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Stabilizing Chaotic Business Cycles Using a Neural Network
Copyright (i) IFAC Computation in Economics, Finance and Engineering: Economic Systems, Cambridge, UK, 1998
Stabilizing Chaotic Business Cycles Using a Neural Network Taisei Kaizoji International Christian University Osawa Mitaka Tokyo, 181 JAPAN Abstract
The purpose of this paper is to propose a bilt-in stabilizer that is, an automatically feedback control through the fine-tuning of government spending and money supply, can stabilize chaotic endogenous business cycles, which would emerge from a standard dynamic IS-LM model, on a unstable economic equilibrium. The nural network controller developed by Konish and Kokame (1995) is used as the bilt-in stabilizer. The bilt-in stabilizer is identified by using a neural network trained by a Back Propagation algorithm. The identification requires the observed economic data but does not require tl}e location of the unstable economic equilibrium point and the local dynamics at the point. Copyright ~ 1998 IFAC Keywords: chaotic business cycles, a bilt-in stabilizer, a neural network.
1. INTRODUCTION
2. A DYNAMIC KEYSIAN MODEL
The time series of many macroeconomic variables have the appearance of irregular fluctuations. The traditional explanation of this is that an essentially stationary economy is subject to random shocks. Important examples of this approach can be found in the works of Lucas and Sargent (1981). They argues that when the economy is subject to a sequence of random shocks, it will behave in away wich resembles real-life business cycles. An altanative way of treating the irregular fluctuations would be to use a model involving non-linear difference (or differential) equations. IT a model of this sort would exhibit chaotic or cyclic behavior, then it could provide an exlanation of the irregular business cycles l . Recent work of this approach are by Day and Shafer (1985), who show how chaotic behavior can emerge in the standard fix price macroeconomic model when induced investment is strong enough. Their dynamic model is reduced to a single first-order difference equation on national product. The chaotic business cycles that emerges in their model are caused by non-linearities of demand for money and investment. The purpose of this paper is to propose a bilt-in stabilizer that is, an automatically feedback control through the fine-tuning of government spending and money supply, can stabilize chaotic endogenous business cycles, which would emerge from a standard dynamic IS-LM model, on a unstable economic equilibrium. The nural network controller developed by Konish and Kokame (1995) is used as the bilt-in stabilizer. The bilt-in stabilizer is identified by using a neural network trained by a Back Propagation algorithm. The identification requires the observed economic data but does not require the location of the unstable economic equilibrium point and the local dynamics at the point.
The model is an discrete-time disequilibrium Keynesian (IS-LM) model, which has often been studied as one of the basic models of endogenous business cycles, is formulated as follows :
=oIFI(Y(t),R(t),G(t)), R(t + 1) - R(t) =02F2(Y(t), R(t), M(t)), Y(t +1) - Y(t)
FI (Y(t), R(t), G(t)) = C(Y(t)) + J(Y(t), R(t)) + G(t) - Y(t) , (3) F2 (Y(t), R(t), M(t))
= L(Y(t), R(t)) -
M(t)
(4)
C(.), J(.), and L(.) are the consumption function , the investment function, and the money demand function. All functions are assumed to be linear in all arguments except the money demand function which is assumed to be log-linear in national product Y(t) and the interest rate R(t). Consumption function : C(Y(t))
1 Useful
J(Y(t), R(t)) 309
(2)
where R(t) denotes the interest rate at time t, Y(t) national product at time t, G(t) government expenditure at time t, and M(t) the nominal money supply at time t. Equations (1) and (2) represent traditional macroeconomic disequilibrium adjustment processes for the commodities and money markets respectively. FI (.) represents excess demand for commodities and services and F2 ( .) represents excess demand for money. 01 and 02 indicate the adjustment speed. In fixed price regimes, each equation will be parametrized by the fixed price level (fixed at unity). We define FI (.) and F2(') as follows:
=0.8 +0.6Y(t)
Investment function : surveys of the DOD-linear economic dynamics literature can be found in Gabisch and Lorenz (1987).
