Because LX and LY cannot be negative, this solution is outside the feasible set.
The values just obtained are inserted into Table 9.3 as solution 1. All other solution values can be calculated in a similar manner and used to complete Table 9.3. It is apparent from the table that not all solutions lie within the feasible space. Only solutions 5, 7, 9, and 10 meet the nonnegativity requirement while also providing a number of nonzero-valued variables that are exactly equal to the number of constraints. These four solutions coincide with the corners of the dual problem's feasible space.
At solution 10, the total implicit value of inputs A, B, and C is minimized. Solution 10 is the optimum solution, where the total implicit value of employed resources exactly equals the $108 maximum profit primal solution. Thus, optimal solutions to primal and dual objective functions are identical.
At the optimal solution, the shadow price for input C is zero, VC = 0. Because shadow price measures the marginal value of an input, a zero shadow price implies that the resource in question has a zero marginal value to the firm. Adding another unit of this input adds nothing to the firm's maximum obtainable profit. A zero shadow price for input C is consistent with the primal solution that input C is not a binding constraint. Excess capacity exists in C, so additional units of C would not increase production of either X or Y. The shadow price for input A of $1.50 implies that this fixed resource imposes a binding constraint. If an additional unit of
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