Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / T / T2 / c

In the $(u,v)$ plane the feasible set lies in the first quadrant above the three lines $v=3/2$, $v=5/2-2u$ and $v=2-(2/3)u$. Its lower boundary has verticesand the region is unbounded upward and to the right. The objective at these vertices is respectively $21$, $43/2$ and $55/2$, so $A$ is optimal for the original supply.

Let the available amount of R be $r\ge 0$. Comparing the three dual vertex objectives gives the optimal valueThus the marginal shadow price of R is $3/4$ below $22/3$, then $3/8$ between $22/3$ and $22$, and zero beyond $22$. At the two breakpoints, optimal dual prices range between the adjacent one-sided prices: $[3/8,3/4]$ and $[0,3/8]$ respectively. At zero supply the right marginal value is $3/4$; dual multipliers on the zero resource can be nonunique. These slope changes describe why the first extra $4/3$ units from the original supply six have the original price, but subsequent units do not.