A simple macro dynamic model with endogenous saving rate: the representative agent model

A simple macro dynamic model with endogenous saving rate: the representative agent model Virginia Sánchez-Marcos Macroeconomics, MIE-UNICAN Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 1 / 59

1 Introduction 2 Sequential markets equilibrium 3 Arrow-Debreu markets equilibrium 4 Efficiency 5 Steady State Equilibrium 6 Mapping the Model to the Data 7 A model with labour augmenting technological progress 8 Recursive Formulation 9 Dynamic Programming 10 The Euler equation in the recursive formulation 11 Dynamics 12 Computation 13 Value Function Iteration 14 Limitations of the representative agent model 15 Limitations of the infinite horizon setup 16 Applications Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 2 / 59

Introduction From Solow to Ramsey (1927) and Cass-Koopmans (1965) Inter-temporal allocation of resources with endogenous saving rate Discrete time Infinite horizon This is one of the workhorse model in economics: the neoclassical growth model or the representative agent model with infinite horizon Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 3 / 59

Sequential markets equilibrium Model ingredients and assumptions Goods in the economy: labor services, n t, capital services, k t, and final good, y t that can be either consumed, c t, or invested, i t. Investment augments the capital stock which depreciates at a constant rate k t+1 = k t (1 δ)+ i t Labor services price is w t and capital services price is r t Representative household/individual/consumer with preferences over consumption and leisure Representative firm that operates with a certain technology Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 4 / 59

Sequential markets equilibrium Model ingredients and assumptions Households owns capital and labor Firms rent capital and labor Markets bring households and firms together Initial stock of capital k 0 No uncertainty and perfect foresight Competitive equilibrium: households and firms take prices as given (prices are beyond their control) Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 5 / 59

Sequential markets equilibrium Households s preferences U(c 0, c 1, c 2,...) = β t u(c t ) t=0 per-period/instantaneous utility function, u(c t ) continuously differentiable strictly increasing, u (.) > 0 strictly concave u (.) < 0 U bounded satisfies Inada conditions: lim c 0u (c) = and lim c u (c) = 0. β (0, 1) household has initial capital k 0 household has a time endowment each period, we normalize it to 1 Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 6 / 59

Sequential markets equilibrium Firms technology production function, Y t = F (K t, N t ) continuously differentiable homogenous of degree 1: Euler Theorem, F (K t, N t) = F K K + F N N strictly increasing in both arguments, F N (K, N) > 0 and F K (K, N) > 0 strictly concave,f KK (K, N) < 0 and F NN (K, N) < 0 satisfies Inada conditions: F (0, N t) = F (K t, 0) = 0 lim K 0 F K (K, N) = lim N 0 F N (K, N) = lim K F K (K, N) = lim N F N (K, N) = 0 δ (0, 1) Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 7 / 59

Sequential markets equilibrium Household s maximization problem max {ct,n t,i t,k t+1} t=0 st β t u(c t ) t=0 c t + i t w t n t + r t k t k t+1 = (1 δ)k t + i t k 0 given c t 0 t t k t+1 A No-Ponzi condition Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 8 / 59

Sequential markets equilibrium The Transversality Condition In an economy with a finite horizon, k T +1 = 0 is an optimality condition An equivalent terminal condition is needed here: Transversality Condition lim λ t k t+1 = 0, where β t u (c t ) = λ t t - it would not be optimal to end up at time t with a positive capital stock if the present value of the marginal utility of terminal consumption were positive. It could, instead, be consumed - prescription of how to behave optimally - very different idea than the no Ponzi game condition - sufficient condition Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 9 / 59

Sequential markets equilibrium Firm s maximization problem subject to max Y t,k t,n t Y t r t K t w t N t Y t = F (K t, N t ) Comment on aggregation Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 10 / 59

Sequential markets equilibrium Markets Households and firms decision must be consistent The market takes care of that through prices Factor markets clearing: n t = N t k t = K t Good market clearing: C t + I t = Y t Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 11 / 59