(1)
=0.2 +0.2Y(t) - O.1R(t)
Money demand function : L(Y(t), R(t)) = exp(5 - R(t) + 0.2Y(t))
Adjustment speed: al =0.5 and a2 =0.7. Assume that the policy makers keeps the government spending and money supply constant until the next section which regards stabilization policies. . G(t) and M(t) are parameterized by the following fixed values
G = 0.5 and M =4,
where UI (t) denotes the discretionary fiscaJ policy and U2(t) the discretionary monetary policy. However, the policy makers can not know the economic structure F(.), so that they must somehow infer a optimal discretionary policy functions, UI (t) and U2(t) from the observed data Y(t) and R( t) . In this paper a neural network is used to identify Ui(t),(i = 1,2). For convinience of analysis let us rewrite the dynamic system (1) and (2) as follows: X(t + 1) = F(X(t)) + AU(t),
(7)
(5)
Under the above specification the economic system described by (1) and (2) has the unique unstable equilibrium (Y', R') where Y' = 5.176 and R' = 4.649, The eigenvalues evaluated at the unstable economic equilibrium (Y',R') are Al = -1.7896 and A2 = 0.889589. Figure 1 shows a time path of national product, Y(t) with the above parameters. The orbit is chaotic appearently. For further analyses of the dynamics see Kaizoji and Chang (1997).
X(t) A =(a.,a2),
= [Y(t),R(tW, U(t)
=[UI(t),U2(t)jT.
The neural network is governed by N
Ui(t) = !(Zi),
Zj = LVijTj , Tj = !(OJ), j=1
OJ = WjIY(t) + Wj2 R(t), !(x) =
y(t) 5.05
2 -1. 1 +exp(-x)
Wj1 is the weight of the connection from Y(t), of the input layer to the j th neuron of the hidden layer, Wj2 the weight of the connection from R( t), of the input layer to the j th neuron of the hidden layer, and Vij the weight of the connection from the j th neuron of the hidden layer to the i th neuron of the output layer. These weights are update to train Ui(t),(i = 1,2) by using Error back propagation algorithm,
5 . 025 5 . 4.915
20
40
60
80
100 120
Time Figure 1. Chaotic business cycles generated from the dynamic Keynsian model.
1
2
= -4 L([1 +!(Zi)][1 - !(Zi)]Vij i=1
3. A BILT-IN STABILIZER
BE x [1 + /(OJ)][l- /(OJ)]Y(t) IJUi(t»'
Figure 1 shows that under the above specification the national products Y(t) fluctuates irregularly below the economic equilibrium level Y' = 5.176 when the policy makers dose not implement any discretionary economic policies. In other words the chaotic business cycles are considered to have negative effect on economic welfare. Hence the policy makers are expected to implement the discrationary economic policies in order to stabilize the chaotic business cycles to the economic equilibrium (Y', R·). The policy makers have two discretionary policy instruments, that is, a fiscal policy and a monetary policy. These discretionary economic policies are defined as follows: G(t) =G+ UI (t) and M(t)
= M + U2(t) .
1
2
= -4 L([1 + /(zi)][l- /(Zi)]Vij i=1
IJE
x [1 + /(OJ)][l- /(OJ)]R(t) IJUi(t)}'
where E is the error function and 11 is the learning rate. Repeating the above updating, the error function E decreases to the minimum value. As it is demonstrated below, Ui (t) ,(i = 1, 2) traind by the above Error back propagation algotiyjm are able to stabilize the chaotic business cycles automatically. Hence Ui(t) is considered as a bilt-in stabilizer. Ui(t) which is identified by the traind neural network is called the bilt~ in stabiIizer below.
(6)
310
The policy makers stabilize the chaotic business cycles. At the same time they try to minimize changes of the government spending and money supply. Therefore the policy makers minimize the following two error functions in order to find optimal bilt-in stbilizers. One of them is an error function Ec corresponding to the distance between X(t + 1) and X(t) . 1
Ec = 2[X(t + 1) - X(tW[X(t + 1) - X(t)].
neurons of the input layer, five neurons of the hidden layer (N = 5), two neurons of the output layer. The learning rate is IJ = 1 and the error weights are kc = 1 and ku =0.9.
y(t)
(8)
Another error function is Eu corresponding to the Euclidean norm of the changes of economic policies AU(t),
Eu
= ~[AU(tW[AU(t)].
(9)
The government makes the stabilization policies Ul (t) and U2 (t) so as to minimizes the error E composed of Ec and Eu, (1O) E = kcEc + kuEu, where kc and ku are the error weights for Ec and Eu, respectively. The differential coefficient aE/aU(t) in Eq. (9) is given by
aE aU(t)
=
aEc aE kc aU(t) + ku aU(t)
= kcAT[F(X(t)) - X(t)] + (kc + ku)AT AU(t),
(11)
Each time the weights are updated using the error back propagation algorithm. After the bilt-in stabilizers is trained enough for the error E to be a minimum value, the bilt-in stabilizer U(t) can be described by
U(t)
= - 1 + k1u/kc A-l[F(X(t)) -
X(t)].
