Chapter 12 Uncertainty Key Concept: contingent consumption plan

1. Chapter 12 Uncertainty • Key Concept: contingent consumption plan • Preferences may depend on the probabilities of states. • Expected utility: u(c1,c2,1, 2)= 1v(c1)+2v(c2). • Risk aversion

2. Chapter 12 Uncertainty • Consider two lotteries • L1: 500,000 (1) • L2: 2,500,000 (0.1), 500,000 (0.89), 0 (0.01) • Which one would you choose? • Two other lotteries • L3: 500,000 (0.11), 0 (0.89) • L4: 2,500,000 (0.1), 0 (0.9) • Again, which one would you choose?

3. Risk is a fact of life. • In addition to lotteries, we face risks when walking across the streets (especially in Taipei), making an investment or even getting married.

4. Since there exist risks, outcomes are not deterministic. So how do we model this situation? • We assume outcomes can be described by a probability distribution. • There are states of nature (s1,s2, …,sn) with probability (1, 2, …,n). • On state i, consumption is ci.

5. Hence, we can think of a contingent consumption plan (c1,c2, …,cn). • We then draw a consumer’s preference on the consumption plan.

6. a contingent consumption plan (c1,c2, …,cn). • The contingent consumption plan is a specification of what will be consumed in different states of nature. • Preference on consumption plan certainly may depend on the probabilities of states, (1, 2, …,n).

7. An example • A lottery ticket costs 1 dollar. • The rule is there will be a winning number drawn from 1-100, each with equal probability.

8. When you buy a ticket, you can choose a number. • If the number you choose matches the winning number, you get 100 dollars. • Otherwise, you get nothing. • Your initial wealth is 200 dollars.

9. How many states are there? • Each state si could be the event that number i is the winning number. So there could be 100 states. • Hence (1, 2, …,n)=(0.01,0.01, …,0.01).

10. If you do not buy any lottery ticket, then your consumption does not depend on the state. • Hence (c1,c2, …,cn)=(200,200, …,200). • If you buy one ticket and choose number 1, then (c1,c2, …,cn)=(299,199, …,199).

11. If you buy two tickets and choose number 1 twice, then (c1,c2, …,cn)=(398,198, …,198). • If you buy one ticket of number 1, another of number 2, then (c1,c2, …,cn)=(298,298,198, …,198).

12. Finally you list all possible consumption plans in your budget set and choose the one you like the best. • When doing so of course it depends on (1, 2, …,n)=(0.01,0.01, …,0.01). • In other words, preference on consumption plan certainly may depend on the probabilities of states, (1, 2, …,n).

13. Another example • Suppose there is a prob of 1-p that some loss D occurs. • Suppose there is an insurance contract which pays the person 1 dollar in exchange for r<1 dollar of premium. • The initial wealth of the person is W.

14. Then we can model the situation as there are two states, (s1,s2) where state 1 is that there is no loss and state is that there is a loss. • Hence (1, 2)=(p,1-p).

15. If the consumer does not buy any insurance, then his consumption would be (W,W-D).

16. If he buys K dollars of insurance, then he has to pay Kr dollars no matter what the state of nature is and will get paid K dollars when there is a loss. • Hence his consumption becomes (W-Kr, W-D+K-Kr).

17. If he buys K dollars of insurance, then his consumption becomes (W-Kr, W-D+K-Kr). • So if we plot c1 on the x, c2 on the y, then his budget line will have the slope of -(1-r)/r.

18. Given we have the budget line and the consumer must have some preference over (c1,c2), so we can derive his optimal consumption plan and therefore determines how much insurance will he buy. • What is insurance really doing?

19. Now what does a “fair” insurance policy mean? • This means on average the insurance company breaks even. • Hence, (1-p)K=Kr or r=1-p. • Intuitively, the premium charged is the probability that the loss will occur.

20. We now turn to discuss the preference over the consumption plans (c1,c2). • As before, MRSc1,c2 =∆c2/∆c1 • If you give me one more unit of c1, how many units of c2 would I be willing to give up to stay indifferent.

21. Intuitively, this certainly depends on how likely I think the two states (s1,s2) are likely to be. • For instance, if I think the state s1 is quite impossible, then I would not be willing to give up many units of c2.

22. This suggests that the preference over the consumption plans (c1,c2) depends also on (1, 2). • Hence in general, we write the utility function representing the preference over the consumption plans (c1,c2) as u(c1,c2,1, 2).

23. Some examples: • u(c1,c2,1, 2)= 1c1+2c2, taking the expected value of the consumption. • u(c1,c2,1, 2)= c11c22 • u(c1,c2,1, 2)= 1ln(c1)+2ln(c2), taking the expected value of ln of the consumption.

24. The utility forms of the first and the third are quite special that we call them having the form of expected utility. • In general, the utility having the form of u(c1,c2,1, 2)= 1v(c1)+2v(c2) is a expected utility function. • I.e., the utility is the expected value of some function v() of consumption.

25. Let us examine whether your utility has the form of expected utility. Suppose you do, then there exists a v() so that: • L1: v(500,000) • L2: 0.1v(2,500,000)+0.89v(500,000)+0.01v(0) • L3: 0.11v(500,000)+0.89v(0) • L4: 0.1v(2,500,000)+0.9v(0)

26. v(500,000) • 0.1v(2,500,000)+0.89v(500,000)+0.01v(0) • 0.11v(500,000)+0.89v(0) • 0.1v(2,500,000)+0.9v(0) • Hence, L1wL2 -0.1v(2,500,000)+0.11v(500,000)-0.01v(0)≥0 • Similarly, L3wL4 -0.1v(2,500,000)+0.11v(500,000)-0.01v(0)≥0 • This is the Allais paradox.

