Evidence that Contradicts Expected Utility, continued

Off we go. We have a little bit of tidying up to do from the last lecture, and then we will pick up in some related material. Last time we closed out a discussion of expected utility theory, and then did a quick application to demand for financial assets. I am pausing on that because one of the questions on the problem set requires a very similar technique. So if you are looking at the problem set and seeing a question that feels in some sense similar, that is the intention. It is intended to give you a walkthrough of this type of exercise.

Then we talked about ways that a person might violate expected utility theory. Not a discussion of things that could hypothetically happen, but rather things that happen with some regularity. Our first one was the Allais paradox, in which a person might plausibly select option A from question one, where the options are A and B, and might then switch to, in air quotes, option D in the second choice. We showed that if we break these numbers down a bit, these are similar types of questions. In fact, they are the same question. Then we discussed the Ellsberg paradox, which always bothers people. I am not quite sure why. In the Ellsberg paradox a person is faced with choosing balls from an urn and faces what is known as ambiguity over two of the colors, black and yellow. Specifically, they do not know exactly how many of 60 out of 90 total balls are black or yellow, and therefore they are faced with an ambiguous proposition, and a person may come up with incoherent probabilities. Lots and lots of you did. Then we got into our example from Kahneman and Tversky, which I will now illustrate.

Did anybody do the exercise I said to do? Nobody did. Man. Pro tip: I am not just saying stuff. I mean, I am just saying stuff, because it is funny for me, but I am not just saying stuff in the sense that you should probably do things when I say to do them. But you do you. It has been my experience that if there is an option of doing zero hours outside of class every week, people will take that option, even if it is not explicitly framed. I would not think of this class as a class where you can succeed and do zero outside of class, but you literally can. You could probably just find somebody else and work off of them for the problem sets, and you will do really badly on the exams. There are people who will take the first exam and just walk out. Pick it up and go, well, I don’t know what to do. So try exercises like this. Don’t worry, I will do it for you.

The example we had was a person who chose, in a first set, from A and B, where the notation omits outcomes with zero value. Let us turn these choices into mathematics. This is the type of exercise we need to get into the routine of doing ourselves.

For problem 3, the pattern we are trying to explain is somebody choosing B over A. Why are we trying to explain that? Because I asked you to raise your hands, and most of you put your hands up for B. So how do I do that in our language? The expected utility of B is 1 times the utility of 3,000 plus $w$. And for A it is 0.8 times $u$ of 4,000 plus $w$, plus 0.2 times $u$ of $w$:

A natural way somebody might make a mistake is omitting that last term. Just because our notation omits it does not mean we can omit it. If you want it to look like the others, you could write it as $0.2\,u(0 + w)$. It is a bit goofy, but it notes that there is a 0.2 chance of getting zero. In the other option, of course, there is no chance of getting zero.

Now the second choice. The predominant pattern is choosing C over D. Just feel that in your squiggly bits. It is at least intuitive to me. What does that look like? Now we have 0.2 $u$ of 4,000 plus $w$, plus 0.8 $u$ of $w$, and this is larger than 0.25 $u$ of 3,000 plus $w$, plus 0.75 $u$ of $w$:

Everybody see how I got to that step? Now the magic happens. I have $0.75\,u(w)$ on the right and 0.8 on the left, so get rid of that bit and turn the 0.8 into 0.05:

Now look at the two inequalities, separated by a line. This is a mathematical equation; we all agree. This is mathematics. So, because this is mathematics, I can divide by a number. Say I divide by four on both sides. I am allowed to divide by whatever I want. The 1 becomes 0.25. The 0.8 becomes 0.2. The 0.2 becomes 0.05:

Anybody notice anything? The two equations are the same, except the inequality is flipped. This is a violation of expected utility theory, even though it is not obviously one. We had to do a little high school algebra, and I will train into you, whether via blood (metaphorically; the red blood on an exam) or through your own proper practice and judicious effort before the exam, ideally, that you have to do this. There is no way you can eyeball this. Maybe you can, but a lot of you didn’t.

