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Normative inference in efficient markets

Abstract

This paper develops a nonparametric method to infer social preferences over policies from prices of securities when agents have non-stationary heterogeneous preferences. We allow for arbitrary efficient risk-sharing mechanisms, formal and informal, and consider a large class of policies. We present a condition on the distribution of aggregate wealth that is necessary and sufficient for the revelation of social preferences over a universal set of policies. We also provide a weaker condition that is sufficient for revelation of social preferences for an arbitrary finite collection of policies.

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Notes

  1. 1.

    As quoted by Plato in Cratylus, 402a.

  2. 2.

    In the analysis, we only require that preferences satisfy concavity and Inada conditions.

  3. 3.

    In a consumption-based model with heterogenous agents, asset pricing theory demonstrates the existence of a positive representative consumer. In particular, given (fixed) efficient equilibrium allocation with strictly concave utilities, there exists a profile of weights such that the equilibrium allocation maximizes the sum of weighted utilities, subject to the constraint that total consumption does not exceed aggregate consumption. The value function of this program rationalizes the observed asset prices in a single-agent economy and thus is often interpreted as a utility of a representative agent. The equilibrium pricing kernel is given by marginal rates of substitution of a representative agent. However, since the weights that define the value function depend on the original equilibrium, the fiction of a positive representative agent does not facilitate comparisons across equilibria as required by normative analyses (see, e.g., Duffie 2010, pp. 10 and 27).

  4. 4.

    In this literature, social preferences are defined according to a cardinal criterion that assumes comparability of agents’ utilities and hence requires particular representations of utility functions that allow Gorman aggregation (see, e.g., Jorgenson 1990). Such normalizations are non-generic and inconsistent with the separable form of von Neumann–Morgenstern utility. Consider a canonical CRRA representation \(U^{i}=E\sum _{t=1}^{\infty }\beta ^{t}(c_{t}^{i})^{1-\sigma }/(1-\sigma )\). Since utility functions are strictly concave, the utility possibility frontier is strictly convex. It follows that for different consumption distributions the corresponding points on the frontier are located on different isoquants (hyperplanes) of utilitarian welfare, and the latter varies in consumption distribution. For the representation \(V^{i}=({U}^{i})^{\frac{1}{1-\sigma }}\), the utility possibility frontier becomes a hyperplane perfectly aligned with some utilitarian isoquant, and utilitarian welfare is invariant to all consumption distributions. Thus, for CRRA preferences, utilitarian analysis requires a \(V^{i}\) representation or its affine transformation identical for all agents. Such a family of cardinal utilities is non-generic within the class of all utility functions representing CARA preferences, and it does not have a separable form postulated by von Neumann and Morgenstern. Analogous considerations hold for other types of Gorman preferences with strictly concave instantaneous utilities.

  5. 5.

    For any profile of implicit prices \(\zeta ^{c}\in \mathcal {W}_{+}\) and level of utility \(U^{i\prime }\in \mathbb {R}_{++}\) expenditure function is

    $$\begin{aligned} e^{i}\left( \zeta ^{c},U^{i\prime }\right) =\min _{c^{i}\in \mathcal {C} _{++}}c_{0}^{i}+\sum _{t\ge 1}E\zeta _{t}^{c}c_{t}^{i}, \end{aligned}$$

    subject to the inequality constraint \(U^{i}\left( c^{i}\right) \ge U^{i\prime }\).

  6. 6.

    De Weerdt and Dercon (2006), Fafchamps and Lund (2003) and Barr and Attanasio (2009) study risk-sharing in villages while Mazzocco (2007) looks at transfers within castes.

  7. 7.

    Note that by this assumption an analysts can infer marginal utility of a representative agent for all realizations of aggregate wealth. Importantly, for our results it would not suffice to know the probability distributions of such marginal utilities.

  8. 8.

    We also make a technical assumption that, whenever an analyst observes a convergent sequence, he can also infer its limit.

  9. 9.

    Centralized financial markets can be a part of an allocative mechanism as well. In such cases, apart from being informative about preferences, they also provide risk-sharing opportunities for the agents. Our formulation, which separates allocation and price determination, is more general, because it allows for all types of risk-sharing mechanisms, and it allows us to make weaker and more realistic assumptions about asset structure that are sufficient for the revelation of preferences, but not for efficiency (Assumption 2).

  10. 10.

    For any process \(x\in \mathcal {C}\), a derivative cash flow is any process \( y(x)\in \mathcal {C}\), where \(y:\mathbb {R}\rightarrow \mathbb {R}\) can be an arbitrary function.

  11. 11.

    One could also define the lower bound for equivalent variation as

    $$\begin{aligned} EV_{c,c^{\prime }}^{-}\equiv \sum _{t=1}^{T}E\int \limits _{C_{t}}^{C_{t}^{+}}v_{t}^{-}(y)\mathrm{d}y-\sum _{t=1}^{T}E \int \limits _{C_{t}^{-}}^{C_{t}}v_{t}^{+}(y)\mathrm{d}y+C_{0}^{\prime }-C_{0}. \end{aligned}$$
    (9)

    With some work, however, one can show that \(EV_{c,c^{\prime }}^{-}=-EV_{c^{\prime },c}^{+}\) and condition \(EV_{c^{\prime },c}^{-}>0\) is equivalent to \(EV_{c,c^{\prime }}^{+}<0\). It follows that the lower bound condition is redundant.

  12. 12.

    For any t, the set is given by the convex hull of set \(\cup _{c\in \mathcal {P}^{F}}S_{t}^{c},\) which is bounded provided that each \(S_{t}^{c}\) is bounded as well.

