## Monday, July 26, 2010

### wheninrome15 on "Can a Soda Tax Save Us From Ourselves?"

Corrections appreciates commentary. A comment by “wheninrome15” was left in response to our post “Can a Soda Tax Save Us From Ourselves?” (June 4th, 2010).   It was thoughtful enough to warrant its own post in response.

The post by wheninrome15 follows:
I'm going to have to side with Mankiw on this one, but his argument is not completely clear, so I can see why you would go in this direction with it. Maybe he is just aiming at a more general audience, but if we're going to do the real deal, we have to address the elephant in the room, namely dynamic inconsistency. Below are my notes from thinking the matter through, hope they will benefit you as well.
To frame this issue, let's first consider a 2-period model with discounting (with 2 periods it doesn't matter what sort of discounting is going on, geometric, hyperbolic or otherwise). The agent maximizes $$u_1(x_1)+du_2(x_2)$$ subject to some budget constraint (say, $$x1+x2=M$$) where d is the discount factor.
In solving this problem, we know that, starting from an allocation of $$u_1(x_1), du_2(x_2)$$, the agent moves an extra dollar to the second period precisely when the transfer causes $$du_2$$ to go up by more than $$u_1$$ goes down. If we are instead thinking of a “multiple agents” framework, then it must be that such bargaining occurs precisely when agent 2, with utility function $$du_2$$, gains more from an extra dollar than agent 1, with utility function u1, loses. So you could think of this as Coasian bargaining, but the 2nd period agent is _not_ someone with utility function $$u_2$$; rather, he has utility function $$du_2$$.
[Sidenote: By the way, in one sense it is an illusion that Coasian bargaining is occurring here. Why are the agents trading with utility functions $$u_1$$ and $$du_2$$ rather than $$u_1$$ and $$u_2$$? The problem is a total unilateral lack of property rights. Agent 1 can steal whatever he wants from agent 2 (provided he has free access to credit, he can even go into debt, which agent 2 will be forced to repay). Agent 2's share is completely determined by agent 1's altruism. If $$d=0$$, for example, then agent 2 is simply screwed, unless we really are thinking of him as an agent with utility function $$du_2=0$$. Another clue is that the outcome is completely independent of the initial assignment of property rights (i.e. period 1 and 2 endowments). But for the present purposes it is actually somewhat useful to continue to suppose Coasian bargaining is occurring, so let's keep it.]
So, when you say that Coasian bargaining will occur, let us be clear that you mean between agents with utility functions $$u_1$$ and $$du_2$$, not $$u_1$$ and $$u_2$$. When Mankiw uses the word “externality,” he does not mean that perfect bargaining isn't taking place, but rather that it is taking place at the exchange rate of 1 to d rather than 1 to 1. Coase does not say what the exchange rate should be, it simply says that, given the exchange rate, trade will occur.
To say that agent 1 imposes an externality on agent 2 is to say that agent 1 is not fully weighing the effect of his actions on agent 2's utility...so how much should he be weighing it? How much, really, should we care about agent 2? Here we are discounting his native utility function $$u_2$$ by a factor of d, but maybe that's just fine...in reality we might think discounting should be going on. Perhaps people are fine with the fact that they don't care about tomorrow as much as they care about today. It would be a poor social planner who made it his goal to eliminate discounting that people really wanted around. It's hard to construct a defensible argument that people shouldn't be geometrically discounting. In a multiperiod model, with discounting $$(1, d, d^2, d^3,...)$$, there is no job for a social planner. But on the other hand it's pretty easy to take issue with hyperbolic discounting and the like. Dynamic inconsistency is rotten, and there is value in helping people to eliminate it [see endnote, but don't read it till you finish this paragraph]. Once you have dynamic inconsistency, everything you're saying simply flies out the window. Coase does not answer the question of what the relative price should be; you have to decide that for yourself, you have to take a stand. In the world of dynamic inconsistency, “the” utility function is no longer well-defined, because it depends on the perspective in time that you choose! You may want a “t=minus infinity” perspective or a “t=10 periods ago” perspective or a “t=now” perspective...but once you pick it you're stuck, you have to evaluate everything from that perspective. It is no longer simply maxing utility, but rather maxing utility w.r.t. time t. Go ahead and treat the agent's weights of ($$1, bd, bd^2, bd^3$$,...) as gospel if you like (that's quasihyperbolic discounting from t=now. The b captures the notion that the agent discounts in the usual geometric way except that he discounts all future periods by an additional factor of $$b<1$$, i.e. he really cares about now), have that be your criterion for how resources ought to be allocated if you like...but all the agent's plans will just fly out the window next period, won't they? If agent 1 wants to stop agent 2 from gorging at agent 3's expense, Coase will not save him! Agent 1 wants the terms of trade to be $$bd$$ to $$bd^2$$ (i.e. 1 to d), but next period they will simply be 1 to $$bd$$. One solution is for agent 1 to precommit, and indeed in a world with perfect information and frictionless and complete commitment, there is no dynamic inconsistency problem. But in reality people are often naive or inert. Thus, the reasoning goes, people can potentially be helped by a soda tax that helps them to resist soda that isn't really maximizing their utility (w.r.t. the point in time that you have decided they really care about). In matters such as these, protecting people from themselves is always always always about dynamic inconsistency. So if your argument does not go there, then you can be sure that it's not getting to the heart of the matter. The point of this is not at all to convince you that soda taxes are a good thing on net; that will come down to Mankiw's last sentence. But that is Mankiw's point too, that it comes down to his last sentence. I do not think his argument stops short of that.
[Endnote: I said that with dynamically consistent discounting, it's hard to argue that a person should do something else. The reason is that, for any proposed alternative, you would be telling them to do something they'd never want to do, no matter what perspective in time they were looking at it from. But with dynamic inconsistency, their decision about what's best depends on their perspective in time, and so in fact you have to make a judgment call about what perspective to call best. And once you pick the perspective, you are forced to concede that the agent -- who does not stick with the perspective you picked (or any perspective, for that matter) -- is doing things that do not maximize his utility.]

