Chapter 12 Ramsey–Cass–Koopmans model - s3-eu-west-1 ...

Chapter 12Ramsey–Cass–Koopmans model

O. Afonso, P. B. Vasconcelos

Computational Economics: a concise introduction

O. Afonso, P. B. Vasconcelos Computational Economics 1 / 33

Overview

1 Introduction

2 Economic model

3 Numerical solution

4 Computational implementation

5 Numerical results and simulation

6 Highlights

7 Main references


Introduction

The Ramsey–Cass–Koopmans (RCK) model, in which the savings rate isendogenous is considered.

Cass (1965) and Koopmans (1965) combined the maximisation for aninfinite horizon, suggested by Ramsey (1928), with Solow–Swan’s capitalaccumulation.The aim is to study whether the accumulation of capital accounts for thelong 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 developmentaffects the transitional dynamics.

A brief introduction to numerical methods to solve boundary value problemswill be provided, following closely Shampine et al. (2003); for a simplesymbolic and numerical computational approach to this chapter seeVasconcelos (2013).


Economic model

The model represents an economy of one sector where households and firmsinteract in competitive markets:

households provide labour services in exchange for wages, consume andaccumulate assets;firms have technical know-how to turn inputs into output, rent capital fromconsumers 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 (wdenotes the wage rate) and capital income (the amount of capital ownedby the household is K , the net real rate of return is R = r + δ).


Economic model

Consumption side

The representative household maximises the present value of an infinite utility,∫ ∞0

u (c(t))e−(ρ−n)tdt (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 aggregateconsumption, 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) isthe flow of labour income and c(t)L(t) is the flow of consumption.


Economic model

Consumption side

Now, denoting by a(t) = A(t)L(t) , per capita assets, results in

a(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 capitaconsumption and expansion of population; the budget constraint appearsby considering market clearing, a(t) = k(t).

Additionally, a non-Ponzi condition must be imposed:

limt→∞

a(t)e−∫ t

0 r(s)−nds = 0 (3)

to ensure that households cannot have exploding debt.


Economic model

Consumption side

The problem is then to maximise (1) restricted by (2) and (3):

maxk(t),c(t)

∫ ∞0

u (c(t))e−(ρ−n)tdt

st.

k(t) = (r(t)− n)k(t) + w(t)− c(t) (4)k(0) > 0 given

limt→∞

k(t)e−∫ t

0 r(s)−nds = 0. (5)


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)

limt→∞

e−(ρ−n)tλ(t)k(t) = 0, (8)

where (8) is the transversality condition, which states that at the end of theprocess the stock of assets should be 0.


Economic model

Consumption side

From (7) results λ(t)λ(t) = − (r(t)− ρ) and by differentiating (6) results the

Euler equationc(t)c(t)

= (r(t)− ρ)σ(t) (9)

where σ(t) = −(u′(c(t)))/(u′′(c(t))c(t)).

Equation (9) can be written as c(t)c(t)

1σ(t) + ρ = r(t),

where the left side is the benefit from the consumption and the right side isthe 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).


Economic model

Productive side

Consider an economy with a representative firm operating in a competitivecontext:

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) = entL(0), L(0) = 1, wheren = L(t)

L(t) ≡dL(t)/dt

L(t) is the labour growth rate.Therefore, firms aim at maximising profit

maxK ,L

F (K (t),L(t))− R(t)K (t)− w(t)L(t),

where R(t) = r(t) + δ is the rental rate of return on capital, and w(t) is thewage rate of effective labour.


Economic model

Productive side

Competitive markets imply that

∂F (K (t),L(t))∂K (t)

= f ′(k(t)) = R(t) (10)

and ∂F (K (t),L(t))∂L(t) = w(t).

Bearing in mind the Euler identity, and since F is homogeneous of degree 1,

K (t)∂F (K (t),L(t))

∂K (t)+ L(t)

∂F (K (t),L(t))∂L(t)

= F (K (t),L(t)).

Dividing both sides by L(t), the result is k(t)f ′(k(t)) + ∂F (K (t),L(t))∂L(t) = f (k(t)).

