Question

Consider the utility functions of three individuals: u(x) = x^1/2, v(x) = ln x, and h(x) = x – 0.01 x², where x represents wealth.

Consider also the following lotteries: X = (w₀ + x₁, w₀ + x_2, w₀ + x_{3; ¼, ½, ¼ )} = (4, 16, 25; ¼, ½, ¼), where w₀ = $2, and lottery Y in which w₁ = $10, so that Y = (12, 24, 33; ¼, ½, ¼). Note that Y = X + (10; 1)

1. Tell whether u(x), v(x) and h(x) are risk averse, risk neutral or risk lovers individuals.

2. Compare individuals u(x) and v(x) with respect to their degree of risk aversion. Tell who is more risk averse.

3. Calculate the risk premium of individuals u(x) and v(x) with respect to lottery X. Did you obtain that the risk premium of v((x) is larger than that of u(x)? Is that result expected? Why?

4. Compare the risk premium of individual u(x) with respect to lotteries X and Y. Did you obtain that the risk premium with respect to lottery X is larger than that with respect to Y? Is that result expected? Why?

5. Compare the risk premium of individual h(x) with respect to lotteries X and Y. Did you obtain that the risk premium with respect to lottery X is larger than that with respect to Y? Is that result expected? Why?

Answer 1

Here Utility functions are representing expected returns / satisfaction levels of three individuals on their wealth, which is represented by three equations [ U(x), V(x) & h(x)]

1. U(x) and V(x) are risk averse because their satisfaction levels are lesser than their wealth and H(x) is risk neutral because its satisfaction level is at par of its wealth. (A) Satisfaction level of U(x) is half of its wealth, (B) Satisfaction level of V(x) is natural log of its wealth, which will always be lesser then the absolute wealth amount and (C) H(x) is almost equivalent to its wealth x, because its satisfaction level is x-0.02, which is 0.98x.

2. If we compare both individuals risk averse level, then from the below given table we can conclude that the at every wealth point, satisfaction level of V is always lower than U. Hence, V will take more risk than U and will have lower risk averse levels.

X (Wealth)	1	2	3	4	5	6	7	8	9	10
Satisfaction Level (U) [ x/2]	0.50	1.00	1.50	2.00	2.50	3.00	3.50	4.00	4.50	5.00
Satisfaction Level (V) [lnx]	0.0	0.7	1.1	1.4	1.6	1.8	1.9	2.1	2.2	2.3

3. Lottery X is giving 3 returns, 4, 16 & 25 with 3 different probability (1/4, 1/2, 1/4) with invested wealth, which are x1, x2 and x3. Now to calculate these 3 wealth amounts, use equations W0 + x1 = 4, W0 + x2 = 16 & W0 + x3 = 25 and we get x1 = 2, x2 = 14 and x3 = 23 provided W0=2 given.

Our risk free return is x/2 for U and lnx for V.

Risk levels (x)	2	14	23
Risk free return for U [x/2]	1.0	7.0	11.5
Risk free return for V [lnx]	0.7	2.6	3.1
Returns from Lottery X	4	16	25
Probability of return	0.25	0.50	0.25
Expected returns from Lottery X	1.00	8.00	6.25
Risk Premium for U	0.00	1.00	-5.25
Risk Premium for V	0.31	5.36	3.11	Higher Risk Premium

4. Now risk levels for Lottery Y can be calculated by using equations: W1 + x1 = 12, W1 + x2 = 24 and W1 + x3 = 33, which will become, x1=2, x2 = 14 and x3 = 23. same as in Lottery X.

Risk levels (x)	2	14	23
Risk Return of U	1.0	7.0	11.5
Returns from Lottery X	4	16	25
Probability of Lottery X return	0.25	0.50	0.25
Expected returns from Lottery X	1.00	8.00	6.25
Returns from Lottery Y	12	24	33
Probability of Lottery Y return	0.25	0.50	0.25
Expected returns from Lottery Y	3.00	12.00	8.25
Risk Premium for U from Lottery X	0.00	1.00	-5.25
Risk Premium for U from Lottery Y	2.00	5.00	-3.25

No, the risk premium in Lottery Y is higher.