Carbon offsets and adverse selection

$($a$)$ Some engineers in Finland have developed a carbon capture technology that costs them nothing

to build, but will capture v $0$ tons of carbon. The engineers know v$,$ but we believe that v is

distributed uniformly on $[0,1].$ If the externality of carbon is $$20/$ton$,$ what is the most that we

$($society$)$ should be willing to pay for the project?

$($b$)$ Now suppose that that the cost of the carbon capture technology is given by some c $0,$

distributed uniformly on $[0,20]$ and uncorrelated with v$.$ What should society be willing to pay for

the carbon capture project? What is the expected value to society of offering this as a payment,

s$,$ to the Finnish engineers in exchange for building the carbon capture device? $($Assume that the

engineers know c$,$ and will reject the offer if s $<$ c$.)$

$($c$)$ Now suppose that the cost c from part $($b$)$ is correlated with v$.$ Specifically, c is no longer

uniformly distributed on $[0,20],$ but instead is given by c $=2$

$320$ v$.$ Continue to assume that

v $$ Unif$(0,1)$ and the carbon externality is $$20/$ton$.$ If we offer the engineers the same contract as

in $($b$),$ what is its expected value? Should society offer the engineers the contract?

$1$

$($d$)$ If we can pay $$2$ to buy satellite data from a private contractor that will allow us to learn the

exact values of v and c$,$ does your answer to $($c$)$ change?

$($e$)$ Finally, suppose that the project will be built no matter what. However, there is a risk $($prob$-$

ability p$)$ that the Finnish carbon capture technology fails catastrophically, causing damage H to

the engineers. The engineers know that, in the event of a catastrophe, the Finnish Government

will pay f to reduce this damage from H to zero. Is this efficient? How much should the Finnish

Government charge the engineers for a permit to undertake this project?