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Econ-49 · economics · Economics

Sequential and recursive equilibrium with a distributional state

LLM status Open unverified

Statement

MA-02 |  Heterogeneous-agent macroeconomics

Published historical question and precise equivalence programme

Definitions and assumptions

Use the incomplete-market economy in MA-01. A sequential equilibrium specifies adapted prices and household consumption/savings plans at every aggregate and individual shock history. Households maximize expected discounted lifetime utility subject to their intertemporal budgets and borrowing bound, and markets clear at every relevant aggregate history. A recursive equilibrium uses the common aggregate state $s_t=(z_t,\mu_t)$ and individual state $(a_t,e_t)$, with the Bellman optimality and distribution-consistency conditions of MA-01.

A sequential equilibrium is representable on this reduced state if its prices, household policies, and conditional continuation values factor through these states: there exist measurable functions $R,w,g,V$ such that the corresponding sequential objects equal these functions at every reachable state, up to probability-null sets. The aggregate transition must also factor through $(z_t,\mu_t,z_{t+1})$. A recursive representation must specify an invariant state domain and consistent continuation choices, rather than only values along one realized path.

Formal question

Characterize conditions under which a sequential equilibrium has such a recursive representation, and under which a recursive equilibrium generates a sequential optimum. Establish an equivalence theorem for a clearly specified equilibrium correspondence, including admissible initial distributions and any needed transversality or uniform-integrability conditions. Determine when auxiliary individual expected continuation values are redundant and when a continuation selection cannot be made as a function of $(z,\mu)$ alone.

Scope and status

Equilibrium existence on the reduced state and representation of every sequential equilibrium are distinct demands. Histories with the same $(z,\mu)$ may support different continuation selections, so Markov dependence cannot simply be assumed. The question retains the entire wealth--efficiency distribution and concerns removal only of the auxiliary continuation coordinate. The source raises the relation as open in 2009; the present formulation separates the logical directions without asserting that all later work leaves them unresolved.

References for the source of the problem

Heathcote, Storesletten, and Violante, Quantitative Macroeconomics with Heterogeneous Households, NBER Working Paper 14768 (2009), https://www.nber.org/system/files/working_papers/w14768/w14768.pdf; verified published version, Annual Review of Economics 1 (2009), Section 5.2, footnote 20, p. 342, https://doi.org/10.1146/annurev.economics.050708.142922. The footnote discusses Jianjun Miao, Competitive equilibria of economies with a continuum of consumers and aggregate shocks, Journal of Economic Theory 128(1) (2006), 274--298, as the earlier enlarged-state result.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Econ-49-A2 · Exploration · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-49

C62-48: Sequential versus reduced-state recursive equilibrium

Outcome: UNRESOLVED as a general programme; a conditional equivalence theorem and a selection obstruction proved.

Tools: web browsing available; Python computation available. Literature checked on 9 October 2026.

1) FORMAL RESTATEMENT

The uploaded question asks which sequential equilibria of the economy called "MA-01" admit representation on the aggregate state $s=(z,\mu)$ and household state $(a,e)$, with prices, policies, continuation values, and the distribution transition all measurable functions of these states. The state $\mu$ is the full cross-sectional distribution, not a finite list of moments.

Missing specification. The nine supplied standalone files do not define MA-01. The minimal correction is to supply utility, production, shock kernels, constraints, initial distributions and transversality, together with the continuation domain. Below we state a bounded Markov specification, not a reconstruction of MA-01.

Quantified representation property. For each relevant aggregate history $h$, let $s(h)=(z(h),\mu(h))$. Let $P_h$ denote the equilibrium price vector and $g_h(a,e)$ the savings policy, defined for every individual state under consideration. A representation requires measurable $P,g,V,\Phi$ on an invariant domain $D$ such that, for every relevant history and every relevant individual state, \[ P_h=P(s(h)),\quad g_h(a,e)=g(a,e,s(h)),\quad V_h(a,e)=V(a,e,s(h)), \] and $s(hz')=\Phi(s(h),z')$ for every possible successor shock. Equalities for policies may be imposed modulo the explicitly chosen individual-state null sets; in our theorem they hold pointwise. Invariance means that every positive-probability successor remains in $D$. A representation on one realized shock path is insufficient.

2) QUICK LITERATURE/CONTEXT CHECK

Cao's primary working paper [Cao] constructs a generalized recursive equilibrium in a specified Krusell--Smith environment. This supports separating state construction and equilibrium existence from the smaller-state representation question. The uploaded historical citations do not themselves identify the missing MA-01 primitives. We therefore do not assert either universal equivalence or its universal failure for that unspecified model.

