CEE5932000testandanswers - CEE593 Prelim Open book November...

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CEE593 Prelim Open book November 20, 2000 1. (20) Assume you are interested in finding the value of x and y that maximizes F(x,y) = 30x – x 2 + 42y – 3y 2 but x + y cannot exceed 10. a) Show how you would do this. b) Show how you would determine the marginal value of F(x,y) with respect to the [sum of x and y at the] optimal solution. c) Show how you would determine the marginal value of F(x,y) with respect to the parameter 30 at the optimal solution. ANSWERS: a) Use Lagrange multipliers: L = 30x – x 2 + 42y – 3y 2 - λ (x+y-10) Note: No slack is needed since x + y wants to be 15 + 7 ideally. Set L/ x = 0; L/ y = 0; L/ ∂λ = 0 and solve simultaneous equations. b) λ which equals F/ x and F/ y c) dF/dp = x where p is the parameter 30. So optimal value of x is the answer. Note: Percent change is [ dF/dp ][ 30/F*] = x*[ 30/F*] were x* and F* are optimal values of x and F(x,y). 2. (40) You are asked to advise a client who likes to gamble. She found a game called “Double your Money” that in each gamble will double her money with 80% probability. However there is a 20% chance in each gamble she will loose her money. The game requires a dollar to begin. Hence if she wins the first gamble she will have $2. If the $2 are gambled a second time, there is an 80% chance she will win another $2, for a total of $4. A third gamble of $4 gives her an 80% chance of winning another $4, for a total of $8. And so on. However each gamble has a 20% chance of losing, and if that happens the total amount she gambles is lost. a)
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CEE5932000testandanswers - CEE593 Prelim Open book November...

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