  13. 13.

    This is a formalization of the Marshallian conjecture about negligible income effects in the context of financial markets. In Principles of Economics, Marshall justifies the assumption of quasilinear utility that underlines money metric analyses by arguing that “expenditure on any one thing, as, for instance, tea, is only a small part of his whole expenditure” (Marshall 1920, p. 842). We demonstrate this conjecture in the context of an infinite-horizon model with heterogenous agents.

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Correspondence to Marek Weretka.

Additional information

The support of National Center for Science (Grant UMO-2017/27/B/HS4/00868) is gratefully acknowledged.

Appendix

Appendix

Before giving the proofs for the results from the paper, we first introduce some notation and state auxiliary results. We say that function \(g_{t}: \mathbb {R}_{++}\rightarrow \mathbb {R}_{++}\) is standard if it is continuously differentiable, satisfies \(g_{t}^{\prime }<0\) and its limits are given by

$$\begin{aligned} \lim _{y\rightarrow 0}g_{t}(y)=\infty \text { and }\lim _{y\rightarrow \infty }g(y)=0. \end{aligned}$$

Any standard function is a bijection, and the family of standard functions is closed under inversion (by the Inverse Function Theorem) and finite summation. Moreover, for any collection of standard functions \( \{g_{t}\}_{t\ge 1}\), preferences given by \(U^{i}(c^{i})=c_{0}^{i}+E \sum _{t=1}^{T}\int _{0}^{c_{t}^{i}}g_{t}(y)\mathrm{d}y\) satisfy the assumptions of Sect. 2.1.

Let \(\{u_{t}^{i}\}_{i}\) be a collection of I twice continuously differentiable utility functions satisfying assumptions of strict monotonicity, strict convexity and Inada conditions (as in Sect. 2.1). Consider the following problem:

$$\begin{aligned} u_{t}(y)=\max _{\left\{ y^{i}\right\} _{i}\in Y\left( y\right) }\sum _{i}u_{t}^{i}\left( y^{i}\right) /\lambda ^{i}, \end{aligned}$$
(10)

where \(Y\left( \cdot \right) \) is a feasible allocation correspondence which, for any \(y\in \mathbb {R}_{++}\), gives

$$\begin{aligned} Y\left( y\right) \equiv \left\{ \{y^{i}\}_{i}\in \mathbb {R}_{+}^{I}|\sum _{i}y^{i} \le y\right\} . \end{aligned}$$

Let \(v_{t}\equiv u_{t}^{\prime }\) be the derivative of the value function of program (10).

Lemma 2

Optimization program (10) satisfies:

  1. 1.

    There exist I functions \(y_{t}^{i}:\mathbb {R}_{++}\rightarrow \mathbb {R}_{++}\) that give unique solutions for any \(y\in \mathbb {R}_{++}\);

  2. 2.

    Index \(v_{t}:\mathbb {R}_{++}\rightarrow \mathbb {R}_{++}\) is a standard function;

  3. 3.

    Marginal rates of substitution for all i coincide, i.e., \( u_{t}^{i\prime }(y^{i}\left( y\right) )/\lambda ^{i}=v_{t}\left( y\right) \).

Proof of Lemma 2

For any \(y\in \mathbb {R}_{++}\), set \(Y\left( y\right) \) is non-empty, convex and compact, while the objective function in (10) is continuous and strictly concave. Thus, the solution to problem (10) exists (Maximum Theorem) and it is unique. Therefore, the argmax functions \(\{y_{t}^{i}(\cdot )\}_{i}\) are well defined (property 1). Under Inada conditions, non-negativity constraints are not binding. From Kuhn–Tucker Theorem, for any \(y\in \mathbb {R}_{++}\) there exists a scalar \( v_{t}\in \mathbb {R}\) such that a solution to problem (10) solves the unconstrained optimization problem

$$\begin{aligned} \max _{\left\{ y^{i}\right\} _{i}\in Y\left( y\right) }\sum _{i}u_{t}^{i}(y^{i})/\lambda ^{i}-v_{t}\cdot \left( \sum _{i}y^{i}-y\right) . \end{aligned}$$
(11)

The first-order (necessary and sufficient) condition implies \(\lambda ^{i}v_{t}=u_{t}^{i\prime }\left( y^{i}\right) \) for each i. Derivative \( u_{t}^{i\prime }\left( \cdot \right) \) is a standard function, and therefore, its inverse \(\tilde{y}_{t}^{i}\left( v_{t}\right) \equiv \left( u_{t}^{i\prime }\right) ^{-1}\left( \lambda ^{i}v_{t}\right) \) and the sum of inverse functions for all agents, \(\sum _{i}\tilde{y}_{t}^{i}(v_{t}): \mathbb {R}_{++}\rightarrow \mathbb {R}_{++}\), are also standard. Function \( v_{t}\left( y\right) \) that assigns corresponding Lagrangian multiplier \( v_{t}\) to each \(y\in \mathbb {R}_{++}\) is implicitly defined by equation \( \sum _{i}\tilde{y}_{t}^{i\ }\left( v_{t}\right) =y\) and thus is an inverse of a standard function and is standard as well (property 2). Finally, the first-order optimization condition for problem (11) implies equality \(u_{t}^{i\prime }\left( y_{t}^{i}\left( y\right) \right) /\lambda ^{i}=v_{t}\left( y\right) \) for any i (property 3). \(\square \)

Lemma 3

For any profile \(c\in \mathcal {P}\), there exists a stochastic process \(\zeta ^{c}\in \mathcal {C}_{++}\) which satisfies \(\zeta _{t}^{c}\overset{a.s.}{=}u_{t}^{\prime i}\left( c_{t}^{i}\right) /\lambda ^{i}\) for any i and \(t\ge 1\).

