Choices under uncertainty

In order to indicate events that have an uncertain outcome we use the term lottery. Every outcome in a lottery has a probability of happening. Probabilities of an outcome are indicated with a number between 0 and 1 and the probability of all the outcomes in the lottery are represented graphically by a probability distribution. Obviously the sum of the probabilities of the possible outcomes sums to 1. You can calculate the value you expect to obtain from the lottery. This value is named the expected value of the lottery and it represents the centre of the probability distribution.

It is computed as the weighted average (using the probabilities of each outcome as weights) of the outcomes of the lottery.

Expected value (EV): (probability of A) x (payoff if A occurs) + …+ (probability of Z) * (payoff if Z occurs)

The variance of the lottery represents instead how much the payoffs deviate from the expected value. It is used to represent how much a lottery is risky. A high variance means that the outcomes of the lottery can be far from the expected value meaning that there might be some outcomes which are much higher than the expected value but, at the same time, there could be some outcomes that are much lower than the expected value. It is computed as the weighted average (using the probabilities of each outcome as weights) of the squared difference from the expected value of each outcome.

Variance = P (A) (payoff (A) – EV)^2 + … + P(Z) (payoff (Z) – EV)^2

DIFFERENT KINDS OF PEOPLE

We can distinguish three different types of consumers according to their attitude toward risk:

• Risk averse

• Risk neutral

• Risk loving

These types of consumers will behave differently when they will face a decision between lotteries. This happens because these consumer derive utility in different ways and therefore have different utility functions. In general, if we give a risk averse person the possibility to choose between a sure amount and a lottery with the same expected value, he will prefer the sure amount. Risk neutral people, as the name suggests, are indifferent between a sure amount and a risky outcome with the same expected value meaning that these kinds of people will evaluate decision considering only expected values of lotteries. A risk averse consumer, instead, will prefer to enter a lottery to a sure amount.

EXPECTED UTILITIES

As we introduced the concept of expected value we can also think about the expect utility a lottery will deliver. It represents the utility you expect to receive when entering a lottery. And it is computed as the weighted average (using the probabilities of each outcome as weights) of the utilities the consumer gets from each outcome and therefore as:

Expected utility (EU): (probability of A) x (utility if A occurs) + …+ (probability of Z) * (utility if Z occurs)

If we consider a sure outcome his expected utility is simply the utility the consumer gets from his outcome and therefore it is represented as point lying on the consumer’s utility function. When you consider lotteries instead, the expected utility is no more a point on the utility function but a point on the line connecting the two outcomes. This is because the consumer, if he enters the lottery, will never get the amount of the expected value but either the lower or higher outcome.

THE UTILITY FUNCTION AND THE REASON BEHIND A CHOICE OF A RISK AVERSE CONSUMER

A risk averse consumer will get more and more utility as the value of the outcome he receives is higher meaning that the utility function is always increasing. However, it does not increase always at the same rate. The utility of a risk averse consumer grows fast at the beginning but then it slows down meaning that the consumer will appreciate more an increase in the outcome when it is low than when it is high. The marginal utility of this kind of people is always positive but decreasing. A function that easily can represent the preferences of a risk averse consumer is U=√x where x represents the value of the outcome. Let’s see with an example how a risk averse consumer makes his choice. Imagine that you offer to this consumer two choices: to get $50 or to play a game where there is 0.5 probability of getting$ 0 and 0.5 probability of getting $100. For the first possibility you can calculate and expected value of: 1∙50=$50 and for the second option the expected value is: 0.5∙0+0.5∙100=$50. As you can see both the options offer the consumer the exact same expected value of $50. We are going to consider as the consumer’s utility function U=√x. If we want to depict the problem graphically we have to represent the utility function for the consumer and the various outcomes offered to him. Outcome A on the graph represents the first possibility for the consumer (the sure amount of $50) while points B and C represent the two different outcomes of the lottery ($0 and $100). As said, both the options have the same expected value (measured on the x-axis) but give the consumer different utilities (measured on the y-axis). The expected utility of the sure amount is represented by point A (because the consumer can really receive that amount). The expected utility of the lottery instead is represent by the point D on the line connecting the two outcomes with a value on the x-axis of $50. As you can see in the graph, because of the preferences of the consumer, the sure amount delivers an utility of √50 = 7.07 while the lottery delivers an expected utility of: [0.5 * √0 ] + [0.5 * √100] = 0.5 * 10 = 5. Therefore, because the sure amount delivers an expected utility which is greater than the lottery, the consumer will choose the sure amount.

