Stability of a discrete-time, macroeconomic disequilibrium model Kaper, B.

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1 Tilburg University Stability of a discretetime, macroeconomic disequilibrium model Kaper, B. Document version: Publisher final version (usually the publisher pdf) Publication date: 982 Link to publication Citation for published version (APA): Kaper, B. (982). Stability of a discretetime, macroeconomic disequilibrium model. (pp. 3). (Ter Discussie FEW). Tilburg: Faculteit der Economische Wetenschappen. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research You may not further distribute the material or use it for any profitmaking activity or commercial gain You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 5. aug. 205

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3 No STABILITY OF A DISCRETETIME, MACROECONOMIC DIS EQUILIBRIUM MODEL. B. Kaper

4 STABILITY OF A DISCRETETIME, MACROECONOMIC DISEQUILIBRIUM MODEL. ABSTRACT. We invesi:.igate the stability of a macroeconomic monetary discretetime model with a constraint on the market for bankcredit. A theorem is proved on asymp 2 totic stability of a piecewise linear discretetime system in R which is not overall linear.

5 ~. INTRODUCTION. In [3] macroeconomic monetary models have been 3eveloped with constraints on the market for bankcredit. In studying the dynamics of these disequilibrium models we met the problem of asymptotic stability of discretetime systems which are not standard in the theory of difference equations, [5]. Similar problems arise in continuoustime disequilibrium problems (c.f. for instance []). Some advancements on that field of research have been made by e.g. Laro~ue [4], v.d. Heuvel [2]. In this paper we present a theorem on asymptotic stability of a piecewise linear discretetime system in ~22 which is not overall linear. This theorem is applicable to a macroeconomic monetary disequílibrium model that has been abstracted from [3]. In order to get a connection as well as possible with the disequilibrium models in Koning we will use the same type of variables.

6 3 2. A MACRO ECONOMIC DISEQUZLIBRIUM MODEL. We consider a macroeconomic monetary model with constraints on the market for bankcredit. The variables of the model represent relative of balanced growth. values on paths If x symbolises the the balancedgrowth value of a variable of the model viation of variable : X e xe In the neighborhood of the equilibrium the first difference ~x : x x is approximately equal growthrate (~x), ) to the difference of the actual and the balanced of the variable x in the past period. extra growthrate of the variable in question If in e then the relative de is X x actual value and x value with respect to the balancedgr.owth value of the the actual (x) deviations of their (cf. This is also known as the appendix). the model a exogeneous variables are zero and there is tween the actual and balancedgrowth value zerovalue at all time. will persist in the equality be the relative variables of the model If any exogeneous variable is given a non zero value relative variables of the model wi leave their zero position. In order to analyse the effect of a permanent pulse on any of the exo geneous variables we will multiply these variables with the Heavisidefunction H, H(x) 0, if x ~ 0, if x ~ 0. The value of all relative variables of the model will be equal to zero if all exogeneous variables are not effective in the model. The following quantities are involved by the model represent ) The the exogeneous part of the variable y national income c(y) demand for consumption i(t) demand for investment goods subscript indicates (the characters in brac;ket.s in question): a retardation of one period.

7 4 rb interest rate of bankcredit bd demand for bankcredit b(q) supply of bankcredit (a: discount rate) s The model is given by the following set of equilibrium and adjustment equations: (2) c Yly t YH (2.2) (2.3) i tlrb f y ~lc t ~2i t2(yy) t3(bd b) t th (2.4) bd Slrb f d2y (2.5) bs Qlrb ah (2.6) rb rb f pl (bd bs ) where b min {bd, bs} 'Phe greek characters provided with a subindex are positive (adjustment) con ;;tants of the mode]. The model will be reduced to a set of first order dif 2 ference equations in R. From (2.) (2.3) we qet an equation for y, ("l.7) y a~ n2tlrb n2t2y n2t3(bd b) f(ny t n2t)h] where a : (ny) n2t2) and bd b min {0, (S f crl)rb d2y ah}. From (2.4) (2.6) we derive an equation for rb, (2.8) rb rb t pl(dlrb } d2y alrb t ah). Let us introduce a new set of variables, xl(n) : Y x2 (n) rb

