Multiple attractors and global bifurcations in a Kaldor-type business cycle model

Abstract

Abstract. We consider a Kaldor-type discrete-time nonlinear business cycle model in income and capital, where investment is assumed to depend both on the difference between normal and current levels of capital stock, and on the difference between the current income and its normal level, through a nonlinear S-shaped increasing function. As usual in Kaldor business cycle models, one or three steady states exist, and the standard analysis of the local stability and bifurcations suggests that endogenous oscillations occur in the presence of only one unstable equilibrium, whereas the coexistence of three equilibria is characterized by bi-stability, the central equilibrium being on the boundary which separates the basins of the two stable ones. However, a deeper analysis of the global dynamic properties of the model in the parameter ranges where three steady states exist, reveals the existence of an attracting limit cycle surrounding the three steady states, leading to a situation of multistability, with a rich and complex dynamic structure.

Animal spirits in the foreign exchange market

“…The set of inequalities (A.2) provide a necessary and sufficient condition to guarantee that 9l 1;2 9 o1 (Agliari et al, 2007;Bischi et al, 2001;Dieci and Westerhoff, 2010). The inequality P 1 ð1Þ 40 is always satisfied under the assumptions made about a and r. From the condition P 1 ðÀ1Þ it follows that b ¼ b L 4ðaÀ2Þ=ar .…”

Hysteresis in the trade cycle

“…Global bifurcations were proven to be present in Ref. [7]. Chaotic dynamics are regularly cited in the literature on discrete-time formulations of the Kaldor model, and were proven to be present in Ref.…”

“…The remaining parameters are the level of expected income ðmÞ, the rate of depreciation of stock ðdÞ and a parameter representing costs of adjustments ðgÞ-all were fixed at the beginning of the analysis, as in Refs. [7,11]. The formulation of the model is complete once a suitable Preisach density is found.…”

Analysis of dynamic regimes in stochastically forced Kaldor model

“…These assumptions are modeled by the function IðY; KÞ ¼ IðYÞ À bK; b > 0. This simple model was discussed from economic point of view in [3][4][5] and widely used in [6][7][8][9][10]. Traditionally, models with a single attractor (equilibrium or limit cycle) are studied.…”