Corollary 3: Besides stability (stable node or stable focus), as identified through
5. Final remarks
t ≈
k and ~ 0.2
−
t ≈
k ,
respectively.
In both cases, the average of the time series is below zero, which is the steady state result in the absence of fluctuations. Thus, one may infer that the cycles are the result of an inefficient response of individuals to the economic conditions they perceive in each moment. The proposed model generates inefficient endogenous business cycles over the benchmark growth models, which do not take in consideration any ability of consumers to respond to the economic behaviour they witness.
5. Final remarks
In this paper, we have asked a simple question: what if agents react to last periods’
economic performance when making consumption decisions today? After all, economists unanimously consider that economic performance is directly attached to consumer confidence. If households are optimistic about the ability of the economy to grow above normal standards, they will be more willing to consume, and therefore the marginal propensity to consume out of income rises. If consumers are pessimistic, the opposite effect should be observed. Thus, we contrast the standard growth model under which agents believe that output converges in the long term to its potential level and act according to such belief, with a framework where agents are essentially myopic in the
sense they react to observable economic results rather than to a far in time benchmark that they do not have any guarantee to hold.
The proposed setup leads to endogenous business cycles that are triggered by households’ behaviour; if agents did not respond to cycles, these would just fade out.
Taking this argument seriously means that countercyclical economic policy may be adequate to attenuate the persistence of economic fluctuations; if agents do not perceive that the economy departs from the potential output level, then they will adjust savings rates obeying only to requisites of optimality under a frictionless scenario. Another major conclusion is that cycles are inefficient, in the sense that the economy will, on average, be worse off with than without them (our numerical examples point in this direction). Cycles penalize long term welfare, and therefore private agents should be educated not to give a strong response to short run fluctuations with regard to their decisions on the allocation of income between consumption and savings.
Appendix
Proof of proposition 1: Section 2 has defined the steady state as the long run locus that satisfies condition k ≡kˆt+ =kˆt =kˆt− =...=kˆt−n
1
1 . Applying this condition to equation (2), one gets
[ ] functional form of the production function
Proof of proposition 2: The proposition states a trivial definition of stability;
thus, this prove only involves clarifying that the presented characteristic polynomial )
(λ
PJ is in fact the polynomial under which condition PJ(λ)=0 allows for the determination of the n+1 eigenvalues of matrix J (λj, j=1,…,n+1).
A characteristic polynomial is generically defined as ) Dividing all terms by the determinant, we continue to have a polynomial that allows for finding exactly the same n+1 solutions; thus, we define
) 1
Replacing the trace, the sums of the principal minors and the determinant that we have computed for our linearized system in the above polynomial, we find the expression in the proposition
Proof of proposition 3: Given the values of the trace and determinant of J, one verifies that 1+Tr(J)+Det(J)>0 and 1-Tr(J)+Det(J)<0, independently of parameter values. Relatively to the condition 1-Det(J)=0, that defines a Neimark-Sacker bifurcation, this is equivalent to the expression in item a) of the proposition. Local stability and local instability conditions are obtained by solving, respectively, Det(J)<1 and Det(J)>1
Proof of proposition 4: For n=2, the trace of the Jacobian matrix is equal to 1 (and, thus, the three eigenvalues associated with the Jacobian matrix cannot be all negative values); the determinant is negative (what implies that an odd number of negative eigenvalues exists). The two previous conditions imply with certainty that
0
Considering simultaneously the three above conditions implies that all the eigenvalues will lie inside the unit circle, that is, we guarantee stability. The three conditions may be rewritten as
(a) 1+Tr(J)+∑M×2(J)+Det(J)>0; (b) 1−Tr(J)+∑M×2(J)−Det(J)>0;
(c) 1−∑M×2(J)+Det(J)⋅
[
Tr(J)+Det(J)]
>0.Conditions (a) and (b) correspond to relations that hold for any admissible combination of parameter values, while condition (c) requires the inequality
0
inequality, one gets:
inequality in proposition 4.
A Neimark-Sacker bifurcation will occur when the following equality is observed:
[
( ) ( )]
0 equivalent to the expression presented in proposition 4 respecting the bifurcation point.Finally, if the system is not stable and we are not over a bifurcation point, then the system will be locally unstable, as referred through point c) of proposition 4
Proof of proposition 5: In the case n=3, one has realized that two of the eigenvalues of matrix J are positive while the other two are negative. This observation allows to compile a set of inequalities that if jointly satisfied imply that local stability holds. These conditions are the following:
i) (1+λ1)⋅(1+λ2)⋅(1+λ3)⋅(1+λ4)>0; first two conditions (i and ii).
