SIMPLE PROOF OF EXISTENCE OF EQUILIBRIUM IN A ONE-SECTOR GROWTH MODEL WITH BOUNDED OR UNBOUNDED RETURNS FROM BELOW

2003 ◽  
Vol 7 (3) ◽  
pp. 317-332 ◽  
Author(s):  
Jorge Durán ◽  
Cuong Le Van
Keyword(s):  
Growth Model ◽  
Simple Proof ◽  
Competitive Equilibrium ◽  
Direct Approach ◽  
Standard Approach ◽  
Existence Of Equilibrium ◽  
Competitive Equilibria

We analyze a Ramsey economy in which gross investment is constrained to be nonnegative. We prove the existence of a competitive equilibrium for the case in which utility need not be bounded from below and the Inada-type conditions need not hold. The analysis is carried out by means of a direct and technically standard approach. This direct approach allows us to obtain detailed results concerning properties of competitive equilibria, and is easily adapted for the analysis of analogous models often found in macroeconomics.

Brauer's Fixed Point Theorem

2021 ◽  
Vol 8 (1) ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Simple Proof ◽  
Exchange Model ◽  
Existence Of Equilibrium ◽  
Economic Applications ◽  
Brouwer’S Fixed Point Theorem ◽  
Brouwer's Fixed Point Theorem

The primary goal of the paper is to deliver a simple proof of equivalence between Brouwer’s fixed point theorem and the existence of equilibrium in a simple exchange model with monotonic consumers. To achieve this end, we discuss some equivalent formulations of Brouwer’s theorem and prove additional ones, that are ’approximating’ in character or seem to be better suited for economic applications than the standard results.

On an “Important Principle” of Arrow and Debreu

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Arnis Vilks
Keyword(s):  
Simple Model ◽  
Technological Progress ◽  
Competitive Equilibrium ◽  
Private Ownership ◽  
Existence Of Equilibrium ◽  
Important Principle

Abstract In their seminal 1954 paper on the existence of competitive equilibrium, Arrow and Debreu state what they call an “important principle”, namely that it is necessary for the existence of equilibrium that every consumer has some asset or can supply some labour service which has a positive price at equilibrium. It does not seem to have been noticed that this claim is incorrect. We provide a very simple model of a private ownership economy with three goods where a competitive equilibrium exists, but consumers who have nothing to sell but their labour end up with zero wealth in equilibrium. As zero wealth must be taken to mean non-survival, and the Arrow–Debreu model is frequently interpreted as assuming that all consumers can survive without trade, we also briefly discuss the issue of non-survival in equilibrium. We finally point out that our example illustrates the possibility that technological progress may result in a situation where the value of work becomes negligible.

scholarly journals Existence of General Competitive Equilibria: A Variational Approach

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
G. Anello ◽  
F. Rania
Keyword(s):  
Variational Inequalities ◽  
Finite Number ◽  
Variational Methods ◽  
Variational Approach ◽  
Existence Result ◽  
Competitive Equilibrium ◽  
Utility Functions ◽  
Locally Lipschitz ◽  
Competitive Equilibria

We study the existence of general competitive equilibria in economies with agents and goods in a finite number. We show that there exists a Walras competitive equilibrium in all ownership private economies such that, for all consumers, initial endowments do not contain free goods and utility functions are locally Lipschitz quasiconcave. The proof of the existence of competitive equilibria is based on variational methods by applying a theoretical existence result for Generalized Quasi Variational Inequalities.