Proof of Lemma 3

We first argue by contradiction that for any policy \(c\in \mathcal {P}\), in each period \( t\ge 1\), the profile of random variables \(\left\{ c_{t}^{i}\right\} _{i}\) solves

$$\begin{aligned} \max _{\{\hat{c}_{t}^{i}\}}E\sum _{i=1}^{I}u_{t}^{i}\left( \hat{c} _{t}^{i}\right) /\lambda ^{i}:\sum _{i}\hat{c}_{t}^{i}\le C_{t}\text { and } \hat{c}_{t}^{i}\ge 0\quad {\text {for all }}i. \end{aligned}$$
(12)

where \(C_{t}\equiv \sum _{i}c_{t}^{i}\). Suppose not. There exist nonnegative random variables \(\left\{ c_{t}^{i\prime }\right\} _{i,t\ge 1}\) such that \( \sum _{i}c_{t}^{i\prime }\le C_{t}\) for all \(t\ge 1\) and

$$\begin{aligned} \sum _{t\ge 1}E\sum _{i=1}^{I}u_{t}^{i}\left( c_{t}^{i\prime }\right) \Big /\lambda ^{i}>\sum _{t\ge 1}E\sum _{i=1}^{I}u_{t}^{i}\left( c_{t}^{i}\right) \Big /\lambda ^{i}. \end{aligned}$$
(13)

For any \(i\ge 2\) define

$$\begin{aligned} c_{0}^{i\prime }\equiv E\sum _{t=1}^{T}u_{t}^{i}\left( c_{t}^{i}\right) \Big /\lambda ^{i}-E\sum _{t=1}^{T}u_{t}^{i}\left( c_{t}^{i\prime }\right) /\lambda ^{i}+c_{0}^{i}, \end{aligned}$$
(14)

and let \(c_{0}^{1\prime }\equiv C_{0}-\sum _{i\ge 2}c_{0}^{i\prime }\). Equation (14) implies \(U^{i}\left( c^{\prime }\right) =U^{i}\left( c\right) \) for all \(i\ge 2\). For agent \(i=1\), Eq. (13) can be rewritten as

$$\begin{aligned} E\sum _{t\ge 1}u_{t}^{1}\left( c_{t}^{1\prime }\right) /\lambda ^{1}-E\sum _{t\ge 1}u_{t}^{1}\left( c_{t}^{1}\right) /\lambda ^{1}> & {} \sum _{i=2}^{I}E\sum _{t\ge 1}\left[ u_{t}^{i}\left( c_{t}^{i}\right) /\lambda ^{i}-u_{t}^{i}\left( c_{t}^{i\prime }\right) /\lambda ^{i}\right] \\= & {} \sum _{i\ge 2}c_{0}^{i\prime }-\sum _{i\ge 2}c_{0}^{i}=c_{0}^{1}-c_{0}^{1\prime }. \end{aligned}$$

Rearranging terms gives \(U^{1}\left( c^{\prime }\right) >U^{1}\left( c\right) \), and hence, c is not efficient, which is a contradiction.

In problem (12), constraints are independent across states, and the value of program (12) cannot exceed state-by-state solution \(\{y_{t}^{i}\left( C_{t}\right) \}_{i}\) , where functions \(\left\{ y_{t}^{i}(\cdot )\right\} _{i}\) are argmax functions as defined in Lemma 2 (property 1). By the same lemma (property 3), for this solution marginal rates of substitution coincide for all agents, \(u_{t}^{i\prime }\left( y_{t}^{i}\left( C_{t}\right) \right) /\lambda ^{i}=v_{t}\left( C_{t}\right) \) in each state. By the uniqueness of the solution for any y,  a profile of nonnegative random variables \(\left\{ c_{t}^{i}\right\} _{i}\), in state \( \omega \) for which \(\{c_{t,\omega }^{i}\}_{i}\ne \left\{ y_{t}^{i}\left( C_{t,\omega }\right) \right\} _{i}\), necessarily gives \(\sum _{i}\left( \lambda ^{i}\right) ^{-1}u^{i}\left( c_{t,\omega }^{i}\right) <\sum _{i}\left( \lambda ^{i}\right) ^{-1}u^{i}\left( y_{t}^{i}\left( C_{t,\omega }\right) \right) \). It follows that \(\left\{ c_{t}^{i}\right\} _{i}\) solves (12) if and only if the probability measure of states for which \(\{c_{t}^{i}\}_{i}\) and \(\left\{ y_{t}^{i}\left( C_{t}\right) \right\} _{i}\) diverge is zero. Define stochastic process \( \zeta ^{c}=\left\{ \zeta _{t}^{c}\right\} _{t\ge 0}\) as \(\zeta _{1}^{c}\equiv 1\) and \(\zeta _{t}^{c}\equiv v_{t}\left( C_{t}\right) \) for \( t\ge 1\). From the former observations, it follows that for any policy \(c\in \mathcal {P}\), the corresponding process satisfies \(\zeta _{t}^{c}\overset{ a.s.}{=}u_{t}^{\prime i}\left( c_{t}^{i}\right) /\lambda ^{i}\) for all i and \(t\ge 1\). \(\square \)

