  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.
