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Econ-186 · economics · computer science · Computational economics

Realistic equilibrium adjustment

LLM status Open unverified

Statement

Original subject group: Economic theory, equilibrium and social choice

Audit classification: Background; proposed extension. The cited paper proves stability results for a different forward-looking model. The delayed rule is an editorial extension; no published statement or residual unsolved status was established.

Definitions and precise research target

Model and research target. Consider a finite pure-exchange economy with agents $i=1,\ldots,n$, goods $g=1,\ldots,m$, positive endowments $\omega_i$, and strictly increasing, strictly concave, twice continuously differentiable utilities $u_i$. Assume demands are unique at every strictly positive normalized price. Let $\Delta_{++}$ be the positive unit price simplex, $z(p)$ aggregate excess demand, and $p^*\in\Delta_{++}$ a competitive equilibrium satisfying $z(p^*)=0$. Fix a delay $\tau\geq0$, diagonal positive adjustment matrix $D$, and projection $\Pi$ onto the simplex tangent space. Study the proposed decentralized rule \[ \dot p(t)=\Pi D z(p(t-\tau)),\qquad p(s)=\phi(s)\quad(-\tau\leq s\leq0), \] where $\phi$ is a continuous positive initial price history. Restrict attention to economies and histories for which the resulting trajectory remains in a specified compact subset of $\Delta_{++}$. Determine economically interpretable necessary or sufficient restrictions on preferences, endowments, $D$, and $\tau$ guaranteeing convergence to equilibrium for every admissible history. Derive sharp delay bounds where possible and construct economies demonstrating their failure outside the guaranteed region. Both local stability and genuinely global convergence must be distinguished.

Definitions and maintained assumptions (editorial unless a source definition is named). Let $\mathbb R_+^m$ be the consumption set, $\Delta_{++}=\{p\in\mathbb R^m:p_g>0,\ \sum_g p_g=1\}$, and \[x_i(p)=\arg\max\{u_i(x):x\in\mathbb R_+^m,\ p\cdot x\le p\cdot\omega_i\},\qquad z(p)=\sum_i(x_i(p)-\omega_i).\] Assume single-valued demands and that $z$ is locally Lipschitz on $\Delta_{++}$; smooth utilities alone do not automatically imply a differentiable demand at consumption corners. Use the Euclidean tangent projection $\Pi=I-\mathbf1\mathbf1^\top/m$, rather than an unspecified price-dependent projection. A history belongs to $C([-\tau,0],\Delta_{++})$, with the supremum norm. An admissible history is one whose unique delay-equation solution exists for all $t\ge0$ and stays in a fixed compact $K\subset\Delta_{++}$. Global convergence means that each admissible solution has a limit $p^\infty$ with $z(p^\infty)=0$; it does not require the same limit for different histories. Local asymptotic stability adds Lyapunov stability in the history norm. An interior rest point of the projected equation is an equilibrium: $\Pi Dz=0$ implies $Dz=c\mathbf1$, and Walras' law gives $c\sum_g p_g/D_{gg}=0$.

Variants and assumption changes

  1. Local delays (editorial). Assume $z$ is $C^1$ near an isolated equilibrium. On the tangent space analyze the roots of $\det(\lambda I-\Pi Dz'(p^*)e^{-\lambda\tau})=0$. Determine the first delay at which a characteristic root reaches the imaginary axis; a linearization conclusion requires the usual noncritical-root hypotheses.
  2. Global domain (editorial). Impose a stated gross-substitutes or monotonicity restriction and first prove that $K$ is forward invariant for all histories taking values in $K$. Then seek a uniform delay bound guaranteeing convergence on that domain. Compactness must be proved or imposed; it is not provided by tangent projection.

References and source audit

  1. Abstract; introduction; §§5–7. Supports the stated research area; exact displayed specification is editorial. https://economics.mit.edu/sites/default/files/inline-files/taton_NBER_new.pdf
  1. Iván Werning; Guido Lorenzoni. 2026. Tâtonnement and Price Setting in General Equilibrium. Abstract; §§5–7, especially §7.2 Proposition 7.2. https://economics.mit.edu/sites/default/files/inline-files/taton_NBER_new.pdf. DOI: https://doi.org/10.3386/w35205.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-186-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-186

Delayed equilibrium adjustment A global small-delay estimate and an invariant-domain obstruction

Problem ID: C62-182. Status: Unresolved; restricted results and reductions.

