For example, a fiscal policy maker may have a prediction as to the value of the fiscal multiplier—the ratio of the effect of a government spending change on GDP to the size of the government spending change—but is not likely to know the exact value of this ratio.
(2) In the presence of multiplier uncertainty, it is no longer redundant to have more policy tools than there are targeted economic variables.
For the simplest possible case,[1] let P be the size of a policy action (a government spending change, for example), let y be the value of the target variable (GDP for example), let a be the policy multiplier, and let u be an additive term capturing both the linear intercept and all unpredictable components of the determination of y.
Then Suppose the policy maker cares about the expected squared deviation of GDP from a preferred value
; then its loss function L is quadratic so that the objective function, expected loss, is given by: where the last equality assumes there is no covariance between a and u. Optimizing with respect to the policy variable P gives the optimal value Popt: Here the last term in the numerator is the gap between the preferred value yd of the target variable and its expected value Eu in the absence of any policy action.
[2] In this case a key result is that, unlike in the absence of multiplier uncertainty, it is not superfluous to have more policy tools than targets: with multiplier uncertainty, the more tools are available the lower expected loss can be driven.
There is a mathematical and conceptual analogy between, on the one hand, policy optimization with multiple policy tools having multiplier uncertainty, and on the other hand, portfolio optimization involving multiple investment choices having rate-of-return uncertainty.
The above discussion assumed a static world in which policy actions and outcomes for only one moment in time were considered.
However, the analysis generalizes to a context of multiple time periods in which both policy actions take place and target variable outcomes matter, and in which time lags in the effects of policy actions exist.