Proof of Lemma 1

Consider policy c and let \(\zeta ^{c}\) be the corresponding implicit prices. Let \( U^{i\prime }\in \mathbb {R}\) be an arbitrary level of utility. By allocative efficiency and Lemma 3, one has \(\zeta _{t}^{c} \overset{a.s.}{=} u_{t}^{i\prime }\left( c_{t}^{i}\right) /\lambda ^{i}\) for all i and \(t\ge 1\). This along with condition \(c_{0}^{i\prime }\equiv \left[ U^{i\prime }-U^{i}\left( c^{i}\right) \right] /\lambda ^{i}+c_{0}^{i}\) gives necessary and sufficient conditions for cost minimization in the problem

$$\begin{aligned} \min _{\hat{c}^{i}\in \mathcal {C}_{++}}\hat{c}_{0}^{i}+\sum _{t}E\zeta _{t}^{c} \hat{c}_{t}^{i}:U^{i}\left( \hat{c}^{i}\right) \ge {U}^{i\prime }, \end{aligned}$$

Observe that for prices \(\zeta ^{c}\) cost minimizing consumption \(c_{t}^{i}\) in periods \(t\ge 1\) is independent from \(U^{i\prime }\). The expenditure function is given by \(e^{i}(\zeta ^{c},U^{i\prime })={c}_{0}^{i\prime }+\sum _{t}E\zeta _{t}^{c}c_{t}^{i}\). Plugging this into the equivalent variation formula (1) gives

$$\begin{aligned} EV_{c,c^{\prime }}^{i}=\left[ U^{i}\left( c^{i\prime }\right) -U^{i}\left( c^{i}\right) \right] \Big /\lambda ^{i}=E\sum _{t\ge 1}\left[ \frac{ u_{t}^{i}\left( c_{t}^{i\prime }\right) }{\lambda ^{i}}-\frac{ u_{t}^{i}\left( c_{t}^{i}\right) }{\lambda ^{i}}\right] +c_{0}^{i\prime }-c_{0}^{i}. \end{aligned}$$

Aggregate equivalent variation \(EV_{c,c^{\prime }}\equiv \sum _{i}EV_{c,c^{\prime }}^{i}\) is then given by

$$\begin{aligned} EV_{c,c^{\prime }}= & {} E\sum _{t\ge 1}\sum _{i}\left[ \frac{u_{t}^{i}\left( c_{t}^{i\prime }\right) }{\lambda ^{i}}-\frac{u_{t}^{i}\left( c_{t}^{i}\right) }{\lambda ^{i}}\right] +\sum _{i}\left[ c_{0}^{i\prime }-c_{0}^{i}\right] =\nonumber \\= & {} E\sum _{t\ge 1}\left[ u_{t}\left( C_{t}^{\prime }\right) -u_{t}\left( C_{t}\right) \right] +C_{0}^{\prime }-C_{0}, \end{aligned}$$
(15)

where in the second inequality we used the fact that, by Lemma 3, policy \(c\in \mathcal {P}\) almost surely solves problem (10) subject to the aggregate consumption constraint for any \( t\ge 1\). The surplus function defined as \(S(c)\equiv \sum _{i}c_{0}^{i}+E\sum _{t\ge 1}u_{t}\left( \sum _{i}c_{t}^{i}\right) \) satisfies, for any pair of policies, \(EV_{c,c^{\prime }}=S(c^{\prime })-S\left( c\right) \). Thus, inequality \(EV_{c,c^{\prime }}\le 0\) holds if and only if \(S\left( c\right) \ge S\left( c^{\prime }\right) \) and surplus function \(S:\mathcal {P}\rightarrow \mathbb {R}\) represents social preferences. It follows that preferences \(\succeq _{*}\) are complete and transitive. Finally, suppose policy c weakly dominates \(c^{\prime }\) in Pareto sense. Then \(U^{i}(c^{i})\ge U^{i}(c^{i\prime })\) for each i and, by monotonicity of the expenditure function in \(U^{i\prime },\) one has \( EV_{c,c^{\prime }}^{i}\le 0\). This implies that \(EV_{c,c^{\prime }}\le 0\) and hence \(c\succeq _{*}c^{\prime }\), i.e., social preferences are Paretian. \(\square \)

Proof of Theorem 1

We first give two lemmas and then prove the theorem. \(\square \)

Lemma 4

The factual pricing kernel in period zero is \(\zeta _{t}^{ \bar{c}}\equiv \left\{ v_{t}(\bar{C_{t}})\right\} _{t=1}^{T}.\)

Proof of Lemma 4

According to Assumption 1 factual consumption \(\bar{c}\) is efficient, and autarky is an equilibrium in financial markets. For any cash flow \(x\in \mathcal {C}\), the first-order optimality condition and no-trade condition jointly imply that

$$\begin{aligned} p_{x}=x_{0}+\sum _{t\ge 1}E\frac{u_{t}^{i\prime }(\bar{c}_{t}^{i})}{\lambda ^{i}}x_{t}=x_{0}+E\sum _{t\ge 1}v_{t}(\bar{C}_{t})x_{t}, \end{aligned}$$
(16)

where the second equality follows from property 3 in Lemma 2, and the fact that the policy almost surely solves problem (10) for any \(t\ge 1\) given aggregate consumption. \(\square \)

Lemma 5

For any \(y>0\), value \(v_{t}(y)\) can be inferred from prices of a \(\bar{C}\)-spanning collection of assets if \(y\in \bar{S}_{t}\).

