ON THE EXISTENCE OF HOPF BIFURCATION IN AN OPEN ECONOMY MODEL

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1 Proceedings of Equadiff , pp ISBN ON THE EXISTENCE OF HOPF BIFURCATION IN AN OPEN ECONOMY MODEL KATARÍNA MAKOVÍNYIOVÁ AND RUDOLF ZIMKA Abstract. In the paper a four dimensional open economy model describing the development of output, exchange rate, interest rate and money supply is analyzed. Sufficient conditions for the existence of equilibrium stability and the existence of business cycles are found. Formulae for the calculation of bifurcation coefficients are derived. Key words. dynamical model, equilibrium, stability, bifurcation, cycles AMS subject classifications. 37Gxx 1. Introduction. In economic theory there are many macroeconomic models describing the development of output Y in an economy. The majority of them are based on the well-known IS-LM model. One of them is the Schinasi s model (see [5]) describing the development of output Y, interest rate R and money supply L s in a closed economy. In [8] the Schinasi s model was extended to an open economy model of the kind Ẏ = α [ I(Y, R) + G + X(ρ) S(Y D, R) T (Y ) M(Y, ρ) ] ρ = β [M(Y, ρ) + C x (R) X(ρ) C m (R)] Ṙ = γ [L(Y, R) L S ] L S = G + M(Y, ρ) + C x (R) T (Y ) X(ρ) C m (R), (1.1) where Y output, ρ exchange rate, R interest rate, L S money supply, I investments, S savings, G government expenditures, T tax collections, X export, M import, C x capital export, C m capital import, L money demand, α, β, γ positive parameters, t time and Y D = Y T (Y ), Ẏ = dy dt, dρ ρ = dt, Ṙ = dr dt, L S = dl S dt. The economic properties of the functions in (1.1) are expressed by the following partial derivatives: I(Y, R) I(Y, R) T (Y ) X(ρ) ρ C x (R) C m (R) S(Y D, R) M(Y, ρ) L(Y, R) S(Y D, R) M(Y, ρ) ρ L(Y, R) < 0. (1.) Department of Mathematics and Descriptive Geometry, Faculty of Wood Sciences and Technology, Technical university Zvolen, Zvolen, Slovakia (kmakovin@vsld.tuzvo.sk). Department of Quantitative Methods and Informatics, Faculty of Economics, Matej Bel University, Banská Bystrica, Slovakia (rudolf.zimka@umb.sk). 9

2 30 K. Makovínyiová and R. Zimka The development of Y, ρ, R and L S in the model (1.1) in a neighborhood of its equilibrium was studied under fixed exchange rate regime in [6] and under flexible exchange rate regime in [] from the point of equilibrium stability and the existence of business cycles. In [4] this model was analyzed under assumption that both ρ and R are flexible. In the paper [4] the