# Suppose that we now consider a variation of the horizon model given in Exercise 8, that…

Suppose that we now consider a variation of the horizon model given in Exercise 8, that explicitly includes one-period borrowing bi at rate rb and one-period lending li at rate rl. The formulation is as follows:

Maximize vN ,

subject to

a) Suppose that the optimal solution includes lending in every time period; that is, l*i > 0 for i = 0, 1, . . . , N − 1. Show that the present value of a dollar in period i is

b) Suppose that there is no upper bound on borrowing; that is, Bi = +∞ for i = 0, 1, . . . , N − 1, and that the borrowing rate equals the lending rate, r = rb = rl. Show that

and that the present value of a dollar in period i is

c) Let wi be the shadow prices on the upper-bound constraints bi ≤ Bi , and let r = rb = rl. Show that the shadow prices satisfy:

d) Assume that the borrowing rate is greater than the lending rate, rb > rl; show that the firm will not borrow and lend in the same period if it uses this linear-programming model for its capital budgeting decisions.

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