Stability and Generalized Competitive Equilibria in a Many-to-Many Gale-Shapley Market Model

2016 ◽  
Vol 63 (3) ◽  
pp. 237-254
Author(s):  
Zbigniew Świtalski
Keyword(s):  
Competitive Equilibrium ◽  
College Admissions ◽  
Supply And Demand ◽  
Type A ◽  
Market Model ◽  
Stable Matchings ◽  
Competitive Equilibria

We define, for some variant of a many-to-many market model of Gale-Shapley type, a concept of generalized competitive equilibrium and show that, under suitable conditions, stable matchings in such a model can be represented as competitive equilibria allocations (and vice versa). Our results are far-reaching generalizations of the “discrete supply and demand lemma” of Azevedo, Leshno (2011) for the college admissions market.Using the results of Alkan, Gale (2003), we also prove a theorem on existence of generalized equilibria in our model.

scholarly journals Competitive Equilibrium and Trading Networks: A Network Flow Approach

2020 ◽  
Author(s):  
Ozan Candogan ◽  
Markos Epitropou ◽  
Rakesh V. Vohra
Keyword(s):  
Network Flow ◽  
Competitive Equilibrium ◽  
Flow Problem ◽  
Comparative Statics ◽  
Bilateral Contracts ◽  
Submodular Flow ◽  
Competitive Equilibria ◽  
Trading Networks ◽  
The Impact ◽  
Shed Light

This paper considers a network of agents who trade indivisible goods or services via bilateral contracts. Under a substitutability assumption on preferences, it is known that a competitive equilibrium exists. In “Competitive Equilibrium and Trading Networks: A Network Flow Approach,” Candogan, Epitropou, and Vohra show how to determine equilibrium outcomes as a generalized submodular flow problem. Existence of a competitive equilibrium and its equivalence to seemingly weaker notions of stability follow directly from the optimality conditions of the flow problem. The formulation enables the authors to perform comparative statics with respect to the number of buyers, sellers, and trades. In particular, they are able to shed light on the impact of new trading opportunities on the equilibrium trades, prices, and surpluses. In addition, they present algorithms for finding competitive equilibria in trading networks and testing stability.

scholarly journals Growth and ideas in a perfectly competitive world: A complement

2021 ◽  
Vol 10 (2) ◽  
pp. 157-163
Author(s):  
Vincent Boitier
Keyword(s):  
Growth Model ◽  
Monopolistic Competition ◽  
Competitive Equilibrium ◽  
Returns To Scale ◽  
Perfect Competition ◽  
Household Economy ◽  
Short Article ◽  
Long Run ◽  
Balanced Growth Path ◽  
The One

In this short article, I build an idea-based growth model with perfect competition in a representative household economy. I obtain significant findings that confirm Boitier (2019). First, a competitive equilibrium, increasing returns to scale, and innovations can be tenable. For that, firms must raise capital from shareholders, and the production function must show decreasing returns to scale in the stock of ideas and in labor. Second, the developed idea-based growth model admits a balanced growth path similar to the one provided in an idea-based growth model with monopolistic competition. Whether innovations are competitive or thrive under monopolistic competition does not constitute an engine-driving long-run growth. Importantly, this reconciles Romer (1990, 2015) with Boldrin and Levine (2008).

Statistical Representations

2013 ◽  
Author(s):  
Lars Peter Hansen ◽  
Thomas J. Sargent
Keyword(s):  
Competitive Equilibrium ◽  
Likelihood Function ◽  
Linear Functions ◽  
State Variables ◽  
Vector Autoregressive ◽  
Competitive Equilibria ◽  
History Of ◽  
Complete History ◽  
Recursive Representation ◽  
Over Time

This chapter describes links between competitive equilibria and autoregressive representations. It shows how to obtain an autoregressive representation for observable variables that are error-ridden linear functions of state variables. In describing how to deduce an autoregressive representation from a competitive equilibrium and parameters of measurement error processes, it completes a key step that facilitates econometric estimation of free parameters. An autoregressive representation is naturally affiliated with a recursive representation of a likelihood function for the observable variables. More precisely, a vector autoregressive representation implements a convenient factorization of the joint density of a complete history of observables (i.e., the likelihood function) into a product of densities of time t observables conditioned on histories of those observables up to time t−1. The chapter also treats two other topics intimately related to econometric implementation: aggregation over time and the theory of approximation of one model by another.

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