Non-implementability of Arrow–Debreu equilibria by continuous trading under volatility uncertainty

Abstract

In diffusion models, a few suitably chosen financial securities allow to complete the market. As a consequence, the efficient allocations of static Arrow–Debreu equilibria can be attained in Radner equilibria by dynamic trading. We show that this celebrated result generically fails if there is Knightian uncertainty about volatility. A Radner equilibrium with the same efficient allocation as in an Arrow–Debreu equilibrium exists if and only if the discounted net trades of the equilibrium allocation display no ambiguity in the mean. This property is violated generically in endowments, and thus Arrow–Debreu equilibrium allocations are generically unattainable by dynamically trading a few long-lived assets.

Appendices

Appendix A: Proofs of Sect. 3

We recall basic results for expected utility economies from [8]. For weights $\alpha ^{i} \ge 0$ and $x>0$, denote by $c_{\alpha }$ the maximizer of $\sum _{i\in \mathbb{I}} \alpha ^{i} u ^{i}(c^{i})$ over $\{c \in \mathbb{R}^{I}_{+}:\sum _{i\in \mathbb{I}} c ^{i} = x\}$. This so-called $\alpha$-efficient allocation $c_{\alpha }$ is characterized by the first-order conditions

$$ \alpha ^{i} \frac{d u^{i}}{d x} (c_{\alpha }^{i})= \alpha ^{j} \frac{d u ^{j}}{d x} (c_{\alpha }^{j}) = : \psi _{\alpha }(x) $$

(A.1)

for agents with strictly positive weights $\alpha ^{i}, \alpha ^{j}>0$; agents $i$ with weight $\alpha _{i}=0$ have $c_{\alpha }^{i}(x)=0$, of course. These equations define implicitly $c_{\alpha }^{i}$ and $\psi _{\alpha }$ as continuous functions of $x>0$ (see [8] for the details).

Set $\Delta =\{ \alpha \in \mathbb{R}^{I}_{+}: \sum _{i} \alpha _{i} =1 \}$ and denote by $\mathbb{O}=(c_{\alpha }(e))_{\alpha \in \Delta }$ the set of efficient allocations in $\mathcal{E}^{P}$. The definition of $\mathbb{O}$ does not depend on the particular $P\in {\mathcal{P}}$ as the $\alpha$-efficient allocations are defined pointwise.

Our strategy of proof is to pick an equilibrium in $\mathcal{E}^{P}$ for an arbitrary $P \in {\mathcal{P}}$ and to show that this equilibrium is an equilibrium in ℰ. In general, equilibrium allocations in $\mathcal{E}^{P}$ are determined only $P$-almost surely. For ℰ, we need, however, that market clearing occurs quasi-surely. The allocations $c_{\alpha }$ in (A.1) are defined pointwise for all $\omega \in \varOmega$. In particular, we have $\sum _{i\in \mathbb{I}} c_{\alpha }^{i}(\omega )= e(\omega )$ for all $\omega$, hence also quasi-surely.

Proof of Theorem 3.1

We first show that the allocations $c_{\alpha }\in \mathbb{O}$ belong to our commodity space ℍ. From $0 \le c_{\alpha }^{i} \le e$, we see that $c_{\alpha }^{i}$ is quasi-surely bounded. As $c_{\alpha }^{i}$ can be written as a continuous function of the aggregate endowment $e$, $c_{\alpha }^{i}$ is also quasi-continuous. Under Assumption 2.5, $e$ is ambiguity-free; as a continuous function of $e$, $c_{\alpha }^{i}$ is also ambiguity-free.

Pick any $P \in {\mathcal{P}}$. Due to our Assumption 2.3, the Assumptions (i)–(iv) in Dana [8] are satisfied. For Assumptions (i) (strict concavity and monotonicity), (ii) (twice continuous differentiability) and (iv) (Inada condition), this is immediate. For Assumption (iii), note that our Bernoulli utility functions are independent of the state $\omega$; by concavity, they are bounded by some linear function. Hence Assumption (iii) of Dana is also satisfied. Since the endowments are bounded away from zero by Assumption 2.3, Assumption (E) in Dana [8] is also satisfied. We can thus apply Theorem 2.5 of Dana [8]: There exist an $\alpha \in \Delta$ and an equilibrium $(\varPsi , c)$ in $\mathcal{E}^{P}$ with $c=c_{\alpha }$ $P$-a.s. and $\varPsi (X)= E^{P}[ \psi _{\alpha }X]$ for $X \in L^{\infty }(P)$. The state price $\psi _{\alpha }$ is a continuous function of $e$, and hence bounded q.s. By (A.1), $\psi _{\alpha }$ is strictly positive. Due to Assumption 2.3, the individual endowments $e^{i}$ are bounded away from zero quasi-surely. Hence we have $\varPsi (e^{i})> 0$ for all $i \in \mathbb{I}$. As a consequence, $\alpha ^{i}>0$ since otherwise $c_{\alpha }^{i}=0$ would be dominated by some strictly positive consumption plan (e.g. choose $x >0$ with $E^{P}[ \psi _{\alpha }x] = \varPsi (e^{i})$. By Assumption 2.3, ${U^{i}(x)=u^{i}(x)}>u ^{i}(0)$).

