APUNTES DE CLASE
CEDE
ISSN 1909-4442
MÉTODOS MÉTO DOS MA MATEM TEMÁTI ÁTICOS COS EN MA MACRO CROECO ECONOM NOMÍA ÍA
Álvaro J. Riascos Villegas
1
SEPTIEMBRE
DE
2006
Centro de Estudios sobre Desarrollo Económico Facultad de Economía Universidad de los Andes
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Serie Apuntes de clase Cede ISSN: 1909-4442 Septiembre de 2006 © 2006, Universidad de los Andes – Facultad de Economía – Cede Carrera 1 No. 18 A – 10, Bloque C Bogotá, D. C., Colombia Teléfonos: 3394949- 3394999, ext. 2400, 2049, 2474.
[email protected] //economia.uniandes.edu.co http: // Ediciones Uniandes Carrera 1 No. 19 – 27, edificio Aulas 6, A. A. 4976 Bogotá, D. C., Colombia Teléfonos: 3394949- 3394999, ext. 2133, Fax: ext. 2158.
[email protected] http//ediciones.uniandes.edu.co Edición, diseño de cubierta, preprensa y prensa digital Proceditor Ltda. Calle 1 No. 27 A – 05. Bogotá, D. C. – Colombia Teléfonos: 2204275, 220 4276, Fax: ext. 102
[email protected] Impreso en Colombia – Printed in Colombia El contenido de la presente publicación se encuentra protegido por las normas internacionales y nacionales vigentes sobre propiedad intelectual, por tanto su utilización, reproducción, comunicación pública, transformación, distribución, alquiler, préstamo público e importación, total o parcial, en todo o en parte, en formato impreso, digital o en cualquier formato conocido o por conocer, se encuentran prohibidos, y sólo serán lícitos en
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Prefacio Esta monografía es una introducción informal a los métodos matemáticos más utilizados en el estudio de economías a lo largo del tiempo. Especí ficamente, esta se centra en los métodos de programación dinámica y el método de Lagrange para resolver problemas de optimización dinámica con o sin incertidumbre y en tiempo discreto. Aquí no hay nada original excepto por el esfuerzo que se ha hecho para hacer de esta monografía un puente agradable de recorrer entre los tratamientos básicos de estas técnicas usualmente relegados a los apéndices de los libros de macroeconomía utilizados en un curso del pregrado y los libros más avanzados que se estudian a nivel de doctorado. Mi deuda con los libros y artículos clásicos en el tema es evidente en muchos de los capítulos y sólo por descuido no están referenciados todos los libros y artículos sobre los que me base para escribir esta monografía. A los autores les pido disculpas y prometo corregir esto en futuras versiones. Muchas personas me han ayudado directa o indirectamente a mejorar esta monografía. En especial, agradezco los innumerables alumnos que padecieron las primeras versiones de estas notas en diferentes cursos de macroeconomía avanzada o de métodos matemáticos en macroeconomía. Me re fiero a alumnos en la Universidad Javeriana, Universidad de los Andes, Universidad del Valle y del Instituto de Matemáticas Puras y Aplicadas de Río de Janeiro. De manera más especí fica, agradezco a Katherine Aguirre, Andrés Arias, Olga Lucia Briñez, Marcela Eslava, Juanita Gonzalez, Franz Hamann, Luisa Estefanía Valdéz, Valdéz, Nini Johanna Serna y Mauricio Villamizar. Cualquier error es mi responsabilidad. Finalmente quiero agradecer al Banco de la República por la disponibilidad de tiempo y el ambiente propicio durante el tiempo que estuve en el departamento de investigaciones económicas donde fue escrita gran parte de esta monografía.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
iv
PREFACE
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Capítulo 1
Economía Dinámica En términos generales la actividad económica no puede modelarse como un proceso estático en el tiempo y en condiciones de certidumbre. En la vida real, este proceso es dinámico e incierto y por lo tanto es natural preguntarse hasta que punto la teoría del equilibrio general, como la formularon Arrow y Debreu permite describir la toma de decisiones económicas más realistas. Es bien sabido que el modelo básico de equilibrio general (Arrow-Debreu en infinitas dimensiones) es lo su ficientemente cientemente general como para abarcar en términos ideales esta situación. En este modelo el espacio de consumo puede tomarse de tal manera que incluya decisiones intertemporales o contingentes a la realización de eventos aleatorios. Basta indexar los diferentes bienes al momento o evento en el cual son consumidos como cualquier otra característica que los define. De hecho, desde el punto de vista teórico este abordaje es muy importante. Sin embargo, también es evidente que este modelo no explota de manera explícita la dimensión temporal del problema ni tampoco reconoce la imposibilidad, bastante real en la práctica, de diversificación del riesgo que resulta de la realización de los eventos aleatorios.1 En este capítulo introduciremos de manera informal los métodos matemáticos que serán utilizados en el resto del libro. En la sección 1.1 estudiaremos el prototipo de modelo que aparece en el estudio de economías dinámicas, el modelo básico de crecimiento económico, que utilizaremos para introducir las dos técnicas principales para resolver modelos dinámicos. El primero pertenece a los métodos de la Programación Dinámica, sección 1.2 y el segundo a los de Control Óptimo (i.e. el método de Lagrange o, más generalmente, el método del Hamiltoniano), sección 1.3. La característica fundamental del primer método, como quedará claro más adelante, es que explota la naturaleza recursiva del problema. Es decir, el hecho de que la estructura del problema de decisión es la misma en todos los períodos. El segundo método, explota la geometría del problema.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
2
CAPÍTULO CAPÍTULO 1. ECONOMÍA ECONOMÍA DINÁMICA DINÁMICA
El modelo básico de crecimiento es un problema en el cual las decisiones no dependen para nada de eventos aleatorios y en el cual éstas son tomadas por un agente representativo (o planificador central) sujeto a una secuencia de restricciones de recursos. Más adelante permitiremos que las decisiones dependan de eventos eventos aleatorios y explicaremos cómo extender los métodos de la Programación Dinámica y Control Óptimo al caso estocástico.
1.1. 1.1.
