Question

This problem takes you through the formal definition of risk aversion / risk loving. Given a lottery P, let E (P) be the expected value of the lottery P. For example, if P= ($10,0.5; $0,0.5), then

Answer 1

• Ann has vNM utility u1(x)= x.

V(P) = 0.5× u1(10) + 0.5 × u1(0)

= 0.5× 10 + 0.5× 0 = 5.

Here E(P)=V(P), is Ann is risk neutral.

Bob has utility u2(x)= log(2x + 1)

V(P)= 0.5 × u2(10) + 0.5× u2(0)

= 0.5 × log(2 × 10 +1) + 0.5× log(2×0 +1)

=0.66 + 0.5 ×0= 0.66

As E(P)= 5 therefore, E(P) >V(P)

Bob is risk averse.

Carl has utility u3(x)= x³.

V(P)= 0.5×u3(10) + 0.5× u3(0)

= 0.5× (10)³ + 0.5 × (0)³

= 500

Here V(P)>E(P).

Carl is risk loving.

• Arrow pratt coefficient is r(x)=–(u"(x)/u'(x)).

Ann has utility u1(x)= x,

u'1(x)=1 and u"1(x)= 0.

So, r1(x)= 0. ( r1(x) is Arrow Pratt coefficient for u1(x) )

Bob has utility u2(x)= log(2x+ 1).

u'2(x)=2/(2x +1). u"2(x)= –4/(2x +1)².

So, r2(x)= 2/(2x +1). ( r2(x) is Arrow Pratt coefficient for u2(x) ).

Carl has utility u3(x)= x³.

u'3(x)= 3x². u"3(x)= 6x.

r3(x)= –2/x. (Here r3(x) is Arrow Pratt coefficient for u3(x) )

So r2(x)>r1(x)>r3(x)

So, Carl is more risk adverse than Ann and Ann has more risk adverse than Bob. So, it supports with previous answers.

• The utility function for Constant Absolute Risk Aversion is u(x)= –e^(–ax), a>0. (It is mentioned that I have taken a instead of rho)

u'(x)= a e^(–ax) and u"(x)= –a² e^(–ax).

r(x)= a, which is a constant.

So, Arrow Pratt coefficient for utility functions is constant. That’s why it is called CARA.

• The utility function for Constant Relative Risk Aversion is u(x)=( x^(1–a))/(1–a).

(It is mentioned that I have used a instead of rho)

u(x)=(x^(1–a))/(1–a).

u'(x)= x^(–a)

u"(x)= –a x^(–a–1).

r(x)= a/x.

Here we see that Arrow Pratt coefficient r(x) is inversely proportional to x. So, as the value of x decreases r(x) increases and vice versa.

Answer is 500