Therefore,∂F (K (t),L(t))

∂L(t)= f (k(t))− k(t)f ′(k(t)) = w(t). (11)


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 onereceived 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

c(t)c(t)

= (f ′(k(t))− δ − ρ)σ(t), (13)

which means that there is a positive relationship between c(t)c(t) and f ′(k(t)).

Equations (12), (13) and the transversality condition show the dynamicbehaviour of the consumption, of the capital and of the per capita GDP.


Economic model

Centralised equilibriumBasically, the central planner maximises (1), considering (12):

maxk(t),c(t)

∫ ∞0

u (c(t))e−∫ t

0 r(s)−ndsdt

st.

k(t) = f (k(t))− (δ + n)k(t)− c(t)k(0) > 0 given

limt→∞

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)

limt→∞

e−(ρ−n)tλ(t)k(t) = 0. (14)


Economic model

Steady state

Considering the Cobb–Douglas production function, f (k(t)) = kα, where α isthe capital share in production, and the Constant Intertemporal Elasticity ofSubstitution, CIES, utility function, u(c(t)) = c(t)(1−θ)−1

1−θ θ 6= 1, the steady stateequilibrium is an equilibrium path in which the capital–labour ratio,consumption and output are constant: c = k = 0{

f ′(k∗)− (δ + ρ) = 0f (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)


Economic model

Golden rule

The golden rule postulates that collectively (or by policy enforcement) thepropensity to save is so that future generations can enjoy the same level ofper capita consumption as initial generations.The golden rule is obtained by maximising c(t) along with k(t) = 0:

maxk(t)

f (k)− (δ + n)k ;

thus, from first–order conditions f ′(k) = δ + n, and for the Cobb–Douglasproduction function,

kgr = (α

δ + n)

11−α .


Economic model

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5

k

c

gross production f(k)depreciation (n+ δ)keq. point

k = 0c = 0golden rulesteady statenet production

Golden rule, equilibrium point, steady stateO. Afonso, P. B. Vasconcelos Computational Economics 16 / 33

Economic model

Saddle pointThe sought solution (c∗, k∗) is a saddle point, and to understand this the localproperties must be accessed. Taylor expansion of first order gives rise to[

c(t)k(t)

]≈ J|(c∗,k∗)

[c − c∗

k − k∗

]where the Jacobian at the equilibrium point is

J|(c∗,k∗) =

∂c(t)∂c

∂c(t)∂k

∂k(t)∂c

∂k(t)∂k

|(c∗,k∗)

=

f ′(k(t))−(δ+ρ)θ

f ′′(k(t))c(t)θ

−1 f ′(k(t))− (δ + n)

|(c∗,k∗)

=

[0 f ′′(k∗)c∗

θ−1 ρ− n

]This matrix has two real eigenvalues, one positive and another negative; thusthe equilibrium point is a saddle point.


Numerical solution

Boundary value problems

Numerical solution of boundary value problems, BVP, that is, the solution of asystem of differential equations where the conditions are specified at morethan one point, is considered.A first–order two–point boundary value problem has the form

y ′(t) =dydt

(t) = f (t , y(t))

with boundary conditionsg(y(t0), y(tT )) = 0,

where f : R× Rm → Rm and g : R2m → Rm.


Numerical solution

Finite difference method

Let us consider a numerical approximation to the solution on a uniform meshof points tj = t0 + jh, j = 0, ...,n − 1, h = tT−t0

n−1 .For simplicity, consider the case where f and g are linear,f (t , y) = p(y)y + q(y) and y(t0) = y0, y(tT ) = yT .Discretising using the trapezoidal rule, yj+1 − yj =

h2 (f (tj , yj) + f (tj+1, yj+1)),

gives rise to the following system of linear equations

1r0 s0

r1 s1. . . . . .

rn−1 sn−11

y0y1y2...

yn−1yn

=

y(t0)u(t1)u2...

un−1y(tn)

,

where rj = −1− h2 p(yj), sj = −1− h

2 p(yj+1) and uj =h2 (p(yj) + p(yj+1)),

together with the two boundary conditions.


