This is a question related to the question: Kuhn-Tucker optimization problem and game theory .The question is: Some cultures emphasize more social interaction more than others. Is there a role for culture in the model?
My solution is:
Maximization of the social function with respect to $t_{12}$ and $t_{21}$
max aln( T - $t_{12}$+ $t_{12}$$t_{21}$) + (1-a)ln (w(T-$t_{12}$)) +aln( T - $t_{21}$+ $t_{12}$$t_{21}$) + (1-a)ln (w(T-$t_{21}$))
and the FOCs are:
a($t_{21}$-1)/(T -$t_{21}$ +$t_{12}$$t_{21}$) + a$t_{21}$/(T -$t_{12}$ +$t_{12}$$t_{21}$)=(1-a)/(T - $t_{21}$)
and
a($t_{12}$)/(T -$t_{21}$ +$t_{12}$$t_{21}$) + a(-1 +$t_{12})$/(T -$t_{12}$ +$t_{12}$$t_{21}$)=(1-a)/(T - $t_{12}$)
But I cannot find the optimal solutions. Please share your ideas with me.