The name has been sitting up there the whole time. Kahneman and Tversky labeled this subproportionality, and they noted that when we think about a larger reward, it has to be less likely. If it were more likely, we would obviously pick that lottery. So: larger, less likely reward. If we scale everything down in probability space by a common number, people are more drawn to the less likely and larger thing. That is what is happening here. People like the certainty of the 3,000. We scale everything down by a factor of four, and suddenly the inequality flips.

If you are so inclined to write things formally (you do not need to memorize this or note it), it says that if outcome $y$ with probability $p$ times $q$ is indifferent to outcome $x$ with probability $p$ (notice that when you take two probabilities and multiply them together, the probability gets smaller), then for any common number we multiply through by, the inequality becomes strict. Where this was an equality, now it becomes an inequality:

Again, if you do not like the mathematics, this is just the notion.

In case you are wondering about the zhuzh of the type of thing you will need to do on an exam: no, I am not going to ask you the names of these, obviously. I don’t care. You will need to use tools like what I just illustrated, on problems you have not necessarily seen before, but of similar flavors.

The reflection effect, unfortunately, I cannot get you on, because if your brain is awake the name illustrates too much. You take two options, say a 45% chance of 6,000 and a 90% chance of 3,000, and you reflect over the y-axis, if you want to think about it like that, or the x-axis, depending on your perspective. Basically, turn all the numbers negative. And often people’s preferences flip. Commonly the observed pattern is B, then C. Why? Because in C there is a chance I get back to even. In D I am almost guaranteed to lose $3,000. That sucks.

So Kahneman and Tversky note this reflection effect. Something akin to it can happen in expected utility. You cannot exactly use the same technique, because of diminishing marginal utility; it takes a little more work to show that this violates expected utility theory, but it does. They show that people’s preferences over losses are the opposite of their preferences over equivalent gains. Put another way, they see risk-averse behavior amongst only gains, that is, when all the numbers are positive, and risk-loving behavior amongst only losses, where the numbers are all negative or zero.

This basic fact is quite universal. In gambling there is what is often called a break-even effect. Somebody at the roulette table, we will use that example again, who is down 20 bucks is more likely to keep playing than the person who is up 20 bucks. At that point it is not diminishing marginal utility or anything like that. That is not it. It is that there is a chance they can get back to zero. We will illustrate this in more detail as we go on.

One I can get you on. All right, you are going to have to raise your hand and answer questions. Consider the following two-stage game. In the first stage there is a probability of 0.75, a three-quarter chance, that the game just ends. Game just ends, whatever. And there is a 25% chance you move on to the second stage. If you reach the second stage, you have a choice between 4,000 with probability 0.8 and 3,000 with probability one. Call these A and B. Raise your hand if you want A. Raise your hand if you want B. Got you all.

We can collapse this into the following.

Everybody, when given this one, was drawn that way. These are the same bets. The exact same bets. I just had this wind-up, the story: well, there is a chance you are going to move on, but then if you do move on… No. That means there is a 20% chance you get 4,000 and an 80% chance you get nothing, or a 25% chance to get 3,000. And in their data, in problem 10, the version framed like this, you get this, versus in the other version, without that framing, you get different answers. Basically, it completely flips. What was 78% of people choosing this, if you frame it independently, you get almost the exact opposite.

Now, I cannot keep doing these, but you get the idea. Another illustration they have: you get a thousand for sure, and in addition choose between a 50-50 of a thousand and five hundred for certain; or you get two thousand for sure, and in addition choose between a 50-50 of lose a thousand and lose five hundred for certain. Hopefully we can all see these are the same question. If not, you are exhibiting the isolation effect.