In broad strokes, your argument as Corrections understands it can be summarized by the following points (in order of their argumentation):

1. Mankiw did not address dynamic inconsistency but it is relevant.
2. We can summarize the intertemporal problem as a problem of negotiation between two agents.
3. That point (2) is an illusion because the first individual has full property rights.
4. Mankiw's point suggests that perfect trade may be taking place, but at the wrong "exchange" rate.
5. We can't make an argument against geometric discounting very well, but we can against hyperbolic discounting.
6. Hyperbolic discounting leads to an "unnecessary" loss in utility.
7. When evaluating utility loss, one must pick a period and discuss utility from that discounted period only.
8. People are naive or intert, and because of this, they may be helped by a soda tax.

Our responses are organized as follows:

• We agree that dynamic inconsistency is an elephant in the room, but not that it is important.
• We argue that the primary vehicle for dynamic inconsistency, hyperbolic discounting, is not suitable because it cannot generate a significant loss in utility for individuals (which is why it is so often used in public policy analysis).
• We argue that dynamic inconsistency generally, and hyperbolic discounting in particular is either 1) secretly hidden among individuals 2) have some semi-hyperbolic discounting along some of their income but not all 3) are constantly starving to death in secret.
• We argue that most empirical time preference literature is not necessarily a product of inconsistency, following a line of argument established by Gary Becker and Casey Mulligan.
• We note that irrationality itself is endogenous, using a paper by Gary Becker and Yona Rubinstein on terrorism and fear to make our main point.
• We disagree that one cannot argue for hyperbolic-style discounting--we believe it makes a great deal of sense when we deal with the idea of tradeoffs-with-certainty and tradeoffs-without-uncertainty.
• Finally we note that hyperbolic discounting may largely be an artifact of the laboratory.
• We conclude with a rule of thumb.

First, we agree that Mankiw did not address dynamic inconsistency, so we did not speak to it. We spoke to a case in which an individual body is composed of many different individuals over time. We further noted that these individuals appeared to all have a degree of income, due to savings. We then suggested that given some sort of exchange between past-and-future individuals is occurring, the two should be on the Pareto Frontier. If they are not, because a future-self smokes, then he would do better to save less, and smoke more, in the "Coaseian" bargain that we suggested. We suggest that this argument remains untouched, and maintain our point that Mankiw is wrong in the point he made, in our reading.

We see your point as related-in-conclusion but actually quite different from Mankiw's current-self future-self terminology. Yours is about a single, specific individual attempting to maximize their total utility.

First, we address hyperbolic discounting. For the uninitiated, economists generally model time preferences as exponential discounting. This suggests that people are as impatient between today and tomorrow as they are between tomorrow and the next day. Hyperbolic discounting suggests something special about today, and that people value the difference between today and tomorrow differently from tomorrow and the next day (less). The difference between the two systems in discrete time is displayed graphically below (click to enlarge; the graphic is deliberately very large in the enhancement). Hyperbolic discounting offers a "now-oriented" bent to discounting.

