# Kuhn-Tucker and optimization (continue)

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}$$t_{21}) + at_{21}/(T -t_{12} +t_{12}$$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.

• If you want to add new information to your question you can always edit it by clicking the edit button under your question. Please do not post new information as a new question, rather edit it into the old one. May 11, 2018 at 19:34

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.

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