Numerical solution

Finite difference method and collocation

This system can be efficiently solved using Gaussian elimination adaptedfor this band format (or through an iterative procedure).

If the functions involved are nonlinear, a nonlinear system of equations mustbe solved, using the Newton method or one of its variants.

The finite difference method just developed is of order 2, due to thetrapezoidal rule used.

Higher–order finite difference methods can be developed considering, forinstance, a Runge–Kutta method.

In a collocation method a solution is computed according to

y(t) =n∑

j=1

xjφj(t),

φj are basis functions defined on [t0, tT ] and x is a n−dimensional vector;to determine x , a grid with n points (collocation points) is considered andthe approximate solution is required to satisfy both the differentialequation and the boundary conditions.O. Afonso, P. B. Vasconcelos Computational Economics 20 / 33

Numerical solution

Boundary value problems in practice

To solve boundary value problemsMATLAB/Octave provide the bvp4c function.

The implemented algorithm can be viewed either as a collocation method oras a finite difference method with a continuous extension.

MATLAB also offers bvp5c.


Numerical solution

Stability of equilibrium points

The solution of Y (t) = AY (t) + B, where Y (t) is an n × 1 vector offunctions, A is an n × n matrix of constants, can be obtained from thesolution of the homogeneous equations and the particular solution.The stability behaviour is ruled by Y (t) = CeAt , and C can be specifiedfor an initial value problem; eA is the matrix exponential, and thuseAt = I + At + 1

2! t2A2 + ....+ 1

n! tnAn + ....

When A results from a linearisation procedure it is a Jacobian matrix.If A is diagonalisable, distinct eigenvalues, then A = UDU−1, whereD = diag(λ1, ..., λn) is the diagonal matrix of the eigenvalues of A andU = [u1 ... un] the matrix of the eigenvectors; thus, Y (t) =

∑ni=1 cieλi tui ,

where ci , i = 1, ..., n, are constants.The equilibrium is asymptotically stable if all eigenvalues have negative realpart 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.


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) andθ = (δ + ρ)/(α ∗ (δ + n + g)− g).



Presentation and parameters

%% RCK model% The Ramsey−Cass−Koopmans model% Implemented by : P .B . Vasconcelos and O. Afonsofunction rckdisp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ;disp ( ’Ramsey−Cass−Koopmans model ’ ) ;disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ;

%% parametersglobal alpha de l t a rho n g the ta kss css k0alpha = 0 . 3 ; % e l a s t i c i t y o f c a p i t a l i n produc t ionde l t a = 0 .05 ; % deprec ia ton ra terho = 0 .02 ; % time preferencen = 0 .01 ; % popu la t ion growthg = 0 .00 ; % exogenous growth ra te o f technologyt he ta = ( de l t a +rho ) / ( alpha ∗ ( de 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 o f s u b s t i t u t i o n ;% s e l e c t the ta so t h a t the saving ra te i s constant s =1/ the ta



Solution

%% Steady s ta te values and shockkss = ( ( de l t a +rho+g∗ t he ta ) / alpha ) ^ ( 1 / ( alpha−1) ) ;css = kss ^ alpha−(n+ de l t a +g ) ∗kss ;k0 = 0.1∗ kss ; % shock at kdisp ( ’ steady−s ta te : ’ )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 knownk = @( t ) ( 1 . / ( ( de l t a +n+g ) .∗ t he ta ) +( k0 .^(1− alpha ) − . . .

