Chapter 3 IS LM in an open economy · Chapter 3 IS–LM in an open economy O. Afonso, P. B. Vasconcelos Computational Economics: a concise introduction O. Afonso, P. B. Vasconcelos

Chapter 3IS–LM in an open economy

O. Afonso, P. B. Vasconcelos

Computational Economics: a concise introduction

O. Afonso, P. B. Vasconcelos Computational Economics 1 / 23

Overview

1 Introduction

2 Economic model

3 Numerical solution

4 Computational implementation

5 Numerical results and simulation

6 Highlights

7 Main references


Introduction

The IS–LM model is extended to a scenario of an open economy, with homeand foreign countries.

The analysis is based on the Mundell–Fleming model arising from thestudies in Mundell (1963) and in Fleming (1962).The aim is to study the effects of both fiscal and monetary policies inopen economy.

Iterative methods for linear systems of equations are introduced.


Economic model

Both economies are described by static equations of the standard IS–LMmodel, where generally:

prices are fixed;the relationships between aggregate variables are established;there is international mobility of assets as well as of goods and servicesthere is an exchange market;imports of one country are the exports of the other.

The model can be used to study the effect of changes either in policyvariables or in the specification of the interaction between endogenousvariables.


Economic model

The general equations that characterise the economies are:product equals aggregate demand at home and foreign countries,Y = C + I + G + NX vs Yf = Cf + If + Gf + NXf ;consumption functions, C = C + c(Y − T ) vs Cf = C f + cf (Yf − Tf );investment functions, I = I − bR vs If = I f − bf Rf ;public spending functions, G = G vs Gf = Gf ;income tax functions, T = T vs Tf = T f ;net exports, NX = NX − j(Y − Yf ) + lE(P f/P) vs NX = −NX ;monetary equilibrium, (M/P) = kY − hR vs (M f/P f ) = kf Yf − hf Rf ;interest rates, R = Rf .


Economic model

Endogenous variables:product, Y and Yf ;consumption, C and Cf ;investment, I and If ;Net exports, NX and NXf ;exchange rate, E ;interest rate, R.

Exogenous variables:government/public spending, G and Gf ;independent/autonomous consumption, C and C f ;independent/autonomous investment, I and I f ;income taxes, T and T f ;money supply, M and M f ;price level (fixed), P and P f .


Economic model

Parameters:0 < c, cf < 1 are the propensities to consume;b, bf > 0 are the interest sensitivities of investment;k , kf > 0 are the output sensitivities of the demand for money;h, hf > 0 are the interest sensitivities of the demand for money;j > 0 is the output sensivity of the net exports in the home country;l > 0 is the real exchange rate sensivity of the net exports in the homecountry.


Economic model

IS curve

The home IS curve shows the continuum of combinations of the interestand production level at which there is equilibrium in the goods andservices market:

Y =1

1− c + j(C + I + NX − cT − bR + jYf + lE(Pf/P). (1)

Representing (Y ,R), respectively, in the x–axis and y–axis, it can bestated that:

IS position depends on exogenous part of (the line) C + I + G + NX ;its slope is negative ∂Y

∂R < 0;points on the left (right) side of the IS curve mean that there is excessdemand (supply) for goods and services.


Economic model

The home LM curve shows the continuum of combinations of the interestand output level at which there is equilibrium in the money market:

MP

= kY − hR. (2)

It can be stated that:LM position depends on M and P;∂Y∂R > 0, since k > 0 and h > 0 (positive slope);points in the left (right) side of the LM curve mean excess money demand(supply).


Numerical solution

Iterative methods are designed to generate a sequence of increasinglyaccurate approximations to the solution.

To obtain a recurrence, matrix A can be partitioned in two: A = M − N.The linear system Ax = d becomes Mx = Nx + d , thus suggesting theiterative process (fixed-point iteration) xk+1 = Gxk + f , G is the iterationmatrix for M nonsingular.This procedure generates a stationary iterative methods since neither G norf depends upon the iteration count.If ||G|| < 1 then I −G is nonsingular and the process converges to thesolution (I −G)x = f for any x0.


