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.