Chapter 12 Ramsey Cass Koopmans model

Transcription

1 Chapter 12 Ramsey Cass Koopmans model O. Afonso, P. B. Vasconcelos Computational Economics: a concise introduction O. Afonso, P. B. Vasconcelos Computational Economics 1 / 33

2 Overview 1 Introduction 2 Economic model 3 Numerical solution 4 Computational implementation 5 Numerical results and simulation 6 Highlights 7 Main references O. Afonso, P. B. Vasconcelos Computational Economics 2 / 33

3 Introduction The Ramsey Cass Koopmans (RCK) model, in which the savings rate is endogenous is considered. Cass (1965) and Koopmans (1965) combined the maximisation for an infinite horizon, suggested by Ramsey (1928), with Solow Swan s capital accumulation. The aim is to study whether the accumulation of capital accounts for the long term growth, in a context in which households choose consumption and savings to maximise utility, intertemporal budget constraint is observed. The increase or decrease of the savings rate with economic development affects the transitional dynamics. A brief introduction to numerical methods to solve boundary value problems will be provided, following closely Shampine et al. (2003); for a simple symbolic and numerical computational approach to this chapter see Vasconcelos (2013). O. Afonso, P. B. Vasconcelos Computational Economics 3 / 33

4 Economic model The model represents an economy of one sector where households and firms interact in competitive markets: households provide labour services in exchange for wages, consume and accumulate assets; firms have technical know-how to turn inputs into output, rent capital from consumers and hire labour services. Moreover, there are a large, but constant, number of identical households; each household is endowed with one unit of labour per period of time; thus, the representative household income comprises labour income (w denotes the wage rate) and capital income (the amount of capital owned by the household is K, the net real rate of return is R = r + δ). O. Afonso, P. B. Vasconcelos Computational Economics 4 / 33

5 Economic model Consumption side The representative household maximises the present value of an infinite utility, 0 u (c(t)) e (ρ n)t dt (1) where ρ is the subjective discount rate, ρ n the effective discount rate, c(t) = C(t)/L(t) is the per capita consumption at t, C(t) is the aggregate consumption, and u is the instantaneous utility function. Denoting by A(t) the asset holdings of the representative household at time t, the following law of motion can be set A(t) = r(t)a(t) + (w(t) c(t))l(t) where r(t) is the risk-free market flow rate of return on assets, w(t)l(t) is the flow of labour income and c(t)l(t) is the flow of consumption. O. Afonso, P. B. Vasconcelos Computational Economics 5 / 33

6 Economic model Consumption side Now, denoting by a(t) = A(t) L(t), per capita assets, results in ȧ(t) = (r(t) n)a(t) + w(t) c(t), (2) per capita assets rises with w(t) + r(t)a(t), and falls with per capita consumption and expansion of population; the budget constraint appears by considering market clearing, a(t) = k(t). Additionally, a non-ponzi condition must be imposed: t lim t a(t)e 0 r(s) nds = 0 (3) to ensure that households cannot have exploding debt. O. Afonso, P. B. Vasconcelos Computational Economics 6 / 33

7 Economic model Consumption side The problem is then to maximise (1) restricted by (2) and (3): max k(t),c(t) 0 u (c(t)) e (ρ n)t dt st. k(t) = (r(t) n)k(t) + w(t) c(t) (4) k(0) > 0 given t lim t k(t)e 0 r(s) nds = 0. (5) O. Afonso, P. B. Vasconcelos Computational Economics 7 / 33

8 Economic model Consumption side This problem can be solved by the current value of the Hamiltonian: H(t, k(t), c(t), λ(t)) = u(c(t)) + λ(t) [(r(t) n)k(t) + w(t) c(t)]. First order conditions give rise to H c (t, k(t), c(t), λ(t)) = u (c(t)) λ(t) = 0 (6) H k (t, k(t), c(t), λ(t)) = λ(t)(r(t) n) = (ρ n)λ(t) λ(t) (7) lim t e (ρ n)t λ(t)k(t) = 0, (8) where (8) is the transversality condition, which states that at the end of the process the stock of assets should be 0. O. Afonso, P. B. Vasconcelos Computational Economics 8 / 33

