We describe an exchange market consisting of many agents with stochastic pref- erences for two goods. When individuals are indifferent between goods, statistical mechanics predicts that goods and wealth will have steady-state gamma distributions. Simulation studies show that gamma distributions arise for a broader class of pref- erence distributions. We demonstrate this mathematically in the limit of large numbers of individual agents. These studies illustrate the potential power of a statis- tical mechanical approach to stochastic models in economics and suggest that gamma distributions will describe steady-state wealths for a class of stochastic models with periodic redistribution of conserved quantities.
The document analyzes the relationship between monetary supply and price level in China using econometric methods. Standard OLS regression shows a positive relationship, but these results are questionable due to potential spurious regression. Time series tests find inconsistent results between different tests on the stationarity of macroeconomic variables. The analysis lays a foundation for further study but more advanced models and tests are needed to fully understand the relationship between money supply and prices in China.
APPLICATION OF THE METHOD OF VARIATION OF PARAMETERS: MATHEMATICAL MODEL FOR ...IJESM JOURNAL
In this paper, a second order wage equation is developed and solved by the method of variation of parameters. The subsequent wage function is then analyzed and interpreted for stability. Speculative parameters, which operate freely dictating employers’ expectations, are included in modeling this equation. The variation of these parameters causes both stability and instability of the wage function depending on circumstances. Where the wage function is exponential, asymptotic stability towards the equilibrium wage rate is observed but where it consists of both exponential and periodic factors, the time path shows periodic fluctuations with successive cycles giving smaller amplitudes until the ripples die naturally. It has been realized that where the wage rate is determined by free market forces of demand and supply, volatility in wage rate may be observed if not controlled. This may increase uncertainties and cause anxiety about investment and employment in the economy. The paper therefore proposes government intervention by creating a middle path in which wage rate is allowed to oscillate freely within a narrow band managed by employers in consultation with the workers under the watch of the government.
Sabatelli relationship between the uncompensated price elasticity and the inc...GLOBMOD Health
Income and price elasticity of demand quantify the responsiveness of markets to changes in income and in prices, respectively. Under the assumptions of utility maximization and preference independence (additive preferences), mathematical relationships between income elasticity values and the uncompensated own and cross price elasticity of demand are here derived using the differential approach to demand analysis.
Citation: Sabatelli L (2016) Relationship between the Uncompensated Price Elasticity and the Income Elasticity of Demand under Conditions of Additive Preferences. PLoS ONE 11(3): e0151390. doi:10.1371/journal.pone.0151390
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0151390
This document discusses the concept of long-run position in a capitalist economy and examines perspectives from Post-Keynesian and Neo-Ricardian schools of thought. It argues that competition is a dynamic process that creates both equilibrium and disequilibrium forces. While competition equalizes profit rates across sectors, it also drives capitalists to constantly improve production and try to outpace rivals. The long-run position should not be seen as a static equilibrium, but rather the center of gravity resulting from these dynamic competitive forces. Investment plays a key role in determining long-run positions by changing the economy's capacity level over time through capital accumulation. Short-run fluctuations in demand are accommodated by changes in capacity utilization, while long-
The aim of this paper is to re-interprete the accelerationist Phillips curve, by studying the effect of the higher-order derivatives of acceleration. We show that a complex dynamic behavior emerges when dealing with a jerk and jounce displacement of price settings, whose simple unfolding leads to a three dimensional vector field which generates a double-scroll chaotic attractor.
Labor welfare based on the intensity of job searchAlexander Decker
This study examines differences in worker welfare based on job search intensity in Makassar, Indonesia. A decomposition analysis found that workers who obtained jobs through word-of-mouth had higher welfare than those using more advanced technologies. Work experience and unemployment duration explained much of the difference, as longer work experience increases wages while longer unemployment lowers reservation wages. Workers searching less intensely expected welfare gains from changes in employment status or education level, while more intense searchers focused on status and education. In general, more intense searching was found to have a negative impact on welfare.
- The document discusses regression analysis using dummy variables to represent qualitative variables like gender, race, or region. Dummy variables take values of 1 or 0 to indicate presence or absence of a quality.
- It cautions that if a qualitative variable has m categories, only introduce (m-1) dummy variables to avoid perfect collinearity. The omitted category serves as the base for comparisons.
- Regression models containing both quantitative and qualitative variables are called ANCOVA models. They control for the effects of quantitative covariates when examining relationships between dummy and dependent variables.
- An example model examines how public school teachers' salaries vary by region and spending per pupil on education, showing the effect of controlling for the quantitative covari
The document analyzes the relationship between monetary supply and price level in China using econometric methods. Standard OLS regression shows a positive relationship, but these results are questionable due to potential spurious regression. Time series tests find inconsistent results between different tests on the stationarity of macroeconomic variables. The analysis lays a foundation for further study but more advanced models and tests are needed to fully understand the relationship between money supply and prices in China.
APPLICATION OF THE METHOD OF VARIATION OF PARAMETERS: MATHEMATICAL MODEL FOR ...IJESM JOURNAL
In this paper, a second order wage equation is developed and solved by the method of variation of parameters. The subsequent wage function is then analyzed and interpreted for stability. Speculative parameters, which operate freely dictating employers’ expectations, are included in modeling this equation. The variation of these parameters causes both stability and instability of the wage function depending on circumstances. Where the wage function is exponential, asymptotic stability towards the equilibrium wage rate is observed but where it consists of both exponential and periodic factors, the time path shows periodic fluctuations with successive cycles giving smaller amplitudes until the ripples die naturally. It has been realized that where the wage rate is determined by free market forces of demand and supply, volatility in wage rate may be observed if not controlled. This may increase uncertainties and cause anxiety about investment and employment in the economy. The paper therefore proposes government intervention by creating a middle path in which wage rate is allowed to oscillate freely within a narrow band managed by employers in consultation with the workers under the watch of the government.
Sabatelli relationship between the uncompensated price elasticity and the inc...GLOBMOD Health
Income and price elasticity of demand quantify the responsiveness of markets to changes in income and in prices, respectively. Under the assumptions of utility maximization and preference independence (additive preferences), mathematical relationships between income elasticity values and the uncompensated own and cross price elasticity of demand are here derived using the differential approach to demand analysis.
Citation: Sabatelli L (2016) Relationship between the Uncompensated Price Elasticity and the Income Elasticity of Demand under Conditions of Additive Preferences. PLoS ONE 11(3): e0151390. doi:10.1371/journal.pone.0151390
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0151390
This document discusses the concept of long-run position in a capitalist economy and examines perspectives from Post-Keynesian and Neo-Ricardian schools of thought. It argues that competition is a dynamic process that creates both equilibrium and disequilibrium forces. While competition equalizes profit rates across sectors, it also drives capitalists to constantly improve production and try to outpace rivals. The long-run position should not be seen as a static equilibrium, but rather the center of gravity resulting from these dynamic competitive forces. Investment plays a key role in determining long-run positions by changing the economy's capacity level over time through capital accumulation. Short-run fluctuations in demand are accommodated by changes in capacity utilization, while long-
The aim of this paper is to re-interprete the accelerationist Phillips curve, by studying the effect of the higher-order derivatives of acceleration. We show that a complex dynamic behavior emerges when dealing with a jerk and jounce displacement of price settings, whose simple unfolding leads to a three dimensional vector field which generates a double-scroll chaotic attractor.
Labor welfare based on the intensity of job searchAlexander Decker
This study examines differences in worker welfare based on job search intensity in Makassar, Indonesia. A decomposition analysis found that workers who obtained jobs through word-of-mouth had higher welfare than those using more advanced technologies. Work experience and unemployment duration explained much of the difference, as longer work experience increases wages while longer unemployment lowers reservation wages. Workers searching less intensely expected welfare gains from changes in employment status or education level, while more intense searchers focused on status and education. In general, more intense searching was found to have a negative impact on welfare.
- The document discusses regression analysis using dummy variables to represent qualitative variables like gender, race, or region. Dummy variables take values of 1 or 0 to indicate presence or absence of a quality.
- It cautions that if a qualitative variable has m categories, only introduce (m-1) dummy variables to avoid perfect collinearity. The omitted category serves as the base for comparisons.