What is happening psychologically (I promised there was going to be some psychology sprinkled in; we are starting to get there) is that you are ignoring, if you succumb to the isolation effect, what are seemingly extraneous parts of the problem. Your brain just kind of turns off for this and acts as if it is not relevant. It is quite relevant. It is very relevant. They say that there are parts of problems in particular, although there are others, where people are prone to ignoring shared components. If two options both have a shared component, add a thousand, or there is a 75% chance you stop, people just go, all right, well, it happens for both of them, so I am just going to treat this problem as if it is in isolation. And that is super wrong, mathematically and by your own choices.

Recall on day one, I believe, I said that we are going to explore circumstances in which people do not maximize their own utility, in some sense. What does that mean? In this instance, I have to think: which of these answers is the right one? I think this one, where you just ask the question straight up, is probably closer to the person’s true preference, versus the one with the long story. But that does require us to take a stance on things that economic theory says should not matter, like a bunch of words. The bunch of words is leading to very different answers, and so we have to explore how those are having a psychological effect on a person.

There is a large literature on such things; they are often known as framing effects. Often these are what people characterize as behavioral economics. They are like, oh, framing stuff, that is behavioral economics. I am basically not going to talk about it other than this example. There are a few reasons. One, they are very hard to study. They are not false. They are hard to study because it depends on the specific context. What counts as extraneous information, what feels like it, depends on what else is on the page. And there is not a very systematic way of doing it. We cannot do things like this. What would we do? You write down the same inequality twice, and then your head would explode or something. Framing effects tend to be not super robust, for that reason. It depends on a lot of contextual information.

That was the end of this deck. We are not at all done, so don’t get excited.

Now we move on. We are going to describe a deeper theoretical problem for expected utility. In my experience it is a somewhat difficult intuition to grasp, so we will return to it a couple of times, and it will feel not right. Notice we are in the 2000s now, so we are getting to modern economics. Rabin and Thaler note that people tend to dislike risky prospects even when they involve an expected gain. The quintessential example of this is that most people, if they actually believe you, if they are actually taking you seriously, and you offer them a coin flip of heads they win 11, tails they lose 10, will not want to flip that coin. That has a positive expected value. It has an expected value of 50 cents. You, as economists: one, you should flip that coin. It is not that much money. And two, you should flip that coin infinite times. If somebody lets you, you should definitely do that.

They use as an illustration that same bet scaled up by a factor of 10: losing $100 and gaining 110. This example I find has resonance even in this room. If I made everybody bring $100 cash to class (unfortunately a thing I cannot do) and said, I am going to give this bet to everybody, I know you have $100 cash on you, do you want to take it? Less than half. For sure.

Now, you can explain this a little bit with our standard visualization. Money on one axis, $u$ of money on the other, and I am hyper-exaggerating the utility function. A concave utility function has the feature that for lots of gambles, if you take a 50-50 gamble and ask, would you rather do that or stay where you are, then because this bit is steeper and that bit is flatter, the expected utility of the gamble is lower than just staying at your wealth. I illustrated something like this before; it is also in the notes on the website. So a concave utility function, remember, is synonymous with risk aversion, a lesson we have described previously.

And therefore the classic explanation of this is, well, the person is risk averse. And they are risk averse enough to throw away a little bit of money. We have done this exercise. As a reminder, suppose the person is facing a lottery, and that lottery is gain 10 with probability 0.5, ah, semicolon, gain zero with probability 0.5. Or they are offered a sure thing of four. Armed with the definition of risk aversion only, you cannot say whether they may or may not take that four. They might. The answer is, it depends on how risk averse they are. They would definitely take five. That is the definition of risk aversion. Anything less than five depends on how risk averse. And so what people say is, look, the fact that they are throwing away a little bit is that they are risk averse. We already knew this.

Matthew Rabin says, wait a second. This does not work. It is wrong on its face, because, as Matthew put it, anything but virtual risk neutrality over modest stakes implies manifestly unrealistic risk aversion over large stakes. Anything but risk neutrality over things like this is going to lead to hilarious conclusions. Now, this puzzled economists for 100-plus years. We were not aware of this fact, so it should not be obvious. But I will illustrate it, and in order to illustrate it, we get a little bit more practice using the functions we have introduced in the class thus far.