9 Economic model Consumption side From (7) results λ(t) λ(t) = (r(t) ρ) and by differentiating (6) results the Euler equation ċ(t) = (r(t) ρ) σ(t) (9) c(t) where σ(t) = (u (c(t)))/(u (c(t))c(t)). Equation (9) can be written as ċ(t) 1 c(t) σ(t) + ρ = r(t), where the left side is the benefit from the consumption and the right side is the benefit due to savings. The non-ponzi condition (3) is implied by the transversality condition (8). The system of differential equations to be solved is formed by (4) and (9). O. Afonso, P. B. Vasconcelos Computational Economics 9 / 33

10 Economic model Productive side Consider an economy with a representative firm operating in a competitive context: Y (t) = F(K (t), L(t)) where the product Y (t) depends on capital K (t) and labour supply L(t). Moreover, total population at instant t is L(t) = e nt L(0), L(0) = 1, where n = L(t) L(t) dl(t)/dt L(t) is the labour growth rate. Therefore, firms aim at maximising profit maxf (K (t), L(t)) R(t)K (t) w(t)l(t), K,L where R(t) = r(t) + δ is the rental rate of return on capital, and w(t) is the wage rate of effective labour. O. Afonso, P. B. Vasconcelos Computational Economics 10 / 33

11 Economic model Productive side Competitive markets imply that F(K (t),l(t)) L(t) F (K (t), L(t)) K (t) = f (k(t)) = R(t) (10) and = w(t). Bearing in mind the Euler identity, and since F is homogeneous of degree 1, F (K (t), L(t)) F(K (t), L(t)) K (t) + L(t) = F (K (t), L(t)). K (t) L(t) F (K (t),l(t)) L(t) Dividing both sides by L(t), the result is k(t)f (k(t)) + = f (k(t)). Therefore, F (K (t), L(t)) = f (k(t)) k(t)f (k(t)) = w(t). (11) L(t) O. Afonso, P. B. Vasconcelos Computational Economics 11 / 33

12 Economic model Decentralised equilibrium Households and firms meet in the market: wages and interest paid by firms are those received by households, the price of the product (normalised to 1) paid by households is the one received by firms. Bearing in mind (10), (11) and a = k, in (2), k(t) = f (k(t)) (δ + n)k(t) c(t), (12) which depicts the dynamic path of k(t). Moreover, from (10) and (9) we get ċ(t) c(t) = (f (k(t)) δ ρ) σ(t), (13) which means that there is a positive relationship between ċ(t) c(t) and f (k(t)). Equations (12), (13) and the transversality condition show the dynamic behaviour of the consumption, of the capital and of the per capita GDP. O. Afonso, P. B. Vasconcelos Computational Economics 12 / 33

13 Centralised equilibrium Economic model Basically, the central planner maximises (1), considering (12): max k(t),c(t) 0 u (c(t)) e t st. 0 r(s) nds dt k(t) = f (k(t)) (δ + n)k(t) c(t) k(0) > 0 given lim t k(t)e (ρ n)t = 0. Thus, the current-value of the Hamiltonian and the first order conditions are H(t, k(t), c(t), λ(t)) = u(c(t))e (ρ n)t λ(t) [f (k(t)) (δ + n)k(t) c(t)] H c (t, k(t), c(t), λ(t)) = u (c(t)) λ(t) = 0 H k (t, k(t), c(t), λ(t)) = (f (k(t)) δ n)λ(t) = (ρ n)λ(t) λ(t) lim t e (ρ n)t λ(t)k(t) = 0. (14) O. Afonso, P. B. Vasconcelos Computational Economics 13 / 33

14 Economic model Steady state Considering the Cobb Douglas production function, f (k(t)) = k α, where α is the capital share in production, and the Constant Intertemporal Elasticity of Substitution, CIES, utility function, u(c(t)) = c(t)(1 θ) 1 1 θ θ 1, the steady state equilibrium is an equilibrium path in which the capital labour ratio, consumption and output are constant: ċ = k = 0 { f (k ) (δ + ρ) = 0 f (k ) = (δ + n) k + c and for the Cobb Douglas and CIES functions { k = ( α δ+ρ ) 1 1 α c = (k ) α (δ + n) k. (15) O. Afonso, P. B. Vasconcelos Computational Economics 14 / 33