- Regression models containing both quantitative and qualitative variables are called ANCOVA models. They control for the effects of quantitative covariates when examining relationships between dummy and dependent variables.
- An example model examines how public school teachers' salaries vary by region and spending per pupil on education, showing the effect of controlling for the quantitative covari
ICLP estudio tamano del mercado de fidelizacion 2010A Meili
first study about global size of Frequent Flyer Programmes and income genereated through mileage sales. Figures brakedowon into continents, presented at the CILA airlines conference in LatAM, May 2010 - in Spanish.
Broad outlines of the policies, programs and procedures that might be investi...Javier Armaolea
This document discusses diversity initiatives at a Spanish company. It outlines 6 key business activities related to diversity: 1) focusing on changing customer needs, 2) developing supplier relationships, 3) setting diversity objectives and goals, 4) strategies for attracting, promoting, and retaining a diverse workforce, 5) human resources policies to support diversity, and 6) training to foster understanding and skills for a diverse workplace. Training is highlighted as an essential component to communicate the company's commitment to diversity and develop skills for maximizing its benefits.
El documento habla sobre una persona que se queja de que su cruz es demasiado pesada y le pide al Señor que la acorte para poder llevarla mejor. Aunque el Señor accede a acortarla, la persona se da cuenta de que la cruz ya no es suficiente para usarla como puente y cruzar.
The document outlines the itinerary and logistical details for a group traveling from Washington D.C. to Madrid, Spain for World Youth Day from August 14-23. The group will fly to Madrid, attend opening ceremonies on August 16, participate in catechetical sessions from August 17-19, attend the vigil with Pope Benedict on August 20 and the closing mass on August 21. The return flight to D.C. is on August 23. The document provides information on meals, transportation, luggage, necessary documents, and schedules of events.
Este documento resume varias noticias locales. La comunidad latina de Bentonville recaudó fondos para el gerente anglosajón de un hotel que padece cáncer. Un maestro de Fayetteville fue arrestado bajo sospecha de tratar de obtener pornografía infantil. Tres personas, incluyendo un pandillero conocido, fueron arrestadas luego de que la policía encontrara metanfetaminas cerca de un bebé durante una parada de tráfico. Arkansas cerró sus puertas a recibir refugiados sirios luego de los at
El documento describe la historia y cultivo del maíz en Asturias. Se introdujo en el siglo XVI y ahora es un alimento básico. Describe las partes de la planta y del maíz, incluyendo usos medicinales. Explica herramientas y procesos para cultivarlo, y platos típicos como tortos, boroña y fariñes. También habla de la esfoyaza, la recolección tradicional del maíz, y coples y refranes relacionados con esta actividad.
The document discusses the growing investment transaction market in central and eastern Europe, noting that while the region generates 20% of European GDP and contains 35% of modern European office space, it only attracts 7% of total real estate investment in Europe. Major trends in 2014 included rising investment volumes across many CEE countries and a large number of office building acquisitions. Going forward, the document suggests interest may grow in secondary cities across the region as investors seek to diversify risks beyond capital cities.
1) Search funnels provide insights into the multi-step process consumers undergo from initial search to conversion.
2) Key metrics like assisted conversions and time lag reveal that the last click attribution model is often misleading as many conversions involve multiple interactions over time.
3) Analyzing top paths, assisted keywords, and time factors between stages allows optimizing budgets and messaging to better assist consumers at each point in the funnel.
Este documento presenta una serie de páginas educativas en línea que ofrecen recursos para docentes y estudiantes. Incluye sitios como Eduteka, que provee estrategias de enseñanza y aprendizaje organizadas por materia; Adivinancero, que contiene adivinanzas clasificadas por tema y juegos; y Frosti, con juegos interactivos sobre diferentes temas académicos. También presenta El Huevo de Chocolate, con actividades como acertijos, canciones y cuentos; y Recursos
El documento presenta comentarios positivos de varios clientes sobre cómo el programa "Mejora del Rendimiento y Organización Personal" les ha ayudado a mejorar su productividad, organización del tiempo y satisfacción laboral mediante la identificación de tareas de alta rentabilidad, establecimiento de metas y mejor delegación y planificación.
Kangdi OEM&ODM chili plaster manufacturerJen Shien
Kangdi OEM&ODM chili plaster manufacturer can help you to relief muacle/back/shoulder/bone/bruises/arthritis/minor aches pain. you can custom capsicum plaster.
soleg group AG is an international distributor and project developer for regenerative energy systems. The company is active in the fields of photovoltaics, solar thermal, heating with wood, Sonnenhaus technology and wind energy. soleg was founded in 1994. Apart from its headquarters in Germany, the company is represented in the Czech Republic, Italy, Austria and Greece
Listado de distribuidores de productos de Evangelización Activa del Padre Ernesto Maria Caro www.evangelizacion.mx Tienda en linea www.integractiva.com.mx
Este documento anuncia las actividades navideñas para niños que se llevarán a cabo en Baby Deli, incluyendo talleres de manualidades recicladas, decoración de galletas, desayunos con Olentzero y los Reyes Magos, y clases de cocina con temática navideña.
Sunamp is a thermal energy storage company headquartered in Edinburgh that has developed modular heat batteries for storing excess heat and electricity from renewable sources. Their core product is the Sunamp Heat Battery, which uses phase change materials to store and discharge heat energy at high power levels. Sunamp is seeking collaborations with Chinese companies in HVAC, energy, and automotive to refine their products for the Chinese market and expand their thermal energy storage solutions.
American Economic Association Some International Evid.docxnettletondevon
This document summarizes an empirical study by Robert Lucas examining the relationship between output and inflation using annual time series data from 18 countries between 1951-1967. Lucas develops an economic model based on the natural rate hypothesis, where supply behavior depends on expected relative prices. The model yields solutions showing inflation and output as distributed lags of nominal output changes. Estimating the model across countries allows testing the natural rate theory's implications regarding the output-inflation relationship.
Dynamic Causal Relationships among the Greater China Stock marketsAM Publications,India
This document examines the dynamic causal relationships between stock markets in Greater China, specifically Mainland China, Hong Kong, and Taiwan. It finds that the Asian financial crisis was a breakpoint, so analyses return and volatility effects over three periods: the full sample period, pre-crisis period, and post-crisis period. Return changes in Mainland China were found to spill over into Hong Kong after the crisis, which then affected returns in Taiwan. Volatility spillover effects between the markets were also examined using a bivariate GARCH model, finding different patterns between the periods.
ICLP estudio tamano del mercado de fidelizacion 2010A Meili
first study about global size of Frequent Flyer Programmes and income genereated through mileage sales. Figures brakedowon into continents, presented at the CILA airlines conference in LatAM, May 2010 - in Spanish.
Broad outlines of the policies, programs and procedures that might be investi...Javier Armaolea
This document discusses diversity initiatives at a Spanish company. It outlines 6 key business activities related to diversity: 1) focusing on changing customer needs, 2) developing supplier relationships, 3) setting diversity objectives and goals, 4) strategies for attracting, promoting, and retaining a diverse workforce, 5) human resources policies to support diversity, and 6) training to foster understanding and skills for a diverse workplace. Training is highlighted as an essential component to communicate the company's commitment to diversity and develop skills for maximizing its benefits.
El documento habla sobre una persona que se queja de que su cruz es demasiado pesada y le pide al Señor que la acorte para poder llevarla mejor. Aunque el Señor accede a acortarla, la persona se da cuenta de que la cruz ya no es suficiente para usarla como puente y cruzar.
The document outlines the itinerary and logistical details for a group traveling from Washington D.C. to Madrid, Spain for World Youth Day from August 14-23. The group will fly to Madrid, attend opening ceremonies on August 16, participate in catechetical sessions from August 17-19, attend the vigil with Pope Benedict on August 20 and the closing mass on August 21. The return flight to D.C. is on August 23. The document provides information on meals, transportation, luggage, necessary documents, and schedules of events.