Proof of Lemma 5

For any \( Y\subset \mathbb {R}\) let \(\Omega _{Y}\equiv \left\{ \omega \in \Omega |\bar{W }_{t,\omega }\in Y\right\} \). For factual consumption in period t, aggregate consumption support is defined as the closure \(\bar{S}_{t}\equiv cl(\tilde{S}_{t}) \), where set \(\tilde{S}_{t}\) is given by

$$\begin{aligned} \tilde{S}_{t}\equiv \left\{ y\in \mathbb {R}|\text { for any }\varepsilon>0 \text {, probability }\pi (\Omega _{\left( y-\varepsilon ,y+\varepsilon \right) })>0\right\} . \end{aligned}$$

Consider an atom, i.e., a value \(y\in \mathbb {R}\) for which \(\pi (\Omega _{\{y\}})>0\). From a \(\bar{W}\)-spanning collection of assets, one can construct a portfolio with a dividend process x that pays \(x_{t,\omega }=1\) in period t and states \(\omega \in \Omega _{\left\{ y\right\} }\), and zero otherwise. Let \(p_{x}\) be the observable price. By Lemma 4, the price is given by

$$\begin{aligned} p_{x}=Ex_{t}v_{t}(\bar{W}_{t})=v_{t}\left( y\right) [\bar{F}_{t}\left( y\right) -\lim _{y^{\prime }\uparrow y}\bar{F}_{t}\left( y^{\prime }\right) ], \end{aligned}$$

where \(\bar{F}_{t}\) is the cdf of factual aggregate consumption. Since \(\bar{ F}_{t}\left( y\right) -\lim _{y^{\prime }\uparrow y}\bar{F}_{t}\left( y^{\prime }\right) >0\), ratio

$$\begin{aligned} v_{t}\left( y\right) =\frac{p_{x}}{\bar{F}_{t}\left( y\right) -\lim _{y^{\prime }\uparrow y}\bar{F}_{t}\left( y^{\prime }\right) }, \end{aligned}$$

is well defined. Price \(p_{x}\) and cdf \(\bar{F}_{t}\) are known to the analyst, and hence, index \(v_{t}\left( y\right) \) is identified.

Next, consider any \(y\in \mathbb {R}\) for which cdf \(\bar{F}_{t}\) is differentiable, and hence, density \(\bar{f}_{t}(y)\equiv \bar{F}_{t}^{\prime }(y)\) is well defined. For any \(\Delta >0\), construct a portfolio with a dividend cash flow x that pays \(x_{t,\omega }=1/\Delta \) in period t and states \(\omega \in \Omega _{(y-\Delta ,y]}\), and zero otherwise. The probability measure of this set is \(\pi (\Omega _{(y-\Delta ,y]})=\bar{F} _{t}(y)-\bar{F}_{t}(y-\Delta )\). Let \(p_{x}(\Delta )\) denote the observed market price. Since for all \(\omega \in \Omega _{(y-\Delta ,y]}\) inequality \( v_{t}(y-\Delta )\ge v_{t}(\bar{C}_{t,\omega })\ge v_{t}(y)\) holds, by Lemma 4 the price of the cash flow x satisfies inequality

$$\begin{aligned} \frac{\pi (\Omega _{(y-\Delta ,y]})}{\Delta }v_{t}(y-\Delta )\ge p_{x}(\Delta )\ge \frac{\pi (\Omega _{(y-\Delta ,y]})}{\Delta }v_{t}(y). \end{aligned}$$

By assumption, \(\bar{F}_{t}\) is differentiable at y, and limit \( \lim _{\Delta \rightarrow 0}\frac{\pi (\Omega _{(y-\Delta ,y]})}{\Delta }= \bar{f}_{t}(y)\) is well defined and the limit price is

$$\begin{aligned} \lim _{\Delta \rightarrow 0}p_{x}(\Delta )=p_{x}(0)=\bar{f}_{t}(y)v_{t}(y). \end{aligned}$$
(17)

Equation (17) shows that value \( v_{t}(y)=p_{x}(0)/\bar{f}_{t}(y)\) is identified if density is nonzero, \( \bar{f}_{t}(y)\ne 0\).

By assumption, the cdf of a random vector \(\left\{ \bar{c}_{t}^{i}\right\} _{i}\) is differentiable except for a countable set of realizations, and so is the cdf of aggregate consumption \(\bar{C}_{t}=\sum _{i}\bar{c}_{t}^{i}\). It follows that density \(\bar{f}_{t}(y)\equiv \bar{F}^{\prime }\left( y\right) \) is well defined for all y except a countable set. Consider any \( y\in \tilde{S}_{t}\). We next argue that for any \(\varepsilon >0\), there exists \(y^{\prime }\in \left( y-\varepsilon ,y+\varepsilon \right) \) for which \(y^{\prime }\) is either an atom or for which density is nonzero \(\bar{ f}_{t}(y^{\prime })>0\), and so \(v_{t}(y^{\prime })\) is identified. Suppose not. Interval \(\left( y-\varepsilon ,y+\varepsilon \right) \) can be partitioned into a countable collection of points for which \(\bar{F}_{t}\) is not differentiable, each with zero mass (no atoms), and a countable collection of open subintervals between these points, each satisfying \(\bar{f }_{t}(y)=0\) for all y in the subinterval and with zero mass. Thus, interval \(\left( y-\varepsilon ,y+\varepsilon \right) \) admits a countable partition where each partition element has zero-probability mass, implying that \(\pi (\Omega _{\left( y-\varepsilon ,y+\varepsilon \right) })=0\), a contradiction to \(y\in \tilde{S}_{t}\).

