You might as well be hung for a sheep as a lamb: the loss function of an agent

Most of those who take macro and monetary policy decisions are agents. The worst penalty which can be applied to these agents is to sack them if they are perceived to have failed. To be publicly sacked as a failure is painful, often severely so, but the pain is finite. Agents thus have loss functions which are bounded above, in contrast to the unbounded quadratic loss functions which are usually assumed for policy analysis. We find a convenient mathematical form for such a loss function, which we call a bell loss function. We contrast the different behaviour of agents with quadratic and bell loss functions in three settings. Firstly we consider an agent seeking to reach multiple targets subject to linear constraints. Secondly we analyse a simple dynamic model of inflation with additive uncertainty. In both these settings certainty equivalence holds for the quadratic, but not the bell loss function. Thirdly we consider a very simple model with one target and multiplicative (Brainard) uncertainty. Here certainty equivalence breaks down for both loss functions. Policy is more conservative than in the absence of multiplicative uncertainty, but less so with the bell than the quadratic loss function.