Este documento resume varias noticias locales. La comunidad latina de Bentonville recaudó fondos para el gerente anglosajón de un hotel que padece cáncer. Un maestro de Fayetteville fue arrestado bajo sospecha de tratar de obtener pornografía infantil. Tres personas, incluyendo un pandillero conocido, fueron arrestadas luego de que la policía encontrara metanfetaminas cerca de un bebé durante una parada de tráfico. Arkansas cerró sus puertas a recibir refugiados sirios luego de los at
El documento describe la historia y cultivo del maíz en Asturias. Se introdujo en el siglo XVI y ahora es un alimento básico. Describe las partes de la planta y del maíz, incluyendo usos medicinales. Explica herramientas y procesos para cultivarlo, y platos típicos como tortos, boroña y fariñes. También habla de la esfoyaza, la recolección tradicional del maíz, y coples y refranes relacionados con esta actividad.
The document discusses the growing investment transaction market in central and eastern Europe, noting that while the region generates 20% of European GDP and contains 35% of modern European office space, it only attracts 7% of total real estate investment in Europe. Major trends in 2014 included rising investment volumes across many CEE countries and a large number of office building acquisitions. Going forward, the document suggests interest may grow in secondary cities across the region as investors seek to diversify risks beyond capital cities.
1) Search funnels provide insights into the multi-step process consumers undergo from initial search to conversion.
2) Key metrics like assisted conversions and time lag reveal that the last click attribution model is often misleading as many conversions involve multiple interactions over time.
3) Analyzing top paths, assisted keywords, and time factors between stages allows optimizing budgets and messaging to better assist consumers at each point in the funnel.
Este documento presenta una serie de páginas educativas en línea que ofrecen recursos para docentes y estudiantes. Incluye sitios como Eduteka, que provee estrategias de enseñanza y aprendizaje organizadas por materia; Adivinancero, que contiene adivinanzas clasificadas por tema y juegos; y Frosti, con juegos interactivos sobre diferentes temas académicos. También presenta El Huevo de Chocolate, con actividades como acertijos, canciones y cuentos; y Recursos
El documento presenta comentarios positivos de varios clientes sobre cómo el programa "Mejora del Rendimiento y Organización Personal" les ha ayudado a mejorar su productividad, organización del tiempo y satisfacción laboral mediante la identificación de tareas de alta rentabilidad, establecimiento de metas y mejor delegación y planificación.
Kangdi OEM&ODM chili plaster manufacturerJen Shien
Kangdi OEM&ODM chili plaster manufacturer can help you to relief muacle/back/shoulder/bone/bruises/arthritis/minor aches pain. you can custom capsicum plaster.
soleg group AG is an international distributor and project developer for regenerative energy systems. The company is active in the fields of photovoltaics, solar thermal, heating with wood, Sonnenhaus technology and wind energy. soleg was founded in 1994. Apart from its headquarters in Germany, the company is represented in the Czech Republic, Italy, Austria and Greece
Listado de distribuidores de productos de Evangelización Activa del Padre Ernesto Maria Caro www.evangelizacion.mx Tienda en linea www.integractiva.com.mx
Este documento anuncia las actividades navideñas para niños que se llevarán a cabo en Baby Deli, incluyendo talleres de manualidades recicladas, decoración de galletas, desayunos con Olentzero y los Reyes Magos, y clases de cocina con temática navideña.
Sunamp is a thermal energy storage company headquartered in Edinburgh that has developed modular heat batteries for storing excess heat and electricity from renewable sources. Their core product is the Sunamp Heat Battery, which uses phase change materials to store and discharge heat energy at high power levels. Sunamp is seeking collaborations with Chinese companies in HVAC, energy, and automotive to refine their products for the Chinese market and expand their thermal energy storage solutions.
American Economic Association Some International Evid.docxnettletondevon
This document summarizes an empirical study by Robert Lucas examining the relationship between output and inflation using annual time series data from 18 countries between 1951-1967. Lucas develops an economic model based on the natural rate hypothesis, where supply behavior depends on expected relative prices. The model yields solutions showing inflation and output as distributed lags of nominal output changes. Estimating the model across countries allows testing the natural rate theory's implications regarding the output-inflation relationship.
Dynamic Causal Relationships among the Greater China Stock marketsAM Publications,India
This document examines the dynamic causal relationships between stock markets in Greater China, specifically Mainland China, Hong Kong, and Taiwan. It finds that the Asian financial crisis was a breakpoint, so analyses return and volatility effects over three periods: the full sample period, pre-crisis period, and post-crisis period. Return changes in Mainland China were found to spill over into Hong Kong after the crisis, which then affected returns in Taiwan. Volatility spillover effects between the markets were also examined using a bivariate GARCH model, finding different patterns between the periods.
The document discusses the AD/AS model, focusing on long run aggregate supply (LRAS).
1) LRAS is a vertical line, representing the theoretical idea that in the long run, changes in the price level do not affect real output. This comes from classical theories about the separation of nominal and real variables.
2) LRAS can shift due to changes in factors of production like labor, capital, and natural resources. A rightward shift increases potential output while a leftward shift decreases it.
3) Factors that can cause LRAS to shift right include increased investment, population growth, and technological advances, while shifts left can occur due to depleted resources or lower investment.
Cascade process and Pareto rule: application to runoff data of two Mexican ri...IJERA Editor
This paper has two objectives: the first is to show that the annual runoff of rivers such as Conchos and Nazas (Mexico), follow the Pareto rule of 80:20 when classes are ordered from largest to less, and can be compared with cascade processes. The second objective is to show that cascade process produce the core which gives rise to fractional integral and therefore to differential equations of fractional order. Finally, we conclude that the Pareto rule is a first approach to saturation described by the complementary characteristic function, and runoff data provide the order of the temporal derivative. Therefore, cascade processes are manifested ubiquitously in nature, and show us a way to evolve towards the imbalanced and become in distribution mechanisms that turn into a transition that destroys old and build new correlations.
This document compares the Black-Scholes model and Merton jump diffusion model for accurately estimating option prices. It estimates parameters for both models using historical stock price and option data from several large companies. The results do not conclusively show that Merton jump diffusion more accurately prices derivatives, but it may better approximate moments from the data. The document provides background on geometric Brownian motion, the Black-Scholes option pricing formula, and the maximum likelihood estimation and EM algorithm used to estimate Merton jump diffusion model parameters.
This document discusses a study that analyzes the relationship between hours worked during school and academic performance using unique data from a college with a mandatory work-study program. A naive OLS regression indicates working more hours is positively associated with better academic performance, but this does not account for endogeneity of hours worked. Instrumental variable estimators can help address endogeneity but finding good instruments is difficult. The study uses new data from Berea College, where all students receive tuition scholarships and must participate in work-study, to better understand how endogeneity may bias estimates of the impact of employment on academics.
1) The document proposes a theory of coupled mode theory to model stock price formation, where bid and ask prices are represented as eigenvalues of a 2x2 price operator with eigenstates of "bid" and "ask".
2) Fluctuations in the matrix elements of the price operator result in changes to the eigenvalues, representing fluctuations in bid and ask prices.
3) The spread, which is the difference between bid and ask prices, can be modeled as the sum of intrinsic and interaction components plus risk components associated with fluctuations in those elements. When the intrinsic and interaction components are small, the spread follows a modified Bessel distribution.
1) The document analyzes the relationship between stock-commodity correlation and business cycles from 1991-2014 using regression analysis. It finds the stock-commodity correlation is positively related to periods of economic weakness, as evidenced by a positive relationship with default spread.
2) Regression models show stock-commodity correlation is serially correlated and has a negative relationship with default spread, indicating higher correlation during recessions. However, the effect of real GDP growth and inflation on correlation is unclear.
3) In conclusion, the findings are consistent with prior research that stock-commodity correlation increases during economic downturns, when firms adjust behaviors and investor pessimism rises.
Garch Models in Value-At-Risk Estimation for REITIJERDJOURNAL
Abstract:- In this study we investigate volatility forecasting of REIT, from January 03, 2007 to November 18, 2016, using four GARCH models (GARCH, EGARCH, GARCH-GJR and APARCH). We examine the performance of these GARCH-type models respectively and backtesting procedures are also conducted to analyze the model adequacy. The empirical results display that when we take estimation of volatility in REIT into account, the EGARCH model, the GARCH-GJR model, and the APARCH model are adequate. Among all these models, GARCH-GJR model especially outperforms others.
