McTaggart and Time-Inconsistency
Someone pointed out that my last post sounded a bit like McTaggart’s argument for the unreality of time. True, I’m usually thinking about McTaggart when I pretend to be thinking about something else. Unless I’m thinking about Spinoza.
But it was a good call. Thinking about time-inconsistency in terms of McTaggart’s famous problem of time helps to bring out what I think is a deep prejudice implicit in macroeconomic models. Effectively, these models are biased heavily towards a conservative, anti-interventionist position. But the bias isn’t plainly visible; it’s hidden behind the odd treatment of time. McTaggart helps us to see this (he helps us to see everything true).
McTaggart’s Problem
McTaggart’s problem goes something like this. First, define two different series of instants or events (it doesn’t matter which): the A-series and the B-series. The A-series is “that series of positions which runs from the far past through the near past to the present, and then from the present through the near future to the far future, or conversely” (The Nature of Existence 2.306). The B-series is “the series of positions which runs from earlier to later, or conversely”.
The crucial point (I believe) in McTaggart’s argument is that the A-series is the time-series viewed from a particular position in the series. It is how time looks to someone at some particular instant. The B-series is the time-series viewed from outside the series — as Peter Geach puts it, it is how God would see the time-series from what Aquinas calls ‘the stronghold of eternity’.
McTaggart’s first point is that there will be conflict over what is past, present, and future from two distinct points of view. As I write this, the Battle of Waterloo is in the past; during the Battle of Waterloo, it was present. If we privilege one point of view as absolutely correct, we abolish change: reality is locked into a permanent present in which the future forever remains future and the past never gets older. On the other hand, if we deny this absolute status to any point of view, we relativise all ascriptions of the predicates past, present, and future to a particular point of view. In absolute terms, all that can exist is the B-series.
For McTaggart, this seemed incoherent. The B-series makes no sense without the A-series. Without past, present, and future, there can be no earlier and later. Thus we cannot say that in absolute terms, independently of all particular points of view, there is only a B-series.
This is where most philosophers don’t follow McTaggart. There doesn’t seem to be anything obviously incoherent about the idea of a time-series whose members bear the relations of earlier and later to each other but to which the predicates past, present, and future do not apply. It might be, however, that McTaggart has a point, as Michael Dummett powerfully argued.
McTaggart and Macroeconomists
In any case, I think that McTaggart provides a nice framework for explaining the economist’s idea of time-inconsistency:
A time-inconsistent strategy, policy, etc. is one that is defined on an A-series but not on a B-series.
Thus, to take one well-known example, suppose a policymaker is working out how much to tax labour versus capital. The best thing to do seems to be to tax capital heavily and labour not at all in the present, but then adjust this in the future. The reason is that if you tax labour now, labourers might withdraw their labour to avoid the tax, thus reducing output, whereas capital is already built: those who own it cannot ‘unsupply’ it to avoid the tax. On the other hand, if you commit to sustaining this policy into the future, producers of capital will suppress the supply of capital to avoid the tax. This is bad overall, since it reduces total production. What is best to do now isn’t the same as what is best to do in the future.
Discussing this example, Edward Prescott writes: “The inconsistency of the optimal solution arises because the optimal tax on labour is zero in the current period and positive in future ones, but eventually future periods become the current one” (21).
Prescott’s point is this. Suppose the state implements the policy: “tax only capital in the present, but not in the future”. At t0, t0 is the present, so the state following the policy will tax only capital at t0. But at t1, t1 is the present, so the state following the policy will tax only capital at t1. And so on for t2, t3, . . . The state will end up taxing only capital at every tn.
The problem is that the meaning of the policy changes from time-position to time-position. At t0, it means: tax only capital at t0 but not at later tn. At t1 it means tax only capital at t1 but not at later tn. These strategies contradict each other. And this is what it means to say the policy is time-inconsistent. Defining a policy only on an A-series will inevitably result in time-inconsistency, since there are as many interpretations of the policy as there are time-positions.
Commitment Technology
The general lesson taken from examples like this in macroeconomics is that unless the state can develop a ‘commitment technology’, it will fall victim to time-inconsistency. We can think of a commitment technology as some external means of fixing the interpretation of a policy. Thus, for instance, the state can, at t0, implement the policy “tax only capital in the present but not in the future” by imposing the capital-only tax at t0 and then making it impossible for itself not to change the tax policy at later tn. Thus when it gets to t1 it can’t implement the policy newly interpreted (tax only capital at t1 but not at later tn).
Why, you might wonder, should the state need such a technology? Couldn’t it just fix the interpretation of the policy ‘in its own mind’? We do this all the time. I might say, e.g., “I’ll lend you money now, but I won’t do so in the future”, and know that the next time you ask sticking to my policy means not lending to you. Even though the referent of “now” has changed, I can stick to my old interpretation of my own policy.
But economists say that the state needs a commitment technology in cases like the one described, because — and here things get bizarre and interesting — at every tn the reason for implementing the policy in the present applies to tn. In the above example, it is true at t0 that existing capital is already there and can’t be unsupplied, so the state can tax it heavily without worrying about effects on incentives. But at t1 the capital that exists at t1 can’t be unsupplied, and at t2 the capital that exists at t2 can’t be unsupplied, and so on. The reason for the capital-only tax never seems to go away.
