Question

Suppose the function u(x) = ln(x) represents your taste over gambles using an expected utility function. Consider a gamble that will result in a lifetime consumption of x0 with probability p, and x1 with probability 1 – p, where x1 > x0.

(a) Are you risk averse? Explain.

(b) Write down the expected utility function.

(c) Derive your certainty equivalent of the gamble. Interpret its meaning. Use the fact that αln(x0) + βln (x1) = ln(x0 α x1 β ).

(d) What is the expected value of the gamble?

(e) What is the risk premium of the gamble?

(f) Let v(x) = u(x) + 100. Does v(x) represent the same taste over gambles as u(x) does? Explain

Answer 1

Ans A)

Arrow Pratt risk aversion for investor if positive then investor is risk averse.

Arrow Pratt Risk aversion coefficient can be calculated as below

-U"(x)/U'(x)

U"(x)=-(1/x^2)

U'(x)=(1/x)

Arrow Pratt risk aversion coefficient is -(-1/x^2)/(1/x)=1/x>0

Hence If we have this log utility then we are risk averse because Arrow Pratt Risk aversion coefficient is positive.

Ans B)

Expected Utility =( probability of lifetime consumption of x0)*(ln(x0))+(1- probability of lifetime consumption of x0)*(ln(x1))

=p*ln(x0)+(1-p)*ln(x1)

Ans C)

Certainty equivalent is the outcome corresponding to expected Utility

CE=inverse of U(E(U))=inverse of U(p*ln(x0)+(1-p)*ln(x1))

=inverse of U(ln((x0)^p*(x1)^(1-p)))

We know that inverse of log function is exp(x)

=Exp(ln((x0^p*x1^(1-p))))=(x0^p*x1^(1-p))

If Y=exp(ln(x)) then ln Y=ln x hence Y=X

exp(ln(x))=x

Certainty equivalent is =(x0^p*x1^(1-p))

Ans D)

Expected value of gamble =Probability of outcome X0*X0+Probability of outcome X1*X1=pX0+(1-p)X1

EMV=X1+p(X0-X1)=X1-p(X1-X0)

Ans E)

Risk premium of gamble =EMV-CE

=X1-p(X1-X0)-X0^p*X1^(1-p)

Ans F)

V"(X)/V'(X)=U"(X)/U'(X)

Arrow Pratt risk aversion is positive for agent with V(x) utility form

Hence this agent is too risk averse

V(x) is linear transformation of U(X) therefore V(X) holds all the properties similar to U(X)..( Properties of v-Neumannn utility)