The stability conditions may be transformed, after some calculus (that is particularly tedious for the third condition), into the following:
i) 1+Tr(J)+∑M×2(J)+Det(J)>0;
ii) 1−Tr(J)+∑M×2(J)−Det(J)>0;
The first two conditions hold under the system’s assumptions; condition iv) is true
for (1 )2
These are the conditions for stability; instability prevails if one of them is not fulfilled.
A Neimark-Sacker bifurcation requires once again the product of eigenvalues to be equal to one, that is, λ1⋅λ2 =1 or λ3⋅λ4 =1 (or both)
Proof of proposition 6: It is straightforward to verify that the balanced growth path is unique and that it corresponds to the capital value in the proposition. For such, just solve the following equation in order to k ,
any equivalent relation. The notion of stability requires those eigenvalues to be above -1 and below 1. Thus, since the expression in the proposition corresponds to referred expression
Proof of proposition 8: Just note that the stability condition 1+Tr(J)+Det(J)>0 is always guaranteed; the left inequality in the proposition
θ − + −δ <ψ a
A
1 holds under 1-Det(J)>0 and the right inequality in the proposition
⋅
< + θ
ψ a
a
1 is satisfied for 1-Tr(J)+Det(J)>0, given the values of the trace and the determinant of the matrix in linearized system (8)
References
Alligood, K. T.; T. D. Sauer and J. A. Yorke (1997). Chaos – an Introduction to Dynamical Systems. New York: Springer-Verlag.
Aloi, M.; H. D. Dixon and T. Lloyd-Braga (2000). “Endogenous Fluctuations in an Open Economy with Increasing Returns to Scale”, Journal of Economic Dynamics and Control, vol. 24, pp. 97-125.
Asada, T. and W. Semmler (1995). “Growth and Finance: an Intertemporal Model.”
Journal of Macroeconomics, vol. 17, pp. 623-649.
Benhabib, J. and R. H. Day (1981). “Rational Choice and Erratic Behaviour.” Review of Economic Studies, vol. 48, pp. 459-471.
Boldrin, M. and L. Montrucchio (1986). “On the Indeterminacy of Capital Accumulation Paths.”, Journal of Economic Theory, vol. 40, pp. 26-39.
Bram, J. and S. Ludvigson (1998). “Does Consumer Confidence Forecast Household Expenditure? A Sentiment Index Horse Race.” Federal Reserve Bank of New York Economic Policy Review, vol. 4, pp. 59-79.
Brock, W. A. and C. H. Hommes (1997). “A Rational Route to Randomness.”
Econometrica, vol. 65, pp.1059-1095.
Brock, W. A. and C. H. Hommes (1998). “Heterogeneous Beliefs and Routes to Chaos in a Simple Asset Pricing Model.” Journal of Economic Dynamics and Control, vol. 22, pp. 1235-1274.
Cazavillan, G.; T. Lloyd-Braga and P. Pintus (1998). “Multiple Steady-States and Endogenous Fluctuations with Increasing Returns to Scale in Production.” Journal of Economic Theory, vol. 80, pp. 60-107.
Cazavillan, G. and P. Pintus (2004). “Robustness of Multiple Equilibria in OLG Economies.” Review of Economic Dynamics, vol. 7, pp. 456-475.
Cellarier, L. (2006). “Constant Gain Learning and Business Cycles.” Journal of Macroeconomics, vol. 28, pp. 51-85.
Chiarella, C.; R. Dieci and L. Gardini (2002). “Speculative Behaviour and Complex Asset Price Dynamics: a Global Analysis.” Journal of Economic Behaviour and Organization, vol. 49, pp. 173-197.
Chiarella, C. and X.-Z. He (2003). “Heterogeneous Beliefs, Risk and Learning in a Simple Asset Pricing Model with a Market Maker.” Macroeconomic Dynamics, vol. 7, pp. 503-536.
Christiano, L. and S. Harrison (1999). “Chaos, Sunspots and Automatic Stabilizers.”
