# What Does it Mean, Mathematically, When we Say That Markets are Efficient? Part Two of an Ongoing Series about Finance

It’s tough to be informed citizen in America without hearing, on regular basis, about how markets are efficient. Indeed, when we go about our day to day affairs, the implications of this efficiency come up in various small ways. We see apartment prices move up and down (and up again) in tandem across the city, despite the fact that no one is calling up every establishment and telling them how to price their units. Likewise, you’ve surely noticed how the grocery store never runs out of fresh produce – though the price of watermelons may jump in the wintertime to prevent shortages.

These are pedestrian and rather trivial examples of a phenomenon that affects all of us on a daily basis – not just through our pocket book, but in the ballot box, where one party in particular (I’ll let you guess which one) has a lot riding on the importance of markets being able to decide the proper allocation of income, healthcare, and investment across society.

But what does it mean that markets are efficient? In this post, I’ll go through a simple but compelling example, taken straight out of a class I TA’d earlier this year for the University of Chicago’s Scott Ashworth. The purpose is to demonstrate how the market (i.e. many people acting in tandem) can actually arrive at the best conclusion without anyone communicating with anyone else. Through the magic of statistics.

Let’s begin (for this, you’ll need to know the basic concept of maximizing through a derivative. You’ll also need to remember that the derivative of the natural log of x is 1/x. Well, actually, now you don’t need to remember it).

1. Let’s take a person. You, for example. You are participating in a very simply market in which you bet on an event. That is, you bet B dollars, and if the event happens, you receive \$1. If it doesn’t, you receive \$0. For obvious reasons, B is going to be between 0 and 1. For now, let’s just assume that it’s somewhere in that range (or domain, for all you/us math nerds).
2. You want to be as happy as possible, and more money makes you happier. So does love, friends, and family. However, in this betting example, you can’t win love, friends, or family. Only money. So let’s assume that you want all the good stuff in this world along with world peace, but when it comes to betting, you’d rather make money than lose it. Fair enough.
3. Given these foundational assumptions, we have an equation, that I’ll took through right after you see it:

Wait don’t leave! This is really simple. The left side speaks for itself. The right side is really made up of two parts: how much you earn if you win your bet, and how much you earn if you lose your bet. Little p is your best guess at the probability that you win. Obviously, if you think about it, p must be greater than 50%; otherwise, you’ll bet on the other guy. So, on average, you’re going to win p percent of the time, and lose (1-p) percent of the time. It’s a weighted average.

What do we take the weighted average of? Well, the two possible outcomes of the election. If the candidate you bet on wins, then your overall life savings (S) and increase by X dollars, where X is the amount of “bets” you made.  In addition, each of those bets cost you the going rate, B, so you lose X times B. On the other hand, if your candidate lost, your bet didn’t pay off, so you have the money in your savings account, S, and you’ve just lost B times X for all the tickets you bought.

Why did we log these two amounts? The answer will be obvious for any econ student,  but for those who prefer to spend their time doing more enjoyable things, you’ll have to trust me on this one. Log is a function that takes whatever is inside it and spits out something else. The larger the thing you feed to log, the larger the result – but, as the number inside log gets larger and larger, the amount the log spits out increases at a smaller and smaller rate. In layman’s terms, more money makes you happy, but more and more money make you happier at a slower and slower rate. You may be familiar with this as the law of diminishing returns.

4. Great! so now we have this equation. Let’s suppose you want to maximize it. That makes sense, right? Everyone wants as much happiness as possible. So let’s do what we’ve done since high school when we’ve wanted to maximize something, and take the derivative with respect to the only thing the better (that’s you) controls: how many tickets you’re going to buy (X).

