 6.4 and observe that $(\alpha ,p)$ corresponds to a competitive market equilibrium $(a,p)$ of the “original” mixed differential information economy $\mathcal{E }$. Thus, a competitive market equilibrium for $\mathcal{E }$ exists.

We now show the details of our proof. Let $(\tilde{\alpha }, p)$ be a quasi equilibrium of the atomless economy $\tilde{\mathcal{E }}^*$, whose existence is proved in Hildenbrand (1974); and let $(\alpha , p)$ be the associated quasi equilibrium of the economy $\mathcal{E }^*$. We now want to prove that $(\alpha ,p)$ is an Arrow-Debreu equilibrium for $\mathcal{E }^*$. To this end, let us denote by $C_ 1^ *$ the set⁸

$$\begin{aligned} C_ 1^ *=\{(t, E)\in T^ *:p\cdot e_{(t,E)}=0\} \end{aligned}$$

and assume that $\mu ^*(C_ 1^ *)\in ]0,\mu ^*(T^ *)[$. Denote by $C_ 2^ *$ the set $T^ *\setminus C_ 1^ *$, by $T_ 1(\omega )$ the set

$$\begin{aligned} T_ 1(\omega )=\{t\in T: (t, F_ t(\omega ))\in C_ 1^ *\} \end{aligned}$$

and by $T_ 2(\omega )$ the set $T_ 2(\omega )= T\setminus T_ 1(\omega )$.

We claim that, in at least one state $\omega \in \varOmega$, both sets $\mu (T_ 1(\omega ))$ and $\mu (T_ 2(\omega ))$ have positive measure. Since the pair $\{T_ 1(\omega ), T_ 2(\omega )\}$ is a partition of $T$, it is enough to show that in at least one state $\omega \in \varOmega$ the set $T_ 1(\omega )$ has measure in the interval $]0,\mu (T)[$.

Assume by contradiction that in each state $\omega$ it is true that $\mu (T_ 1(\omega ))\in \{0,\mu (T)\}$. Define the two sets

$$\begin{aligned} A=\{\omega \in \varOmega : \mu (T_ 1(\omega ))=0\} \end{aligned}$$

and

$$\begin{aligned} B=\varOmega {\setminus } A. \end{aligned}$$

Then $A$ is not empty, otherwise from $\mu (T_ 1(\omega ))=\mu (T)$ for each $\omega$ it would follow $\mu ^ *(C_ 1^ *)=\mu ^*(T^*)$ a case that is excluded by strict positivity of initial endowments in each state. On the other hand, the case in which the set $B$ is empty is excluded by the assumption $\mu ^*(C_ 1^ *)> 0$. Let $\omega \in B$. Then $\mu (T_ 1(\omega ))=\mu (T)$ means that for almost all agents $t\in T$

$$\begin{aligned} p\cdot e_{(t, F_ t(\omega ))}=\sum _{\omega ^{\prime }\in F_ t(\omega )}p(\omega ^{\prime })\cdot e_ t(\omega ^{^{\prime }})=0. \end{aligned}$$

Now observe that from $\omega \in B$ it follows that $F_ i(\omega )\subseteq B$, for each information type $i\in I$. If not, there would exist $i\in I$ and $\bar{\omega }\in F_ i(\omega )$ such that $\bar{\omega }\notin B$, that is $\mu (T_ 2(\bar{\omega }))=\mu (T)$, and hence

$$\begin{aligned} \sum _{\omega ^{^{\prime }}\in F_ i(\bar{\omega })}p(\omega ^{\prime })\cdot e_ t(\omega ^{\prime })>0\quad \text{ for }\,\, \text{ almost }\,\,\text{ all }\,\, t\in T. \end{aligned}$$

Since for each agent $t$ with information type $i$, $F_ i(\bar{\omega })= F_ i(\omega )= F_ t(\omega )$, we have a contradiction.

Also observe that from $\omega \in A$ it follows that $F_ i(\omega )\subseteq A$, for each information type $i\in I$. If not, there would exist $i\in I$ and $\bar{\omega }\in F_ i(\omega )$ such that $\bar{\omega }\in B$ that is, by the previous argument, $F_ i(\omega )\subseteq B$ and this is impossible.

