Is forward-looking inflation targeting destabilizing? The role of policy's response to current output under endogenous investment

In sticky price models with endogenous investment, virtually all monetary policy rules that set a nominal interest rate in response solely to future inflation induce real indeterminacy of equilibrium. Applying the Samuelson-Farebrother conditions, we obtain a necessary and su?cient condition for local real determinacy, which reveals that increasing price stickiness or letting policy respond also to current output may help ensure a unique equilibrium. We find that the first channel by itself has a quantitatively negligible effect and almost all strict inflation-targeting rules lead to indeterminacy, whether with higher price stickiness or overall stickiness by incorporating firm-specific capital, sticky wages, or both. The effect of the second avenue depends on labor supply elasticity and stickiness. With high labor supply elasticity and price stickiness, indeterminacy is much less likely to occur as policy also responds to output. With estimated labor supply elasticity or empirically reasonable price stickiness, policy's response to output helps little in ensuring determinacy; even incorporating firm-specific capital makes only a marginal improvement. Incorporating sticky wages, on the other hand, greatly enhances the role of policy's response to output in ensuring determinacy. With both sticky wages and firm-specific capital incorporated, even a tiny response of policy to current output can render equilibrium determinate for a wide range of response of policy to future inflation.