Sequential markets equilibrium Definition. Given K 0, a SM competitive equilibrium for this { economy consist of a sequence of prices {ˆr t, ŵ t } t=0 and a sequence of quantities ĉ t, î t, ˆk } t+1, ˆn t for { t=0 the household and a sequence of quantities for the firm ˆKt+1, ˆN } t, Ŷt such that: { 1 Given k 0 > 0 and {ˆr t, ŵ t } t=0 the sequence ĉ t, î t, ˆk } t+1, ˆn t solves the t=0 maximization problem of the household above { } 2 Given {ˆr t, ŵ t } t=0 the sequence ˆK t, ˆN t, Ŷ t solves the maximization t=0 problem of the firm above 3 Markets clear Y t = C t + I t t=0 n t = N t k t = K t Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 12 / 59

Sequential markets equilibrium Characterization of equilibrium Household s problem u (c t ) = βu [c t+1 )(r t+1 + (1 δ)] c t = w t + r t k t k t+1 + (1 δ)k t Euler equation Feasibility condition Additionally, the transversality condition is required lim λ t k t+1 = 0 t Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 13 / 59

Sequential markets equilibrium Characterization of equilibrium Firm s problem F N (K t, 1) = w t F K (K t, 1) = r t Note that in equilibrium: Y t = r t K t + w t N t National Accounts Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 14 / 59

Sequential markets equilibrium Characterization Market clearing Y t = C t + I t n t = N t k t = K t Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 15 / 59

Arrow-Debreu markets equilibrium Household s maximization problem max {ct,n t,i t,k t+1} t=0 st β t u(c t ) t=0 p t [c t + i t ] p t [w t n t + r t k t ] t=0 t=0 k t+1 = (1 δ)k t + i t t k 0 given c t 0 Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 16 / 59

Arrow-Debreu markets equilibrium Firm s maximization problem max {Kt,N t} t=0 p t [F (K t, N t ) w t N t r t K t ] t=0 Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 17 / 59

Arrow-Debreu markets equilibrium Definition. An AD competitive equilibrium for this economy consists { of a sequence of prices {ˆp t, ˆr t, ŵ t } t=0 and a sequence of quantities ĉ t, î t, ˆk } t+1, ˆn t t=0 for the household and a sequence of quantities for the firm { ˆKt+1, ˆN t, Ŷt such that: 1 Given k 0 > 0 and {ˆp t, ˆr t, ŵ t } t=0 the sequence {c t, i t, k t+1, n t } t=0 solves the maximization problem of the household above 2 Given {ˆp t, ˆr t, ŵ t } t=0 the sequence {K t+1, N t, Y t } t=0 solves the maximization problem of the firm above 3 Markets clear Y t = C t + I t } t=0 n t = N t k t = K t Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 18 / 59

Efficiency Pareto Optimal Allocations Definition. An allocation {c t, k t, n t } t=0 is feasible if for all t 0 F (n t, k t ) = c t + i t = c t + k t+1 (1 δ)k t c t 0, k t 0, 0 n t 1 Definition. An allocation {c t, k t, n t { } t=0 is Pareto } efficient if it is feasible and there is no other feasible allocation ĉ t, ˆk t, ˆn t such that t=0 β t u(ĉ t ) > β t u(c t ) t=0 t=0 Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 19 / 59

Efficiency Social Planner Problem subject to max {c t,k t+1} t=0 β t u(c t ) t=0 F (k t, n t ) = c t + k t+1 (1 δ)k t c t 0, k t+1 0 Additionally, a no-ponzi game condition should be added Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 20 / 59

Efficiency Social Planner Problem max {kt+1} t=0 t=0 β t u(f (k t ) k t+1 ) st 0 k t+1 f (k t ) = F (k t, 1) + (1 δ)k t k 0 given The solution must satisfy u (c t ) = βu (c t+1 )f (k t+1 ) t Example lim λ t k t+1 = 0 t Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 21 / 59

