Incentive compatibility of rational expectations equilibrium in large economies: a counterexample

Abstract

This note provides a counterexample for a large economy in which a rational expectations equilibrium (REE) does not possess the desirable property of incentive compatibility for each agent. The key point here is that the REE equilibrium price depends on the private information of every individual agent. Thus, we propose to focus on those REE prices that depend only on the macro states and are not influenced by individual agents’ private information. Such a REE will be incentive compatible.

1 Introduction

Radner (1979) and Allen (1981) extended the finite agent Arrow–Debreu–McKenzie economy to allow for asymmetric information, where each agent is characterized by a random utility function, random initial endowment, and private information with a prior. The equilibrium notion that Radner puts forward is called rational expectations equilibrium (REE), which is an extension of the deterministic Walrasian equilibrium of the Arrow–Debreu–McKenzie model. According to the REE, each individual maximizes her interim expected utility conditioned on her own private information as well as the information generated by the equilibrium price.

In a REE, the private information possessed by individuals can influence the equilibrium price. This leads to the incentive compatibility issue as individuals may misreport their own private information in their favor. One would imagine that this issue should not arise in a large economy (an economy with an atomless measure space of agents) as each individual is negligible in size compared to the whole population. This note shows the contrary. We provide a counterexample for a REE in a large economy, where every agent can manipulate her private information for a higher utility.

The key point is that even if each individual is negligible in terms of size, her influence on the equilibrium price can nevertheless be significant if we allow the REE price to depend on all the information at the individual level. This suggests that considering an arbitrary REE in a large economy may not lead to desirable results. One should pay attention to those REE prices that capture the meaning of perfect competition, for example, those prices that depend only on the macro states and are not influenced by individual agents’ private information. We may also point out that it may be too much to require an equilibrium price to reveal all the private information in a large economy. Our counterexample in Sect. 4 precisely has this problem.

This note is organized as follows. In Sect. 2, we present the economic model, and the notions of REE and incentive compatibility. The main result is stated in Sect. 3 with its proof in Sect. 4. Some concluding remarks are provided in Sect. 5.

2 The economic model

In this section, we define the notions of a private information economy, REE and incentive compatibility by following the model presented in Sun et al. (2012).

2.1 Private information economy

We consider an atomless probability space¹ $(I, \mathcal{I }, \lambda )$ as the space of agents. Each agent receives a private signal $q \in T^0=\{q_1, q_2, \dots , q_L\}$. $\mathcal{T }^0$ is the power set of $T^0$. A signal profile $t$ is a function from $I$ to $T^0$. For $i \in I$, $t(i)$ (also denoted by $t_i$) is the private signal of agent $i$ while $t_{-i}$ is the restriction of $t$ to the set $I \setminus \{i\}$. Let $(T, \mathcal{T }, P)$ be a probability space that models the uncertainty associated with the private signal profiles for all the agents.² For simplicity, we shall assume that $(T, \mathcal{T })$ has a product structure so that $T$ is the product of $T_{-i}$ and $T^0$, while $\mathcal{T }$ is the product $\sigma$-algebra of $\mathcal{T }^0$ and a $\sigma$-algebra $\mathcal{T }_{-i}$ on $T_{-i}$. For $t \in T$ and $t_i^{\prime } \in T^0$, we shall adopt the usual notation $(t_{-i}, t_i^{\prime })$ to denote the signal profile whose value is $t_i^{\prime }$ for $i$ and $t_j$ for $j \ne i$.

The private signal process $f$ is a function from $I \times T$ to $T^0$ such that $f(i, t) = t_i$ for any $(i, t) \in I \times T$. For each $i \in I$, let ${\tilde{t}_i}$ be the projection mapping from $T$ to $T^0$ with ${\tilde{t}_i}(t) = t_i$.