3) ATTACK PLAN

Proof routes. Prove Bellman verification on a fully specified invariant state space; derive a fibre-consistency criterion for a given sequential equilibrium; impose uniqueness of individual maximizers as a sufficient way of eliminating policy-selection dependence on history.

Disproof routes. Look for two histories with the same $(z,\mu)$ and different prices; alternatively exploit ties in optimal saving choices while keeping the distribution unchanged; finally test whether unbounded tails invalidate the transversality step. The clean construction below uses the second route.

Landscape and tools. Bellman contraction identifies values; measurable selection defines policies; pushforwards express distribution consistency; fibre factorization tests collisions; conditional iteration verifies deviations; discounted tail bounds justify limits; and telescoping budgets construct the obstruction.

4) WORK

Explicit bounded specification for the theorem.
Let the asset interval $A$ be compact, and let the efficiency set $E$ and aggregate-shock set $Z$ be finite. Set $X=A\times E$, and let $D\subset Z\times\mathcal P(X)$ be a Borel invariant aggregate-state domain. Time is $t\in{\mathbb{N}}=\{0,1,2,\ldots\}$, with discount factor $0<\beta<1$. At state $(x,s)$ the household chooses an action $a'$ in a nonempty feasible set $\Gamma(x,s)$, receiving a measurable reward $\ell(x,s,a')$ with $|\ell|\le M$. Given Borel price functions $P$ and an aggregate transition $\Phi$, a specified Markov probability kernel $Q(dx',ds'\mid x,s,a')$ describes the next individual and aggregate states. The individual is a price taker: a unilateral deviation does not change $P$ or $\Phi$.

Assume the Bellman operator \[ (\mathcal BV)(x,s)=\sup_{a'\in\Gamma(x,s)} \left\{\ell(x,s,a')+\beta\int V(x',s')Q(dx',ds'\mid x,s,a')\right\} \] maps bounded Borel functions to bounded Borel functions and its fixed point has a Borel maximizing selector. These are explicit measurability/attainment assumptions, satisfied in appropriate compact continuous control specifications but not automatic for arbitrary Borel primitives. Define market clearing, firm optimality, and resource feasibility as part of the declared economy. Distribution consistency for deterministic saving policy $g$ is, for each possible $z'$, \[ \mu'(da',de')=\int_{A\times E}\delta_{g(a,e,s)}(da') P_E(de'\mid e,z,z')\,\mu(da,de), \] where $P_E$ is the declared efficiency kernel. Both equilibrium notions in this theorem are distributional: cross-sectional states are conditional laws satisfying this equality. No additional agent-indexed continuum implementation or unproved exact law of large numbers is invoked.

Theorem (Recursive verification and a conditional converse): Under the bounded specification, the following statements hold.

(a) Suppose Borel $P,g,V,\Phi$ satisfy Bellman optimality with $g$ maximizing, distribution consistency, and all declared market and firm conditions on $D$. Then for every initial $s_0\in D$ they induce a sequential competitive equilibrium, optimal against all admissible history-dependent household deviations.

(b) Fix one initial aggregate state and all of its relevant finite aggregate histories. Suppose a sequential equilibrium specifies optimal household continuations for every individual state at every such history. Suppose its prices, policies, and successor aggregate states are constant on fibres of the state map $h\mapsto s(h)$, and the primitive household feasible sets, rewards, and shock kernels depend only on the stated Markov states. Then it has a recursive representation on the invariant set of all these reachable aggregate states. Its continuation values necessarily factor through the same states. This converse does not assert existence of such a fibre-consistent sequential equilibrium.

Proof: For any bounded $V,W$, taking the difference of the two suprema gives $\|\mathcal BV-\mathcal BW\|_\infty\le\beta\|V-W\|_\infty$. For completeness, starting from zero the successive Bellman iterates have successive differences at most $M\beta^j$. The geometric series makes them uniformly Cauchy, their limit is bounded Borel, and the displayed Lipschitz inequality makes the limit a fixed point. If two fixed points existed, their distance would be at most $\beta$ times itself, so they would coincide. The fixed point satisfies $\|V\|_\infty\le M/(1-\beta)$.

