The issue is there isn't really a second dimension
there -- it's all value that can be measured in money. There is a number of dollars at which it actually does make sense to log the forest. The problem is that the government allows it happen for much less than that amount.
Setting explicit prices may help in theory, but there are two major problems. The first is that the price is inherently subjective. There are no market transactions to look at to see what "forest continues to exist" is currently trading for. It's the existing state and self-perpetuates without anybody having to do or pay anything until such time as someone actively expends resources to change it. So at best you're speculating, and choosing the wrong value -- whether too low or too high -- is costly. Too low and we haven't solved the existing problem, too high and we can't have a transportation system because no one can mine steel to build trains or cars or ships. So accuracy matters, but accuracy is expensive. Getting the valuation right is a detail-oriented labor-intensive process.
And in theory we already have this. It's supposed to be the job of our representatives in the legislature to do this. Which brings us to the second problem, which is that because the value is subjective, we still have the collective action problem. The logging company lobbies to have the value of the forest set low, the diffuse general public doesn't strongly object, and we're back to square one.
One of the strong options to solve at least some of this is to require all budgetary issues to be put up for public referendum. Not just "entire government budget, yea or nay" but at the line item level. The government proposes to sell the forest to the logging company for ten million dollars, yea or nay? Then the public votes nay and the legislature has to come up with the money some other way.
Naturally that isn't very popular with lobbyists because then the public tends to vote down the harmful but politically expedient revenue generation methods, so there is always strong lobbying pressure not to do things like that, and we're back to the collective action problem. It also has the general direct democracy problem of people not having enough time to evaluate issues carefully, but in principle there shouldn't have to be that many different or complex revenue generation methods. The existing tax code is unnecessarily complex and people could plausibly make time to understand a much simpler one -- which would also remove a large number of the existing harmful carve outs for groups with strong lobbyists.