15 Economic model Golden rule The golden rule postulates that collectively (or by policy enforcement) the propensity to save is so that future generations can enjoy the same level of per capita consumption as initial generations. The golden rule is obtained by maximising c(t) along with k(t) = 0: maxf (k) (δ + n)k; k(t) thus, from first order conditions f (k) = δ + n, and for the Cobb Douglas production function, k gr α = ( δ + n ) 1 1 α. O. Afonso, P. B. Vasconcelos Computational Economics 15 / 33

16 Economic model c gross production f(k) depreciation (n + δ)k eq. point 0.5 k = 0 ċ = 0 golden rule steady state net production k Golden rule, equilibrium point, steady state O. Afonso, P. B. Vasconcelos Computational Economics 16 / 33

17 Economic model Saddle point The sought solution (c, k ) is a saddle point, and to understand this the local properties must be accessed. Taylor expansion of first order gives rise to [ ] [ ] ċ(t) c c J k(t) (c,k ) k k where the Jacobian at the equilibrium point is J (c,k ) = ċ(t) c k(t) c ċ(t) k k(t) k f (k(t)) (δ+ρ) θ (c,k ) f (k(t))c(t) θ = 1 f (k(t)) (δ + n) [ ] = 0 f (k )c θ 1 ρ n (c,k ) This matrix has two real eigenvalues, one positive and another negative; thus the equilibrium point is a saddle point. O. Afonso, P. B. Vasconcelos Computational Economics 17 / 33

18 Numerical solution Boundary value problems Numerical solution of boundary value problems, BVP, that is, the solution of a system of differential equations where the conditions are specified at more than one point, is considered. A first order two point boundary value problem has the form with boundary conditions y (t) = dy (t) = f (t, y(t)) dt g(y(t 0 ), y(t T )) = 0, where f : R R m R m and g : R 2m R m. O. Afonso, P. B. Vasconcelos Computational Economics 18 / 33

19 Numerical solution Finite difference method Let us consider a numerical approximation to the solution on a uniform mesh of points t j = t 0 + jh, j = 0,..., n 1, h = t T t 0 n 1. For simplicity, consider the case where f and g are linear, f (t, y) = p(y)y + q(y) and y(t 0 ) = y 0, y(t T ) = y T. Discretising using the trapezoidal rule, y j+1 y j = h 2 (f (t j, y j ) + f (t j+1, y j+1 )), gives rise to the following system of linear equations 1 r 0 s 0 r 1 s =, r n 1 s n 1 1 y 0 y 1 y 2.. y n 1 y n y(t 0 ) u(t 1 ) u 2.. u n 1 y(t n ) where r j = 1 h 2 p(y j), s j = 1 h 2 p(y j+1) and u j = h 2 (p(y j) + p(y j+1 )), together with the two boundary conditions. O. Afonso, P. B. Vasconcelos Computational Economics 19 / 33

20 Numerical solution Finite difference method and collocation This system can be efficiently solved using Gaussian elimination adapted for this band format (or through an iterative procedure). If the functions involved are nonlinear, a nonlinear system of equations must be solved, using the Newton method or one of its variants. The finite difference method just developed is of order 2, due to the trapezoidal rule used. Higher order finite difference methods can be developed considering, for instance, a Runge Kutta method. In a collocation method a solution is computed according to y(t) = n x j φ j (t), j=1 φ j are basis functions defined on [t 0, t T ] and x is a n dimensional vector; to determine x, a grid with n points (collocation points) is considered and the approximate solution is required to satisfy both the differential equation and the boundary conditions. O. Afonso, P. B. Vasconcelos Computational Economics 20 / 33

21 Numerical solution Boundary value problems in practice To solve boundary value problems MATLAB/Octave provide the bvp4c function. The implemented algorithm can be viewed either as a collocation method or as a finite difference method with a continuous extension. MATLAB also offers bvp5c. O. Afonso, P. B. Vasconcelos Computational Economics 21 / 33

