Lecture 7: Risk Preferences I

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FRANK SCHILBACH: All right. Welcome to lectures seven and eight. We're going to talk about risk preferences. Hello? We're going to talk about risk preferences. In particular, sort from the perspective of expected utility, which is sort of the classical way of economics to view this. This is going to take one and 1/2, perhaps two lectures. Broadly speaking, we're going to look at what does economics usually assume. How does economics think about risk preferences, about choices involving risk?

How do we think about sort of measuring risk preferences? What are some of the implications and what are some of the limits of risk preferences in terms of what we can explain and what we cannot explain? And then in the following lectures we're going to have an alternative model of reference dependence preferences where essentially you're going to relax or change some of the underlying assumptions on how to measure preferences related to risk.

Problem set two will be posted soon-- later this week. I'm sure you can't wait for this. A reminder, late submissions will not be accepted. This is always like a commitment problem because people come up with all sorts of good excuses why they submitted the problem set late and then I kind of feel bad about it. So I'm here with committing to not accepting any late submissions for whatever reasons unless you have sort of a medical excuse or sort of an excuse notice. So no my PDF was corrupted and all sorts of other good reasons. I was a student once as well and I know students are very creative.

We will post previous problems sets, midterms, and finals for you to practice overall. I think of the problem sets less of a way for testing you or testing sort of whether you can do it or not, but rather as a way for you to practice things, whether you have understood the materials in parts of the past problem sets. And midterms and finals will sort of help you with that. As usual, please ask questions on Piazza or come to office hours.

So what we're going to talk about is, broadly speaking risk aversion. How do economists think about choices involving risk? Then again, I sort of outline sort of the very simple or the basic model of-- main workhorse model of economics-- is to think about choices involving risk, which is the expected utility model. We're going to then think about how do we measure risk preferences, the underlying preference parameters that sort of are embedded in this model. We're going to lead then to some absurd implications in particular. So discrepancy-- how people tend to think about small scale and large scale risk aversion.

What I mean by that-- essentially small gambles that involve a few dollars versus really large scale choices that involve thousands of dollars. And what I'm going to show you, very similar to some degree to how we think about time preferences where the typical exponential discounting model cannot explain both short run and long run time preference decisions that people make. That's just a sort of a calibration-- really very hard to do. Similarly, the expected utility model has problems with reconciling small scale and large scale choices that people make. I'll tell you that in more detail.

But in short, the summary is if people are risk averse when it comes to small gambles, that implies that they're absurdly risk averse coming from large gambles if you sort of take that model seriously. And so that's kind of like not really true in reality. And so then we sort think about how to relax those kinds of assumptions. OK. So first let's think about what kinds of choices and decisions do in fact involve risk and uncertainty. What examples do we have in your life? What is risky or what would involve uncertainty? Yes?

AUDIENCE: Whether or not [INAUDIBLE].

FRANK SCHILBACH: Right. Whether or not to get education. Whether or not to study and so on. Because the reward often is like uncertain. Right? You might get a job. You might not get a job. You might do really well in college. You might not. And so on and so forth. So the costs-- you might like it, you might not like it, and so on. It's not clear. So the costs and benefits are uncertain. There might be a recession when you graduate. And so on and so forth. So the returns and costs are both uncertain. Yes?

AUDIENCE: Making large purchases, like buying [INAUDIBLE]. Because you don't know [INAUDIBLE].

FRANK SCHILBACH: Yeah, exactly. You might buy a house or you might think about buying or renting more broadly. And there it depends a lot on essentially what's happening to the housing market. If the housing market goes up or down, the choice to buy a house versus renting is very different. Right? So if the housing market goes up, most likely you should probably buy a house. If the housing market actually happens to tank, the [INAUDIBLE] [? at ?] [? least ?] it's a bad idea to do so. Yes?

AUDIENCE: [INAUDIBLE] [? renters ?] insurance.

FRANK SCHILBACH: Right. So the two choices that we have so far were essentially choices involving risk where you sort of essentially decide to do something where the outcomes are uncertain or risky in some way.

Now, what you're saying, in some sense, is like, a bunch of other choices, potentially, are risk mitigation strategies. So you have, essentially, certain risk in your life. And you have choices where you could say, I could buy insurance. Or I could sort make other choices that mitigate or reduce the risk that I'm exposed to. And one sort of canonical example of that is purchasing any type of insurance but, in particular, renter's insurance, as you mentioned. Yes.