This lecture note introduces classical macroeconomic theory, which examines the linkages between interest rates, money, output, and inflation. It will first cover classical monetary theory, which assumes money is neutral and its supply only determines prices. It then presents a basic classical model of the real economy with aggregate supply and demand. Equilibrium output and interest rates are determined by the intersection of these curves. Money enters by facilitating transactions but does not affect real variables. Money demand depends on nominal income and interest rates, determining equilibrium money holdings and the price level.
This document provides an overview of different types of microeconomics analysis:
1) Micro statics refers to the equilibrium between economic variables at a single point in time. It shows the relationship between price and quantity at a moment when supply and demand curves intersect.
2) Micro dynamics analyzes how an initial equilibrium breaks down and a new equilibrium is established over time as prices and quantities change. It shows the disequilibrium process.
3) Comparative micro statics compares different equilibriums at different points in time as conditions change, but does not analyze the process of moving between equilibriums. It is concerned with explaining different equilibrium states rather than the transition between them.
Money, income, and prices in bangladesh a cointegration and causality analysisAlexander Decker
This document summarizes a study that examines the causal relationship between money, income, and prices in Bangladesh from 1972/73 to 2009/10. It finds:
1) Money, income, and prices are cointegrated, indicating a long-run relationship.
2) Bivariate analysis finds bidirectional causality between money and income, supporting neither Keynesians nor Monetarists. It finds unidirectional causality from money to prices, supporting Monetarists.
3) Trivariate analysis also finds bidirectional causality between money and income conditional on prices, and unidirectional causality from money to prices conditional on income.
So monetary policy should consider feedback effects between money and income
This document presents a discrete valuation methodology for swing options using a forest model approach. It develops numerical implementations of swing options on one-factor and two-factor mean-reverting underlying processes using binomial trees. It establishes convergence via finite difference methods and considers qualitative properties and sensitivity analysis. The methodology values swing options as a system of coupled European options and allows for various discrete models of the underlying process.
- The document analyzes forecasting volatility for the MSCI Emerging Markets Index using a Stochastic Volatility model solved with Kalman Filtering. It derives the Stochastic Differential Equations for the model and puts them into State Space form solved with a Kalman Filter.
- Descriptive statistics on the daily returns of the MSCI Emerging Markets Index ETF from 2011-2016 show a mean close to 0, standard deviation of 0.01428, negative skewness, and kurtosis close to a normal distribution. The model will be evaluated against a GARCH model.
This document is a thesis that examines potential cointegration among biotech stocks. It begins with an introduction that provides background on cointegration and discusses how little is known about cointegration of individual stocks. The document then reviews relevant theoretical frameworks for unit root testing and cointegration testing. It also discusses characteristics of biotech stocks and criteria for stock valuation. The empirical analysis will apply unit root and cointegration tests to selected biotech stock data. The results may provide insights into whether these stocks share a common stochastic trend.
This document presents a novel approach for combining individual realized volatility measures to form new estimators of asset price variability. It analyzes 30 different realized measures estimated from high frequency IBM stock price data from 1996-2007. It finds that a simple equally-weighted average of the realized measures is not outperformed by any individual measure and that combining measures provides benefits by incorporating information from different estimators. Optimal linear and multiplicative combination estimators are estimated and none of the individual measures are found to encompass all the information in other measures, further supporting the use of combination estimators.
PREDICTIVE EVALUATION OF THE STOCK PORTFOLIO PERFORMANCE USING FUZZY CMEANS A...ijfls
The aim of this paper is to investigate the trend of the return of a portfolio formed randomly or for any
specific technique. The approach is made using two techniques fuzzy: fuzzy c-means (FCM) algorithm and
the fuzzy transform, where the rules used at fuzzy transform arise from the application of the FCM
algorithm. The results show that the proposed methodology is able to predict the trend of the return of a
stock portfolio, as well as the tendency of the market index. Real data of the financial market are used from
2004 until 2007.
This document describes a stock price model based on the voter model from statistical physics. The model represents stock investors on a lattice who can be in a buying, selling, or neutral position based on interactions with neighboring investors. The stock price is then derived from the positions of investors. Computer simulations of the 1D model are presented, showing the fluctuations of generated stock prices and returns. The document also discusses properties found in real stock market data, such as power law distributions and volatility clustering, that the model aims to replicate.
Similar to Statistical Equilibrium Wealth Distributions in an Exchange Economy with Stochastic Preferences (20)
Viral Protein Structure Predictions - Consensus StrategyKeiji Takamoto
This document proposes a "consensus model" approach to predict the structure of viral surface proteins using multiple sequences. It generates structural decoys for 16 diverse sequences of 5 viral proteins using Rosetta. It clusters the decoys and identifies a "consensus cluster" representing the common structure shared among the diverse sequences. Evaluation on benchmark proteins shows this approach can capture the overall protein fold despite genetic distance between sequences. This consensus structure may help design immunogens to elicit broadly neutralizing antibodies against viruses.
Monovalent Cations Mediate Formation of Native Tertiary Structure of Tetrahym...Keiji Takamoto
Monovalent cations like sodium ions can induce the formation of native tertiary structure in the Tetrahymena thermophila ribozyme in the absence of divalent ions like magnesium. Using hydroxyl radical footprinting and analytical ultracentrifugation, the researchers monitored the formation of individual tertiary contacts and global compaction of the ribozyme as a function of sodium ion concentration. They found that most native tertiary contacts form at high sodium ion concentrations, though a few contacts near where divalent ions bind require magnesium. Both native and non-native tertiary contacts can form at low sodium ion concentrations. The results suggest that monovalent ions can push the ribozyme into the native folded state by forming native
Radiolytic Protein Footprinting with Mass Spectrometry to Probe the Structure...Keiji Takamoto
Structural proteomics approaches using mass spectrometry are in- creasingly used in biology to examine the composition and struc- ture of macromolecules. Hydroxyl radical–mediated protein foot- printing using mass spectrometry has recently been developed to define structure, assembly, and conformational changes of macro- molecules in solution based on measurements of reactivity of amino acid side chain groups with covalent modification reagents. Accurate measurements of side chain reactivity are achieved using quanti- tative liquid-chromatography-coupled mass spectrometry, whereas the side chain modification sites are identified using tandem mass spectrometry. In addition, the use of footprinting data in conjunc- tion with computational modeling approaches is a powerful new method for testing and refining structural models of macromolecules and their complexes. In this review, we discuss the basic chemistry of hydroxyl radical reactions with peptides and proteins, highlight various approaches to map protein structure using radical oxidation methods, and describe state-of-the-art approaches to combine com- putational and footprinting data.
Principles of RNA compaction : insights from equilibrium folding pathway of p...Keiji Takamoto
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2. 418 SILVER, SLUD, AND TAKAMOTO
asset prices. Yet classical economic theory describes deterministic transi-tions
of the detailed states of an economic system which balance supply
and demand in response to changes in preference and in information (e.g.,
in expectations of future returns). This dichotomy between (largely)
deterministic detailed transitions and stochastic observed aggregates in
economics [1, 2] is analogous to that in physics between motions of New-tonian
particles and macro- level observations of Brownian motion and
thermodynamic variables [3–5]. In both economics and physics, the limit-ing
behavior for superposed effects of large numbers of individuals
(particles) is due to probabilistic central limit theorems. The first such
theoretical results asserted Gaussian-distributed displacements [1–4]. More
realistic models in physics, going beyond Gaussian observations, have
incorporated interactions among particles [5]. In economics, statistical
description of aggregates based on probabilistic limit theorems has been
broadened to include infinite-variance laws and stable-increments processes
[6–8]. Economic models leading to non-Gaussian distributions often
involve complex or heterogeneous interactions among market participants
[9–11], but results with true statistical-mechanics flavor, deriving such
behavior from detailed microeconomic-level mechanisms, are rare.
One early attempt to incorporate thermodynamic and statistical
mechanical reasoning in economics [12] was largely confined to analogy
without detailed mathematical content. Another approach [13], which we
largely follow, formulates the problem of economic market equilibrium as
that of finding a time-invariant, nondegenerate probability distribution in a
high-dimensional state space describing the fortunes of a large population
of firms and individuals. The authors of [13] propose that micro-economic
production decisions be treated deterministically while market
fluctuations based on changing preferences and exogenous influences be
considered a stochastic process. Reference [2] provides one of the few
cases of analysis with simplifying large-population approximations leading
to true invariant distributions of a large state-space Markov model in eco-nomics.
