You have to invest exactly $40 for one year. You can invest among projects A, B, C, D, and E. The following table shows the marginal return on each dollar invested in these projects. For example, if you invest just $1 in project A, you will get back $2 after one year. If however, you invest $3 in project A, you will get back $2+$1.9+$1.8=$5.7 after one year. Assume you can invest in multiples of $1 in any project.

(a) How should you allocate your investment across the projects to maximize the total amount you get back after one year? You cannot invest negative amount in a project. You may invest in all projects or only some projects.

(b) Now suppose you do not have any cash but own all shares of rms A, B, C, D, and E. Each rm has $8 in cash. Firm A has project A, rm B has project B, and so on. Each rm chooses how much to invest in its project. If it invests more than the cash it has, it raises the additional investment needed from you. If it invests less than the cash it has, it pays out the excess cash to you. The cash ow resulting from investments will be returned to you after one year. Since you do not have any cash, you can provide any additional investment to a rm only from cash dividends you get from another rm. If you keep some cash without investing, it will not grow (so for example, if you keep $5 under mattress, that will remain $5 after one year). Firms want to maximize the total cash you get after one year. For each firm, determine the capital it will raise or the dividend it will pay. Ignore taxes.

Dollar invested TOUR 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Marginal Return Project A Project B Project C Project D Project E 2 1.6 2.2 2.4 1.05 1.9 1.55 2.05 2.2 1.05 1.8 1.5 1.9 2 1.05 1.7 1.45 1.75 1.8 1.05 1.6 1.4 1.6 1.6 1.05 1.5 1.35 1.45 1.4 1.05 1.4 1.3 1.3 1.2 1.05 1.3 1.25 1.15 1 1.05 1.2 1.2 1 1 0.8 1.05 1.1 1.15 0.85 0.6 1.05 1 1 1.1 0.7 0.5 1.05 0.9 1.05 0.55 0.5 1.05 0.8 1 0.5 0.5 1.05 0.7 0.95 0.5 0.5 1.05 0.6 0.9 0.5 0.5 1.05 0.5 0.85 0.5 0.5 1.05 0.5 0.8 0.5 0.5 1.05 0.5 0.75 0.5 0.5 1.05 0.5 0.7 0.5 0.5 1.05 0.5 0.65 0.5 0.5 1.05 Dollar invested TOUR 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Marginal Return Project A Project B Project C Project D Project E 2 1.6 2.2 2.4 1.05 1.9 1.55 2.05 2.2 1.05 1.8 1.5 1.9 2 1.05 1.7 1.45 1.75 1.8 1.05 1.6 1.4 1.6 1.6 1.05 1.5 1.35 1.45 1.4 1.05 1.4 1.3 1.3 1.2 1.05 1.3 1.25 1.15 1 1.05 1.2 1.2 1 1 0.8 1.05 1.1 1.15 0.85 0.6 1.05 1 1 1.1 0.7 0.5 1.05 0.9 1.05 0.55 0.5 1.05 0.8 1 0.5 0.5 1.05 0.7 0.95 0.5 0.5 1.05 0.6 0.9 0.5 0.5 1.05 0.5 0.85 0.5 0.5 1.05 0.5 0.8 0.5 0.5 1.05 0.5 0.75 0.5 0.5 1.05 0.5 0.7 0.5 0.5 1.05 0.5 0.65 0.5 0.5 1.05