Efficiency Social Planner Problem Theorem. Given the assumptions above a sequence {k t+1 } t=0 that satisfies the Euler Equation and the Transversality Condition is a solution to the planner problem. Proof Why to solve this problem? WELFARE THEOREMS How to solve it? NUMERICAL METHODS: DYNAMIC PROGRAMMING and others Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 22 / 59

Efficiency Proof Imagine {k t+1 } t=0 satisfying Euler Equation and lim t λ t k t+1 = 0 with c t = F (k t, n t ) k t+1 + (1 δ)k t Define W (k t, k t+1 ) = u(c t ) = u(f (k t, n t ) k t+1 + (1 δ)k t ) Note that Euler Equation can be written as W 2 (k t, k t+1 ) = βw 1 (k t+1, k t+2 ) Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 23 / 59

Efficiency Proof } Let {ˆk t+1 t=0 be another feasible allocation with k 0 = ˆk 0 Then for {k t+1 } t=0 to be a solution of the SPP D = lim T T β t (W (k t, k t+1 ) W (ˆk t, ˆk t+1 )) = t=0 has to be non-negative Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 24 / 59

Efficiency Proof Since W is concave D lim T T β t (W 1 (k t, k t+1 )(k t ˆk t ) + W 2 (k t, k t+1 )(k t+1 ˆk t+1 )) t=0 Given k 0 = ˆk 0 we can re-arrange the expression as follows T 1 D lim β t (W 2 (k t, k t+1 ) + βw 1 (k t+1, k t+2 ))(k t+1 ˆk t+1 )+ T t=0 +β T W 2 (k T, k T +1 )(k T +1 ˆk T +1 ) Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 25 / 59

Efficiency Proof Using Euler equation this simplifies to D lim β T W 2 (k T, k T +1 )(k T +1 ˆk T +1 ) T That (using Euler equation again) we can rewrite as D lim T β T +1 W 1 (k T +1, k T +2 )(k T +1 ˆk T +1 ) Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 26 / 59

Efficiency Then D lim T β T +1 W 1 (k T +1, k T +2 )k T +1 Note that: D lim T β T +1 u (c T +1 )[F k (k T +1, n T +1 ) + (1 δ)]k T +1 Note that β T +1 u (c T +1 )[F k (k T +1, n T +1 ) + (1 δ)] = β T u (c T ) D lim T λ T k T +1 The expression on the right is exactly the Transversality condition, so D lim T λ T k T +1 = 0 Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 27 / 59

Efficiency Welfare Theorems First Welfare Theorem: the allocations c t and i t and k t+1 obtained solving the competitive equilibrium also solve the planner s problem Second Welfare Theorem: if the allocations c t and i t and k t+1 solve the planner s problem, there exist prices r t, w t that support these allocations as a competitive equilibrium. So we can solve the planner s problem and then find the equilibrium prices to build a competitive equilibrium However, the Theorems fail under some conditions: externalities, public goods, taxes... Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 28 / 59

Steady State Equilibrium Definition. A steady state equilibrium is a competitive equilibrium in which allocations are constant over time. c t+1 = c t = c k t+1 = k t = k Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 29 / 59

Mapping the Model to the Data Calibration We need to choose functional forms and parameter values in order to find numerical solutions for our model economy The main idea is to select the model economy parameters in such a way that it is consistent with the data statistics in certain dimensions The origins of calibration are related to Real Business Cycle models (Kydland and Prescott (1982)) In RBC literature parameter values are identified in such a way that the balanced growth path statistics of the model economy is consistent with the data once aggregates shocks are switched off (in this particular case parameters may be obtained algebraically) However, sometimes the calibration and computation problem is a join process, as we need solving the model in order to calibrate it: then parameters are found to minimize the distance between data and model statistics Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 30 / 59

Mapping the Model to the Data Calibration Steps Selecting of functional forms production function utility function Building consistent measures of the data statistics: see Cooley, T. (1995), Ch.1. Solving the system of equations (may be non-linear) Estimation versus calibration: (2007) Fernà ndez-villaverde, J. (2010) or Canova, F. Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 31 / 59

