An alternative formulation of the undiscounted problem developed in Exercise 8 is to maximize the total earnings on the projects rather than the horizon value. Earnings of a project are defined to be the net cash flow from a project over its lifetime. The alternative formulation is then:

Let x* j for j = 1, 2, . . . , J solve this earnings formulation, and suppose that the funds constraints are binding at all times i = 0, 1, . . . , N for this solution.

 a) Show that x *j for j = 1, 2, . . . , J also solves the horizon formulation given in Exercise 8.

 b) Denote the optimal shadow prices of the funds constraints for the earnings formulation as y’i fori = 0, 1, 2, . . . , N. Show that yi = 1 + y’i for i = 0, 1, 2, . . . , N are optimal shadow prices for the funds constraints of the horizon formulation.

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