We also would like to include another source of uncertainty in our model—the macro level uncertainty. Let $S = \{s_1, s_2, \dots , s_K \}$ be the set of all possible macro states of nature and $\mathcal{S }$ the power set of $S$. The $S$-valued random variable $\tilde{s}$ on $T$ models the macro level uncertainty. For each macro state $s \in S$, denote the event $(\tilde{s} = s) = \{t \in T: \tilde{s} (t)= s\}$ that $s$ occurs by $C_s$. The probability that $s$ occurs is $\pi _s = P(C_s)$. Without loss of generality, assume that $\pi _s > 0$ for each $s \in S$. Let $P_s$ be the conditional probability measure on $(T, \mathcal{T })$ when the random variable $\tilde{s}$ takes value $s$. Thus, for each $B \in \mathcal{T }$, $P_s(B) = P(C_s \cap B)/\pi _s$. It is obvious that $P = \sum _{s \in S} \pi _s P_s$.

The private signal process $f$ is essentially pairwise independent conditioned on the macro state of nature $\tilde{s}$. In other words, for $\lambda$-almost all $i \in I$, $\tilde{t}_i$ and $\tilde{t}_j$ are independent conditioned on $\tilde{s}$ for $\lambda$-almost $j \in I$. With this independence assumption, we need to work with a joint agent-probability space $(I \times T, \mathcal{I } \boxtimes \mathcal{T }, \lambda \boxtimes P)$ that extends the usual measure-theoretic product $(I \times T, \mathcal{I } \otimes \mathcal{T }, \lambda \otimes P)$ of the agent space $(I, \mathcal{I }, \lambda )$ and the probability space $(T, \mathcal{T }, P)$, and retains the Fubini property, which is called a Fubini extension.³ The process $f$ is assumed to be $\mathcal{I } \boxtimes \mathcal{T }$-measurable.

The common consumption set for all the agents is the positive orthant $\mathbb{R }_+^m$. Let $u$ be a function from $I \times \mathbb{R }_+^m \times T$ to $\mathbb{R }_+$ such that for any given $i \in I$, $u(i, z, t)$ is the utility of agent $i$ at consumption bundle $z \in \mathbb{R }_+^m$ and signal profile $t \in T$. For any given $(i, t) \in I \times T$, we assume that $u(i, z, t)$ (also denoted by $u_{(i, t)}(z)$ or $u_i(z, t)$)⁴ is continuous, monotonic in $z \in \mathbb{R }_+^m$.⁵ For each $z \in \mathbb{R }_+^m$, $u_z (\cdot , \cdot )$ is a $\lambda \boxtimes P$-integrable function on $I \times T$.

In our model, the initial endowment of an agent depends on her private signal. The initial endowment profile $e$ is a function from $I \times T^0$ to $\mathbb{R }_+^m$ such that for $(i, q) \in I \times T^0$, $e(i, q)$ is the initial endowment of agent $i$ when her private signal is $q$. We assume that for each $q \in T^0$, $e(\cdot , q)$ is $\lambda$-integrable over $I$, and $\int _I e(i, q) d\lambda$ is in the strictly positive cone $\mathbb{R }^m_{++}$.⁶

Formally, the private information economy is denoted by

$$\begin{aligned} \mathcal{E } = \left\{ I \times T, u, e, f, (\tilde{t}_i, i \in I), \tilde{s}\right\} . \end{aligned}$$

2.2 Rational expectations equilibrium

As usual, a price is a normalized nonnegative vector $p$ in $\Delta _m$, where $\Delta _m$ is the unit simplex of $\mathbb{R }_+^m$. A price process $\tilde{p}$ is a measurable function from $T$ to $\Delta _m$. For each $t \in T$, $\tilde{p}(t)$ is the price when the signal profile is $t$. For notational simplicity, the letter $p$ will be used both for a price and a price process. The terms “price” and “price process” are used synonymously in this note.

An allocation $x$ to be a measurable mapping from $I \times \Delta _m \times T^0$ to $\mathbb{R }_+^m$. For each $(i, p, q) \in I \times \Delta _m \times T^0$, $x(i, p, q)$ is the consumption bundle of agent $i$ when the price is $p$ and her derived signal is $q$.

Since an agent’s initial endowment is contingent on her private signal $q \in T^0$, we denote the budget set for agent $i$ by $B_i(p, q)$ when the price is $p$ and her private signal is $q$. Hence, $B_i(p, q) = \{z \in \mathbb{R }_+^m: p \,z \le p \, e(i, q)\}$.