Yeah?

AUDIENCE: For farmers, the crop they choose to grow.

FRANK SCHILBACH: Yes, so essentially, call that-- in development in particular, or development economics in particular, there's lots of issues of production choices that people make. This could be like what crops to grow, whether you should buy a machine, whether you should start a business, and so on. So there's all sorts of choices in terms of what business to go into, what kinds of specifics, what product to sell if you have a business.

And in the farmer case, what crops should you grow? Should you buy fertilizer? Should you use other inputs and so on and so forth. Should you do [INAUDIBLE] there's a bunch of different other-- should you intercrop? There's a bunch of different other choices that you could make that essentially involve risks, because the outcome, essentially, is uncertain, because essentially, the season might be good or bad.

In the farmer's case, if you, for example, purchase fertilizer or a certain-- for example, you could purchase or use drought-resistant crops. Of course, if there's no drought, then that's not really that helpful. But if there's a drought, that's really high return to do. Yes? Yes?

AUDIENCE: Creating an investment portfolio.

FRANK SCHILBACH: Creating a--

AUDIENCE: Investment portfolio.

FRANK SCHILBACH: Yeah, so investing your money into a different-- in the stock market or in other sort of bonds or the like-- how should you invest the money? That's kind of related, to some degree, to the renting versus buying a house. You can think of buying as a house as one asset, one potential very sort of illiquid asset that you could buy. Similarly, you could buy stocks. You could buy bonds. You could keep just cash and so on. And there, that very much depends on, the return depends on things that are out of your hand, which essentially is just what the stock market might do.

So I think once you think about choices involving risks, essentially almost any choice in your life actually is risky to some degree or uncertain, ranging from going to college, doing problem sets, studying for exams. Which exams should you study for? Which questions are people going to ask?

Health decisions-- should you invest in your health or not? For a lot of diseases that people might get, it's often very uncertain. So even if you're smoking a lot, not every smoker falls sick or the like. No, just the risk of cancer and other diseases increases. That sort of financial investments, dating choices, and so on and so forth-- even friendship choices are, in some sense, risky. If you want, riding a bicycle-- that's very sort of small choices-- wearing a helmet versus not.

Essentially, almost anything in your life, if you think about what are the outcomes, what are the different choices that you have, and what are the outcomes associated with those choices, almost all of those choices are associated with uncertainty and a sense of you're not quite sure. Is the return going to be high or low? Then as I said before, in addition to that, there's sort of risk mitigating strategies and a sense of you have a lot of choices in a lot of issues that are associated with risk. And now you can choose to reduce your exposure to risk by purchasing insurance or, for example, also by avoiding certain behaviors, right?

So if you're worried about being robbed, for example, in a certain part of town, you might choose to go through that part of town. And that's a risky thing to do. Or you can have risk mitigating strategies where you just don't leave your house, or just go in different ways, or just never go into certain areas, which essentially are ways to protect you, to reduce your risk exposure overall. Any questions?

OK, so now let me sort of just tell you three broad, stylized facts that we're going to try and explain or try to tackle. And that's kind of what economics is trying to do. So the first question is-- the first thing that comes to mind when you think about economics and modeling choices involving risk is risk aversion. How do we think about risk aversion? Or what is risk aversion? Why are people averse to risk? Yeah.

AUDIENCE: Risk aversion is the tendency to avoid bets that are more risky, even though they will still ostensibly benefit you the same.

FRANK SCHILBACH: Mhm. So one-- you said, essentially, if there are certain bets that involve risk, where the expected values or an expectation you're going to do pretty well-- perhaps better than some safe outcomes-- people tend to avoid those kinds of bets. And that we might call risk aversion. So that's exactly right. But now, why are people doing that? What are the underlying reasons for that? Yes.

AUDIENCE: Potentially involved is loss aversion, which in the readings, I believe, some studies by [INAUDIBLE] and others mentioned that, showed that we had aversions specifically to losing money or utility rather than risk in and of itself. But that might be wrong.

FRANK SCHILBACH: Right. So one part would be to say, people might sort of lose or gain money in certain gambles. And what you're saying is, essentially, people might not treat the losses the same as they treat the gains. And then you might sort of decline certain gambles or certain risks. You're just worried about losing out, and you put a lot of value on that. That's exactly right. We're going to talk about this next week in a lot more detail. But what are some other reasons why people might not want to engage in risk? Yes.