For (a), use the kernels and policy to define the sequential laws starting from any $s_0\in D$. Existence of a process with these laws is the usual iterated-kernel construction on standard Borel spaces; its hypotheses hold because the domains and kernels here are Borel. Bellman optimality implies, under any admissible deviation, \[ V(X_t,S_t)\ge \ell(X_t,S_t,a'_{t}) +\beta{\mathbb{E}}[V(X_{t+1},S_{t+1})\mid\text{the household history at }t]. \] This conditional inequality holds even if the chosen action uses additional past information, since it holds for every feasible action at the current state and the transition is Markov conditional on that action. Iterating and taking expectations gives \[ V(x_0,s_0)\ge{\mathbb{E}}\sum_{t=0}^{N-1}\beta^t\ell_t +\beta^N{\mathbb{E}} V(X_N,S_N). \] The last term has absolute value at most $\beta^NM/(1-\beta)$ and tends to zero. The discounted reward series converges absolutely, dominated by $M/(1-\beta)$. Under $g$ every Bellman inequality is equality, so $g$ attains the upper bound and is optimal against every such deviation. By the declared pushforward equality and invariance, all subsequent distributions and aggregate states agree with the recursive construction. The stipulated market and firm conditions hold at every reached state. This proves (a).

For (b), finite $Z$ and one initial state give a countable set of finite aggregate histories and hence a countable reachable state set $D_0$. Include every positive-probability successor, not merely a single realized path. Fibre consistency defines $P(s)$, $g(x,s)$, and $\Phi(s,z')$ independently of the chosen representative history. Enumerate the histories and use the first representative of each state. Each $g(\cdot,s)$ is measurable by the sequential specification, and countability of $D_0$ makes the combined function Borel on $X\times D_0$: preimages are countable unions of its measurable slices times singleton states. Prices and transitions are measurable for the same reason.

The resulting fixed-price, fixed-transition household control problem is the same at any two histories in a fibre: feasible actions, rewards, and conditional transition laws coincide at every subsequent step. The Bellman argument already proved supplies its unique bounded optimal value $V$. Sequential optimality for every individual state therefore identifies each $V_h(x)$ with $V(x,s(h))$. Moreover its policy must attain the Bellman maximum. Otherwise at some current state choosing a maximizing action and then its Borel optimal continuation would strictly improve upon the declared sequential continuation, contradicting optimality. Distribution consistency and markets transfer directly from all the sequential histories. This proves the representation on $D_0$. ∎

What the fibre condition says, and does not say.
For a given sequential equilibrium, any state representation necessarily gives identical prices and policies whenever both the aggregate and individual states coincide. The theorem makes that necessary condition sufficient on the stated countable reachable domain when the other Markov primitives and continuation requirements hold. On an uncountable prescribed domain of initial states, mere set-theoretic constancy on fibres is not enough: Borel factor maps, or a separate measurable-factorization theorem with verified hypotheses, are also needed.

If prices and aggregate transitions already factor and each fixed-state Bellman maximizer is unique, sequential optimality forces policy fibre consistency. This is a useful sufficient condition, but strict concavity of utility alone is not a proof that equilibrium prices or aggregate transitions factor. Nor does the theorem construct an equilibrium or prove uniqueness of the aggregate equilibrium.

Example (Same full distribution, different optimal policies): Consider the following fully specified auxiliary economy, not asserted to equal the missing MA-01. There is a unit continuum of households, one deterministic efficiency and aggregate state, labor supply one per household, $A=[0,1]$, $\beta=1/2$, and $u(c)=c$. Production is $F(K,L)=2K+2L$, with full capital depreciation. Competitive gross capital return and wage are both $2$; firms earn zero profit at these prices and any demanded nonnegative inputs maximize profit. Household budgets are \[ c_t=2a_t+2-a_{t+1},\qquad a_{t+1}\in[0,1]. \] Every such asset path is feasible, with $1\le c_t\le4$. For every $N$, \[ \sum_{t=0}^{N-1}2^{-t}c_t =2a_0+4(1-2^{-N})-2^{-(N-1)}a_N. \] This follows by expanding the budget sum: each interior asset appears with coefficients $2\cdot2^{-t}$ and $-2^{-(t-1)}$, which cancel. Bounded assets make the last term vanish, so every feasible path has utility $2a_0+4$ and is optimal.

Initialize the full distribution at $\mu=(\delta_0+\delta_1)/2$. At even dates use $g_t(a)=a$; at odd dates use $g_t(a)=1-a$. Both maps preserve $\mu$. Thus $K_t=K_{t+1}=1/2$ at all dates. Aggregate consumption is $2K_t+2-K_{t+1}=5/2$, gross investment is $1/2$, and output is $3$, so goods and factor markets clear. The preceding household argument proves sequential optimality, including off-support individual states if their policies are extended by the same formulas.

