In this paper, we discuss the relationship between three time lags (the consumption lag, the investment decision lag and the gestation lag) and oscillatory economic fluctuations in the Keynesian IS-LM system. We first confirm that in the absence of time lags, the monotone convergence to the unique equilibrium is observed. Next, we demonstrate that, in the Keynesian IS-LM system, in the existence of the investment decision lag and the gestation lag, oscillatory fluctuations are generated around the equilibrium if the gestation lag is relatively long, while in the existence of the consumption lag and the gestation lag, oscillatory behaviors may not occur. We then conclude that the existence of time lags may be one of the major causes of oscillatory fluctuations.
Allen, R. G. D. 1959. Mathematical Economics. 2nd ed. London: Macmillan.
Andronov, A. A., A. A. Vitt, and S. E. Khaikin. 1966. Theory of Oscillators. Oxford: Pergamon Press.
Asada, T. 1987. “Government Finance and Wealth Effect in a Kaldorian Cycle Model.” Journal of Economics 47 (2): 143–166.
Asada, T. 1997. Macrodynamics of Growth and Cycles. Tokyo (in Japanese): Nihon Keizai Hyouronsha.
Asada, T., and H. Yoshida. 2001. “Stability, Instability and Complex Behavior in Macrodynamic Models with Policy Lags.” Discrete Dynamics in Nature and Society 5: 281–295.
Asea, P. K., and P. J. Zak. 1999. “Time-to-build and Cycles.” Journal of Economic Dynamics and Control 23: 1155–1175.
Bellman, R., and K. L. Cooke. 1963. Differential-Difference Equations. New York: Academic Press.
Blinder, A. S., and R. M. Solow. 1973. “Does Fiscal Policy Matter?” Journal of Public Economics 2: 319–337.
Brock, W. A. 1991. “Hicksian Non-Linearity.” In Value and Capital: Fifty Years Later, edited by L. W. McKenzie, and S. Zamagni, 310–330. London: Macmillan.
Chang, W. W., and D. J. Smyth. 1971. “The Existence and Persistence of Cycles in a Non-linear Model: Kaldor’s 1940 Model Re-examined.” Review of Economic Studies 38 (1): 37–44.
Chang, W. W., and D. J. Smyth. 1972. “Stability and Instability of IS-LM Equilibrium.” Oxford Economic Papers 24 (3): 372–384.
Coddington, E. A., and N. Levinson. 1955. Theory of Ordinary Differential Equations. New York: McGraw Hill.
Frisch, R., and H. Holme. 1935. “The Characteristic Solutions of a Mixed Difference and Differential Equations Occurring in Economic Dynamics.” Econometrica 3 (2): 225–239.
Gallegati, M., L. Gardini, T. Puu, and I. Sushko. 2003. “Hicks’ Trade Cycle Revisited: Cycles and Bifurcations.” Mathematics and Computation in Simulations 63: 505–527.
Goodwin, R. M. 1951. “The Non-linear Accelerator and the Persistence of Business Cycles.” Econometrica 19 (1): 1–17.
Grammatikopoulos, M. K., E. A. Grove, and G. Ladas. 1986. “Oscillations of First-order Neutral Delay Differential Equations.” Journal of Mathematical Analysis and Applications 120: 510–520.
Györi, I., and G. Ladas. 1991. Oscillation Theory of Delay Differential Equations: With Applications. Oxford: Oxford University Press.
Hale, J. 1976. Theory of Functional Delay Differential Equations. New York: Springer-Verlag.
Hansen, A. H. 1949. Monetary Theory and Fiscal Policy. New York: McGraw-Hill.
Harrod, R. F. 1937. “Mr. Keynes and Traditional Theory.” Econometrica 5 (1): 74–86.
Hicks, J. R. 1937. “Mr. Keynes and the ‘Classics’: A Suggested Interpretation.” Econometrica 5 (2): 147–159.
Hicks, J. R. 1950. A Contribution to the Theory of the Trade Cycle. Oxford: Clarendon Press.
Hirsch, M. W., and S. Smale. 1974. Differential Equations, Dynamical Systems, and Linear Algebra. San Diego, California: Academic Press.
Hommes, C. 1993. “Periodic, Almost Periodic and Chaotic Behavior in Hicks’ Non-linear Trade Cycle Model.” Economics Letters 41: 391–397.
Hommes, C. 1995. “A Reconsideration of Hicks’ Non-linear Trade Cycle Model.” Structural Change and Economic Dynamics 6: 435–459.
James, R. W., and M. H. Belz. 1938. “The Significance of the Characteristic Solutions of Mixed Difference and Differential Equations.” Econometrica 6 (4): 326–343.
Kaldor, N. 1940. “A Model of the Trade Cycle.” Economic Journal 50 (197): 78–92.
Kalecki, M. 1935. “A Macrodynamic Theory of Business Cycles.” Econometrica 3 (3): 327–344.
Kalecki, M. 1937. “A Theory of the Business Cycle.” Review of Economic Studies 4 (2): 77–97.
Keynes, J. M. 1936. The General Theory of Employment, Interest and Money. London: Macmillan.
Klein, L. R. 1947. The Keynesian Revolution. New York: Macmillan.
