American Journal of Applied Sciences, 2012, 9 (12), 1944-1952 ISSN: 1546-9239 ©2012 Science Publication doi:10.3844/ajassp.2012.1944.1952 Published Online 9 (12) 2012 (http://www.thescipub.com/ajas.toc)
Computational Techniques for Stability Analysis of a Class of Dynamic Economic Models
George E. Halkos and Kyriaki D. Tsilika
Department of Economics, Laboratory of Operations Research, University of Thessaly, Korai 43, 38 333, Volos, Greece
Received 2012-01-10, Revised 2012-11-06; Accepted 2012-12-12 ABSTRACT Modern microeconomics and macroeconomics study dynamic phenomena. Dynamics could predict future states of an economy based on its structural characteristics. Dynamic models in discrete time discrete state economic systems may take the form of one single linear difference equation or a system of linear difference equations. In this study we use Xcas and Mathematica as software tools in order to generate results concerning the dynamic properties of the solutions of the difference equation(s) and, determine whether an economic equilibrium exists. Our computational approach does not require solving the difference equation(s) and makes no assumptions for initial conditions. The results provide quantitative information based on the qualitative properties of the mathematical solutions rather than on their quantifiable ones. The relevant output of CAS software is created in a way as to be interpreted without the knowledge of advanced mathematics. The computer codes are fully presented and can be reproduced as they are in computational-based research practice and education.
Keywords: Economic Equilibrium Exists, Computational-Based Research, Mathematical Models, Dynamic Economic System, Fundamental Dynamic Equation
1. INTRODUCTION multiplier-acceleration interaction model, inflation- unemployment model in discrete time, dynamic market Dynamic economic analysis is to determine whether, models, macroeconomic and macroeconometric models. given sufficient time, economic variables tend to In multi equation models, difference equations are converge to certain equilibrium (steady state) values. Time often combined into a single fundamental dynamic is considered as a discrete variable, meaning that any equation. A simple second or third order difference variable undergoes a change only once within a period of equation usually does not suffice to explain cycles and time. Thus, mathematical models used consist of one single other fluctuation phenomena that economic activity or a simultaneous set of n-order linear difference equations. generates. As models become larger their dynamic Linearity is not restrictive in economic applications since it behavior becomes more difficult and less may be imposed on a model through means of a first order straightforward. Therefore, a proper usage of computer taylor approximation. software off-loads manual solving procedures that require In discrete time discrete state dynamic economic heavy mathematical background. Studies towards this et al systems, only a subset of elements is given, these are the direction are made in textbooks as (Amman ., 1996; state variables of the economy such as the capital stock Huang and Crooke, 1997; Tesfatsion and Judd, 2006; and asset position. Other variables may be free to take any Miranda and Fackler, 2004; Varian, 1996) and papers as value even in the initial period; these are typically flow (Kendrick and Amman, 1999; Al-Rawi et al ., 2007). variables such as consumption and labor. Classic discrete The study is organized as follows. Materials and time discrete state economic models are the Cobweb Methods section introduces the concepts of stability and model with memory of several periods, Samuelson asymptotic convergence and sets the theoretical Corresponding Author: George E. Halkos, Department of Economics, University of Thessaly, Korai 43, 38 333, Volos, Greece Tel: 0030 24210 74920 Fax: 0030 24210 74772
Science Publications AJAS 1944 George E. Halkos and Kyriaki D. Tsilika / American Journal of Applied Sciences, 9 (12) (2012) 1944-1952