THE UTILITY FUNCTION AND THE REASON BEHIND THE CHOICE OF A RISK NEUTRAL CONSUMER

The utility a risk neutral consumer gets increases with the value of the outcome he receives always at the same pace. Therefore, both when he earns a little amount of money and when he earns a lot his utility will grow in the same way. Because of that, the utility function of this kind of consumers is represented with a straight line. The general utility function of this kind of consumers is: U = a+bI where a can either be 0 or a positive number and b must be positive (because otherwise the utility function would be flat, meaning that he is indifferent whether he earns a lot or nothing at all, or downward sloping meaning that the more you offer to the consumer the less he is happy). Therefore, the marginal utility is always constant. We can ask this kind of consumer to choose like we did with the risk averse one, so to choose between a sure amount of $50 and a lottery with payoffs $0 and $100 with the same probability. We represent, as we did before, the outcomes on the utility function of the consumer. However, contrarily to what happens in the cases of the other kinds of consumers, If you consider two outcomes of a lottery, the line connecting them (where we need to represent the expected utility of the lottery) coincides with the consumer’s utility function. This means that neither the utility function, nor that line is higher than the other indicating that the consumer is indifferent between the two. If you want to explain that algebraically let’s suppose the utility function of the consumer is U = 5x where x is, as always, the value of the outcome. The sure amount will give the consumer an expected utility of 250 (simply 1, which represents the probability, times the utility derived from $50 which is 250) and so will the lottery because expected utility is equal to: 0.5∙0+0.5∙5∙100 = 250.

THE UTILITY FUNCTION AND THE REASON BEHINF THE CHOICE OF A RISK LOVING CONSUMER

The utility of a risk loving consumer increases slowly when he receives a low value but it increases much faster when he receives more. The marginal utility of this kind of people is positive and increasing. The preferences of a risk loving consumer can be represented for example by the function U = x^2 where x represents the value of the outcome. Let’s examine what this consumer will choose between a sure amount of $50 (A) and a lottery with prices $0 (B) and $100 (C) with equal probability. As we did for the previous types of consumer we can represent the three possibilities on the graph with points along the utility function . Notice that, this time, the line connecting the two outcomes of the lottery is higher than the utility function. The expected value of the sure amount is simply 1∙50^2 = 2500 (A) while the expected value of the lottery is: 0.5∙0^2 + 0.5∙100^2 = 5000 (D) . As expected the consumer will choose to enter the lottery and not to take the sure amount.

HOW TO CONVINCE A RISK AVERSE CONSUMER TO CHOOSE A LOTTERY OVER SAFE OUTCOMES

Consider once again the choice with the point of view of a risk averse consumer. Remember that the consumer chose the sure amount to the lottery having the same expected value. But, what happens if we give the consumer the possibility to choose between a sure amount and a lottery having an higher expected value? In order to keep the problem as the one analysed before, we can ask ourselves how much should we reduce the sure amount so that the consumer prefers the lottery to it. We define the minimum difference in order to make the consumer indifferent among the choices as the risk premium. As we reduce the sure amount offered to the consumer, we move along the utility function to the left and we stop when the expected utility of the lottery (point D) has the same height. In our case the expected utility of the lottery for our consumer is 5 therefore we have to find on our utility function the x corresponding to the y value 5. Because utility was defined as U = √x, we need to replace U with 5 and find the x. So: 5 = √x and therefore x = 25. This means that, because of his preferences, our consumer is indifferent whether he chooses the sure amount of $25 or a lottery with expected value of $50.