Given a consumption bundle $z \in \mathbb{R }_+^m$, a signal $q \in q^0$ and a price $p$, the interim (conditional) expected utility of agent $i$ is defined as follows:

$$\begin{aligned} U_i(z|p, q) = E\{u(i, z, t) | \tilde{p} = p, \tilde{t}_i = q\}. \end{aligned}$$

(1)

In the REE, an agent updates her belief on the distribution of signal profiles based on her signal and observation of the equilibrium price. She computes her expected utility with the updated belief and aims to maximize the interim expected utility subject to her budget constraint. The formal definition of the REE is given below.

Definition

(Rational Expectations Equilibrium (REE)) A REE for the private information economy $\mathcal{E } = \{I \times T, u, e, f, (\tilde{t}_i, i \in I), \tilde{s}\}$ is a pair of an allocation and a price process $(x^*, p^*)$ such that:

• 1. $x^*$ is feasible, i.e., $\int _I x^*(i, p^*(t) , t_i) d\lambda = \int _I e(i, t_i) d\lambda$ for $P$-almost all $t \in T$;

• 2. for $\lambda$-almost all $i \in I$ and for $P$-almost all $t \in T$, $x^*(i, p^*(t), t_i)$ is a maximizer of the following problem:

  $$\begin{aligned} \begin{array}{ll} \max \,\,\, &{} U_i(z| p^*(t), t_i) \\ \mathrm{subject} \; \mathrm{to}\,\,\, &{} z \in B_i(p^*(t), t_i). \end{array} \end{aligned}$$

The following notion of incentive compatibility says that an agent cannot increase her interim expected utility by mis-reporting her private signal.

Definition 2

(Incentive Compatibility) A REE $(x^*, p^*)$ is said to be incentive compatible if for $\lambda$-almost all $i \in I$,

$$\begin{aligned} U_i(x^*(i, p^*(t), t_i) |p^*(t), t_i) \ge U_i(x^*(i, p^*(t_{-i}, t_i^{\prime }), t_i)|p^*(t_{-i}, t_i^{\prime }), t_i) \end{aligned}$$

holds for $P$-almost all $t \in T$ and for all $t_i^{\prime } \in T^0$.

3 Incentive compatibility: a counterexample

The following result shows that it is not true that under a REE in a large economy, the agents will automatically report their signals truthfully. The key idea in such a result is that the REE price $p^*$ can be influenced by the private signal of each individual agent; the details are given in the next section.

Proposition 1

There exists a REE $(x^*, p^*)$ in a large private information economy $\mathcal E ^p$, where the private signals are independent of each other, and the macro state function $\tilde{s}$ can be regarded as constant, such that every agent has an incentive to mis-report her signal.

4 Proof of Proposition 1

In this section, we need to use some very basic nonstandard analysis. The reader can refer Loeb et al. (2000). Fix an infinitely large even hyperinteger $n \in {}^* \mathbb N _\infty$. Let $I$ be $\{1, 2, \dots , n\}$ with its internal power set $\mathcal{I }_0$ and internal counting probability measure $\lambda _0$ on $\mathcal{I }_0$ with $\lambda _0(A) = |A|/|I|$ for any $A \in \mathcal{I }_0$, where $|A|$ is the internal cardinality of $A$. Let $(I, \mathcal{I }, \lambda )$ be the Loeb space of the internal probability space $(I, \mathcal{I }_0, \lambda _0)$, which will serve as the space of agents for the large private information economy considered here.

Let $T^0 \!= \!\{0, 1\}$ be the signals for individual agents, and $T$ the set of all the internal functions from $I$ to $T^0$ (the space of signal profiles). Let $\mathcal{T }_0$ be the internal power set on $T$, $P_0$ the internal counting probability measure on $(T, \mathcal{T }_0)$ (i.e., the probability weight for each $t \!=\! (t_1, t_2, \dots , t_n) \!\in \! T$ under $P_0$ is $1/2^n$), and $(T, \mathcal{T }, P)$ the corresponding Loeb space.

