chapter 2 is–lm in a closed economy

Chapter 2IS–LM in a closed economy

O. Afonso, P. B. Vasconcelos

Computational Economics: a concise introduction

O. Afonso, P. B. Vasconcelos Computational Economics 1 / 25

Overview

1 Introduction

2 Economic model

3 Numerical solution

4 Computational implementation

5 Numerical results and simulation

6 Highlights

7 Main references


Introduction

The standard IS–LM model, Investment Saving–Liquidity preference Moneysupply, is a macroeconomic set-up that illustrates the relationship betweenreal output (production) and interest rates in the goods and services market(IS curve) and in the money market (LM curve).

When the IS curve crosses the LM curve, the equilibrium is achieved.In this model the economy is described by static equations, assumingthat prices are fixed and that the relationships between aggregatevariables such as production, consumption, investment,government/public spending and money supply are well established.

MATLAB/Octave is used to solve the model, exploring LU factorisation, inclosed economy (autarky case).


Economic model

The set-up of the typical IS–LM model specifies relationships amongaggregate variables.This model can be used to study the effect of changes either in policyvariables or in the specification of the interaction between endogenousvariables.Model:

product equals aggregate demand, Y = C + I + G;consumption function, C = C + c(Y − T );investment function, I = I − bR;public spending function, G = G;income taxes function, T = T ;monetary equilibrium, M

P = kY − hR;money supply function, M = M.


Economic model

Endogenous variables:product, Y ;consumption, C;investment, I;interest rate, R.

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

Parameters:c is the propensity to consume, 0 < c < 1;b > 0 is the interest sensitivity of investment;k > 0 is the output sensitivity of the demand for money;h > 0 is the interest sensitivity of the demand for money.


Economic model

IS curve

The IS curve shows the continuum of combinations of the interest andproduction level at which there is equilibrium in the goods and servicesmarket:

Y =1

1 − c

(C + I + G − cT − bR

). (1)

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

IS position depends on C + I + G;∂Y∂R < 0, since 0 < 1− c < 1 (negative slope);points on the left (right) side of the IS curve mean that there is excessdemand (supply) for goods and services.


Economic model

LM curve

In the monetary–financial market let us consider two types of assets:money (currency in circulation – notes and coins – and available depositsin banks) and the others.The LM curve shows the continuum of combinations of the interest andoutput 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

Mathematically, the model is represented by a system of linear equations,Ax = d , where

A is the coefficient matrix of the endogenous variables,d is the right-hand side vector with the coefficients of the exogenousvariables, andx is the solution vector.

In Chapter 1, Gaussian elimination was introduced to tackle this problem.In this chapter, some insights on stability of the numerical method areoffered.Furthermore, LU decomposition is introduced, which allows to solve fordifferent right hand sides at a less computational cost.


Numerical solution

Gaussian elimination to solve Ax = d , where A is n × n, to be numericallystable, must be implemented carefully.To eliminate all entries below the k th position of the k th column-vector inmatrix A, the product Mk A is performed, where

Mk =

1 · · · 0 0 · · · 0...

. . ....

.... . .

...0 · · · 1 0 · · · 00 · · · − ak+1k

akk1 · · · 0

.... . .

......

. . ....

0 · · · − ankakk

0 · · · 1

;

Mk is the elementary elimination matrix, akk is the pivot, and aikakk

,i = k + 1, ...,n, are the multipliers.


Numerical solution

Applying this procedure to all columns in A and multiplying both sides ofAx = d by nonsingular matrices Mj , j = 1, · · · ,n − 1, the resulting system

MAx = Md ,

where M = Mn−1...M2M1, can be cheaply solved by back-substitution.The LU factorisation of A is

LU decomposition

A = LU, (3)

where L = M−1 = M−11 ...M−1

n−1 is unit lower triangular and U = MA isupper triangular.To compute Ax = d , two triangular systems of equations, Uy = d andLx = y , must be solved.The factorisation is independent of the right-hand side, thus the samefactorisation can be reused to solve linear systems for different values ofd .


Numerical solution

A problem occurs if the pivot element is zero. In this case one must applya row interchange between row k and any row i , i > k , having a nonzeroentry in column k .Moreover, to minimise the propagation error, one should choose for pivotthe element aik , i = k , ...,n with the largest magnitude.This process is called partial pivoting and allows for a stableimplementation of Gaussian elimination.A matrix performing a row interchange, permutation matrix, between rowsi and j has the form of an identity matrix with rows i and j permuted.Letting P = Pn−1...P2P1,

PA = LU. (4)

MATLAB/Octave:LU factorisation: [L,U]=lu(A) or [L,U,P]=lu(A)solution x : x=U\(L\d) or x=U\(L\P*d).


Computational implementation

The following baseline values are considered:C = 55, c = 0.63,I = 75, b = 1500, k = 0.6, h = 2700, G = G = 200,M = M = 200, T = T = 110 and P = P = 1.