8 We get a S firstorder system in ~2z: xl(nt) an2t2 x(n) (2.9) bzxl(n) ar)2t x2(n) f(dl x2(nfl) Pld2 xl(n) f al)x2(n) f(8p Assuming that all variables were equal get at period ~,x2() arl2t3 min ap)x2(n) f PaH(n). to zero up to the zeroth period we {0, ~} PiQ This will be taken as the initial value of the replaced by function then might be sists of two linear subsystems whereas overall QH(n)} t a(rly t n2t)h(n) ~xl () Heaviside ar2t3 min {0, first order system the value. the system itself System is ( 2.9). ( 2.9) 3ubsystem, if d2xi(n) x(nfl) Alx(n) f(dl t ol)x2(n) a ~ 0 t bi where A bl a~2t2 t arl2t3ó2 an2t pcs2 (8fQ)K~ [ an2t3a t a ;ny f p~t ), c~q~ T and x(n) Subsystem 2, : [xl(n), if 82x(n) x(nfl) A2x(n) x2(n~l T ; t(dltal)x2(n) a~ 0 t b2 where A`',ind arl~ ~ ~G ar~2 t p S~ (8t6)P con ofcourse not linear: ar2t3(slfai) The

9 6 b2 ~a(ny f Pla]. n~t). The equilibrium position x of the overall system (2.9) is just equal common equilibrium solution of both subsystems provided it exists to the (c.f, the appendix), x(i A.)lb, i or 2. linear transformation to system Firially we apply a y(ni i x(n which transforms (2.9) (2.9), x, into a homogeneous (ntl) an2t2y(n) 2(nfl) c~cs2y~(n) firstorder system in R2, a~2tly2(n) min {0. S2y(n~ f (Slfal)y2(n)} (2.0) t (SPQlpl)y2(n) or equivalently into two linear subsystems, if d2y (n) (dltal)y2(n) f ~(nfl) Aly(n), ~(ntl) A2y(n). ~ 0 else The minimum function in (2.0) implies continuity of the right hand side of In the next section it will be shown that the 0 solution of such a 2 will be asymptotically stable if homogeneous pieccwi:;e linear system in ik (2.0). both subsystems art asymptotically stable. CONCLUSION: systems The equilibrium of (2.0) are asymptotically stable, Itr A,I i Or equivalently, ~ det A, i ~, is asymptotically stable i.e. if i E{,2} the equilibrium position of the model asymptotically stable if if both sub (2.) (2.6) is

10 f ap2t.s~ t ~an7t2 (2.) 7 (dltol)p~ t ar~2t 3~ FI~ol) ) PS~ ~ ~(a~2t2 f an2t3ó2)([ótq~p) ~ar~ltg t (6to ) I~ ~ NUMERICAL EXAMPLE: If we take the following set of coefficients tyie conditions (2.) (l. l~) (2.) (2.6) del is then ~(a~2t2) ( asymptotically stable: y t i~ S ii F~ exog. (a (0. 3) ). and [ 6}~~ p ) (2.2) f an2tle~lu2 in the ~ ~nodel are satisfied and the mo

11 3. f3 2 IN R. PIECEWISELINEAR DISCRETE DYNAMIC SYSTEMS We ccnsei~.lr~r 2 R (3.) tlr~~ lcl l.uwinc~ autonomous i~iecewiselinear diiyerence equation in Axn ~tl where A: A, i if x E C., i i E I C, be closed cones n. n {,2,...,n}, in R2 with vertices in the origin, with disjoint U C. ~t2; Let the numbering of the cones around the i origin be anticlockwise interiors, Ci ~~ Ci} We will prove ' {a ~ifllu~i.f~ ~ a z 0}, i E In, ntl asymptotic stability of the zero s~~lution of system 0 is an asymptotically stable solution of each of the x ~fl : (3.) if linear subsystems A.x i~ and the function in the right hand side of (3.) is continuous. THEOREM. Consider system (3.2) (3.). I tr A. Ii If ~ det A, i ~, Ki E I, n and (3.3) Ai ~i Ai ~ then system PROOF. The (3.) is asymptotically stable. the concept of a Liapunov We will make use of function (c.f. [5]). form of the Liapunov function is based on the one constructed by Laroque for piecewise linear differential V(x) where e Let V: R is positive 2 ~ R, : det2(x, Ax] is defined by (c.f. [4]). function has been introduced by van den Heuvel A refinement of Laroque's his thesis ( 2]. systems (3.5). definite on R2. in defined by t tlixll4 Because of (3.3) Relative to the V is a continuous system (3.) define function that.