assertion on the existence of business cycles was formulated under assumption of the existence of the Liapunov bifurcation constant a. In the present paper a formula for the calculation of this Liapunov bifurcation constant a is found supposing that the functions I(Y, R) and S(Y D, R) are nonlinear with respect to Y and linear with respect to R, and other functions in (1.1) are linear with respect to all their variables except government expenditures G, which are constant. In the end of the paper an example illustrating the achieved results is presented.. Analysis of the model (1.1). Using denotation the model (1.1) has the form B(Y, R) = I(Y, R) S(Y D, R) C(R) = C m (R) C x (R) D(Y ) = G T (Y ) F (Y, ρ) = X(ρ) M(Y, ρ), Ẏ = α [B(Y, R) + D(Y ) + F (Y, ρ)] ρ = β [F (Y, ρ) + C(R)] Ṙ = γ [L(Y, R) L S ] L S = D(Y ) F (Y, ρ) C(R), (.1) where on the base of (1.) B(Y, R) C(R) D(Y ) F (Y, ρ) F (Y, ρ) ρ and the sign of B(Y,R) can be positive or negative or zero according to the properties of I(Y, R) and S(Y D, R). Assume: 1. The functions I(Y, R), S(Y D, R), T (Y ), C(R), D(Y ), F (Y, ρ), L(Y, R) are of the kind I(Y, R) = i 0 + i 1 Y i3 R, C(R) = c 0 + c 3 R, S(Y D, R) = s 0 + s 1 (Y D ) + s 3 R, D(Y ) = d 0 d 1 Y, Y D = Y T (Y ), F (Y, ρ) = f 0 f 1 Y + f ρ, T (Y ) = t 0 + t 1 Y, L(Y, R) = l 0 + l 1 Y l 3 R, with positive coefficients i 1, i 3, s 1, s 3, t 1, c 3, d 1, f 1, f, l 1, l 3.. The model (.1) has an isolated equilibrium E = (Y, ρ, R, L S ), Y ρ R L S > 0. Remark 1. A sufficient condition for the existence of an isolated equilibrium E of the model (.1) was presented in [4]. Consider an isolated equilibrium E = (Y, ρ, R, L S ) of (.1). After the transformation Y 1 = Y Y, ρ 1 = ρ ρ, R 1 = R R, L S 1 = L S L S,

3 Equadiff-11. On the existence of Hopf bifurcation in an open economy model 31 the equilibrium E goes into the origin E1 = (Y1 = 0, ρ 1 = 0, R1 = 0, L S 1 = 0) and the model (.1) takes the form Ẏ 1 = α [B(Y 1 + Y, R 1 + R ) + D(Y 1 + Y ) + F (Y 1 + Y, ρ 1 + ρ )] ρ 1 = β [F (Y 1 + Y, ρ 1 + ρ ) + C(R 1 + R )] Ṙ 1 = γ [L(Y 1 + Y, R 1 + R ) L S 1 L S 1 ] L S 1 = D(Y 1 + Y ) F (Y 1 + Y, ρ 1 + ρ ) C(R 1 + R ). (.) Performing Taylor expansion of the functions on the right-hand side in (.) at the equilibrium E1 we get the model [ 4 ] Ẏ 1 = α [(b 1 d 1 f 1 )Y 1 + f ρ 1 b 3 R 1 ] + α a k Y1 k + O( Y 1 5 ) ρ 1 = β (f 1 Y 1 f ρ 1 c 3 R 1 ) Ṙ 1 = γ (l 1 Y 1 l 3 R 1 L S 1 ) L S 1 = (f 1 d 1 )Y 1 f ρ 1 c 3 R 1, k= (.3) where b 1 = B(Y,R ), b 3 = i 3 + s 3, a k = k B(Y,R ), k =, 3, 4. k