1 Modelo dinámico con tasa ahorro exógena

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1 Master en Economía. Instrumentos del Análisis Económico. Macroeconomía Hoja de Ejercicios Modelo dinámico con tasa ahorro exógena 1. Imagine dos países que se encuentran en su senda de crecimiento equilibrado. En el país A la producción per capita es 10 veces mayor que en el país B. Suponga que α = 0.3 y que ambos comparten el mismo estado de la tecnología y la misma tasa de depreciación. Se observa que el tipo de interés en el país B es 5 veces superior al del país A. Es esto coherente con las implicaciones del modelo de Solow? 2. Explique cómo los siguientes hechos afectan al capital por trabajador tanto en el largo plazo como a lo largo de la transición en el modelo de Solow-Swan: (a) La destrucción de un 30% del stock de capital debido a un huracán. (b) Un incremento permanente en la tasa de inmigración. (c) Un incremento permanente en la tasa de participación en el mercado laboral. (d) Un incremento permanente en la tasa de depreciación. (e) Un incremento temporal de la tasa de ahorro. (f) Un incremento permanente en la tasa de ahorro. 3. Imagine una economía en la que en cada período t se produce con una función de producción Cobb-Douglas, F (K t, L t ) = K α t L t 1 α a partir del capital, K t, y del trabajo existentes, L t. La población crece a una tasa constante e igual a n. En esta economía se dedica a la inversión una fracción s de la renta de cada período. El capital se deprecia en el proceso productivo a una tasa δ. (a) Escriba la tasa de crecimiento del capital por trabajador en el equilibrio de esta economía. (b) Cuál es la tasa de crecimiento del PIB en el estado estacionario? (c) Calcule la elasticidad de la producción por trabajador con respecto a la tasa de ahorro en el estado estacionario. De acuerdo con este modelo, qué predice que ocurra con la producción por trabajador si se produce un incremento de la tasa de ahorro del 10% y sabe que el peso de la renta del capital sobre el PIB en esta economía asciende a 0.3? (d) Los datos muestran que los países ricos tiene una renta por trabajador aproximadamente 10 veces superior a la de los países pobres y que el capital por trabajador es aproximadamente 25 veces superior en los países ricos que en los pobres, es esto coherente con las implicaciones del modelo económico que ha desarrollado en los apartados anteriores? (suponga que los países se encuentran en su estado estacionario). 1

2 4. Imagine una economía en la que la tecnología viene representada por F (K t, G t ) = K α t G 1 α t donde K t es el capital agregado de dicha economía y G t es el gasto público en infraestructuras. Imagine que dicho gasto público se financia con un impuesto proporcional τ sobre la renta agregada de la economía. El tamaño de la población es L t y crece a una tasa exógena n. (a) Escriba la condición de equilibrio presupuestario del gobierno y la condición de equilibrio en el mercado de bienes. (b) Encuentre una expresión para la tasa de crecimiento del capital por trabajador en esta economía. (c) Podemos pensar en este modelo como una re-interpretación del modelo AK? 5. Use data from Penn World TAble 9.0 (section Expert data access) of GDP (rgdpna), capital (rkna) and labor (emp) for Spain and the US to calculate annual growth rate of total factor productivity under the assumption that the production function is F (K, L) = AK 1/3 L 2/3. Calculate the contribution of annual growth rate of GDP per worker of the different inputs over the periods , , , Modelo dinámico con tasa ahorro enógena 2.1 Horizonte Finito, equilibrio parcial 1. Imagine un hogar que vive 60 períodos (años). El hogar ingresa w t unidades de bien de consumo en cada uno los primeros 45 años de vida, durante el resto de su vida sus ingresos son 0. Suponga que existe un mercado financiero en el que se puede ahorrar y pedir prestado a un tipo de interés r. El hogar deriva utilidad del consumo en cada uno de los períodos de vida, descontando el futuro a una tasa anual β. La utilidad que el hogar obtiene del consumo en cada período viene dada por la función de utilidad u(c j ) = c1 σ j 1 σ. (a) Escriba el problema del hogar utilizando una sola restricción presupuestaria. (b) Obtenga una expresión para el nivel de consumo óptimo del hogar en cada período j. (c) Dibuje el perfil de consumo y renta de ciclo vita de este individuo si β = 0.98 y el tipo de interés en la economía es el 4%. 2.2 Horizonte infinito, equilibrio general Growth model with externality (Romer, JPE 1986). Let k t be the firm s capital and K t be aggregate capital in the economy. Assume that the production function for each firm is f (k t, K t ) = kt α Kt λ, where the term Kt λ reveals the presence of a positive externality. Assume α, λ > 0 and α + λ < 1. Capital depreciates at the rate δ (0, 1]. There are many small firms, all alike; each firm takes K t as given. We normalize the number of both firms and agents to 1. Households own capital. The lifetime utility of a representative households is given by t=0 βt u (c t ). 2

