Stability

The analysis of a dynamic system is widespread in economic theory. As time passes, economic variables change. A fundamental issue is whether the system is stable. As time passes, do the economic variables converge to a steady state, either of constant values or of constant growth? Typically, economic thinking supposes stability, so to analyze whether the system is stable is fundamental. Below we investigate the stability of constant coefficients, linear difference and differential equations via the von Neumann–Morgenstern theorem of the alternative, applied in a Hermitian Euclidean Jordan algebra. The analysis connects the eigenvalues of the dynamic system with the existence of a Lyapunov linear transformation. The approach to stability by Lyapunov is to define a function that is zero at the steady state but positive elsewhere. One defines the function so away from the steady state it declines as time passes. Under general conditions, it follows that the steady state is stable. Set the analysis in the complexification X of a Euclidean space and the Hermitian Euclidean Jordan algebra HX of self-adjoint linear transformations on the complexification. We look for some positive linear transformation X in the complexification such that for any nonzero xt the quadratic $$\left \langle \boldsymbol {x}_{t},{\textit {{\sf { {X}}}}}\boldsymbol {x}_{t}\right \rangle$$ is positive and declines each period. The stability alternative is a restatement of the von Neumann–Morgenstern alternative derived previously. For a square linear transformation A on HX, exactly one of the following has a solution: Stable Alternative AX≪0,X≫0; Not Stable Alternative A⊤X*⪰0,X*≻0. We apply this theorem of the alternative to study stability. For the difference equation, one choice of A applies. For the differential equation, a different choice applies. However the same two alternatives apply to either. In each case, the stable alternative states that there exists a Lyapunov transformation, so the dynamic system is stable. We interpret the not stable alternative in terms of eigenvectors and eigenvalues.