From the previous observations, it follows that for any \(y\in \tilde{S}_{t}\) there exits a sequence \(\left\{ y_{n}\right\} _{n}\) that converges to y, such that values \(\left\{ v_{t}(y_{n})\right\} _{n=0}^{\infty }\) are identified. By continuity of \(v_{t}\), the sequence converges to limit \( v_{t}(y)\) which can be inferred by an analyst, and value \(v_{t}(y)\) is identified. Finally support \(\bar{S}_{t}=cl(\tilde{S}_{t})\) is defined as the union of all limit points for all convergent sequences in \(\tilde{S}_{t}\) , and the analogous identification argument extends for the entire support \( \bar{S}_{t}\).\(\square \)

Before concluding the proof of the theorem, we make several observations. Let \(\left\{ v_{t}\right\} _{t\ge 1}\) be data generating indices. From Lemma 5, it follows that restrictions \(\{v_{t}|_{ \bar{S}_{t}}\}_{t\ge 1}\) are observable by the analyst (can be inferred from prices of \(\bar{W}\)-spanning securities). By Eq. (16), restrictions \(\{v_{t}|_{\bar{S}_{t}}\}_{t\ge 1}\) are also sufficient for factual prices of arbitrary cash flows, and hence, prices of assets beyond \(\bar{C}\)-spanning securities are redundant in normative inference. Let \(\left\{ v_{t}^{+},v_{t}^{-}\right\} _{t\ge 1}\) be bounds derived from \(\{v_{t}|_{\bar{S}_{t}}\}_{t\ge 1}\) and let \(\left\{ \tilde{v} _{t}\right\} _{t\ge 1}\) be a collection of arbitrary standard functions satisfying \(v_{t}^{+}\ge \tilde{v}_{t}\ge v_{t}^{-}\) for all \(t\ge 1\). Since on the support \(\tilde{v}_{t}|_{\bar{S}_{t}}=v_{t}^{+}|_{\bar{S} _{t}}=v_{t}|_{\bar{S}_{t}}\), indices \(\left\{ \tilde{v}_{t}\right\} _{t\ge 1}\) give rise to the same pricing kernel, a single-agent mechanism with preferences \(U^{1}(c^{1})=c_{0}^{1}+E\sum _{t=1}^{T}\int _{0}^{c_{t}^{1}} \tilde{v}_{t}\left( y\right) \mathrm{d}y\) rationalizes the factual prices of assets. We next prove the sufficiency and the necessity of the full support condition.

(if) Suppose \(\bar{S}_{t}=\mathbb {R}_{+}\) for all \(t\ge 1\). By Lemma 5, index \(v_{t}\) can be inferred for the entire domain \(\mathbb {R}_{++}\) for all \(t\ge 1\). For any \(c,c^{\prime }\in \mathcal {P}\), equivalent variation (15) can be written as a sum of expected integrals of observable index \(v_{t}\) and formula (2) expresses a unique value of \(EV_{c,c^{\prime }}\) in terms of observables. Thus, social preferences are revealed for the universal set, \(\mathcal {P}\).

(only if) Fix data generating \(\left\{ v_{t}\right\} _{t\ge 1}\), and suppose for some \(t^{*}\ge 1\) one has \(\bar{S}_{t^{*}}\ne \mathbb {R }_{+}\). Complement \((\bar{S}_{t^{*}})^{Com}\) is open in \(\mathbb {R}_{+}\) and hence \((\bar{S}_{t^{*}})^{Com}\ne \left\{ 0\right\} \). Therefore, there exists a \(y>0\) and \(\varepsilon >0\) such that \(\left( y-\varepsilon ,y+\varepsilon \right) \subset (\bar{S}_{t^{*}})^{Com}\cap \mathbb {R} _{++}\).

Consider weakly decreasing functions \(\tilde{v}_{t^{*}}^{RA}:\mathbb {R} _{++}\rightarrow \mathbb {\bar{R}}_{+}\) and \(\tilde{v}_{t^{*}}^{RN}: \mathbb {R}_{++}\rightarrow \mathbb {\bar{R}}_{+}\) defined as follows. For all \(y^{\prime }\le y-\varepsilon \), the functions coincide with upper bound \( \tilde{v}_{t^{*}}^{RA}\left( y^{\prime }\right) =\tilde{v}_{t^{*}}^{RN}\left( y^{\prime }\right) \equiv v_{t^{*}}^{+}(y^{\prime })\), while for \(y^{\prime }\ge y+\varepsilon \) let \(\tilde{v}_{t^{*}}^{RA}\left( y^{\prime }\right) =\tilde{v}_{t^{*}}^{RN}\left( y^{\prime }\right) \equiv v_{t^{*}}^{-}(y^{\prime })\). On the interval \( (y-\varepsilon ,y+\varepsilon )\) function \(\tilde{v}_{t^{*}}^{RA}\left( y^{\prime }\right) \) is defined as

$$\begin{aligned} \tilde{v}_{t^{*}}^{RA}(y^{\prime })\equiv v_{t^{*}}^{+}(y)+\frac{ v_{t^{*}}^{-}(y)-v_{t^{*}}^{+}(y)}{2\varepsilon }(y^{\prime }-y+\varepsilon ). \end{aligned}$$

The index is strictly decreasing on the interval, and therefore, the function represents (locally) risk-averse preferences. Index \(v_{t^{*}}^{RN}\) on the considered interval exhibits constant marginal utility

$$\begin{aligned} \tilde{v}_{t^{*}}^{RN}(y^{\prime })\equiv \frac{v_{t^{*}}^{-}(y)+v_{t^{*}}^{+}(y)}{2}. \end{aligned}$$

and associated preferences are (locally) risk neutral.