El Mode Modelo lo Bás Básico ico de de Crec Crecim imie ien nto
El problema que queremos estudiar es el del crecimiento de una economía cuando sus agentes deben determinar de manera óptima cuánto consumen y cuánto ahorran en cada instante del tiempo. La parte ahorrada en cada momento se puede invertir en acumulación de capital para el período siguiente, permitiéndole aumentar su producción y por lo tanto sus posibilidades de consumo. Un modelo sencillo de crecimiento en un ambiente determinístico (el modelo básico de crecimiento) pero que a la vez ilustra plenamente los métodos de la Programación Dinámica podría especificarse de la siguiente manera (Ramsey [1928], Cass [1965] y Koopmans [1965]). Supongamos que existen una gran cantidad de agentes idénticos, con vidas infinitamente largas y que en cada período del tiempo deben decidir cómo utilizar el único bien de consumo que se produce en esta economía. Sea yt la cantidad producida de este bien durante el período t utilizando como único insumo la cantidad de capital kt . Suponemos que todas las variables son per capita capita y que la población población no crece. crece. Las posibilidades de producción de esta economía las representamos a través de una función de producción neoclásica f (ver Apéndices), de tal manera que yt = f (kt ). Hacemos las hipótesis habituales sobre f : f es una función continua, estrictamente creciente, estrictamente cóncava y f (0) = 0. Además supondremos que Lim f 0 (k) → ∞ (condición de Inada), y Lim f 0 (k) → 0. k→0 k→∞ En el período t el producto se divide entre el consumo en este período ct , y la inversión bruta it . Es decir: ct + it = f (kt ), ct , kt ≥ 0 para todo t
(1.1)
Asumimos que el capital se deprecia a una razón constante δ ∈ [0, 1]; luego la dinámica del capital es: kt+1 = (1 − δ )kt + it
(1.2)
Además, suponemos que cada agente tiene una misma cantidad de capital inicial k0 ≥ 0. Finalmente, los agentes tienen preferencias U sobre todas las secuencias de consumo; asumimos que dichas preferencias son de la siguiente forma:
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
1.2. PROGRAM PROGRAMACI ACIÓN ÓN DINÁMICA DINÁMICA
3
Donde u(ct ) representa la utilidad del consumo en el mismo instante t, y β es un factor que descuenta la utilidad de consumo en el futuro. Implícitamente, hemos supuesto que los agentes le dan más importancia al presente. Otra posible interpretación de β es que representa la probabilidad de no morir de un período perío do a otro. Así mismo, podríamos considerar el mismo problema con un beta diferente cada período (β t ∈ (0, 1)), pero por simplicidad no consideramos ese caso. La pregunta que queremos responder en esta economía es: ¿Dado un k0 ≥ 0, cómo deben consumir los agentes a través del tiempo (c0 , c1 ,...) de tal forma que maximicen su utilidad 1.3, sujeto a la restricción de recursos 1.1 y a la ecuación 1.2 que describe la evolución de la única variable de estado de esta economía, kt ?. Esta terminología relacionada con variables de estado, será clara más adelante. Suponemos que u satisface las mismas propiedades que f y que además, es acotada (esta última garantiza que 1.3 tiene sentido para toda secuencia (c0 , c1 ,...)). Formalmente el problema es: ∞
m´ ax ax
X
β t u(ct ), β ∈
(0, 1)
t=0
kt+1 ct + it
= (1 − δ )kt + it = f (kt ), ct , kt ≥ 0 para todo t.
Secuencial. Este problema lo llamamos el Problema Secuencial.
1.2. Progra Programa mación ción Dinám Dinámica ica Supongamos que dado k0 nosotros conseguimos resolver el problema anterior anterior y el valor máximo de la función objetivo lo denotamos por v (k0 ) (v se llama la función valor ). Imaginemos ahora el problema del agente representativo un período más tarde. Dada una cantidad de capital inicial k1, el agente representativo debe resolver un problema con la misma estructura del anterior y su valor máximo máximo debe ser v(k1 ), de esta manera, si interpretamos el coeficiente β como un factor que trae a valor presente las utilidades futuras del agente , entonces
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
4
CAPÍTULO CAPÍTULO 1. ECONOMÍA ECONOMÍA DINÁMICA DINÁMICA
o equivalentemente, m´ ax ax{u(c0 ) + βv ((1 − δ )k0 + f (k0 ) − c0 )} c0
s.a : 0 ≤ c0 ≤ f (k0 ) + (1 − δ )k0
Luego, v debe satisfacer: v (k0 ) = max a´x{u(c0 ) + βv ((1 − δ )k0 + f (k0 ) − c0 )} c0
s.a
0 ≤ c0 ≤ f (k0 ) + (1 − δ )k0
:
Obsérvese que en la expresión anterior no hay nada especial con el argumento en los períodos 0 y 1 que no pueda ser usado en los períodos n y n + 1 para cualquier n. Luego debe ser que para todo período n se cumple la ecuación funcional: v(kn ) = max a´x {u(cn ) + βv ((1 − δ )kn + f (kn ) − cn )} cn
s.a
(1.4)
0 ≤ cn ≤ f (kn) + (1 − δ )kn
:
Así, si conociéramos la función v, entonces la secuencia óptima {cn } quedaría caracterizada como las soluciones al último problema para cada uno de los períodos. La formulación anterior, en términos de una ecuación funcional, pone en evidencia evidencia el carácter carácter recursivo recursivo del problema problema.. La ecuación ecuación 1.4 la llamarem llamaremos os la ecuación de Bellman . Olvidándonos de los subíndices, vemos que si k es el monto de capital en un cierto período, entonces el consumo óptimo c en el mismo período es la solución al problema: v(k) = max a´x u(c) + βv ((1 − δ )k + f (k ) − c)) c
s.a
:
(1.5)
0 ≤ c ≤ f (k) + (1 − δ )k
Este problema lo llamamos el Problema Funcional asociado al Problema Secuencial con el que iniciamos esta sección. Visto de esta forma, el problema
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
5
1.2. PROGRAM PROGRAMACI ACIÓN ÓN DINÁMICA DINÁMICA
en dar una estructura especial al conjunto de las funciones continuas y acotadas (una métrica que lo haga un espacio métrico completo 3 ) y probar que el operador T es una contracción 4 . De esta manera podemos aplicar el Teorema del Punto Fijo para Contracciones y así obtener tanto la existencia de una función v que resuelva el problema (4), como un método para encontrarla dado por este mismo teorema. Sin entrar en mayores detalles por ahora, utilizaremos el Teorema del Punto Fijo para contracciones para resolver el ejemplo siguiente. Informalmente este teorema nos dice que si T es una contracción, contracción, entonces existe una única v, punto fi jo de T y además, dado cualquier v0 en el dominio T se tiene: l´ım T n (v0 ) = v
n→∞
El método de programación dinámica afirma que, bajo ciertas condiciones, v (k0 ) es el valor máximo del problema secuencial cuando el capital inicial es k0 . Adicionalmente, Adicionalmente, si h es la función definida sobre los stocks de capital k tal que c = h(k) es el consumo que resuelve el problema de maximización: m´ ax ax {u(c) + βv ((1 − δ )k + f (k ) − c))} c
s.a : 0 ≤ c ≤ f (k ) + (1 − δ )k
para todo k, entonces la secuencia {kt, ct } definida por kt+1 = f (kt ) − h(kt ) + (1 − δ )kt , ct = h(kt ) es una secuencia que resuelve el problema secuencial. La función h la llamamos la función de política . El siguiente caso particular del modelo básico de crecimiento ilustra las ideas principales. Ejemplo 1 (Brock y Mirman [1972]). Sea u(ct) = log(ct ), f (kt ) = ktα donde α ∈ (0, 1) y δ = 1 (la función u no satisface todas las hipótesis impuestas anteriormente (en particular, no es acotada); sin embargo, más adelante veremos como adaptar la teoría a este caso) entonces el problema de crecimiento discutido hasta ahora se reduce a: ∞
m´ ax ax {ct }
X t=0
β t log(ct )
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
6
CAPÍTULO CAPÍTULO 1. ECONOMÍA ECONOMÍA DINÁMICA DINÁMICA
Para resolver este problema utilizaremos el método dado por el Teorema del Punto Fijo para contracciones. Sea v0 = 0 , entonces: v1 (k) = max a´x{log(c)} c
0 ≤ c ≤ kα
:
s.a
este problema tiene la solución de esquina c = kα , luego v1 (k) = α log(k). Para calcular v2 resolvemos el problema: v2 (k ) = max a´x{log(c) + βα log(k α − c)} c
que lo reseulve c =
kα 1+βα 1+βα
0 ≤ c ≤ kα
:
s.a
, luego
v2 (k) = α(1 + βα )log(k ) + βα log
Ahora de igual forma podemos deducir que:
µ ¶ βα 1 + βα
2 2
− log(1 + βα ).