Then $\{A, B\}$ is a measurable partition of $\varOmega$ and the algebra generated by this partition is contained in ${\mathcal{F }}_ i$, for each $i\in I$, contradicting the assumption that $\bigwedge _{i}\mathcal{F }_i=\{\emptyset , \varOmega \}$. This proves our claim, that is there exists at least one state $\omega \in \varOmega$ in which $\mu (T_1(\omega ))\cdot \mu (T_2(\omega ))>0$.

Let us denote by $a$ the allocation of $\mathcal{E }$ corresponding to the quasi equilibrium allocation $\alpha$ of $\mathcal{E }^*$ and apply the condition $(I)$ to the partitions $\left\{ T_ 1(\omega ), T_ 2(\omega )\right\} _{\omega \in \varOmega }$ and to the allocation $a$. Let us denote by $b$ the allocation defined by $(I)$ in $\mathcal{E }$ and let $\beta$ be the corresponding allocation in $\mathcal{E }^ *$. Then for each $\omega \in \varOmega$ and for each $t\in T_2(\omega )$ we have

$$\begin{aligned} V_{(t,E)}(\beta _{(t,E)})\ge V_{(t,E)}(\alpha _{(t,E)}), \end{aligned}$$

and in at least one state $\omega _0 \in \varOmega$ the inequalities is strict for each $t\in T_ 2(\omega _{0})$ and $(t, F_ t(\omega _{0}))\in C^ *_ 2$. Hence there exists a coalition $C^*_ 0\subseteq C^ *_ 2$ for which $p\cdot \beta _{(t, E)}>p\cdot e_{(t, E)}$, while for the remaining type-agents $(t,E)\in C_ 2^ *\setminus C^*_ 0$, by continuity and monotonicity $p\cdot \beta _{(t, E)}\ge p\cdot e_{(t, E)}$.

Since for each state $\omega \in \varOmega$, ${\int _{C^ *_ 1}e_{(t,E)}(\omega )\,\mathrm{d}\mu ^*=\int _{T_ 1(\omega )}e_{t}(\omega )\, \mathrm{d}\mu }$, and

$$\begin{aligned} \int \limits _{T_ 1(\omega )}e_ t(\omega )\,\mathrm{d}\mu +\int \limits _{T_ 2(\omega )}a_ t(\omega )\,\mathrm{d}\mu \ge \int \limits _{T_ 2(\omega )}b_ t(\omega )\,\mathrm{d}\mu \end{aligned}$$

then

$$\begin{aligned} \int \limits _{C^ *_ 1}e_{(t,E)}\,\mathrm{d}\mu ^*+\int \limits _{C^ *_ 2}\alpha _{(t,E)}\,\mathrm{d}\mu ^* \ge \int \limits _{C_ 2^ *}\beta _{(t, E)}\,\mathrm{d}\mu ^*. \end{aligned}$$

It follows that

$$\begin{aligned} p\cdot \int \limits _{C^ *_ 2}\alpha _{(t,E)}\,\mathrm{d}\mu ^*&= p\cdot \int \limits _{C^ *_ 1}e_{(t,E)}\,\mathrm{d}\mu ^*+p\cdot \int \limits _{C^ *_ 2}\alpha _{(t,E)}\,\mathrm{d}\mu ^*\\&\ge p\cdot \int \limits _{C_ 2^ *}\beta _{(t,E)}\,\mathrm{d}\mu ^*> p\cdot \int \limits _{C_ 2^*}e_{(t,E)}\,\mathrm{d}\mu ^*\\&\ge p\cdot \int \limits _{C_ 2^ *}\alpha _{(t,E)}\,\mathrm{d}\mu ^*, \end{aligned}$$

which is a contradiction. The previous argument shows that the case $\mu ^*(C_ 1^*)\in (0,\mu ^*(T^ *))$ is excluded. Since $\mu ^*(C_ 1^*)=\mu ^*(T^ *)$ is excluded by the assumption of strictly positive initial endowments, then $\mu ^*(C_ 1^*)=0$ and hence $(\alpha , p)$ is an Arrow-Debreu equilibrium. Coming back to the original mixed market $\mathcal{E }$, from Remark 6.4, one can deduce that the corresponding allocation $a$ is a competitive market equilibrium of $\mathcal{E }$. Hence, the set $\mathrm{CME}(\mathcal{E })$ is non empty. $\square$