(12)
The derivation of Eq. (11) is demonstrated by Konishi and Kokame (1995). The dynamics of the economic system (7) at the economic equilibrium point, X· is approximately governed by W+ 1) =H~(t) + AU(t), (13)
60
80
100
120
Time Figure 2. Stabilization of chaotic busines cycles on the unstable economic equilibrium. 4. CONCLUDING REMARKS A bilt-in stabilizer is proposed for stabilizing chaotic business cycles. The bilt-in stabilizer is identified by using the observed economic data. The identification does not require the information of the dynamically economic system such as the location of the unstable economic equilibrium and the local dynamics at the point. Therefore the policy makers implements the stabilization policies in order to stabilize chaotic business cycles using the obseved data. It remains to be seen how the bilt-in stabilizer can be extended to more general and perusable frameworks including the labor market and the international trade.
[1] Day, Richard H. and Wayne Shafer, (1985), Keynesian Chaos, Journal 0/ Macroeconomics, 7, 3, 277-295. [2] Gabisch, G., and H. W. Lorenz, (1987) Business Cyclys Theory, New York: Springer-Verlag. [3] T. Kaizoji and C-S Chang (1997), "Chaotic Business cycles and the Stabilization Policy in a Dynamic Keynesian Model" , the Journal 0/ Social Science, Vol.35, pp.41- 56. [4] K. Konishii and H. Kokame (1995), "Stabilizing and tracking unstable focus points in chaotic systems using a neural netwoek", Physics Letters A 206,203-210. [5] Lucus, R. E., Jr. and T. J. Sargent (eds.) (1981) Rational Expectations and Econometric Practice, Minneapolis: University of Minnesota Press.
((=ku/kc), (14)
where I is the 2 x 2 unit matrix. Substituting Eq. (13) in Eq. (12), the linearized dynamics become
W + 1) = 1: ( ((H + I)~(t).
40
5. REFERENCE
where W) = X(t) -X·. The bilt-in stabilizer U(t) which is identified by the trained neural network can be also approximately given by
U(t)=-I:(A-l(H-I)~(t),
20
(15)
Therefore the economic equilibrium becomes stable under the appropreate rate of the error weights ku / kc· Figure 2 shows a stabilized chaotic orbit of Y(t) by the bilt-in stabilizer. The neural network consists of two 311
Stabilizing Chaotic Business Cycles Using a Neural Network Taisei Kaizoji International Christian University Osawa Mitaka Tokyo, 181 JAPAN Abstract
The purpose of this paper is to propose a bilt-in stabilizer that is, an automatically feedback control through the fine-tuning of government spending and money supply, can stabilize chaotic endogenous business cycles, which would emerge from a standard dynamic IS-LM model, on a unstable economic equilibrium. The nural network controller developed by Konish and Kokame (1995) is used as the bilt-in stabilizer. The bilt-in stabilizer is identified by using a neural network trained by a Back Propagation algorithm. The identification requires the observed economic data but does not require tl}e location of the unstable economic equilibrium point and the local dynamics at the point. Copyright ~ 1998 IFAC Keywords: chaotic business cycles, a bilt-in stabilizer, a neural network.
1. INTRODUCTION
2. A DYNAMIC KEYSIAN MODEL
The time series of many macroeconomic variables have the appearance of irregular fluctuations. The traditional explanation of this is that an essentially stationary economy is subject to random shocks. Important examples of this approach can be found in the works of Lucas and Sargent (1981). They argues that when the economy is subject to a sequence of random shocks, it will behave in away wich resembles real-life business cycles. An altanative way of treating the irregular fluctuations would be to use a model involving non-linear difference (or differential) equations. IT a model of this sort would exhibit chaotic or cyclic behavior, then it could provide an exlanation of the irregular business cycles l . Recent work of this approach are by Day and Shafer (1985), who show how chaotic behavior can emerge in the standard fix price macroeconomic model when induced investment is strong enough. Their dynamic model is reduced to a single first-order difference equation on national product. The chaotic business cycles that emerges in their model are caused by non-linearities of demand for money and investment. The purpose of this paper is to propose a bilt-in stabilizer that is, an automatically feedback control through the fine-tuning of government spending and money supply, can stabilize chaotic endogenous business cycles, which would emerge from a standard dynamic IS-LM model, on a unstable economic equilibrium. The nural network controller developed by Konish and Kokame (1995) is used as the bilt-in stabilizer. The bilt-in stabilizer is identified by using a neural network trained by a Back Propagation algorithm. The identification requires the observed economic data but does not require the location of the unstable economic equilibrium point and the local dynamics at the point.