Suppose you have wealth 20,000, and you turn down the opportunity of a 50-50 bet to win 110, lose 100. We all think this is a reasonably plausible setup, and that is kind of important. Yeah? Sure? Okay. So given that this is a reasonably plausible setup, that an actual human could actually do this, let us just work through the math. Imagine that this person has a CRRA utility function, and ask, what values of $\rho$ are consistent with that? That is, how large does $\rho$ have to be?

Recall from the last lecture that $\rho$ in this utility function has the nice feature of capturing the degree to which a person does not like risk. If Al and Bob have the same wealth, but Al has a larger coefficient of risk aversion, this guy, $\rho$, then he is going to be more risk averse than Bob. Notice, linguistically, I smuggled something in that your brain may not have caught. I had to qualify that with if Al and Bob have the same amount of wealth. If Al is Jeff Bezos and Bob is me, me Bob, and they have the same or different $\rho$, it is going to be a much more complicated comparison. However, we have some numbers here. So what would we do? Well, historically, members of this class would throw up their hands, do nothing, get sad, and get a bad grade, but you are better than the past people. Why? Because, I don’t know, we will find out. You tell me.

What you would do is say, let us plug stuff in. Ask when the expected utility of our current wealth is greater than or equal to the expected utility of this lottery. That is when 20,000 to the $1 - \rho$ over $1 - \rho$, our current wealth, is greater than or equal to the expected utility of the lottery, and the lottery is characterized by a 0.5 chance of 20,110 and a 0.5 chance of 19,900. So we take our 0.5, and we do 20,110 to the $1 - \rho$ over $1 - \rho$, plus 0.5 times 19,900 to the $1 - \rho$ over $1 - \rho$:

You can multiply through, and then you have to do logs, and this is the type of torture you get to do on problem set two, but fortunately, for now, I will just tell you the answer. It is just a numeric question. You can have anything solve it. I think I mentioned this, but as a reminder, there are no calculators on the exams. There is a tiny, tiny, tiny bit of arithmetic. You have to divide numbers by two. You can do that in your head. I believe in you. And if you don’t: imagine a question was worth 10 points, and you had something like the expression above somewhere, some intermediate step in your answer. I would give you nine points, and then I would circle it, and I would go something like, no, no, no, me so sad, please, why, no, no. And you laugh, but I will probably do something like that, and depending on how late in the exam you do it, you may get a longer and longer stream of my consciousness. So, don’t worry about it, is the point. This would still get you most of the points and a little bit of hilarious self-shame.

Now, to solve this, with a little work you can just isolate $\rho$, and you get $\rho$ of 18. Plus some decimals. Whatever, who cares, I’m bored, what are we talking about? This class is all making me hot or sleepy or both. Remember where we are headed. I said, not five minutes ago, that anything but seeming risk neutrality over modest stakes will lead to manifestly ridiculous behavior over large stakes. Paraphrasing. I am just going to round this to 19. This is just for rounding. It does not matter. It is not misleading. This is a very general conclusion.

Let us play a game. Let us change it. Rather than solving for $\rho$, because that is annoying, let us take a given $\rho$ and ask how I feel about different lotteries. We start with a $\rho$ of 19, because it is easier to write 19, and we change our lottery to lose $Y$, win $X$. Because I changed it to 19, I will accept it if, when I lose 100, I gain at least 110. Or 111. It is no longer 110, because I rounded it up to 19. This should not feel particularly troubling. It seems like perfectly reasonable human behavior. It is a little risk averse, but it is fine.

If I double the possible loss from 100 to 200, the gain I need to have goes up by more. It is not just doubling. Now the potential gain I need in order to offset that loss is about 250.¹ The intuition, hopefully, if I did not erase it, is still staring you in the face over here in the concave utility function. As I make this loss bigger, we are in the steep part of the utility function. So as I move this little dotted line toward the origin, then, because that is the flat part, I have to move farther out this way. Make sense?

