Question

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. max ?_{j=1}^n cjxj + yT - ?_{j=1}^n a1jxj + y1 ? b1 - ?_{j=1}^n atjxj - (1 + r)yt-1 + yt ? bt for t = 2,..., T 0 ? xj ? 1 for j = 1,..., n. (i) Write the dual of the above linear program. (ii) Solve the dual linear program found in (i). [Hint: Note that some of the dual variables can be computed by backward substitution.] (iii) Write the complementary slackness conditions. (iv) Deduce that the first T constraints in the primal linear program hold as equalities. (v) Use the complementary slackness conditions to show that the solution obtained by setting xj = 1 if cj + ?_{t=1}^T (1 + r)^{T-t} atj > 0, and xj = 0

          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. max ?_{j=1}^n cjxj + yT - ?_{j=1}^n a1jxj + y1 ? b1 - ?_{j=1}^n atjxj - (1 + r)yt-1 + yt ? bt for t = 2,..., T 0 ? xj ? 1 for j = 1,..., n. (i) Write the dual of the above linear program. (ii) Solve the dual linear program found in (i). [Hint: Note that some of the dual variables can be computed by backward substitution.] (iii) Write the complementary slackness conditions. (iv) Deduce that the first T constraints in the primal linear program hold as equalities. (v) Use the complementary slackness conditions to show that the solution obtained by setting xj = 1 if cj + ?_{t=1}^T (1 + r)^{T-t} atj > 0, and xj = 0

$Consider the following investment problem over T years, where the objective is to maximize the value of the investments in year T. We assume a perfect capital market with the same annual lending and borrowing rate r > 0 each year. We also assume that exogenous investment funds bt are available in year t, for t = 1,..., T. Let n be the number of possible investments. We assume that each investment can be undertaken fractionally (between 0 and 1). Let atj denote the cash flow associated with investment j in year t. Let cj be the value of investment j in year T (including all cash flows subsequent to year T discounted at the interest rate r). The linear program that maximizes the value of the investments in year T is the following. Denote by xj the fraction of investment j undertaken, and let yt be the amount borrowed (if negative) or lent (if positive) in year t. max ?j=1^n cjxj + yT - ?j=1^n a1jxj + y1 ? b1 - ?j=1^n atjxj - (1 + r)yt-1 + yt ? bt for t = 2,..., T 0 ? xj ? 1 for j = 1,..., n. (i) Write the dual of the above linear program. (ii) Solve the dual linear program found in (i). [Hint: Note that some of the dual variables can be computed by backward substitution.] (iii) Write the complementary slackness conditions. (iv) Deduce that the first T constraints in the primal linear program hold as equalities. (v) Use the complementary slackness conditions to show that the solution obtained by setting xj = 1 if cj + ?t=1^T (1 + r)^T-t atj > 0, and xj = 0$

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