A note on the “unique” business cycle in the Keynesian theory

Abstract

In this paper, we explore the existence and "uniqueness" of a limit cycle in the Keynesian theory. In a model with the simplest (linear) Keynesian consumption function and the logistic investment function based upon the profit principle, we establish the existence of a periodic orbit (irrespective of the speed of quantity adjustment) and, with the help of the theory on generalized Liénard systems, verify the uniqueness of it for the case in which the speed of quantity adjustment is large enough.

On dynamics in a medium-term Keynesian model

“…Condition (25) means that the marginal propensity to invest (including the indirect marginal effect through a change in the real rate of interest) evaluated at the unique equilibrium I * Y + I * ρ e R * Y is less than that to save S * Y and implies that the Keynesian stability condition, which is a standard assumption in macroeconomics (cf. Marglin and Bhaduri [18]), holds at equilibrium.…”

“…where 25 This model is denoted by "Model (K 2 )." For a transformation of Model (K 2 ), the following vector z = (z 1 , z 2 , z 3 , z 4 ) T is introduced:…”

“…24 Throughout this paper, the superscript T represents the transpose operator. 25 Throughout this paper, the over-line ( * ) represents the complex conjugate.…”

“…The stability of limit cycles can also be confirmed by proving (not numerically but analytically) the uniqueness, as well as existence, of a limit cycle. For recent studies on economic dynamics taking this approach, see Murakami [24,25,26,27]. 8 For mathematical derivations of the Keynesian (or non-Walrasian) investment and saving functions, see Murakami [22,23].…”