Private and Social Optimum with externality in this example

Firstly, I wanted to know if we may be consider as an externality, a situation such that a customer A bought a bike to a seller B, not properly checked - and which may provoke some accidents.

Obviously there is some damage here, not fixed by the price, because otherwise the bike would have been cheaper I guess, but is it not just an asymmetry information ? I am really hesitating.

Then, my two other questions are related to practical exercise:

• If I have a function such that $U(x_1,x_2)= C\ln(x_1-10)+D\ln(x_2)$. Can we agree on the fact that the function is NOT a Cobb-Douglas one? I thought that because if I take the exponential then I have something like $(x_1-10)^C(x_2)^D$ and although $C=D=1/2$, if I can increase by two $x_1$ and $x_2$ I don't double my utility. But don't know if that is enough to prove it.

• Then in the second part, it's about how to internalize the social cost of an externality in order to reach the optimum. The context is quite basic: I pollute the environment in order to produce something, that I sell to: $p$. The cost of polluting is $2q^2$. The individual inverse demand function is $p=-q +100$. I did the calculation in order to find in private optimum how much good will be produced, found the how much good will be produced in social optimum if we take in account the social cost of 10\$ per unit.

Basically, I found that with private optimum we produce 20 and with social one, 18. Then, I don't manage to find if we may reach 18 (by computing the damage on the seller, we will receive for every product 10 dollars less) - the social preferred level of production - by creating a tax? I guess a pigouvian one. I know that implementing a tax is not always the best solution because sometimes the externalities can be internalize by one of the two sides less costly, but I'm quite lost.

• Please ask separate questions in separate posts. – Giskard Sep 19 '17 at 9:06