To understand why hyperbolic discounting might be of concern to economists, we can examine how two individuals who have ten time periods to consume 100 units of a good might behave. We give one exponential discounting with $$\beta=.95$$. This indicates that he would be indifferent between consuming 1 unit of good today or .95 units tomorrow. Similarly, we give another with quasi-hyperbolic discounting .95203 and the .96, deliberately calibrating it for comparability with our exponential discounting--they choose to consume the same amount on the first day. However, because in each period, the hyperbolic discounter is newly-impatient upon arrival at a new period, and initially they consume more than they planned in the first period. Using log-preferences, this is displayed graphically below (click to enlarge).

We might notice a few things. First, that the red line, the hyperbolic plan, is different than the green line, what actually happens. The second is how, with reasonable parameters (log-preferences and the aforementioned discount rates) and over a long span (ten periods, or years as calibrated), the deviation from optimal plan is not very large. Indeed, if we were to calculate how much our hyperbolic discounter should pay to stay on his optimal plan, it would be .0322 units, of his 100 units. That is, in this reasonable example calibrated for a ten-year horizon, individuals lose 0.03% of their initial income due to hyperbolic discounting. The sheer smallness of this number suggests that economists concerning themselves with this loss in public policy without first examining the much larger government waste must have ulterior motives.

But let us forego our concerns about the minute nature of hyperbolic discounting and instead note that even were it not small, any person without hyperbolic discounting should be hunting for these individuals with as much passion as the harpies who hunted an Aeschylean Orestes in The Eumenides. Why should we hunt these people with great passion?

Say they have a $$\beta$$ of .95 and a $$\delta$$ of .975. Therefore, in exchange for 1.026 units of period three, I buy one unit of period two consumption from you. Time passes one period, and I can now sell you the one unit of period two consumption for 1.08 units of period three, netting myself a profit of .05 in the third period without sacrificing any consumption.

Of course, this does not have to be done with only one unit--it can be done with your entire fortune. Because the relative prices between future periods are changing for hyperbolic discounters, any arbitrager may come along and, over the course of two periods, consensually exchange all of a hyperbolic discounter's fortune away from him--a hyperbolic discounter and his money are soon parted.

This leads us to conclude that either a) hyperbolic discounters do not exist b) they are well hidden within the population (and thus safe from such traders or c) they have starved in the streets, silently, bereft of the money that was once theirs.

We might further note that time preferences are endogenously determined, as Gary Becker and Casey Mulligan do in their 1997 Quarterly Journal of Economics paper "The Endogenous Determination of Time Preference." They note that by endogenizing discount rates, "it appears possible to explain with a model of rational behavior many assertions in the literature that are claimed to imply irrational choices." Their list includes the fact that people are not equally patient, that income is associated with higher consumption growth, and the relationship between schooling and consumption growth, relationships previously identified as irrational.

Finally, and in light of our previous point, we should note that any lingering inefficiencies will be small. It is difficult to see any market failure preventing individuals from investing in their own time preferences, in individuals requiring the government to provide that good, especially given information asymmetry going the wrong way. At any place where there should be great inefficiencies due to "irrationalities", we should also see great individual effort to overcome the irrationalities.

This is seen, for example, in the paper "Fear and Response to Terrorism: An Economic Analysis" by Gary Becker and Yona Rubinstein, who identify the role of fear in economic behavior, finding that individuals overcome their terror (they study suicide bombers and the use of buses by Israelis) more when they have incentives to do so. So too, do we expect individuals who lose the most because they are irrational to invest the most in changing their own behavior. We might add that subsidizing "irrationality" though a series of government nudges and shoves encourages the opposite type of meta-behavior.

Let us forego the ideas that hyperbolic discounting cannot generate large losses in utility with reasonable parameters, or that they provide arbitrage opportunities, or that individuals will optimally invest in controlling their irrationality and that there appear to be no market failures. Instead, let us note that even without these, hyperbolic discounting might have a real economic rationale. Specifically, there is a reason that today and tomorrow and tomorrow and the next day are different. The decisions we make are assured between today and tomorrow, where we have no such assurances in the next day.

Corrections only sees hyperbolic discounting literature applied in the literature on public choice, perhaps because only public choice examines long enough time periods for hyperbolic discounting to become important (for example, discussing global warming). In this case, there is an obvious reason why individuals should be today-focused. Specifically, because tomorrow has a degree of uncertainty that is not present today, but is present in all future periods. Let us imagine, for the sake of argument, that who is in power politically is an i.i.d. process. Then in making decisions about what legislation to pass, and what to have it depend on, whoever is passing the law should "trust" themselves more than future generations of lawmakers, because they have a smaller chance at being in power in all future periods (and due to our i.i.d. nature, the same, nonunity chance of being in power in all periods). This is a case in which hyperbolic discounting is "rational," though inefficient due to public choice incentives, rather than having anything to do with discounting.

