Hello, welcome back. In life, there are plenty of situations where individuals, households or firms face two or more choices. For example, you might have to decide, how much time you want to allocate to watching these videos versus watching your favorite TV show. Or how do you decide how much to consume today, versus how much to save for retirement, so that you can consume in the future? I, for example, live as a non-vegetarian in a vegetarian household, so often I am faced by the decision of how much kale I would like to eat versus how much steak. That is, of course, if I have a choice. Or in the case of a portfolio, how much risk would you be willing to take for what amount of reward, right? How would you evaluate the trade-off between say, holding treasury bills versus a very risky strategy that is akin to picking up nickels in front of an oncoming steamroller, right? A very risky strategy. So in this lecture, we're going to be talking about how we make decisions when we have choices. Specifically, we're going to be talking about preferences, how economists use preferences to describe these choices and the trade-offs embedded in these choices. Finally, we're going to talk about utility functions and how utility functions help us represent preferences, in order to rank choices and make optimal decisions or optimal choices. So we use preferences to describe the trade-offs in choices. Preferences tell us how individuals make the trade-offs among different choices. What are your preferences, for example, over consumption today versus consumption tomorrow? All right, your preference will tell us that trade-ff, that trade-off specific to you or your preference over how much time you want to spend working versus how much time you want to spend playing. So preferences are unique to each individual. So when we think about our portfolio choices, for example, how much risk you might be willing to take for what kind of reward is specific to you. Your answer to that question maybe very different than mine because we may have different preferences. Now, since we can't really observe preferences, we use the concept of utility to measure how satisfied an individual is with her choices. So you can think of utility as a numerical index that describes preferences or it's an index that help us rank different choices, right? Simply put, utility helps convey the notion of how you feel, right? So the higher the utility, the better you feel. We construct utility functions to give us a systematic way of coming with a measure of that index, to rank the different choices. So you can think of the choices as the inputs to a function, and the utility function given as a measurement of how good we feel or how satisfied we feel with the choices we make. All right, of course the actual numerical value has no meaning, it's simply a way to rank the choices. All right, so here's an example, we can write down a utility function as a function of wealth. All right, gives us a way to measure how satisfied we are with the level of wealth we have, so we could write that as U(W) or a utility of wealth. Now, the investor's utility is not to final wealth itself, but the utility function translates the amount of wealth, the level of wealth, into a subjective numerical utility index. All right, so it just tells us how satisfied or how good we feel about the level of wealth. So let me give you a graphical example of utility function of wealth, might look like this. So on the X-axis I'm going to have, let's say, increasing wealth, right? And on the Y-axis I'm going to have utility of wealth. I'm not going to really label it in numbers on a scale because it doesn't really matter, right? So what might a utility function of wealth look like? Well, for example, it might look something like this. All right, so what does this utility function tell us? Well, first of all, for one, it increases with the level of wealth, right? Well, this is probably a reasonable way to describe investors' preferences, right? How we feel about the level of wealth. Because we generally prefer more to less, so we get a greater level of utility from higher levels of wealth than less. Now second, notice how I do the utility function, it is concave, right? And the concavity is a measure of how much you value that extra dollar of wealth. All right, so when we don't have a lot wealth, right? When we have little wealth, you're probably going to get much greater utility from going from having only $1 to having $2, right? That extra dollar is worth a lot to you, right? But if you had, for example, $100,000, all right, going from $100,000 to $100,001 doesn't really make that much difference, right? So when the investors are poor, towards the left of the axis, the slope is really steep. When wealth is high, on the right hand side, right, the utility is much more flat. Reflecting the fact that you already have lots of wealth, so that extra dollar does not give you as much utility, all right? So in economics we call this the diminishing marginal utility, right? As you increase wealth, the utility you get from an extra dollar goes down. All right, so notice that the slope of the utility function changes as wealth increases, right? High marginal utility, that is this area, right? Is when we really value that extra dollar, when we hurt the most if it's taken away, right? So we can call this bad times for us, right? Bad times are the times where the utility curve is really, really steep, when an investor hurts the most. Good times, on the other hand, is when the utility curve is flat and the margin utility is low, right? Such that having that extra dollar does not make it a big difference. All right, so in this lecture, we talked about how preferences describe trade-offs in choices. We use utility to represent investors' preferences. We construct the utility function to come up with an index that measures how satisfied an individual or an investor is with their choices, all right? The utility function also tells us what defines an investor's bad times. That is, in this example, the bad times are when the investor values that extra dollar the most. Now, going forward, you will hear me often talk about what constitutes an investor's bad times. You will see that optimal asset allocation will actually boil down to identifying what your, or your clients' and investors' bad times rare, all right? And accepting those bad times in exchange for an extra reward or a premium.
