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Andrea Loi

Publications and source records attributed to Andrea Loi.

At least 37 records · Page 2Linked to original sources
arXiv2023

Endowments, patience types, and uniqueness in two-good HARA utility economies

This paper establishes a link between endowments, patience types, and the parameters of the HARA Bernoulli utility function that ensure equilibrium uniqueness in an economy with two goods and two impatience types with additive separable preferences. We provide sufficient conditions that guarantee uniqueness of equilibrium for any possible value of $γ$ in the HARA utility function $\fracγ{1-γ}\left(b+\frac{a}γx\right)^{1-γ}$. The analysis contributes to the literature on uniqueness in pure exchange economies with two-goods and two agent types and extends the result in [4].

econ.TH↗
arXiv2023

Uniqueness of equilibrium and redistributive policies: a geometric approach to efficiency

This paper examines the relationship between resource reallocation, uniqueness of equilibrium and efficiency in economics. We explore the implications of reallocation policies for stability, conflict, and decision-making by analysing the existence of geodesic coordinate functions in the equilibrium manifold. Our main result shows that in an economy with M = 2 consumers and L goods, if L coordinate functions, representing policies, are geodesics on the equilibrium manifold (a property that we call the finite geodesic property), then the equilibrium is globally unique. The presence of geodesic variables indicates optimization and efficiency in the economy, while non-geodesic variables add complexity. Finally, we establish a link between the existing results on curvature, minimal entropy, geodesics and uniqueness in smooth exchange economies. This study contributes to the understanding of the geometric and economic properties of equilibria and offers potential applications in policy considerations. Keywords: Uniqueness of equilibrium, redistributive policies, geodesics, equilibrium manifold, equilibrium selection, curvature, geodesics.

econ.TH↗
arXiv2023

Ricci iterations of well-behaved Kähler metrics

We introduce a large class of canonical Kähler metrics, called in this paper well-behaved, extending metrics induced by complex space forms. We study Kähler--Ricci iterations of well-behaved metrics on compact and non-compact Kähler manifolds. That is, we are interested in well-behaved metrics for which the iteration of the Ricci operator is a multiple of a Kähler metric, i.e., $ρ_ω^k=λΩ$. In particular, when $k=1$, under some condition on the maximal domain of definition of canonical coordinates, we show that $λ$ is forced to be positive. Moreover, for arbitrary $k$, we prove two additional results. Namely, if $ω$ and $Ω$ are induced by a flat metric, then $ω$ is Ricci-flat. Finally, if a Kähler-Ricci soliton $Ω$ arises as Kähler--Ricci iteration of a metric $ω$ induced by a complex space form, then the Kähler--Ricci soliton is forced to be trivial, that is, Kähler--Einstein. These three theorems extend well known results on Kähler--Einstein metrics to higher iterations of the Ricci operator and a larger class of metrics.

math.DG↗
arXiv2023

Immersions into Sasakian space forms

We study immersions of Sasakian manifolds into finite and infinite dimensional Sasakian space forms. After proving Calabi's rigidity results in the Sasakian setting, we characterise all homogeneous Sasakian manifolds which admit a (local) Sasakian immersion into a nonelliptic Sasakian space form. Moreover, we give a characterisation of homogeneous Sasakian manifolds which can be embedded into the standard sphere both in the compact and noncompact case.

math.DG↗
arXiv2022

Equilibrium selection: a geometric approach

In this paper we propose a geometric approach to the selection of the equi- librium price. After a perturbation of the parameters, the new price is selected thorough the composition of two maps: the projection on the linearization of the equilibrium man- ifold, a method that underlies econometric modeling, and the exponential map, that associates a tangent vector with a geodesic on the manifold. As a corollary of our main result, we prove the equivalence between zero curvature and uniqueness of equilibrium in the case of an arbitrary number of goods and two consumers, thus extending the previous result by [6].

econ.TH↗
arXiv2022

On holomorphic isometries into blow-ups of $\mathbb C^n$

We study the Kähler-Einstein manifolds which admits a holomorphic isometry into either the generalized Burns-Simanca manifold $(\tilde {\mathbb C}^n, g_S)$ or the Eguchi-Hanson manifold $(\tilde {\mathbb C}^2, g_{EH})$. Moreover, we prove that $(\tilde {\mathbb C}^n, g_S)$ and $(\tilde {\mathbb C}^2, g_{EH})$ are not relatives to any homogeneous bounded domain.

math.DG↗
arXiv2022

Embeddings of metric Boolean algebras in $\mathbb{R}^{N}$

A Boolean algebra $\A$ equipped with a (finitely-additive) positive probability measure $m$ can be turned into a metric space $(\A , d_{m})$, where $d_{m}(a,b)= m ((a\wedge\neg b)\vee(\neg a\wedge b))$, for any $a,b\in A$, sometimes referred to as \emph{metric Boolean algebra}. In this paper, we study under which conditions the space of atoms of a finite metric Boolean algebra can be isometrically embedded in $\mathbb{R}^{N}$ (for a certain $N$) equipped with the Euclidean metric. In particular, we characterize the topology of the positive measures over a finite algebra $\A$ such that the metric space $(\mathsf{At}(\A), d_m)$ embeds isometrically in $\mathbb{R}^{N}$.

math.LO↗
arXiv2022

Some characterizations of the complex projective space via Ehrhart polynomials

Let $P_{λΣ_n}$ be the Ehrhart polynomial associated to an intergal multiple $λ$ of the standard symplex $Σ_n \subset \mathbb{R}^n$. In this paper we prove that if $(M, L)$ is an $n$-dimensional polarized toric manifold with associated Delzant polytope $Δ$ and Ehrhart polynomial $P_Δ$ such that $P_Δ=P_{λΣ_n}$, for some $λ\in \mathbb{Z}^+$, then $(M, L)\cong (\mathbb{C} P^n, O(λ))$ (where $O(1)$ is the hyperplane bundle on $\mathbb{C} P^n$) in the following three cases: 1. arbitrary $n$ and $λ=1$, 2. $n=2$ and $λ=3$, 3. $λ=n+1$ under the assumption that the polarization $L$ is asymptotically Chow semistable.

