A seller owns a single unit of an invisible good. It is worth $v(s)$ to a buyer, where $s ∈ R^+$ is the quality of the good. The function $v(·)$ is strictly increasing and strictly concave with $v(0) = 0$. Quality is chosen by the seller, and the cost of providing a good of quality $s$ is given by a function $c(s)$, which is strictly increasing and strictly convex with $c(0) = c '(0) = 0$ and $c'(0) < v'(0)$. (Implicit in this description is the assumption that the buyer’s preferences are quasilinear is some good other than the one being traded.)
There are many parts to this question, but I'm most concerned about the step up. Which asks write down the first-order conditions to find the pareto optimal quality that maximizes the sum of the utilities and confirm that it is positive.
I'm thinking that the utility functions are as follows:
Where $x$ is dollars spent on anything besides the good that is being traded (like the composite commodity theorem)
The utility of the seller would just be the value they can sell the good for minus it's cost?
This results in the following equation:
Which doesn't make sense because of the assumptions of the first derivatives in the statement of the problem. So I'm not quite sure what I am doing wrong. Any assistance would be greatly appreciated. Thanks in advance!