A model with labour augmenting technological progress Details Economy with population growth and labour augmenting technological progress N t = (1 + n) t, N 0 = 1 Y t = AK α t ((1 + g) t N t ) 1 α Feasibility constraint is C t + K t+1 (1 δ)k t = AK α t ((1 + g) t N t ) 1 α Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 32 / 59

A model with labour augmenting technological progress Social Planner Problem s.t. max {c t,k t+1} t=0 β t u(c t ) t=0 C t + K t+1 (1 δ)k t = AK α t ((1 + g) t N t ) 1 α Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 33 / 59

A model with labour augmenting technological progress Variables per capita c t = C t C t = N t (1 + n) t y t = k t = Y t (1 + n) t K t (1 + n) t Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 34 / 59

A model with labour augmenting technological progress Variables per efficient unit of labor c t = ỹ t = k t = C t (1 + g) t (1 + n) t = c t (1 + g) t Y t (1 + g) t (1 + n) t = y t (1 + g) t K t (1 + g) t (1 + n) t = k t (1 + g) t Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 35 / 59

A model with labour augmenting technological progress Reformulation Budget constraint C t (1 + g) t (1 + n) t + K t+1 (1 + g) t (1 + n) t (1 δ)k t (1 + g) t (1 + n) t = AK t α ((1 + g) t N t ) 1 α (1 + g) t (1 + n) t c t + k t+1 (1 + g)(1 + n) k t (1 δ) = A k α t Utility function u(c t ) = c1 σ t 1 σ = ( c t(1 + g) t ) 1 σ 1 σ = (1 + g) t(1 σ) c t 1 σ 1 σ β = β(1 + g) (1 σ) Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 36 / 59

A model with labour augmenting technological progress Reformulation s.t. max { c t, k t+1} t=0 β t u( c t ) t=0 c t + k t+1 (1 + g)(1 + n) k t (1 δ) = A k α t First order conditions β t u ( c t ) = λ t (1 + g)(1 + n)λ t = λ t+1 (Aα k α 1 t+1 + (1 δ)) Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 37 / 59

A model with labour augmenting technological progress Balance growth path There exists a BGP c t = c t+1,..., k t = k t+1 Assuming u(c) = logc and β = 1 and that ρ, g, n are small 1 + ρ ( ) 1 1 α αa k = ρ + g + n + δ c = A k α (1 + g)(1 + n) k + k(1 δ) = A k α [g + n + δ] k ỹ = A k α ĩ = [g + n + δ] k (1 + g)(1 + n) = β(1 + g) 1 σ (Aα k α 1 + (1 δ)) Note that r = Aα k α 1 (see next slide) Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 38 / 59

A model with labour augmenting technological progress Balance growth path All per capita variables grow at constant rate g in the BGP What happen to wages and interest rates? max AK α t ((1 + g) t N t ) 1 α w t N t r t K t w t = A(1 α)(1 + g) t K α t ((1 + g) t N t ) 1 α = (1 + g) t (1 α)a k α r t = AαKt α 1 ((1 + g) t N t ) 1 α = αa k α 1 Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 39 / 59

A model with labour augmenting technological progress Parameters identification, US data (Q) Parameters: α, A, δ, n, β, σ, g Some are chosen arbitrarily: A = 1, σ = 1 The rest are selected so that the model long-run implications match long-run average observations from the data Data Targets Data Value Parameters (i) Annual Population Growth Rate= 1.1% n = 0.27% (ii) Annual GDP Growth Rate= 2.1% g = 0.55% (iii) Capital Share*= [0.25,0.4] α = [0.25,0.4] (iv) I Y = ĩ ỹ = [g + n + δ] K Y δ = I /Y K/Y n g δ = 1.6% I Y = 25% K Y = A2.6/Q10.4 (v) 1 + r δ = (1 + g)(1 + n)(1 + ρ) ρ = 1%** Annual r = 1% *How to compute proprietor s income? How to impute rental income for owner-occupied housing? Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 40 / 59

