Chapter 9 Solow - Amazon Web Services

Chapter 9Solow

O. Afonso, P. B. Vasconcelos

Computational Economics: a concise introduction

O. Afonso, P. B. Vasconcelos Computational Economics 1 / 27

Overview

1 Introduction

2 Economic model

3 Computational implementation

4 Numerical results and simulation

5 Highlights

6 Main references


Introduction

In 1956, Robert Solow published a seminal paper on economic growth anddevelopment.

The proposed model, also known has the Solow–Swan, ignores someimportant aspects of macroeconomics, such as short-run fluctuations inemployment, and makes several assumptions to describe the long-runpath of the economy.It remains highly influential even today and despite its relative simplicityconveys a number of very useful insights about the dynamics of thegrowth process.It is also worth teaching from a methodological perspective.

Numerical methods for solving differential equations will be introduced and theEuler method implemented to solve initial value problems (Dahlquist andBjörck (2008) and Süli and Mayers (2003)).


Economic model

This is a model of capital accumulation in a pure production economy.Countries produce and consume only a single, homogeneous good(output, GDP or real income).There are no prices because there is no need for money.Everyone works all the time, saves a fixed portion of income, invests andowns the firm (consumer side is not modelled).There is no government, thus no taxation nor subsidies.It is a closed economy model.

This model captures the pure impact that savings (investment) has on thelong-run standard of living (per capita income).The model is built around two equations:

a production function anda capital accumulation equation.


Economic model

Production function

The neoclassic production function is assumed to have the Cobb–Douglasform:

Y (t) = F (K (t),L(t)) = K (t)αL(t)1−α (1)

where K (t) is capital input, L(t) is labour input and α, between 0 and 1, is thecapital share in production.

This production function is neoclassic sinceit exhibits constant returns to scale,presents positive and diminishing marginal returns to factor accumulation,satisfies the Inada conditions.


Economic model

Production function

In per capita terms, the production function (1) is:

Y (t)L(t)

=K (t)αL(t)1−α

L(t)⇔ y(t) = k(t)α (2)

where y = Y (t)L(t) and k(t) = K (t)

L(t)From (1), taking logs and differentiating with respect to time on both sides,denoting by Y (t) and L(t) the derivatives of Y and L with respect to t , theresults is

Y (t)Y (t)

= αK (t)K (t)

+ (1− α) L(t)L(t)

.

The growth rate of per capita output is then simply

y(t)y(t)

= αk(t)k(t)

.

Thus, the source of increase in per capita output is the capital deepening.


Economic model

Capital accumulation

The second key equation describes the path of capital accumulation

K (t) = Y (t)− C(t)− δK (t) (3)

i.e. the addition to the capital stock each period, K (t), depends positively onsavings and negatively on depreciation, which takes place at rate δ.Since a fraction s of output is saved, Y (t)− C(t) = sY (t), the labour inputgrows at rate L(t)

L(t) = n, the path of per capita capital accumulation is

k(t) = sy(t)− (δ + n)k(t). (4)

Thus, the change in per capita capital each period is determined by threeterms: per capita investment, sy(t); per capita depreciation, δk(t); andpopulation growth, nk(t).This is a differential equation and, for an initial stock of capital k0, itdefines an initial value problem.


Economic model

Transitional dynamics: the Solow diagramThe Solow diagram has two curves, plotted as functions of k(t), and can beused to understand per capita output evolves over time

f(k)

sf(k)

(δ+n)k

k∗

consumption

investment

Solow diagram


Economic model

Transitional dynamics: the Solow diagram

The curve y(t) = f (k(t)) = k(t)α is the production function, and the curvesy(t) = sk(t)α depicts the amount of per capita investment, which shiftsy(t) down by the factor s.In turn, the line (δ + n)k(t) represents the amount of new per capitainvestment required to keep the amount of per capita capital constant.When the difference between the curve sy(t) and the line (δ + n)k(t) ispositive (negative) the economy is increasing (decreasing) its per capitacapital: capital deepening (widening) is occurring.when sy(t) = (n + δ)k(t), so that k(t) = 0, the amount of per capitacapital remains constant, and such a point is a steady state.

To sum up, we have the following.From the diagram it is found the steady-state value of per capita capital.Then, from the production function, the also constant steady-state valueof output per worker is obtained.In turn, the steady state per capita consumption is given by the differencebetween steady state per capita values of output and investment.O. Afonso, P. B. Vasconcelos Computational Economics 9 / 27

Economic model

Steady state

Since, in steady state, k(t) = 0, (4) and (2) can be used to solve thesteady-state values of per capita capital and per capita output; thus,

k(t) = sk(t)α − (δ + n)k(t)⇒ 0 = sk∗α − (δ + n)k∗

and so the steady-state values are:

k∗ =(

sδ + n

) 11−α

(5)

y∗ =(

sδ + n

) α1−α

. (6)

Ceteris paribus, countries that have high savings/investment rates will tend tobe richer since they accumulate more per capita capital and, as a result, havemore per capita output. In turn, countries that have high population growthrates will tend to be poorer.


