Subgame Perfect Nash equilibrium: two stage game Charitable organizations are ranked according to the proportion of their total budget that is spend on programs(what directly goes to the recipients of their services) relative to money spent on administration(e.g salaries paid to its staff). Suppose there is a charitable organization which spends P on its programs and S on administration (salaries). So its total budget is B = S + P and a proportion P/B goes to programs. The manager of the charity cares about the charity's reputation (ranking) but also cares about his salary. So suppose he maximizes $Q =(P/B)^a$ $S^{1-a}$ ,$ 0 &lt;a&lt; 1 $. 

The only donors to the charity are players 1 and 2, who care about the total amount of money that goes to the recipients of the charity's services. Donor k has a budget W and a utility function 
$U_k$ = $c_k$P, where $c_k$ is the dollar amount spent on a private good. Let player k's donation to the charity be $D_k$, k = 1 ,2 .
The timing of actions is as follows: In stage 1 the donors choose $D_1$ and $D_2$ simultaneously, and in stage 2 , the charity then chooses S and P. 

a) Find the sub-game perfect equilibrium of this game. 

b) Someone clams that the donors give less in total to the charity as the efficiency level of the charity (from their standpoint) increases. Verify the validity of this claim in this simple model and give the intuition for your answer. 

c)Now suppose that in addition to the charitable organization above, there is another charity which is identical to first charity except that its manager's objective function is $Q =(P/B)^b$ $S^{1-b}$ ,$ 0 &lt;b&lt; 1$. Call these charities a-type and b-type. Assume that a >b . Denote player k's donation to charity j as $D_k^j$, k=1 ,2 and j=a ,b. What is the  sub-game perfect Nash equilibrium of the game? (Hint: write down the donors' Kuhn-Tucker conditions). 

What I did is that

For a) I start from the 2nd stage game and maximize the charity's utility function 

max $(P/B)^a$ $S^{1-a}$ 

s.t B = S+ P

The unconstrained problem becomes:
max $(P/B)^a$ $(B-P)^{1-a}$  

Taking FOC wrt P  we obtain P*= aB

Then going in state 1, I determine the Best response function.Donor 1 wants to maximize

max (W -D1)(D1 +D2) 

The best response function for donor 1 is D1 = (W- D2)/2 .
Doing the same for donor 2, the besr response function is   D2 = (W- D1)/2 
Solving the optimal choices are D1*=D2*= W/3 .
Going back to stage 1 and substituting , P*=2W/3 , B*=2W/3a and S*=2W(1-a)/3a 
This is the Nash equilibrium. 

For part b)

The d Q/ d(P/Q) = a $( P/ B)^{a-1}$ $S^{1-a}$ > 0

and $d ^2$Q/d $(P/Q)^2$ = a(a-1)$(P/B)^{a-2}$ $S^{1-a}$ &lt; 0 

But I stuck and I do not know how to interpret it.

For part c) the donors' Kuhn-Tucher conditions  are :

$D_1^a$ + $D_2^a$ $\geq$ $D_1^b$ + $D_2^b$    

$D_1^a$ + $D_2^a$ $\geq 0 $ 

$D_1^b$ + $D_2^b$ $\geq 0 $ 

But if I plug them into the donor's maximization problem I do not end up to a result.

Any help will be appreciated.
Thank you
In part a, if $B = D_1 + D_2$, then the SGPE should be

$\left\lbrace D_1 = \frac{W}{3},\ D_2 = \frac{W}{3}, \left\lbrace  P = \alpha (D_1 + D_2), \ S = (1 - \alpha) (D_1 + D_2) \right\rbrace \right\rbrace$

Don't say $P = \alpha \frac{2W}{3}$. That's an action, and the second stage best respond should a strategy (function) to make the equilibrium subgame perfect.