The aggregate state $(z,\mu)$ never changes. At an individual state $a=0$, which has mass $1/2$ at every date, saving is $0$ at even dates and $1$ at odd dates. No single time-independent function $g(a,e,z,\mu)$ can equal both. Hence this sequential equilibrium has no policy representation on the proposed reduced state. Its value nevertheless does factor: $V(a)=2a+4$. Adding that already determined continuation value as another coordinate does not distinguish the two policy choices.

5) VERIFICATION

The example uses the full distribution, not a moment approximation. Its failure is a policy-selection collision on a positive-mass individual state, not a null-set disagreement. All rewards and assets are bounded, so no transversality problem drives the failure. Linear utility is deliberate: a claim restricted to strictly concave MA-01 preferences is not disproved by this auxiliary economy. The finite check in the script verifies that both pushforwards of $(1/2,1/2)$ are $(1/2,1/2)$ and that $5/2+1/2=3$.

In the theorem, compactness is not misused to imply measurable selection for arbitrary data; the needed Bellman measurability and maximizing selector are explicit assumptions. Invariance includes all positive-probability aggregate branches. With unbounded rewards, the proof would need convergence of the discounted utility series and $\beta^N{\mathbb{E}}|V(X_N,S_N)|\to0$ for the equilibrium and every deviation used in verification. An Euler equation alone does not supply those limits.

6) FINAL

{LABEL: UNRESOLVED.}

Strongest checked partial result. On the declared bounded Markov specification, recursive equilibria induce sequential equilibria for every initial state in the invariant domain, and a given full-continuation sequential equilibrium admits a reduced-state representation under the explicit fibre and measurability conditions. The auxiliary economy proves that optimal continuation values can be state-dependent while selected optimal policies are not.

Exact first gap for the original programme. After the missing MA-01 primitives are fixed, prove that a full sequential equilibrium correspondence admits a Borel continuation selection whose prices and successor aggregate distributions depend only on $(z,\mu)$, and whose household policies are consistent across repeated states. Neither household Bellman contraction nor existence of some sequential equilibrium proves this selection assertion.

Next three moves. (1) Prove fibre consistency of MA-01 continuation prices and laws. (2) Establish measurable selection on an invariant domain covering the required initial distributions. (3) Verify transversality for the actual utility and borrowing constraints.

Likely minimal obstruction. Repeated aggregate states may support distinct continuation equilibria, not merely household ties. The two-atom example isolates this selection issue without disproving an unspecified stricter economy.

References

  • [Cao] Dan Cao. Recursive Equilibrium in Krusell and Smith (1998). Author working paper, title page dated September 2016, hosted in the AEA 2018 meeting materials. The version checked is the 72-page working paper, especially the abstract and introduction; the citation does not equate its expanded-state result with an unrestricted reduced-state theorem. https://www.aeaweb.org/conference/2018/preliminary/paper/Ty859AnT.

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Econ-49-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-49

Sequential and recursive equilibrium with a distributional state A countable-history representation criterion with verification

Problem ID: C62-48. Status: Unresolved; rigorous restricted results.

Short recap and scope

The target distinguishes existence of some recursive equilibrium on the full distributional state from representation of a particular sequential equilibrium. We prove a necessary and sufficient collision criterion on the invariant set reachable from a countable collection of initial distributions, and a verification theorem allowing all adapted household deviations. Bounded utility is an explicit additional restriction. No existence theorem for the general incomplete-market model is inferred.

Heathcote, Storesletten and Violante [hsv], Section 5.2, footnote 20, discuss the historical question of a reduced state. Miao [miao] proves an enlarged-state recursive characterization involving continuation values and sequential equilibria with the same payoffs. Such a representation need not reproduce the same sequential allocation on the smaller state. The results here establish an elementary exact criterion for a specified equilibrium family, not a replacement for Miao's existence theorem.