Intuitively, of course, this is wrong. If the state taxes only capital all the time, the capitalists will get wise to the game, stop supplying capital, and use labour instead. This will reduce overall production, which is not what the state wants. But what does it mean to say that the capitalists will get wise to the game? Why don’t they think like this: “it doesn’t matter if the state taxes present capital; as long as it doesn’t tax future capital there’s no reason for us not to supply more.” And then why don’t they reinterpret this at every instant, so that the antecedent always holds true, even if the state imposes the time-inconsistent policy? The state never breaks its commitment not to tax future capital, since the meaning of “future” keeps changing to be one step ahead of the tax.
Well of course the capitalists would not reason like this. They know that when they reason, at t0, that they should supply more capital only if the state doesn’t preserve its tax policy into the future, they mean: only if the state doesn’t preserve the policy from t1 (or some later position) onwards. And they don’t change this interpretation even when they get to t1 (or whichever later position).
But then if they can fix their interpretation without a ‘commitment technology’, why can’t the state do so? It seems that the capitalists are given cognitive access to the B-series in a way that the state is not. The capitalists can see the difference between (1) a policy that applies at some tn but not at tk for k>n and (2) a policy that applies only to the present, with “the present” continually reinterpreted so as to apply to every tn. If the state could see this, it would see that (1) is in its interests but (2) is not, and thus it would avoid time-inconsistency just by reasoning properly, without recourse to a ‘commitment technology’.
Economist’s Bias
In this thinking, the state is assumed to be incapable of doing what the capitalists (or more generally the private agents in the model) are assumed to be perfectly capable of doing, namely thinking in terms of the B-series — or, more specifically, fixing an interpretation of “past”, “present”, and “future” in order to apply a policy defined in these terms.
Private agents are constantly assumed to be capable of thus fixing their interpretations. Take discount rates, for instance, which are all over the models. An agent is supposed to apply some discount factor, θ, to an expected stream of income. A dollar today is worth more than a dollar tomorrow, a dollar the day after tomorrow is worth more than a dollar tomorrow, etc. — so long as the agent is at all impatient.
Now when an agent is calculating the utility to her of some stream of incomes, she adds up the income streams, I, for each time, t, applying the discount factor, θ, as follows:
But notice here that the dimension of t is the B-series. The time-positions are arranged in an absolute series running from 0 to N. If the agent were thinking in terms of the A-series, she would need not one utility function, but several, each representing a different value for each different time-position selected as the “present” (p). Thus:
The fact that the agent has only one utility-function here implies that she is able to fix her interpretation of “present dollar”, “future dollar”, etc. But if she can do so, why can’t the state? Why can’t the state fix its interpretation of “present” and “future” in selecting the policy “tax capital only in the present but not in the future”, so as to achieve the genuinely beneficial once-off tax policy?
This assumed difference in capacities between the state and private agents is not, as far as I know, ever explained in these models, because the models don’t acknowledge the assumption. In models like the ones in the Prescott article I cited, the point is obscured behind a confusion. The confusion is between the problem of time-inconsistency — which is what all the verbal examples are about — and a different problem: that of policy-invariance. The state’s mistake, in selecting a time-inconsistent policy, is held to be that it assumes the behaviour of agents to be indifferent to what policy is selected.
But this is a different and independent problem. The state could assume policy-sensitivity of the agents and still fall into the time-inconsistency trap. Consider that in the example above, the state does assume that capitalists will change their behaviour based on tax policy: that’s the whole point of choosing the policy to tax capital only now but not in the future. The problem is that the state keeps reinterpreting “now” and the agents do not.
Prescott gives Robert Lucas’s model (or really the schema for a model), in which the behaviour of private agents (d_t) is policy-sensitive. The general state of the economy at t is x_t, the policy active at t is u_t, and ϵ_t is any external shocks at t. Then we have a ‘law of motion’, which takes the economy from one general state into the next, as a function of all the other factors:
The policy rule determines policy at a time based on the state of the economy at that time:
The decision rule for private agents is defined as:
The delta_pi term here “corresponds to the behavioural equations of econometric models, but indexed by the policy rule” (19).
Notice that this model leaves no room for time-inconsistency, since so far we have only a B-series. That is, we have a series (t, t+1, t+2, . . .) on which earlier and later can be defined (t+n is earlier than t+k iff n<k; conversely for later) but past, present, and future cannot be defined. To get time-inconsistency into the policy, we might again break the policy rule up into sub-rules applying for each interpretation of “present”:
This will then get recursively defined for x_t+1, x_t+2, etc. — at each step there will be as many policy rules as there are possible time-positions to be interpreted as “the present”.
The key point is that if the policy rule is broken up into time-inconsistency this way, the decision rule of the private agents will inherit the time-inconsistency! After all, that decision rule is “indexed by the policy rule”: now it will have to be indexed to the whole set of policy rules corresponding to different interpretations of the present (values of p).
This is not the situation in the verbal examples. There (as we saw in the example above) the policy rule is supposed to be time-inconsistent but the behavioural rule of the private agents is not. To represent that situation would be difficult in a model of Lucas’s general form. Economists, I think, overlook the need to represent it, because they don’t realise it’s what they’re assuming.
If they did realise this, they might see that the crucial assumption — policymakers think time-inconsistently but private agents don’t — needs justification. Or they might give it up. But they’re prevented from confronting this by a confusion between two separate issues: time-inconsistency and policy-invariance. I don’t think that these relate in the way that macroeconomists tend to assume.
But, as ever, I might have completely missed the point. At least I got to think about McTaggart, and now you did too, or do too, or will do too.
