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 $x^*$
$$ L= v(x_i,t)-cx_i +\mu [x_i-0]$$
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: $\mu \ge 0$
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.