Kuhn Tucker optimization problem

Question

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 $x^*$

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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.

Answer 1

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.