The delayed target and its source

Let $D$ be a positive diagonal matrix, and let aggregate excess demand $z$ satisfy Walras' law $p\cdot z(p)=0$ at every positive price. Write $\Pi=I-\mathbf1\mathbf1^\top/m$ and $F(p)=\Pi Dz(p)$. The proposed rule is $\dot p(t)=F(p(t-\tau))$ on the positive price simplex. The paper by Lorenzoni and Werning [lw], especially Section 7.2, Proposition 7.2, studies a different forward-looking system. Its stability theorem does not apply to this delayed equation. The results below concern the catalogue's editorial extension directly.

We prove a sufficient convergence condition for every trajectory in a declared compact domain, and show why requiring a common compact convex domain to be invariant for every continuous history in that domain is generally impossible when $\tau>0$. This distinction is material for economic applications: compactness cannot be obtained merely by projecting the velocity onto the simplex tangent space.

A quantitative sufficient condition

Let $p^*$ be an interior competitive equilibrium and let $K$ be a compact subset of the positive simplex containing $p^*$. On $K$ suppose \begin{equation}\label{wsref-0-62-6f-75-6e-64-73} \langle p-p^*,F(p)\rangle\leq-a\|p-p^*\|^2, \qquad \|F(p)-F(q)\|\leq L\|p-q\| \end{equation} for all $p,q\in K$, with $a,L>0$. Assume a given continuous history in $K$ generates a global solution remaining in $K$. These are explicit additional restrictions on projected excess demand, not automatic consequences of strict concavity or gross substitutes.
Theorem (Convergence for admissible histories): Under (bounds), every such solution converges to $p^*$ if \[ 0\leq\tau<a/L^2. \] For $\tau>0$, put $x(t)=p(t)-p^*$ and $C_0=\frac12\|x(\tau)\|^2+ \frac{L^2\tau}{2}\int_{-\tau}^{\tau}\|x(v)\|^2\,dv$. Then for every $R\geq\tau$, \[ (a-L^2\tau)\int_\tau^R\|x(t)\|^2\,dt\leq C_0. \]
Proof: Translate $p^*$ to zero and write $G(x)=F(p^*+x)$, so $G(0)=0$. For $t\geq\tau$, the solution is differentiable along $[t-\tau,t]$ and \[ x(t)-x(t-\tau)=\int_{t-\tau}^tG(x(v-\tau))\,dv. \] For $V(t)=\frac12\|x(t)\|^2$, add and subtract $G(x(t))$ to get \begin{align*} V'(t)&\leq-a\|x(t)\|^2 +L\|x(t)\|\|x(t-\tau)-x(t)\|\\ &\leq-a\|x(t)\|^2 +L^2\|x(t)\|\int_{t-2\tau}^{t-\tau}\|x(v)\|\,dv\\ &\leq-\left(a-\frac{L^2\tau}{2}\right)\|x(t)\|^2 +\frac{L^2}{2}\int_{t-2\tau}^{t-\tau}\|x(v)\|^2\,dv. \end{align*} The last step applies $2uv\leq u^2+v^2$ inside the integral. On integrating from $\tau$ to $R$ and changing the integration order, each $v$ belongs to an interval of outer times of length at most $\tau$. Thus the double integral is at most $\tau\int_{-\tau}^{R}\|x(v)\|^2\,dv$. Split this integral at $\tau$ and use $V(R)\geq0$ to obtain the claimed estimate.

The positive coefficient makes $\int_\tau^\infty\|x(t)\|^2dt$ finite. Compactness and continuity of $F$ bound $\|x'(t)\|$, so $x$ is uniformly continuous for $t\geq0$. If $x$ did not converge to zero, there would be $\epsilon>0$ and times tending to infinity at which its norm exceeds $\epsilon$. Uniform continuity supplies intervals of a common positive length on which the norm exceeds $\epsilon/2$. Taking a disjoint subsequence of those intervals contradicts square integrability. For $\tau=0$ the direct inequality $V'\leq-2aV$ gives $V(t)\leq e^{-2at}V(0)$ and convergence. ∎