The model is an discrete-time disequilibrium Keynesian (IS-LM) model, which has often been studied as one of the basic models of endogenous business cycles, is formulated as follows :
=oIFI(Y(t),R(t),G(t)), R(t + 1) - R(t) =02F2(Y(t), R(t), M(t)), Y(t +1) - Y(t)
FI (Y(t), R(t), G(t)) = C(Y(t)) + J(Y(t), R(t)) + G(t) - Y(t) , (3) F2 (Y(t), R(t), M(t))
= L(Y(t), R(t)) -
M(t)
(4)
C(.), J(.), and L(.) are the consumption function , the investment function, and the money demand function. All functions are assumed to be linear in all arguments except the money demand function which is assumed to be log-linear in national product Y(t) and the interest rate R(t). Consumption function : C(Y(t))
1 Useful
J(Y(t), R(t)) 309
(2)
where R(t) denotes the interest rate at time t, Y(t) national product at time t, G(t) government expenditure at time t, and M(t) the nominal money supply at time t. Equations (1) and (2) represent traditional macroeconomic disequilibrium adjustment processes for the commodities and money markets respectively. FI (.) represents excess demand for commodities and services and F2 ( .) represents excess demand for money. 01 and 02 indicate the adjustment speed. In fixed price regimes, each equation will be parametrized by the fixed price level (fixed at unity). We define FI (.) and F2(') as follows:
=0.8 +0.6Y(t)
Investment function : surveys of the DOD-linear economic dynamics literature can be found in Gabisch and Lorenz (1987).
(1)
=0.2 +0.2Y(t) - O.1R(t)
Money demand function : L(Y(t), R(t)) = exp(5 - R(t) + 0.2Y(t))
Adjustment speed: al =0.5 and a2 =0.7. Assume that the policy makers keeps the government spending and money supply constant until the next section which regards stabilization policies. . G(t) and M(t) are parameterized by the following fixed values
G = 0.5 and M =4,
where UI (t) denotes the discretionary fiscaJ policy and U2(t) the discretionary monetary policy. However, the policy makers can not know the economic structure F(.), so that they must somehow infer a optimal discretionary policy functions, UI (t) and U2(t) from the observed data Y(t) and R( t) . In this paper a neural network is used to identify Ui(t),(i = 1,2). For convinience of analysis let us rewrite the dynamic system (1) and (2) as follows: X(t + 1) = F(X(t)) + AU(t),
(7)
(5)
Under the above specification the economic system described by (1) and (2) has the unique unstable equilibrium (Y', R') where Y' = 5.176 and R' = 4.649, The eigenvalues evaluated at the unstable economic equilibrium (Y',R') are Al = -1.7896 and A2 = 0.889589. Figure 1 shows a time path of national product, Y(t) with the above parameters. The orbit is chaotic appearently. For further analyses of the dynamics see Kaizoji and Chang (1997).
X(t) A =(a.,a2),
= [Y(t),R(tW, U(t)
=[UI(t),U2(t)jT.
The neural network is governed by N
Ui(t) = !(Zi),
Zj = LVijTj , Tj = !(OJ), j=1
OJ = WjIY(t) + Wj2 R(t), !(x) =
y(t) 5.05
2 -1. 1 +exp(-x)
Wj1 is the weight of the connection from Y(t), of the input layer to the j th neuron of the hidden layer, Wj2 the weight of the connection from R( t), of the input layer to the j th neuron of the hidden layer, and Vij the weight of the connection from the j th neuron of the hidden layer to the i th neuron of the output layer. These weights are update to train Ui(t),(i = 1,2) by using Error back propagation algorithm,
5 . 025 5 . 4.915
20
40
60
80
100 120
Time Figure 1. Chaotic business cycles generated from the dynamic Keynsian model.
1
2
= -4 L([1 +!(Zi)][1 - !(Zi)]Vij i=1
3. A BILT-IN STABILIZER
BE x [1 + /(OJ)][l- /(OJ)]Y(t) IJUi(t»'
Figure 1 shows that under the above specification the national products Y(t) fluctuates irregularly below the economic equilibrium level Y' = 5.176 when the policy makers dose not implement any discretionary economic policies. In other words the chaotic business cycles are considered to have negative effect on economic welfare. Hence the policy makers are expected to implement the discrationary economic policies in order to stabilize the chaotic business cycles to the economic equilibrium (Y', R·). The policy makers have two discretionary policy instruments, that is, a fiscal policy and a monetary policy. These discretionary economic policies are defined as follows: G(t) =G+ UI (t) and M(t)
= M + U2(t) .
