Risk Aversion - June 14

[Amit Goyal]

Assume utility is increasing in wealth levels.
Consider a consumer who has the following wealth levels:
w1 = 10, w2 = 6,  w3 = 4 and w4= 0.  She faces two lotteries:
(0,1/2,1/2,0) and  (1/2,0,0,1/2). If she is a risk-
averse, expected utility maximizer, rank and explain her preferences
over the two lotteries.

[Chinni18]

In first lottery, expected wealth is 3+2=5
And in 2nd it is 5+0=5
So she should be indifferent between the two lotteries, is that right? I am really not sure!

[anon_econ]

can't we simply say that the expected wealth is the same in both cases and the variance is lower in the first case so she would prefer the first one? like in this figure 0.5u(4)+0.5u(6)>0.5u(0)+0.5u(10)?

[Chinni18]

Ooo yes I never even thought of variance! Then I agree she'll prefer the first

[aditi5000]

exp returns Lottery 1 = 3+2 =5 and lotter 2 = 5

risk of lottery 1  =  0.5( 6-5)^2 + 0.5(4-5)^2= 0.5+0.5=1 = 1

risk of lottery2 = 0.5(10-5)^2+ 0.5(0-5)^2 = 25 = 5

So she prefers 1 !

[anon_econ]

Sir r we right?

[AJ]

ya, i am getting the same answer..
she will prefer the first one..

But, i just assumed U(x)= sqrt(x) ... which can b the utility function of a risk averse individual..

and found expected utilities.

@Amit sir, will the method of finding expected wealth and variance, AND finding some expected utility always give same answer...???

[Amit Goyal]

Well the answers you all got is correct. But the written arguments supporting the claim are not precise. The graph that Vasudha plotted is the precise reason why he will prefer lottery 1 to lottery 2. More formally, we do in the following way: a risk averse individual has a concave utility function over money. And
Expected utility from lottery 1
= 0.5 u(6) + 0.5 u(4)
= 0.5 u(0.6(10) + 0.4(0)) + 0.5 u(0.4(10) + 0.6(0))
≥  0.5 [ 0.6 u(10) + 0.4 u(0) ] + 0.5 [ 0.4 u(10) + 0.6 u(0) ]   [By concavity of u(.)]
= 0.5 u(10) + 0.5 u(0)
= Expected utility from lottery 2

[ritu]

thanx amit sir,vasudha,aditi...:)
i have never met power girls like vasudha and aditi.....good going girls:)

[aditi5000]

You are the awesome one Ritu !!