Ladas, G., and Y. G. Sficas. 1986. “Oscillations of Neutral Delay Differential Equations.” Canadian Mathematical Bulletin 29: 438–445.
Lange, O. 1938. “The Rate of Interest and the Optimum Propensity to Consume.” Economica 5 (17): 12–32.
Lin, X., and H. Wang. 2012. “Stability Analysis of Delay Differential Equations with Two Discrete Delays.” Canadian Applied Mathematical Quarterly 20 (4): 519–533.
Lorenz, H. W. 1987. “Goodwin’s Nonlinear Accelerator and Chaotic Motion.” Journal of Economics 47 (4): 413–418.
Marglin, S. A., and A. Bhaduri. 1990. “Profit Squeeze and Keynesian Theory.” In The Golden Age of Capitalism: Reinterpreting the Postwar Experience, edited by A. Marglin, and J. B. Schor, 153–186. Oxford: Clarendon Press.
Matsumoto, A. 2009. “Note on Goodwin’s 1951 Nonlinear Accelerator Model with an Investment Delay.” Journal of Economic Dynamics and Control 33: 832–842.
Matsumoto, A., and F. Szidarovszky. 2010. “Continuous Hicksian Trade Cycle Model with Consumption and Investment Time Delays.” Journal of Economic Behavior and Organization 75: 95–114.
Matsumoto, A., and F. Szidarovszky. 2015a. “Nonlinear Multiplier-accelerator Model with Investment and Consumption Delays.” Structural Change and Economic Dynamics 33: 1–9.
Matsumoto, A., and F. Szidarovszky. 2015b. “Nonlinear Cournot Duopoly with Implementation Delays.” Chaos, Solitons and Fractals 79: 157–165.
Meade, J. E. 1937. “A Simplified Model of Mr. Keynes’ System.” Review of Economic Studies 4 (2): 98–107.
Metzler, L. A. 1941. “The Nature and Stability of Inventory Cycles.” Review of Economics and Statistics 23: 113–129.
Modigliani, F. 1944. “Liquidity Preference and the Theory of Interest and Money.” Econometrica 12 (1): 45–88.
Murakami, H. 2014. “Keynesian Systems with Rigidity and Flexibility of Prices and Inflation-deflation Expectations.” Structural Change and Economic Dynamics 30: 68–85.
Murakami, H. 2015. “Wage Flexibility and Economic Stability in a Non-Walrasian Model of Economic Growth.” Structural Change and Economic Dynamics 32: 25–41.
Phillips, A. W. 1954. “Stabilisation Policies in a Closed Economy.” Economic Journal 64 (254): 290–323.
Puu, T. 2007. “The Hicksian Trade Cycle with Floor and Ceiling Dependent on Capital Stock.” Journal of Economic Dynamics and Control 31: 575–592.
Reddaway, W. B. 1936. “The General Theory of Employment, Interest and Money.” Economic Record 12: 28–36.
Robertson, D. H. 1936. “Some Notes on Mr. Keynes’ General Theory of Employment.” Quarterly Journal of Economics 51: 168–191.
Samuelson, P. A. 1939. “Interactions between the Multiplier Analysis and the Principle of Acceleration.” Review of Economics and Statistics 21 (2): 75–78.
Sasakura, K. 1996. “The Business Cycle Model with a Unique Stable Limit Cycle.” Journal of Economic Dynamics and Control 20: 1763–1773.
Semmler, W. 1986. “On Nonlinear Theories of Economic Cycles and the Persistence of Business Cycles.” Mathematical Social Sciences 12: 47–76.
Timlin, M. F. 1942. “Long-run Effects of Fiscal and Monetary Policy on Aggregate Demand.” Keynesian Economics. Toronto: University of Toronto Press.
Tobin, J., and W. Buiter. 1976. “Long-run Effects of Fiscal and Monetary Policy on Aggregate Demand.” In: Monetarism, edited by J. L. Stein, 273–309. North-Holland, New York.
Torre, V. 1977. “Existence of Limit Cycles and Control in Complete Keynesian System by Theory of Bifurcations.” Econometrica 45 (6): 1457–1466.
Tsuzuki, E. 2015. “Dynamic Analysis of Two Policy Lags in a Kaldorian Model.” Discrete Dynamics in Nature and Society. Article ID: 927138.
Varian, H. R. 1979. “Catastrophe Theory and the Business Cycle.” Economic Inquiry 17: 14–28.
Yoshida, H. 2003. Theory of Business Cycles: Non-linear Dynamic Approach. Nagoya (in Japanese): Nagaya University Press.
Yoshida, H., and T. Asada. 2007. “Dynamic Analysis of Policy Lag in a Keynes-Goodwin Model: Stability, Instability, Cycles and Chaos.” Journal of Economic Behavior and Organization 62: 441–469.
Corresponding author: Hiroki Murakami, Research Fellow (PD), Japan Society for the Promotion of Science; Graduate School of Economics, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
SNDE recognizes that advances in statistics and dynamical systems theory can increase our understanding of economic and financial markets. The journal seeks both theoretical and applied papers that characterize and motivate nonlinear phenomena. Researchers are required to assist replication of empirical results by providing copies of data and programs online. Algorithms and rapid communications are also published.