Numerical solution

Stationary iterative methods

Most usual methods:Richardson, MR = I, NR = I − A;Jacobi, MJ = D, NJ = −E − F ;Gauss–Seidel, MGS = D − E , NGS = F ;Successive Over–Relaxation, MSOR = D−$E , NSOR = (1−$)D +$F ,

with

D the diagonal of A,−E the strictly lower triangular part of A,−F the strictly upper triangular part of A.

A backward Gauss–Seidel is obtained by just replacing E by F , and SOR with$ = 1 recovers the (forward) Gauss–Seidel method.


Numerical solution

Nonstationary iterative methods

To solve Ax = d for A, n × n, sparse and large, a common technique is toproject the problem on a subspace Km(A), m� n, of much smallerdimension than that of A, and to extract from x0 +Km(A) an approximatesolution, imposing conditions on the residual.Krylov subspace methods seek the approximate solution fromKm(A) ≡ span{r0,Ar0, ...,Am−1r0}, for r0 = d − Ax0 the initial residual.


Numerical solution

Nonstationary iterative methods

Different classes of nonstationary iterative methods can be deriveddepending on the conditions imposed.

The minimum residual approach: find xm for which ‖b − Ax‖2 is minimal overKm(A). The well-known GMRES (Generalised Minimum RESidual) methodis based on this approach.The Ritz–Galerkin approach: find xm for which the residual is orthogonal toKm(A). The CG (Conjugate-Gradient) method for symmetric definite-positivematrices belongs to this approach.The Petrov–Galerkin approach: find xm for which the residual is orthogonalto a subspace different from Km(A). Among several methods, the most usedmay be the BiCGstab (stabilised Bi–Conjugate Gradient) iterative method.


Computational implementation

The following baseline values are considered:C = C f = 55, I = I f = 47, NX = −10,G = Gf = 150, M = M f = 210,T = T f = 150 and P = P f = 1;c = cf = 0.63, b = bf = 1500, k = kf = 0.6, h = hf = 2700, j = 0.1, l = 10;In the present case, countries have the same dimension and there is perfectmobility under R = Rf .



Presentation, parameters and exogenous variables

%% Mundell−Fleming Model% An open Economy i n the shor t−medium run : the Mundell−Fleming Model% Implemented by : P .B . Vasconcelos and O. Afonsodisp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ;disp ( ’ IS−LM model : open economy ’ ) ;disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ;

%% parameters ( f stands f o r f o r e i g n )c = 0 .63 ; b = 1500; k = 0 . 6 ; h = 2700;c f = 0 .63 ; b f = 1500; k f = 0 . 6 ; h f = 2700;j = 0 . 1 ; l = 10;

%% exogenous v a r i a b l e sC_bar = 55; I_bar = 47;Cf_bar = 55; I f _ b a r = 47; NX_bar = −10;G_bar = 150; M_bar = 210; T_bar = 150; P_bar = 1 ;Gf_bar = 150; Mf_bar = 210; Tf_bar = 150; Pf_bar = 1 ;



Endogeneous variables

%% endogenous v a r i a b l e s% Y, Yf , p roduc t ion ; C, Cf , consumption% I , I f , investment ; NX, NXf , net expor ts% R, Rf , i n t e r e s t ra te ; E, exchange ra te

%% mat r i x rep resen ta t i on o f the model : Ax=d% A, c o e f f i c i e n t mat r i x% Y C I NX R Yf Cf I f NXf Rf EA = [ . . .

1 −1 −1 −1 0 0 0 0 0 0 0 % Y=C+ I +G+NX−c 1 0 0 0 0 0 0 0 0 0 % C=C_bar+c (Y−T)

0 0 1 0 b 0 0 0 0 0 0 % I =I_bar−b∗Rk 0 0 0 −h 0 0 0 0 0 0 % M/P=k∗Y−h∗R0 0 0 0 0 1 −1 −1 −1 0 0 % Yf=Cf+ I f +Gf+NXf0 0 0 0 0 −c f 1 0 0 0 0 % Cf=Cf_bar+ c f ( Yf−Tf )0 0 0 0 0 0 0 1 0 bf 0 % I f = I f_ba r−bf∗Rf0 0 0 0 0 k f 0 0 0 −hf 0 % Mf / Pf= k f ∗Yf−hf∗Rfj 0 0 1 0 − j 0 0 0 0 − l ∗Pf_bar / P_bar % NX=NX_bar− j (Y−Yf ) + l ∗E∗Pf /P0 0 0 1 0 0 0 0 1 0 0 % NX=−NXf0 0 0 0 1 0 0 0 0 −1 0 ] ; % R=Rf

% x = [Y,C, I ,NX,R, Yf , Cf , I f , NXf , Rf ,E ] ’ , vec to r o f the endogeneous v a r i a b l e s



Compute and show the solution

% d , vec to r o f the exogeneous v a r i a b l e sd = [ G_bar ; C_bar−c∗T_bar ; I_bar ; M_bar / P_bar ; . . .