22 Numerical solution Stability of equilibrium points The solution of Ẏ (t) = AY (t) + B, where Y (t) is an n 1 vector of functions, A is an n n matrix of constants, can be obtained from the solution of the homogeneous equations and the particular solution. The stability behaviour is ruled by Y (t) = Ce At, and C can be specified for an initial value problem; e A is the matrix exponential, and thus e At = I + At + 1 2! t 2 A n! t n A n +... When A results from a linearisation procedure it is a Jacobian matrix. If A is diagonalisable, distinct eigenvalues, then A = UDU 1, where D = diag(λ 1,..., λ n) is the diagonal matrix of the eigenvalues of A and U = [u 1... u n] the matrix of the eigenvectors; thus, Y (t) = n i=1 c ie λ i t u i, where c i, i = 1,..., n, are constants. The equilibrium is asymptotically stable if all eigenvalues have negative real part and unstable if at least one has positive real part. For a 2 2 Jacobian matrix, the equilibrium point is a node when both eigenvalues are real, a saddle when eigenvalues are real but with opposite signs, a focus when eigenvalues are complex conjugate. O. Afonso, P. B. Vasconcelos Computational Economics 22 / 33

23 Computational implementation The following baseline values are considered: α = 0.3, δ = 0.05, ρ = 0.02, n = 0.01, g = 0.00 (to recover the baseline model) and θ = (δ + ρ)/(α (δ + n + g) g). O. Afonso, P. B. Vasconcelos Computational Economics 23 / 33

24 Computational implementation Presentation and parameters %% RCK model % The Ramsey Cass Koopmans model % Implemented by : P. B. Vasconcelos and O. Afonso function rck disp ( ) ; disp ( Ramsey Cass Koopmans model ) ; disp ( ) ; %% parameters global alpha d e l t a rho n g t h e t a kss css k0 alpha = 0. 3 ; % e l a s t i c i t y of c a p i t a l i n production d e l t a = ; % depreciaton r a t e rho = ; % time preference n = ; % p o p u l a t i o n growth g = ; % exogenous growth r a t e of technology t h e t a = ( d e l t a +rho ) / ( alpha ( d e l t a +n+g ) g ) ; % inverse i n t e r t e m p o r a l e l a s t i c i t y of s u b s t i t u t i o n ; % s e l e c t t h e t a so t h a t the saving r a t e i s constant s =1/ t h e t a O. Afonso, P. B. Vasconcelos Computational Economics 24 / 33

25 Computational implementation Solution %% Steady s t a t e values and shock kss = ( ( d e l t a +rho+g t h e t a ) / alpha ) ^ ( 1 / ( alpha 1) ) ; css = kss ^ alpha (n+ d e l t a +g ) kss ; k0 = 0.1 kss ; % shock at k disp ( steady s t a t e : ) f p r i n t f ( k = %8.6 f, c = %8.6 f \ n, kss, css ) ; disp ( shock : ) ; f p r i n t f ( i n i t i a l value f o r k : k0 = %8.6 f \ n, k0 ) ; %% Exact s o l u t i o n % f o r t h i s model an a n a l y t i c a l s o l u t i o n i s known k t ) ( 1. / ( ( d e l t a +n+g ). t h e t a ) +( k0.^(1 alpha ) / ( ( d e l t a +n+g ). t h e t a ) ) exp( (1 alpha ).... ( d e l t a +n+g ). t ) ). ^ ( 1. / ( 1 alpha ) ) ; c t ) (1 1./ t h e t a ). k ( t ). ^ alpha ; O. Afonso, P. B. Vasconcelos Computational Economics 25 / 33

26 Computational implementation Solution %% Approximate s o l u t i o n using bvp4c ( matlab s o l v e r ) % the RCK model i s a BVP problem nn = 100; i f ~ exist ( OCTAVE_VERSION, b u i l t i n ) s o l i n i t = b v p i n i t ( linspace ( 0, nn, 5 ), [ ] ) ; % MATLAB else s o l i n i t. x = linspace ( 0, nn, 5 ) ; s o l i n i t. y = 0.5 ones ( 2, length ( s o l i n i t. x ) ) ; % Octave end s o l = s o l i n i t ) ; i f ~ exist ( OCTAVE_VERSION, b u i l t i n ) x i n t = linspace ( 0, nn, 5 0 ) ; S x i n t = deval ( sol, x i n t ) ; % MATLAB else x i n t = s o l. x ; S x i n t = s o l. y ; % Octave end O. Afonso, P. B. Vasconcelos Computational Economics 26 / 33

