# Quality Provision Game

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:

$u_B(x,s)=x+v(s)$

Where $x$ is dollars spent on anything besides the good that is being traded (like the composite commodity theorem)

$u_S(s)=v(s)-c(s)$

The utility of the seller would just be the value they can sell the good for minus it's cost?

Thus, $max_s(u_B+u_S)=v'(s)+v'(s)-c'(s)=0$

This results in the following equation:

$2v'(s)=c'(s)$

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!

• I realized that I made a mistake in reading the problem (though I read it several times) and the condition about the derivatives is only at zero. And thus the final equality would be satisfied by only a positive $s$ due to the concavity of the two functions. I would still appreciate if anyone checked my work to make sure it is reasonable/correct. – jlang Dec 1 '16 at 3:14

Maybe I'm missing some information about this model, but the intuitive interpretation to me is that $$u_B = v(s) - p$$ and $$u_S = p - c(s)$$ where $p$ is the price paid by the buyer for the good.

Pareto optimality is obvious, and it's easy to check that the Pareto optimal quality is positive.

• Yes, I think I was over complicating this problem. Their utilities should definitely just be the surplus either individual gets after the transaction. – jlang Dec 1 '16 at 19:16
• What do you think of this? Consider the following game. First, the buyer makes an offer of p dollars. After observing this offer, the seller either accepts or rejects the offer, and then decides what quality to provide. If the seller accepts an offer and provides quality of s, the buyer’s utility (= surplus) is v(s) − p and the seller’s is p − c(s). If the seller rejects the offer and provides quality of s, then the buyer’s utility is 0 and the seller’s is −c(s). I am getting that the seller will accept any offer and produce quality zero. Thus, the buyer will offer a price of zero. – jlang Dec 1 '16 at 19:16
• That sounds correct. – Theoretical Economist Dec 1 '16 at 21:13