 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2007/t/2/c/ii/solution

Let the third profit coefficient be

c_{3}

. For the same basis, the resource dual prices solve

2 y_{1} + y_{2} = 3

and

5 y_{1} + 3 y_{2} = c_{3}

y_{1} = 9 - c_{3}, y_{2} = 2 c_{3} - 15.

y_{1}, y_{2} \geq 0

and

3 y_{1} + 2 y_{2} \geq 4

. These inequalities are

c_{3} \leq 9

c_{3} \geq 15/2

and

c_{3} \geq 7

, respectively. Hence

15/2 \leq c_{3} \leq 9.

The production vector is unchanged because the resource constraints are unchanged. Equality at a boundary can allow additional optimal solutions.

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