framework that ensures stability for a single, linear, constant where, r i are real or complex roots of the characteristic n n-1 n-2 coefficient difference equation. In addition, eigen- polynomial a 0r +a 1r +a 2r +…+a n = 0. decomposition and asymptotic stability are discussed for Stability Conditions II. Next we present a multivariable economic models expressed in matrix form. determinental expression of necessary and sufficient Results’ section proposes computer codes for stability stability conditions. conditions for single difference equations and for systems of n Schur Theorem. The real polynomial f (x) = a 0 x difference equations in Mathematica and Xcas. Then, n-1 n-2 +a 1x + a 2x +…+a n = 0 is called Schur stable if its asymptotic behavior in convergent cases is predicted. Some roots x are |x |<1. The condition |x |<1 holds if and only applications are used to perform computations’ potential i i i if the n determinants ∆i (i = 1,…,n) are positive. Given: into several selected dynamic economic models. Finally,
Discussion section concludes the study. a0 0 ...0 a nn1 a− ... a 1
a10 a ... 0 0 a n2 ... a 2. MATERIALS AND METHODS ...... a a ...a 0 0 ... a ∆ = n1n2− − 0 n n a 0 ... 0 a a ...a When the solution of a dynamic economic system n 01 n1− an1n− a ... 0 0 a 0 ...a n2− follows a restrictive non-explosive path this is what ...... characterizes economic equilibrium. The structure of a12n a ...a 0 0 ... a 0 the mathematical model that insures existence of equilibrium has been studied among others in (Batra, The determinants ∆ are Eq. 2: 2006; Blanchard and Kahn, 1980; Folsom et al ., 1976; i
Gu et al ., 2003; Hamza and Khalaf-Allah, 2007; Jury, 1974). In economic applications stability appears to be a0 0a nn1 a − aa aa0a a property with diachronic interest as shown in studies ∆=0n ∆= 10 n ∆ et al 1, 2 ,...,n (2) like (Chaumpsaur ., 1977) up to recent studies of aan0 a0aa n 01 (Ratto, 2008; Gomes, 2010). an1n− a 0a 0 The solution that follows a convergent time path is considered stable. Figure 1a-d depict the motion of variables during a given period. Specifically, Fig. 1 a, b For a detailed analysis see (Chiang, 2006; Jury, 1974; Neumann, 1979). depict unstable solutions that belong to models with explosive behaviors. Figure 1c depicts a solution that 2.2. Stability Conditions For Systems Of Linear follows a convergent time path with dampened Constant Coefficient Difference Equations oscillations; Fig. 1d depicts solutions with a non- oscillatory convergent time path. Stability Conditions III. Consider a system of linear constant coefficient first order difference equation in uk 2.1. Stability conditions for Linear Constant Eq. 3: Coefficient Difference Equations u= Au (3) Stability Condition I. The general solution of a n-th k+ 1 k order linear difference equation a0yt+n-1+a 2 yt+n-1+a 2yt+n- Where: +…a y = g (t) consists of two parts, the complementary 2 n t u = A vector of date k economic variables solution y and the particular solution y , so that y = y k c p t c A = Some square matrix +y . We are interested in the limiting behavior of the p complementary function yc. We say that the general By extending the vector uk and relabeling some lagged solution is stable if and only if lim yc = 0 . Hence as t→∞, t→∞ variables we can transform a higher order difference the behavior of the general solution of the difference equation system into a first order difference equation equation is essentially that of the particular function. For (Chiang, 2006). It turns out that the eigenvalues of A help a n-th order linear difference equation with constant determining whether an equilibrium exists. After k steps coefficients, a necessary and sufficient condition for the there are k multiplications of the transformation A = PDP -1 particular solution to be stable is that Eq. 1: and the solution of (3) is the solution of Eq. 4:
< = k k− 1 ri 1,i 1,...,n (1) uk1+ = Au 1 = PDPu 1 (4)
Science Publications 1945 AJAS George E. Halkos and Kyriaki D. Tsilika / American Journal of Applied Sciences, 9 (12) (2012) 1944-1952
(a)
(b)
(c)
(d)
Fig. 1. About here (a) Unstable oscillatory behavior (b) Unstable non-oscillatory behavior (c) Stable oscillatory behavior (d) Stable non-oscillatory behavior