We now claim that $(\varPsi , c)$ is an equilibrium in ℰ. Note that $\varPsi$ is well defined on $\mathbb{H} \subseteq L^{\infty }(P)$. Since $\sum _{i \in \mathbb{I}} c_{\alpha }^{i} (\omega ) = e(\omega )$ for all $\omega$, $c_{\alpha }$ clears the market for every $\omega \in \varOmega$, hence quasi-surely. The budget constraint $\varPsi (c^{i})=\varPsi (e^{i})$ is satisfied because $(\varPsi , c)$ is an Arrow–Debreu equilibrium in $\mathcal{E}^{P}$. It remains to show that $c^{i}_{\alpha }$ maximizes utility in ℰ subject to the budget constraint. Let $b$ be budget-feasible for agent $i$. As $c_{\alpha }$ is an Arrow–Debreu equilibrium in the expected utility economy $\mathcal{E}^{P}$, we have $E^{P}[ u^{i}(c_{\alpha }^{i})] \ge E^{P} [u^{i}(b)]$. As $c_{\alpha }^{i}$ is ambiguity-free, we have $U^{i}(c_{\alpha }^{i})=E^{P} [u^{i}(c_{\alpha }^{i})]$. Therefore,

$$ U^{i}(b) \le E^{P} [u^{i}(b)] \le E^{P} [u^{i}(c_{\alpha }^{i})] = U ^{i}(c_{\alpha }^{i}). $$

 □

As a preparation for the proof of Theorem 3.2, we now show that the allocations in $\mathbb{O}$ are also efficient in the Knightian economy ℰ. This is not obvious as in general, different measures in $\mathcal{P}$ could be the “worst case” measure for different agents, and efficient allocations would thus depend on those worst-case measures. Our result hinges on Assumption 2.5.

Proposition A.1

1. Under Assumption 2.5, the efficient allocations in the Knightian economy ℰ coincide with the allocations in $\mathbb{ O} =(c_{\alpha })_{\alpha \in \Delta }$. Each $c_{\alpha }$ is ambiguity-free.

2. Under Assumption 2.6, each $c_{\alpha }$ is full insurance.

Proof

Let $b\in \mathbb{O}$ be an efficient allocation. It is well known that $b$ maximizes the weighted sum of utilities $\sum _{i\in \mathbb{I}} \alpha ^{i} U^{i}(b^{i})$ for some $\alpha \in \Delta$; compare [8, Proposition 2.3]. Set $\varGamma = \{ \omega \in \varOmega : \exists i \in \mathbb{I}\mbox{ with }b^{i}(\omega ) \neq c^{i} _{\alpha }(\omega )\}$. As the Bernoulli utility functions $u^{i}$ are strictly concave by Assumption 2.3 and by the definition of $c_{\alpha }$, we have

$$ \sum _{i\in \mathbb{I}} \alpha ^{i} u^{i}\big(b^{i}(\omega )\big) < \sum _{i\in \mathbb{I}} \alpha ^{i} u^{i}\big(c_{\alpha }^{i}(\omega ) \big) $$

for all $\omega \in \varGamma$. If $\varGamma$ is not a polar set, there is $P \in {\mathcal{P}}$ with $P[\varGamma ]>0$ and we have

$$ E^{P}\bigg[ \sum _{i\in \mathbb{I}} \alpha ^{i} u^{i}(b^{i})\bigg] < E ^{P} \bigg[\sum _{i\in \mathbb{I}} \alpha ^{i} u^{i}(c_{\alpha }^{i}) \bigg] . $$

Since $c_{\alpha }$ is ambiguity-free, $E^{P}[ \sum _{i\in \mathbb{I}} \alpha ^{i} u^{i}(c_{\alpha }^{i})] = \sum _{i\in \mathbb{I}} \alpha ^{i} U^{i}(c_{\alpha }^{i})$. On the other hand, by ambiguity aversion,

$$ \sum _{i\in \mathbb{I}} \alpha ^{i} U^{i}(c_{\alpha }^{i})\le E^{P} \bigg[\sum _{i\in \mathbb{I}} \alpha ^{i} u^{i}(b^{i})\bigg], $$

and we obtain a contradiction. We thus conclude that $\varGamma$ is a polar set and therefore $b=c_{\alpha }$ quasi-surely.

