The last question is-If any project received zero fun thing then it must have lower alpha than any project that gets some funding. This is from the book- Optimization in Economic Theory by Avinash K. Dixit

Exercise 3.3: Investment Allocation A capital sum C is available for allocation among n investment projects. If the non-negative amount x; is allocated to project j for j = 1, 2, ... n, the expected return from this portfolio of projects is n j=1 The allocation is to be chosen to maximize this. Find the first-order necessary conditions from the Kuhn-Tuck Theorem. Define n n H = E (a;/B;), K = (1/ B;). j=1 j=1 Show that (i) If C > H, then a part of the total sum available is left unused. (ii) If a; > (H -C)/ K for all j, then every project will receiv some funding. (iii) If any project receives zero funding, then it must have