Let us keep going. If the potential loss is 500, now I need to have double the gain. A person will now only accept this if the potential gain is nearly a thousand. This is starting to get rather risk averse. Notice this is a person who is not bankrupted by this. We are not at the origin of this function, literally; nowhere close. They have twenty thousand dollars; we are taking five hundred. They are not pleased, but it is one fortieth of their overall wealth. This is a thing that humans actually spend. Humans spend one fortieth of their overall wealth all the time. They buy cars and houses, and the house is usually significantly more than one fortieth of one’s wealth. At 750, it becomes about 5,800. We are 8x-ing it.

And here is a question. If we go to 1,000, how large does $X$ need to be? It is an actual question. Make a guess. Okay, 10K. Other guesses? Just shout them out. 15, 8, 12. So you are seeing an intuition. In everybody’s guesses, you recognize the pattern; you have extrapolated from a pattern that something is getting a little bit funky. The part that our brains do not do very well is that the correct answer here is infinity. There is no possible gain. You cannot set $X$ sufficiently large. This function has to be getting so flat this way that this steep bit cannot be overcome.

Now that feels weird. That feels like a very strange thing. Notice, I am not bankrupting this person. Every human being on the planet would take a gamble, lose one twentieth of your wealth to gain infinity dollars. Coin flip. Every single one. I cannot do this in class, of course. The point is that what we have been working with is bad to the core.

Expected utility theory has this feature because everything rests on a concave utility function, and that foundation is shaky. It is shaky in the following way. It has to do with where we started. If a person is risk averse over a small amount, then they are going to be more and more risk averse as we blow it up. Another way to think about it: who here had a TI-83 or TI-86 or something in high school? Or still does? Okay, enough of you. So you have got your screen on your calculator, and you take some curve, and you zoom in, and you zoom in, and you zoom in, and you zoom in, and you eventually get a line. This zoom-in is stakes of minus 10, plus 11. On the scale of 20,000, of course it is. I have zoomed in a ton. This is 100 times zoomed in relative to 1,000. So if there is curvature at all at this level of zoom, imagine it is this curvy at this level, then when I zoom out and add two zeros, it is going to look like that. It is going to just be kinked. And this part is so flat that I can keep going that way forever, and it is still going to be flat.

Now this is a very counterintuitive fact, counterintuitive enough that it made one person very, very famous. And it has nothing to do with that $\rho$ example. That is just a numerical example I am giving you so I can give an example tied to this class. Any concave utility function has this feature. Here is another example. If Johnny is a risk-averse expected utility maximizer, that is, he has a concave utility function (we have hopefully memorized that; I am going to keep cueing you; you need to memorize that this is synonymous with this), what does that mean? It means that Johnny would rather have the expected value of a lottery than play out that lottery. So imagine that Johnny is a risk-averse expected utility maximizer, and that for any initial wealth, Johnny will reject a 50-50 gamble of lose 100, gain 110. Then consider a 50-50 gamble of lose a thousand, gain $X$. What is the minimum $X$ he would accept? It is infinity. It is infinity for any wealth. Johnny can even be Jeff Bezos. If Jeff Bezos would turn this down, then Jeff Bezos is going to turn down ridiculous stuff for a thousand dollars.

And that is not a description of humans. That is a description of mathematics. And that description of mathematics, if we try to apply it to people, is going to lead to misleading conclusions. Keep in mind, $1,000 losses are incredibly regular. If you get into a fender bender, it is going to be $1,000. It is going to be more than $1,000, probably. If you have a health issue, it is going to be significantly more than $1,000, and you, or your parents, more likely, are paying a significant annual premium to mitigate that risk. Well, this says that in some sense the demand for insurance should be astronomically high. If people were behaving according to expected utility theory, health risks put you way out on this steep part of the utility function. If you said, look, your current wealth is here, you just have to take a little bit of money from that, people would happily take that. They would pay a lot for health insurance. We see a degree of that, but not nearly the degree that is implied by people’s actual real-world behavior over things like gain 110, lose 100.