2
v3 (k ) = α(1 + βα + β α )log(k ) + β α log
µ
¶
µ ¶ βα 1 + βα
+
βα + β 2 α2 βα β α ( + )log 1 + βα + β 2 α2 2 − log(1 + βα + β α2 ) − β log(1 + βα ) 2 2
µP ¶
luego en general vemos que vn (k) = An + α
n−1 i=0
(βα )i log(k ), donde An es
una constante que debemos determinar. Se sigue que v(k) = A + 1−αβα log(k) donde A es una constante que podemos encontrar simplemente observando que v debe satisfacer: v (k ) = max a´x{log(c) + βv (k α − c)} c
s.a
Es decir,
:
0 ≤ c ≤ kα
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
7
1.2. PROGRAM PROGRAMACI ACIÓN ÓN DINÁMICA DINÁMICA
Una vez resuelto el problema de la existencia, el siguiente paso es describir de la manera manera más precisa precisa posible las propieda propiedades des de la función valor valor v y de la trayectoria trayectoria óptima {kt } asociada a la trayectoria óptima del consumo. Si bien algunas características de la solución van a depender de la forma particular de las funciones de producción y utilidad, otras características muy importantes se cumplen de manera más general como por ejemplo, la unicidad de la solución óptima y la estabilidad de las trayectorias . Supongamos que el problema secuencial (excepto por el valor del capital inicial) tiene una solución de estado estacionario. Esto es, una solución en la que todas las variables crecen a una tasa constante e igual a cero, por ejemplo kt = k ∗ para todo t. El problema de estabilidad consiste en saber si, comenzando con un valor inicial de capital k0 diferente al valor k∗ , la solución óptima al problema secuencial converge con el tiempo a la solución de estado estacionario. Es decir, si kt → k∗ cuando t → ∞. En el ejemplo anterior tenemos que existen dos soluciones de estado esta1 cionario. Una corresponde a k∗ = 0 y la otra a k∗ = ( βα ) 1 α . En el primer caso, es fácil ver que no se cumple la propiedad de estabilidad mientras que en el segundo, como α ∈ (0, 1) si se cumple la propiedad de estabilidad. Posteriormente Posteriormente estudiaremos esta propiedad para un problema más general. Ejemplo 2 (Long y Plosser [1983]) Consideremos el modelo básico de crecimiento con oferta laboral. Sea nt ∈ (0, 1) la cantidad de trabajo que ofrece el agente representativo y lt la cantidad de tiempo que dedica al ocio. Supongamos que lt + nt = 1. Sea u(ct , lt ) = θ log(ct ) + (1 − θ) log( log(lt ); f (kt , nt ) = ktα nt1−α y δ = 1 . Así, el problema secuencial es: −
∞
M ax s.a kt+1
X
β t (θ log(ct ) + (1 − θ)log(1 − nt ))
t=0
: = ktαnt1−α − ct
El problema funcional asociado es: v(k) = max a´x{θ log(c) + (1 − θ)log(1 − n) + βv (kα n1−α − c)} c,n
(1.6)
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
8
CAPÍTULO CAPÍTULO 1. ECONOMÍA ECONOMÍA DINÁMICA DINÁMICA
Reemplazando en la restricción obtenemos: c=k
α
∙ θ(1
− α)
1 − θα
¸
1−α
y por lo tanto la función v1 (k) es:
Ã
v1 (k ) = θ log k α
∙ θ(1
− α)
1 − θα
¸ ! 1−α
µ
θ(1 − α) + (1 − θ)log 1 − 1 − θα
¶
=
= αθ log k + A1
donde A1 es una constante. Teniendo v1 (k) podemos calcular v2 (k): v2 (k ) = max a´x{θ log(c) + (1 − θ)log(1 − n) + βαθ log(kα n1−α − c) + βA(1.7) 1} c,n
s.a
0 0
: 6 6
c 6 k α n1−α n61
Ahora, podemos derivar las condiciones de primer orden con respecto a c y n : [c] :
θ
−
βθα kα 1 α
=0
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
9
1.3. EL MÉTODO MÉTODO DE LAGRAN LAGRANGE GE
Ahora, debemos veri fi car car que efectivamente este candidato es el correcto; reemplazando en la ecuación 1.6 y de nuevo escribiendo las condiciones de primer orden obtenemos: [c] : [n] :
θ βαθ = c (1 − αβ ) (kα n1−α − c) βαθ (1 − α)kα n−α 1−θ = 1−n (1 − αβ ) (k α n1−α − c)
Simpli fi cando cando encontramos estos candidatos a las funciones de política: n =
θ(1 − α) 1 − α (θ + β − θβ )
c = (1 − αβ )k
α
µ
(1.8)
θ(1 − α) 1 − α (θ + β − θβ )
¶
1−α
(1.9)
Finalmente, reemplazando en la ecuación 1.6 obtenemos:
"
αθ log k + A = θ log (1 − αβ )k α 1 − αβ
µ
θ (1 − α) 1 − α (θ + β − θβ )
¸
¶ #
θ(1 − α) +(1 − θ)log 1 − + 1 − α (θ + β − θβ )
∙
µ ¶ µ
αθ β log[kα 1 αβ
1
θ(1 − α) (θ β θβ )
1−α
¶
1−α
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
10
CAPÍTULO CAPÍTULO 1. ECONOMÍA ECONOMÍA DINÁMICA DINÁMICA
de Euler o más generalmente como el Método de Lagrange (este a su vez, es un caso particular de la teoría del control óptimo) 5 . Una vez más utilizaremos el modelo básico de crecimiento para ilustrar las ideas principales de este método. Obsérvese que el problema de optimización en el modelo básico de crecimiento se puede escribir de forma equivalente como: ∞
m´ ax ax {ct }
s.a
:
X
β t u (f (kt ) − kt+1 + (1 − δ )kt )
t=0
0 ≤ kt+1 ≤ f (kt ) + (1 − δ )kt k0 dado.