Journal of Monetary Economics, vol. 44, pp. 3-31.
Croushore, D. (2005). “Do Consumer-Confidence Indexes Help Forecast Consumer Spending in Real Time?” The North American Journal of Economics and Finance, vol. 16, pp. 435-450.
Coury, T. and Y. Wen (2005). “Global Indeterminacy and Chaos in Standard RBC Models.” University of Oxford and Cornell University working paper.
Day, R. H. (1982). “Irregular Growth Cycles.” American Economic Review, vol. 72, pp.406-414.
De Grauwe, P. and M. Grimaldi (2005). “Heterogeneity of Agents, Transactions Costs and the Exchange Rate.” Journal of Economic Dynamics and Control, vol. 29, pp.
691-719.
Deneckere, R. and S. Pelikan (1986). “Competitive Chaos.” Journal of Economic Theory, vol. 40, pp. 13-25.
Dion, D. P. (2006). “Does Consumer Confidence Forecast Household Spending? The Euro Area Case.” MPRA Paper, University Library of Munich, Germany.
Dosi, G.; G. Fagiolo and A. Roventini (2006). “An Evolutionary Model of Endogenous Business Cycles.” Computational Economics, vol. 27, pp. 3-34.
Elaydi, S. (2000). Discrete Chaos. Boca Raton, FL: Chapman & Hall / CRC Press.
Gaunersdorfer, A. (2000). “Endogenous Fluctuations in a Simple Asset Pricing Model with Heterogeneous Agents.” Journal of Economic Dynamics and Control, vol.
24, pp. 799-831.
Goenka, A. and O. Poulsen (2004). “Factor Intensity Reversal and Ergodic Chaos.”
Working paper 04-13, Aarhus School of Business, Department of Economics.
Goeree, J. K. and C. H. Hommes (2000). “Heterogeneous Beliefs and the Non-linear Cobweb Model. Complex Nonlinear Dynamics and Computational Methods.”
Journal of Economic Dynamics and Control, vol. 24, pp. 761-798.
Goh, K. L. (2003). “Does Consumer Confidence Forecast Consumption Expenditure in New Zealand?” Treasury Working Paper Series 03/22, New Zealand Treasury.
Grandmont, J. M. (1985). “On Endogenous Competitive Business Cycles.”
Econometrica, vol. 53, pp. 995-1045.
Guo, J. T. and K. J. Lansing (2002). “Fiscal Policy, Increasing Returns and Endogenous Fluctuations.” Macroeconomic Dynamics, vol. 6, pp. 633-664.
Hommes, C. H.; J. Sonnemans; J. Tuinstra and H. van de Velden (2005). “A Strategy Experiment in Dynamic Asset Pricing.” Journal of Economic Dynamics and Control, vol. 29, pp. 823-843.
Huang, W. (2005). “On the Statistical Dynamics of Economics.” Journal of Economic Behavior and Organization, vol. 56, pp. 543-565.
Huang, W. and R. H. Day (2001). “On the Statistical Properties of Ergodic Economic Systems.” Discrete Dynamics in Nature and Society, vol. 6, pp. 181-189.
Li, T. and J. Yorke (1975). “Period Three Implies Chaos.” American Mathematical Monthly, vol. 82, pp. 985-992.
Lloyd-Braga, T.; C. Nourry and A. Venditti (2006). “Indeterminacy in Dynamic Models: When Diamond Meets Ramsey.” Journal of Economic Theory, forthcoming.
Lorenz, H.-W. (1997). Nonlinear Dynamical Economics and Chaotic Motion, 2nd edition, Berlin and New York: Springer-Verlag.
Lux, T. and M. Marchesi (2000). “Volatility Clustering in Financial Markets: a Micro-Simulation of Interacting Agents.” International Journal of Theoretical and Applied Finance, vol. 3, pp. 675-702.
Manfredi, P. and L. Fanti (2004). “Cycles in Dynamic Economic Modelling.” Economic Modelling, vol. 21, pp. 573-594.
McNabb, B. and K. Taylor (2002). “Business Cycles and the Role of Confidence:
Evidence from Europe.” Discussion Papers in Economics nº 02/3, University of Leicester.
Medio, A. and M. Lines (2001). Nonlinear Dynamics: a Primer. Cambridge, UK:
Cambridge University Press.