AUDIENCE: I'd rather be definitely OK than either [? to ?] [? the ?] [? point ?] between super successful or die.

FRANK SCHILBACH: Mhm. And why is that?

AUDIENCE: [INAUDIBLE] diminishing marginal utility, [? maybe. ?]

FRANK SCHILBACH: Right. So one part is-- and that's exactly how economists think about this-- is diminishing marginal utility. That's to say, if you're really poor, getting your first dollar has really high value to you-- the reason being that now, essentially, otherwise you would starve, or you can't eat and so on. The value of whatever you purchase with that dollar is really high, because you just have nothing otherwise.

Now, if I give you a million dollars and then another dollar, then essentially the additional dollar that I give you after the million is just not doing very much. So the marginal utility of that dollar is low. And that's sort of diminishing marginal utility of wealth. That's exactly how economists talk about this. We're going to get back to that in a lot of detail. Yeah?

AUDIENCE: I think if you look a little more at extreme examples, [INAUDIBLE] if I lose so much money, I won't be able to afford my expenses or something. So it doesn't matter what is [? created ?] [? by ?] [INAUDIBLE] lose a lot [INAUDIBLE]

FRANK SCHILBACH: Right. So there might be sort of certain minimum standards, in some sense, that people have over their outcomes. Or you say, for example, you really want to have a place to sleep or you want to have some meal. And essentially, if you're below that threshold, essentially your marginal utility anywhere below that is really high, because you really want to get over that threshold. And so you might avoid certain choices or investments or the like. If there's even a small chance of getting below your threshold, you might be really averse to that. But what are some more perhaps psychological reasons why people don't like risks? Yeah?

AUDIENCE: [INAUDIBLE] [? potential ?] [INAUDIBLE]

FRANK SCHILBACH: Mhm. So that's right. That's what people do. We're going to also talk about that. But why do people not like the risk? Or some people like, actually, risk. But what is the issue about having a lot of exposure? Why do people purchase insurance, for example? Yes?

AUDIENCE: Some amount of uncertainty [INAUDIBLE] [? risk ?] [? aversion, ?] [INAUDIBLE] the higher the uncertainty of the outcome, [INAUDIBLE]

FRANK SCHILBACH: Right. So there's one part that's what you said, which is there's diminishing marginal utility of wealth. And this is exactly how we're going to model this and how economists think about this. However, there's other things involved, which is just things like anxiety or uncertainty. People might just not-- suppose a storm is coming up. I might purchase insurance. And if I purchase insurance, like flood insurance, or the like for a house, I might just feel much-- I might sleep better at night and so on and so forth.

I might just feel better about concerns-- or like health insurance, for example, might lower anxiety and so on, because people are just not worried constantly about like falling ill or like disasters happening. And that's beyond potentially any diminishing marginal utility of wealth. That sort of anxiety, stress, worries, and so on. Any other reasons? Yeah?

AUDIENCE: I think even beyond people wanting to not be stressed and anxious, there's also the element of, if you're buying insurance to smooth your consumption between different states of the world, it makes it a lot easier to plan for states after [INAUDIBLE] If you're--

FRANK SCHILBACH: Well, that's interesting.

AUDIENCE: [INAUDIBLE]

FRANK SCHILBACH: Yes. That's exactly right. So in some sense, if you have insurance, if you reduce risks in the states of the world, you can sort of, essentially, exclude a bunch of bad states of the world. For example, suppose you buy health insurance, suppose you have flood insurance, and so on. A lot of bad things that might happen you might be able to deal with. And you don't have to necessarily have a contingency plan of what if I fall sick and then go bankrupt, and all sorts of other bad things happen.

Or what if I-- there's flood insurance. I can't pay the bills and then have to leave my house and so on. So essentially, it makes planning easier, in part because it's just sort of easier in the mind. People feel more comfortable. And part-- it's actually computational or just easier to do.

So I added some reasons here. There's a bunch of different reasons. I think it's important to understand that economists have modeled risk aversion as diminishing marginal utility of wealth. That's a very simple way of doing this. It captures a lot of things, but perhaps not all of them.

And so here's some reasons that I mentioned. Sort of contingent planning becomes harder if you have risk. That's what Maya just said.

People are worried or stressed or anxious when they have lots of uncertainty. People might feel regret over missed opportunity. That's like, if I offer you some insurance right now, and you might say, well, it's actually unlikely that anything bad happens. But in case something really bad happens and you had the chance to actually avoid it, then you might feel particularly bad not just because the outcome is bad, but because it offered it to you and you didn't accept it.