Proposition (A derivative condition and equilibrium uniqueness): Suppose $K$ is convex, $F$ is continuously differentiable on a neighborhood of $K$ within the price hyperplane, and for every tangent vector $v$ and $p\in K$, \[ v^\top\frac{F'(p)+F'(p)^\top}{2}v\leq-a\|v\|^2, \qquad\|F'(p)\|\leq L. \] Then (bounds) holds. Furthermore $p^*$ is the only rest point in $K$, and every interior rest point of the projected equation is a competitive equilibrium.
Proof: Integrate the derivative of $F$ along the segment from $q$ to $p$. The operator norm bound gives Lipschitz continuity. Taking $q=p^*$ and an inner product with $p-p^*$ gives the dissipation inequality; the skew-symmetric part has zero quadratic form. A second rest point would violate that inequality with $a>0$. Finally $\Pi Dz(p)=0$ means $Dz(p)=c\mathbf1$. Walras' law yields $0=p\cdot z(p)=c\sum_g p_g/D_{gg}$. The sum is strictly positive, so $c=0$ and $z(p)=0$. ∎

This condition can be checked from price derivatives of aggregate demand and announced adjustment speeds. Deriving it from broad primitive classes of preferences remains a substantive restriction, not a proved consequence of the maintained economic assumptions.

Why a history-uniform compact convex invariant set fails

Theorem (Invariant-domain obstruction): Let $K$ be a nonempty compact convex subset of a finite-dimensional affine space, and let $F$ be continuous and locally Lipschitz on a neighborhood of $K$, taking values in its linear ambient space. Fix $\tau>0$. If the equation $\dot p(t)=F(p(t-\tau))$ stays in $K$ for all $t\geq0$ for every continuous initial history in $K$, then $F(y)=0$ for every $y\in K$.
Proof: If $F(y)=v\ne0$ at some $y\in K$, choose $x\in K$ maximizing the linear functional $p\mapsto v\cdot p$; compactness assures attainment. By convexity the straight-line history from $y$ at time $-\tau$ to $x$ at time zero lies entirely in $K$. The resulting solution has right derivative $p'(0)=F(y)=v$. Hence \[ v\cdot p(t)=v\cdot x+t\|v\|^2+o(t)>v\cdot x \] for all sufficiently small positive $t$, contradicting maximality of $x$ if $p(t)\in K$. This contradiction applies to every nonzero $F(y)$. ∎

Applied within the simplex hyperplane, this theorem shows that the catalogue's variant asking for invariance for all $K$-valued histories cannot hold for a nontrivial compact convex $K$ and a nonzero delayed adjustment field. It does not rule out admissible subsets of histories, nonconvex domains, or bounded individual trajectories. Theorem Theorem (Convergence for admissible histories) therefore explicitly conditions on admissibility, exactly as the main formulation permits.

A sharp spectral boundary in the symmetric linear case

The following statement concerns characteristic roots and exact linear solutions only, and does not invoke an unproved nonlinear linearization principle.
Proposition (Scalar and symmetric characteristic-root bound): For $\dot x(t)=-c x(t-\tau)$ with $c>0$, the characteristic equation $\lambda+c e^{-\lambda\tau}=0$ has no root of nonnegative real part when $c\tau<\pi/2$. At $c\tau=\pi/2$ it has roots $\pm ic$ and the bounded nonconvergent real solution $x(t)=\cos(ct)$. For a real symmetric positive-definite matrix $A$ on the price tangent space, $\dot x=-A x(t-\tau)$ has no nonnegative-real-part characteristic root when $\tau\lambda_{\max}(A)<\pi/2$, and has a bounded nonconvergent solution at equality.
Proof: Write a scalar root as $\lambda=x+iy$ with $x\geq0$. Taking modulus gives $|\lambda|=c e^{-x\tau}\leq c$, hence $|y|\tau\leq c\tau$. The real part of the equation gives $x=-c e^{-x\tau}\cos(y\tau)$. If $c\tau<\pi/2$, the cosine is strictly positive, contradicting $x\geq0$. The case $\tau=0$ is included directly. At equality substitution verifies $\lambda=ic$ and its conjugate. Direct differentiation verifies the cosine solution: $-c\cos(c(t-\tau))=-c\sin(ct)$. An orthogonal eigenbasis diagonalizes the symmetric vector equation into scalar equations with coefficients the eigenvalues of $A$. Apply the scalar result to each coordinate; at equality choose the cosine solution along a largest-eigenvalue vector. ∎

These modal examples show the relevant inverse-speed delay scale, but are not themselves constructed pure-exchange economies. The sufficient nonlinear bound $a/L^2$ can be more conservative than the symmetric spectral threshold. Constructing primitive economies with sharp global delay failures, proving admissibility for economically meaningful history classes, and obtaining necessary global conditions remain open in this attempt.

References

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