Economics: Expected Value, Risk, and Uncertainty

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In this lesson, we'll introduce how economists describe people's behavior under uncertainty or the way people make choices in the face of risk. Let's suppose I offer you a gamble. I'm going to flip this coin and if it comes up heads, I'll pay you $2.00. If it comes up tails, on the other hand, I'll pay you nothing. How much would you be willing to pay to play this game with me? Well, let's figure out, first of all, what the expected value of this gamble is. The expected value is the average return if you play the game over and over and over again. The expected value is calculated as follows.
First of all, take the odds of each outcome and multiply those odds by the value of that outcome. So, for instance, the odds of the coin coming up heads are 50 percent or one-half. The value in that case would be $2.00, the amount of money that I would pay you. The odds of the coin coming up tails are also one-half and your payment, in that case, would be zero. You get nothing according to the rules of this game. So, if you multiply the odds one-half times the outcome, $2.00 and add that to the odds one-half multiplied by zero dollars, you get the expected value of this gamble, which in this case is equal to $1.00. Think of this $1.00 this way. If you played this game with me over and over and over and over again, your average payoff would approach $1.00.
Now, that's the expected value of the gamble. How much would you be willing to pay me to play this game? Well, some of you might be willing to pay $1.00 to play the game. In that case, you are what we call risk neutral. A risk neutral agent values a gamble at its expected value. You're the people who would pay me a dollar to play the coin toss game. Some of you might be willing to pay me even more than $1.00, maybe a $1.05 or $1.25 to play this game. If your willingness to pay is greater than the expected value, we say that you are risk inclined or risk loving. These are people who are willing to pay more than the expected value to take a risk. These are people who are excited about the prospect of winning and value that more than the possibility of the loss.
Now, many of you out there would not be willing to pay me a dollar. Your maximum willingness to pay for this gamble may be only 75 cents, or 50 cents, or 25 cents. This is what we call risk aversion. People who are not willing to pay the expected value of a gamble are described as risk averse. For a risk averse agent, the risk of losing is valued more heavily than the possibility of winning and therefore the value of the gamble to that person is less than the average return, less than the expected value. Now, it's kind of interesting why it is that people will buy insurance and also go to Las Vegas and gamble on a vacation.
Sometimes we act as though we are risk averse. Whenever you buy insurance on your house, you're paying someone to take the risk away from you and usually what you are paying is greater than the expected value of your loss. There's a very small probability that your house will actually burn down. But, if you multiply that probability by the value of your loss, in the case of a fire, you'll get the expected value of the loss to you. The fact that you're willing to pay more than that, that you're willing to give up more money than you expect to loss in order to be protected from the risk is an indication of your risk aversion.
On the other hand, if you buy a lottery ticket and go and gamble in Las Vegas, you are paying more to play that game than you are expecting to win. That's because you're focused on the possibility of a big return is enticing you into the game. You're behaving as a risk lover. If there is a million to one chance of winning the lottery and the return on winning is a million dollars, then the expected value of a lottery ticket is $1.00. If you're willing to pay $2.00 for a lottery ticket, it's because the jackpot excites you and the risk of losing your $2.00 is not as important to you. You're behaving as a risk lover.
So, sometimes we'll behave as a risk lover, in one aspect of our lives, but as risk averse in other aspects. Most of us are risk averse when it comes to the possibility of losing our income or losing our house or losing something else that is of great value. On the other hand, we're likely to gamble with small amounts, because we enjoy the prospect of winning a big jackpot. So, the way economists describe risk attitudes is with three concepts. People who are excited about risk would be called risk lovers or people who are risk inclined. These are people who are willing to pay more than the expected value in order to take a gamble. People who are neutral about risk, who value a gamble at its expected value are called risk neutral and finally, most of us, with at least large amounts of value, are risk averse. We would be willing to pay more than the expected value of a loss to be protected from the loss and less than the expected value of a gain in order to have the chance of winning. We're the people who value the coin toss game at 50 cents or 75 cents, but never at $1.25.
This is the notion then of how economists think about risk behavior. This idea of risk behavior is the basis for a theory about how socks are priced, based on their expected value, how insurance is priced and people's willingness to pay to avoid risk becomes the price of insurance. Also, it has to do with how much extra you would have to pay one of your employees if you put her on a sales commission. Once her income is uncertain, she'll need some kind of compensation to make her willing to take that risk. So the theory of risk behavior is used frequently in Economics to describe financial markets, to describe a sales plan, to describe insurance markets. It's a useful concept.
So, for a quick review. To calculate the expected value of a gamble, take the probability of each outcome and multiply it by the value of that outcome, add them up and you get the expected value. Now, to consider how much an agent is willing to pay to take that gamble, you need to know about the agent's risk attitude. If the agent is risk inclined, he'll pay more than the expected value. If the agent is risk neutral, he'll pay exactly the expected value and if the agent is risk averse, he's willing to pay some number that's less than the expected value.
Market Failures
Uncertainty
Understanding Expected Value, Risk, and Uncertainty Page [2 of 2]