27. So if you choose 1 over 2 but 4 over 3 (this happens to a lot of lab subjects) or you choose 2 over 1 but 3 over 4, then your preference cannot be represented by the expected utility function. • There is nothing wrong about it, it just shows that the expected utility function cannot accurately represent your preference.

28. In u(c1,c2,1,2)= 1c1+2c2= 1v(c1)+2v(c2) where v(c)=c • In u(c1,c2,1,2)= 1ln(c1)+2ln(c2)= 1v(c1)+2v(c2) where v(c)=lnc • We can think of v(c) as the utility of certain consumption. • In this sense, then u() is the expected utility of consumption (c1, c2).

29. Utility function u() of this particular form is called a von Neumann-Morgenstern utility function or an expected utility function. • The function v() is called the Bernoulli function by some.

30. If we have an expected utility function u and we multiply it by some positive constant a and add a constant b so f(u)=au+b, • then F≡f(u) is also an expected utility function.

31. f(u)=au+b, then F≡f(u) is also an expected utility function. • F(c1,c2,1,2) • =f(u(c1,c2,1,2)) • =au(c1,c2,1, 2)+b • =a(1v(c1)+2v(c2))+b • =1(av(c1)+b)+2(av(c2)+b) • =1f(v(c1))+2f(v(c2)) • so F is also an expected utility function.

32. This kind of transformation, multiplying a positive constant and adding a constant, is called a positive affine transformation. • Turns out that given an expected utility function, if you apply a positive affine transformation, then you get another expected utility function. Moreover, any other kind of transformation will destroy the expected utility property.

33. The most important property characterizing the expected utility is the independence assumption.

34. Independence • For instance, if u(c1,c2,c3,1,2 ,3)≥u(c1’,c2’,c3,1,2 ,3) then • u(c1,c2,d3,1,2 ,3) ≥u(c1’,c2’,d3,1,2 ,3).

35. If u(c1,c2,c3,1,2 ,3)≥u(c1’,c2’,c3,1,2 ,3) then • u(c1,c2,d3,1,2 ,3) ≥u(c1’,c2’,d3,1,2 ,3). • We can think of this as with prob 3, state 3 occurs. • In the first case, c3 is the outcome while in the second d3. • However, it does not matter what the outcome in state 3 is.

36. In state 3, some common outcome will occur. • Since it is common, it will not affect preference. • Hence preference is determined solely by the fact that 1v(c1)+2v(c2) ≥1v(c1’)+2v(c2’). • This has some flavor of independence.

37. Before, (c1,c2,c3) is consumed at the same time. • So it may be the case that when consuming c3,we prefer (c1,c2) to (c1’,c2’) • while when consuming d3, our preference is reversed.

38. It is different now because if state 3 occurs, states 1 or 2 will not occur. • Since • u(c1,c2,c3,1,2,3)= 1v(c1)+2v(c2)+3v(c3), • MRS12 • =MU1/MU2 • =1v’(c1)/2v’(c2) • which does not depend on c3.

39. Is it reasonable? • going to Venice (V) • watching a movie (M) about Venice • staying home (H). • However, I may prefer • (0.99)V+(0.01)H to (0.99)V+(0.01)M • because the latter entails disappointment, another outcome of the lottery.

40. Moreover, comparing L1 (500,000, 1) to L2 (2,500,000, 0.1; 500,000, 0.89; 0, 0.01) I may choose L1 because there is a possibility that I will regret that I should have chosen otherwise if I have chosen L2 and 0 is realized. • On the other hand, there is no such clear-cut regret potential exists between L3 (500,000, 0.11; 0, 0.89) to L4 (2,500,000, 0.1; 0, 0.9) • Regret is on a choice not made.

41. A person who prefers the expected value of a gamble to the gamble itself is a risk averter.

42. When a person is indifferent, he is risk neutral. • Finally, if a person prefers a gamble to the expected value of the gamble, then he is risk loving. • Draw a figure with v() concave, linear, convex to illustrate on an expected utility maximizer.

43. Fig. 12.2

44. Fig. 12.3

45. Go back to the insurance example. Assume a risk averse expected utility maximizer. • Then on an indifference curve, when c1 is greater, c2 has to lower.

46. |MRS12|=MU1/MU2=1v’(c1)/2v’(c2), since v() is concave • when c1 is greater and c2 is lower • v’(c1)/v’(c2) is lower • |MRS12| is smaller • So we have the usual convex to the origin indifference curves.

47. If insurance is fair, i.e. r=1-p, then at optimum, it must be • |MRS12| • =pv’(c1)/[(1-p)v’(c2)] • =(1-r)/r. • So v’(c1)=v’(c2) or c1=c2. • Facing a fair insurance, a risk averse, expected utility maximizer will choose to fully insure.

48. On the other hand, if the insurance company makes some profit, then r>1-p and p>1-r. • So v’(c1)/v’(c2) • =(1-r)(1-p)/pr<1. • Since v’’<0 • c2< c1. • Wealth when the loss occurs is not as high. Not fully insured.

49. At the 45 degree line, since c1=c2, with an expected utility function, |MRS12|=p/(1-p) the relative likelihood ratio of state 1 to state 2.

50. Look at another example. Suppose a consumer has wealth w and is considering to invest some amount x in a risky asset. • The asset has a return rate of rg in the good state and -rb in the bad state. Good state occurs with probability p and bad with 1-p.