1 . / ( ( de l t a +n+g ) .∗ t he ta ) ) ∗exp(−(1−alpha ) . . .. ∗ ( de l t a +n+g ) .∗ t ) ) . ^ (1 . / ( 1 − alpha ) ) ;

c = @( t ) (1−1./ the ta ) .∗ k ( t ) . ^ alpha ;



Solution

%% Approximate s o l u t i o n using bvp4c ( matlab so l ve r )% the RCK model i s a BVP problemnn = 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 ( l inspace (0 , nn , 5 ) , [ 0 . 5 0 . 5 ] ) ; % MATLABelse

s o l i n i t . x = l inspace (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

endso l = bvp4c (@ode_bvp ,@bcs, s o l i n i t ) ;i f ~exist ( ’OCTAVE_VERSION ’ , ’ b u i l t i n ’ )

x i n t = l inspace (0 , nn ,50 ) ; Sx in t = deval ( sol , x i n t ) ; % MATLABelse

x i n t = so l . x ; Sx in t = so l . y ; % Octaveend



Plot the solution

%% Plo t both the a n a l y t i c a l and numer ical s o l u t i o n from bvp4csubplot (2 ,1 ,1 ) ; hold onezp lo 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 kplot ( x i n t , Sx in t ( 1 , : ) , ’ r . ’ ) ; % p l o t approx . by bvp4c f o r kplot (0 , k0 , ’ go ’ ) ; % p l o t i n i t i a l value f o r ki 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 ’ ) ;

elselegend ( ’ 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



Plot the solution

subplot (2 ,1 ,2 ) ; hold onezp lo 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 cplot ( x i n t , Sx in t ( 2 , : ) , ’ r . ’ ) % p l o t numer ica l approx imat ion f o r ci 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 ’ ) ;

elselegend ( ’ exact ’ , ’ bvp4c ’ , ’ l o c a t i o n ’ , ’ SouthEast ’ ) ;xlabel ( ’ t ’ ) ; ylabel ( ’ c ’ ) ;

end



RCK system

% Boundary cond i t i onsfunction res = bcs ( ya , yb )% Y=[ k ( t ) ; c ( t ) ] , ya f o r i n i t i a l cond i t i ons and yb f o r f i n a l onesglobal kss k0res = [ 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 nfunction dydt=ode_bvp (~ , y )% Y=[ k ( t ) ; c ( t ) ]global alpha de l t a rho n g the tadydt = [ . . .

( y ( 1 ) ^ alpha−y ( 2 )−(n+ de l t a +g ) ∗y ( 1 ) ) ; . . . %ode : k( y ( 2 ) ∗ ( alpha∗y ( 1 ) ^ ( alpha−1)−( de l t a +rho+g∗ t he ta ) ) / t he ta ) ; . . . %ode : c

] ;


Numerical results and simulation

0 10 20 30 40 50 60 70 80 90 100

2

4

6

8

t

(1/((δ+n+g) θ)+(k01−α

−1/((δ+n+g) θ)) exp(−(1−α) (δ+n+g) t))1/(1−α)

k

exactbvp4ck0 initial value

0 10 20 30 40 50 60 70 80 90 100

0.8

1

1.2

1.4

t

(1−1/θ) k(t)α

c

exact

bvp4c

Transition dynamics to steady state


Numerical results and simulation

0 2 4 6 8 10 12 140.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

k

cc = 0

k = 0

Phase diagram, RCK model


Highlights

The RCK model, was formulated by Ramsey (1928) followed later byCass (1965) and Koopmans (1965). Tjalling Koopmans won the 1975Nobel Memorial Prize in Economic Sciences.The model is used to analyse the performance of the economy as therational behaviour of utility maximising individuals. It is considered abuilding block of dynamic general equilibrium models.This chapter introduces the bvp4c and bvp5c MATLAB functions tosolve boundary value problems for ordinary differential equations.


Main references

References

L. F. Shampine, I. Gladwell, S. ThompsonSolving ODEs with MATLAB.Cambridge University Press, 2003.

D. CassOptimum growth in an aggregative model of capital accumulation.The Review of economic studies, 233–240, 1965.

T. C. KoopmansOn the Concept of Optimal Economic Growth.The Economic Approach to Development Planning, 225–287, 1965.

F. P. RamseyA mathematical theory of saving.The economic journal, 543–559, 1928.

P. B. VasconcelosEconomic growth models: symbolic and numerical computations.Advances in Computer Science: an International Journal, 2(6): 47–54,2013.O. Afonso, P. B. Vasconcelos Computational Economics 33 / 33

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