To determine which utility functions have the expected utility property, we must check if they are linear in the probabilities [tex]\(\pi_1\)[/tex] and [tex]\(\pi_2\)[/tex]. This property means that the utility function can be written in such a form where changes in [tex]\(\pi_1\)[/tex] and [tex]\(\pi_2\)[/tex] lead to proportionate changes in the overall utility. Let's go through each utility function step-by-step:
1. Utility Function (a)
[tex]\[
u(c_1, c_2, \pi_1, \pi_2) = a (\pi_1 c_1 + \pi_2 c_2)
\][/tex]
- This form is clearly linear in [tex]\(\pi_1\)[/tex] and [tex]\(\pi_2\)[/tex]. Each probability [tex]\(\pi_1\)[/tex] and [tex]\(\pi_2\)[/tex] is multiplied by the respective constants [tex]\(c_1\)[/tex] and [tex]\(c_2\)[/tex] and then summed up, scaled by [tex]\(a\)[/tex].
- Hence, it satisfies the expected utility property.
2. Utility Function (b)
[tex]\[
u(c_1, c_2, \pi_1, \pi_2) = \pi_1 c_1 + \pi_2 c_2^2
\][/tex]
- Here, while [tex]\(\pi_1 c_1\)[/tex] is linear, [tex]\(\pi_2 c_2^2\)[/tex] is not linear in [tex]\(\pi_2\)[/tex] because [tex]\(c_2^2\)[/tex] implies a quadratic term.
- Therefore, this function does not satisfy the expected utility property.
3. Utility Function (c)
[tex]\[
u(c_1, c_2, \pi_1, \pi_2) = \pi_1 \ln c_1 + \pi_2 \ln c_2 + 17
\][/tex]
- This utility function is linear in [tex]\(\pi_1\)[/tex] and [tex]\(\pi_2\)[/tex] since the logarithmic terms [tex]\(\ln c_1\)[/tex] and [tex]\(\ln c_2\)[/tex] are constants with respect to the probabilities [tex]\(\pi_1\)[/tex] and [tex]\(\pi_2\)[/tex]. The constant term [tex]\(17\)[/tex] does not affect the linearity.
- Hence, this function satisfies the expected utility property.
Based on this analysis, the utility functions that have the expected utility property are:
(a) [tex]\( u(c_1, c_2, \pi_1, \pi_2) = a (\pi_1 c_1 + \pi_2 c_2) \)[/tex]
(c) [tex]\( u(c_1, c_2, \pi_1, \pi_2) = \pi_1 \ln c_1 + \pi_2 \ln c_2 + 17 \)[/tex]
So, the final answer is:
[tex]\[
[1, 3]
\][/tex]