From Proposition 2.2 of Dana [8] (see also the discussion above), we know that $c_{\alpha }$ is a continuous function of the aggregate endowment $e$. Under Assumption 2.5, $c_{\alpha }$ is thus ambiguity-free, and under Assumption 2.6, $c_{\alpha }$ is quasi-surely constant, or full insurance. □

Proof of Theorem 3.2

Let $(\varPsi , c)$ be an Arrow–Debreu equilibrium of ℰ. By the first welfare theorem, $c$ is efficient. By Proposition A.1, there exists $\alpha \in \Delta$ with $c=c_{\alpha }$ and $c_{\alpha }$ is ambiguity-free. This proves part 1 (a). If we impose even Assumption 2.6, Proposition A.1 shows that $c_{\alpha }$ is full insurance, proving part 2 (a).

For part 1 (b), note that we have $\alpha ^{i}>0$ for all $i \in \mathbb{I}$, as individual endowments are strictly positive. Otherwise, $c^{i}_{\alpha }=0$ which is dominated by the strictly positive individual endowment $e^{i}$ (Assumption 2.3). Due to the first order condition of individual utility maximization, any equilibrium price functional $\varPsi$ is collinear with some supergradient of $U^{i}$ at $c^{i}_{\alpha }$. For any $i\in \mathbb{I}$, the set of supergradients contains all linear functionals of the form $\frac{d }{d x} u^{i}(c^{i}_{\alpha })\cdot P$, where $P$ is a minimizer in the set of priors. Since $u^{i}(c^{i}_{\alpha })$ is ambiguity-free, $E^{P}[u^{i}(c^{i}_{\alpha })]$ is constant on $\mathcal{P}$, and hence every element in $\mathcal{P}$ is a minimizer of the multiple prior expected utility. From (A.1), $\alpha ^{i}\frac{d }{d x} u^{i}(c^{i}_{\alpha })=\psi _{\alpha }$, which is independent of $i$ and ambiguity-free by part 1 (a).

By Assumption 2.3, the marginal utilities $\frac{d }{d x} u^{i}$ are continuous and decreasing. Since $c_{\alpha }$ is a continuous function of the aggregate endowment $e$ and since $e$ is bounded away from zero, $\psi _{\alpha }$ is quasi-continuous and bounded. Since $e$ is also bounded and the marginal utilities $\frac{d }{d x} u^{i}$ are decreasing, $\psi _{\alpha }$ is also bounded away from zero. We thus obtain a price functional of the form

$$ \varPsi (b)=E^{P}[\psi _{\alpha }b] $$

such that $(c,\varPsi )$ is an Arrow–Debreu equilibrium.

For part 2 (b), note that $\psi _{\alpha }$ is constant since $c_{\alpha }$ is full insurance. Hence, without loss of generality, we can replace the price functional $\varPsi (b)=E^{P} [\psi _{ \alpha } b]$ by $\varPsi (b)=E^{P}[b]$. □

Appendix B: Beyond the Bachelier model

The Bachelier model we presented allows negative values of the price process. Theorem 4.2 is still valid when our $G$-Brownian motion $B=B^{+} +B^{-}$ of the Bachelier model is decomposed into the positive part $B^{+}$ and negative part $B^{-}$. The trading strategies are then given by $\theta ^{k,+}_{t}=\theta ^{k}_{t}1_{\{B^{k}_{t} \geq 0\}}$ and $\theta ^{k,-}_{t}= -\theta ^{k}_{t}1_{\{B^{k}_{t}< 0\}}$, where $\theta ^{k}$ denotes the fractions invested in the $k$th uncertain assets of Theorem 4.2. In the same fashion, as mentioned in Sect. 5 of Duffie and Huang [17], the number of assets becomes $2 d +1$.

Theorem 4.2 is still valid if we replace the process $B$ with a symmetric $\mathbb{E}$-martingale of the form $M=M_{0}+ \int _{0}^{\cdot }V_{t} \mathrm{d}B_{t}$ such that $V_{t} \in \mathbb{H}^{d\times d}_{+}$ with $V^{ij}_{t}=0$ and $V^{ii}$ is q.s. bounded away from zero. Indeed, it suffices to show that every stochastic integral of the form $\int _{0}^{T} \theta _{s} \mathrm{d}B_{s}$ for some $\theta \in \mathcal{M}$ can be written as $\int _{0}^{T} \theta _{s} \mathrm{d} B_{s}=\int _{0}^{T} \theta ^{M}_{s} \mathrm{d}M_{s}$ for some suitable $\theta ^{M}\in \mathcal{M}$. The proof then follows the same lines as the proof of Theorem 4.2, by substituting $B$ with $M$. The obvious candidate is $\theta ^{M,k}_{t}= \theta ^{k}_{t} / V^{kk}_{t}$. We need to show that the stochastic integral $\int _{0}^{T} \theta ^{M}_{t} dM _{t}$ is well defined. Note that $\theta ^{M}\in \mathcal{M}$ because $V^{kk}$ is bounded away from zero q.s.