There's sort of disappointment relative to expectations. This is essentially getting into territory of losses and gains. When people have certain expectations, they have an expectation to have a certain income and the like. Now, if bad stuff happens, they fall below those expectations and perceive those outcomes then as losses compared to the status quo or the expectations that they had.

Again, economists think about this as diminishing marginal utility of wealth. That's a very simple way of modeling this. And we're going to see kind of what the limitations of those are.

Now, I'm going to show you the three stylized facts about the world. And then we're going to discuss kind of how to perhaps model that. So the first one is essentially very simple.

People are risk averse in various ways. And one sort of basic fact is that lots of people buy insurance. People are willing to pay to purchase insurance that gives you essentially money or an expectation less money than you pay, right?

So fair insurance, as economists would model is or think about it, is to say if I purchase insurance that pays you the expected value of the insurance. So if I pay a premium for an insurance, there's a probability of something bad happening and then sort of a loss or some insurance payment that I get in case something bad happens. Fair insurance would be the premium is essentially, in expectation, the same as the expectation of the loss, which is the probability of the loss occurring and the actual payment that I get from the insurance.

Of course, the insurance industry wants to make money. So the insurance industry will not offer fair insurance, but less than fair insurance. Essentially, there's a price for purchasing insurance. Lots of people are willing to pay for insurance. People are willing to pay money to get insurance to reduce the risk or the exposure to risks that they have.

There's social security in various ways where essentially people sort of insure themselves or society insures them for old age or not being able to take care of themselves. You could argue that's perhaps also due to present bias or other paternalistic reasons. But surely like society, in some sense, is helping people insure themselves against potential states of the world where they might be in need.

There's very sort of other-- sorry-- institutions, including sort of extended families, informal insurance in developing countries, sharecropping, and so on and so forth. What's sharecropping? Yes.

AUDIENCE: When you go multiple crops at the same time.

FRANK SCHILBACH: No. That's intercropping or the like, but there might be referred to that as well. Yeah.

AUDIENCE: It's when the person rents out their land for you work on and you get all the benefits from the land, but [? they ?] [INAUDIBLE] [? rent ?] [INAUDIBLE].

FRANK SCHILBACH: Right, exactly. So it's essentially these kinds of arrangement, which are often somebody has land. Somebody rents out the land. And then you have to pay them something back and then can keep some of the output and so on. And often that's essentially some way essentially reducing risk.

So there's various sort of institution. But broadly speaking, we think, in many situations, people don't like risk. And they look for ways to reduce the exposure to risks.

There's also these informal insurance arrangements, which are things like people get together. And they sort of help each other. Whenever something bad happens, one person is then being helped by everybody else. And that's sort of replacing some sort of formula for insurance schemes and particularly in developing countries.

Second, risk reduction has its price. That is to say people are willing to take on risk if the return is high enough. So another way to put this is, if you want to purchase insurance, usually you have to pay for it. The insurance industry makes a lot of money.

Put differently, people are willing to take on some risk if things get cheaper, right? For example, if you think about buying a car, you could buy a super safe car with all sorts of safety features. Not everybody does that.

The reason is because cheaper cars are less safe. Cars are cheaper. So you're might just sort of willing to take on some risk in some situations for some price, right?

When you think about starting a restaurant, restaurants essentially fail all the time. Yet people are always sort of willing to do it. Presumably, the reason is because if you actually succeed, you're going to make quite a bit of money.

So if there's a high expected return, people are willing to take on some risk. People put in money into the stock market, so they increase their risk exposure. The reason why they do that is because, in expectation, they make quite a bit of money. But of course, that often entails risk.

So one way to think about the entire sort of finance industry is risk intermediation. Essentially, there are some businesses that have a lot of risk. And that risk is sort of offloaded to investors. And the investors accept that risk. They say, I'm willing to take on that risk, but only for a good return.

So I'm not going to take on risk from you if I'm not, in expectation, making money. But often, essentially, there's a trade off between. And this is-- well, if you've taken finance classes and so on, that's kind of obvious. There's a trade between risk and expected return.

In what situations are people actually willing to take on risk for its own sake or just-- so I told you people are risk averse. And that's true in most situations. But where are people actually willing to take on risks? Yes.

AUDIENCE: [INAUDIBLE] casino?