Para simplificar un poco la notación introduciremos las siguientes de finiciones. Sea r(kt , kt+1 ) = u (f (kt) − kt+1 + (1 − δ )kt ) y Γ(kt) = {k ∈ R : 0 ≤ k ≤ f (kt) + (1 − δ )kt } entonces el problema del modelo básico de crecimiento es equivalente a: ∞
m´ ax ax
{ct ,kt }
s.a
:
kt+1
∈
X t=0
Γ (kt )
k0 dado.
β t r(kt , kt+1)
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
11
1.4. EJERCICIOS EJERCICIOS Y SOLUCION SOLUCIONES ES
En los próximos capítulos discutiremos una interpretación de esta condición. Intuitivamente, a lo largo de la trayectoria óptima el valor presente de la utilidad marginal de una unidad adicional de capital en el in finito no debe ser positiva. Ejemplo 3 (Brock y Mirman [1972] una vez más). Aplicando las ecuaciones de Euler al ejemplo de Brock y Mirman obtenemos: −
1 kt∗α − kt∗+1
α−1 αkt∗+1 + β ∗α =0 kt+1 − kt∗+2
(1.11)
Es fácil obtener la solución de estado estacionario a partir de esta ecuación en diferencias fi nitas. nitas. Sin embargo, no es obvio cómo utilizar la condición de transversalidad, ecuación 1.10, para resolver esta ecuación por fuera del estado estacionario. De hecho, más adelante dedicaremos un buen esfuerzo a explotar la condición de transversalidad. transversalidad. Por el momento, basta con observar que la anterior k ecuación se puede reducir a una ecuación de primer orden. Sea xt = ktt+1 α , entonces el sistema es equivalente a: ∗
∗
µ
1 − xt+1 1 − xt
¶
xt = αβ
y la condicion de transversalidad se puede reescribir como: lim β t
t→∞
α
1 − xt
= 0.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
12
CAPÍTULO CAPÍTULO 1. ECONOMÍA ECONOMÍA DINÁMICA DINÁMICA
Derivando c∗ con respecto a β obtenemos: α
∂c ∗ α(βα ) 1 α = (1 − β ) > 0 ∂β β (1 (1 − α) −
Derivando β con respecto a r obtenemos: ∂β = −(1 + r )−2 < 0 ∂r
De estas dos ecuaciones se desprende que el nivel de consumo del estado estacionario es una función decreciente de la tasa de interés. Ejercicio 2 Mostrar formalmente que en el ejemplo de Brock y Mirman.
1. Se cumple cumple la propie propiedad dad de estabili estabilidad dad par para a k∗ > 0 pero no para k∗ = 0 . 2. Si el capital capital inicial esta por por debajo debajo del capital capital de estado estacion estacionario, ario, la tasa de crecimiento del capital es una función decreciente del nivel de capital. Esta es una ilustración de la hipótesis de convergencia en la teoría del crecimiento crecimiento 3. Supong Supongamo amoss que que f (kt ) = Aktα donde A es una constante. Mostrar que la dinámica óptima del capital es kt+1 = βαAktα y la función de política es ct = (1 − βα )Aktα . Utilizando dos valores diferentes de A ilustrar la hipótesis de convergencia condicional: controlando por diferentes parámet-
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
13
1.4. EJERCICIOS EJERCICIOS Y SOLUCION SOLUCIONES ES
Para usar una notación similar a la del capítulo, podemos replantear el problema secuencial de la siguiente manera: ∞
m´ ax ax
X
β t ln(ct )
t=0
0 ≤ ct ≤ xt y xt+1 = xt − ct
Donde ct es la cantidad de pastel que nos comemos en t. El problema funcional es: v (xt ) = max a´x{ln(ct ) +βv (xt − ct )} 0 ≤ ct ≤ xt
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
14
CAPÍTULO CAPÍTULO 1. ECONOMÍA ECONOMÍA DINÁMICA DINÁMICA
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Bibliografía
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
16
BIBLIOGRAFÍA
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Capítulo 2
Programación Dinámica: El caso determinístico
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
CAPÍTULO 2. 18CAPÍTULO
PROGRAM PROGRAMACI ACIÓN ÓN DINÁMICA: DINÁMICA: EL CASO DETERMIN DETERMINÍSTICO ÍSTICO
producción de las firmas y consecuentemente, el nivel de consumo o inversión, en este caso las variables de control, que los agentes pueden escoger. Esto se
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
19 Condición 2 β ≥ 0 y para todo x0 ∈ X , existe M x0
∈ R tal que
P ∞
β t r(xt , ut ) ≤
t=0
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
CAPÍTULO 2. 20CAPÍTULO
PROGRAM PROGRAMACI ACIÓN ÓN DINÁMICA: DINÁMICA: EL CASO DETERMIN DETERMINÍSTICO ÍSTICO
sup {r(x0 , u0 )+β v(g(x0 , u0 ))}. Nuevamente, olvidándonos de los subíndices, u0 ∈Γ(x0 )
e
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
21 Anotación 2 Intuitivamente l´ım β t v (xt ) > 0 implica que se están subutilizant→∞
do los recursos cuando la dinámica factible es { } Si l´ β t ( ) < 0
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
CAPÍTULO 2. 22CAPÍTULO
PROGRAM PROGRAMACI ACIÓN ÓN DINÁMICA: DINÁMICA: EL CASO DETERMIN DETERMINÍSTICO ÍSTICO
0 podemos concluir que
P ∞
β t r (x∗t , u∗t ) ≥ v(x∗0 ) luego por la desigualdad anteri-
t=0
e
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
58
BIBLIOGRAFÍA
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Capítulo 4
Economía Dinámica: el caso estocástico En la teoría desarrollada hasta este momento hemos excluido por razones de simplicidad el carácter incierto sobre el cual se toman la mayoría de las decisiones económicas. Por esto queremos decir que, en la mayoría de los casos, cuando los diferentes agentes económicos se ven en la obligación de tomar una decisión ellos desconocen por lo menos parcialmente, el ambiente ambiente económico. Por ejemplo, las decisiones en el campo de la agricultura dependen estrechamente del comportamiento climático. Siendo éste un factor impredecible, los agentes no tienen otra alternativa que tomar sus decisiones contingentes a la realización de estos eventos aleatorios. Las decisiones en el mercado bursátil son también altamente altamente inciertas. Comprar o no acciones depende del comportamiento futuro de los precios, que desde el punto de vista de los agentes, son bastante impredecibles. Igualmente en la industria, muchas decisiones de inversión dependen de la tasa de interés o la tasa de cambio, variables altamente impredecibles. Por esta razón, debemos buscar otra forma de modelar el comportamiento racional de los agentes (en el sentido de que estos maximizan una función de utilidad que refleja sus preferencias sobre las diferentes alternativas), y que lleve en consideración el carácter contingente (o condicional) con el que los agentes deben tomar sus decisiones. Una alternativa es suponer que los agentes maximizan la utilidad esperada de sus decisiones, o por ejemplo, el bene ficio esperado en el caso de una firma. Aquí desarrollamos este punto de vista y comenzaremos con la estructura general de estos problemas.1 El primer punto a discutir, que es de vital importancia para nuestro estudio, es la forma de modelar los eventos aleatorios2 . Sin entrar en detalles, supongamos que tenemos un espacio de probabilidad (Ω, z, P ), donde Ω representa representa el conjunto de todos los acontecimientos o sucesos posibles que puedan tener algu1 Estas
notas están basadas en el capítulo (2) de Stokey-Lucas [1989].
