arXiv subjects

Leandro Lyra Braga Dognini

Publications and source records attributed to Leandro Lyra Braga Dognini.

8 recordsLinked to original sources

arXiv2026

Stochastic Non-Tâtonnement Processes and the Attraction Principle

I introduce stochastic non-tâtonnement processes and argue that they are a natural outcome of general equilibrium theory. To this end, I revisit classical demand theory to define two diffeomorphisms: the normalized Walrasian demand and a map that flattens indifference hypersurfaces. These diffeomorphisms are applied to the three canonical manifolds in the consumption domain (i.e., the indifference and the offer hypersurfaces, and the trade hyperplane) to analyze their images in the normalized and flat domains, and their relations to the set of Pareto optimal allocations are discussed. Then, I derive, for arbitrary non-tâtonnement processes, an attraction principle based on the dynamics of marginal rates of substitution seen on the floor of the flat domain. This principle motivates the definition of stochastic non-tâtonnement processes and leads to a stochastic version of the First Welfare Theorem. When utility functions are attractive and sharp, a class of stochastic non-tâtonnement processes, called common density processes, is particularly well suited for Monte Carlo simulations, which can be used to compute the distribution of trade outcomes on the contract curve and to model price stickiness.

econ.TH↗

arXiv2026

Depth Projections and the Local Nature of the Cass Criterion

This paper defines the depth projections of the indifference and offer hypersurfaces, and of the transformation frontier. These projections are functions conceived to measure curvature (thus are not numbers, as the Gaussian curvature is) and their second-order behavior is governed by the Hessians of the expenditure and profit functions. In particular, up to second order, the depth projection of the offer hypersurface is always exactly twice that of the indifference hypersurface. In an overlapping generations economy with production, these projections lead to the derivation of general statements for the necessity and sufficiency of the Cass criterion in the form $\sum^{\infty}_{t=1}1/(\Vert p_{t}\Vert\sum_{h\in G_{t}}\Vert c^{h}\Vert)=\infty$, which allows unbounded dynamics for the demography and for per capita endowments, and reconciles the Cass criterion with the Balasko--Cass--Shell relabeling algorithm. Furthermore, the assumptions behind these statements reveal the local nature of the Cass criterion: only the behavior of the indifference hypersurfaces and transformation frontiers in a neighborhood of the equilibrium allocation matters.

econ.TH↗

arXiv2026

Selection of Efficient Monetary Equilibria Through Aggregate Real Savings-Based Taylor Rule

This paper uses the generalized Cass criterion $\sum^{\infty}_{t=1}(\Vert p_{t}\Vert\sum_{h\in G_{t}}\Vert e^{h}_{t}\Vert)^{-1}=\infty$ to extend the results from Dognini (2026) regarding the existence of efficient monetary equilibria on consumption-loan overlapping generations economies. These results reveal that if the economy is prone to savings, then monetary equilibria will emerge in a pure non-stationary general equilibrium model with heterogeneous households, thus providing a solution to the Hahn (1965) problem. It is also proved that, in prone-to-savings economies, non-vanishing relative aggregate real savings fully characterize efficient monetary equilibria. I use this result to show that a Taylor rule based on an inflation ceiling and a relative aggregate real savings floor can be used to control the price level and lead the economy towards an efficient monetary equilibrium. In contrast, a Taylor rule based solely on an inflation target is able to control the price level but generally leads the economy towards an inefficient monetary equilibrium.

econ.TH↗

arXiv2026

Existence and Calculation of Optimal Monetary Equilibria on Overlapping Generations Economies