FRANK SCHILBACH: Right. Lots of people actually go to casinos. And here, the expected return is actually a negative. You know, you're going to lose money on average. Unless you're sort of a smart MIT student who can count cards in poker or something, you're going to lose money essentially.

So there must be some form of some preference or some desire to take on risk in some situations because you cannot just-- this is not like investing in the stock market where, on average, you're going to get money. If you go to the casino, on average you're going to lose money. Now, you could say it's so much fun and so on. There could also be addiction. There could be also self-control issues and so on.

Or there could be something about people's beliefs or preferences that induce them to take on risk. But notice that's different from what we discussed before. That's not really consistent with risk aversion because people choose to increase the risk that they're exposed to. Any other-- yeah.

AUDIENCE: Buying lottery tickets.

FRANK SCHILBACH: Exactly. People are buying lots of lottery tickets. Similar, they're doing lots of sports betting. And again, what you're doing here is, to be very clear, on average you're going to lose money and that you're going to increase risk or the exposure to risk.

And there's some questions on why people are doing this. We're going to get back to this. We're going to not talk about this today, but I sort of want to flag that. While people are risk averse in many, in almost, nearly all situations of the world of important choices that you encounter, there are some choices where people are, in fact, exposing themselves to risk in addition to exposure to risks that actually has high expected value. Any questions on this?

OK. So now, we're going to talk about the expected utility and sort of how do economists think about how should we behave. And that's a normative model, how the economists think about how people should behave when it comes to choices involving risk. So what is expected utility?

What does the model assume? Or what is it about? Yeah.

AUDIENCE: It's the utility of each state of the world multiplied by the probability of that state of the word occurring.

FRANK SCHILBACH: Right, exactly. So the assumption here is there's different states of the world. There's good things and bad things may happen.

You might get a job. You might not get a job. You might get a good grade, bad grade, and so on. You can sort of partition the world or anything that's going to happen in the future into different states of the world.

We can associate a utility, so an outcome, and an associated utility with that state of the world, right? If you get a good job, you get a high income. And if you don't get a job, you get a low income or whatever or unemployment insurance or whatever. And there are certain utilities associated with these outcomes.

Expected utility, now, is essentially say, OK, now for each of these states of the world, there's a probability of the state of the world happening. And I'm going to use essentially the weighted average, which essentially is weighting each state with their associated probability and then using the associated utilities for each state. And the expectations of those utilities is what's expected utility.

And that's very complicated. I'm going to sort of get [? to where ?] [? that ?] [? is ?] said in more words than necessary. Let me sort of get back to this.

So first, the thing about expected monetary value-- so suppose there's a gamble. And this is not a very simplified. There's a gamble. I'll call it G over two states of the world. State 1 occurs with probability p and yields monetary payoff x. State 2 occurs with probability 1 minus p and yields monetary payoff y.

Now, the expected monetary value-- now, this is not expected utility. This expected monetary value. This is how much money you get in expectation-- is essentially just the weighted average of those two things sort of weighted by the probabilities. So that's p times x plus 1 minus p times y. That's just the expected value of how much money you're going to get from this gamble.

Now, this is just a definition of fair gamble. And this is what economists tend to use a lot. A fair gamble is one with a price equal to its expected monetary value, right?

So if I ask you would you like to take this gamble, you're going to ask kind of, is this a fair gamble? Essentially, that's just asking about is it paying the expected value of this in terms of money.

I put monetary in parentheses because I could also provide you a fair gamble of apples. And then it would just be the expected value of apples. That would be also a fair gamble potentially.

Now, what's the expected utility of this gamble? Well, now, you need to have a utility function for each of these outcomes. So if my utility function is u of x, this kind of how much utility I get in the state of actually getting x or y.

So-called xi is where i is the state of the world. So essentially, now, it's the weighted average of p i times u of xi, so in this case, p times u of x plus y minus p times u of y. Any questions on this so far?

And so when you now think about evaluating gambles using an expected utility, if your utility function is linear, then you're going to essentially decide the same way as if you were just evaluating the expected monetary value, right? And if not, if the utility function is concave and convex, then essentially people are potentially risk averse. So how economists think about expected monetary value is the history of that used to be, the first theory that people wrote down was, this was a model how people thought people should behave.

There was a normative model that people wrote at some point and said, rational people should essentially maximize monetary payouts, which is say, if the expected monetary value of a certain gamble is high, you should accept it. Or if it's higher than its price, then you should accept it, otherwise not.