So this is a tricky result, and the logic comes over final wealth outcomes. Rabin shows that this operates for any concave utility function. We are going to come back to this. But the point is that any small-stakes curvature implies ridiculous large-stakes curvature. And a lot of what happens in the wedge between economics and behavioral economics is that somebody smart takes the model very seriously, as we have done in this class. You take the initial model of behavior and say, let us see what it predicts. Let us see what it predicts in a variety of situations. Good theory, whether it is physics or economics, makes predictions in a variety of situations. And all theories are wrong, so it is okay to be a little wrong. Theories are abstractions from the world that we live in. But this is really wrong. It is really, really, really wrong, and it is really wrong in ways that we care about. It is going to make bad predictions over every market, like insurance, that actually exists and that people do real economic work in. So, whatever.

Now, if you are so inclined, you can think about this in terms of the marginal utility. The marginal utility is the first derivative, and marginal utility decreasing is the second derivative. And this basically says that the second derivative is really negative. That means it is really, really curvy, but it is really, really curvy in ways that are counterfactual. It just cannot be that curvy for actual humans’ preferences. This is called the calibration theorem, and it highlights that small-stakes risk aversion is just not coming from that particular channel.

Now, Nick Barberis, Ming Huang, and Dick Thaler did this as close to for real as you possibly can, over even larger stakes. Larger stakes make this easier for the standard model to explain. And yet they find that most MBA students turn down a real gamble of flip a coin, lose 500, gain 550. And they did this (it was fantastic; I think Dick still does it every now and then) after teaching them that if you are a manager in a company, you should take this gamble. If you are a fiduciary, that is, you have a legal responsibility to maximize revenue for your company, you should definitely take this gamble. This gamble is not going to bankrupt your company, and on average it is going to make you 25 bucks, so you should flip that coin. That is the logic I have picked on a couple of times, inherent in effective altruism, this notion that we should try to maximize expected value as a social, normative thing, something we ought to do. But one, people who are literally taught that, five minutes later, do not do it. So it is not a very good description of humans. And two, if you scale these numbers up enough, you get into bankruptcy risk. If that is $550 billion versus lose $500 billion, it is technically positive expected value, but good luck. It is a coin flip.

Another point is that MBA students have plenty of money. They should not be bothered, particularly, by losing $500, in some abstract sense. This particular example is not as stark as the gain 110, lose 100, but it basically is. The same person would turn down a coin flip of gain 88 trillion, lose 10,000. Come on. It is not positive infinity; we have gotten better, I guess, but not by much. I am, of course, reasonably confident that everybody would take that. Everybody in this room would find all of their friends, if they did not have $10,000, and run together as fast as possible, if you actually thought somebody was going to pay you this, and take it. Of course you would.

So I have introduced, in my mind (your minds are allowed to go wherever they want), what I believe is essentially the death knell of expected utility theory. First, we have seen some behavior that is inconsistent with the theory. So not only are we seeing behavior, we are seeing theoretical reasons why the core is just never going to quite work, if what we are trying to explain is that people tend to be risk averse over relatively modest stakes, things like $10 or $20, $100, even $500. That $500 example, I challenge you to ask your friends. Ask somebody after this class: if you were actually serious, would you take a gain $550, lose $500 coin flip with me right now? I think a lot of people would hesitate. Everybody would be like, I am not going to take that, ugh, I don’t want to lose 500 bucks.