There are several references, and a growing physics literature, with
statistical mechanics flavor in economic modeling, which describe equilibria
via maximum ‘‘entropy’’ configurations subject to constraints such as fixed
total wealth [14–16].
In this paper, we adopt a combined theoretical and simulation approach
to a simple model in which a fixed supply of goods is traded among indi-viduals
whose preferences for goods are determined stochastically. A
central feature of our model is that individuals periodically change their
preferences for goods independently, according to a distribution of pref-erences
in the population. The changes in aggregate demand translate
in the next time-period into changes in the market-clearing price. As
individuals adjust the mix of assets they hold to satisfy their changing
3. EQUILIBRIUM WEALTH DISTRIBUTIONS 419
preferences, the value of their goods waxes and wanes according to how
well, by chance, they anticipate future demand.
Using four separate lines of argument, of increasing generality, we show
that the model results in a probabilistic steady state, independent of initial
conditions, with gamma distributions of goods and wealths. The first is a
simulation study, which confirmed the steady-state behavior of the model
under various assumed preference distributions and showed that wealths
and amounts of goods behaved like gamma random variables with param-eters
depending only on the preference distribution. The second line of
argument follows the heuristic of [14] from statistical physics in calculat-ing
maximum ‘‘entropy’’ configurations subject to the constraint of fixed
total wealth. This argument is applicable only when individuals are indif-ferent
between goods, in which case we, like [14], treat all microstates as
equiprobable because all microstates accessible from each wealth-configu-ration
are. The third line of argument is a simple consequence of a theorem
of [17]: gamma-distributed wealths are implied by the assumption that in
steady state, wealths are jointly independent with ratios independent of
their sums. Our final argument, a mathematical development along the
lines of the approach to statistical mechanics in [4], shows how the gamma
distribution for wealths follows necessarily from the assumption that the
individual wealths have steady-state distributions characteristic of a set of
independent identically distributed variables conditionally given their sum.
The generality of our result that wealths have steady-state gamma distribu-tions,
across our family of models, suggests that it may be applicable to a
variety of stochastic models in which conserved quantities are periodically
re-balanced.
2. THE MODEL
For simplicity, we begin by considering a market consisting of N indi-viduals
and two goods, A and B, the total number of units of each being
conserved respectively at aN and bN. At the start, we assign to individuals
(indexed by i) certain amounts ai0 of good A and bi0 of good B. At all
integer times t, individuals hold amounts of good A and B respectively
denoted ait and bit , and measure their total wealths in terms of an external
numéraire, such as gold or inflation-adjusted specie, the total quantity of
which is conserved, at wN. We use such a‘‘gold standard’’ in order to
maintain symmetry in the treatment of goods A and B, and with a view to
allowing many more goods in further developments of the model. A
somewhat simpler and less artificial approach, which we also follow in
defining notations, is to view units of A as numéraire, with Jt denoting the
time-dependent value of one unit of good B in units of good A at time t.
4. 420 SILVER, SLUD, AND TAKAMOTO
Initially, the holdings ai0 , bi0 are assigned; thereafter, at each time-period
t 1, individuals determine fractions of their wealths
wit=ai, t−1+Jtbi, t−1 (1)
to allocate to good A, with the remainder allocated to good B. This is
equivalent to maximizing over ait ¥ [0, wi, t−1] an individual Cobb–Douglas
[18] utility function of the form
Uit(ait)=(ait)fit (wit −ait)1−fit, (2)
where fit is a doubly indexed sequence of independent and identically dis-tributed
random variables with values in the interval [0, 1]. The maxi-mization
of (2) results in the assignment
fitwit=ait
ˇ (3)
(1−fit) wit=Jtbit .The so-far-undetermined price Jt is uniquely determined from variables at
time t−1 together with {fit} via the market-clearing constraint
N
C
i=1
N
ait=aN, C
i=1
bit=bN. (4)
This constraint immediately yields
N
C
i=1
N
fitwit=aN, C
i=1
(1−fit) wit=JtbN (5)
from which there follows, upon substituting (1) for wit and solving for Jt ,
N
Jt=C
i=1
N
(1−fit) ai, ;t−1C
i−1
fitbi, t−1 . (6)
Substituting this expression for Jt back into (3) leads to the expression
ait=fitai, t−1+fitbi, t−1 ;j (1−fjt) aj, t−1/;j fjtbj, t−1
4 (7)
bit=(1−fit)[bi, t−1+ai, t−1 ;j fjtbj, t−1/;j (1−fjt) aj, t−1].Thus, individuals’ holdings of goods (7), wealth (1), and prices (6) at time t
can be expressed in terms of their previous holdings (ai, t−1, bi, t−1) and the
N random variables {fit}, making this a Markov process discrete in time
with 2N continuous state-variables ait and bit (which satisfy two linear
constraints, making 2N−2 the effective state dimension).
5. EQUILIBRIUM WEALTH DISTRIBUTIONS 421
The equations determining new state from old have the alternative
expression (6) together with
Rait
bit S=R fit
(1−fit)/JtSR1
Jt S− Rai, t−1
bi, t−1 S (8)
or, after premultiplying the last equation by the row-vector (1, Jt+1),
wi, t+1=1fit+
Jt+1
Jt
(1−fit)2wit . (9)
If values are expressed in units of a ‘‘money’’ numéraire of which there is
a total quantity wN, the aggregate relations (5) imply that the total of
(a+bJt) N units of numéraire A at time t has value wN, so that 1 unit of
A has monetary value w/(a+bJt) at time t, and all prices in units of A are
converted to money prices through multiplication by the factor
w/(a+bJt). The wealth wit of agent i at time t in terms of units of A is
then equivalent to the ‘‘money wealth:’’
mit — witw/(a+bJt).
The simulation results in the next section are presented in terms of
money-wealths. As will be seen in Section 6 below, when N is large the
ratio w/(a+bJt) is distributionally indistinguishable from the constant 1,
and thereore mit and wit have asymptotically the same distribution.
3. SIMULATION RESULTS
ki
ki
ki
We began by examining, in computer simulations with two assets (A and
B) and N=1000 individuals, whether our model with uniform preference
distribution leads to a steady state and a recognizable limiting distribution
for individual wealths. In our simulations, the assets were initially either
equally divided (ai0=bi0=1/N for all i) or alternatively 99.9 % of A and B
were initially in the hands of one individual with the remaining assets
equally divided. We characterized the distribution of goods and wealth
every 50 cycles by evaluating their first five moments, i.e. the average over
individuals of a, b, and money-wealths m, for k=1 to 5. After about
10 ·N (i.e., 10,000) cycles, the moments appeared to have reached steady-state
values that were independent of the initial conditions. This steady
state is summarized in Table I, which gives average moments and standard
deviations, where averages were taken over 200 cycles (every 50th cycle
from cycle 15,050 to 25,000). Since the moments were essentially identical for
6. 422 SILVER, SLUD, AND TAKAMOTO
TABLE I
Measured and Predicted Values of Weighted kth Moments and nk for Steady-State Goods and
Money Wealth Distributions Arising from Uniform Preferences
Weighted kth moment nk computed from moments
k Theory Simulation Theory Simulation
Mean St. dev. Mean St. dev.
Goods
1st 1 1.00
2nd 2 1.97 0.06
3rd 6 5.79 0.54 1 1.06 0.14
4rd 24 26.6 4.9 1 1.14 0.27
5th 120 111 48 1 1.31 0.41
Wealths
1st 1 1.00
2nd 1.5 1.48 0.03
3rd 3 2.90 0.17 2 2.09 0.24
4rd 7.5 7.14 0.9 2 2.14 0.41
5th 22.5 21.1 5.0 2 2.30 0.63
Note. Theoretical values in columns 2 and 5 are n−kC(n+k)/C(n) and n for the gamma dis-tribution
(n=c=1) for goods and (n=c=2) for wealths. Simulation results are mean and
standard deviation for weighted moments in columns 3 and 4, and nk (see text for definition)
in columns 6 and 7, calculated over every 50th cycle of the data from 15,050 to 25,000.
the ai and bi distributions, as expected by symmetry, only one set of
moments is listed (goods distribution). The moments scaled as a−k for
goods and w−k for wealths; to eliminate these scaling factors, Table I gives
weighted moments, equivalent to choosing a=w=1. The Gamma(n, c)
probability density, with shape-parameter n, scale parameter c, and mean
n/c, is defined as p(x)=(cn/C(n)) xn−1 exp(−xc) for x > 0, and we found
that the moments of the empirical goods distribution were in excellent
agreement with those of Gamma(1, 1/a), while the empirical moments of
(monetary) wealths agreed closely with those of Gamma(2, 2/w) (Table I,
column 3).