Let $(I \times T, \mathcal{I }_0 \otimes \mathcal{T }_0, \lambda _0 \otimes P_0)$ be the internal product probability space of $(I, \mathcal{I }_0, \lambda _0)$ and $(T, \mathcal{T }_0, P_0)$. Let $(I \times T, \mathcal{I } \boxtimes \mathcal{T }, \lambda \boxtimes P)$ be the Loeb space of the internal product $(I \times T, \mathcal{I }_0 \otimes \mathcal{T }_0, \lambda _0 \otimes P_0)$, which is indeed a Fubini extension of the usual product probability space by Keisler’s Fubini Theorem (see, for example, Loeb et al. 2000). Let $f(i, t) = t_i$. Then, the private signals $f_i$ are independent of each other. In other words, we can assume that the macro state function $\tilde{s}$ is constant.

The common consumption space is the nonnegative orthant $\mathbb R ^2_+$. For agent $i \in I$ with $i$ odd, let her endowment $e_i =(1, 0)$, and her utility function $u_i(x_1, x_2)$ be $\left( x_1^{-2} + \left( {12 \over 37}\right) ^3 x_2^{-2} \right) ^{-{1\over 2}}$ for positive $x_1$ and $x_2$ and zero otherwise. For agent $i \in I$ with $i$ even, let her endowment $e_i =(0, 1)$, and her utility function $u_i(x_1, x_2)$ be $\left( \left( {12 \over 37}\right) ^3 x_1^{-2} + x_2^{-2} \right) ^{-{1\over 2}}$ for positive $x_1,\, x_2$, and zero otherwise.⁷ Given a price system $(p_1, p_2)$, agent $i$’s demand is $\bigg (p_1 \left[ p_1 + {12 \over 37} p_2 ({p_1 \over p_2})^{1/3} \right] ^{-1}, \, {12 \over 37}({p_1 \over p_2})^{1/3} p_1 \Big [p_1 + {12 \over 37} p_2 ({p_1 \over p_2})^{1/3} \Big ]^{-1}\bigg )$ if $i$ is odd, and $\bigg ({12 \over 37}({p_2 \over p_1})^{1/3} p_2 \left[ p_2 + {12 \over 37} p_1 ({p_2 \over p_1})^{1/3} \right] ^{-1}, \, p_2 \Big [p_2 + {12 \over 37} p_1 ({p_2 \over p_1})^{1/3} \Big ]^{-1}\bigg )$ if $i$ is even.

It can be checked that $p^1 = ({1 \over 2}, {1 \over 2})$ and $p^2 = ({64 \over 91}, {27 \over 91})$ are equilibrium prices of such a large deterministic economy.⁸ Under the equilibrium price $p^1$, agent $i$’s demand and utility are, respectively, $x^1_i =({37 \over 49}, {12 \over 49})$ and $u^1_i =({37 \over 49})^{3/2}$ when $i$ is odd, and $x^1_i= ({12 \over 49}, {37 \over 49})$ and $u^1_i= ({37 \over 49})^{3/2}$ when $i$ is even. Under the equilibrium price $p^2$, agent $i$’s demand and utility are, respectively, $x^2_i = ( {148 \over 175}, \, {64 \over 175})$ and $u^2_i = ({148 \over 175})^{3/2}$ when $i$ is odd, and $x^2_i = ( {27 \over 175}, \, {111 \over 175})$ and $u^2_i = ( {111 \over 175} )^{3/2}$ when $i$ is even. It is clear that ${111 \over 175} < {37 \over 49} < {148 \over 175}$.

Now, for the corresponding private information economy, define a price process

$$\begin{aligned} p^* (t) = \left\{ \begin{array}{l@{\quad }l} {p^1} &{} \text{ if } \sum _{j=1}^n t_j \text{ is } \text{ odd, }\\ p^2 &{} \text{ if } \sum _{j=1}^n t_j \text{ is } \text{ even. } \end{array}.\right. \end{aligned}$$

(2)

Let $x^*(i, p, q)$ be $x^1_i$ if $p=p^1$, and $x^2_i$ if $p=p^2$. Then $(x^*, p^*)$ is a REE.