Consider policies \(c,c^{\prime }\in \mathcal {P}\) that differ only in terms of consumption in period \(t^{*}\) and period zero. For policy c, aggregate consumption in \(t^{*}\) is deterministic, \(\bar{C}_{t^{*}}= \tilde{y}\), and for policy \(c^{\prime }\) consumption is a mean preserving spread; i.e., it takes two values \(\tilde{y}-\varepsilon \) and \(\tilde{y} +\varepsilon ,\) each with probability \(\frac{1}{2}\). Note that \(c,c^{\prime }\in \mathcal {P}\). Let

$$\begin{aligned} L\equiv \frac{1}{2}\left( \ \int \limits _{\tilde{y}-\varepsilon }^{\tilde{y}} \tilde{v}_{t^{*}}^{RA}(y)\mathrm{d}y-\int \limits _{\tilde{y}}^{\tilde{y} +\varepsilon }\tilde{v}_{t^{*}}^{RA}(y)\mathrm{d}y\right) , \end{aligned}$$

be the loss of surplus due to higher consumption variance under policy \( c^{\prime }\) in period \(t^{*}\), assuming risk-averse preferences. For policy c period zero consumption is then \(\bar{C}_{0}=0\), while for policy \(c^{\prime }\) it is \(C_{0}^{\prime }=L/2\).

For \(t\ne t^{*}\) let the two indices coincide with the data generating one, i.e., \(\tilde{v}_{t}^{RA}=\tilde{v}_{t}^{RN}=v_{t}\). Since \( C_{0}^{\prime }\) compensates for only half of the loss, an agent with utility \(U^{RA}(c^{i})=c_{0}^{i}+E\sum _{t=1}^{T}\int _{0}^{c_{t}^{i}}\tilde{v} _{t}^{RA}\left( y\right) \mathrm{d}y\) prefers a safe policy, i.e., \( U^{RA}(C)>U^{RA}(C^{\prime })\), while for risk-neutral utility \( U^{RN}(c)=c_{0}+E\sum _{t=1}^{T}\int _{0}^{c_{t}}\tilde{v}_{t}^{RA}\left( y\right) \mathrm{d}y\) the inequality is reversed, \(U^{RA}(C)<U^{RA}(C^{\prime })\). Observe that utility functions \(\tilde{v}_{t^{*}}^{RA}\) and \(\tilde{v} _{t^{*}}^{RA}\) are not continuously differentiable and hence are not standard. However, each of the two indices can be arbitrarily closely approximated by standard functions, such that \(v_{t}^{+}\ge \tilde{v} _{t}\ge v_{t}^{-}\), which preserve the inequalities in the utilities. A single-agent mechanism with such approximations rationalizes observed factual prices (for any t, indices \(\tilde{v}_{t}\) satisfy \(v_{t}^{+}\ge \tilde{v}_{t}\ge v_{t}^{-}\)), and in one mechanism the social ranking is \( c\succ c^{\prime }\), while in the other it is \(c^{\prime }\succ c\). Thus, prices of securities cannot reveal social preferences for set \(\mathcal {P}\) given \(\left\{ v_{t}\right\} _{t\ge 1}\). \(\square \)

Proof of Proposition 1

By strict monotonicity of \(v_{t}\left( y\right) \) in y, one has \(v_{t}^{+}\ge v_{t}\ge v_{t}^{-}\ge 0\) for all \(t\ge 1\), and hence, \(EV_{c,c^{\prime }}^{+}\ge EV_{c,c}\). Thus, \(EV_{c,c^{\prime }}^{+}\le 0\) implies \( EV_{c,c^{\prime }}\le 0\) and hence \(c\succeq _{*}c^{\prime }\). \(\square \)

Proof of Proposition 2

Fix a finite collection of policies \(\mathcal {P}^{F}\). Suppose for each \(t\ge 1\) set \( \bar{S}_{t}\) is \(\varepsilon \)-dense in \(\mathbb {R}_{++}\) where \( 0<\varepsilon <\min _{c\in \mathcal {P}^{F},t\ge 1}\inf S_{t}^{c}\). Since each \(S_{t}^{c}\) is bounded away from zero, such \(\varepsilon \) exists. For any t, define recursively a countable subset of \(\bar{S}_{t}\), denoted by \( \{y_{t}^{n}\}_{n=0}^{\infty }\) as follows. For \(n=0\), choose \(y_{t}^{0}\in \bar{S}_{t}\) for which \(0<y_{t}^{0}\le \varepsilon \). Given \(y_{t}^{n}\in \bar{S}_{t}\), let \(y_{t}^{n+1}\in [y_{t}^{n}+\varepsilon ,y_{t}^{n}+3\varepsilon ]\). By \(\varepsilon \)-denseness of \(\bar{S}_{t}\), such collection \(\{y_{t}^{n}\}_{n=0}^{\infty }\) exists. Moreover, by construction, for all elements in the collection the distance between any two elements is no smaller than \(\varepsilon \), while the distance between any two adjacent elements is no larger than \(3\varepsilon \). Consider observable statistic \(EV_{c,c^{\prime }}^{-}\) defined in Footnote 11. For any pair of policies \(EV_{c,c^{\prime }}^{-}\le EV_{c,c^{\prime }}\) and hence \(EV_{c,c^{\prime }}^{+}-EV_{c,c^{\prime }}\le EV_{c,c^{\prime }}^{+}-EV_{c,c^{\prime }}^{-}\). The difference between the two statistics is given by

$$\begin{aligned} EV_{c,c^{\prime }}^{+}-EV_{c,c^{\prime }}^{-}=\sum _{t=1}^{T}E\int \limits _{\min (C_{t},C_{t}^{\prime })}^{\max (C_{t},C_{t}^{\prime })}\left( v_{t}^{+}(y)-v_{t}^{-}(y)\right) \mathrm{d}y. \end{aligned}$$
(18)