The linear approximation matrix of (.3) is α(b 1 d 1 f 1 ) αf αb 3 0 A(α, β, γ) = βf 1 βf βc 3 0 γl 1 0 γl 3 γ. (.4) f 1 d 1 f c 3 0 The characteristic equation of A(α, β, γ) is λ 4 + a 1 (α, β, γ)λ 3 + a (α, β, γ)λ + a 3 (α, β, γ)λ + a 4 (α, β, γ) = 0, (.5) where a 1 = α( b 1 + d 1 + f 1 ) + βf + γl 3, a = γc 3 + βγf l 3 + α(βf ( b 1 + d 1 ) + γ(b 3 l 1 + l 3 ( b 1 + d 1 + f 1 ))), a 3 = αγ(βf (l 1 (b 3 + c 3 ) + l 3 ( b 1 + d 1 )) c 3 ( b 1 + d 1 + f 1 ) b 3 (f 1 d 1 )), a 4 = αβγd 1 f (b 3 + c 3 ). As we are interested in equilibrium stability and in the existence of cycles in the model (.3) it is suitable to find such values of parameters α, β, γ at which the equation (.5) has a pair of purely imaginary eigenvalues λ 1 = iω, λ = iω, and the rest two eigenvalues λ 3, λ 4 are negative or have negative real parts. We shall call such values of parameters α, β, γ as critical values of the model (.3) and denote them α 0, β 0, γ 0. Mentioned types of eigenvalues are ensured by the Liu s conditions ([3]): a 1 a a 3 a 4 (.6) 3 = (a 1 a a 3 )a 3 a 1 a 4 = 0. (.7)

4 3 K. Makovínyiová and R. Zimka The conditions (.6) are satisfied if b 1 + d 1 0, α > α = c 3 l 3 ( b 1 + d 1 + f 1 ) + b 3 l 1, ( b 1 + d 1 + f 1 )c 3 + (f 1 d 1 )b 3 < 0. The relation (.7) can be expressed in the form [ ] a (1) (β, γ)β + a () γ α + where in the power of (.8) there is [ b (1) (β, γ)β + b () (γ)γ a () = ( b 1 + d 1 + f 1 )[b 3 (f 1 d 1 ) ] α + c (1) (β, γ)β + c () γ = 0, +c 3 ( b 1 + d 1 + f 1 )][ b 3 l 1 l 3 ( b 1 + d 1 + f 1 )] c () = c 3 l 3 [b 3 (f 1 d 1 ) + c 3 ( b 1 + d 1 + f 1 )] < 0. Take γ at any positive level and denote it γ 0. Then a () γ 0 > 0 and c () γ 0 < 0. Denote A(β) = a (1) (β, γ 0 )β + a () γ 0, B(β) = b (1) (β, γ 0 )β + b () (γ 0 )γ 0, C(β) = c (1) (β, γ 0 )β + c () γ 0. (.8) Take β such small that A(β) > 0 and C(β) < 0. Denote this β as β 0. Then the equation A(β 0 )α + B(β 0 )α + C(β 0 ) = 0 has two roots α 1, = B(β 0) ± { (B(β 0 )) 4A(β 0 )C(β 0 ) α1 < 0 = A(β 0 ) α > 0, and the function 3 (α) = A(β 0 )α + B(β 0 )α + C(β 0 ) has the following form (Fig..1): Fig..1. Denote α = α 0. The triple (α 0, β 0, γ 0 ) is the critical triple of the model (.3). Thus for given specific values of β 0, γ 0 it is always possible to find such a value α 0 that the condition (.7) is satisfied. From Figure 1 we get in a small neighborhood of α 0 : If α > α 0 then 3 (α) = (a 1 a a 3 )a 3 a1 a 4 > 0. If α < α 0 then 3 (α) = (a 1 a a 3 )a 3 a1 a 4 < 0. (.9) If α = α 0 then 3 (α) = (a 1 a a 3 )a 3 a1 a 4 = 0.