That’s not too bad. We simply remember that the derivative of the natural log of x is 1/x. The rest follows from basic calculus. To maximize, we set the equation equal to 0.
5. Now, we rearrange:

All that’s happened is we’ve put XB on the left side and everything else on the right. You’ll see why in a moment.
6. Now we have something that’s getting pretty cool:

What’s happened here, you ask? Simple: we’ve taken the X number of tickets that you purchased, and added up all the X’s that the market has purchased. We need to do this because the market isn’t just composed of you. Instead, we’re assuming that there are n people in the market, and each one (who we’ll call i) is maximizing their happiness.  So we take the formula you arrived at in steps 5, and we multiply both sides by X. The left side now, instead of containing your X number of tickets, contains all the tickets everyone has purchased. The right side is just like the right side in step 5, but summed up across all n people. You’ll notice the subscript i now appears besides every p. It’s there because now, everyone has their own best guess at the likelihood that your candidate will win. Some people might think he’ll lose – in that case, their p is less than 0.5.
7. Below, we just rearrange the right side of the equation from step 6, taking out the constants from the sigma summation:
8. This is a very important step, and one that introduces economics into what has, until now, been a mathematical exercise:

We’ve taken the right side from step 7, and set it equal to 0. Why? Take a look at the left side of Step 6. That’s all the tickets everyone has purchased, times the amount they bet on each one. That’s the total value of what market will buy, given the parameters on the right side. When is this situation stable? Simple: when the market doesn’t want to purchase any more tickets – i.e., when the price, B, of each ticket causes neither buyers nor sellers to want to act. Just like the price of watermelons at your grocery store is stable because no one wants to sell them for a lower price and no one wants to buy them at a higher price, so to the market is stable when the buyers and sellers are stable – i.e. when the market is in equilibrium. Since the left side of the equation in part 6 is still equal to the rearranged right side of the equation in step 8, we can simply carry through the logic and set the equation equal to 0. The market is now in equilibrium.
9. Here’s the magic part. S is your savings – the amount of money you had before you went in to buy a ticket. Let’s assume that you’re not homeless and unemployed (in which case you probably shouldn’t be betting on elections to begin with) and that S is greater than 0. Since B is greater than 0, that means that the only way for the left hand side to multiply out to zero is for everything in the sigma summation to be equal to 0. It’s a mathematical property. So let’s reduce our equation to that:
10. Now we’re at the home stretch:

Let’s split that up:

Since B (the price of the ticket) is the same for everyone, we simply have n times B. Move that term to one side, and…
11. Voila

Now, It may not like much, but the equation above is the underpinning of capitalism. And that would be the law of large numbers. The equation we’ve just arrived at says that B, the price of each ticket, is exactly equal to the average of each person’s bet about how likely the candidate is to win – or more generally, it is the sum of everyone’s best guess. For a second, that may seem like a trivial conclusion – but it’s not. It says that if everyone in the market is doing what comes most naturally – maximizing their happiness – then the price will reflect the all the information contained by everyone participating in the market.

Why does this matter? For starters, think about how much work it is to run an election. You have to hire tends of thousands of attendants, open up countless voting places, and – in many countries – give workers valuable time off to cast their vote. What the equation above shows (and what websites like intrade model every day) is that simply asking people to bet harnesses the power of statistics to make a tremendously precise representation of their preferences, which only grows in precision as the number of voters increases. If that sounds like voter polling, that’s because it is. But it also the way prices work in our economy – whether it’s the price of Apple stock or the price of an apple.

*Some caveats worth mentioning at the close. First, the estimate of B is only as good as average of people’s guesses. That works great for an election – but not so well when you’re asking people who don’t much knowledge about something (we can very well be collectively stupid). Second, the model above is not precisely how stock prices work, because those aren’t binary. It is, however, how futures and intrade work, and the stock market works analogously. Finally, there is a very important assumption here that one could easily miss: the assumption that people begin with the same wealth and have equal votes. Whenever you’re dealing with money, the wealthy are far more likely to be able to afford big risks – they are far less risk averse than the average middle class family. But that’s a topic for another day.