The declared restricted household problem

Let $z,e$ have finite spaces, aggregate transition $q(z'\mid z)$, and conditional individual transition $q(e'\mid e,z,z')$. Assets belong to a Borel interval $A$. The reduced aggregate state is $s=(z,\mu)$, where $\mu$ is a probability on $A\times E$ with finite first moment. The capital and labor aggregates, prices and budget are those of the catalogue incomplete-market model: \begin{align*} K&=\int a\,d\mu>0, & L&=\int e\,d\mu>0,\\ R&=1-\delta+F_K(z,K,L), & w&=F_L(z,K,L),\\ c&=Ra+we-a'>0. \end{align*} Feasible savings satisfy the borrowing bound and any additional declared asset constraint. Assume all relevant controls and kernels are Borel. Impose exact conditional aggregation: for every feasible Borel savings rule $g$, common state $s=(z,\mu)$, next aggregate shock $z'$, Borel set $B\subseteq A$ and efficiency $e'$, the realized cross-sectional next measure satisfies almost surely \[ \mu'(B\times\{e'\})= \sum_e\int_A {\mathbf 1}\{g(a,e,s)\in B\} q(e'\mid e,z,z')\,\mu(da,\{e\}). \] This conditional continuum law of large numbers is an assumption on the population probability construction, not a conclusion from individual transition probabilities alone. For the verification theorem impose a uniform bound $|u(c)|\leq M$ on every feasible consumption under every admissible deviation, and $\beta\in(0,1)$. This holds, for example, on a declared compact consumption interval bounded away from zero with continuous utility; it is not automatic for logarithmic utility near zero.
Theorem (Recursive verification against history-dependent plans): Let $D$ be an invariant Borel domain of aggregate states. Suppose Borel prices, a Borel transition $\mathcal T(s,z')\in D$, a feasible savings rule $g(a,e,s)$ and a bounded Borel function $V$ satisfy, at every feasible individual state and every $s\in D$, \[ V(a,e,s)=\sup_{a'\in B(a,e,s)} \left\{u(R(s)a+w(s)e-a')+ \beta\sum_{z',e'}q(z'\mid z)q(e'\mid e,z,z') V(a',e',\mathcal T(s,z'))\right\}, \tag{1} \] with $g$ attaining the supremum. Suppose $\mathcal T$ is the population pushforward induced by $g$ and the efficiency kernel, and goods and capital markets clear on $D$. Under the stated conditional continuum law of large numbers, these objects generate a sequential competitive equilibrium from every $s_0\in D$. The function $V$ is the actual lifetime-utility supremum over all feasible adapted household plans, not only Markov plans.
Proof: Recursively apply $q$ and $\mathcal T$ to construct the aggregate path; for each household use the declared individual kernel and $g$. Their Borel kernels determine finite-history laws consistently and hence the path law. Distribution consistency and the continuum law of large numbers give the specified cross-sectional distribution at each date; the assumed pointwise clearing identities give market clearing. For any household's adapted feasible plan, the aggregate path remains fixed by price-taking. Conditional on its history, the continuation kernel in (1) is the stated one. Bellman optimality gives \[ V(a_0,e_0,s_0)\geq {\mathbb{E}}\left[\sum_{t=0}^{T-1}\beta^t u(c_t) +\beta^T V(a_T,e_T,s_T)\right]. \] This follows by induction, multiplying each conditional inequality by $\beta^t$ and cancelling consecutive continuation terms. Bounded $V$ makes the final term vanish uniformly. The utility bound makes the discounted sum absolutely integrable and permits its limit inside expectation. Thus $V$ bounds the utility of every adapted plan. For the plan $g$, each Bellman inequality is equality; the same limit shows its utility equals $V$. Feasibility, optimality and clearing are exactly the sequential equilibrium conditions. ∎

When a given sequential family factors through the state

Take a nonempty countable set of initial aggregate states and, for each one, an infinite-horizon sequential equilibrium specified at every positive- probability finite aggregate history. Shocks have finite support, so the union $\mathcal H$ of these histories is countable. Include continuations after every positive-probability next shock, not just one realized path. Let $s(h)=(z(h),\mu(h))$ and $D=\{s(h):h\in\mathcal H\}$. Equip the probability-measure state space with its usual Borel structure; its singletons are Borel, and thus every subset of countable $D$ is Borel.

The sequential equilibrium is assumed comprehensive: at each aggregate history $h$, a Borel current policy $g_h(a,e)$ is specified and is part of a sequentially optimal continuation for every feasible individual state, including states of zero current population mass. Prices $(R_h,w_h)$ and aggregate next-state maps $s(hz')$ are also specified. Individual state is sufficient for idiosyncratic transition probabilities; the exogenous kernels above are independent of earlier history conditional on their stated arguments. All household continuation utilities obey the uniform bound used in the preceding theorem.

