<p>Player $1$'s utility maximization problem is the following :</p>

<p>$$\max_{0 \leq t_{12} \leq T} \ \ \left(T-t_{12} + t_{12}t_{21}\right)^a \left(w(T-t_{12})\right)^{1-a}$$</p>

<p>An equivalent way to solve the above problem is maximizing the $\log$ of the objective i.e. one can solve </p>

<p>$$\max_{0 \leq t_{12} \leq T} \ \ a\ln\left(T-t_{12} + t_{12}t_{21}\right) +(1-a)\ln  \left(w\right) + (1-a)\ln  \left(T-t_{12}\right) $$</p>

<p>Solving it we get the best response function of player 1 as</p>

<p>\begin{eqnarray*} t_{12} = \begin{cases} 0 &amp; \text{if } t_{21} \leq\frac{1}{a} \\ \frac{T(at_{21}-1)}{(t_{21}-1)} &amp; \text{if } t_{21} &gt; \frac{1}{a} \end{cases} \end{eqnarray*}</p>

<p>Likewise, the best response function of player 2 is</p>

<p>\begin{eqnarray*} t_{21} = \begin{cases} 0 &amp; \text{if } t_{12} \leq  \frac{1}{a} \\ \frac{T(at_{12}-1)}{(t_{12}-1)} &amp; \text{if } t_{12} \geq  \frac{1}{a} \end{cases} \end{eqnarray*}</p>

<p>Now you can solve the best responses to get the Nash Equilibria.</p>

CC BY-SA 4.0
2018-06-13T13:35:27.177Z
Amit I solve the exercise but I cannot get the same results. Is there anything that I do wrong? Thank you in advance.
Stefanos Makridis
CC BY-SA 4.0
2018-06-13T15:42:14.563Z
@StefanosMakridis You're right. I've updated it.
Amit
CC BY-SA 4.0
2018-05-11T19:14:06.560Z
Thank you so much for your answer. This is an understandable approach. I also have a question that it is related to that exercise. I copy the link.https://economics.stackexchange.com/questions/21945/kuhn-tucker-and-optimization-continue
Stefanos Makridis
CC BY-SA 4.0
2018-05-11T03:40:00.520Z
2018-06-13T15:39:02.810Z
2018-06-13T15:39:02.810Z
Amit
<p>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 $p=1$.</p>

<blockquote>
  <p>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.</p>
</blockquote>

<p><strong>My solution :</strong> </p>

<p>$L = s_i^a c_i^{1-a}  - \lambda(c_i + s_i - (T-s_i)w) + \mu (c_i + s_i - T)$</p>

<p>FOC: </p>

<p>$\frac{\partial L}{\partial \lambda} =  c_i + s_i - (T - s_i)w = 0$ </p>

<p>$\frac{\partial L}{\partial s_i} = a s_i^{a-1} c_i^{1-a} - \lambda(1+w) +\mu = 0$ </p>

<p>$\frac{\partial L}{\partial c_i} = (1-a) s_i^a c_i^{-a} - \lambda +\mu = 0$ </p>

<p>$\frac{\partial L}{\partial \mu} = c_i + s_i - T  = 0 $</p>

<p>By following this procedure, I cannot come to the solution. 
Please share your ideas with me.</p>

<p>Solving that I obtain :<a href="https://i.sstatic.net/rd69m.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/rd69m.png" alt="Exercise"></a></p>

<microeconomics><mathematical-economics><consumer-theory><self-study>
Optimization problem with Kuhn-Tucker conditions
Stefanos Makridis
# Optimization problem with Kuhn-Tucker conditions

(Asked by: username_0 on 2018-05-09)

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.

Solving that I obtain :[![Exercise](https://i.sstatic.net/rd69m.png)](https://i.sstatic.net/rd69m.png)

**Tags:** microeconomics, mathematical-economics, consumer-theory, self-study

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## 1 answers

### Answer by username_1 on 2018-05-11. Score: 4 | Accepted answer

Player $1$'s utility maximization problem is the following :

$$\max\_{0 \leq t\_{12} \leq T} \ \ \left(T-t\_{12} + t\_{12}t\_{21}\right)^a \left(w(T-t\_{12})\right)^{1-a}$$

An equivalent way to solve the above problem is maximizing the $\log$ of the objective i.e. one can solve

$$\max\_{0 \leq t\_{12} \leq T} \ \ a\ln\left(T-t\_{12} + t\_{12}t\_{21}\right) +(1-a)\ln \left(w\right) + (1-a)\ln \left(T-t\_{12}\right) $$

Solving it we get the best response function of player 1 as

\begin{eqnarray\*} t\_{12} = \begin{cases} 0 & \text{if } t\_{21} \leq\frac{1}{a} \\ \frac{T(at\_{21}-1)}{(t\_{21}-1)} & \text{if } t\_{21} > \frac{1}{a} \end{cases} \end{eqnarray\*}

Likewise, the best response function of player 2 is

\begin{eqnarray\*} t\_{21} = \begin{cases} 0 & \text{if } t\_{12} \leq \frac{1}{a} \\ \frac{T(at\_{12}-1)}{(t\_{12}-1)} & \text{if } t\_{12} \geq \frac{1}{a} \end{cases} \end{eqnarray\*}

Now you can solve the best responses to get the Nash Equilibria.

> **Comments:**
> **username_0**: Thank you so much for your answer. This is an understandable approach. I also have a question that it is related to that exercise. I copy the link.https://economics.stackexchange.com/questions/21945/kuhn-tucker-and-optimization-continue  
> **username_0**: Amit I solve the exercise but I cannot get the same results. Is there anything that I do wrong? Thank you in advance.  
> **username_1**: @StefanosMakridis You're right. I've updated it.  

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