GPT 6 Astra Ultra attempt on Econ-69
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Imperfect information: a recursive benchmark and failure of posterior-mean closure
Problem:D83-67. Status: Partial results; the complete catalogue target remains unresolved. Problem and scope
The catalogue asks for equilibrium and an adequate recursive state with non-Gaussian signals and dispersed information. We give a finite-state public-information benchmark, then a common-prior counterexample to using only the posterior mean of productivity. Since even the public-information subcase defeats that proposed statistic, it cannot be universally sufficient in the larger family. The broader motivation is the imperfect-information research agenda surveyed by Mankiw and Reis [MR].A one-period competitive market
A unit mass of identical price-taking households has terminal endowment $e(X)=1+X$, where $X$ takes values in a finite subset of $[0,H]$. There is a zero-net-supply security paying $X$ and a riskless numeraire. At relative security price $p\in[0,H]$, a household chooses $a\in[-\bar a,\bar a]$ and consumes \[ c=1+X+a(X-p),\qquad 0<\bar a<1/(H+1). \] The restriction makes consumption strictly positive at every state and candidate price. Preferences are $\mathbb E_b[\log c]$ under the common posterior $b$. The initial security endowment is zero. The security purchase is financed in the numeraire; no other intertemporal transfer is permitted. A normalized two-price vector is $(1,p)/(1+p)$, so working with relative $p$ does not change the catalogue's price normalization.Theorem (Equilibrium price from the entire posterior): For every posterior $b$, the price \begin{equation}
p(b)=\frac{\mathbb E_b[X/(1+X)]}{\mathbb E_b[1/(1+X)]}\label{wsref-0-65-71-3a-70-72-69-63-65}
\end{equation} and allocation $a=0$ form a competitive equilibrium. The price is the only relative price that can support market clearing. If $b$ gives positive probability to two distinct values of $X$, each household's optimal holding at this price is uniquely zero.
Proof: The denominator is positive and the ratio is a weighted average of $X$, hence lies in $[0,H]$. Utility is differentiable and concave in $a$, with derivative at zero \[
\mathbb E_b\left[\frac{X-p}{1+X}\right].
\] Equation (eq:price) makes this derivative zero, so concavity makes zero optimal. The second derivative is $-\mathbb E_b[(X-p)^2/c^2]$, strictly negative when the payoff is nonconstant. Market clearing follows because all households hold zero. At any other price the derivative at zero is nonzero. Concavity and identical preferences then imply that every optimum has the same strict sign, contradicting zero aggregate holdings. In the degenerate payoff case $X=x$ almost surely, $p=x$ makes every holding optimal and $a=0$ clears; prices above or below $x$ again produce demands of a common strict sign. ∎
Proposition (Equal posterior means, different equilibrium prices): There is one finite common-prior economy with two positive-probability public signals, both implying $\mathbb E[X\mid Y]=1$, whose equilibrium prices are respectively $1$ and $1/2$.
Proof: Take prior probabilities $(1/4,1/2,1/4)$ on $X=0,1,2$. The public signal is $Y=A$ when $X=1$ and $Y=B$ otherwise. At $A$, the posterior is concentrated on $1$, giving price $1$. At $B$, it puts equal weight on $0$ and $2$. Both posterior means equal $1$, but at $B$ the numerator of (eq:price) is $1/3$ and its denominator is $2/3$, giving price $1/2$. Both signals are already public, so conditioning on the equilibrium price provides no additional information. The comparison therefore respects Bayes' rule and price observation; it is not based on incompatible subjective beliefs. ∎
A dynamic sufficient statistic in a restricted economy
Let $X_t$ be a finite-state Markov chain with known matrix $P$. Before the date-$t$ security market, all households see a finite-valued signal $Y_t$ with likelihood $\ell_y(x)$ conditional on $X_t=x$. Signals are conditionally independent given the state path. The period-$t$ market settles after trading, and its realized endowment and security payoff publicly reveal $X_t$. There is no saving across dates, so lifetime expected utility is the discounted sum of these period utilities. The actual pretrade public information is generated by $(Y_1,\ldots,Y_t,X_1,\ldots,X_{t-1})$. Only positive-probability observation histories are used in conditional formulas.Proposition (Exact finite-dimensional public state with settlement observations): The pretrade posterior vector \[
b_t(x)=\Pr(X_t=x\mid Y_1,\ldots,Y_t,X_1,\ldots,X_{t-1}),
\] together with calendar date for a finite horizon, determines the equilibrium price, aggregate allocation, and its own conditional next-period law. After settlement at date $t$, the posterior of $X_t$ becomes a point mass at the observed $X_t$. For a next signal $y$ define \[
K_y(x)=\sum_z P(x,z)\ell_y(z),\qquad
B_{x,y}(x')=\frac{P(x,x')\ell_y(x')}{K_y(x)}
\quad\text{when }K_y(x)>0.
\] The next pretrade posterior equals $B_{X_t,Y_{t+1}}$. Conditional on $b_t$, its law assigns mass $b_t(x)K_y(x)$ to $B_{x,y}$, summing masses when distinct pairs give the same posterior.
Proof: At settlement the common endowment $1+X_t$ reveals the current state, so the next prediction uses the row $P(X_t,\cdot)$, rather than an unupdated hidden-state filter. The Markov property and conditional independence then give $K_y$ and $B_{x,y}$ by Bayes' rule. Before settlement, $X_t$ has law $b_t$; multiplying by the conditional signal probability gives the claimed transition law, a function only of $b_t$. Period equilibrium is given by (eq:price) and zero holdings. Since controls cannot transfer resources across dates or affect states, signals or public settlement information, periodwise optimality maximizes the discounted sum. Prices are functions of the already public pretrade posterior and do not add information. This verifies all three sufficiency requirements while including the information conveyed by realized consumption and payments. ∎
Remaining gap
This construction permits non-Gaussian signals but removes dispersed private information, private saving, and endogenous information acquisition. Under private signals, the distribution of beliefs and beliefs about others may matter, and public prices can reveal information not present in each agent's signal history. The displayed finite filter cannot simply be assigned independently to each agent in such an economy. The counterexample establishes a specific obstruction to mean closure; it does not prove that every richer model requires an infinite-dimensional state.References
- [MR] N. G. Mankiw and R. Reis, Imperfect Information and Aggregate Supply, Handbook of Monetary Economics, Chapter 5; author manuscript, especially Sections 7.2, 7.4 and 7.5. https://personal.lse.ac.uk/reisr/papers/10-HofME.pdf. The market, calculations and counterexample here are self-contained benchmarks and are not attributed as results of that survey.