Science Publications 1946 AJAS George E. Halkos and Kyriaki D. Tsilika / American Journal of Applied Sciences, 9 (12) (2012) 1944-1952
Where: difference equation is stable. For a direct answer to this P = A matrix with columns the eigenvectors and stability test, we define stabilitytest1 function in Xcas, characteristic vectors of the matrix A with arguments the characteristic polynomial (poly) of D = The Jordan matrix of A verifying D = P−1AP the difference equation and its variable (var). (Strang, 1988) 5G p.264) stabilitytest1 function returns «stable» for systems with equilibrium state(s) in case where all determinants (2) The necessary stability conditions for the system (3) have positive signs and «unstable» for systems that have are Eq. 5: explosive behavior otherwise:
−n < trA < n stabilitytest1(poly,var):=if([seq(sign(schurseries(poly,var −1 < D < 1 (5) )[[k]]),k=1..degree(poly,var))]==makelist(1,1,degree(pol y,var))) stable; else unstable;
Where: In Xcas environment, stability conditions (5) are trA = The trace of the coefficient matrix Α examined using stabilitytest2 function taking system’s n = The number of variables (Strang, 1988) 5J) coefficient matrix as argument. stabilitytest 2 function returns «stable» for systems with equilibrium state(s) in 3. RESULTS case where conditions (5) are satisfied and «unstable» for systems that have explosive behavior otherwise:
3.1. All Electronic Files are Available on Request stabilitytest2(x):=if (abs(trace(x)) Xcas is a Computer Algebra System available free In case equilibrium exists, we create steadystate in http://www-fourier.ujf- function with arguments system’s coefficient matrix (a) and grenoble.fr/~parisse/giac.htmlIn Xcas programming the system’s initial state in a column matrix form. environment we create schurseries function, with steadystate function calculates the asymptotic state of the arguments the characteristic polynomial (poly) of the system in a column matrix form. Steadystate function uses difference equation and its variable (var). schurseries Xcas’ built in function matpow which calculates the function calculates the determinant series (2). The k=100000 power of a matrix by jordanization, as shown in (4): computer codes are: steadystate(a,initialstate):=approx(matpow(a,100000))*i A11(poly,var):=matrix(degree(poly,var),degree(poly,var) nitialstate ,(j,k)->if(j polynomial (poly) of the difference equation and its A21(poly,var):=matrix(degree(poly,var),degree(poly,var) variable (var). charvalues function generates the list of ,(j,k)->if(j Science Publications 1947 AJAS George E. Halkos and Kyriaki D. Tsilika / American Journal of Applied Sciences, 9 (12) (2012) 1944-1952 absolute values of the characteristic roots is less than 1 t-2 to period t-1 and r is a parameter. If 0≤r≤1, suppliers and «unstable» for systems that have explosive behavior expect the next price change, Et-1 (p t)-pt-1, to be in the otherwise: opposite direction of the previous price change, ∆Pt-2. If -1≤r≤0, suppliers expect the next price change to be in stabilitytest[poly_,var_]:=If[Max[Table[Abs the same direction as the previous price change. The [Root[poly,i]],{i,1,Length[Solve[poly D demand function is given by Qt= A + Bp t . Equilibrium ==0,var]],1}]]<1,stable,unstable] S= D occurs if Qt Q t . Substitutions lead to the following steadystate function in Xcas calculates the asymptotic second-order difference equation: k k -1 behavior of the system uk+1 = A u1 = PD P u1 after k=100000 steps. Let us define now steadystate function in g(1− r) gr (f − a) Mathematica. Mathematica’s steadystate function provides pt2+− p t1 + − p t = greater calculative precision, since the asymptotic behavior b b b of the system is estimated for power k→∞. The first argument of steadytate function in We will test the stability for several values of g, b, r, Mathematica contains system’s coefficient matrix (a) and a, f. The example is taken from (Hoy et al ., 2011) p.826. the second argument system’s initial state (initial) in a In Mathematica we get a list of the absolute values column matrix form. steadytate function calculates the of the characteristic roots by