1
2
= -4 L([1 + /(zi)][l- /(Zi)]Vij i=1
IJE
x [1 + /(OJ)][l- /(OJ)]R(t) IJUi(t)}'
where E is the error function and 11 is the learning rate. Repeating the above updating, the error function E decreases to the minimum value. As it is demonstrated below, Ui (t) ,(i = 1, 2) traind by the above Error back propagation algotiyjm are able to stabilize the chaotic business cycles automatically. Hence Ui(t) is considered as a bilt-in stabilizer. Ui(t) which is identified by the traind neural network is called the bilt~ in stabiIizer below.
(6)
310
The policy makers stabilize the chaotic business cycles. At the same time they try to minimize changes of the government spending and money supply. Therefore the policy makers minimize the following two error functions in order to find optimal bilt-in stbilizers. One of them is an error function Ec corresponding to the distance between X(t + 1) and X(t) . 1
Ec = 2[X(t + 1) - X(tW[X(t + 1) - X(t)].
neurons of the input layer, five neurons of the hidden layer (N = 5), two neurons of the output layer. The learning rate is IJ = 1 and the error weights are kc = 1 and ku =0.9.
y(t)
(8)
Another error function is Eu corresponding to the Euclidean norm of the changes of economic policies AU(t),
Eu
= ~[AU(tW[AU(t)].
(9)
The government makes the stabilization policies Ul (t) and U2 (t) so as to minimizes the error E composed of Ec and Eu, (1O) E = kcEc + kuEu, where kc and ku are the error weights for Ec and Eu, respectively. The differential coefficient aE/aU(t) in Eq. (9) is given by
aE aU(t)
=
aEc aE kc aU(t) + ku aU(t)
= kcAT[F(X(t)) - X(t)] + (kc + ku)AT AU(t),
(11)
Each time the weights are updated using the error back propagation algorithm. After the bilt-in stabilizers is trained enough for the error E to be a minimum value, the bilt-in stabilizer U(t) can be described by
U(t)
= - 1 + k1u/kc A-l[F(X(t)) -
X(t)].
(12)
The derivation of Eq. (11) is demonstrated by Konishi and Kokame (1995). The dynamics of the economic system (7) at the economic equilibrium point, X· is approximately governed by W+ 1) =H~(t) + AU(t), (13)
60
80
100
120
Time Figure 2. Stabilization of chaotic busines cycles on the unstable economic equilibrium. 4. CONCLUDING REMARKS A bilt-in stabilizer is proposed for stabilizing chaotic business cycles. The bilt-in stabilizer is identified by using the observed economic data. The identification does not require the information of the dynamically economic system such as the location of the unstable economic equilibrium and the local dynamics at the point. Therefore the policy makers implements the stabilization policies in order to stabilize chaotic business cycles using the obseved data. It remains to be seen how the bilt-in stabilizer can be extended to more general and perusable frameworks including the labor market and the international trade.
[1] Day, Richard H. and Wayne Shafer, (1985), Keynesian Chaos, Journal 0/ Macroeconomics, 7, 3, 277-295. [2] Gabisch, G., and H. W. Lorenz, (1987) Business Cyclys Theory, New York: Springer-Verlag. [3] T. Kaizoji and C-S Chang (1997), "Chaotic Business cycles and the Stabilization Policy in a Dynamic Keynesian Model" , the Journal 0/ Social Science, Vol.35, pp.41- 56. [4] K. Konishii and H. Kokame (1995), "Stabilizing and tracking unstable focus points in chaotic systems using a neural netwoek", Physics Letters A 206,203-210. [5] Lucus, R. E., Jr. and T. J. Sargent (eds.) (1981) Rational Expectations and Econometric Practice, Minneapolis: University of Minnesota Press.
((=ku/kc), (14)
where I is the 2 x 2 unit matrix. Substituting Eq. (13) in Eq. (12), the linearized dynamics become
W + 1) = 1: ( ((H + I)~(t).
40
5. REFERENCE
where W) = X(t) -X·. The bilt-in stabilizer U(t) which is identified by the trained neural network can be also approximately given by
U(t)=-I:(A-l(H-I)~(t),
20
(15)
Therefore the economic equilibrium becomes stable under the appropreate rate of the error weights ku / kc· Figure 2 shows a stabilized chaotic orbit of Y(t) by the bilt-in stabilizer. The neural network consists of two 311