Economic model

Golden rule

The optimal capital accumulation leads to the golden rule savings rate thatmaximises the steady-state level of consumption.The per capita consumption c(t) is given by c(t) = (1− s)y(t) and s can bewritten as s = y(t)−c(t)

y(t) .

At the steady state, k(t) = 0, that is, sy∗ = (δ + n)k∗.The latter can be written in terms of c(t), as c∗ = y∗ − (δ + n)k∗.To find k that maximises c, the first order condition, dc

dk = 0, produces

kgr =

(α

δ + n

) 11−α

, (7)

considering A = 1. Note that k∗ > kgr whenever s > α.


Economic model

Golden rule

f(k)

sf(k)

(δ+n)k

sgr

f(k)

k∗kgr

Golden rule savings rate


Economic model

Linear approximation

The Solow model has a closed solution and can be demonstrated that theconvergence process is (globally) stable.Taking (4), considering k(t) = φ(k(t)), a first order Taylor approximation fork(t), being φ(k(t)) at least twice differentiable, is given by

k(t) ' φ(k∗) + dφ(k(t))dk(t) |k∗(k(t)− k∗)

'(

sf ′(k∗)− (n + δ)

)(k(t)− k∗)

since φ(k∗) = 0.Thus, whenever k(t) < k∗ (k(t) > k∗) then k(t) > 0 (k(t) < 0) and physicalcapital will accumulate (diminish) towards the steady state.


Economic model

Differential equation

Since most differential equations are not analytically soluble, numericalsolution of ordinary differential equations is a fundamental technique.A differential equation involves an unknown function, y(t), and its derivatives.

A first–order ordinary differential equation, ODE, has the form

y(t) =dydt

(t) = f (t , y(t)) (8)

where f : R× R→ R and y(t) : [t0, tT ] ⊂ R→ R.Equation (8) is non-autonomous but often economic problems aretime-autonomous

y(t) =dydt

(t) = f (y(t)). (9)

The solution of (8) (or (9)) is a family of functions determined by aconstant.


Economic model

Initial value problems

A particular solution is computed by requiring that it goes through a specificpoint, the initial condition, (t0, y0 = y(t0)).The problem specified both by (8) (or (9)) and the initial condition is called aninitial value problem, IVP.

Solving the IVP is to predict the path that a quantity will take during acertain time interval, given the initial quantity.Problems involving ODEs of a higher order can be reduced to a system offirst ODE equations by introducing new variables.Seldom do these equations have solutions that can be expressed in aclosed form or the analytical form is often too cumbersome; solutiontechniques are generally unable to deal with large and nonlinear systemsof equations that arise in real problems.


Economic model

Initial value problems

To solve a continuous problem in a computer, a discretisation process isrequired. Initial value problems can be numerically solved using finitedifference methods and recursive procedures.

The numerical procedures are based on approximations y0, y1, · · · , yT tothe exact solution y(t0), y(t1), · · · , y(tT ) at the grid points:t0 < t1 < · · · < tT .The distances hn = tn − tn−1, n = 1, · · · ,T , are called step sizes and, forsimplicity, equal step sizes, that is, uniform grids, where h = (tT − t0)/T ,are considered.The aim is, starting with the initial value y0 = y(t0), to find yn, whichapproximates y(tn), by recurrence relations in such a way that the valueof yn+1 could be stated as a function of yn.


Economic model

Euler method

From (8), the slope of the tangent line at (t0, y(t0)) can be computed;taking a small step along the computed tangent, another point (t1, y1) isreached, from where again a tangent can be computed;perform this process iteratively until (tT , yT );a polygonal curve y0, · · · yT is computed, approximating the soughtsolution: y(t0), · · · y(tT ) at points t0, · · · tT .The approximation error can be reduced considering small step sizes, h.This is know as the

Euler method

yn+1 = yn + hf (tn, yn),n = 0,1, · · · ,T . (10)

It is a first-order method since the error of the final result, global error, isproportional to h.The error in a single step, local error, is proportional to h2. High-ordernumerical methods will be explored in the following chapters.


Economic model

It is relevant for the stability of (linearised) economic models to get someinsight on the solution of linear differential equations.The solution of y(t) = ay(t) + b can be obtained from the solution of thehomogeneous equations (b = 0) and the particular solution.The stability behaviour is ruled by the former.

Since et satisfies y(t) = y(t), one should try eλt .Then, we get the result that λ = a and thus y(t) = ce−at , c constant(which can be determined for an initial value problem).The stability of the solution depends on the sign of λ: if λ < 0 then thesolution converges to the steady state.