writing: asymptotic behavior of the system in a column matrix form: G = 13;b = -16;r=-3/13;a = 60;f = 2; steadystate[a_,initial_]:=Simplify[Jordan charvalues[x^2-g/b (1-r) x-g/b r,x]//N Decomposition[a][[1]].Limit[MatrixPower {1.16144, 0.161438} [JordanDecomposition[a][[2]],k],k →∞]. Inverse[JordanDecomposition[a][[1]]]]. initial//MatrixForm Alternatively we decide stability using: Generalizing previous output, we create stabilitytest[x^2-g/b (1-r) x-g/b r,x] unstable distributionk function, with arguments system’s coefficient matrix (a) and system’s initial state (initial). For different values of g,b,r,a,f, Mathematica’s Distributionk function calculates the system state in stability test gives: step/time k, in a column matrix form. Steadystate and distributionk functions in Mathematica are based on the G = 13;b = -16;r = 3/13;a = 60;f = 2; k k −1 equality A = PD P and use Mathematica’s built in charvalues[x^2-g/b (1-r) x-g/b r,x]//N function JordanDecomposition which calculates a {0.433013,0.433013} similarity matrix and the Jordan canonical form of a stabilitytest[x^2-g/b (1-r) x-g/b r,x] square matrix: stable distributionk[a_,initial_]:=Simplify[Jordan 3.4. The Cobweb Model for 3 Periods Decomposition[a][[1]].MatrixPower[Jordan The Cobweb model could give a higher order Decomposition[a][[2]],k].Inverse difference equation, i.e.: [JordanDecomposition[a][[1]]]].initial//MatrixForm 18yt3++ 9y t2 + − 5y t1 + −= 2y t 0 Let us examine existence of equilibrium in a number of economic models. We apply Xcas’ programmed function schurseries: 3.3. The Cobweb Model schurseries(18*x^3+9*x^2-5*x-2,x) The supply function for period t is 320, 97216, 18439680 QS = F + GE (p) where QS is the quantity supplied and t t1t− t Xcas’ stability test gives: Et-1 (P t) is the price that suppliers in period t-1 expected to prevail in period t. Assuming that Et-1 (p t) = P t-1 -r∆Pt-2 stabilitytest1(18*x^3+9*x^2-5*x-2,x) where ∆Pt-2 = P t-1-pt-1-Pt-2is the price change from period stable Science Publications 1948 AJAS George E. Halkos and Kyriaki D. Tsilika / American Journal of Applied Sciences, 9 (12) (2012) 1944-1952 = − + 3.5. A Macroeconometric Model Yt 3.0716Y t1− 3.6561Y t2 − 2.0850Y− 0.5585Y + 0.0535Y The following macroeconomic model is a quarterly t3− t4 − t5 − +1.1427G − 2.5300G + model constructed by Kmenta and Smith. The equations t t− 1 were estimated with quarterly data over the period 1954- 1.3779Gt2−+ 0.5853G t3 − − 0.7463G t4 − 1963. The example is taken from (Pindyck and +0.1784Gt− 5 + 0.3168M t − Rubinfeld, 1998). 0.7499M+ 0.6253M − 0.2000M t1− t2 − t3 − ∧ +0.0082Mt4− + 0.0046M t5 − Ct=− 1.7951 + 0.1731Y t + 0.421(L t +0.048Lt − 0.1065L t1− + 0.0580L t2 − −0.7275L ) + 0.7275C t1− t1 − +0.0246L − 0.0314L ∧ t3− t4 − d = − + It 2.5624 0.4411r t 0.1381(S t1− +0.075Lt− 5 + 0.1034t − 0.2050(t − 1) ∧ +0.1267(t − 2) − 0.0192 −S ) + 0.0237t + 0.8917I d t2− t1 − −− −− ∧ (t 3) 0.0032(t 4) 0.5113 r It= 3.6083 − 0.5127r t + 0.1267(S t1− ∧ In Mathematica we get a list of the absolute values − + + r St2− ) 0.0218t 0.6483I t1 − of the characteristic roots by writing: ∧ i I= 3.0782 − 0.8934r + 0.3713(S t t t1− charvalues[x^5-3.0716*x^4+3.6561*x^3-2. ∧ i 085*x^2+0.5585*x-0.0535,x] −St2− ) + 0.450t + 0.3178I t1 − ∧ {0.207946, 0.593134, 0.593134, = + rt 13.8928 0.261Y t 0.855162, 0.855162} −0.1501M + 0.0588M t t− 1 Alternatively we check stability conditions using: = +d +++ r i Yt C tttt I I I G t i St= Y t − I t stabilitytest[x^5-3.0716*x^4+3.6561 *x^3-2.085*x^2+0.5585*x-0.0535,x]stable Lt= M t + TD t Where: Testing stability in Xcas we get: Y = Gross national product C = Consumption expenditures schurseries(x^5-3.0716*x^4+3.6561* Id = Producer’s outlays on durable plant and x^3-2.085*x^2+0.5585*x-0.0535,x) equipment 1, 0.84, 0.24, 0.005, 2e-06 Ir = Residential construction i Or: I = Increase in inventories G = Government purchases of goods and services plus stabilitytest1(x^5-3.0716*x^4+3.6561*x^ net foreign investment 3-2.085*x^2+0.5585*x-0.0535,x) S = Final sales of goods and services stable T = Time in quarters (first quarter of 1954 = 0) R = Yield on all corporate bonds, percent per annum 3.6. Some