Gf_bar ; Cf_bar−c f ∗Tf_bar ; I f _ b a r ; Mf_bar / Pf_bar ; . . .NX_bar ; 0 ; 0 ] ;

%% compute the endogenous va r i a b l e sx = gmres (A, d , [ ] , 1 e−10,11) ;

%% show the s o l u t i o ndisp ( ’ computed endogenous va r i a b l e s : ’ )f p r i n t f ( ’ product home / f o r e i g n : %7.2 f %7.2 f \ n ’ , x ( 1 ) , x ( 6 ) )f p r i n t f ( ’ consumption home / f o r e i g n : %7.2 f %7.2 f \ n ’ , x ( 2 ) , x ( 7 ) )f p r i n t f ( ’ investment home / f o r e i g n : %7.2 f %7.2 f \ n ’ , x ( 3 ) , x ( 8 ) )f p r i n t f ( ’ net expor ts home / f o r e i g n : %7.2 f %7.2 f \ n ’ , x ( 4 ) , x ( 9 ) )f p r i n t f ( ’ i n t e r e s t ra te home / f o r e i g n (%%) : %7.2 f %7.2 f \ n ’ , . . .

x ( 5 ) ∗100 ,x (10) ∗100)f p r i n t f ( ’ exchange ra te : %7.2 f \ n ’ , x (11) )


Numerical results and simulation

computed endogenous variables:product home/foreign : 389.81 389.81consumption home/foreign : 206.08 206.08investment home/foreign : 33.73 33.73net exports home/foreign : -0.00 0.00interest rate home/foreign (%): 0.88 0.88exchange rate : 1.00



Impact of a decrease in taxes by 20

T from 150 to 130change in home / foreign countriesin product : 8.96 8.96in consumption : 18.24 5.64in investment : -2.99 -2.99in net exports : -6.30 6.30in interest rate : 0.00 0.00



Impact of an increase in government spending by 20

G from 150 to 170change in home / foreign countriesin product : 14.22 14.22in consumption : 8.96 8.96in investment : -4.74 -4.74in net exports : -10.00 10.00in interest rate : 0.00 0.00



Impact of an increase in money spending by 20

M from 210 to 230change in home / foreign countriesin product : 24.57 -8.77in consumption : 15.48 -5.52in investment : 2.92 2.92in net exports : 6.17 -6.17in interest rate : -0.00 -0.00


Highlights

This model was developed in the early 1960s. Due to this development,Robert Mundell won the Nobel Prize in Economics in 1999.The Mundell–Fleming model enables the discussion of the equilibrium inan open economy and the impact of changes in fiscal and monetarypolicies in this context.Iterative methods (stationary and nonstationary) to compute approximatesolutions of linear systems are exposed, emphasising GMRES, BiCGstaband PCG (Preconditioned CG).


Main references

M. Burda and C. WyploszMacroeconomics: a European textOxford University Press (2009)

J. W. DemmelApplied Numerical Linear AlgebraSIAM (1997)

R. J. GordonMacroeconomicsPearson Education (2011)

C. T. KelleyIterative Methods for Linear andNonlinear EquationsSIAM (1995)

N. G. MankiwMacroeconomicsWorth Publishers (2009)

Y. SaadIterative methods for sparse linearsystemsSIAM (2003)

M. FlemingDomestic financial policies underfixed and under floating exchangeratesStaff Papers-InternationalMonetary Fund, 369–380 (1962)

R. MundellInflation and real interestJournal of Political Economy,71(3): 739–773 (1963)


Documents

Chapter 3 IS LM in an open economy · Chapter 3 IS–LM in an open economy O. Afonso, P. B. Vasconcelos Computational Economics: a concise introduction O. Afonso, P. B. Vasconcelos