<p>Here is the Lagrangian function for the stated optimization problem:
$$\mathcal{L}(x_i, t) = v(x_i, t) - cx_i + \mu x_i$$</p>

<p>Necessary conditions for optimality:</p>

<p>$$\frac{\partial\mathcal{L}}{\partial x_i} =\frac{\partial v}{\partial x_i} - c + \mu = 0 $$
and
$$x_i\geq 0, \ \mu \geq 0, \ \mu x_i = 0$$</p>

<p>To solve it, consider the following cases :</p>

<ul>
<li><p>$x_i &gt; 0$</p>

<p>$x_i &gt; 0 \rightarrow \mu = 0 \rightarrow \dfrac{\partial v}{\partial x_i} - c = 0$</p>

<p>If there exists $x_i^* &gt; 0$ such that $\dfrac{\partial v}{\partial x_i}\Big\vert_{x_i=x_i^*} - c = 0$, then $x_i = x_i^*$ solves the problem.</p></li>
<li><p>$x_i = 0$</p>

<p>$x_i = 0 \rightarrow \mu = c - \dfrac{\partial v}{\partial x_i}$</p>

<p>If $c - \dfrac{\partial v}{\partial x_i}\Big\vert_{x_i=0} \geq 0$, then $x_i = 0$ solves the problem.</p></li>
</ul>

CC BY-SA 4.0
2018-05-10T00:45:01.400Z
Thanks a lot Amit. I have one more question if you want to take a look. Thanks. https://economics.stackexchange.com/questions/21908/question-about-budget-constraint-and-utility-maximization
studentp
CC BY-SA 4.0
2018-05-08T13:12:55.023Z
2018-05-08T13:12:55.023Z
Amit
<p>Assume that the player I select $x_i \ge 0$ at constant cost level $c&gt;0$</p>

<p>The payoff function for player i is </p>

<p>$$v(x_i,t)-cx_i$$</p>

<p>where $t$ is technology parameter. </p>

<p>The function v(.)is twice continuously differentiable increasing and strictly concave in $x_i$. </p>

<p>$v(0,t)=0$</p>

<p>$\partial v(0,t)/\partial _i&gt;c$ </p>

<p>$\partial v(x_i,t)/\partial _i&lt;c$ </p>

<p>I want to maximize this payoff with respect to $x_i$. But the question strictly emphasize to use Kuhn Tucker method and to state and discuss the slackness conditions. </p>

<p>And I need to find the solution of this problem, say $x^*$</p>

<hr>

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

<p>$$ L= v(x_i,t)-cx_i +\mu [x_i-0]$$</p>

<p>First order condition </p>

<p>$$(\partial v(x_i,t)/\partial _i)-c+\mu=0$$</p>

<p>Kuhn Tucker condition</p>

<p>$$\mu [x_i-0]=0$$ for $\mu \ge 0$</p>

<p><em>Case 1</em>: $\mu \ge 0$</p>

<p>Then, $x_i=0$</p>

<p><em>Case 2</em>: $\mu = 0$</p>

<p>Then,  $x_i=0$
$$(\partial v(x_i,t)/\partial _i)-c=0$$</p>

<p>However, the question gives that $$(\partial v(x_i,t)/\partial _i)-c&lt;0$$</p>

<p>This is a contradiction I think. Therefore, I don’t think my solution is true. And I cannot write the $L$ function with two constraints but I know that Kuhn Tucker method requires at least two constraints. </p>

<p>Please share your ideas with me. Thanks a lot. </p>

<microeconomics><mathematical-economics><consumer-theory><self-study>
Kuhn Tucker optimization problem
studentp
# Kuhn Tucker optimization problem

(Asked by: username_0 on 2018-05-08)

Assume that the player I select $x\_i \ge 0$ at constant cost level $c>0$

The payoff function for player i is

$$v(x\_i,t)-cx\_i$$

where $t$ is technology parameter.

The function v(.)is twice continuously differentiable increasing and strictly concave in $x\_i$.

$v(0,t)=0$

$\partial v(0,t)/\partial \_i>c$

$\partial v(x\_i,t)/\partial \_i<c$

I want to maximize this payoff with respect to $x\_i$. But the question strictly emphasize to use Kuhn Tucker method and to state and discuss the slackness conditions.

And I need to find the solution of this problem, say $x^\*$

---

**My solution:**

$$ L= v(x\_i,t)-cx\_i +\mu [x\_i-0]$$

First order condition

$$(\partial v(x\_i,t)/\partial \_i)-c+\mu=0$$

Kuhn Tucker condition

$$\mu [x\_i-0]=0$$ for $\mu \ge 0$

*Case 1*: $\mu \ge 0$

Then, $x\_i=0$

*Case 2*: $\mu = 0$

Then, $x\_i=0$
$$(\partial v(x\_i,t)/\partial \_i)-c=0$$

However, the question gives that $$(\partial v(x\_i,t)/\partial \_i)-c<0$$

This is a contradiction I think. Therefore, I don’t think my solution is true. And I cannot write the $L$ function with two constraints but I know that Kuhn Tucker method requires at least two constraints.

Please share your ideas with me. Thanks a lot.

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


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## 1 answers

### Answer by username_1 on 2018-05-08. Score: 2 | Accepted answer

Here is the Lagrangian function for the stated optimization problem:
$$\mathcal{L}(x\_i, t) = v(x\_i, t) - cx\_i + \mu x\_i$$

Necessary conditions for optimality:

$$\frac{\partial\mathcal{L}}{\partial x\_i} =\frac{\partial v}{\partial x\_i} - c + \mu = 0 $$
and
$$x\_i\geq 0, \ \mu \geq 0, \ \mu x\_i = 0$$

To solve it, consider the following cases :

* $x\_i > 0$

  $x\_i > 0 \rightarrow \mu = 0 \rightarrow \dfrac{\partial v}{\partial x\_i} - c = 0$

  If there exists $x\_i^\* > 0$ such that $\dfrac{\partial v}{\partial x\_i}\Big\vert\_{x\_i=x\_i^\*} - c = 0$, then $x\_i = x\_i^\*$ solves the problem.
* $x\_i = 0$

  $x\_i = 0 \rightarrow \mu = c - \dfrac{\partial v}{\partial x\_i}$

  If $c - \dfrac{\partial v}{\partial x\_i}\Big\vert\_{x\_i=0} \geq 0$, then $x\_i = 0$ solves the problem.

> **Comments:**
> **username_0**: Thanks a lot username_1. I have one more question if you want to take a look. Thanks. https://economics.stackexchange.com/questions/21908/question-about-budget-constraint-and-utility-maximization  

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