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= $100 and a square root utility function; i.e., U(W) = W0.5, and 2) an otherwise identical person with initial wealth W= $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 1000.5 = 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 2000.5 = 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.

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