Recursive Formulation Social Planner s problem We can rewrite the problem above as follows W (k 0 ) = max k 1 s.t. 0 k 1 f (k 0), β k 0 given max {k t+1} t=1 s.t 0 k t+1 f (k t) k 1 given u(f (k 0 ) k 1 ) + β t 1 u(f (k t ) k t+1 ) t=1 and then W (k 0 ) = max u(f (k 0 ) k 1 ) + βw (k 1 ) 0 k 1 f (k 0 ), k 0 given Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 41 / 59

Recursive Formulation Social Planner s problem v(k) = max k u(f (k) k ) + βv(k ) k [0, f (k)] Γ(k) This is called Bellman equation Numerical algorithms can be used to solve it Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 42 / 59

Recursive Formulation Social Planner s Problem Stationary problem: all information about the past that bears on current and future decisions is summarized by k. The structure of the choice problem that a decision maker faces is identical at every point in time We can omit time subscript, the only relevant information is initial k Note that the problem with a finite horizon is not stationary, it matters how many periods are left, in the infinite horizon case the remaining horizon is always the same, the only thing that changes period by period is the initial capital stock State variables: those that summarize the position of the system before the current period decisions are made. Control variables: choices Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 43 / 59

Recursive Formulation Social Planner s Problem k : state variable, it determines what allocations are feasible from today onwards k : control variable, decided by the social planner v(k) : discounted lifetime utility of the representative agent from the current period onwards if the social planner is given capital stock k at the beginning of the current period and allocates consumption across time optimally. It is current utility plus the discounted lifetime utility from tomorrow onwards How to solve this? It is a functional equation (FE) Its solution is a function v() solving the FE and an optimal policy function k = g(k). Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 44 / 59

Dynamic Programming A brief summary of results Under the assumptions we have made for preferences and technology it can be proved that If a function represents the value of solving the sequential problem, then this function solves the Bellman equation. If a function solves the Bellman equation, then it gives the value of the optimal program in the sequential formulation. If a sequence solves the sequential program, it can be expressed a decision rule that solves the Bellman equation. If a decision rule solves the recursive formulation, it generates sequences that solve the sequential problem. Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 45 / 59

Dynamic Programming A quick summary of results Furthermore, the following can be proved There exists v () that solves the Bellman equation and the solution is unique (Contracting Mapping Theorem) lim n T n (v 0 ) = v, where (Tv) = max k Γ(k) u(k, k ) + βv(k ), Γ(k) being the feasible set. This would be the base for the Value Function Iteration method. v is strictly concave, strictly increasing and differentiable g(k) is increasing Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 46 / 59

The Euler equation in the recursive formulation v(k) = u(k, g(k)) + βv(g(k)) g(k) : u 2 (k, g(k)) + βv (g(k)) = 0 k : u 1 (k, g(k)) + u 2 (k, g(k))g (k) + βv (g(k))g (k) = v (k) u 1 (k, g(k)) = v (k) Envelope theorem v (g(k)) = u 1 (g(k), g(g(k))) βu 1 (g(k), g(g(k))) = u 2 (k, g(k)) Euler equation Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 47 / 59

Dynamics Convergence Given our assumptions about preferences and technology the following properties of the law of motion for capital k = g(k) are satisfied g(k) is continuous and single-valued for all k g(k) is strictly increasing g(0) = 0 There exists k such that g(k) k for all k > k, k is greater than (f ) 1 (1/β) Speed of convergence Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 48 / 59

Dynamics Empirical evidence Differences of GDP across countries Conditional convergence Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 49 / 59

Computation Methods to Solve In general the type of economic model we have presented have not analytical solution, so we use numerical approximations to the solution. There are several methods Guess and Verify Linear Quadratic Approximation return function is assumed to be quadratic and the law of motion is linear certainty equivalence principle is enforced on the solution (only the mean matters, but not the variance) Gauss-Seidel Value Function Iteration Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 50 / 59

