Using an analytically tractable heterogeneous agent New Keynesian model, we show that whether incomplete markets resolve New Keynesian “paradoxes” depends on the cyclicality of income risk. Incomplete markets reduce the effectiveness of forward guidance and multipliers in a liquidity trap only with procyclical risk. Countercyclical risk amplifies these “puzzles.” Procyclical risk permits determinacy under a peg; countercyclical risk may generate indeterminacy even under the Taylor principle. By affecting the cyclicality of risk, even “passive” fiscal policy influences the effects of monetary policy.

We show that in heterogeneous agent New Keynesian (HANK) economies with countercyclical risk, the natural interest rate is endogenous and co-moves with output, leaving the economy susceptible to self-fulfilling fluctuations. Unlike in representative agent New Keynesian models, the Taylor principle is not sufficient to guarantee uniqueness of equilibrium in HANK if risk is even mildly countercyclical: Multiple bounded equilibria exist, no matter how strongly monetary policy responds to changes in inflation. For an active monetary policy to eliminate self-fulfilling fluctuations, it must stabilize the endogenous natural rate fluctuations. Alternatively, a passive monetary and active fiscal regime can also eliminate equilibrium multiplicity.

We study optimal monetary policy in an analytically tractable heterogeneous agent New Keynesian model with rich cross-sectional heterogeneity. Optimal policy differs from a representative agent benchmark because monetary policy can affect consumption inequality, by stabilizing consumption risk arising from both idiosyncratic shocks and unequal exposures to aggregate shocks. The trade-off between consumption inequality, productive efficiency, and price stability is summarized in a simple linear-quadratic problem yielding interpretable target criteria. Stabilizing consumption inequality requires putting some weight on stabilizing the level of output, and correspondingly reducing the weights on the output gap and price level relative to the representative agent benchmark.