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

<p>For (1) the function is valued at 3.78821 and for (2) the function is valued at 3.68444. </p>

<p>So it seems that (1) is the solution for the maximum. </p>

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

<p>Function<br>
<a href="https://i.sstatic.net/P7y2c.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/P7y2c.jpg" alt="Function"></a></p>

<p>Contour Set ($t_{12}$ on X axis):  </p>

<p><a href="https://i.sstatic.net/z74Vj.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/z74Vj.jpg" alt="enter image description here"></a></p>

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

CC BY-SA 4.0
2018-08-09T18:37:43.007Z
2018-08-09T18:37:43.007Z
erik
<p>This is a question related to the question: <a href="https://economics.stackexchange.com/questions/21902/optimization-problem-with-kuhn-tucker-conditions">Kuhn-Tucker optimization problem and game theory</a> .The question is: Some cultures emphasize more social interaction more than others. Is there a role for culture in the model?   </p>

<p>My solution is:</p>

<p>Maximization of the social function with respect to $t_{12}$ and $t_{21}$ </p>

<p>max aln( T - $t_{12}$+ $t_{12}$<em>$t_{21}$) + (1-a)ln (w(T-$t_{12}$)) +aln( T - $t_{21}$+ $t_{12}$</em>$t_{21}$) + (1-a)ln (w(T-$t_{21}$))  </p>

<p>and the FOCs are:</p>

<p>a($t_{21}$-1)/(T -$t_{21}$ +$t_{12}$<em>$t_{21}$) + a$t_{21}$/(T -$t_{12}$ +$t_{12}$</em>$t_{21}$)=(1-a)/(T - $t_{21}$)</p>

<p>and</p>

<p>a($t_{12}$)/(T -$t_{21}$ +$t_{12}$<em>$t_{21}$) + a(-1 +$t_{12})$/(T -$t_{12}$ +$t_{12}$</em>$t_{21}$)=(1-a)/(T - $t_{12}$)</p>

<p>But I cannot find the optimal solutions. Please share your ideas with me. </p>

CC BY-SA 4.0
2018-05-11T19:34:36.130Z
If you want to add new information to your question you can always [edit it](https://economics.stackexchange.com/posts/21923/edit) 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.
Giskard
<microeconomics><mathematical-economics><self-study>
Kuhn-Tucker and optimization (continue)
Stefanos Makridis
# Kuhn-Tucker and optimization (continue)

(Asked by: username_0 on 2018-05-11)

This is a question related to the question: [Kuhn-Tucker optimization problem and game theory](https://economics.stackexchange.com/questions/21902/optimization-problem-with-kuhn-tucker-conditions) .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.

**Tags:** microeconomics, mathematical-economics, self-study
> **Comments:**
> **username_1**: If you want to add new information to your question you can always [edit it](https://economics.stackexchange.com/posts/21923/edit) 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.  

---

## 1 answers

### Answer by username_2 on 2018-08-09. Score: 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  
[![Function](https://i.sstatic.net/P7y2c.jpg)](https://i.sstatic.net/P7y2c.jpg)

Contour Set ($t\_{12}$ on X axis):

[![enter image description here](https://i.sstatic.net/z74Vj.jpg)](https://i.sstatic.net/z74Vj.jpg)

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


---