3 (a) Find a formula for the steady state aggregate capital that solves the social planner problem. (b) Formulate a competitive equilibrium in this model. Find the steady state aggregate capital. (c) Is the competitive equilibrium Pareto optimal? Provide an economic interpretation. Home production.(benhabib, Rogerson and Wright, JPE 1991) Consider an economy in which individuals enjoy utility from consumption of a market good, c m t, and from consumption of home production, c h t. Each individual is endowed with 1 unit of time and has a technology to transform her time endowment in home production. Instantaneous utility function is β t [ γ log c m t + (1 γ) log c h t + log(1 l m t l h t ] where l m t is the fraction of time devoted to market production adn l h t is the fraction of time devoted to home production. Home production technology is given by c h t = (l h t ) φ Market technology is given by Y t = K α t (L m t ) 1 α (a) Solve the social planner problem of this economy. (b) Find equilibrium prices for capital and labour and characterize the steady state equilibrium. (c) Use the following information to calibrate the parameters in this economy: K Y = 4, i Y = 0.3 and lm = 0.3 and l h = 0.1. wl m Y = 0.6, Consider an economy in which individuals derive utility from c t and leisure l t, according to the following utility function β t {(1 µ) log(c t ) + µ log(l t )}...where 0 < µ < 1 t=0 where β < 1. Assume individuals are endowed with 1 unit of productive time each period. Production is obtained from labour and capital according to the following production function F (k t, n t ) = K α t L 1 α t. (a) Solve the social planner problem for this economy. (b) Write down the equations that characterize an steady state equilibrium of this economy. (c) Calibrate model parameters (δ, β, α y µ.) using data for Italy from Penn World Table 9.0. Use the average value of statistics for the period : delta annual rate of depreciation labsh labor share, csh i ratio of investment to GDP and avh annual hours per person. Assume hours endowment of individuals is

4 Guess and Verify Method. In the representative agent model assume u(c) = log(c) and F (k, n) = k α n 1 α. Verify that the form of the value function of the recursive problem of the social planner is of the type v(k) = A+Bln(k) looking for the parameter values A y B that satisfy the function equation. Assume δ = 1. Consider an economy in which households have utility function u(c) = log(c) and firms operate with the following production function F (k, 1) = Ak α. Parameter values are β = 0.95, A = 1, α = 0.35 and δ = Use value function iteration with a grid of 500 equally spaced points from 0.2 k to 1.2 k (where k is the steady state capital) to solve the recursive planner s problem. Simulate the economy for 100 periods starting from k 0 = 0.5 k and plot the realizations of the optimal paths for capital, consumption, output, real wage and rate of return of capital. Cash in advance. Consider an economy in which households receive utility in each period according to [u(c t ) + v(1 N t )], where (C t is consumption and (1 N t ) is leisure. In order to consume they must hold cash (money), so that the nominal value of consumption must be less or equal cash holdings, P t C t M t, where P t the price of the consumption good. The government increases money supply at rate µ in each period, M t+1 = M t (1 + µ), to fund transfers to households, T R. There are firms that hire labor to produce a consumption good. Households can issue bonds in each period B t. Each bond that is issued in t pays 1 unit of consumption in t + 1 and its nominal price is denoted by Q t. Denote with W t the nominal wage. (a) Write down first order conditions of the household and firm problem. (b) How is inflation in a steady state equilibrium in which consumption is constant? (c) Assume that utility is logarithmic and find steady state consumption in this economy. Is money neutral? 1. Growth model with externality (Romer, JPE 1986). Let k t be the firm s capital and K t be aggregate capital in the economy. Assume that the production function for each firm is f (k t, K t ) = kt α Kt λ, where the term Kt λ reveals the presence of a positive externality. Assume α, λ > 0 and α + λ < 1. Capital depreciates at the rate δ (0, 1]. There are many small firms, all alike; each firm takes K t as given. We normalize the number of both firms and agents to 1. Households own capital. The lifetime utility of a representative households is given by t=0 βt u (c t ). (a) Find a formula for the steady state aggregate capital that solves the social planner problem. (b) Formulate a competitive equilibrium in this model. Find the steady state aggregate capital. (c) Is the competitive equilibrium Pareto optimal? Provide an economic interpretation. 4

