A Class of Nonlinear Second Order Difference Equations from Macroeconomics
نویسنده
چکیده
where the constant A,, = Co + I0 + Go represents the sum of the minimum consumption, the “autonomous” investment and the fixed government spending in period n, and Y, is the output--GNP or national income-in period n. The net investment amount in the same period is given as Z, = ac( Y,I Y,-,). The constant c E (0, 1) represents Keynes ’ “marginal propensity to consume” or the MPC, while the coefficient CY > 0 is the “accelerator”. The linear model above improved the earlier Keynesian models and resulted in substantial new research. However, this model was soon found to be unsatisfactory, since CYC can exceed unity in typical economic settings (see, [2, Chap. 91). This fact results in exponentially divergent solutions for the linear equation, which of course, is not observed in reality. Certain nonlinear models were subsequently proposed to resolve this anomaly. For instance, rather than the linear Keynesian consumption C(Y) = cY + C’, , Samuelson considered a nonlinear consumption function [3]. Samuelson’s assumptions amount to a third order difference equation and an MPC which is itself a decreasing&n&on of output. Some years later, Hicks proposed a model in which consumption was linear, but investment and output were both piecewise linear [4]. Hicks’ model results in a second order difference equation with constant MPC, but the accelerator is not defined (or could mathematically be set equal to zero) for a certain range of output differences; the model also happens to be nonautonomous because of a time dependent “hard ceiling” on output (as well as the induced investment “floor”). There have been several other models such as continuous time models of Goodwin and Kaldor, or the more recent stochastic models which we have not mentioned because of the attributes italicized here; [5] contains brief discussions of some of these models and comprehensive bibliography.
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