A well-known feature of overlapping generations economies is that the First Welfare Theorem fails and equilibrium may be inefficient. The Cass (1972) criterion furnishes a necessary and sufficient condition for efficiency, but it does not address the existence of efficient equilibria, and Cass, Okuno, and Zilcha (1979) provide nonexistence examples. A closely related question (known as the Hahn (1965) problem) deals with the existence of monetary equilibria. In this paper, I provide sufficient conditions for the existence of optimal monetary equilibria on consumption-loan, non-stationary overlapping generations economies without durable, dividend-paying assets, cash-in-advance constraints, wealth-transfer mechanisms, or transaction costs. Essentially, the economy must be prone to savings. Furthermore, I develop an algorithm to find these optimal monetary equilibria as the limit of nested compact sets. These compact sets are the result of a backward calculation through equilibrium equations departing from the set of optimal monetary equilibria of well-behaved tail economies.

econ.TH↗

arXiv2026

A Closer Look on the Influence of Constraints Upon the Optimization of the Nonadditive Entropic Functional $S_{q}$

The thermal-equilibrium canonical distribution is currently obtained by maximizing the Boltzmann-Gibbs-von Neumann-Shannon entropy $S_{BG}(p)=k\sum^{W}_{i=1}p_{i}\ln 1/p_{i}$ constrained to $\sum^{W}_{i=1}p_{i}=1$ and $\sum^{W}_{i=1}p_{i}\,e_{i}=U$, $e_{1}\leq\ldots\leq e_{W}$ being the energies of the $W$ possible states and $U\in[e_{1},e_{W}]$ their mean value. We revisit a generalized version of this optimization problem grounded in the nonadditive entropy $S_{q}(p)=k\,(\sum^{W}_{i=1}p_{i}^{q}-1)/(1-q)$ (frequently, though not necessarily, $q\in(0,1)$; $S_1=S_{BG}$), and the constraint $\sum^{W}_{i=1} p_{i}^{q^{\prime}}e_{i} / \sum^{W}_{i=1}p_{i}^{q^{\prime}}=U$, $q^{\prime}>0$. Sufficient conditions for existence, strict positivity, and uniqueness of solutions are derived, along with a theorem that enables their closed-form calculation. We apply these results to deepen the understanding of the two standard cases in the literature ($q^{\prime}=1$ and $q^{\prime}=q$), as well as of a new one ($q^{\prime}=2-q$). We prove that these standard cases are the only ones yielding optimizing probability distributions of $q$-exponential form. Furthermore, we define an effective temperature $T_{q,q^{\prime}}$ through a Clausius-like relation $1/T_{q,q^{\prime}}=\partial S_{q} / \partial U$ and derive a Helmholtz-like energy $F_{q,q^{\prime}}=U-T_{q,q^{\prime}}S_{q}$, with the former grounding the validity of the $0^{th}$ Principle of Thermodynamics within this generalized statistical mechanics. Finally, we show that the case with a linear constraint (i.e., $q^{\prime}=1$) with $q\in(0,1)$ (i) preserves the Third Law of Thermodynamics; (ii) can be used to model classical many-body Hamiltonian systems with arbitrarily-ranged interactions; and (iii) resembles features of low-dimensional nonlinear dynamical systems at the edge of chaos.