Now, it turns out that, in practice, that's not descriptively accurate. That's just not how people behave on the world. And in fact, and the reason being that people are risk averse in most situations.

When using, now, the expected monetary value, essentially they use the expected money value as a definition for risk neutrality. If somebody is risk neutral, if somebody doesn't care at all about risk in some situation, that person essentially just is maximizing the expected monetary value, right? So a decision maker is-- and this is a definition-- is risk neutral if, for any lottery G, she is indifferent between G and getting the expected monetary value G for sure. And so the decision maker is risk neutral if the utility function is linear, OK?

Essentially, the more money you get, then there's no diminishing marginal utility of money. Now, what's risk aversion then? A decision maker is risk averse if, for any lottery G, she prefers getting the expected monetary value G for sure rather than taking G. And the person is risk loving if the person rather has the lottery than the expected monetary value for sure.

These are just definition. That's just the way how economists think about risk. That's just definition, defining how we think about risk aversion and risk lovingness if you want.

Now, let me give you just a very simple example. Suppose a person with wealth, $10,000 is offered a gamble. The gamble is you can gain $500 with a 50% chance and lose $400 with a 50% chance. Will you accept this gamble? How do we do this now?

Suppose I'm just maximizing the expected monetary value. What am I going to do? Yeah.

AUDIENCE: You take [INAUDIBLE] something [INAUDIBLE]?

FRANK SCHILBACH: Right. So what I'm going to do is I'm going to just look at what's my expected monetary value of accepting your lottery, which is 0.5, which is the probability of a loss times 9,600. This is 10,000 minus the 400 that I lose plus 0.5 times 10,000 plus 500, which gives me 10,050.

If I reject the lottery, I'm just where I am before. Now, the risk neutral decision maker will reject the gamble, in fact, irrespective of the initial wealth. Because, essentially, everything is linear, so you just drop out the wealth. You can just look at what's the expected value regardless of how much money the person has.

AUDIENCE: So you mean [INAUDIBLE] [? accepting ?] [INAUDIBLE]?

FRANK SCHILBACH: Yes, sorry. That's a typo. Yes, sorry. That's a typo. Yes, thank you.

Yeah. So exactly, the expected monetary value is higher than the status quo, so you accept the gamble. And that doesn't depend on the initial wealth. OK. So now, what's the expected utility maximizer do? And how does an expected utility maximizer think about this? Yes?

AUDIENCE: In their calculation, rather than weighting the 9,600 and the 10,500, they'll weight the utility value.

FRANK SCHILBACH: Exactly. So now, we need the utility function. What's the utility of 9,600, the utility of 10,500, and the utility of 10,000? Now, will she accept the gamble?

Well, now, it depends essentially on the utility function. What's the shape of that utility function look like? In particular, it depends on the concavity of the utility function.

So what do I mean by the concavity of the utility function? This is concave function. What do I mean by that? What's the definition? Yes.

AUDIENCE: [INAUDIBLE]

FRANK SCHILBACH: Right. So one definition is a second derivative is negative. That's exactly right. That's true if the function is twice differentiable.

We have a slightly different definition that's slightly more general because it doesn't depend on differentiability. But essentially, it's to say it's the following definition, if the utility of a convex combination of two outcomes-- I'm going to tell you about this in a second-- is larger than the convex combination of the utility of those outcomes. What do I mean by that?

Suppose you have an outcome x and a utility associated with that that's u of x. Suppose you have an outcome y and a utility of u of y associated with that. And now, suppose you have a convex combination of x and y, which is just p is a probability of p times x plus 1 minus p times y. That's in the middle of x and y. So I take a weighted average of x and y that adds up to 1.

Now, if I have the utility associated with that convex combination, that's u of px plus 1 minus p times y. That's just the utility associated with that. Now, if I draw a line between those two graphs and look at what's the convex combination of p, so what's p of u of x plus 1 minus p of u of y?

That's essentially the weighted average of the utilities associated with x and y. Now, the question is, is the average of the utility of the average higher or lower than the average of the utilities? Can somebody explain this in words of what I just said? Or what do I mean? Can somebody repeat this? Yes?

AUDIENCE: I mean, technically, we're just trying to see if you take two points in the [INAUDIBLE] and draw a line between them, would the line be [? below ?] [? the curve? ?]

