Abstract
The hypothesis that aggregate fluctuations may represent an endogenous feature of dynamic competitive economies with incomplete markets has been advanced in several papers.1 The role of incomplete markets seems essential for the appearance of cycles in one sector models.2 Becker and Foias (1987) and Woodford (1988a) have pointed out that the elasticity of substitution of the production function plays a fundamental role in the existence of cyclic equilibrium paths. In those papers cycles are generated if the substitutability between capital and labor is not too great. Heterogeneity of households is also a crucial component of their fluctuation theories.
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References
See Bewley (1986), Becker and Foias (1987), and Woodford (1988a) . Boldrin and Woodford (1990) and Guesnerie and Woodford (1992) survey those equilibrium models.
Dechert (1984) shows that the one sector Ramsey model exhibits a monotonic aggregate capital stock along an optimal program for a wide variety of technologies. Benhabib, Jafarey, and Nishimura (1988) show that standard turnpike properties may be recovered for a class of recursive preferences in Pareto optimal growth with many consumers.
See Boyd (1990) and Hernández (1991) for additional research on this model. Bewley (1986) seems the first to draw attention to market incompleteness as a potential source of equilibrium fluctuations.
See Woodford (1988b) for a critique of the closely related cash-in-advance models where laborers choose not to borrow due to high discount rates.
Ramsey equilibrium programs were shown to exist with general utility functions by Becker, Boyd and Foias (1991) .
See Becker and Foia (1987). Henández (1991) recovers this result in the case where all households have a common discount factor.
Becker and Foias (1987) provide an ad hoc construction of an eauilibrium 2-cycle.
This is also the intuition for the 2-cycle example built by Becker and Foias (1987). The intertemporal tolerance for consumption variation and discounting also enter the analysis.
Majumdar and Mitra (Chapter 3) examine changes in the utility and production functions in a one-sector Ramsey model with wealth effects. They demonstrate the possibility of topological chaos for an open set of production and felicity functions that are the economic primitives of the model.
Becker and Foias (1987) give a special version of these condtions for the case of a cycle of period 2.
We have dropped the subscript on δ since the meaning is clear.
See equation (3.1) below for a p</b>recise definition of Φ.
See Becker and Foias (1990), Section IV.
If there is only one household, then the model is equivalent to the standard Ramsey optimal growth model with a one sector technology. Optimal capital sequences are always monotonic in that framework.
The intertemporal tolerance is the reciprocal of the Arrow-Pratt measure of absolute risk aversion. This is of course a measure of the degree of concavity of u. Hence the tolerance function also serves the same purpose. The intertemporal elasticity of snbstitution, A(y)/y, is a more commonly used measure of sensitivity to intertemporal consumption variation.
The units of measurement in (3.4A) are output per unit of capital.
Other useful references on center manifold theory may be found in Carr (1981) and Ruelle (1989).
See Becker and Foias (1987) for the underlying intuition.
This should not lead to any confusion since the analysis applies only to the map.
This Taylor series might not converge.
This follows from the Implicit Function theorem.
That is, the fourth derivative is very much larger than 1.
The tangent spaces are given by the space Co2) RR+) of twice continuously differentiable real-valued functions defined on R+ having compact support. This space is a strict inductive limit of Banach spaces. Since this structure does not play any economic role in our analysis we will leave all the functional-geometric details to the mathematically minded reader.
lndeed, ℘k is a submanifold of ℘ of codimension 1, that is defined by one equation. For example, a surface in R3 is a codimension 1 submanifold. Hence, a codimension 1 submanifold in ℘ signifies the submanifold is a “rich” class of functions
Notice that the second inequality in Condition (D) implies (C).
See Aubin (1977, p. 52) for details of this construction.
We retain the use of the notation f’ , f” etc. for the unperturbed production (and income) functions when the meaning is clear.
We note that Berndt (1976) notes the time series estimates of the elasticity of substitution generally provide lower estimates than 1 for that parameter. Indeed, he notes those estimates tend to be centered in the range 0.1 to 0.2.
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Becker, R.A., Foias, C. (2000). The Local Bifurcation of Ramsey Equilibrium. In: Optimization and Chaos. Studies in Economic Theory, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04060-7_5
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DOI: https://doi.org/10.1007/978-3-662-04060-7_5
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