Fix an agent $i \in I$ with $i$ odd. Consider the signal profile $t \in T$ with $\sum _{j=1}^n t_j$ odd (the probability of such an event is half). Then, $x^*(i, p^*(t), t_i) = x^1_i$ with utility $u^1_i =({37 \over 49})^{3/2}$. However, if agent $i$ mis-reports her type from $t_i$ to $t_i^{\prime } = 1 - t_i$, then $1 - t_i + \sum _{j\ne i}^n t_j$ is even, and her consumption would become $x^*(i, p^*(t_{-i}, t_i^{\prime }), t_i) = x^2_i$ with a strictly higher utility $u^2_i = ( {148 \over 175})^{3/2}$. Since $u_i$ does not depend on $t$, the interim (conditional) expected utility $U_i(z|p, q)$ as in Eq. (1) is always $u_i(z)$. Hence, for $t \in T$ with $\sum _{j=1}^n t_j$ odd and $t_i^{\prime } = 1 - t_i$, we have

$$\begin{aligned} U_i(x^*(i, p^*(t), t_i) |p^*(t), t_i) = u_i(x^*(i, p^*(t), t_i)) = \left( {37 \over 49}\right) ^{3/2}, \end{aligned}$$

which is strictly less than

$$\begin{aligned} U_i(x^*(i, p^*(t_{-i}, t_i^{\prime }), t_i)|p^*(t_{-i}, t_i^{\prime }), t_i) = u_i(x^*(i, p^*(t_{-i}, t_i^{\prime }), t_i)) = \left( {148 \over 175}\right) ^{3/2}. \end{aligned}$$

Next, fix an agent $i \in I$ with $i$ even. Consider the signal profile $t \in T$ with $\sum _{j=1}^n t_j$ even (the probability of such an event is half). Then, $x^*(i, p^*(t), t_i) = x^2_i$ with utility $u^2_i = ( {111 \over 175})^{3/2}$. However, if agent $i$ mis-reports her type from $t_i$ to $t_i^{\prime } = 1 - t_i$, then $1 - t_i + \sum _{j\ne i}^n t_j$ is odd, and her consumption would become $x^*(i, p^*(t_{-i}, t_i^{\prime }), t_i) = x^1_i$ with a strictly higher utility $u^1_i = ({37 \over 49})^{3/2}$. Hence, for $t \in T$ with $\sum _{j=1}^n t_j$ even and $t_i^{\prime } = 1 - t_i$, we have

$$\begin{aligned} U_i(x^*(i, p^*(t), t_i) |p^*(t), t_i) = u_i(x^*(i, p^*(t), t_i)) = \left( {111 \over 175}\right) ^{3/2}, \end{aligned}$$

which is strictly less than

$$\begin{aligned} U_i(x^*(i, p^*(t_{-i}, t_i^{\prime }), t_i)|p^*(t_{-i}, t_i^{\prime }), t_i) = u_i(x^*(i, p^*(t_{-i}, t_i^{\prime }), t_i)) = \left( {37 \over 49}\right) ^{3/2}. \end{aligned}$$

Therefore, every agent can mis-report her private signal to obtain a strictly higher utility under an event with probability $1/2$. This means that the incentive compatibility condition fails for every agent.

5 Concluding remarks

A REE in a large economy is naturally expected to be incentive compatible as each individual in such an economy is intuitively negligible. This note provides a counterexample showing the contrary. In the counterexample, the equilibrium price flips as an individual switches from one private signal to another unilaterally, resulting in a better utility for the individual. We believe that it may be too much to require an equilibrium price to reveal the private information of every individual agent in a large economy. Therefore, we propose that one should restrict the attention to those equilibria whose prices capture the meaning of perfect competition, for example, prices that depend only on macro state of nature, as in Sun et al. (2012) and Sun and Yannelis (2008). When the REE price depends only on the macro states, incentive compatibility is not an issue since an individual agent’s private signal cannot influence the macro states.⁹