# General Equilibrium with Price Adjustments—A Dynamic Programming Approach

## Abstract

**:**

## 1. Introduction

## 2. The Model Economy

#### 2.1. Individual Optimization: The Primal Problem and the Dual Problem

#### 2.2. The Dynamic Optimization Program for the Economy as a Whole

## 3. Price Dynamics and Lyapunov Stability of the Economy

#### Conditions for Lyapunov Stability of the Economy

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Mirowski, P. More Heat than Light, Economics as Social Physics, Physics as Nature’s Economics; Cambridge University Press: Cambridge, UK, 1989. [Google Scholar]
- Arrow, K.J. An Extension of the Basic Theorems of Classical Welfare Economics. Berkeley Symp. Math. Stat. Probab.
**1951**, 2, 507–532. [Google Scholar] - Walras, L. Elements of Pure Economics; Routledge: London, UK, 2003; (1874). [Google Scholar]
- Arrow, K.J.; Debreu, G. Existence of an Equilibrium for a Competitive Economy. Econometrica
**1954**, 22, 265–290. [Google Scholar] [CrossRef] - von Neumann, J.V. A Model of General Economic Equilibrium. Rev. Econ. Stud.
**1945**, 13, 1–9. [Google Scholar] [CrossRef] - Smale, S. Dynamics in General Equilibrium Theory. Am. Econ. Rev.
**1976**, 66, 288–294. Available online: https://www.jstor.org/stable/1817235 (accessed on 15 May 2022). - Scarf, H. Some Examples of Global Instability of the Competitive Equilibrium. Int. Econ. Rev.
**1960**, 1, 157–172. [Google Scholar] [CrossRef][Green Version] - Gale, D. A Note on Global Instability of Competitive Equilibrium. Nav. Res. Logist. Q.
**1963**, 10, 81–87. [Google Scholar] [CrossRef] - Samuelson, P. Foundations of Economic Analysis; Harvard University Press: Cambridge, MA, USA, 1947. [Google Scholar]
- Mas-Colell, A.; Whinston, M.; Green, J. Microeconomic Theory; Oxford University Press: Oxford, UK, 1995. [Google Scholar]
- Hahn, F.H.; Negishi, T. A Theorem on Non-Tatonnement Stability. Econometrica
**1962**, 30, 463–469. [Google Scholar] [CrossRef] - Champsaur, P.; Cornet, B. Walrasian exchange processes. In Economic Decision Making, Games, Econometrics and Optimisation; Elsevier Sciences Publishers B.V.: Amsterdam, The Netherlands, 1990. [Google Scholar]
- Balasko, Y. Out-of-equilibrium price dynamics. Econ. Theory
**2007**, 33, 413–435. Available online: http://www.jstor.org/stable/27822608 (accessed on 15 May 2022). - Bonnisseau, J.-M.; Nguenamadji, O. Discrete Walrasian exchange process. Econ. Theory
**2013**, 52, 1091–1100. Available online: http://www.jstor.org/stable/23470168 (accessed on 15 May 2022). - Goeree, J.K.; Lindsay, L. Market design and the stability of general equilibrium. J. Econ. Theory
**2016**, 165, 37–68. [Google Scholar] [CrossRef][Green Version] - Mandel, A.; Gintis, H. Decentralized Pricing and the equivalence between Nash and Walrasian equilibrium. J. Math. Econ.
**2016**, 63, 84–92. [Google Scholar] [CrossRef][Green Version] - Fisher, F.M. Disequilibrium Foundations of Equilibrium Economics (Econometric Society Monographs); Cambridge University Press: Cambridge, UK, 1983. [Google Scholar] [CrossRef]
- Bala, V.; Majumdar, M. Chaotic Tatonnement. Econ. Theory
**1992**, 2, 437–445. Available online: https://www.jstor.org/stable/25054664 (accessed on 15 May 2022). - Danziger, L. A Dynamic Economy with Costly Price Adjustments. Am. Econ. Rev.
**1999**, 89, 878–901. Available online: http://www.jstor.org/stable/117163 (accessed on 15 May 2022). - Bentes, S.R.; Menezes, R.; Mendes, D.A. Long memory and volatility clustering: Is the empirical evidence consistent across stock markets? Phys. A Stat. Mech. Its Appl.
**2008**, 387, 3826–3830. [Google Scholar] [CrossRef][Green Version] - Ran, M.; Tang, Z.; Chen, W. The Influence of Trading Volume, Market Trend, and Monetary Policy on Characteristics of the Chinese Stock Exchange: An Econophysics Perspective. Open Phys.
**2019**, 17, 985–998. [Google Scholar] [CrossRef] - von Mises, L. Die Wirtschaftsrechnung im sozialistischen Gemeinwesen. Arch. Für Soz
**1920**, 47, 86–121. [Google Scholar] - von Hayek, F. Individualism and Economic Order; University of Chicago Press: Chicago, IL, USA, 1948. [Google Scholar]
- Lange, O. On the Economic Theory of Socialism: Part One. Rev. Econ. Stud.
**1936**, 4, 53–71. [Google Scholar] [CrossRef] - Bellman, R. Dynamic Programming; Princeton University Press: Princeton, NJ, USA, 1957. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lindgren, J.
General Equilibrium with Price Adjustments—A Dynamic Programming Approach. *Analytics* **2022**, *1*, 27-34.
https://doi.org/10.3390/analytics1010003

**AMA Style**

Lindgren J.
General Equilibrium with Price Adjustments—A Dynamic Programming Approach. *Analytics*. 2022; 1(1):27-34.
https://doi.org/10.3390/analytics1010003

**Chicago/Turabian Style**

Lindgren, Jussi.
2022. "General Equilibrium with Price Adjustments—A Dynamic Programming Approach" *Analytics* 1, no. 1: 27-34.
https://doi.org/10.3390/analytics1010003