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
60
CAPÍTULO CAPÍTULO 4. ECONOMÍA ECONOMÍA DINÁMICA: DINÁMICA: EL CASO ESTOCÁSTIC ESTOCÁSTICO O
na relevancia para la actividad económica; z representa los eventos (conjuntos de sucesos) que pueden ocurrir y P es la probabilidad probabilidad (objetiva) (objetiva) con la que se 3 realizan estos eventos . Ahora, estos resultados posibles, resumidos en el conjunto Ω, deben tener una manifestación muy particular en el ambiente económico bajo consideración. Más concretamente, debemos pensar en la forma como esos resultados afectan el marco analítico sobre el que se va trabajar. La forma usual de hacerlo, es a través de variables aleatorias definidas sobre este espacio de probabilidad. Más especí ficamente, a través de un proceso estocástico {θt } , que en cada instante t y para cada realización ω ∈ Ω, nos dice cómo afecta éste, nuestro marco analítico (el lector podrá encontrar una discusión más detallada en el Apéndice). Para el tipo de problemas que consideraremos, el efecto de estas realizaciones se manifiesta en la dinámica que siguen las variables de estado. En los ejemplos veremos de manera precisa la forma como pueden manifestarse. En general, el problema secuencial en un ambiente estocástico tiene típicamente la forma:
"X ∞
sup E s.a xt+1 ut x0
t
#
β r (xt , ut )
t=0
: = g(xt , ut , θt+1) ∈ Γ (xt ) ∈ X, dado,
donde xt ∈ Rn , ut ∈ Rm, X ⊂ Rn , θt ∈ Rl , r es una función de X × Rm en R, g es una función de X × Rm × Ω en X y Γ es una correspondencia de X × Ω en Rm. Mantenemos la interpretación usual de las funciones, pero es necesario especificar la estructura del problema de decisión en cada período. Las variables x en este problema van a resumir el ambiente económico completo sobre el cual se toman las decisiones. Estas son las variables de estado que pueden ser de dos tipos: estados endógenos y estados exógenos (probablemente aleatorios) y que, cuando sea necesario los distinguiremos de la siguiente forma. Los estados endógenos los denotaremos por xt ∈ Rns y los estados exógenos los denotaremos denotaremos n e por zt ∈ R donde n = ns + ne. Las variables θt+1 son variables aleatorias 3 Se puede hacer una distinción importante entre riesgo e incertidumbre que tiene origen en los escritos de Keynes [1921] y Knight [1921]. Fundamentalmen Fundamentalmente te la idea consiste en distinguir una situación de riesgo, donde la realización de un evento es aleatoria pero con distribución conocida como por ejemplo, el resultado resultado de tirar unos dados no sesgados, y una de incertidumbre, en donde la distribución es desconocida como por ejemplo, el resultado de una carrera de caballos. Keynes y Knight argumentaban que en las mayoría de las decisiones económicos era mucho más importante la segunda forma de incerteza. En este libro no haremos tal distinción pues siempre invocaremos la hipótesis de expectativas racionales para resolver nuestros modelos. Una implicación de ésta es que la probabilidad (subjetiva) que los agentes económicos utilizan para determinar la incerteza de los eventos es, en equilibrio, la misma que la probabilidad verdadera (objetiva) con la que estos ocurren. Este es uno de los supuestos básicos de la hipótesis de expectativas racionales. Las consecuencias económicas de distinguir entre estas dos formas de incerteza es una área
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
61 exógenas que asumimos son independientes e idénticamente distribuidas y que son la fuente de la incertidumbre de la economía. Anotación 17 Por el momento supondremos que el estado inicial x0 es un valor especí fi co co de X . Sin embargo, más adelante vamos a generalizar al caso en que la información inicial sobre los estados estada dada en la forma de una distribución inicial conocida.
Asociado al problema secuencial, tenemos el siguiente problema funcional: v (xt ) =
sup ut ∈Γ(xt )
{r(xt , ut ) + βE t [v(g (xt , ut , θt+1)]} ,
donde E t[.] denota el valor esperado dada la información hasta el período t (más concretamente, la información al comenzar el período t). En nuestro caso, esta información corresponde al conocimiento de x0 , x1,...,xt . Obsérvese que el conocimiento en t de las variables de estado hasta t supone implícitamente el conocimiento de todos los controles u0 , u1,...,ut−1 (hasta t − 1) y de todas las variables exógenas θ0 , θ1 ,...,θt (hasta t). Al finalizar el período, ut es conocido. Como puede sospecharse a partir de esta formulación, la teoría de la programación dinámica en el caso estocástico se desarrolla de manera análoga al caso determinístico. Esto es cierto con relación a la equivalencia entre los dos problemas problemas bajo ciertas ciertas condicion condiciones, es, al método método iterativ iterativoo e incluso, incluso, a los métodos numéricos. Como el análisis formal de la teoría, es ligeramente más complicado que el caso determinístico y requiere de una formación sólida en teoría de la probabilidad en lo que sigue, procederemos de manera informal. En particular, supondremos que el problema funcional siempre tiene solución y que el supremo se realiza como un máximo. Por lo tanto de ahora en adelante escribiremos el problema con el operador de maximización. Todas las características mencionadas presentan di ficultades técnicas más complejas que en el caso determinístico pero no dejan de estar estrechamente relacionadas. Con el objeto de familiarizar más al lector con los problemas estocásticos, los siguientes ejemplos se muestran como generalizaciones naturales de los ejemplos tratados en las notas anteriores. Por ahora, vale la pena resaltar una primera diferencia importante con las ideas desarrolladas en los capítulos anteriores: el análogo al estado estacionario de los modelos determinísticos. El primer ejemplo nos servirá como una introducción al concepto de estado estacionario y la propiedad de estabilidad en el caso estocástico. El ejemplo típico es, una vez más, el modelo básico de crecimiento. Supongamos que la incertidumbre en la economía se re fleja en cambios aleatorios en el sector productivo de la economía. Uno puede pensar en el caso de que el sector productivo dependa de condiciones climáticas o en el caso en que al interactuar muchos agentes con información incompleta y asimétrica sobre las condiciones del mercado, estos tomen decisiones en una dirección u otra, que en el agregado
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
62
CAPÍTULO CAPÍTULO 4. ECONOMÍA ECONOMÍA DINÁMICA: DINÁMICA: EL CASO ESTOCÁSTIC ESTOCÁSTICO O
producción de acuerdo a la siguiente especi ficación: yt = zt f (kt ) ,
donde {zt } es una secuencia de variables aleatorias i.i.d.4 Es decir, independientes e idénticamente distribuidas. Así, el problema del agente representativo es: ∞
m´ ax ax s.a kt+1 ct , kt
X
β t u (ct )
t=0
: = zt f (kt ) − ct + (1 − δ ) kt ≥ 0 k0 , z0 dados.
