Exercise 2.13 Consider the following investment problem over T years, where the objective is to maximize the value of the investments in year T. We assume a perfect capital market with the same annual lending and borrowing rate r>0 each year. We also assume that exogenous investment funds bt are available in year t, for t=1,,T. Let n be the number of possible investments. We assume that each investment can be undertaken fractionally (between 0 and 1). Let atj denote the cash flow associated with investment j in year t. Let cj be the value of investment j in year T (including all cash flows subsequent to year T discounted at the interest rate r) The linear program that maximizes the value of the investments in year T is the following. Denote by xj the fraction of investment j undertaken, and let yt be the amount borrowed (if negative) or lent (if positive) in year t maxj=1ncjxj+yTj=1na1jxj+y1j=1natjxj(1+r)yt1+yt0xjb1btfort=2,,T1forj=1,,n. (i) Write the dual of the above linear program. Exercise 2.13 Consider the following investment problem over T years, where the objective is to maximize the value of the investments in year T. We assume a perfect capital market with the same annual lending and borrowing rate r>0 each year. We also assume that exogenous investment funds bt are available in year t, for t=1,,T. Let n be the number of possible investments. We assume that each investment can be undertaken fractionally (between 0 and 1). Let atj denote the cash flow associated with investment j in year t. Let cj be the value of investment j in year T (including all cash flows subsequent to year T discounted at the interest rate r) The linear program that maximizes the value of the investments in year T is the following. Denote by xj the fraction of investment j undertaken, and let yt be the amount borrowed (if negative) or lent (if positive) in year t maxj=1ncjxj+yTj=1na1jxj+y1j=1natjxj(1+r)yt1+yt0xjb1btfort=2,,T1forj=1,,n. (i) Write the dual of the above linear program