Empirical Applications M = Money supply, i.e., demand deposits plus currency outside banks Application 1 TD = Time deposits in commercial banks Multinational companies in the U.S., Japan and L = Money supply plus time deposits in commercial Europe have assets of $4 trillion. At the start, $2 banks (representing liquid wealth) trillion are in the U.S. and $2 trillion in Europe. Each year ½ of the U.S. money stays home, ¼ goes to both All the variables except for t and r are measured in Japan and Europe. For Japan and Europe ½ stays billions of 1958 dollars and the variables G, M, TD and t home and ½ is sent to the U.S. Find the limiting are taken to be exogenous. distribution of the $4 trillion as the world ends. Find d The original model by substituting C t-1, S t-1, S t-2, I t-1, the distribution at year k. (The application is taken r i I t-1, I t-1, results in the fundamental dynamic equation: from (Strang, 1988) ex. 5.3.11 p. 273). Science Publications 1949 AJAS George E. Halkos and Kyriaki D. Tsilika / American Journal of Applied Sciences, 9 (12) (2012) 1944-1952 In Xcas environment we define: stabilitytest2 (a) "stable" b:=[[1/2,1/2,1/2],[1/4,1/2,0],[1/4,0,1/2]] steadystate(a,[[0],[5]]): stabilitytest2(b) 3 “stable” 2.0 steadystate(b,[[2],[0],[2]]): In Mathematica environment we define: 2.0 1.0 a:={{0.8,0.3},{0.2,0.7}} 1.0 steadystate[a,{{0},{5}}]: In Mathematica environment we define: 3. b:={{1/2,1/2,1/2},{1/4,1/2,0},{1/4,0,1/2}} 2. Steadystate[b,{{2},{0},{2}}]: distributionk[a,{{0},{5}}]: 2 5(− 0.60.5k + 0.61. k ) 1 k k 5(0.60.5+ 0.41. ) 1 Application 3: Consider two economies: Distributionk[b,{{2},{0},{2}}]: C= 0.6Y C = 0.8Y 1t 1,t1− 2t 2,t1− 1k+++ 2k 3k1 k 1k +++ 2k 3k 0+++−− 0 0 (220 40 + 70 + 30) I=+ 5 0.2Y I =+ 10 0.25Y 1t 1,t1− 2t 2,t1− 2 M=+ 10 0.1Y M =+ 5 0.3Y 1 k 1k+ 2k + 3k + 4k + 1t 1,t1− 2t 2,t1− ++−(2 20 40 + 50 + 50 + 40 ) 2 X= M X= M 1t 2t 2t 1t 1 k 1k+ 2k + 3k + 4k + −− 1k Y=++− C I X M Y =++− C I X M (10−++ 0 30 ++ 0 40)2 + 1t 1t 1t 1t 1t 2t 2t 2t 2t 2t 2 k k 1kk+ 2kk + 3k + 2k − (20−+++ 2 302 + 0 2 + 30 )2 +− In the present model exports of one economy are 1k+++ 2kk + 3kk + + 4kk + imports of the other. Given that Y10 = Y 20 = 100, find the (0 02 02 02) asymptotic behavior of Y1,t , Y 2,t , which results after 1 k 1k+ 2k + 3k + (1−− 0 0 + 60 + 40 ) increasing the autonomous part of investment of the two 2 economies by ∆I1=10 and ∆I2=20 respectively. 1 k 1k+ tk + 1k − +++(1 0 50 + 60 + 2 ) The two macroeconomic models result in the system 2 of difference equations: −−+2k 1k 2kk + 3k1k ++ +2 (0 + 02 + 02) Y− 0.7Y − 0.3Y = 10 1,t+ 1 1t 2t →∞ Y− 0.1Y − 0.75Y = 15 Application 2: Find the limiting values of yk, z k, K , 2,t+ 1 1t 2t where: An initial solution of the system is the solution of = + yk1+ 0.8y k 0.3z k the homogeneous system: = + zk1+ 0.2y k 0.7z k Y= 0.7Y + 0.3Y 1,t+ 1 1t 2t = = assuming y0 0,z 0 5. Y= 0.1Y + 0.75Y 2,t+ 1 1t 2t In Xcas environment we define: Which gives the complementary solution yc, part of the total solution yt = y c+y p of the nonhomogeneous a:=[[0.8,0.3],[0.2,0.7]] system. Science Publications 1950 AJAS George E. Halkos and Kyriaki D. Tsilika / American Journal of Applied Sciences, 9 (12) (2012) 1944-1952 Testing the stability of the complementary solution Specifically, it examines whether they follow a convergent in Xcas: time path or not. In case convergence is assured, the asymptotic state of the system is given, taking into stabilitytest2([[0.7,0.3],[0.1,0.75]]) account its initial value. Stability results come from “stable” conditions depending on characteristic roots and/or determinental expressions of coefficient matrices of We conclude that income vector Yt for both difference equations and not by the initial values of some economies reaches a steady state as t→∞. elements. Steady state predictions apply Jordan Decomposition method. All results are generated by 4. DISCUSSION computer codes in both free computer algebra system Xcas and commercial mathematical package Mathematica. Computer packages provide solutions of the characteristic equations, letting the user compute their 6. 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