FRANK SCHILBACH: Right. Is the line above or below the curve? In this case, the line is below the curve. Now, what that means is, if I give you two outcomes and I say, would you rather have-- you could have x and y, or would you rather have the some weighted average of x and y? Now, the question is, what's the utility associated with this average of x and y? That's the thing that you see here on the left upper side is u of p of x plus 1 minus p times y.

That's essentially the utility of the weighted average. Is that larger or smaller than the weighted average of the utility, which is the thing that I show you here below? And what you see is, in this case-- and this is because the line is exactly as I say-- it's below the utility function.

If the line is below the utility function, that means essentially that the utility of the weighted average is higher than the weighted average of the utilities, which means essentially the function is concave. And that means essentially that the person is risk averse, as we call it. You'd rather have the average than the spread of the two outcomes. Any questions on this or comments?

OK. So you can look at this in detail but essentially it's a simple definition. So now, expected utility says essentially the following. It says a risk averse person rejects all fair gambles. And again, fair gambles are gambles that pay you the expected monetary value.

And the reason why that person does that is because the expected utility is lower than the utility of the expected monetary value. Essentially, as you just said, it's because the straight line is below the utility function. And that's essentially exactly the same definition here. Therefore, a risk averse person who has a concave utility function rejects all fair gambles.

Now, what does expected utility theory then say? Well, it says risky options are valued by doing three things. One is you have to define utility over final outcomes. And this is sort of getting back what you were saying earlier.

People might be worried about losses or gains or the like. We're assuming all of this away. We're just saying, their final outcomes-- how much money you have, what grades you have, how many kids you have, and so on, these are final outcomes, things that sort of where an absolute value is defined.

There's a utility associated with those outcomes. It's not about you expected more money or less money or the like. That's completely irrelevant. We're just looking at final outcomes.

How much money do you end up actually getting? And we're associating some utility with that. That's assumption number one, or that's the first thing one does.

Second is you weight these utilities for each outcomes by its probability. Essentially, we know what the probabilities are for each of those outcomes. We're going to take the weighted average of these utilities.

And then we sort of adding them up. And then by adding them up, essentially we can evaluate all sorts of lotteries. And then we just compare those lotteries either with some fixed amount of money or some other lotteries, the outcomes that we might get.

Now, there's two key implicit assumptions that are really important. One is only final outcomes matter. It doesn't matter what you expected in advance. It doesn't matter what you thought you might get. And it doesn't matter what you had yesterday.

All of these things are completely irrelevant in the simplest form of expected utility unless you sort of have information or the like. Only final outcomes matter. And then there is linearity and probabilities.

That's to say we put weight on the different types of states of the world relative to what the probability of those states are. So it cannot be that, if something is twice as likely, you should put twice as much weight on that in your evaluation of the outcomes. It cannot be that this is non-linear in certain ways. There's linearity in probabilities. Any questions on this?

So now, let me get back to what I said previously. What are we assuming away here? What are the things that are not in here? Yeah.

AUDIENCE: [INAUDIBLE]

FRANK SCHILBACH: Exactly. So essentially, all the other things that we said previously we are assuming away, for example, things like anxiety over certain outcomes, worries, stress, and so on. I'm also assuming away regret. I'm assuming away the gains and losses.

So essentially, anything we said before is essentially just stripped away and simplified in some sense and saying, we can explain a lot of behaviors just using the concavity of the utility function. Now, I have one example for you. And I think, I encourage, you sort of for any of these kinds of assumptions or kind of functions or things that you see in economics, it's worth sort of looking out in the world what people are actually doing and trying to see is that actually compatible with people, with behavior that we see in the world. And here's sort of one example.

So I actually don't think this is irrational behavior. And that's actually a good example of so what we might confuse with irrational behaviors. So I guess what we see is that expected utility has a lot of trouble explaining this behavior, right?

Because essentially, you spent like $5 on those lotteries. I'm offering you $10. So you get twice as many tickets.

So your probability of winning will be twice as high. Presumably, you prefer winning or losing. So, therefore, you should obviously take that deal unless there's some transaction costs and the like.

People are not doing that. The main reason that's mentioned here is regret. Now, what the person was saying here is these are perhaps irrational decisions.

I actually don't think that's right. Essentially, it's just we cannot rationalize the decision that we see with expected utility in the sense that it looks like the person behaves in irrational ways, but the person may just have regret aversion, something that essentially is not in the utility function. We sort of modeled it in the wrong way. And sort of, by not capturing this, we might miss certain behaviors.