5 Equadiff-11. On the existence of Hopf bifurcation in an open economy model 33 Consider a critical triple (α 0, β 0, γ 0 ) of the model (.3). Further we shall investigate the behavior of Y 1, ρ 1, R 1 and L S1 around the equilibrium E 1 with respect to parameter α, α (α 0 ε, α 0 + ε), ε and fixed parameters β = β 0, γ = γ 0. Performed considerations enable us to answer the question about the stability of the equilibrium E 1 of the model (.3) in a small neighborhood of the critical value α 0. Theorem 1. Let the conditions (.8) be satisfied and let (α 0, β 0, γ 0 ) be such a critical triple of the model (.3) that A(β 0 ) > 0 and C(β 0 ) < 0. Then: 1. If α 0 α then at every α > α the equilibrium E 1 is asymptotically stable.. If α 0 > α then at every α > α 0 the equilibrium E 1 is asymptotically stable and at every α < α < α 0 the equilibrium E 1 is unstable. Proof. The conditions (.8) guarantee that a 1 a a 3 and a 4 > 0 what together with (.9) at α > α α 0 means that the Routh-Hurwitz necessary and sufficient conditions for all the roots of the equation (.5) to have negative real parts ( 1 (α) > 0, (α) 3 (α) 4 (α) > 0) are satisfied. If α < α < α 0 then 3 (α) < 0 what means that the Routh-Hurwitz conditions are not satisfied. To gain the bifurcation equation of the model (.3) it is suitable to transform (.3) to its partial normal form on invariant surface. After the shift of α 0 into the origin by relation α 1 = α α 0 the model (.3) takes the form Ẏ 1 = α 0 [(b 1 d 1 f 1 )Y 1 + f ρ 1 b 3 R 1 ] + (b 1 d 1 f 1 )Y 1 α f ρ 1 α 1 b 3 R 1 α 1 + α 0 a k Y1 k + α 1 a k Y1 k + O( Y 1 5 ) k= ρ 1 = β 0 (f 1 Y 1 f ρ 1 c 3 R 1 ) Ṙ 1 = γ 0 (l 1 Y 1 l 3 R 1 L S 1 ) L S 1 = (f 1 d 1 )Y 1 f ρ 1 c 3 R 1. k= (.10) Consider the matrix M which transfers the matrix A(α 0, β 0, γ 0 ) into its Jordan form J. Then the transformation x = My, x = (Y 1, ρ 1, R 1, L S1 ) T, y = (Y, ρ, R, L S ) T takes the model (.10) into the model Ẏ = λ 1 Y + F 1 (Y, ρ, R, L S, α 1 ) ρ = λ ρ + F (Y, ρ, R, L S, α 1 ) Ṙ = λ 3 R + F 3 (Y, ρ, R, L S, α 1 ) L S = λ 4 L S + F 4 (Y, ρ, R, L S, α 1 ), (.11) where ρ = Y, F = F 1, and F 3, F 4 are real (the symbol - means complex conjugate expression in the whole article; for the sake of simplicity we suppose that λ 3, λ 4 are real and not equal). Theorem. There exists a polynomial transformation Y = Y 3 + h 1 (Y 3, ρ 3, α 1 ) ρ = ρ 3 + h (Y 3, ρ 3, α 1 ) R = R 3 + h 3 (Y 3, ρ 3, α 1 ) L S = L S3 + h 4 (Y 3, ρ 3, α 1 ), (.1)

6 34 K. Makovínyiová and R. Zimka where h j (Y 3, ρ 3, α 1 ), j = 1,, 3, 4, are nonlinear polynomials with constant coefficients of the kind h j (Y 3, ρ 3, α 1 ) = ν (m1,m,m3) Y m1 3 ρ m 3 α m3 1, j = 1,, 3, 4, h = h 1, m 1,m,m 3 with the property h j ( α 1 Y 3, α 1 ρ 3, α 1 ) = m 1,m,m 3 ν (m1,m,m3) ( α 1 ) k Y m1 3 ρ m 3, k 4, which transforms the model (.11) into its partial normal form on invariant surface Ẏ 3 = λ 1 Y 3 +δ 1 Y 3 α 1 +δ Y 3 ρ 3 +U (Y 3, ρ 3, R 3, L S3, α 1 )+U (Y 3, ρ 3, R 3, L S3, α 1 ) ρ 3 = λ ρ 3 + δ 1 ρ 3 α 1 + δ Y 3 ρ 3 + U + U