Since index \(v_{t}^{+}\) is non-increasing and \(v_{t}^{-}\) is non-decreasing in the support size, the difference \(EV_{c,c^{\prime }}^{+}-EV_{c,c^{\prime }}^{-}\) is no larger than the analogous difference derived from a subset of the support, \(\{y_{t}^{n}\}_{n=0}^{\infty }\subset \bar{S}_{t}\). For the latter subset, the index bounds \(v_{t}^{+},v_{t}^{-}\) are given by step functions and, in period t and state \(\omega \), the difference in money metric welfare can be written as a sum of “rectangular” areas,

$$\begin{aligned} \int \limits _{\min (C_{t},C_{t}^{\prime })}^{\max (C_{t},C_{t}^{\prime })}\left( v_{t}^{+}(y)-v_{t}^{-}(y)\right) \mathrm{d}y\le \sum _{n=0}^{\infty }\left( y_{t}^{n+1}-y^{_{n}}\right) \times \left( v_{t}\left( y_{t}^{n}\right) -v_{t}\left( y_{t}^{n+1}\right) \right) . \end{aligned}$$
(19)

By construction, \(\left( y_{n+1}-y_{n}\right) \le 3\varepsilon \) and also \( \sum _{n=0}^{\infty }v_{t}\left( y_{t}^{n}\right) -v_{t}\left( y_{t}^{n+1}\right) =v_{t}\left( y_{t}^{0}\right) \). Thus, the inequality can be written as

$$\begin{aligned} \int \limits _{\min (C_{t},C_{t}^{\prime })}^{\max (C_{t},C_{t}^{\prime })}\left( v_{t}^{+}(y)-v_{t}^{-}(y)\right) \mathrm{d}y\le 3\varepsilon \sum _{n=0}^{\infty }\left( v_{t}\left( y_{t}^{n+1}\right) -v_{t}\left( y_{t}^{n}\right) \right) \le 3\varepsilon v_{t}\left( y_{t}^{0}\right) . \end{aligned}$$
(20)

Define positive scalar \(\alpha \equiv \max _{t\ge 1}|v_{t}(y_{t}^{0})|/3T<\infty \). Using this and (18) gives

$$\begin{aligned} EV_{c,c^{\prime }}^{+}-EV_{c,c^{\prime }}^{-}\le 3\varepsilon \sum _{t}v_{t}\left( y_{t}^{0}\right) \le \varepsilon \alpha . \end{aligned}$$
(21)

Fix \(\varepsilon =\frac{1}{2}\min _{c,c^{\prime }\in \mathcal {P} ^{F}}\left| EV_{c,c^{\prime }}\right| /\alpha \). Consider any pair \( c,c^{\prime }\in \mathcal {P}^{F}\). Observe that

$$\begin{aligned} EV_{c,c^{\prime }}^{+}-EV_{c,c^{\prime }}\le EV_{c,c^{\prime }}^{+}-EV_{c,c^{\prime }}^{-}\le \frac{1}{2}\min _{c,c^{\prime }\in \mathcal { P}^{F}}\left| EV_{c,c^{\prime }}\right| , \end{aligned}$$

and hence \(EV_{c,c^{\prime }}^{+}\le EV_{c,c^{\prime }}+\frac{1}{2} \left| EV_{c,c^{\prime }}\right| \) and it follows that inequality \( EV_{c,c^{\prime }}<0\) implies \(EV_{c,c^{\prime }}^{+}<0\). Since policies in \( \mathcal {P}^{F}\) are non-indifferent, \(EV_{c,c^{\prime }}\ne 0\), either \( EV_{c,c^{\prime }}^{+}<0\) or \(EV_{c^{\prime },c}^{+}<0\) holds and the preference relation for pair \(c,c^{\prime }\) is revealed. Since this is true for all pairs of policies in \(\mathcal {P}^{F}\), social preferences are revealed provided that supports \(\bar{S}_{t}\) are \(\varepsilon -\)dense. \(\square \)

Proof of Proposition 3

We use the well-known fact that an inverse of a strictly decreasing and convex bijection is convex, and hence, a subset of convex standard functions is closed under inversion and summation. Consider definitions used in the proof of Lemma 2. Observe that function \(\tilde{ y}^{i}(v_{t})\) is the inverse of a strictly decreasing and convex (standard) function \(u_{t}^{i\prime }\left( \cdot \right) \), and hence, it is strictly decreasing and convex. Thus, the sum \(\sum _{t}\tilde{y}_{t}^{i}(v_{t})\) is also strictly decreasing and convex. Finally function \(v_{t}\left( y\right) \) is an inverse of \(\sum _{t}\tilde{y}_{t}^{i}(v_{t})\), and hence, it is convex. \(\square \)

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Weretka, M. Normative inference in efficient markets. Econ Theory 68, 787–810 (2019). https://doi.org/10.1007/s00199-018-1144-6

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Keywords

  • Social preferences
  • Normative predictions
  • Asset prices

JEL Classification

  • D43
  • D53
  • G11
  • G12
  • L13