# The Prevalue Function

Despite aggressive self-promotion of my back-catalog of blog posts, I have been relatively slow to promote my academic papers on my blog. I will make an effort to right that imbalance by sprinkling in some blog posts about my academic papers.

By the way, while I give my blog tender-loving care constantly, I make very little effort to tend my university websites. The go-to source for my academic career (including my CV) is not any of my university websites, but this post, which I keep updated. Also, see my blog bio at the link right under the blog banner and motto.

My paper “The Effect of Uncertainty on Optimal Control Models in the Neighbourhood of a Steady State” is one you may not be aware of. You can see the abstract and a link to an ungated version above. One of the most useful things in the paper is a bit of terminology: a name for what will become the value function *after *optimization. Below is text of the section defining the “prevalue function”:

### Christening the “prevalue function” *F*(*k*, *x*, *ω*)

In many different contexts, close analysis of the Bellman equation requires careful attention to the properties of the function underneath the maximisation operator on the right-hand side of the Bellman equation. Therefore, the right-hand side of the Bellman equation before maximisation deserves a name. Let me propose the name “prevalue function”. To make the case that this is a reasonable coinage, let me make the analogy to the terminology of physiology, in which a hormone that is converted into another hormone is given the prefix “pro-”. For example, “prothrombin” is converted into thrombin (one of the key agents in blood clotting). Similarly, if maximising a function over *x* yields the value function, then the original function that is converted into the value function by that maximisation can be called the “prevalue function”. There are two justifications for using *pre*—instead of *pro*— in the phrase “prevalue function”. First, “value” is of Latin rather than Greek derivation, so it is appropriate to prefix it with the Latin “pre-” instead of the Greek “pro-”. Second— and more importantly—political uses of “values” would give the word “provalue” other possible meanings that could be confusing—or at least distracting.

On usage, let me suggest that the phrase “prevalue function” will be most useful if it can be used in a flexible way for the object to be maximised on the right-hand side of any Bellman equation, whether a continuous-time or discrete-time Bellman equation, and regardless of how the Bellman equation is arranged. In other words, regardless of whether terms unaffected by the maximisation operator are put on the left- or right-hand side of the equation, and regardless of whether the equation is divided through by a term unaffected by the maximisation operator, the function under the maximisation operator can be called “the prevalue function”. (Thus, turning the “prevalue function” into the value function may require some other simple operations beyond maximising over the control variable vector.)