Ṙ 3 = λ 3 R 3 + V (Y 3, ρ 3, R 3, L S3, α 1 ) + V (Y 3, ρ 3, R 3, L S3, α 1 ) L S 3 = λ 4 L S3 + W (Y 3, ρ 3, R 3, L S3, α 1 ) + W (Y 3, ρ 3, R 3, L S3, α 1 ), (.13) where U (Y 3, ρ 3, 0, 0, α 1 ) = V (Y 3, ρ 3, 0, 0, α 1 ) = W (Y 3, ρ 3, 0, 0, α 1 ) = 0 and U ( α 1 Y 3, α 1 ρ 3, α 1 R 3, α 1 L S3,α 1 )=V ( α 1 Y 3, α 1 ρ 3, α 1 R 3, α 1 L S3,α 1 ) = W ( α 1 Y 3, α 1 ρ 3, α 1 R 3, α 1 L S3, α 1 ) = O( α 1 ) 5. The resonant terms δ 1 and δ in the model (.13) are determined by the formulae δ 1 = F 1 α 1, δ = 1 λ F 1 F F 1 ρ 6λ 1 ρ F + 1 λ 1 F 1 ρ F ρ 1 λ 3 F 1 F 3 ρ + 1 F 1 F 3 (λ 1 λ 3 ) ρ 1 F 1 F 4 1 F 1 F 4 + λ 4 Ls ρ (λ 1 λ 4 ) ρ Ls F 1 ρ, where all derivatives in δ 1 and δ are calculated at E 1 and α 1 = 0. Proof. Differentiating (.1) in the power of (.11) and (.13) we get the equations for the determination of the individual terms of the polynomials h j, j = 1,, 3, 4, and the resonant terms δ 1, δ by standard step by step procedure. As the whole record of this procedure is rather long we are omitting it. The model (.13) takes in polar coordinates Y 3 = re iϕ, ρ 3 = re iϕ the form ṙ = r(ar + bα 1 ) + Ũ (r, ϕ, R 3, L S3, α 1 ) + Ũ (r, ϕ, R 3, L S3, α 1 ) ϕ = ω + cα 1 + dr + 1 r [Φ (r, ϕ, R 3, L S3, α 1 ) + Φ (r, ϕ, R 3, L S3, α 1 )] Ṙ 3 = λ 3 R 3 + Ṽ 1 (r, ϕ, R 3, L S3, α 1 ) + Ṽ 1 (r, ϕ, R 3, L S3, α 1 ) L S 3 = λ 4 L S3 + W 1 (r, ϕ, R 3, L S3, α 1 ) + W 1 (r, ϕ, R 3, L S3, α 1 ), (.14) where a = Re δ, b = Re δ 1.

7 Equadiff-11. On the existence of Hopf bifurcation in an open economy model 35 The behavior of solutions of the model (.14) around its equilibrium for small parameters α 1 depends on the signs of the constants a, b. It is known that to every constant solution of the bifurcation equation ar +bα 1 = 0 a periodic solution of (.14) corresponds (see for example [1]). The following theorem ([4]) gives a sufficient condition for the negativeness of the coefficient b. Theorem 3. Let the assumptions of Theorem 1 be satisfied. Let in addition α 0 > α and the critical value β 0 be such that c 3 β 0 f l 3 > 0. Then the value of the coefficient b is negative. Analyzing the model (.14) and taking into account all transformations which have been done to get (.14) we can formulate on the base of Poincaré-Andronov-Hopf bifurcation theorem (see for example [7]) the following statement. Theorem 4. Let the assumptions of Theorem 1 be satisfied. Let in addition α 0 > α and the critical value β 0 be such that c 3 β 0 f l 3 > 0. Then: 1. If a > 0 then the equilibrium E 1 of the model (.3) is unstable also at the critical triple (α 0, β 0, γ 0 ) and to every α > α 0 there exists an unstable limit cycle.. If a < 0 then the equilibrium E 1 of the model (.3) is asymptotically stable also at the critical triple (α 0, β 0, γ 0 ) and to every α < α < α 0 there exists a stable limit cycle..1. Numerical example. Take the functions I, S, T, C, D, F, L from the model (1.1) in the form I = Y 0.3R, S = (Y D ) + 0.3R, Y D = 0.7Y + 0., T = Y, C = R, D =.7 0.3Y, F = Y ρ, L = + 0.8Y 0.R. Then (1.1) has the following form [ Ẏ = α 0.6 ] Y 0.01(0.7Y + 0.) 