We also performed simulations using nonuniform preference distribu-tions.
In this case, the simulation results suggested that the steady-state
goods and wealth distributions were still closely approximated by gamma
distributions, although the parameter n was no longer an integer. We per-formed
simulations with two types of preference distributions: truncated
Gaussians, and polynomial densities of the form p(f) 3 (min(f, 1−f))c
7. EQUILIBRIUM WEALTH DISTRIBUTIONS 423
for 0 < f < 1, where c=1, 2, 3, 4, 5, −1/2, −2/3, −3/4, and −4/5. The
Gamma(n, n/w) distribution with mean w and shape parameter n has as
kth moment OmkP=(n/w)−k C(n+k)/C(n). It follows that the weighted
ratio of moments (1/w)OmkP/Omk−1P increases by 1/n when k increases by
1. In the simulations, the values of nk calculated from the weighted ratios of
moments were fairly constant for successive values of k (Table I, column 6,
and Table II), which suggests that the distributions are close to Gamma.
TABLE II
Estimates of nk, Calculated from Differences in Successive Moment Ratios
Preference dist’n Goods dist’n Wealth dist’n
Mean St. dev. Mean St. dev.
Gaussian, s=0.125
n1 8.03 0.69 17.4 2.34
n2 8.13 0.85 16.7 1.77
n3 8.05 1.08 16.8 1.79
p(f)=12f2, 0 [ f [ 0.5
n1 4.56 0.41 9.03 0.76
n2 4.49 0.66 8.90 1.00
n3 4.45 0.95 8.53 1.13
p(f)=4f, 0 [ f [ 0.5
n1 2.50 0.26 4.82 0.32
n2 2.51 0.46 4.67 0.64
n3 2.63 0.68 4.60 1.06
Gaussian, s=0.25
n1 2.14 0.23 4.30 0.29
n2 2.17 0.38 4.41 0.47
n3 2.31 0.57 4.62 0.67
Gaussian, s=0.5
n1 1.28 0.16 2.50 0.19
n2 1.36 0.27 2.56 0.35
n3 1.56 0.42 2.72 0.59
p(f)=1/`8f, 0 < f [ 0.5
n1 0.45 0.09 0.88 0.12
n2 0.53 0.16 0.98 0.22
n3 0.69 0.25 1.17 0.35
p(f)=(108f2)−1/3, 0 < f [ 0.5
n1 0.31 0.07 0.58 0.10
n2 0.38 0.13 0.67 0.19
n3 0.52 0.19 0.86 0.31
Note. ‘‘Gaussian’’ with indicated standard deviation s is truncated, i.e., conditioned to lie in
[0, 1]. The power-law densities were all taken symmetric about 0.5.
8. 424 SILVER, SLUD, AND TAKAMOTO
FIG. 1. Comparison of theoretical and simulated wealth distributions for some of the
preference distributions shown in Tables I and II. Values of n were taken from the simulation
results as in Table II.
Figure 1 shows graphically, in the form of slightly smoothed histograms
plotted in Excel overlaid with gamma densities with mean w and shape
factors n taken from Tables I and II, that the steady-state wealth distribu-tions
from the simulation data were very close to gamma. While the simu-lation
results are suggestive, they do not prove that the wealths asymptoti-cally
follow gamma distributions. Hence, we sought theoretical arguments.
4. HEURISTIC ARGUMENT COUNTING MICROSTATES
The nonlinearity of Eq. (7) makes the exact analysis of this system
intractable. However, for the particular case of a uniform preference dis-tribution
for fit for all i and t, with density h(f)=1 for 0 < f < 1, the
asymptotic behavior of the system for large N and t can be roughly under-stood
by arguing as in [14]. In the following heuristic argument, there is a
close analogy between the distribution of wealth in the economic system
9. EQUILIBRIUM WEALTH DISTRIBUTIONS 425
and the distribution of energy among molecules of an ideal gas in statistical
mechanics. In the latter case all ‘‘microstates’’ (specifications of the energy
of each molecule) with the same total energy are deemed equally likely.
One considers the number of ways W in which one can have Nj molecules
with energy Ej , subject to the constraints that the total energy and number
of molecules are fixed. The constraints are handled via Lagrange mul-tipliers
and the distribution of energy is determined by maximizing the
function
G(Nj, Ej)=log W−l1 1C
j
Nj−N2−l2 1C
j
NjEj −E2
with respect to the Nj (see [19]). In the case of the economics model with
uniform preferences, all ways of dividing an individual’s wealth between
goods A and B are equally likely. If wealth is measured in discrete units
(e.g., dollars), the number of ways of allocating money-wealth mi between
two goods is proportional to mi+1 (zero dollars to good A, or one, or two,
... up to mi dollars). Hence, the number of ways one can have N1 individ-uals
with wealth m1 , N2 individuals with wealth m2 , etc., is W(Nj, mj)=
N!Pj((mj+1)Nj/Nj!). The factorials account for the fact that different
combinations of individuals in a set of wealth bins represent different
microstates. There are N! permutations of individuals, but this overcounts
the number of microstates because permutations of individuals within a
wealth bin do not represent different microstates; the Nj ! terms in the
denominator correct for this overcounting. The Nj can be made indepen-dent
variables by the introduction of Lagrange multipliers l1 and l2 corre-sponding
to the constraints that the number of individuals and the total
wealth are constant. Maximizing the function G(Nj, mj)=log W(Nj, mj)−
l1(;j Nj−N)−l2(;j Njmj−Nw) with respect to the Nj , and then solving
for l1 and l2 from the constraint equations, leads, in the limit as Nj Q .
and sums are replaced by integrals, to the Gamma(2, 2/w) density
p(m)=(2/w)2 m·exp(−2m/w) for wealths. For the case of n goods instead
of two goods, this generalizes to Gamma(n, n/w). The derivation of the
distribution of goods (ait or bit) among individuals follows the same argu-ment
except that the term analogous to (mj+1)Nj is omitted because there
is no degeneracy in the way the goods can be allocated; this leads to
W(Nj, mj)=N!/PjNj ! and to a Gamma(1, 1/a) density p(m).
5. A QUALITATIVE ARGUMENT
In this section, we apply a purely probabilistic result to our model, under
an appealing qualitative assumption, to derive the gamma distribution for
10. 426 SILVER, SLUD, AND TAKAMOTO
steady-state wealths mit . The assumption is that arbitrary subsets of finitely
many wealth-variables at a time behave as though a steady-state distribu-tion
exists (for large t), and that
(a) the wealth-variables in the subset are asymptotically independent
(for large numbers N of agents and large time-index t), i.e., the total-wealth
asymptotically for large N imposes no constraint upon them, and
(b) the random mechanism of wealth-partition asymptotically makes
the total wealth in a sub-population of individuals {i1, i2, ..., ik} approxi-mately
independent of the way in which that wealth is subdivided among
the individuals in that sub-population.
Then the conclusion is that no distribution for the wealth-variables except
gamma is possible.
In many economic contexts, agents form coalitions or groupings. Here,
groups of finitely many agents are artificial, since the underlying model
contains no mechanism for cooperation in the setting of preferences.
(However, such a mechanism would be interesting to include in later
refinements of the model.) The assumption (a)–(b) above says that the
aggregate wealth for a finite grouping of agents fluctuates statistically (in
steady state) in a way which is statistically unrelated to the fluctuating par-tition
of the grouping’s wealth among its individual members.
The underlying probabilistic result is the following theorem of [17]:
Theorem 1. If X, Y are independent positive random variables such that
X/(X+Y) and X+Y are independent, then X, Y are respectively distributed
as Gamma(n1 , c), Gamma(n2 , c) for positive constants n1 , n2 , c.
