After graduating from UCR with a stellar degree in economics, Jack is going to work at WorldWideWidgets. He gets to design his own compensation package in terms of the balance of base salary and commission. Base salary without commission is $20,000. Jack knows his monthly sales are random, being equally likely to be $10,000 or $35,000. He is to choose his commission level ß, 0 = ß = 1; given this choice, Jack's monthly income will be Sß + 20, 000(1 - ß), where S is the realized level of sales that actually happens. Jack is an expected-utility decision maker, with utility function u(y) = sqrt(y), where y is the realized monthly income Jack receives.
Calculate the utility-maximizing commission rate for Jack to choose.