Mitra, T.; K. Nishimura and G. Sorger (2005). “Optimal Cycles and Chaos.” Cornell University, Kyoto University and University of Vienna working paper.
Nishimura, K.; G. Sorger and M. Yano (1994). “Ergodic Chaos in Optimal Growth Models with Low Discount Rates.” Economic Theory, vol. 4, pp. 705-717.
Nishimura, K. and M. Yano (1994). “Optimal Chaos, Nonlinearity and Feasibility Conditions.” Economic Theory, vol. 4, pp. 689-704.
Nishimura, K. and M. Yano (1995). “Nonlinear Dynamics and Chaos in Optimal Growth: an Example.” Econometrica, vol. 63, pp. 981-1001.
Onozaki, T.; G. Sieg and M. Yokoo (2000). “Complex Dynamics in a Cobweb Model with Adaptive Productive Adjustment.” Journal of Economic Behavior and Organization, vol. 41, pp. 101-115.
Onozaki, T.; G. Sieg and M. Yokoo (2003). “Stability, Chaos and Multiple Attractors: a Single Agent Makes a Difference.” Journal of Economic Dynamics and Control, vol. 27, pp. 1917-1938.
Sarkovskii, A. N. (1964). “Coexistence of Cycles of a Continuous Map of a Line Into Itself.” Ukranichkii Matematicheskii Zhurnal, vol. 16, pp. 61-71.
Schmitt-Grohé, S. (2000). “Endogenous Business Cycles and the Dynamics of Output, Hours, and Consumption.” American Economic Review, vol. 90, pp. 1136-1159.
Semmler, W. (1994). Business Cycles: Theory and Empirical Methods. Dordrecht:
Kluwer.
Souleles, N. S. (2004). “Expectations, Heterogeneous Forecast Errors, and Consumption: Micro Evidence from the Michigan Consumer Sentiment Surveys.”
Journal of Money, Credit and Banking, vol. 36, pp. 39-72.
Stutzer, M. J. (1980). “Chaotic Dynamics and Bifurcations in a Macro-Model.” Journal of Economic Dynamics and Control, vol. 2, pp. 353-376.
Westerhoff, F. H. (2004). “Multiasset Market Dynamics.” Macroeconomic Dynamics, vol. 8, pp. 596-616.
Figures
Figure 1 – Local dynamics in the model without optimization (n=1).
Figure 2 – Local dynamics in the model without optimization (n=2).
Det(J)
)
2(J M×
∑
Bifurcation point
+ − +
+
⋅
+ 2
1 2 2 ) 2 1 (
2 µ
µ µ
+ − +
+
− 2
1 2 2
2 µ 2 µ
Stability area
Det(J)=1 Det(J)
Tr(J)
1+Tr(J)+Det(J)=0 1-Tr(J)+Det(J)=0
1
Figure 3 – Bifurcation diagram (A, k~t
) for the model without optimization (n=1).
Figure 4 – Bifurcation diagram (A, k~t
) for the model without optimization.
A closer look, for 0.253<A<0.276 (n=1).
Figure 5 – Lyapunov exponents in the model without optimization (n=1).
Figure 6 – Attractor for the model without optimization (A=0.25; n=1).
Figure 7 – Attractor for the model without optimization (A=0.33; n=1).
Figure 8 – Basin of attraction for the model without optimization (A=0.33; n=1).
The area in black is the basin.
Figure 9 – Time series of k~t
for the model without optimization (A=0.25; n=1).
Figure 10 – Bifurcation diagram (A,k~t
) for the model without optimization (n=4).
Figure 11 – Local dynamics in the model with optimization (n=1).
Figure 12 – Local dynamics in the model with optimization, for a selected array of parameter values and a varying A (n=1).
1 1.049
Det(J)=1 Det(J)
Tr(J) A=0
2.791
A→+∞
1.628
Det(J)=1 Det(J)
Tr(J)
1+Tr(J)+Det(J)=0 1-Tr(J)+Det(J)=0
1
Figure 13 – Bifurcation diagram (A,k~t
) for the model with optimization (n=1).
Figure 14 – Attractor for the model with optimization (A=0.2593; n=1).
Figure 15 – Time series of k~t
for the model with optimization (A=0.2593; n=1).
Figure 16 – Basin of attraction for the model with optimization (A=0.2593; n=1).
The area in black is the basin.