Value Function Iteration The idea We start from any function v 0 then the sequences defined by: v j+1 (k) = max u(f (k) 0 k f (k) k ) + βv j (k ) converges to the solution v of the planner s problem as j Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 51 / 59

Value Function Iteration Algorithm More specifically: 1. Guess v 0 (k) = 0 2. Solve v 1 (k) = max u(f (k) k ) + βv 0 (k ) 0 k f (k) The optimal solution is k = g 1 (k) = 0 k 3. Then 4. Now we go for v 2 v 2 (k) = 5. Iterate and it will converge to v v 1 (k) = u(f (k)) max u(f (k) k ) + βv 1 (k ) 0 k f (k) Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 52 / 59

Value Function Iteration Discrete state space Define a grid for k K = (K 1, K 2,...K m ) Define a matrix M as follows M = u [f (K 1 ) K 1 ] u [f (K 1 ) K 2 ]... u [f (K 1 ) K m ]) u [f (K 2 ) K 1 ] u [f (K 2 ) K 2 ]... u [f (K 2 ) K m ]... u [f (K m ) K 1 ] u [f (K m ) K 2 ]... u [f (K m ) K m ] Eliminate unattainable cells by setting M il = 1000 if K l > f (K i ) Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 53 / 59

Value Function Iteration Step by step 1. Guess an initial column vector V j R m, set j = 0 2. Given V j and M, compute V j+1 as V j+1 = max [ M + β(v j e) T ] where T denotes the transpose, e = [1, 1,..., 1] is a row vector of size m filled with ones, and the max is taken by rows. You need to compute G j = arg max { M + β(v e) T }, where G j is a column vector of m components, in which G {1, 2,..., p} indicates the number of the column which maximizes row i 3. Compute V j+1 V j. If the distance is bigger than the tolerance criterion ε, return to step 2 with j = j + 1. Otherwise, the procedure ends with V = V j+1 4. Simulations: starting from k 0 = K i, we can obtain the optimal sequence of capital as follows k 1 = K l with l = G i, k 2 = K h with h = G l and so on. Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 54 / 59

Value Function Iteration Some comments Of course we may have more than one state variable in the problem: for example, if we introduce exogenous aggregate uncertainty in productivity, F (K, L) = zk α L 1 α, then we will have an additional state variable. Then the space of states will be N k N z Finite horizon economies: age is an state variable Above we only allow agents choosing one of the capital values in the grid, however we can improve the numerical solution using interpolation Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 55 / 59

Value Function Iteration Extension to endogenous labor supply Social Planner Problem s.t. max {c t,n t,k t+1} t=0 Labour supply in the steady state How to choose ψ? β t [logc t ψn t ] t=0 C t + K t+1 (1 δ)k t = AK α t N 1 α t 1 n = α ψ 1 + [ ] δ (1 α) ρ + 1 Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 56 / 59

Limitations of the representative agent model Some comments The representative agent model allow us to address lot of questions (growth, business cycles, asset pricing), however It cannot be used to address questions related to income and wealth distribution It may be the case that the distribution of agents across states matters for the response of the economy to policy reforms, then, the response of an average household may be different from the average response of different types of agents (policy functions are not linear in the state variables) So heterogenous agents models are required in those cases - initial wealth heterogeneity - idiosyncratic shocks to productivity Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 57 / 59

Limitations of the infinite horizon setup Some comments There are some economic questions of interest that should be addressed in a finite horizon framework: pensions You will study the Overlapping Generations Model, Diamond (1965) In the simplest version of the model there is heterogeneity in age The model has some theoretical properties that are interesting Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 58 / 59

Applications Some applications Growth Business Cycles: fluctuations of aggregate variables Fiscal Policy: effect of taxes and public expenditure, optimal taxation Monetary Policy: short and long-run effects of monetary policies Macroeconomics (MIE-UNICAN) A simple macro dynamic model with endogenous saving rate: the representative Virginia Sánchez-Marcos agent model 59 / 59