Now, we're going to talk a lot about-- not about lotteries right now. We're going to get back to this a little bit in terms of why people engage in risk. But I just want to be clear on what we're assuming here. We're assuming a lot of stuff away, and I want you to be aware of that.

There's another question which actually the question or the video did not sort of try to tackle, which is why are people playing these lotteries in the first place. Why engage in the lotteries in the first place? In some sense, that wasn't clear either.

Again, we're going to get back to that. But the point of the video was to show you that, in some sense, these are a bunch of assumptions that are in the expected utility model. Not all of these assumptions are right. And you know, we want to be sort of aware of that.

But let me sort of just summarize what I just told you. And this, in some sense, is recap of 14.01 if you want, which I think in part you also discussed in recitation. Or there's like a handout from 14.01 that you can look at to study it in more details.

So many important economic choices involve risk and people are risk averse in many contexts. The expected utility model is a workhorse model of the economics for studying risk. And the way it's done is, essentially, one takes the weighted average of utilities from final outcomes. That's what matters for assessing outcomes.

OK. And so, now, we're going to see, OK, now taking that model very seriously, what can be explained? And what are perhaps the limits of doing so? And sort of risk aversion comes solely and exclusively from the concavity of the utility function. There's no other reasons to avoid risk. Then essentially your utility function is concave.

OK. So now, how do we measure risk? And that's, again, sort of definitions that economists use. When you think about risk aversion, how do you measure this?

Well, you measure it essentially through the concavity of the utility function, which is, as you were saying earlier, it's coming from the second derivative of the utility function. There's two main measures that economists use. There's sort of the absolute or the coefficient of absolute risk aversion. We call that r.

It's essentially taking the second derivative, which tends to be negative. So we take the negative of that. We scale it by the first derivative.

That's essentially to make it insensitive to if you multiply the utility function by a constant. Presumably, that doesn't change anything to your choices. So you risk aversion should not change. And, therefore, we sort of have to normalize or we divide by the first derivative.

A second version of that, or an alternative version, is the coefficient of relative risk aversion, which essentially is-- we call it-- gamma. Gamma is x, the wealth outcome that we look at times r times the coefficient of absolute risk aversion. It's the elasticity of the slope of the utility function, which I've written out here.

And sort of one very nice property of this-- and again, that's a definition. There's not much to argue with this. This is just how economists measure this. One nice property of this is, if you look at portfolio models or the like, one implications of constant relative risk aversion, which I'm going to show you a function of in a bit, is that people with constant relative risk aversion invest a constant share of their wealth in risky assets regardless of their level of wealth.

That's a main sort of result from a finance or portfolio models. In some sense, that's sort of irrelevant for you. These are just definitions in the sense of, if you wanted to measure risk aversion, this is what economists have used mostly, OK?

Now, if I give you this definition, how would you actually measure this? If you wanted to know my risk aversion, how would you do that? And so let me give you actually a utility function here.

So let me give you actually two utility functions. Here's just examples of one example of the constant absolute risk aversion function that you see here. Or a Constant Relative Risk Aversion, CRRA, this is what we mostly use.

So that's just the definition of a function that has the property that it has constant relative risk aversion. You can sort of verify that. We're going to focus here on CRRA functions, which are sort of what economists mostly use. So now, if I told you this is my utility function, my totally function looks like this, now how would you estimate my risk aversion?

AUDIENCE: [? I ?] can give you two gambles and then [INAUDIBLE]?

FRANK SCHILBACH: Right. So could you give me essentially choices of outcomes that have essentially some uncertainty or different risk involved. And then I'll give you the choices. If I say I prefer one over the other, can you tell what my gamma is?

AUDIENCE: No.

FRANK SCHILBACH: What can you tell me? Or what can you say?

AUDIENCE: I guess you can say the [? extent, ?] but maybe not [INAUDIBLE] value [INAUDIBLE].

FRANK SCHILBACH: Yeah. You can put some bounds on it, right? So I'm going to show you this in a second. But essentially, if I say I prefer one option over the other, you're going to have gamma on the left-hand side, gamma on the right-hand side, and give you some equations, essentially some inequality.

And then if you solve that equation, you're going to get some bound on gamma that sort of essentially tells you below or above. Or my risk aversion must be below or above a certain number. What else could we do? Yes?

AUDIENCE: [? Ask ?] [? them ?] [? when they're ?] [INAUDIBLE]?