It is worth mentioning that the behaviour is different from the discrete case(eλt vs λt ): for discrete time both sign and modulus of λ are relevant tounderstand the equations behaviour.


Computational implementation

The following baseline values are considered:s = 0.4, A = 1, α = 0.3, δ = 0.1 and n = 0.01.



Presentation and parameters

%% Solow model% Neoc lass ica l growth model ( exogenous growth model )% Implemented by : P .B . Vasconcelos and O. Afonsodisp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ;disp ( ’ Solow model : exogenous growth model ’ ) ;disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ;

%% parameterss = 0 . 4 ; % savings ra teA = 1; % t e c h n o l o g i c a l progress ( Hicks n e u t r a l )alpha = 0 . 3 ; % c a p i t a l share i n produc t ionde l t a = 0 . 1 ; % deprec ia t i on ra ten = 0 .01 ; % popu la t ion growth ra tef p r i n t f ( ’ s A alpha de l t a n \ n ’ ) ;f p r i n t f ( ’ %6.2 f %6.2 f %6.2 f %6.2 f %6.2 f \ n ’ , s ,A, alpha , de l ta , n ) ;



Solution

%% steady−s ta te and numer ica l s o l u t i o nodesolow = @( t , k ) s∗A∗k^ alpha−( de l t a +n ) ∗k ;kss = f s o l v e (@( k ) odesolow ( [ ] , k ) ,10) ;[ t , y ] = my_euler ( odesolow , [ 0 , 1 0 0 ] , 0 . 5∗ kss , 0 . 0 1 ) ;plot ( t , y ) ;i f ~exist ( ’OCTAVE_VERSION ’ , ’ b u i l t i n ’ )

% l a b e l s f o r MATLABxlabel ( ’ $t$ ’ , ’ 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% l a b e l s f o r Octavexlabel ( ’ t ’ ) ; ylabel ( ’ k ’ ) ;

end



Euler method

function [ t , y ] = my_euler ( f , tspan , y0 , h )%( progress ive ) Euler method wi th tspan =[ t0 t f ] and f i x e d% step s ize to solve the IVP y ’= f ( t , y ) , y ( t0 ) =y0% inpu t :% f : f u n c t i o n to i n t e g r a t e ( f =@( t , y ) )% tspan : i n t e g r a t i o n i n t e r v a l [ t0 , t f i n a l ]% y0 : i n i t i a l c o n d i t i o n a t t0% h : ( constant ) step s ize% output :% t : s p e c i f i c t imes used% y : s o l u t i o n evaluated at t

t0 = tspan ( 1 ) ; t f = tspan ( 2 ) ;t = t0 : h : t f ; y = zeros (1 , length ( t ) ) ;y ( 1 ) = y0 ; yn = y0 ; tn = t0 ;for n = 1: length ( t )−1

yn = yn+h∗ f ( tn , yn ) ;y ( n+1) = yn ;tn = tn+h ;

endend


Numerical results and simulation

0 20 40 60 80 1003

3.5

4

4.5

5

5.5

6

6.5

t

k

Transition dynamics to steady state



Depreciation rate from 0.1 to 0.15

0 50 1003

3.5

4

4.5

5

5.5

6

6.5

t

k

0 5 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

k

k

(δ+n)*k

(δnew

+n)*k

δ=0.1

δnew

=0.15

Transition dynamics and Solow diagram: variation on δ



Technology from A = 1 to Anew = 1.1

0 50 1003

3.5

4

4.5

5

5.5

6

6.5

7

7.5

t

k

0 5 10 150

0.5

1

1.5

2

2.5

k

k

sAkα

sAnew

kα

A=1

Anew

=1.1

Transition dynamics and Solow diagram: variation on A


Highlights

The Solow model was developed in 1956 by the Nobel Laureate of 1987,Robert Solow.The Solow model shows how an economy changes over time until it getsto steady state.Savings rates determine the level of per capita capital (and output) levelin steady state. The higher the savings rate, the higher the level of percapita capital (output) level.Population growth has a negative impact on capital accumulation. Thehigher the population growth rate, the lower level of capital (output)steady state.However, neither factors can explain sustained economic growth asobserved in developed countries. In the current model, per capita capital(output) is invariant in steady state.This chapter introduces the Euler method, the simplest numerical methodto solve initial value problems.


Main references

G. Dahlquist, Å BjörckNumerical methods in scientific computing, volume 1.Society for Industrial Mathematics, 2008

E. Süli, D. .F. MayersAn introduction to numerical analysis.Cambridge University Press, 2003.

R. SolowA Contribution to the Theory of Economic Growth.Quarterly Journal of Economics, 70(1): 65–94, 1956.


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Chapter 9 Solow - Amazon Web Services