Endogenous Business Cycles and Dynamic Inefficiency

Abstract

This paper explores how the occurrence of local indeterminacy and endogenous business cycles relates to dynamic inefficiency, as defined by Malinvaud (1953), Phelps (1965) and Cass (1972). We follow Reichlin (1986) and Grandmont (1993) by considering a two-period OLG model of capital accumulation with labor-leisure choice into the first-period of agents' life and consumption in both periods. We first show that local indeterminacy and Hopf bifurcation are necessarily associated with a capital-labor ratio that is, at steady state, larger than the Golden Rule level. Consequently, paths converging asymptotically towards the steady state are shown to be dynamically inefficient, as there always exists another trajectory that starts with the same initial conditions and produces more aggregate consumption at all future dates. More surprising, however, is our main result showing that stable orbits, generated around a dynamically inefficient steady state through a supercritical Hopf bifurcation, may, in contrast, be dynamically efficient.

Local indeterminacy under dynamic efficiency in a two-sector overlapping generations economy

“…If b is too close to 0, i.e. if the model is close to the aggregate formulation, we get the same conclusion as Cazzavillan and Pintus [2,3]: local indeterminacy under dynamic efficiency can never arise with any positive elasticity of capital-labor substitution.…”

“…Under Assumption 1, T (k, y, ) is homogeneous of degree one, concave and C 2 over K. 2 Denoting w the wage rate, r the gross rental rate of capital and p the price of investment good, all in terms of the price of the consumption good, we derive from the envelope theorem that…”

On efficiency and local uniqueness in two-sector OLG economies

“…The period-2 cycles generated by the flip bifurcation are either saddle-point stable if the bifurcation is super-critical, or locally unstable if the bifurcation is sub-critical. As a result, contrary to Cazzavillan and Pintus[8], locally indeterminate periodic cycles cannot occur in this configuration.…”

“…Let us denote κ t = k t /ℓ t the capital-labor ratio at time t ≥ 0 and letκ ≡ k(ℓ)/ℓ be the maximal admissible value of κ. We can then redefine the set of admissible paths given by (8) as follows…”

“…Considering an aggregate OLG model augmented to include endogenous labor, Cazzavillan and Pintus[8,9] also show that local indeterminacy is ruled out if the steady state is characterized by an under-accumulation of capital.…”