cond-mat.stat-mech↗

arXiv2026

Generalized Algebra Grounded on Nonadditive Entropies

The class of $N$-body complex systems with total number of microscopic states given by $W(N) \sim ν^{N^γ}\;(ν>1, \,γ> 0)$ can be thermostatistically handled with the nonadditive entropic functional $S_δ(\{p_{i}\}) = k\sum_{i=1}^W p_i \Bigl(\ln \frac{1}{p_i} \Bigr)^δ\;(δ>0,\,S_1=S_{BG})$, $S_{BG}=k\sum_{i=1}^W p_i \ln \frac{1}{p_i}$ being the Boltzmann-Gibbs functional. Indeed, $S_{δ=1/γ}(\{1/W(N)\})=k[\ln W(N)]^{\frac{1}γ} \propto N$, as mandated by thermodynamics. Another wide class is that with $W(N) \sim N^ρ\;(ρ>0)$ and a generalized statistical mechanics grounded on the nonadditive entropic functional $S_q(\{p_{i}\})=k\sum_{i=1}^W p_i \ln_q \frac{1}{p_i} \;(q\in \mathbb{R},\;S_1=S_{BG})$, with $\ln_q z =\frac{z^{1-q}-1}{1-q}\; (z\geq0,\;q\in\mathbb{R},\;\ln_1 z=\ln z)$, satisfactorily handles such systems with $q=1-1/ρ$. Furthermore, for this class, the size of the corresponding admissible phase space is characterized by $\ln_q (x\otimes_q y) =\ln_q x + \ln_q y,\, x,y\geq1,\,q\leq 1$, and the $q$-product $x\otimes_q y=[x^{1-q}+y^{1-q}-1]^{\frac{1}{1-q}}_{+}\;(x\otimes_1 y=xy)$ also leads to the definition of a $q$-algebra. The entropic functional $S_{q,δ}(\{p_{i}\})=k\sum_{i=1}^W p_i \Bigl(\ln_q \frac{1}{p_i} \Bigr)^δ\;(q\in\mathbb{R},δ>0)$ unifies both cases above: $S_{q,1}=S_q$, $S_{1,δ}=S_δ$ and $S_{1,1}=S_{BG}$. In this paper, we generalize the $q$-algebra associated with $S_{q}$ to a new one associated with $S_{q,δ}$, namely the $(q,δ)$-algebra.

cond-mat.stat-mech↗

arXiv2026

The Design of Optimally Balanced Pay-as-you-go Social Security Systems

This paper bridges social security design and general equilibrium theory to conceive optimally balanced pay-as-you-go systems. The design is based on the backward calculation algorithm from Dognini (2025), which is used to find optimal monetary equilibria of prone-to-savings non-stationary overlapping generations economies with heterogeneous households. In particular, this algorithm makes the design applicable for reforming pay-as-you-go systems in countries undergoing demographic transitions. Due to households balanced budgets under equilibrium prices (i.e., Walras' law), these optimally balanced pay-as-you-go systems resemble the well-known notional accounts systems. The design is illustrated in a simplified framework using the past and forecast demographic and productivity dynamics of Brazil, China, India, Italy, and the United States from 1950 to 2070.

econ.TH↗

arXiv2025

Generalized Boltzmann-Gibbs Distribution and the Electronic Partition Function Paradox

This paper generalizes the entropy maximization problem leading to the Boltzmann-Gibbs distribution through the nonadditive entropy $S_{q,s}(p)=k_{s}\sum^{W}_{i\geq1}p_{i}\ln_{q}1/p_{i}$, $q\in(0,1)$, which is a rescaled version of $S_{q}$ \cite{Tsallis1988} by a factor $k_{s}=k^{q}(e_{\max}/(W^σ-1))^{1-q}$, $σ>0$, varying according to the underlying energy spectrum and satisfying $k_{s}\rightarrow k$ (Boltzmann constant) as $q\rightarrow 1$. The maximization problem based on $S_{q,s}$ is used to derive analytical generalizations of the Boltzmann-Gibbs distribution for the case of an energy spectrum that uniformly approaches a continuous, unbounded limit with a common degeneracy, the harmonic oscillator, and the one-dimensional box. Furthermore, I demonstrate that this generalized problem yields a two-tier model with finite structural parameters $β$ for the hydrogen atom in free space and, therefore, can be used to circumvent the Electronic Partition Function Paradox and obtain a family of well-defined thermodynamic behaviors indexed by $q\in(0,1)$. In particular, for $q=0.5$, the specific heat of the free hydrogen atom becomes the Boltzmann constant. Finally, it is shown that all the limiting processes involved in these four cases lead naturally to the same definition of the scale factor $k_{s}$ that characterizes $S_{q,s}$ (``s'' stands, in this context, for ``spectrum'') in order to grant finite, smooth, macroscopically observable temperature values that are related to the entropic functional by $\partial S_{q,s}/\partial U=1/\vert T\vert^{q}_{\pm}$, which recovers $\partial S_{BG}/\partial U=1/T$ \cite{Clausius1865} as $q\rightarrow1$.

math-ph↗