This theorem of Lukacs [17] was cited for economic relevance in a
rather different context by Farjoun and Machover [13, pp. 63–72]. They
were interested in the random variables profit-rate R and labor cost rate Z
per unit of capital per unit time in an economy, obtained by aggregating
over all firms. They adduced Theorem 1 to conclude that these variables
are approximately Gamma distributed because they found it qualitatively
and empirically plausible (after some discussion) that the ratio R/Z of the
gross ‘‘return to capital’’ R over ‘‘return to labor’’ Z should fluctuate
approximately independently of R+Z (the total return excluding rent).
Farjoun and Machover recognized that to apply Lukacs’s Theorem in this
setting, R and Z would have to be approximately independent, but they did
not attempt to justify this independence.
The formal result which we derive from Theorem 1 is as follows.
Theorem 2. Suppose that the array of wealth variables wit (implicitly
also indexed by the number of agents N in our model) is such that for large
11. EQUILIBRIUM WEALTH DISTRIBUTIONS 427
N, t the joint distribution of each finite nonrandom set {i1, ..., ik} of the
random variables (wi1, t, wi2, t, ..., wik, t) converges (in the usual sense of
pointwise convergence of the joint distribution functions to a proper joint dis-tribution
function at continuity points of the latter), and that under the limit-ing
distribution of (W1 , ...,Wk), the variables Wj are positive and jointly
independent, and
Sk —W1+···+Wk
is independent of 1W1
Sk
,
W2
Sk
, ...,
Wk−1
Sk 2.
Then all of the limiting Wj variables are gamma distributed with the same
scale parameter c > 0.
Proof. For each fixed j=1, ..., k, X=Wj and Y=;k
i=1 Wi−Wj are
independent random variables such that X/(X+Y) is independent of
X+Y. Thus Theorem 1 implies that Wj and ;k
i=1 Wi−Wj are gamma dis-tributed
with the same scale parameter c > 0. Since it is well known that the
sum of independent gamma variables with common scale parameter is
gamma with the same scale, ;k
i=1 Wi is gamma with scale parameter c,
which is therefore the same for all j, and our proof is complete. L
The limitation of this approach is that the appealing assumption of
independence of sums and ratios of wealths of subsets is not readily dedu-cible
from our model.
6. PROBABILISTIC THEORY
For discrete-time Markov processes with continuous state-spaces, such
as the 2N−2 dimensional process defined in (4) and (7) above, there is a
fully developed but not very well-known theory of long-run asymptotics in
terms of equilibrium or stationary distributions, for which see the book
[20]. Under various sets of conditions, which are not easy to apply in the
present context of the process defined by Eq. (7), such continuous-state
processes can be shown to possess a unique stationary distribution which
can be found by solving a high-dimensional integral equation. The
challenge for application of this structure to economic theory is to obtain
simple approximate conclusions for observables such as the distributions or
moments of the wealth-variables wit or mit .
The mathematical argument of this section proceeds in three main steps.
Because of space limitations, we describe the main steps but omit some
mathematical details. We assume throughout that for large N and t
1
N
N
C
i=1
w2
it(1−fit)k
Q m2, k, k=0,1,2 (A.1)
12. 428 SILVER, SLUD, AND TAKAMOTO
with probability 1, where the constant limits m2, k do not depend upon N or
t (but definitely do depend on k and the distribution of fit ). Denote
m2 — m2, 0= lim
N, t Q .
N
N−1 C
i=1
w2
it , s2=Var(fit).
A further technical assumption required throughout is that the density
h(f) — pf(f) has finite third moment and has Fourier transform abso-lutely
integrable on the real line.
First we apply a refinement of Central Limit Theory (the Edgeworth
expansion [21], pp. 533ff, as generalized for non-iid summands in [22]) to
establish the form of the conditional density of normalized price deviations
Xt+1 —`N (Jt+1 −J) up to terms of order N−1/2, where
J=a(1−m)/(bm), m=E(fit).
Second, we apply related limit theory along with an auxiliary assumption
(A.2 below) to provide, again up to terms of order N−1/2, the joint density
of (Xt, w1t, f1t). Third, we show that gamma is the only possible form for
asymptotic marginal distributions for w1t which is time-stationary within
the model (7), that is, for which the marginal distribution of w1, t+1 within
(Xt+1, w1, t+1, f1, t+1) is the same up to order 1/`N as that of wit .
For the precise formulation of the first step, consider N large, and cal-culate
from equations (6), (5), and (3), that
Jt+1
Jt
;j (1−m(fj, t+1 −m)) fjtwjt
;j (fj, t+1 −m+m)(1−fjt) wjt
=
J
Jt 31−
=
1
aN(1−m)
C
j
(fj, t+1 −m) fjtwjt 4
×31+
1
bNmJt
C
j
(fj, t+1 −m)(1−fjt) wjt 4−1
.
Now, applying Theorem 19.3 of [22] to the nonidentically distributed but
conditionally independent summands, given {fjt, wjt}N
j=1, yields the
conclusion that the conditional density (at y) of Xt+1 given (Xt, {wjt, fjt})
(at Xt=x) is, asymptotically for large N,
bm
s`m2
f 1 ymb
s`m2 211+
c3(y)
`N 2exp 1 bx
`N
q2(y)2+o(N−1/2), (10)
13. EQUILIBRIUM WEALTH DISTRIBUTIONS 429
where f is the standard normal density, c3 is a cubic polynomial, q2 is a
quadratic, and o(N−1/2) denotes terms of order less than N−1/2. In particu-lar
Xt —`N (Jt −J) %D
N10,
s2m2
b2m22. (11)
Thus, the central limit theorem implies that for large N, prices will be
normally distributed, and further shows how the variance of prices depends
on the mean and variance of preferences and the variance of wealths. In
conformity with the assertion (11) of asymptotic normality, we remark that
when histograms were made from Jt variables simulated (with N 100,
and fit ’ Uniform[0, 1]) from the model under study, these histograms
looked approximately normal, but the histograms of the variables log Jt
looked still more symmetric and therefore closer to normality. Further-more,
as expected, in the limit as N Q ., the prices become fixed. Since
price fluctuations drive the redistribution of wealth, it may seem puzzling
that the initial wealth distribution is significantly altered for very large N.
The explanation is that the number of time-steps required to reach the new
steady state increases with N, and is of the order of N according to the
simulation results.
The next step relies on an auxiliary assumption restricting the form of
the conditional density of (w1t, f1t) given Xt (or equivalently, given Jt ),
namely:
(A.2) The variables wit have a ‘‘nonlattice distribution [22, condition
(19.29)] with third moments bounded uniformly in N, t; and asymptoti-cally
for large N, t, conditionally given Jt , the 2-vectors (wit, wit fit)OE behave
as N independent identically distributed random vectors subject to
N
R wit
i=1 N−1 C
wit fit S=Ra+bJt
a S. (12)
This assumption can be compared with, but is quite different in spirit from,
the independence hypothesis of Theorem 2. In our model, the conditions
(12) impose a specific dependence on the wealths wit . However, as the
number N of agents gets large, nonrandom finite subsets of them may have
wealths which are hardly affected by the constraints (12) and may therefore
be approximately independent, as assumed in Theorem 2. The alternative
approach of (A.2) is to argue that the variables wit may still behave inde-pendently
conditionally given the sum-constraints. This latter approach is
motivated as in classical statistical mechanics (cf. [4]), where the exact
equilibrium distribution of particle positions and momenta is intractable
but the individual phase-space coordinates behave like independent
14. random variables conditionally given a sum-condition fixing energy and
possibly some macroscopic integrals of the motion of a dynamical system.
By use of the same non-iid Edgeworth-expansion result of [22] cited
above, Assumptions (A.2) plus (A.1) lead to the following technical
conclusion: For a family of positive differentiable functions gN on [0, .)
such that the integrals > w3gN(w) dw are uniformly bounded, and for some
functions c1 , c2 , c3 which may depend upon x,
pw1t, f1t | Xt (w, f | x)=h(f) gN(w) 11+
`N 2+o(N−1/2),
c1w+c2w(f−m)+c3
(13)
.0
where the 10
final o term is understood as a function rN(f, w, x) such that for
each x, as N gets large, N1/2 >>w3 |rN(f, w, x)| dw df Q 0.
