/raming and LLAs This problem works through a sequence of framing exercises.

a] Ralph produces a single product, with quantity denoted x. Profit is given by the expressionx(lO - .sx), andcapacity is constrained soO:s; x:s; 10. Determine Ralph's optimal output.

b] A new customer arrives on the scene. Let y denote the quantity of output Ralph produces for this second customer. This customer is a mirror image of the first, so Ralph'sproblem is now to select quantities x and y to maximize profit ofx(lO - .sx)

+ y(lO - .sy), subject to a capacity constraint of 0 :s; x + y:s; 10. Determine Ralph's optimal output of each product, i.e., x and y. You should find x = y = 5.

c] Ralph likes to keep things simple, and enjoys working with single product decision frames. It tu ms out that the optimal x can be located in this case by maximizing any of the following functions:

x(lO - .sx) + [50 - .5x2];

x(lO - .sx) + [_.5x2]; or x(lO - .sx) + [-5x].

Verify this claim. Thencarefully explain why each function allows us to identify the optimal choice of y. Can you relate this to cost allocation?

d] Now suppose Ralph must immediately decide on the quantity of the first product (x); after this decision has been implemented, Ralph wiIl leam whether demand for the second product materializes. If it does, and if Ralph suppIies yunits of the second product, total profitwiIl be x(lO - .sx) + y(lO - .5y). Naturally, we stiIl require x + y :s; 10. Let a denote the probability demand for the second product materiaIizes. So Ralph' s problem is now to maximize expected profit of x(lO - .5x)

+ ay(lO - .5y), subject to a capacity constraint of 0 :s; x + y :s; 10. The solution is x

= 10/(1 +

a) and y = 10 - x. (x now denotes the immediate choice of first product quantity, and y the choice of second product quantity provided demand materiaHzes.)
How do you interpret this solution?
e] FinalI y, go back to Ralph' s penchant for keeping things simple. It tums out the optimal x can be located here by maximizing any of the following functions:
x(lO - .5x) + a[50 - .5x2];
x(lO - .5x) + a[-.5x2]; or x(10 - .5x) + [-lOax/(1 + a)].
Verify this claim. Then carefully relate each function to its eounterpart in the initial story (where a = 1).AppendixLO1