Kuhn-Tucker and optimization (continue)

Question

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.

Answer 1

I tried this for a=0.5, T=10, w=2 and I get the following real solutions for t12 and t21:
(1) $t_{12}=t_{21}=2.8165$
(2) $t_{12}=t_{21}=1.1835$

For (1) the function is valued at 3.78821 and for (2) the function is valued at 3.68444.

So it seems that (1) is the solution for the maximum.

If it helps, here is the function and the contour sets at those parameter values.

Function

Contour Set ($t_{12}$ on X axis):

By the way, are your FOCs correct? For the FOC with t12, I have: $-\frac{(1 - a)}{(T - t_{12})} + \frac{a (-1 + t_{21})}{(T - t_{12} + t_{12} t_{21})} + \frac{a t_{21}}{( T - t_{21} + t_{12} t_{21})} = 0$