Environmental economists who do CGE modeling continue to introduce a damage parameter such that the damage to the capital stock is a deterministic function of a polynomial of global average temperature.
As shown in the author's equation (2), the capital stock of an economy next period can be written as:
K_t+1 = (1-Damage)*(1-Depreciation)*K_t + Investment_t
Intuitively, if you own a valuable asset at time t, this asset will depreciate a little over the course of the year and due to climate change it will suffer some damage and so your asset position at the start of the next year reflects this negative adjustment of your original asset position (that it suffers depreciation and climate damage) and then you can offset this by investing more in the asset.
Norhaus and Dietz and Stern write out the Damage function = f(Temperature).
This deterministic f() and time invariant function is written at a point in time as:
Damage_t = 1 - 1/(1+ b1*Temperature + b2*temperature*temperature)
What I don't like about this approach is the lack of serious discussion that b1 and b2 converge to zero over time. The whole adaptation micro-foundational approach as discussed in my Climatopolis book and in the work of economists such as Richard Tol and by Desmet and Rossi-Hansberg is that the economy reorganizes itself in anticipation of climate change so that that b1 and b2 converge to zero. This reorganization is not costless but it is taking place and it isn't modeled in this mechanical and deterministic set of equations.
The point of Greenstone et. al. is that over the 20th century in the U.S that b1 and b2 have converged to zero for urban activity. Given that the vast bulk of U.S GNP is based on urban activity this highlights how the temperature impact on our economy has shrunk over time.
This research approach violates the Lucas Critique as reduced form parameters are treated as structural parameters. A more rigorous and scientific research program would be to explicitly link b1 and b2 to real investments in the economy and to the spatial organization of economic activity. Such investments occur under uncertainty but under the expectation that we have incentives to reorganize economic activity.
The existing Nordhaus model appears to ignore the rational expectations hypothesis. The agents in the model should be able to solve out for the expected future damage costs because they know the posited damage function of how temperature maps into damage to capital and they know how GDP growth maps into temperature growth. Anticipating future damage to the capital stock under the status quo policy should nudge forward looking investors to seek out investments that are less risk and in my spatial setting are on "higher ground".
Intuitively, if you expect to face a 30% tax in 10 years on capital (because of Mother Nature not the IRS), if you continue to invest in the same old assets (i.e. coastal housing) that you have always invested in --- then you have strong incentives to seek out less risky assets to invest in. In my model, investors can substitute away from capital along the coast and now seek out projects on "higher ground". The introduction of cities into these models would completely change the damage equation presented above and highlights that it is not a structural relationship and thus any predictions based on such a equation are likely to be of low value. Instead, today's damage function is highly likely to be an upper-bound on future damage because of the constant re-organization of the economy to reduce the impact of temperature on our quality of life.