math.DG↗
arXiv2022

Holomorphic isometries into homogeneous bounded domains

We prove two rigidity theorems on holomorphic isometries into homogeneous bounded domains. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of a homogeneous bounded domain is trivial, i.e. Kähler-Einstein. In the second one we prove that a homogeneous bounded domain and the flat (definite or indefinite) complex Euclidean space are not relatives, i.e. they do not share a common Kähler submanifold (of positive dimension). Our theorems extend the results proved in [A. Loi, R. Mossa, Proc. Amer. Math. Soc. 149 (2021), no. 11, 4931-4941] and [X. Cheng, Y. Hao, Ann. Global Anal. Geom. 60 (2021), no. 1, 167-180] respectively.

math.DG↗
arXiv2021

Robustness of statistical models

A statistical structure $(g, T)$ on a smooth manifold $M$ induced by $(\tilde M, \tilde g, \tilde T)$ is said to be {\em robust} if there exists an open neighborhood of $(g,T)$ in the fine $C^{\infty}$-topology consisting of statistical structures induced by $(\tilde M, \tilde g, \tilde T)$. Using Nash--Gromov implicit function theorem, we show robustness of the generic statistical structure induced on $M$ by the standard linear statistical structure on ${\R}^N$, for $N$ sufficiently large.

math.ST↗
arXiv2021

Risk aversion and uniqueness of equilibrium: a polynomial approach

We study the connection between risk aversion, number of consumers and uniqueness of equilibrium. We consider an economy with two goods and $c$ impatience types, where each type has additive separable preferences with HARA Bernoulli utility function, $u_H(x):=\fracγ{1-γ}\left(b+\frac{a}γx\right)^{1-γ}$. We show that if $γ\in \left(1, \frac{c}{c-1}\right]$, the equilibrium is unique. Moreover, the methods used, involving Newton's symmetric polynomials and Descartes' rule of signs, enable us to offer new sufficient conditions for uniqueness in a closed-form expression highlighting the role played by endowments, patience and specific HARA parameters. Finally, new necessary and sufficient conditions in ensuring uniqueness are derived for the particular case of CRRA Bernoulli utility functions with $γ=3$.

econ.TH↗
arXiv2021

On canonical radial Kaehler metrics

We prove that a radial Kaehler metric g is Kaehler-Einstein if and only if one of the following conditions is satisfied: 1. g is extremal and it is associated to a Kaehler-Ricci soliton; 2. two different generalized scalar curvatures of g are constant; 3. g is extremal (not cscK) and one of its generalized scalar curvature is constant.

math.DG↗
arXiv2021

Kaehler Ricci solitons induced by infinite dimensional complex space forms

We exhibit families of non trivial (i.e. not Kaehler-Einstein) radial Kaehler-Ricci solitons (KRS), both complete and not complete, which can be Kaehler immersed into infinite dimensional complex space forms. This result shows that the triviality of a KRS induced by a finite dimensional complex space form proved in [12] does not hold when the ambient space is allowed to be infinite dimensional. Moreover, we show that the radial potential of a radial KRS induced by a non-elliptic complex space form is necessarily defined at the origin.

math.DG↗
arXiv2021

Minimal entropy and uniqueness of price equilibria in a pure exchange economy

We introduce uncertainty into a pure exchange economy and establish a connection between Shannon's differential entropy and uniqueness of price equilibria. The following conjecture is proposed under the assumption of a uniform probability distribution: entropy is minimal if and only if the price is unique for every economy. We show the validity of this conjecture for an arbitrary number of goods and two consumers and, under certain conditions, for an arbitrary number of consumers and two goods.

econ.TH↗
arXiv2020

Strictly regular and cscK metrics

A Kaehler metric $g$ with integral Kaehler form is said to be partially regular if the partial Bergman kernel associated to mg is a positive constant for all integer m sufficiently large. The aim of this paper is to prove that for all n\geq 2 there exists an n-complex dimensional manifold equipped with strictly partially regular and cscK metric g. Further, for n\geq 3, the (constant) scalar curvature of g can be chosen to be zero, positive or negative.

math.DG↗
arXiv2020

Extremal Kaehler metrics induced by finite or infinite dimensional complex space forms

In this paper we address the problem of studying those complex manifolds $M$ equipped with extremal metrics $g$ induced by finite or infinite dimensional complex space forms. We prove that when $g$ is assumed to be radial and the ambient space is finite dimensional then $(M, g)$ is itself a complex space form. We extend this result to the infinite dimensional setting by imposing the strongest assumption that the metric $g$ has constant scalar curvature and is well-behaved (see Definition 1 in the Introduction). Finally, we analyze the radial Kaehler-Einstein metrics induced by infinite dimensional elliptic complex space forms and we show that if such a metric is assumed to satisfy a stability condition then it is forced to have constant non-positive holomorphic sectional curvature.

math.DG↗
arXiv2020

$η$-Einstein Sasakian immersions in non-compact Sasakian space forms

The aim of this paper is to study Sasakian immersions of (non-compact) complete regular Sasakian manifolds into the Heisenberg group and into $ \mathbb{B}^N\times \mathbb{R}$ equipped with their standard Sasakian structures. We obtain a complete classification of such manifolds in the $η$-Einstein case.

math.DG↗