During tonight’s lecture, on page 27 we revisited the risky investment problem that was initially mentioned on page 5. The scenarios we considered involved 1) a person with initial wealth W_{0 }= $100 and a square root utility function; i.e., U(W) = W^{0.5}, and 2) an otherwise identical person with initial wealth W_{0 }= $200. We discovered under scenario 1, that Investment A has higher expected utility than Investment B:
W(0) | 100 | |||
P | 50 | |||
Investment A | ||||
State | p(s) | X(a,s) | W(s) = W(0)-P(A)+X(a,s) | U(W(s)) |
Low | 0.5 | 50 | 100 | 10 |
High | 0.5 | 150 | 200 | 14.14 |
Expected | 100 | 150 | 12.07 | |
value | ||||
Investment B | ||||
State | p(s) | X(b,s) | W(s) = W(0)-P(B)+X(b,s) | U(W(s)) |
Low | 0.5 | 0 | 50 | 7.07107 |
High | 0.5 | 240 | 290 | 17.03 |
Expected | 120 | 170 | 12.05 | |
value |
Furthermore, since the utility of not investing under scenario 1 is 100^{0.5 }= 10, the optimal choice is to invest in Investment A.
Under scenario 2, Investment B has higher expected utility than Investment A:
W(0) | 200 | |||
P | 50 | |||
Investment A | ||||
State | p(s) | X(a,s) | W(s) = W(0)-P(A)+X(a,s) | U(W(s)) |
Low | 0.5 | 50 | 200 | 14.1421 |
High | 0.5 | 150 | 300 | 17.32 |
Expected | 100 | 250 | 15.73 | |
value | ||||
Investment B | ||||
State | p(s) | X(b,s) | W(s) = W(0)-P(B)+X(b,s) | U(W(s)) |
Low | 0.5 | 0 | 150 | 12.2474 |
High | 0.5 | 240 | 390 | 19.75 |
Expected | 120 | 270 | 16.00 | |
value |
Since the utility of not investing investing under scenario 2 is 200^{0.5 }= 14.14, the optimal choice is to invest in Investment B.
The reason why the otherwise identical wealthier person favors the riskier investment is because her utility has the property of decreasing risk aversion (actually, more technically, decreasing absolute risk aversion). Behaviorally, this implies that as one grows wealthier, she becomes less risk averse, even if her utility function remains the same. What drives this result is that as she grows wealthier, the utility consequence of a given risk declines. Thus the poorer person in this example takes Investment A because it is less risky, whereas the wealthier (otherwise identical) investor takes Investment B because the increment in expected value (of $20) is worth assuming the additional risk associated with Investment B.