Rabin and Thaler offer the following:

So you may recall that a few lectures ago I initially wrote down expected utility theory as a sum of probabilities times utilities of the outcomes, and put a little squiggle over it, because it is not right. We said no, no, that is not right. To do actual expected utility theory, you sum things up including the wealth. This is what the theory was written down as in the 1930s, when this was all formalized. This is what they wrote down. And Rabin and Thaler say, well, wait a second. Maybe we should go back to something closer to the version without the $w$. Maybe the notion is that, much like in the isolation effect, that example where I said there is a three-quarters chance the game ends and all your brains said, all right, well, I will just ignore that part, maybe we kind of just ignore this part. It is common to all of the lottery assessments. There are lots of lotteries; I am going to constantly have to put the $w$ in, over and over and over again. And it seems psychologically plausible, and indeed seems to happen, that people assess changes in wealth levels rather than absolutes.

And so we are going to build (in fact, we are not, but Danny Kahneman and Amos Tversky are going to build) a theory that is centered on these two features: specifically, that losses loom larger than equal-sized gains, and that preferences exhibit a sensitivity toward changes rather than absolute levels. That is what we will do with our remaining minutes. How are we doing? Good? Great? Groovy? Okay? Mediocre? Bad? Sad? Mad? Glad? I can keep doing this. It is going to be long-term, guys. This theory is called prospect theory. Prospect theory is a made-up term. Danny says he made it up, but then Amos says he made it up.

For the previous one, aren’t we also considering the wealth? When I think about it, if I do minus 10, plus 11, and then it is flattening out: if we just did the axis based on wealth, can’t we just realize how much change in wealth is happening? Why is that not working?

I think you have an intuition, and it is a good question for a couple of reasons. One reason is that I suspect a part of you feels like it is intuitive to just ignore the wealth and say, why don’t we just do it that way the whole way? It is plus 11, minus 10; let us just assess the lottery in that way. But the verbal question was, can we use it with the wealth, and that clause is slightly imprecise. I am not picking on you. We all ask things that way. What I think you mean is, why don’t we center this at the wealth? The utility function somehow centered around the wealth. If it is still a globally concave function, the same conclusions apply. So that does not do it. You cannot just take a globally concave function and shift it. How do I know this? Because two lectures ago I told you the numbers don’t matter. The shape of the function matters, but its location on the axes does not. We traditionally draw utility functions in the positive numbers. But if I take it and put all the information over here, where all the numbers are negative, that is a valid utility function. It is totally valid. It will give you all the right answers. This person will behave according to expected utility theory; nothing wrong. All the numbers are negative, and it looks kind of weird. So I cannot just do a recentering with a globally concave function.

So, hold your horses. What does it mean to think about changes rather than absolutes? Literally, it means this is an absolute, and this is a change. An absolute is $w + x$. A change is $x$. We can formalize that, as I will in a moment. When we eventually get there, we are going to do a bunch of different stuff. I will describe an introduction, then present the full model of prospect theory. We are in this weird spot where I will probably do intro-y stuff and then postpone the second half to early next week.

Before we get into it, I want to remind you of the helpful tip I presented last time, or two times ago, I cannot remember. Some concepts are hard to get your head around, and using extreme cases is a good way to think about stuff. For many of you, if you have opened the problem set (now would be a good time to be able to answer yes to that), problem three might feel like there is some difficulty there. When in doubt, try thinking about extreme cases: a person who is infinitely risk averse, and a person who is risk neutral. Notice that the two extremes are not infinitely risk averse and risk loving. There is an asymmetry there. We do not want to flip over it. The definition only offers that a person is a tiny bit different from risk neutral, but how she actually behaves will depend on how risk averse, as we just illustrated, and that can lead to some important things.

So try using extremes. Think about limit cases, and think about what the limits are. For instance, in our handy-dandy $w^{1-\rho}/(1-\rho)$ function, $\rho$ is defined as being in zero to infinity. Its limit is not minus infinity; it cannot be. So what do these mean? If we put in zero, as we previously discussed, we recover risk neutrality.

