So take a look at this graph of wealth versus utility. This is one of the most common graphs that you will see in economics. Particularly in microeconomics, which is the study of how we behave as individuals. And I think most of us are intuitively comfortable with the idea that while more money is better, the increasing incremental dollars become less significant as each new dollar arrives. So, if we have no money and suddenly we make $100, that's huge once we've got that $100, the next $100 is not as big a deal. It still makes us happier, but it's not, doesn't give us such a boost in happiness as the first 100. Now, notice that I'm using the word happy. On the graph, the vertical axis says utility. That's the economist's word for happiness. I prefer to use the word happiness because utility seems like a very cold word, and what we're thinking about here is literally what does make us happy. Because it's what makes us happy that will give us the incentives to go and work more. Now, the other thing that's interesting about utility or happiness is there's no quantitative scale the way there is a quantitative scale for wealth. We measure wealth unit by unit in dollars. But there's no unit called a util or a happy that we can measure on the vertical axis so all we can really look at are relative changes in happiness that come with earning more money. So incremental increases, the marginal utility of money, is decreasing each additional dollar gives us more happiness, but not as much more happiness as the dollar before. So let's play a little game here, and think about what makes us happy in terms of whether or not we want more money. So ask yourself what you would pay, how much money you would be willing to spend to play the following game. Here is the game, you are going to toss a fair coin if it lands heads, you'll get $15. If it lands tails, you'll get $5. What's the expected value of the game? It's $10. Where's that coming from? The probability of getting 15 is a half. The probably of getting 5 is a half. Multiply each outcome by it's probability, add those together and we get an expected value of $10. Another way to think about expected value is to ask yourself, if I played this game. A hundred times, what do I expect I would end up with? Well, after 100 times you probably are going to be right around or close to the $1,000 mark, 100 times the expected value. Now, ask yourself if the expected value is ten, but there is this variance around ten, how much money would you be willing to spend to play this game? Ask yourself, what's the utility or what happiness would you get from playing this game? The utility that you would get, U(G), the utility of the game, is half of the utility that you get from one outcome. That's for $15 plus half of the utility that you get from the other outcome. So just think for a second. How much would you be willing to pay to play this game? So I just asked you the question, how many dollars would you be willing to pay to play that game? Let's look at this in the context of our risk aversion graph. We've got the $5 mark and the $15 mark on the horizontal axis. Look at where those go to on the vertical axis. In particular, look at the difference between the U(15) and the U(5). Look at the range that you have there. It's a smaller range than the distance between 5 and 15 on the horizontal axis. Again this is part of the fact that each incremental dollar is worth a little bit less to us in happiness or utility terms. We can look at this mathematically. Take a look at the bottom of the slide, we've got the utility of the gain for us is half of the utility of 5, plus half of the utility of 15. And look at where that is on the vertical axis, the utility of the gain is worth less to us than the utility of the expected value of the gain, which as you can see is U(10). This is essentially the definition of risk aversion. But our utility for the outcomes individually is a little bit less than if we were looking at the combination of those outcomes together. That's risk aversion. Now, what we're going to see in divergence terms is that people are not always risk averse. And they don't always estimate the probabilities correctly. And that's where behavioral finance comes in. We'll see where there are predictable ways in which people make errors on both of those aspects.