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In a paper published in AER in 1999, the following argument was made concerning the linearization of a nonlinear system at a steady state:
“The law of motion of the economy can be described locally by the matrix … H .. This matrix applied to any initial deviation … from the zero-tax, equal-wealth steady state describes the deviation implied in the next period. It is straightforwardt to check that this matrix has one eigenvalue equal to one and one positive and less than one. The eigenvalue equal to one indicates the steady-state indeterminacy we expected, and the fact that the remaining eigenvalue is less than one says that the system is nonexplosive.”
The paper cites no mathematical work on local behavior of stationary points in dynamical system to support its contention that unit root plus stable root in the linearization implies stability of the nonlinear system. In fact, there is no such citation! Moreover, anyone who took a serious dynamic economics course in the late 1970’s would have seen the Hopf bifurcation literature of those days and know that this argument is false.
The same paper defined equilibrium to a function Psi(A, tau) that satisfied a nonlinear fixed-point problem. The paper never said what function space was assumed to contain Psi(A, tau), whether the operator defining the iterations for the fixed-point problem was continuous in an appropriate topology for that function space, nor anything else necessary to even properly formulate the fixed-point problem. The “math” looks like dynamic programming, but there is no attempt to show that the critical operator is a contraction in some useful metric.
I was surprised by AER’s interest in publishing nontrivial and original work in nonlinear functional analysis. However, readers of AER should go elsewhere to learn anything about nonlinear functional analysis.