Now we start with some intuitions about what it means to assess changes rather than absolutes. I promised I would answer the question, and here I am. I am going to use The Simpsons, because I am old as hell. Also because The Simpsons used to be funny. It is not what it used to be. Moe is the bartender. He says, if you want to signal me, use this bird call, and then he makes a bird call. And then an eagle swoops down and starts pecking him in the face. And he says, ow, ow, not the face. I cannot do the Moe voice, but you get the idea. And then the eagle starts pecking him in the groin. And he says, oh, oh, okay, okay, the face. And then the eagle switches back to pecking him in the face. And he says, whoa, that actually feels good after the crotch.

That is basically what it means to attend to contrast rather than absolutes. These are both painful sensations. But the contrast means that the sensation feels different. This is a dumb example. Of course it is. It is not that dumb. Here is a physical reality of how human eyesight works. The rods and cones in your eyes are very sensitive. They can see a single photon. They cannot exactly see it. It is a fun test: you can flash a single photon into a person’s eye, and they cannot see anything, it is not like you see a flash, but a person will respond better than chance that something happened. The reason they cannot recover anything more is that your vision is largely dictated by your brain, not by your eyes. However, if I make this room half as light as it currently is, that is, there are half as many photons flying around, you could still read your book. In fact, outside there are probably twice as many photons as in here. The sun is very bright. We cannot really appreciate how bright it is, because our brain counters that a lot. Our brain does a lot of this processing, and our brain is so good at attending to contrast and normalizing to the range that is local to our surroundings that we can do things like see in a very dark room, very, very, very dark, and see in regular daylight, where you are operating over possibly two or three orders of magnitude or more.

In virtually all physical, physiological, and psychological instances, people’s responses tend to reflect adaptation, change, and contrast rather than absolute outcomes. This is nearly universally true.

Mathematically, I am saying that something psychologically closer to the version without the $w$ feels more right than the version with it. Where we will get to is that we use a different function in there. Call it $f$. We will explore its properties. But we have two little steps. Step one is acknowledging this reality, which I will call reference-dependent preferences, because there is a reference point about which we attend to adaptation, change, and contrast. This is often implicit: the average brightness of the room, the average amounts of money that we experience in our lives, things like this. Sometimes it can be explicit. And our feelings (it is all about the feels), or the corollary of those, our choices, are also reference dependent.

So consider a modified utility function, where we think about something, again, call it $u$, call it whatever letter you like, it is a function, call it $f$: some function that takes two arguments. The traditional utility function we have used in expected utility has a single argument, a monetary outcome. It is a sum, but the sum is the sole argument. We are going to say no, we are going to take two different arguments. One is going to be the changes, and the other is going to be some reference level, which is used to assess those changes:

This will be clear in a minute. When in doubt, as I move forward, especially next week, if you want to think about the reference point or reference level, think about your current wealth, or the status quo. How things currently are. That is a little loose. It is not exactly formalized. We will formalize it. But it is a good intuition, and it is not misleading whatsoever.

The idea of reference-dependent preferences, and its representation in prospect theory, is the bigger of the two successes in behavioral economics that form the spine of this course. The first bit, prospect theory, forms the first third. The second bit, procrastination, or the tendency to prefer things now rather than later, forms the second. So you are essentially going to get the greatest hits of behavioral economics.

All my slides are screwed up right now, because I do not know exactly where I am going to end every day. Where did I end today? Here. So on Tuesday we will discuss prospect theory in all of its detail. It is not that hard. We are going to change this function, and then we will explore its applications over the coming lectures, discovering along the way some of its key features.

Reminder: the problem set is due in class on Tuesday. The next one will go out right after it, and it is due, I believe, two weeks after that. The timing of its due-ness has to do with making it due before the exam. So, as I warned, it is the longest one of the entire term. It is the whole point. You are going to learn via pain about spacing your efforts over time. You have thus far failed, but if you fail the next time, it will be more painful. Office hours will begin in a few minutes. Feel free to come any time between now and 5:30. Otherwise, have a nice weekend. And whenever you turn in your stuff, please staple it, as I mentioned before. That is it. See you later.