12 9 ~I(x) V(Ax) V(x). Then V(x) det2[ Ax, A2x] t eu Axll 4 detz[ x, Ax] ell xll 4 ) {det2(aj) } det2[ x, Ax] t e{iiaxu4 Uxll4} 2 The expression det [x, Ax] equals zero if x is a real eigenvector of A. Consider the case that for some index i E I A. has an eigenvalue with a real n i eigenvector y that belongs to Int(Ci). By virtue of (3.2) we have ~~~ ~. Choose Y such that 0 ~ Y ~ a4. Then the set K defined by Y Ky : {x E R2~IIAxll4 Ilxll4 ~yllxll4} rl Int (Ci) contains y. Ky is an open cone. If the eigenvector of A, just equals q, theii by i i the continuity property of A~ is also an eigenvector of Ai ~qi Ai ~i Ai ~~ In that case in the definition of K the intersection should be taken with Int (C, U C ). i i qi Let V be the set of all eigenvectors of A satisfying the above conditions, v: {y I H i E In:[ AiY ay, a E IR,y ~ OJ nj y E ci~ }. To each ~ E V an open cone K can be assigned. Define Y K : U K. ye V y Then by assumption (3.2) and the definition of K we have for each x E K (3.4) V(x) ~ FyUxu4. Let us determine next the scalars a, f3, and e a: min {det2[ x, Ax] I Uxll, x~ K} (3 : max {IlAxll ~ IIxU } ) CayleyHamilton theorem: A2 tr(a) A det(a) I.

13 0 and (3.5) Note that a ~: 2S4 [ the scalars max idet 2 (Ai) I a and t3 exist by functions on compact sets. V(x) ~{det2(a~) i E In}], continuity of the respective the For each x~ K we have }aiixb4 t 2S4( i max {det2(ai) I i E In}).,~4x4. and hence (3.6) V(x) ~ laidet2(a.) 2 ~ From the definition of V and }IIxq4, the inequalities (3.4) that V is a Liapunov function in the classical case: tive definite. Then 0 is and (3.6) V(x) (globally) asymptotically stable. and we may conclude V(x) are posi~

14 Appenr~ices.. Let us denote the actual and balanced growth repectively g x e. in the past period by g rate Then and ( ltg) x xe ( fge) xe The first difference!~x 2. The ~xe X X Q e variable x f ge. e. is ( gge ) g4e. are equivalent if equilibrium positions of the linear subsystems holds (a.l) xxe ~ e of the relative there the relation (IAe)lbl Let us define A, A (IA2)lb2 b, where A A2 f A and bl bz f b, an23(dl}vl) an23s2 0 0 and b [ar2t3o, Relation (a.l) bz ~ b will 0]T. reduced into the successively be following relations: (IA2A)(IA2)lb2 or eventually b A(IA2)b2. The last relation can be 3. The equilibrium position xl x2 where checked by straiqht is given by Q{(StQ)Pla(nYfnZl) QlíPid2a(~Yfn7i) f forward substitution. x[xl, x2] anztlpla}, Pla(lfar2t2)}, T,

15 2 R( t an2t2)(dfqi)pl t an2tlpld2. In t}re numerical example the equilibrium is given by x(u.2, 0.24). The equilibrium quantities of the model are co.u9, i 0.24, y0.2, b bd b~ 0.3h, rb 0.24.

16 3 ~~ E~kalbar J.C., The stability of nonwalrasian processes, Ecvnometrica 48 (980), ~2~ FIeuvel, P. van den, The stability of a macroeconomic system with quantity constraints (98), Thesis 'Pechnische Hogeschool, Eindhoven, the Netherlands. [3) Koning, J.H., Kredietrantsoenering en onevenwichtigheid (982), Thesis, Tilburg University, the Netherlands. ~4~ C,aroque, G., Notes and Comments, A comment on "Strahle Spillover amonc~ Subst.i.tut:es", Iteview of Economic Studies (9fi), xz VII7, 35'~36. [5] Lasalle, J.P., Stabi.lity theory for difference equations, Studies in ordinary different.ial equations Ed. J. Hale, publ. by M.A.A., 977.

