# Finding subgame perfect equilibrium

The main part of the question is as follows (if you cannot read this, I can immediately write it)

My question is how to find the subgame perfect Nash equilibrium for the both cases $\bar{g_2}\ge \bar{g_1}$ and $\bar{g_2}< \bar{g_1}$

My attempt is

For firm 2

$$max [f_2(G)-c_2(g_2)]$$ $$max [\beta ln(g_1+g_2)-g_2]$$

By FOCs

$$\beta / (g_1+g_2)- 1=0$$

$$g_2=\beta -g_1$$

For firm 1

$$max [f_2(G)-c_2(g_2)]$$ $$max [\theta ln(g_1+\beta -g_1)-g_1]$$

I cannot get FOC with respect to $g_1$

At this point I’m stuck. How can I proceed this solution?

And I have another question

For the same question, I assume the following part Again I need to find subgame perfect Nash equilibrium of this game where player 1 contributes a positive amount of public good.?

Sorry but I could not this part completely. Any helps will be appreciated.

Since I could not understand such type of questions for the SPE, I asked lots of time. Thanks.

First verify that $\overline{g}_1 = \theta$ and $\overline{g}_2 = \beta$. We'll now find Subgame perfect equilibrium for all possible values of $(\theta, \beta)$ satisfying $\theta > \beta> 1$.

To do so, we first maximize player 2's payoff with respect to his contribution taking as given player 1's contribution: \begin{eqnarray*} \max_{g_2 \geq 0} & \ \beta\ln (g_1+g_2)- g_2 \end{eqnarray*}

and we get the best response strategy of player 2 as a function of player 1's contribution:

\begin{eqnarray*} g_2 = \max(\beta - g_1, 0) \end{eqnarray*}

Next, solve player 1's payoff maximization problem taking as given player 2's strategy

\begin{eqnarray*} \max_{g_1 \geq 0} & \ \theta\ln (g_1+g_2)- g_1 \\ \text{s.t.} & \ g_2 = \max(\beta - g_1, 0)\end{eqnarray*}

and we get \begin{eqnarray*} g_1^* = \begin{cases} 0 & \text{if } \theta < e\beta \\ \theta & \text{if } \theta \geq e\beta\end{cases} \end{eqnarray*}

Consequently, the contribution of player 2 in a subgame perfect outcome is \begin{eqnarray*} g_2^* = \begin{cases} \beta & \text{if } \theta < e\beta \\ 0 & \text{if } \theta \geq e\beta\end{cases} \end{eqnarray*}

For the next one, the cost of player 2 is \begin{eqnarray*} c_2(g_2) = \begin{cases} g_2 & \text{if } g_1 \geq \overline{g}_1 = \theta \\ \lambda g_2 & \text{if } g_1 < \overline{g}_1 = \theta\end{cases} \end{eqnarray*}

First verify that in this case $\overline{g}_1 = \theta$ and $\overline{g}_2 = \dfrac{\beta}{\lambda}$. We'll now find Subgame perfect equilibrium for all possible values of $(\theta, \beta, \lambda)$ satisfying $1 <\theta \leq \dfrac{\beta}{\lambda} < \beta$.

To do so, we first maximize player 2's payoff with respect to his contribution taking as given player 1's contribution: \begin{eqnarray*} \max_{g_2 \geq 0} & \ \beta\ln (g_1+g_2)- c_2(g_2) \end{eqnarray*}

and we get the best response strategy of player 2 as a function of player 1's contribution:

\begin{eqnarray*} g_2 = \begin{cases} \dfrac{\beta}{\lambda} - g_1 & \text{if } g_1 < \overline{g}_1 = \theta \\ \max\left({\beta} - g_1, 0\right) & \text{if } g_1 \geq \overline{g}_1 = \theta \end{cases} \end{eqnarray*}

Next, solve player 1's payoff maximization problem taking as given player 2's strategy \begin{eqnarray*} \max_{g_1 \geq 0} & \ \theta\ln (g_1+g_2)- g_1 \\ \text{s.t.} & \ g_2 = \begin{cases} \dfrac{\beta}{\lambda} - g_1 & \text{if } g_1 < \overline{g}_1 = \theta \\ \max\left({\beta} - g_1, 0\right) & \text{if } g_1 \geq \overline{g}_1 = \theta \end{cases}\end{eqnarray*}

and we get \begin{eqnarray*} g_1^* = \begin{cases} 0 & \text{if } \lambda < e \\ \theta & \text{if } \lambda \geq e\end{cases} \end{eqnarray*}

Consequently, the contribution of player 2 in a subgame perfect outcome is \begin{eqnarray*} g_2^* = \begin{cases} \dfrac{\beta}{\lambda} & \text{if } \lambda < e \\ \beta - \theta & \text{if } \lambda \geq e\end{cases} \end{eqnarray*}

• Thank you for your great answer. Your solution was really helpful to understand. Also can you look at my another solution. We have tried to solve it but really stack. And we need your help. Thank you so much. economics.stackexchange.com/questions/22121/… May 24 '18 at 0:29

Note, there is a mistake in FOC---it should be $$\beta/(g_1+g_2) - 1 =0,$$ so the response function of firm $2$ is $$g_2 = \beta - g_1.$$

This means that a unit increase in a public good contributed by firm $1$ causes firm $2$ to decrease its contribution by the same unit. Then, notice that firm $1$, who correctly anticipates such a (optimal) response by firm $2$, is better off by not contributing anything (hint: $g_1$ only affects $C(g_1)$ up to $g_1 \in [0,\beta]$).

I guess you can take from here to solve the second part