Kydland, Prescott, and Backwards Causation

In 1977, Finn Kydland and Edward Prescott published an article, “Rules Rather than Discretion: The Inconsistency of Optimal Plans”. The article had significant influence over policymaking and helped to win Kydland and Prescott the 2004 Nobel Memorial Prize.

One thing the article achieved was to kill off hope for successful economic planning based on optimal control theory. It begins boldly:

Optimal control theory sounds pretty good to many people. What’s wrong with choosing the best decision at each point in time? There is a worry about unintended consequences, of course, but later KP make it clear that this isn’t their worry:

KP are raising a problem with the strategy of trying to choose the best policy, given the current situation and possible effects on the entire future. What could possibly be left out of account there? KP’s answer is: effects on the past!

Are KP really suggesting a theory of backwards causation? I want to argue that they are. But the matter is confused by the reference to rational expectations. Where expectations are very, very rational, they converge perfectly with the actual future. And so, counterfactually, if the future were different, then the rational expectations would be different. This counterfactual dependence of past expectations upon future eventualities is easily confused with backward causation.

Think of Newcomb’s Problem. The one-boxer makes what looks like a valid induction from the Predictor’s demonstrated abilities: If I choose the one box now, that will mean that the Predictor put the big reward into the one box. The two-boxer thinks that the one-boxer, in arguing this way, is sailing dangerously close to a theory of backward causation. Still, doesn’t the one-boxer have a point?

Something similar, I think, is going on with KP. They begin by setting up a model of what they call “consistent” policy:

Keep that in mind. A consistent policy is one in which the policy at each time is made, taking as given previous decisions. Now:

According to KP, the problem lies here:

Now, you might ask, if 2 is a later period than 1, then how can π2 have effects upon x1? That looks a lot like backwards causation. The implicit reply is that expectations are rational. Here is where things become Newcombish. When agents decide on x1, they know that the policy in period 2 will be π2. So if, counterfactually, π2 were relevantly different, the agents would choose a different x1. The future determines the past.

But this explanation makes nonsense of KP’s argument. The consistent policy was defined as one in which policy is made at each time, with previous choices taken as given. In other words, the definition of consistency entails a tensed reading of statements like “Policy π2 is chosen”. Policies are chosen at particular times, so that there can be times at which a statement of that sort is not true and others where it is true.

KP’s mathematical model, however, allows no room for this indispensable tensing. Equations have no tense. This is another case in which I wish economists would use more logic (e.g. tense logic) and less mathematics.

According to the equations, π2 must be already determined when x1 is being determined, since the function that determines x1 has π2 as an input. Now we might say that x1 is determined timelessly, outside of both periods. But if so then there is no possibility of the consistent policy in which π2 is determined at period 2, taking x1 as given. If x1 is given, then it has already been determined, and if it has been determined then π2 must already have been determined as well.

The problems here rapidly become theological in character. A variable determined outside of time cannot have a variable determined in time as an input. From the citadel of eternity, it is impossible to wait and see what eventuates. Thus if π2 is determined in time, x1 must also be.

But if we have π2 determined at period 2 and x1 determined at period 1, then there is no way to make sense of the idea of the policy problem at all. At period 2, the policy problem is supposed to be to find the π2 that maximises the social optimality function, given π1 and x1. But since x1 is given, π2 is given: the function is already maximised and, for that matter, minimised, since there are no independent variables. And again there can be no consistent policy, in which each πt is decided at that t and not before.

As far as I can see, the only way to make sense out of KP’s claim is as follows. Assume that x1 is determined at period 1 from some ‘default’ value for π2. At period 2, policy can change the value of π2 and thus the value of x1. This is straightforward backwards causation. Now we can distinguish between the consistent and the optimal policy. The consistent policy will fail to actually maximise S(x1, x2, π1, π2), because it will fail to take into account the effects on x1 wrought by π2. The optimal policy will successfully maximise S(x1, x2, π1, π2) by taking into account the impact of the future on the past.

But this leads us into a sea of nonsense. The vector (π1,π2) is meant to timelessly represent the policy choices made at periods 1 and 2. But if the value of π2 changes from its default value to a different value, when does it do so? Not at period 2, since in setting π2 to the default value we have already determined that the policy choice at period 2 is for the default value. Not at a later period, since this is two-period model. We could perhaps try to introduce a new time dimension along which changes to the past, present, and future can alike be represented, but then we’re very soon off to the races with J.W. Dunne.

The problem is that the notion of backwards causation is very hard to render coherent. If KP are going to use it, they owe us a developed tense logic capable of rendering it sensible — a few equations representing un-tensed mathematical propositions won’t do the trick.

What about the basic insight that KP are credited with formally expounding? This much is true: where it is the case that A&B is worse than A on its own, it is nevertheless optimal to adopt a policy of doing B when A occurs, if — and only if — adopting such a policy actually prevents the future occurrence of A.

But that is something the optimal control theorist can happily concede. The optimal control theorist says: when choosing at t, take as given all choices made up until t and decide on the policy with the best expected outcome from t onwards. This is what KP call a consistent policy. But if the calculation determining it is done well it will naturally include any disincentive or similar effects the policy will have on future choices. Why then do KP declare the consistent policy suboptimal? Read their explanation again:

Since disincentive effects are part of “the effect of this decision upon the entire future”, they cannot be what the consistent policymaker ignores. Rather, the consistent policymaker ignores the possibility of backwards causation: future policy affecting current decisions, through the ‘expectations mechanism’.

Am I taking the ‘expectations mechanism’ too literally? Do KP mean only that policymakers should think about how policy made now affects what people now and in the future will expect future policy to be? No. They can’t mean that, since the consistent policy that they critique does, by their own admission, take all current and future effects into account, including effects on current and future expectations.

What such a policy doesn’t take into account is the effect the future can have on the present — the effect that what happens at a later time can have on what happens at a previous time. And it sounds pretty sensible to leave that out of account, since the alternative is to believe in backwards causation.

One last point: in Economy and the Future, Jean-Pierre Dupuy proposes to advocate one-boxing the Newcomb problem by invoking a notion of non-causal dependence, which he calls “counterfactual determination”. While later causes cannot bring about earlier effects, later events can counterfactually determine previous events. By choosing one box, one counterfactually determines that the Predictor has put the big prize in it; by choosing two, one counterfactually determines that the Predictor has left the big prize out.

To me, this just looks like backwards causation wearing a semantic disguise. And so does “the expectations mechanism”. Dupuy and KP, to my mind, make the same mistake. It is possible to perfectly predict the future, but this does not show that something in the future has caused or ‘determined’ the prediction. All it shows is that features of the present, detectable by the predictor, are such as to guarantee a certain future.

A wise policymaker will take into account the effects her policy will have on such features. These will include the expectations people have of future policy which then dictate their behaviour. But taking these into account is perfectly consistent with policy based on optimal control theory. So either KP’s argument is invalid or backwards causation is both intelligible and real.