## Abstract

In the following model we propose a simple change to the nonlinear Hicksian trade cycle model of 1950 through just internalizing capital stock. This brings no alien elements into the model, it just makes explicit what is there already, i.e., investment, considering that capital is the cumulative sum of successive investments. This makes it possible to tie the “floor” disinvestment to capital stock through its depreciation rate. The consequence is that one can dispense with the exogenous floor (constant, or growing) altogether. Through capital accumulation the model produces an endogenous growth trend, more explicitly, growth cycles around a trend. Thus also the Hicksian autonomous growth trend can be dispensed with, and the model becomes self contained. A problem then may seem to be that without these exogenous trends the growing variables, income and capital, cannot be reduced to stationarity through trend elimination. A new method, proposed by the author 55 years ago, which we call relative dynamics, replaces the growing income by the income growth factor and the growing capital by the capital to income ratio, and these appear as stationary time series, predominantly periodic The change removes arbitrary assumptions, such as equality of growth rates for the exogenous trends, in autonomous expenditures and the investment floor. This seems to be good as at second thought the floor level apparently must be decreasing rather than growing when capital accumulates. The change also produces both trend **and** cycles on its own, which the original multiplier-accelerator model cannot, and further reduces periodic growth rates from 50–100% in the original model to more realistic 0.2–10%.

## References

- Allen RGD (1956) Mathematical economics. Macmillan, LondonGoogle Scholar
- Duesenberry J (1950) Hicks on the trade cycle. Q J Econ 64:464–476Google Scholar
- Gallegati M, Gardini L, Puu T, Sushko I (2003) Hicks’s trade cycle revisited: cycles and bifurcations. Math Comput Simul 63:505–527Google Scholar
- Gandolfo G (1985) Economic dynamics: methods and models. North-Holland, AmsterdamGoogle Scholar
- Goodwin RM (1951) The nonlinear accelerator and the persistence of business cycles. Econometrica 19:1–17Google Scholar
- Hicks JR (1950) A contribution to the theory of the trade cycle. Oxford University Press, OxfordGoogle Scholar
- Hommes CH (1991) Chaotic dynamics in economic models. Wolters-Noodhoff, GroningenGoogle Scholar
- Palander T (1953) On the concepts and methods of the “Stockholm School”. International economic papers no. 3, MacmillanGoogle Scholar
- Puu T (1963) A graphical solution to second order homogeneous difference equations. Oxf Econ Pap 15:53–58CrossRefGoogle Scholar
- Puu T (1987) Complex dynamics in continuous models of the business cycle. In: Lecture notes in economics and mathematical systems, vol 293. Springer, Berlin, pp 227–259. ISBN 3-540-18183-0Google Scholar
- Puu T (2007) The Hicksian trade cycle with floor and ceiling dependent on capital stock. J Econ Dyn Control 31:575–592CrossRefGoogle Scholar
- Puu T, Gardini L, Sushko I (2005) A multiplier-accelerator model with floor determined by capital stock. J Econ Behav Organ 56:331–348CrossRefGoogle Scholar
- Rau N (1974) Trade cycle: theory and evidence. Macmillan, LondonCrossRefGoogle Scholar
- Samuelson PA (1939) Interactions between the multiplier analysis and the principle of acceleration. Rev Econ Stat 21:75–78CrossRefGoogle Scholar
- Sushko I, Gardini L, Puu T (2004) Tongues of periodicity in a family of two-dimensional maps of real Möbius type. Chaos Solitons Fractals 21:403–412CrossRefGoogle Scholar
- Sushko I, Gardini L, Puu T (2010) Regular and chaotic growth cycles in a Neo-Hicksian floor/ceiling model. J Econ Behav Organ 75:77–99CrossRefGoogle Scholar
- von Haberler G (1937) Prosperity and depression. Harvard University Press, CambridgeGoogle Scholar