0.35Y ρ 0.6R ρ = β ( 0.05Y ρ + 0.R 0.576) Ṙ = γ (0.8Y 0.R L S + ) L S = 0.5Y 0.03ρ 0.R (.15) The equilibrium of (.15) is E = (Y = 9, ρ = , R = , L S = ), where the values of Y and L S are taken in 10 milliards. After the transformation of E into the origin and Taylor expansion the model (.15) has the form Ẏ 1 = α( 0.341Y ρ 1 0.6R 1 ) [ + α Y Y ] 9331 Y O( Y 1 5 ) ρ 1 = β (0.05Y ρ 1 0.R 1 ) Ṙ 1 = γ (0.8Y 1 0.R 1 L S 1 ) L S 1 = 0.5Y ρ 1 0.R 1. (.16)

8 36 K. Makovínyiová and R. Zimka The matrix of linear approximation of this model is A(α, β, γ) = 0.341α 0.03α 0.6α β 0.03β 0.β 0 0.8γ 0 0.γ γ = The eigenval- Take β 0 = 1, γ 0 = 1. Then α 0 = 5( ) ues of the matrix A(α 0, β 0, γ 0 ) are. λ 1. = i, λ. = i, λ3. = , λ4. = The formulae for the calculation of the resonant constants δ 1 and δ, which are introduced in Theorem, give the values The bifurcation coefficients are δ 1. = i, δ. = i. a = Re δ. = 0.000, b = Re δ1. = , and the partial normal form of the model (.16) on invariant surface for critical triple (α 0, β 0, γ 0 ) = (0.6774, 1, 1) in polar coordinates is ṙ = r(0.000r α 1 ) + Ũ (r, ϕ, R 3, L S3, α 1 ) + Ũ (r, ϕ, R 3, L S3, α 1 ) ϕ = α r + 1 r [Φ + Φ ] Ṙ 3 = R 3 + Ṽ 1 (r, ϕ, R 3, L S3, α 1 ) + Ṽ 1 (r, ϕ, R 3, L S3, α 1 ) L S 3 = 0.086L S3 + W 1 (r, ϕ, R 3, L S3, α 1 ) + W 1 (r, ϕ, R 3, L S3, α 1 ). The value of α from (.8) is α. = On the base of the values a and b we get. the following result: To every value of parameter α > α 0 = the equilibrium of the model (.15) is asymptotically stable and there exists an unstable limit cycle. To every value of parameter α (0.3648, ) this equilibrium is unstable. REFERENCES [1] Yu. N. Bibikov, Multi-frequency non-linear oscillations and their bifurcations, The Publishing House of the Saint Petersburg University, Saint Petersburg, 1991, (in Russian). [] P. Fellnerová and R. Zimka, On Stability and Fluctuations in Open Economy Model under Flexible Exchange Rate Regime, Prace Naukowe Akademii Ekonomicznej we Wroclawiu, 1036 (004), [3] W. M. Liu, Criterion of Hopf Bifurcations without using Eigenvalues, Journal of Mathematical Analysis and Applications, 18 (1994), [4] K. Makovínyiová and R. Zimka, On Fluctuations in an Open Economy Model, Proceeding of the 7 th International conference Applications of Mathematics and Statistics in Economy, University of Economics Prague, Faculty of Informatics and Statistics, Professional Publishing, Prague (004), [5] G. J. Schinasi, Fluctuations in a Dynamic, Intermediate-Run IS-LM Model: Applications of the Poincaré-Bendixon Theorem, Journal of Economics Theory, 8 (198), [6] L. Striežovská and R. Zimka, Cycles in an Open Economy Model under Fixed Exchange Rate Regime with Pure Money Financing, Operations Research and Decisions, Quarterly 3 4, Wroclaw (00), [7] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, Berlin, [8] E. Zimková and R. Zimka, The Stability of an Open Economy Model, Mundus Symbolicus 10, Economic University, Prague (00),