Finally, we note, as Glenn Harrison and Morten Igel Lau do in "Is the Evidence for Hyperbolic Discounting in Humans Just An Experimental Artefact?," that while individuals participating in laboratory experiments appear to prefer to be paid in the present rather than the future, this may be a function of rational payment uncertainty rather than any discounting. Indeed, in this case, hyperbolic discounting in the lab in this situation is actually optimal behavior.

A number of the points we make hold for any form of dynamic inconsistency. We chose hyperbolic discounting because it was both mentioned in the post and is the most common form of dynamic inconsistency.

In light of these arguments, Corrections recalls a the opening of George Stigler's "The Intellectual and the Market Place"

The Intellectual has never felt kindly toward the market place; to him it has always been a place of vulgar men and of base motives. Whether this intellectual was an ancient Greek philosopher, who viewed economic life as an unpleasant necessity which should never be allowed to become obtrusive or dominant, or whether this intellectual is modern man, who focuses his scorn on gadgets and Madison Avenue, the basic similarity of view has been pronounced.
Corrections sees the relationship between the intellectual and individual freedom to be similar. Whether it is Rousseau's belief that individuals must be "forced to be free" (end of Section 7), or Plato's belief in a static, unchanging society where popular music, theatre, and poetry are banned, and the right to raise one's own children is eliminated (in The Republic), or Isaiah Berlin's corruption of the phrase "liberty" in Two Concepts of Liberty to argue a perverted liberty that required an obligation from one's fellow men beyond non-interference, intellectuals have rarely been friends of liberty.

It is in the light of a would-be tyrant that all paternalists should be seen, Corrections notes. As we have noted before, paternalists are would-be slavemasters with a smile.  History, economic and otherwise, as well as the economic theory of public choice, help us see this relationship. Individuals are very good at running their own lives, and are miserable, if not genocidal, when running the lives of others.  The ambivalence of intellectuals to a soda tax is simply a symptom of the hostility of intellectuals to the market place, and to liberty.

1. Thanks for the response! I just wanted to clarify something in reply.

First, to reveal a bit more of my position, I think that hyperbolic discounting itself is far too simplistic. I would say that, yes, a given person discounts different things differently. In particular, I think people are very much subject to what you might call "visceral impulses." They smell a warm pie and all of a sudden they are willing to pay a lot for it right now, even if they would have paid very little were they offered a chance to lock in their offer in advance. Or they see a place they used to do heroin and it makes it particularly hard to resist relapsing, even though they would have chosen differently were they able to lock their behavior in earlier that year.

Of course we can cook up the parameters to make inconsistency arbitrarily costly, or arbitrarily similar to regular geometric discounting. The question is, what do we actually think characterizes reality? I would argue for a very case-by-case answer to that question, and I would argue that in some cases, inconsistency is potentially devastating. From the perspective of t=newyears when the addict made his resolution to quit, this little bit of dynamic inconsistency has just destroyed his life. If that's the perspective I care about, then I am going to conclude that dynamic inconsistency can be terribly costly, and I may wish to help him overcome it. Dynamic inconsistency does not become important only in the very long term; rather, I would say you only need a 3-period model to bring out the worst bits of it.

So, wouldn't we see arbitreurs rushing out of the woodwork to take advantage of these people? Yes! They are called heroin dealers, and by all accounts they do a pretty good job of squeezing every last penny out of the addict through mutually consentual transactions, and often these inconsitent people do die and if they're famous you may even hear about them. Dynamic inconsistency could also explain the little "impulse buy" items lining the check-out aisles. This is a chocolateer inducing a spike in the consumer's demand curve and then extracting surplus which, depending on your perspective, may or may not be artificial.

In this sort of world, an individual arbitreur is limited in what he can take by the case-by-case nature of the discounting. A consumer might want more pie than he wants to have wanted, but he doesn't want 500 pies. And money itself is probably not terribly subject to non-geometric discounting, although it could be induced if people are up against a borrowing constraint and so they actually need more money to enable the consumption they are inconsistent about. It's worth noting that people in this latter situation _can_ be taken advantage of with, say, credit cards with high penalties for failure to pay off the balance.

Here is what I believe. I think that we live in a world where many people are eternally planning to quit smoking "tomorrow." I think that we live in a world where mail-in-rebates WORK on people.

[continued in next comment]

2. [comment continued]