17 4 IN 98 REEDS VERSCHENEN: 0.. J.J.A. Moors Inadmissibility of linearly invariant estimators in truncated parameter spaces 0.2. H. Peer De mathematische structuur J. Klijnen van conjunctuurstructuurmodellen en een rekenprocedure voor numerieke simulatie van deze modellen 0.3. H. Peer Macro economic policy options in nonmarkt structures jan. jan. febr J. van Mier ~vergelijkinger en operatoren maart 0.5. A.L. Hempenius 0.6. R.J.M. Heuts 0.7. B. Kaper 0.8. R.M.J. Heuts and R. Willemse 0.9. J.P. Heesters 0. J.P. Heesters Definities van gemiddelde factorproductiviteiten en bezettingsgraad in een jaargangenmodel voor industriële sectoren, met een toepassing voor de sector Chemische Industrie Asymptotic Robustness of Prediction Intervals of Arima Models by Deviations of Normality Some aspects of differential equations with discontinuous righthand sides Impulse response patterns for various dynamic time teries models Aankleden of uitkleden? Een kritische beschouwing van de honorering van de huisarts vrij beroepsbeoefenaar Aankleden of uitkleden? Een kritische beschouwing van de honorering van de medisch specialist vrij beroepsbeoefenaar ten opzichte van de ambtenaar maart mei juli juni sept. okt.. Dr. G.P.L van Roij Rentearbitrage, valutaspeculatie en wisselkoersen nov. 2. J. Glombowski A Comment on Sherman's Marxist Cycle Model revised version 3. Drs. W.A.M. de Lange Deeltijdarbeid op de Katholieke H.A.C. de ConinckMerckx Hogeschool Tilburg M.R.M. Turlinas M.C.M. Puyk nov. nov.

18 5 4. Drs, w.a.m. de Lange Tabellenboek bij het Onderzoek L.H.M. Bosch 'Deeltijdarbeid op de Katholieke M.C.M. Turlings Hogeschool Tilburg' nov. 5. H. Peer Economische groei en uitputtelijke grondstoffer. nov.

19 6 IN 982 REEDS VERSCHENEN: O. W. van Groenendaal 02. M.D. Merbis 03. F. Boekema 04. P.T.W.M. Veugelers 05. F. Boekema 06. P. van Geel 07. J.H.M. Donaers, F.A.M, van der Reep Building and analyzing an jan econometric model with the use of a hybrid computer; part I. System properties of the jan. interplay model Decentralisatie en regionaal maart sociaaleconomisch beleid Een monetaristisch model voor maart de Nederlandse economie Morfologie van de "Wolstad", april Over het ontstaan en de ontwikkeling van de ruimtelijke geleding en struktuur van Tilburg. Over de (on)mogelijkheden mei van het model van Knoester. De betekenis van het monetaire beleid voor de Nederla.~dse eccnomie, presentatie van een analyse aan de hand van een eenvoudig model mei 08. R.M.J. Heuts The use of nonlinear transformation in ARIMAMOdels when the data are nongaussian distributed juni 09. B.B. van der Genugten 0. J. Roemen. J. Roemen 2. M.D. Merbis prelimi juli On the compensator Part I Problem formulation and naries Asymptotic normality of least squares estimators ín autoregressive linear regression models. j~i Van koetjes en kalfjes Z juli Van koetjes en kalfjes II juli 3. P. Slangen Bepaling van de optimale beleidsparameters voor een stochastisch kasbcheersprobleem met continue controle aug. 4. M.D. Merbis Linear Quadratic Gaussian Dynamic Games aug.

20 7 5. P. Hinssen Een kasbeheermodel onder J. Kriens onzekerheid sept. J. Th. van Lieshout 6. A. Hendriks en "Van Bedrijfsverzamelgebouw T, van der BijVeenstra naar Bedrijvencentrum" okt. 7. F.W.M. Boekema Industriepolitiek, Regionaal A.J. Hendriks beleid en Innovatie okt. L.H.J. Verhoef

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