5 2. Home production.(benhabib, Rogerson and Wright, JPE 1991) Consider an economy in which individuals enjoy utility from consumption of a market good, c m t, and from consumption of home production, c h t. Each individual is endowed with 1 unit of time and has a technology to transform her time endowment in home production. Instantaneous utility function is β t [ γ log c m t + (1 γ) log c h t + log(1 l m t l h t ] where l m t is the fraction of time devoted to market production adn l h t is the fraction of time devoted to home production. Home production technology is given by c h t = (l h t ) φ Market technology is given by Y t = K α t (L m t ) 1 α (a) Solve the social planner problem of this economy. (b) Find equilibrium prices for capital and labour and characterize the steady state equilibrium. (c) Use the following information to calibrate the parameters in this economy: wlm = 0.6, K Y = 4, i Y = 0.3 and lm = 0.3 and l h = Consider an economy in which individuals derive utility from c t and leisure l t, according to the following utility function β t {(1 µ) log(c t ) + µ log(l t )}...where 0 < µ < 1 t=0 where β < 1. Assume individuals are endowed with 1 unit of productive time each period. Production is obtained from labour and capital according to the following production function F (k t, n t ) = K α t L 1 α t. (a) Solve the social planner problem for this economy. (b) Write down the equations that characterize an steady state equilibrium of this economy. (c) Calibrate model parameters (δ, β, α y µ.) using data for Italy from Penn World Table 9.0. Use the average value of statistics for the period : delta annual rate of depreciation labsh labor share, csh i ratio of investment to GDP and avh annual hours per person. Assume hours endowment of individuals is Y 4. Consider a representative agent economy in which the consumer lives an infinite number of periods and enjoy utility from consumption and leisure according to the following utility function u(c, l) = logc + ψlog(1 n) (time endowment in each period is normalized to 5

6 1). The representative agent discount the future at rate β. There is a government in the economy that has to finance a stream of public expenditure {g t } t=0 by using an income tax τ t in each period. We assume that public budget has to be balanced in every period. There is no allowance for depreciation. In this economy a firm operates using the following technology y t = kt α, n 1 α t ) (a) Define a sequential market competitive equilibrium (SME) for this economy (b) Find the optimal allocation for a Social Planner who has to set aside G t units of output in each period. (c) Is the SME efficient? (d) Calibrate this model economy (β, ψ, α, τ, δ) to match the following targets: (i) two thirds of income goes to labor, (ii) government expenditure is equal to 20% of GDP, (iii) investment rate is equal to 15%, (iv) the annual after-tax return to capital, net of depreciation, is 4% and (v) people devote to work one third of their time endowment. 5. Guess and Verify Method. In the representative agent model assume u(c) = log(c) and F (k, n) = k α n 1 α. Verify that the form of the value function of the recursive problem of the social planner is of the type v(k) = A + Bln(k) looking for the parameter values A y B that satisfy the function equation. Assume δ = Consider an economy in which households have utility function u(c) = log(c) and firms operate with the following production function F (k, 1) = Ak α. Parameter values are β = 0.95, A = 1, α = 0.35 and δ = Use value function iteration with a grid of 500 equally spaced points from 0.2 k to 1.2 k (where k is the steady state capital) to solve the recursive planner s problem. Simulate the economy for 100 periods starting from k 0 = 0.5 k and plot the realizations of the optimal paths for capital, consumption, output, real wage and rate of return of capital. 7. Cash in advance. Consider an economy in which households receive utility in each period according to [u(c t )+v(1 N t )], where (C t is consumption and (1 N t ) is leisure. In order to consume they must hold cash (money), so that the nominal value of consumption must be less or equal cash holdings, P t C t M t, where P t the price of the consumption good. The government increases money supply at rate µ in each period, M t+1 = M t (1 + µ), to fund transfers to households, T R. There are firms that hire labor to produce a consumption good. Households can issue bonds in each period B t. Each bond that is issued in t pays 1 unit of consumption in t + 1 and its nominal price is denoted by Q t. Denote with W t the nominal wage. (a) Write down first order conditions of the household and firm problem. (b) How is inflation in a steady state equilibrium in which consumption is constant? (c) Assume that utility is logarithmic and find steady state consumption in this economy. Is money neutral? 6