States of the World

States of the World In this section we introduce the notion that in the future the outcome in the world is not certain. Plus we introduce some related concepts.

Possibilities We will assume in our analysis in this chapter that tomorrow either one of two things will happen, and wealth (or income) might be different depending on what does occur. So, wealth depends on which state of the world occurs. An example of this notion might be that tomorrow it either rains or shines. Wealth may depend on which results. On the next screen let’s see a graph we will use to do some analysis in this area.

States and wealth Wealth at state 2 What is being measured in the graph is the wealth that would occur at each state of the world. Only one state occurs, but we plot the wealth that would occur in each state as an ordered pair. Wealth at state 1

example If there is no fire your wealth will be 100, but if there is a fire your wealth will be reduced to 40 (60 damage and assuming you have no insurance.) Maybe you buy insurance for 20 – no fire wealth is 80, fire wealth is 100-60-20 +60 = 80. Wealth at fire (80, 80) (100, 40) Wealth at no fire

Some notation: si = state i, where i=1, 2 wi = wealth in state i, pi = probability of state i, and (E1, E2) is the endowment point – which you would have if you bet nothing. Odds Suppose you have an opportunity to bet on s2 and are given odds of q2 to 1. If you bet B on s2 you win q2B if s2 occurs, but you lose B if s1 occurs. If you can take either side of the bet, you can bet on s1 but must grant odds of q2 to 1. If you bet B on s1 You win B is s1 occurs, but you lose q2B if s2 occurs.

The budget line given this opportunity is w2 = E2 + q2E1 – q2w1,where if you bet on s2, w1 = E1 – B and w2 = E2 +q2B. and if you bet on s1, w1 = E1 + B and w2 = E2 – q2B. Example: You are given opportunity to bet on tails and are given odds of 2 to 1. You have $100. The budget line is w2 = 100 + 2(100) – 2w1. If you bet $25 on tails, w1 = 100 – 25 =75, w2 = 100 + 2(25) =150. If you bet $25 on heads, w1 = 100 + 25 = 125, w2 = 100 – 2(25) = 50.

Example: You are given opportunity to bet on a fire (by buying insurance) and are given odds of 3 to 1. s1 = no fire and s2 = fire. E1=100, E2=40 The budget line is w2 = 40 + 3(100) – 3w1. If you bet $15 on fire, w1 = 100 – 15 =85, w2 = 40 + 3(15) =85. w2 (fire) This is a graph of the budget line. 85 40 85 100 w1 (no fire)

Expected value Recall in the example we had a “basket” of wealth = 100 if no fire occurred and wealth = 40 if a fire occurred. The expected value of this point or basket is the average value of the wealth outcomes, with wealth weighted by the probability of occurrence of each state. To continue with the example, say the probability of a fire is .25 and the probability of no fire is .75. The expected value of the basket (100, 40) is .75(100) + .25(40) = 75 + 10 = 85. Each basket or point in the graph has an expected value.

Iso-expected value lines The expected value, EV, of a point in the graph is EV = p1w1 + p2w2, where the p’s are the probabilities in each state and the w’s are the wealth values in those states. We could rewrite this equation as w2 = (EV/p2) – (p1/p2)w1. This is a line. If there are many points that have the same expected value, then all the points would be on the line. We would call the line an iso-expected value line (iso means same in Latin, I believe.) On the next screen we see a bunch of iso-expected value lines, with lines farther out from the original having a greater expected value. Notice the slope of a line is the ratio of probabilities.

Iso-expected value lines Each downward sloping line has the same expected value all along the curve. Wealth, state 2 45 degree line Wealth, state 1

Fair Odds Odds are said to be fair odds if the expected value of any bet is the same as the expected value of not betting at all. Back to the insurance example, Expected value of no bet .75(100) + .25(40) = 85 Expected value of $15 bet on fire .75(100 – 15) + .25(40 + 3(15)) = .75(85) + .25(40 + 3(15)) = 85 So odds of 3 to 1 are fair odds.

When an individual is offered fair odds the budget line coincides with an iso-expected value line. Now, on slide 7 we had an iso-expected value line for the insurance example. If odds are different from fair odds, then this line would not be budget line. When odds are 3 to 1 we have fair odds and that line is the budget line. Say odds are 3.5 to 1 on a fire. The budget would be w2 = 40 + 3.5(100) – 3.5w1. If you bet 15 on a fire w1 = 100 – 15 = 85, w2 = 40 + 3.5(15) = 92.5

In this example the odds are better than fair odds, the budget line in this case is steeper than at fair odds. Both lines go through the endowment point (100, 40). w2 (fire) 92.5 85 40 85 100 w1 (no fire)

Riskiness On slide 10 we put in a 45 degree line. The significance of this line can be thought of by considering a point on the line. The individual would have the same wealth which ever state of the world occurs. This point is then one where the individual knows with certainty what wealth they will have next period. Moving away from the 45 degree line along an iso-expected value curve would mean those points are more risky relative to the 45 degree line point because the variation between the two wealth values at those points gets larger.

example You have a dollar. A game of chance is available. If heads comes up you get 50 cents ( and then you have 1.50), but if tails comes up you lose 50 cents (and then you have .50). Heads will come up 50 % of the time and thus so will tails. If you do not bet, EV = .5(1) + .5(1) = 1. If you accept the bet EV = .5(1.50) + .5 (.5) = 1. The same expected value occurs here. The no bet has the same wealth in either state and would thus be on the 45 degree line. If the bet is accepted there is a greater amount of variation between the two wealth possibilities.

Risk Aversion – A person is risk averse when the person always prefers the least risky among baskets with the same expected value. So, when odds are fair the budget line is the iso-expected value line. The point on the 45 degree line has the same wealth in either case, so it is least risky. In the insurance example the endowment point was (100, 40). Wealth - fire 45 degree line Wealth - No fire (E1, E2)

Gambling at favorable odds Say you are risk averse and have $5. Say you have an opportunity to bet on the toss of an unbiased coin. If the odds on tails are 1 to 1, then some points on the budget line are i) If the bet is $1, w1=4, w2 = 6. The expected value of this beat is (1/2)4 + (1/2)6 = 5, which is the expected value of not betting at all. So the fair odds are 1 to 1. ii) If the bet is $2 w1 = 3, w2 = 7. We see these points on the next slide.

Wealth - tails Note the points are (5, 5) (4, 6) (3, 7) and when odds are 1 to 1 the points all have the same expected value. This means (5, 5) is preferred because the values are least variable. This establishes the indifference curves as seen on next slide. 7 6 5 3 4 5 Wealth - heads

Wealth - tails So, at fair odds this person will not bet. But if odds are better than fair, the person may bet some. Let’s do that next. Note when the odds change only the budget line changes – here rotates around endowment. 7 6 5 3 4 5 Wealth - heads

Wealth - tails If the odds are better, like 3 to 1 on tails then the person will see that small bets can lead to more utility, but if the risk averse person has to make large bets, they would be worse off and thus not make the bet. Note small and large depend on the initial wealth value. 7 6 5 3 4 5 Wealth - heads

States of the World