27 Plot the solution Computational implementation %% P l o t both the a n a l y t i c a l and numerical s o l u t i o n from bvp4c subplot ( 2, 1, 1 ) ; hold on e z p l o t ( k, [ 0, nn ] ) ; % p l o t a n a l y t i c a l s o l u t i o n f o r k plot ( x i n t, S x i n t ( 1, : ), r. ) ; % p l o t approx. by bvp4c f o r k plot ( 0, k0, go ) ; % p l o t i n i t i a l value f o r k i f ~ exist ( OCTAVE_VERSION, b u i l t i n ) legend ( exact, bvp4c, $k_0$ i n i t i a l value, l o c a t i o n, SouthEast ) ; set ( legend, I n t e r p r e t e r, l a t e x ) ; xlabel ( $t$, I n t e r p r e t e r, LaTex ) ; ylabel ( $k$, I n t e r p r e t e r, LaTex ) ; else legend ( exact, bvp4c, k_0 i n i t i a l value, l o c a t i o n, SouthEast ) ; xlabel ( t ) ; ylabel ( k ) ; end O. Afonso, P. B. Vasconcelos Computational Economics 27 / 33

28 Plot the solution Computational implementation subplot ( 2, 1, 2 ) ; hold on e z p l o t ( c, [ 0, nn ] ) % p l o t a n a l y t i c a l s o l u t i o n f o r c plot ( x i n t, S x i n t ( 2, : ), r. ) % p l o t numerical approximation f o r c i f ~ exist ( OCTAVE_VERSION, b u i l t i n ) legend ( exact, bvp4c, l o c a t i o n, SouthEast ) ; xlabel ( $t$, I n t e r p r e t e r, LaTex ) ; ylabel ( $c$, I n t e r p r e t e r, LaTex ) ; else legend ( exact, bvp4c, l o c a t i o n, SouthEast ) ; xlabel ( t ) ; ylabel ( c ) ; end O. Afonso, P. B. Vasconcelos Computational Economics 28 / 33

29 Computational implementation RCK system % Boundary c o n d i t i o n s function res = bcs ( ya, yb ) % Y=[ k ( t ) ; c ( t ) ], ya f o r i n i t i a l c o n d i t i o n s and yb f o r f i n a l ones global kss k0 res = [ ya ( 1 ) k0 ; yb ( 1 ) kss ] ; % D i f f e r e n t i a l system s p e c i i f i c a t i o n function dydt=ode_bvp ( ~, y ) % Y=[ k ( t ) ; c ( t ) ] global alpha d e l t a rho n g t h e t a dydt = [... ( y ( 1 ) ^ alpha y ( 2 ) (n+ d e l t a +g ) y ( 1 ) ) ;... %ode : k ( y ( 2 ) ( alpha y ( 1 ) ^ ( alpha 1) ( d e l t a +rho+g t h e t a ) ) / t h e t a ) ;... %ode : c ] ; O. Afonso, P. B. Vasconcelos Computational Economics 29 / 33

30 Numerical results and simulation 8 (1/((δ+n+g) θ)+(k0 1 α 1/((δ+n+g) θ)) exp( (1 α) (δ+n+g) t)) 1/(1 α) k exact bvp4c k 0 initial value t (1 1/θ) k(t) α 1.2 c exact bvp4c t Transition dynamics to steady state O. Afonso, P. B. Vasconcelos Computational Economics 30 / 33

31 Numerical results and simulation ċ = k = 0 c k Phase diagram, RCK model O. Afonso, P. B. Vasconcelos Computational Economics 31 / 33

32 Highlights The RCK model, was formulated by Ramsey (1928) followed later by Cass (1965) and Koopmans (1965). Tjalling Koopmans won the 1975 Nobel Memorial Prize in Economic Sciences. The model is used to analyse the performance of the economy as the rational behaviour of utility maximising individuals. It is considered a building block of dynamic general equilibrium models. This chapter introduces the bvp4c and bvp5c MATLAB functions to solve boundary value problems for ordinary differential equations. O. Afonso, P. B. Vasconcelos Computational Economics 32 / 33

33 Main references References L. F. Shampine, I. Gladwell, S. Thompson Solving ODEs with MATLAB. Cambridge University Press, D. Cass Optimum growth in an aggregative model of capital accumulation. The Review of economic studies, , T. C. Koopmans On the Concept of Optimal Economic Growth. The Economic Approach to Development Planning, , F. P. Ramsey A mathematical theory of saving. The economic journal, , P. B. Vasconcelos Economic growth models: symbolic and numerical computations. Advances in Computer Science: an International Journal, 2(6): 47 54, O. Afonso, P. B. Vasconcelos Computational Economics 33 / 33