I have no idea how to answer for this question. Could you explain how to solve t

Question

I have no idea how to answer for this question.

Could you explain how to solve these problems?

Calculate the expected utility of a person who has wealth W = $10000, faces a potential loss of C = $5000 with probability and has a utility index u(x) over money:

(i) = 0.01 and u(x) = x2;

(ii) = 0.01 and u(x) = x;

(iii) = 0.1 and u(x) = 2x/10000;

(iv) = 0.1 and u(x) = ln(x) with the natural base

(v) = 0.1 and u(x) = log10(x)

with the base 10 Which of the above utility indexes exhibit risk aversion? Find the certainty equivalents and risk premia of the gambles in each of the above cases. Explain the similarity between cases (iv) and (v).

Explanation / Answer

Condition for risk aversion is that the utility function should be concave and for a function to be concave its second deravitive should be negative.

(i) u(x) = x2; u''(x) = 2 which is positive so not risk averse

(ii) u(x) = x; u''(x) = -1/4x^3/2 which is negative so risk averse

(iii) u(x) = 2x/10000; u''(x) = 0 which is positive so not risk averse

(iv) u(x) = ln(x); u''(x) = -1/x^2 which is negative so risk averse

(v) u(x) = log10(x); u''(x) = -1/x^2*ln(10) which is negative so risk averse

(i) = 0.01 and u(x) = x2;

U(x) = .01*5000*5000 + .99*10000*10000 = 99250000

CE = x^2 = 99250000 so x = 9962.43

Risk Premium = 10000 - 9962.43 = 37.57

(ii) = 0.01 and u(x) = x;

U(x) = .01*sqrt(5000) + .99*sqrt(10000) = 99.707

CE = sqrt(x) = 99.707 so x = 9941.51

Risk Premium = 10000 - 9941.51 = 58.49

(iii) = 0.1 and u(x) = 2x/10000;

U(x) = .1*(-2-5000/10000 + .9*(-2-10000/10000) = -2.95

CE = -2-x/10000 = -2.95 so x = 9500

Risk Premium = 10000 - 9500 = 500

(iv) = 0.1 and u(x) = ln(x) with the natural base

U(x) = .1*ln(5000) + .9*In(10000) = 9.14

CE = Ln(x) = 9.14 so x = 9330.33

Risk Premium = 10000 - 9330.33 = 669.67

(v) = 0.1 and u(x) = log10(x)

U(x) = .1*log(5000) + .9*Iog(10000) = 3.97

CE = Log(x) = 3.97 so x = 9330.33

Risk Premium = 10000 - 9330.33 = 669.67

There is similarity in case (iv) and case (v) as we can see that certainty equivalents and risk premia of the gambles are same.

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