Define the collision conditions: whenever $s(h)=s(\widetilde h)$, \begin{align} (R_h,w_h)&=(R_{\widetilde h},w_{\widetilde h}), \tag{C1}\\ g_h(a,e)&=g_{\widetilde h}(a,e) \quad\text{for every feasible }(a,e), \tag{C2}\\ s(hz')&=s(\widetilde h z') \quad\text{for every }z'\text{ with }q(z'\mid z(h))>0. \tag{C3} \end{align} Equality on all feasible individual states is stronger than population- almost-everywhere equality and is needed for the declared comprehensive representation. Under the catalogue's factor-price formula (C1) is automatic. Given distribution consistency, (C2) also implies (C3).

Theorem (Exact reduced-state representation on the reachable domain): The specified comprehensive sequential family admits a single stationary recursive representation $(R,w,g,V,\mathcal T)$ on $D$, agreeing with its current policies, prices, aggregate transitions and continuation utilities at every supported history and every feasible individual state, if and only if (C1)--(C3) hold. The resulting value satisfies $|V|\leq M/(1-\beta)$, equals the actual sequential household optimum, and generates an equilibrium from every initial state in $D$.
Proof: Necessity follows by evaluating each single-valued recursive function at the same state in two histories. For sufficiency choose for each $s\in D$ one representative history $h_s$. Define prices and policy from this history, and set $\mathcal T(s,z')=s(h_s z')$ for supported shocks. For unsupported shocks choose any element of $D$; their values do not affect expectations. Conditions (C1)--(C3) make all supported definitions independent of the representative. Countability of $D$, together with the Borel individual policies on each slice, proves measurability. The domain is invariant by its definition from all supported finite histories.

Starting from $(a,e,s)$, follow $g$ under these kernels and define $V$ to be its expected discounted utility. Finite-horizon expected sums are Borel by finite iteration of Borel kernels, and their tails are bounded uniformly by $M\beta^T/(1-\beta)$. Their limit is therefore Borel and bounded. The collision conditions imply by induction that the recursive path and each original continuation have identical conditional laws: their current policies agree, then (C3) agrees on next aggregate states, and the individual transition kernel agrees by the primitive Markov assumption. Thus this $V$ equals every corresponding sequential continuation utility, including at zero-mass individual states.

For any feasible current $a'$, deviate once and then follow the assigned optimal continuation at the next aggregate history and individual state. Comprehensive sequential optimality makes the value of this deviation at most $V(a,e,s)$. Its value is exactly the expression inside the braces of (1), by the just-established continuation equality. Following $g$ gives equality, by its definition as the achieved continuation value. Hence (1) holds and $g$ attains it. Population consistency and market clearing are inherited from the representatives and are well-defined on collisions. The preceding verification theorem proves sequential optimality and equilibrium from every $s\in D$. ∎

Corollary (What state enlargement must distinguish): If two supported histories have the same reduced state but different current policies at a feasible individual state, no comprehensive recursive representation on $(z,\mu)$ can reproduce both continuations. Any additional coordinate $v(h)$ that succeeds in reproducing them must take different values at those histories. Conversely, if the map $h\mapsto s(h)$ is injective, the collision conditions hold vacuously (apart from primitive consistency), and the theorem gives a recursive representation on its countable reachable domain.
Proof: A function evaluated at identical arguments cannot equal two different policies. If an enlarged coordinate also agrees, its arguments remain identical and the same contradiction applies. With injective $s$, each state has one representative history, so all collision requirements are automatic and the representation theorem applies. ∎

The last observation does not give a continuous or computationally useful representation: a countable history tree may be encoded in a countable set of distinct distributional states with very irregular neighboring policies. It also does not give a common representation across all uncountably many admissible initial distributions.

Verification, relation to the target, and remaining gap

The proof checks household deviations at zero-mass individual states; checking only the equilibrium population path would not establish the Bellman supremum. Boundedness is exactly what removes the terminal continuation term; with unbounded utility it must be replaced by an appropriate transversality or uniform-integrability condition for every comparison plan. The collision obstruction is a logical criterion, not an asserted example of multiplicity in the catalogue economy. The first unresolved economic step is to prove collision consistency, or construct its failure, from the specified incomplete-market primitives without assuming a preselected comprehensive family. Extending the measurable construction to an uncountable invariant domain requires a measurable selection argument not supplied here. The full equivalence and existence programme remains unresolved.

References

Completion Estimate

30%

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