Subgame Perfect Nash equilibrium: two stage game

Question

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 <a< 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 <b< 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)$(P/B)^{a-2}$ $S^{1-a}$ < 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

Answer 1

We'll first find manager's strategy. Manager of the charity chooses $S$ and $P$ by solving the following problem :

\begin{eqnarray*} \max_{S, P} & \ \frac{P^a S^{1-a}}{B^a} \\ \text{s.t.} & \ P+S = B \end{eqnarray*} where $B = D_1+ D_2$. Solving it we get the manager's strategy as a function of donations $D_1, D_2$ as : \begin{eqnarray*} P &=& aB = a(D_1+D_2)\\S &= &(1-a)B = (1-a)(D_1+D_2) \end{eqnarray*} Given the charity's strategy, donors 1 and 2 choose donations $D_1^*$ and $D_2^*$ in such a way that

• $D_1^*$ solves \begin{eqnarray*} \max_{c_1, D_1} & \ c_1a(D_1+D_2^*) \\ \text{s.t.} & \ c_1 + D_1 = W \end{eqnarray*}
• $D_2^*$ solves \begin{eqnarray*} \max_{c_2, D_2} & \ c_1a(D_1^*+D_2) \\ \text{s.t.} & \ c_1 + D_2 = W \end{eqnarray*}

Solving them we get $D_1^* = D_2^* = \dfrac{W}{3}$. Consequently, manager's response strategy yields that $P^* = \dfrac{2aW}{3}$ is spent on programs in the equilibrium outcome. Utility of both the donors in equilibrium is equal to $\dfrac{4aW^2}{9} $.

However, the equilibrium we found is not efficient. If we consider an alternative donation plan $D_1' = D_2' = \dfrac{W}{2}$, the amount spend on programs will go up to $P' = aW$. Utility of both the donors will now be $\dfrac{aW^2}{2}$ which is higher than before. Also, manager will be better off because he received higher donations.

Now you can try (c) yourself.

Answer 2

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.