Presentation, parameters and exogenous variables

%% IS−LM model% A Closed Economy i n the shor t−medium run% Implemented by : P .B . Vasconcelos and O. Afonsodisp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ;disp ( ’ IS−LM model : c losed economy ’ ) ;disp ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ;

%% parametersc = 0 .63 ; % marginal p ropens i t y to consumeb = 1500; % s e n s i b i l i t y o f the investment to the i n t e r e s t ra tek = 0 . 6 ; % s e n s i b i l i t y o f the money demand to the producth = 2700; % s e n s i b i l i t y o f the money demand to the i n t e r e s t ra te

%% exogenous v a r i a b l e sC_bar = 55; % autonomous consumptionI_bar = 75; % autonomous investmentG_bar = 200; % government spendingT_bar = 110; % tax on incomeM_bar = 200; % money supplyP_bar = 1; % pr i ce l e v e l ( f i x e d i n the shor t−run )disp ( ’ p o l i c y exogenous va r i a b l e s : ’ )f p r i n t f ( ’G = %d ; T = %d ; M = %d \ n ’ ,G_bar , T_bar , M_bar ) ;



Endogenous variables and linear system

%% endogenous v a r i a b l e s% Y, product% C, consumption% I , investment% R, i n t e r e s t 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 RA = [ 1 −1 −1 0 % Y=C+ I +G

−c 1 0 0 % C=C_bar+c (Y−T)0 0 1 b % I =I_bar−bRk 0 0 −h ] ; % M/P=kY−hR

% x = [Y ;C; I ;R] , vec to r o f the endogeneous v a r i a b l e s% 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 ] ;



Compute, show and plot the solution

%% compute the endogenous va r i a b l e s[ L ,U] = lu (A) ; % LU f a c t o r i z a t i o nx = U\ ( L \ d ) ; % s o l u t i o n o f the l i n e a r system

%% 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 , Y : %6.2 f \ n ’ , x ( 1 ) ) ;f p r i n t f ( ’ consumption , C: %6.2 f \ n ’ , x ( 2 ) ) ;f p r i n t f ( ’ investment , I : %6.2 f \ n ’ , x ( 3 ) ) ;f p r i n t f ( ’ i n t e r e s t ra te (%%) , R: %7.2 f \ n ’ , x ( 4 ) ∗100) ;

%% p l o t the s o l u t i o n : IS−LM diagram (R aga ins t Y)% generate a g r i d on Y; i n t h i s case , s ince t h i s i s a l i n e% only 2 po in t s would have been requ i redY = 0.95∗x ( 1 ) :1 .05∗ x ( 1 ) ;% IS : I =Y−C−G and I =I_bar−bR expressed i n terms of RC = C_bar+c ∗ (Y−T_bar ) ; I =Y−C−G_bar ; IS =( I_bar−I ) / b ;% LM: M_bar / P_bar=kY−hR expressed i n terms of RLM = 1/ h∗ ( k∗Y−M_bar / P_bar ) ;plot (Y , IS , ’−−b ’ ,Y,LM, ’ r ’ )t i t l e ( ’ IS and LM curves ’ ) ; legend ( ’ IS ’ , ’LM ’ ) ;xlabel ( ’ product , Y ’ ) ; ylabel ( ’ i n t e r e s t ra te , R ’ ) ;


Numerical results and simulation

------------------------------------------------------------IS-LM model: closed economy------------------------------------------------------------exogenous variables:G = 200; T = 110; M = 200; P = 1computed endogenous variables:

product, Y: 528.64consumption, C: 318.74investment, I: 9.90interest rate (%), R: 4.34



500 510 520 530 540 550 5600.036

0.038

0.04

0.042

0.044

0.046

0.048

0.05IS and LM curves

product, Y

inte

rest ra

te, R

IS

LM

IS–LM diagram



Fiscal policy: government/public authorities decreasetaxes from 110 to 100

We just need to change the vector d followed by a new call to the solver.shock: new value for T = 100.00new computed endogenous variables:




500 510 520 530 540 550 560 5700.036

0.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052


product, Y

inte

rest ra

te, R

IS

LM

new IS

new LM

Decrease in T



Fiscal policy: government/public authorities increasepublic spending from 200 to 210

shock: new value for G = 210.00new computed endogenous variables:




500 510 520 530 540 550 560 5700.036

0.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052


product, Y

inte

rest ra

te, R

IS

LM

new IS

new LM

Increase in G



Monetary policy: monetary authorities decrease themoney supply from 200 to 180

shock: new value for M = 180.00new computed endogenous variables:




480 490 500 510 520 530 540 550 5600.036

0.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052


product, Y

inte

rest ra

te, R

IS

LM

new IS

new LM

Decrease in M


Highlights

The IS–LM model, embodying Keynes’s view, was developed in 1937 bythe Nobel Laureate of 1972, John Hicks.The model allows to study the effects of different fiscal and monetarypolicies on output and interest rates, as well as on consumption andinvestment.LU factorisation, as part of the solution of systems of linear equations, isintroduced along with the notion of stability and conditioning. Stabilitymeasures the quality of the algorithm with respect to the problemwhereas conditioning is a measure on the data with respect to theproblem where the data is going to take part.


Main references

R. BarroMacroeconomics, A ModernApproachThompson (2008)

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

W. Carlin and D. SoskiceMacroeconomics, Imperfections,Institutions and PoliciesOxford University Press (2006)

G. Dahlquist and Å BjörckNumerical methods in scientificcomputingSociety for Industrial Mathematics(2008)

J. W. DemmelApplied Numerical Linear AlgebraSociety for Industrial Mathematics(1997)

R. J. GordonMacroeconomicsPearson Education (2011)

N. G. MankiwMacroeconomicsWorth Publishers (2009)

D. RomerAdvanced MacroeconomicsMcGraw-Hill/Irwin (2006)


chapter 2 is–lm in a closed economy

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