En este ejemplo, la variable de estado endógena es kt , la variable de estado exógena es zt, la variable de control es ct y la fuente de incertidumbre es la misma variable de estado zt . Luego, para expresar el problema en exactamente la misma forma que el problema secuencial de arriba, introducimos una variable θ t = zt y de esta manera la función de transición g la podemos identi ficar como: (kt+1, zt+1) = g (kt , zt , ct , θt+1 ) = (zt f (kt ) − ct + (1 − δ ) kt , θt+1)
y Γ(kt , zt ) =
{ct : 0 ≤ ct ≤ zt ktα + (1 − δ ) kt }
Anotación 18 Si suponemos, como usualmente se hace en la literatura, que log (zt) sigue un proceso autorregresivo de primer orden: log (zt) = ρlog (zt−1 ) + θt entonces la función de transición g la podríamos identi fi car car como: (kt+1, log(zt+1)) = g (kt , zt , ct , θt+1) = ( zt f (kt ) − ct + (1 − δ ) kt , ρ log(zt ) + θt+1)
De igual manera que en el caso determinístico, en general, la solución de este problema requiere de métodos numéricos. Por esta razón, utilizaremos la especificación de Brock y Mirman para poder resolver explícitamente el modelo e ilustrar las ideas principales. Ejemplo 12 (Brock y Myrman [1972], el caso estocástico). Supongamos que el capital se deprecia completamente al fi nal nal de cada período ( δ δ = 1), que la α función de producción es de la forma f (kt ) = zt kt donde α ∈ (0, 1) y que la función de utilidad es logarítmica. Así, nuestro problema se transforma en: ∞
m´ ax ax s.a kt+1
X
β t log(ct )
t=0
: = zt ktα − ct k dado.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
63 Ahora, de manera informal, si utilizáramos el método iterativo para encontrar la función valor, no es difícil sospechar, después de un par de iteraciones, que un buen candidato a ser la función valor es: v(k, z ) = a + b log(k) + c log(z ), donde a, b y c son constantes que debemos determinar. Nos proponemos ahora veri fi car car que en efecto ésta es la forma de la función valor. De la ecuación funcional, sabemos que debe cumplirse que: a+b log(kt )+c log(zt ) =
sup
0≤ct ≤zt ktα
{log(ct ) + βE t [a + b log(zt ktα − ct ) + c log(zt+1)]}
Claramente, la solución a este problema debe ser interior. Las condiciones de primer orden implican que el consumo óptimo es: ct =
zt ktα 1 + βb
Sustituyendo en la ecuación de Bellman es fácil ver que a = 1−1 β ln(1 − αβ ) + αβ α 1 (1−β)(1−αβ) αβ) ln(αβ ), b = (1−αβ) αβ) y c = (1−αβ) αβ) son constantes que hacen nuestro candidato a función valor satisfacer la ecuación de Bellman. Luego, la función de política es: ct = (1 − βα )zt ktα
y la dinámica óptima del capital esta dada por: kt+1 = βαz t ktα
(4.1)
Ahora, como punto de referencia para pensar con relación al problema de estabilidad bajo incertidumbre, nos referiremos al ejemplo anterior. Puesto que la dinámica del capital es un proceso estocástico, no es del todo claro en qué sentido es que el capital converge a un capital de “estado estacionario”. Una posibilidad natural, es que la distribución que caracteriza la dinámica del capital en cada instante φt , “converja” en algún sentido que debemos especi ficar, a una distribución φ, que proponemos como la distribución que caracteriza el estado estacionario, invariante a la dinámica de éste (esto es ciertamente más general que suponer que la convergencia es a una distribución concentrada concentrada en un punto). punto). Es decir, si el estado inicial de la economía es una realización de la distribución
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
64
CAPÍTULO CAPÍTULO 4. ECONOMÍA ECONOMÍA DINÁMICA: DINÁMICA: EL CASO ESTOCÁSTIC ESTOCÁSTICO O
donde G es la distribución del choque tecnológico. Ahora, sea H (a, b) = P (kt+1 ≤ b | kt = a) . La función de transición H la podemos interpretar como la probabilidad que en t + 1 el capital sea menor o b igual a b dado que en t el capital era a. Obsérvese Obsérvese que H (a, b) = G( αβa α ), que es una distribución conocida, luego podemos expresar la distribución del capital en t + 1 como: φt+1 (b) =
Z
(4.2)
H (a, b) dφt (a)
En el lenguaje de la teoría de la probabilidad, decimos que la dinámica óptima del capital es un Proceso de Markov con función de transición H (a, b) . La ecuación 4.2 es el análogo, en términos de la distribución distribución del stock de capital, de la ecuación 4.1. Usando este lenguaje, la propiedad de estabilidad puede enunciarse como: Existe una distribución φ, “límite” de las distribuciones {φt } , tal que: φ (b) =
Z
H (a, b) dφ (a) .
