# The discounted formulation of the capital-budgeting problem described in the previous exercise…

The discounted formulation of the capital-budgeting problem described in the previous exercise can be stated as:

represents the discounted present value of the cash flows from investing at unit level in project j, and λi for i = 1, 2, . . . , N are the shadow prices associated with the funds-flow constraints. Suppose that we wish to solve the above discounted formulation of the capital-budgeting problem, using the discount factors determined by the optimal solution of the previous exercise, that is, setting:

Show that the optimal solution x ∗ j for j = 1, 2, . . . , J , determined from the undiscounted case in the previous exercise, is also optimal to the above discounted formulation, assuming:

[Hint: Write the optimality conditions for the discounted problem, using shadow-price values s λ∗ i = 0 for i = 1, 2, . . . , N. Does x*j (j = 1, 2, . . . , J ) satisfy these conditions? Do the shadow prices on the upper bounding constraints for the discounted model differ from those of the undiscounted model?]

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