The argument that normal-time policy-relevant fiscal multipliers should be presumed to be very small can be made more general. Optimizing central banks will be expected to offset shifts in discretionary fiscal policy—and thus lead to multi-plier estimates near zero—under relatively unrestrictive conditions. Consider a government with an objective function u defined on a set of economic outcomes Π, and economic laws Γ:M → Π leading from a monetary policy m in M to an economic outcome ϕ in Π. Then if ϕ' is the optimal outcome, a central bank will choose an element m' such that Γ(m') = ϕ'. The adoption of a different fiscal policy g will change the relationship from monetary policies to economic outcomes to a different Γg. However, assume that the domain M and the range Π of Γ is the same as the domain and range of Γg. Then an optimizing central bank will choose a monetary policy m'' so that Γg(m'') = ϕ'. It will engage in full fiscal offset.
Thus there are two key requirements for this fiscal offset to hold:
- The optimal outcome ϕ' must be in the set Πg of outcomes attainable by monetary policy under the counterfactual fiscal as well as in the set of outcomes Π.
- No superior economic outcome ϕ"' such that u(ϕ"') > u(ϕ') that is not attainable under the baseline set of possible outcomes Π can be in the counterfactual set of attainable outcomes Πg.