1 Dynamic models with exogenous saving rate

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1 Macroeconomics, UNICAN-MIE 1 Dynamic models with exogenous saving rate 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 más la depreciación 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, output por trabajador y el consumo 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, L t, K t ) = AKt α L 1 α t K ρ t donde K t es el capital agregado de dicha economía. Suponga que la población es constante e igual a L. (a) Escriba una expresión de la tasa de crecimiento del capital productivo per capita en esta economía bajo el supuesto de que α + ρ = 1. (b) Imagine dos países que comparten los mismos valores de A, δ, α, s, pero uno de ellos tiene una población superior al otro, qué predice este modelo sobre sus tasas de crecimiento? (c) Cómo se comportaría esta econmía si α + ρ < 1? 2 Dynamic model with endogenous saving rate 2.1 Finite Horizon, partial equilibrium 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. 2.2 Neoclassical growth model 2. Consider the social planner problem of the neoclassical growth model with endogenous labour supply max {c t,k t,l t} t=0 β t u(c t, l t ) t=0 subject to f(k t, n t ) c t + k t+1 l t + n t = 1 c t 0, k t, l t, n t 0 k 0 given Show that the set of feasible allocations is convex. 2

3 3. Two sector economy. Consider an economy with an infinitely lived consumer and a productive technology with two sectors: one sector produces consumer goods and the other produces investment goods. Precisely, the Pareto optimal allocations are described by the following maximization problem: max {c t, k t+1, i t} t=0 β t u (c t ) t=0 s.t. c t = (kt c ) ac ( i c t + i k t = A kt k ) ak kt+1 c = (1 d c ) kt c + i c t ( kt+1 k = 1 d k) kt k + i k t i c t, i k t 0 k k 0, k c 0 > 0 are given. Here, subscripts and superscripts c and k denote parameters or variables in the consumption and the capital sectors, respectively; a c and a k determine the marginal productivity of capital, a c, a k (0, 1); d k and d c are the depreciation rates of two types of capital, d k, d c (0, 1]; and A is the relative productivity in two sector. The first two constraints are the production function in each sector; the second two constraints show that new capital can be allocated to either sector. Nevertheless, once a capital good has been used in a given sector, it cannot be reallocated to the other sector. Assume that the non-negativity constraints for investments are not binding in the optimal solution. (a) Argue that the problem is recursive. Identify the control variables, the state variables, the transition and return functions. (b) Write down the FOCs for optimality by using the Lagrangian. (c) Find formulas that determine the steady state equilibrium. (d) Assume that u (c) = log (c) ; a c = a k ; d c = d k = 1. Find a solution for any initial condition for the capital stock. (Hint: try to convert this problem into the Ramsey s model). (e) Describe a market economy with two firms and one consumer such that, in equilibrium, the solution to the problem is equivalent to the optimal solution described by FOCs found in (b). Notice that consumption and investment goods are produced by using different technologies; therefore, it is necessary to introduce the relative prices between these commodities. Furthermore, due to decreasing returns to scale, the two production firms earn positive profits after paying for capital. To distribute these profits, assume that the agent owns the firms and gets dividends. 3

4 4. 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, n t, K t ) = kt α n 1 α t 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 and labour (each household is endowed with one unit of time in each period). 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. 5. Consider the following problem: max {c t, k t+1, i t} t=0 t=0 β t (c t ηc t 1 ) γ+1 γ + 1 s.t. c t + i t = kt α k t+1 = (1 d) k t + i t k 0, c 1 > 0, given. (Notice that past consumption affects current utility. If η > 0, it means that the agent is unhappy when current consumption is low relative to past consumption; if η < 0 it is interpreted as if consumption provides happiness for more than one period.) (a) Is the problem recursive? If so, say what are the control variables, the state variables, the transition function, the return function. Propose two different definitions of state and control variables. (b) Write down the Bellman equation for one of the definitions of state and control variables, proposed in (a). (c) Derive the Euler equation from the Bellman equation. 6. Learning by doing. Imagine an infinitively lived representative consumer who enjoys utility from consumption and leisure, so her lifetime utility is given by t=0 βt u(c t, l t ), with 0 < β < 1, u c > 0, u l > 0, u cc < 0 and u ll < 0. The consumer is endowed with one unit of time in each period. Consumer s efficient units of labor, that we denote by h t, are the product of households s labour supply, (1 l t ) and household s human capital, x t. Consumer s human capital evolves as a function on labour supply, in particular, 4

5 x t+1 = f(x t, 1 l t ). In the economy there is a linear technology that transforms aggregate efficient units of labour, H t, into consumption goods (denote the marginal return of an efficient unit of labour by A). The representative consumer consumes her labour income in each period. We normalize the number of both firms and agents to 1. (a) Write down the consumer s maximization problem as a dynamic programming problem. Identify state and control variables. (b) Provide an interpretation of the equation that characterizes the optimal allocation of leisure. (c) What is the equilibrium wage per efficient unit of labour? (d) Write down the equations that characterize a steady state equilibrium for this economy. 7. 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 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. 8. 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 5

6 Market technology is given by where z t follows an AR(1) Y t = e zt K α t (L m t ) 1 α z t+1 = ρz t + ɛ t+1 with ɛ t iid from a normal distribution with zero mean and variance σ 2 ε. (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 (steady state, z t = 0): wlm = 0.6, K Y Y = 4, i Y = 0.3 and lm = 0.3 and l h = 0.1. (d) What can you say about the volatility of hours in this economy with respect to the economy without household production? 9. 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. 11. Consider the representative agent model 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 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 taxing labour income τ t in each period. However, in order to finance public deficit, government may issue debt in each period (bonds) that has to be repaid in the following period. Denote by q t the price of a bond issued in period t that pays 1 unit of consumption good in period t + 1. Denote by B t the amount of debt issued in period t In this economy a firm operates using the following technology Y t = Kt α, Nt 1 α ). Assume there is an initial capital K 0 as well as an initial outstanding government debt B 0. (a) Write down the household problem. (b) Write down government budget constrain. (c) Find the equations that characterize a SME equilibrium in this economy. 6

7 (d) What can be said about the relationship between the bond equilibrium price and the return to capital in this economy? 7