Optimization problem with Kuhn-Tucker conditions

Question

Consider a game with two players where each player $i=1,2$ has preferences $u_i=s_i^a c_i^{1−a}$, where $c_i$ is consumption and $s_i$ is social interaction. $s_i$ is given by $s_i=t_i+t_{ij}\times t_{ji}$ , where $t_i$ is the time spent by player $i$ alone and $t_{ij}$ is the time player $i$ spends with player $j$. Player $i$ has to decide how much of his or her time $T$ to allocate between work, having time alone, $t_i$, and social interaction $t_{ij}$. Assume that for each hour, player $i$ works, that he or she earns the wage $w$ and assume that the price of the consumption good $c_i$ is normalized to $p=1$.

Carefully define the optimization problem for player 1. Write down the Kuhn-Tucker conditions and discuss these conditions. Explain why player 1 faces a strategic situation. Find the best-response functions for player 1 and 2. Graph these functions.

My solution :

$L = s_i^a c_i^{1-a} - \lambda(c_i + s_i - (T-s_i)w) + \mu (c_i + s_i - T)$

FOC:

$\frac{\partial L}{\partial \lambda} = c_i + s_i - (T - s_i)w = 0$

$\frac{\partial L}{\partial s_i} = a s_i^{a-1} c_i^{1-a} - \lambda(1+w) +\mu = 0$

$\frac{\partial L}{\partial c_i} = (1-a) s_i^a c_i^{-a} - \lambda +\mu = 0$

$\frac{\partial L}{\partial \mu} = c_i + s_i - T = 0 $

By following this procedure, I cannot come to the solution. Please share your ideas with me.

Solving that I obtain :

Answer 1

Player $1$'s utility maximization problem is the following :

$$\max_{0 \leq t_{12} \leq T} \ \ \left(T-t_{12} + t_{12}t_{21}\right)^a \left(w(T-t_{12})\right)^{1-a}$$

An equivalent way to solve the above problem is maximizing the $\log$ of the objective i.e. one can solve

$$\max_{0 \leq t_{12} \leq T} \ \ a\ln\left(T-t_{12} + t_{12}t_{21}\right) +(1-a)\ln \left(w\right) + (1-a)\ln \left(T-t_{12}\right) $$

Solving it we get the best response function of player 1 as

\begin{eqnarray*} t_{12} = \begin{cases} 0 & \text{if } t_{21} \leq\frac{1}{a} \\ \frac{T(at_{21}-1)}{(t_{21}-1)} & \text{if } t_{21} > \frac{1}{a} \end{cases} \end{eqnarray*}

Likewise, the best response function of player 2 is

\begin{eqnarray*} t_{21} = \begin{cases} 0 & \text{if } t_{12} \leq \frac{1}{a} \\ \frac{T(at_{12}-1)}{(t_{12}-1)} & \text{if } t_{12} \geq \frac{1}{a} \end{cases} \end{eqnarray*}

Now you can solve the best responses to get the Nash Equilibria.