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

The payoff function for player i is


where $t$ is technology parameter.

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


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

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

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.


1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.