En este caso, decimos que la distribución distribución φ es invariante a la función de transición H. La distribución φ es nuestro análogo estocástico al estado estacionario en el caso determinístico. Ejemplo 13 (Brock y Mirman [1972], el caso estocástico una vez más) A diferencia del ejemplo anterior en donde nos preguntabamos por la distribución invariante del stock de capital ahora, por simplicidad, calcularemos la distribución invariante del logaritmo del stock de capital. Luego supongamos que ln (zt ) secuencia µt , σt2 t=0, de N 0, σ 2 . Vamos a mostrar que ln (kt ) N (µt , σ t2 ) y que la secuencia =0,1... medias y varianzas convergen a constantes µ y σ2 respectivamente . La conjetura natural, que más tarde formalizaremos, es que la distribución de ln (kt ) converge en algún sentido a una distribución invariante φ N µ, σ2 (obsérvese que φ sería la distribución invariante del logarítmo del stock de capital. Anteriormente encontramos que la dinámica óptima del capital es:
¡ ¢
∼
©¡ ¢ª
∼
∼
¡ ¢
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
65 Si reemplazamos hacia atrás sucesivamente hasta llegar a la primera observación de k, k0 , encontramos la siguiente expresión:
ÃX ! t
ln kt+1 =
i
α
t
ln(βα ) +
i=0
X
αi ln(zt−i ) + αt+1 ln k0
(4.3)
i=0
Tomando el valor esperado de esta ecuación y teniendo en cuenta la distribución de zt obtenemos:
ÃX ! "ÃX ! X "X # ÃX ! t
αi
E [ln kt+1] =
ln(βα ) + αt+1 ln k0
(4.4)
i=0
Ahora, a partir de la ecuación 4.3, la varianza V de ln kt+1 es: t
V [ln kt+1]
= V
t
αi
ln(βα ) + αt+1 ln k0 +
i=0 t
= V
#
αi ln(zt−i )
i=0
αi ln(zt−i )
i=0
y como zt son variables aleatorias i.i.d, entonces cov [ln(zt ), ln(zt−i )] = 0 para todo i = 6 0 . Luego t
α2i σ 2
V [ln kt+1] =
i=0
Las ecuaciones 4.3, 4.4 y 4.5 implican que ln(kt) ∼ N (µt , σt2) donde µt
=
σ t2
=
¡ ¢ ³ ´
1
1 − αt+1 ln(βα ) + αt+1 ln k0 1−α 1 2(t+1) σ2 1 − α2(t 1 − α2
Nótese que si α ∈ (0, 1) entonces:
l´ım µt
t→∞
l´
2
=
ln(αβ ) 1−α σ2
(4.5)
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
66
CAPÍTULO CAPÍTULO 4. ECONOMÍA ECONOMÍA DINÁMICA: DINÁMICA: EL CASO ESTOCÁSTIC ESTOCÁSTICO O
Anotación 19 Sin embargo, obsérvese que log[k∗ ] = E [φ]. Ejemplo 14 (Control Óptimo Lineal. Basado en Sargent [1987]). Consideremos el problema:6
"X
#
∞
sup E s.a xt+1
β t (xt0 Qxt + ut0 Rut + 2xt0 W ut )
t=0
: = Axt + Bu t + εt+1 x0 dado,
donde εt es un proceso estocástico i.i.d con media cero y matriz de varianzacovarianza Σ. El lector puede veri fi car car fácilmente que la función valor de este problema es: v(x, ε) = x0 P x +
β
tr(P Σ), 1 − β donde tr denota la traza de la matriz y P es la misma que teníamos en el
problema determinístico (i.e la solución al problema de Riccati). De otra parte, la función de política es: ut = −(R + βB 0 P B )−1 (βB 0 P A + W 0 )xt = −F xt
Luego, la dinámica óptima ésta dada por: xt+1 = ( A − BF )xt + εt+1.Obsérvese que la función de política es la misma función de política del caso determinístico. Esta propiedad propiedad la llaman los l os economistas economistas el principio de equivalencia determinística y es una característica muy particular de los problemas lineales-cuadráticos y no una propiedad general de los problemas de optimización dinámica estocástica. El resultado depende de tres características de este ejemplo: La función retorno es cuadrática, la dinámica de transición de las variables de estado es lineal y E [εt+1 | xt ] = 0 uctuaciones económicas. 7 Ejemplo 15 El costo en bienestar de las fl uctuaciones El agente representativo:
"X ∞
E
1−σ − t ct
β
1
1
#
)
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
67 tasa de crecimiento del consumo. La componente exp(− 12 σz2 )zt es la responsable de las fl uctuaciones uctuaciones del consumo. De esta manera tenemos una interpretación clara de los parámetros µ y σz2 . Ahora, de fi namos namos el bienestar de una economía 2 para parámetros dados λ, µ y σz , como:
"X # ∞
W (λ,µ,σ z2 ) =
E
β t U (ct )
t=0
El costo en bienestar de pasar de una tasa de crecimiento µ0 a µ es el valor de λµ que resuelve: W (λu , µ , σ z2 ) = W (0, µ0 , σ z2 )
No es difícil demostrar que para preferencias logarítmicas:9 λu =
µ ¶ 1 + µ0 1+µ
β
1−β
−1
El costo en bienestar de pasar de una economía con volatilidad 0 a σz2 , es el valor de λ que resuelve: W (λσ , µ , σ z2 ) = W (0, µ, 0)
Este valor se puede calcular como:
⎡ P¡ (1 + λ )(1 + µ) exp( E ⎣ ∞
t=0
σ
t
1−σ
− ((1 + µ)t )
¢
1−σ − 12 σ z2 )zt
=0
⎤ ⎦
1 (1 + λσ )1−σ exp(− σz2 (1 − σ ))E [exp((1 − σ )log zt )] = 1 2 1 1 ⇒ (1 + λσ )1−σ exp(− σz2 (1 − σ )) exp( exp( σ z2 (1 − σ )2 ) = 1 2 2 1 ⇒ (1 + λσ )1−σ exp(− σz2 ((1 − σ ) − (1 − σ )2 )) = 1 2 1 2 1−σ ((1 + λ ) p( )) =1 ⇒
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
68
CAPÍTULO CAPÍTULO 4. ECONOMÍA ECONOMÍA DINÁMICA: DINÁMICA: EL CASO ESTOCÁSTIC ESTOCÁSTICO O
σz ( %)
Deve Develo lopi ping ng
Indus Industr tria iall
G7
4.13
2.25
2.11
σ
1 0.09 0.025 0.02 5 0.45 0.1 0.1 10 0.9 0.2 0.2 Costo en bienestar bienestar en términos términos de consumo consumo anual ( %) Los mayores valores que se observan corresponden al coe fi ciente más alto de fi ciente aversión al riesgo. En este caso, lo que la tabla nos indica es que para los países en desarrollo el costo puede puede estar alrededor alrededor del 0.9 % y para para los industrializados alred alreded edor or del 0.2 %. Si bien estas magnitude magnitudess no resulta resultann ser desprec despreciable iables, s, sí son bastante inferiores a los efectos de cambiar de tasa de crecimiento, o a otros costos como por ejemplo, el costo de un nivel de in fl ación ación anticipado entre el 10 % y 30 %. Más adelante adelante calcular calcularemos emos este costo para para el caso caso colombiano colombiano..10
La versión estocástica del método de Lagrange es una aplicación de la fórmula de Benveniste y Scheinkman al caso estocástico. Informalmente: ∂v (xt ) ∂r (xt , h(xt )) = + βE t ∂x i ∂x i
"X n
k=1
#
∂v (g (xt , h(xt ), θt+1)) ∂g k (xt , h(xt ), θt+1) , ∂x k ∂x i
y las condiciones de primer orden del problema funcional son:
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
69 y, ∂r (xt , zt , h(xt , zt )) +βE t ∂u j
"X ns
k=1
#
∂v (xt+1 , zt+1) ∂g x,k (xt , zt , h(xt , zt ), θt+1) = 0 , j = 1 ,...m ∂x k ∂u j
donde (xt+1, zt+1) = (gx (xt , zt , h(xt , zt), θt+1), gz (zt , θt+1)). ∂v (xt ,zt ) Sea λi,t = ∂v( , i = 1,...ns entonces estas dos ecuaciones se pueden ∂x i reescribir como: ∗
¸
∂r (x∗t , zt , u∗t ) ∂g x (x∗t , zt , u∗t , θt+1) − λi,t = 0, i = 1 ,...ns (4.6) + βE t λt+1 · ∂x i ∂x i
∙
¸
∂r (x∗t , zt , u∗t ) ∂g x (x∗t , zt , u∗t , θt+1) + βE t λt+1 · = 0, j = 1 ,...m ∂u j ∂u j
∙
y la dinámica de las variables de estado endógenas: x∗t+1 = gx (x∗t , zt , u∗t , θt+1)
Si λt ≥ 0 y xt ≥ 0, la condición de Transversalidad es: l´ım β t λt · x∗t = 0
t→∞
(4.7)
(4.8)
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
70
4.1. 4.1.
