Stochastic Equilibrium the Lucas Critique and Keynesian Economics

In this paper, a mathematically rigorous solution overturns existing wisdom regarding New Keynesian Dynamic Stochastic General Equilibrium. I develop a formal concept of stochastic equilibrium. I prove uniqueness and necessity, when agents are patient, with general application. Existence depends on appropriately specified eigenvalue conditions. Otherwise, no solution of any kind exists. I construct the equilibrium with Calvo pricing. I provide novel comparative statics with the non-stochastic model of mathematical significance. I uncover a bifurcation between neighbouring stochastic systems and approximations taken from the Zero Inflation Non-Stochastic Steady State (ZINSS). The correct Phillips curve agrees with the zero limit from the trend inflation framework. It contains a large lagged inflation coefficient and a small response to expected inflation. Price dispersion can be first or second order depending how shocks are scaled. The response to the output gap is always muted and is zero at standard parameters. A neutrality result is presented to explain why and align Calvo with Taylor pricing. Present and lagged demand shocks enter the Phillips curve so there is no Divine Coincidence and the system is identified from structural shocks alone. The lagged inflation slope is increasing in the inflation response, embodying substantive policy trade-offs. The Taylor principle is reversed, inactive settings are necessary, pointing towards inertial policy. The observational equivalence idea of the Lucas critique is disproven. The bifurcation results from the breakdown of the constraints implied by lagged nominal rigidity, associated with cross-equation cancellation possible only at ZINSS. There is a dual relationship between restrictions on the econometrician and constraints on repricing firms. Thus, if the model is correct, goodness of fit will jump.