Question

B3 (10 marks). Agatha values an uncertain income y, according to the expected utility index EU-y-rVar() where y is the mean income, Var(y) is the variance and r measures Agatha's degree of risk aversion. Agatha rents a plot of land runs and, in return for a fixed rent F which she pays to a landlord, she gets to keep a share B of the output z. It follows that her income is given by Suppose, for simplicity, the output is either 220 or 180 with equal probability (a) If r = 흖, F = 10 and β = , coinpute Agatha's mean income and her expected utility index (b) Keeping pose another landlord offers Agatha a similar contract with an output share of β , and a fixed rent of F-28. Should she accept the offer? 10

Answer 1

a)

Probability of x being 220=p1=1/2

Probability of x being 180=p2=1/2

Let us calculate y if x=200

$y1=-F+\beta x=-10+(1/10)*220=12$

Let us calculate y if x=180

$y2=-F+\beta x=-10+(1/10)*180=8$

Mean income=p1*y1+p2*y2=(1/2)*12+(1/2)*8=10

V arzancely ) = 〉 P(X-X)-(1/2) * (12-10)24(1/2) * (8-10)2-4

So,

$EU=\overline{y}-rVar(y)=10-(1/4)*4=9$

b)

Probability of x being 220=p1=1/2

Probability of x being 180=p2=1/2

Let us calculate y if x=200

$y1=-F+\beta x=-28+(1/5)*220=16$

Let us calculate y if x=180

$y2=-F+\beta x=-28+(1/5)*180=8$

Mean income=p1*y1+p2*y2=(1/2)*16+(1/2)*8=12

Variance(y)-〉·p(X-X)2ー(1/2) * (16-12)4 (1/2) * (8-12)2-16

So,

EU-y-rVar(y)-12-(1/4) * 16-8

Expected utility index has decreased. She will accept the offer.