CAPÍTULO CAPÍTULO 4. ECONOMÍA ECONOMÍA DINÁMICA: DINÁMICA: EL CASO ESTOCÁSTIC ESTOCÁSTICO O
Ejer Ejercic cicios ios y Solu Solucio cione ness
Ejercicio 16 El problema es el siguiente:
"X # ∞
m´ ax ax E s.a ct + kt+1
β t ln ct
t=0
:
zt ktα k0 , z0 dado,
6
y donde {zt } es i.i.d con ln(zt )˜N [0, σ2 ]. Utilizar el algoritmo de Howard para resolver el problema (Ayuda: Suponga que la dinámica óptima del capital es de la forma kt+1 = a0 (zt ktα ) donde a0 es una constante. Solución 12 Considerando el algorítmo de Howard, primero proponemos una dinámica óptima del capital: kt+1 donde a0
= a0 (ktα zt ) ∈ [0, 1] es una constante
De acuerdo al algoritmo, tenemos:
"
∞
#
(1)
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
71
4.1. EJERCICIOS EJERCICIOS Y SOLUCION SOLUCIONES ES
Como α ∈ (0, 1) :
µ µ ¶¶
ln kt = ln a0
1 − αt 1−α
t−1
+ αt ln k0 +
αi ln (zt−1−i )
i=0
Reemplazando esta expresión en la ecuación (3), J 0 (·) =
X
ln((1 − a0 )) + ln(z0 ) + 1 − β
"X à µ µ ¶¶ ∞
αE 0
β t ln a0
t=0
1 − αt 1−α
t−1
+ αt ln(k0 ) +
X
!#
αi ln(zt−1−i )
i=0
Tomando esperanzas y teniendo en cuenta que α y β son menores que uno, podemos podemos simpli fi car car algebráicamente esta expresión hasta encontrar: βα ln(a0 ) α ln k0 ln(1 − a0 ) J 0 (·) = + + + 1 − β (1 − αβ ) (1 − β ) 1 − αβ
Sean: A0 =
ln A(1−a0 ) 1−β
+
βα ln(Aa ln(Aa0 ) (1−αβ)(1 αβ)(1−β ) ;
A1 =
2−αβ 1−αβ
J 0 (·) = A0 + A1 ln z0 +
µ ¶
2 − αβ ln(z0 ) 1 − αβ
entonces:
α ln(k0 ) 1 − αβ
Para continuar con el algoritmo de Howard debemos maximizar:
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
72
CAPÍTULO CAPÍTULO 4. ECONOMÍA ECONOMÍA DINÁMICA: DINÁMICA: EL CASO ESTOCÁSTIC ESTOCÁSTICO O
1. Prob Probar ar que las traye trayectori ctorias as óptimas óptimas son de la forma: ct = π 1 zt ktα, kt+1 = π 2 zt ktα , donde π 1 y π 2 son constantes. 2. Mostrar Mostrar que el consumo consumo y el producto producto siguen siguen un proc proceso eso autoreg autoregre resivo sivo de orden 2 (i.e., AR(2)). Solución 13 El problema secuencial es:
"X ∞
Max E 0 kt+1
=
#
β t (θlnct + (1 − θ)ln(1 − nt ))
t=0 α 1−α zt kt nt − ct ,
k0 , z0 dados.
El problema funcional asociado es: v(k, z ) = M ax{θln(c) + (1 − θ )ln(1 − n) + βE t [v (k 0 , z 0 )]} c,n
0 6 c 6 zk α n1−α , 0 6 n61 k0 = zk α n1−α − c ln (z 0 ) = ρln (z ) + θ0
Iterando:
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
73
4.1. EJERCICIOS EJERCICIOS Y SOLUCION SOLUCIONES ES
luego: v1 (k, z ) = θln (z ) + αθln (k ) + (1 − α) θln
Sea A1 = (1 − α) θln v2 (k, z ) =
h i (1−α)θ 1−αθ
+ (1 − θ) ln
∙ (1
− α) θ
1 − αθ
h i 1−θ 1−αθ
¸
+ (1 − θ)ln
∙1
−θ
1 − αθ
¸
entonces:
M ax {θln(c)+(1−θ)ln(1−n)+βE [θln (z 0 ) + αθln(k 0 ) + A1 ]} α 1 −α 0 6 c 6 zk n 06n61
donde k0 = zk α n1−α −c y como ln(z 0 ) = ρln(z )+ θ0 entonces E [ln(z 0 )] = ρln(z ).
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
74
CAPÍTULO CAPÍTULO 4. ECONOMÍA ECONOMÍA DINÁMICA: DINÁMICA: EL CASO ESTOCÁSTIC ESTOCÁSTICO O
donde a y A son constantes que debemos determinar. Sustituyendo en la ecuación funcional y con un poco de álgebra se obtiene: A =
"
µ
θ (1 − α) θln (1 − αβ ) 1 − β 1 + α (θβ − θ − β )
1
+
1 1 − β
(1 − θ) ln
∙1
− θ − αβ + αθβ
1 + α (θβ − θ − β )
"µ
¸
¶ # 1−α
βαθ θ (1 − α) ln αβ + 1 − β (1 − αβ ) 1 + α (θβ − θ − β )
1
a=
(1
θ βρ ) (1
αβ )
¶ # 1−α
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Bibliografía [1] [1] Carvaja Carvajal, l, A. y A. Rias Riasco cos. s. 2005 2005.. Nota Notass de Clas Clasee sobr sobree Prob Probab abil ilid idad ad.. http://www.webpon http://www.webpondo.org/ariascos/File do.org/ariascos/Files/Probabilidad s/Probabilidad/ln2.pdf /ln2.pdf
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
Trusted by over 1 million members
Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.
APUNTES DE CLASE CEDE
1
SEPTIEMBRE
2006
DE
EL CENTRO DE ESTUDIOS SOBRE DESARROLLO ECONÓMICO-CEDE- de la Facult Fac ultad ad de Eco Econom nomía ía de la Uni Univer versida sidad d de losAnde losAndes s se fun fundó dó en 195 1958, 8, conel obj objeti etivo vo de realiz realizar ar invest investigacio igaciones nes económi económicas cas tantoteóricascomo empíri empíricas. cas. Actualmente, las áreas de interés para el CEDE son: Macroeconomía y Sector Financiero, Evaluación Socioeconómica Socioeconómica de Proyec Proyectos, tos, Economí Economía a Ambiental, Econom Economía ía Agrícola, Demografía, Educación, Salud, Economía Laboral, Economía Regional y