In fact, the derivation of (13) from (A.2) shows that
bx
s22
c1=
, c2=
−mbx
s2m2
, c3=−m1c1 , (14)
where we define for convenience
m1=a/m, s22
— m2 −m21
.
Formula (13) turns out to be precisely what is needed for the third step
of our derivation of gamma-distributed wealths. It is much less stringent
than (A.2) in restricting only the joint behavior of (w1t, f1t, Xt). It can be
understood as a first-order Taylor-series correction for large N to the con-ditional
independence of w1t and f1t given Xt and (12). That the variables
w, f appear in this correction only through terms proportional to w, wf is
plausible because (12) is the model’s restriction on independence of w1t , f1t .
It can further be derived from (13) by use of (A.1), (5) and symmetric
permutability of the agents within the model (7), that c1 , c2 , c3 in (13) are
necessarily linear in x and have the form (14). Thus, (13) can replace A.2 as
the fundamental assumption describing the way in which wealths behave
independently (to order N−1/2) despite having a fixed sum.
Our third step is to derive and solve the stationary equation expressing
the requirement that the joint density of (w1, t+1, Xt+1) be the same up to
terms of order 1/`N as the joint density of (w1t, X). By (10), (11), and
(13), letting pXt (x) and pXt+1(y) respectively denote the approximately
normal densities of Xt and Xt+1, we find that the joint density of
(Xt, w1t, f1t, X1, t+1) is, up to terms of smaller order than N−1/2,
pXt (x) gN(w) h(f) pXt+1(y) exp 3 bx
5w−m1
22
`N s−
+q2(y)64,
wm(f−m)
s2m2
(15)
430 SILVER, SLUD, AND TAKAMOTO
15. EQUILIBRIUM WEALTH DISTRIBUTIONS 431
where q2(y) is quadratic as in (10). However, as a function of
(Xt, w1t, f1t, Xt+1), by definition of Xt and Xt+1 together with (9),
w1, t+1=11+1J+Xt+1/`N
J+Xt/`N
−12(1−f1t)2w1t . (16)
Our task is to compare, up to terms of order 1/`N , the joint distribu-tion
of (w1t, Xt) obtained from (15) with that for (w1, t+1, Xt+1) obtained
from (13) and (14), which at (w*, y) is
gN(w*) pXt+1(y) 11+
s22
`N 2+o(N−1/2).
by(w*−m1)
We do this by calculating E(k(w1, t+1, Xt+1) in two ways, with respect to
the two density forms just given, and equating them, where k(w, y) is an
arbitrary smooth and compactly supported function. With (16) substituted
for w1, t+1, this gives
FFFF k(w 11−f)
y−x
J`N
, y2pXt+1(y) pXt (x) h(f) gN(w)
31+
bx
`N 5q2(y)+
1
s22
1w 11+(1−f)
y−x
`N 2−m1 2
−
s2m2 11+(1−f)
(f−m) mw
y−x
J`N264dx df dy dw
=FF pXt+1(y) gN(w) 31+
`N 4k(w, y) dy dw.
by(w−m1)
s22
Now subtract from both sides of the last equation the term
FF pXt+1(y) gN(w) k(w, y) dy dw,
multiply through by `N, and use the approximate identity > xpXt (x) dx=
O(N−1/2), to find after using smoothness of k to translate a difference
quotient into a derivative plus remainder,
m
FF pXt+(y) gN(w) 51−1J
(y−m1) w “k
“w
(w, y)+k(w, y)
b
s22
(w−m1) m6dy dw
% b
s22
FF pXt+1 (y), gN(w) y(w−m1) k(w, y) dy dw,
16. 432 SILVER, SLUD, AND TAKAMOTO
where % means that the difference between the two sides converges to 0 as
N Q .. This implies that
FF pXt+1(y) gN(w) 5(y−m)(w−m1) k(w, y)
−
(1−m) s22
bJ
(y−m) w “k
“w
(w, y)6dy dw % 0.
Since k is an arbitrary smooth compactly supported function, and since
integration by parts says that
F wgN(w) “k
“w
(w, y) dw=−F k(w, y)(wg−
N(w)+gN(w)) dw,
it follows that, up to terms converging to 0 as N Q .,
FF pXt+1(y) gN(w) 5(y−m)(w−m1) k(w, y)
(1−m) s22
+
bJ
N(w)+gN(w))6dy dw % 0.
(y−m) k(w, y)(wg−
Since k is an arbitrary smooth compactly supported function on
[0, .)×[0, 1], we can equate 0 to the integrand of the last integrals to
obtain the approximate equation
gN(w) 1m1(w−m1)
s22
+12=−wg−
N(w). (17)
All solutions to this equation have the form
log g(w)=−F 1m1
s22
1−m21
+
/s22
w 2dw
=C−
m1
s22
w+1m21
s22
−12log w
22
22
21
for some constant C, and the only such solution g which is a probability
density on [0, .) is the Gamma(m/s, m1/s) density, which has mean m1
and variance s22
=m2 −m21
.
Thus, the equality up to order N−1/2 terms of (expectations with respect
to) the distributions of (w1t, Xt) and of (w1, t+1, Xt+1) has been shown to
imply that the wealth variables w1, t+1 must be approximately gamma. The
17. EQUILIBRIUM WEALTH DISTRIBUTIONS 433
second moment of w1t is not predicted from this development, although it
is connected to the limiting variance of the variables `N(Jt+1 −J) through
s2m2
b2m2 .
Asymptotic variance of `N (Jt+1 −J)=
7. DISCUSSION
We simulated the evolution of artificial markets in which individuals buy
and sell goods in order to achieve a certain fraction f of their wealth in
each type of good. In our model, the prices of goods were determined
endogenously to equate fixed supply with variable demand. In contrast to
an analysis in which individual preferences are fixed, we randomized pref-erences
periodically, with preference probabilities determined by various
distribution functions. The purpose of the repeated randomization was to
see how stochastic forces would alter the distribution of goods and wealth.
While goods and wealth were continuously exchanged among individuals,
their distributions came to an equilibrium, independent of starting condi-tions,
determined by the form of the distribution of preferences for good A
versus B.
Remarkably, the equilibrium densities of goods and wealth had the same
mathematical form p(x)=cnxn−1e−cx/C(n) for all preference distributions
tested. While this form was suggested for the case of uniform preferences,
it was not obvious that it should hold for non-uniform preferences.
Nevertheless, we showed by an extension of the central limit theorem that
if individual wealths behave in steady state as identically distributed, inde-pendent
variables conditioned on their fixed sum, then all preference func-tions
satisfying reasonable continuity conditions lead to steady-state
gamma distributions for wealth. The mathematical derivation (Eq. (11))
shows how the shape factor n depends on the mean and variance of the
preference distribution and the variance of successive price ratios. The
latter is clearly determined by the preference distribution, but we do not, at
present, have a method other than simulation to calculate it. While theory
predicts the asymptotic behavior for large N, it is remarkable that simula-tions
with as few as 1000 individuals led to equilibrium distributions very
close to the asymptotic limit. Thus, characteristics of the ‘‘thermodynamic
limit’’ may already be apparent in systems with modest numbers of indi-viduals.
We do not propose the model outlined here as an accurate description of
human behavior in economic markets. Rather, we view the model as an aid
to thinking about how statistical phenomena may influence economic out-comes
such as the distribution of wealth. In the model, inequality in wealth
18. 434 SILVER, SLUD, AND TAKAMOTO
is seen to be a consequence of stochastic forces even when all individuals
are equivalent. Wealth disparity increases, the greater the diversity in pref-erences,
and decreases, the larger the number of goods traded. Since the
shape parameter n of the gamma distribution corresponds to the number of
goods traded in the case of uniform preferences, nonuniform preference
functions can be viewed as having different effective numbers of goods
traded, with more heterogeneous (U-shaped) preferences corresponding to
a reduced number of goods. The similarities between the economics model
and the statistical treatment of an ideal gas in physics suggests ways in
which the economics model might be modified to take into account
interactions between individuals that allow correlation among their pref-erences.
The method used to derive probabilistically the form of the steady
state depends on the particular feature of the transition mechanism that
individual agents’ wealths evolve independently and symmetrically except
for rebalancing in each generation. Such a rebalancing mechanism seems
characteristic of price-allocation within mathematical economics but can
arise also in neural-network or machine-learning models.
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