# 4.2. Damage function estimates

The Working Group says little about the estimates of economic damages from climate change, except to call for additional research. Implicitly, it adopts without question the approach taken by each model; this alone can explain the difference between the SCC estimates from FUND and DICE (Ackerman and Munitz 2011).

DICE assumes that as temperatures rise, an increasing fraction of output is lost to climate damages. We will use *D* for damages as a fraction of the GDP that would be produced in the absence of climate change; *R* = 1 – *D* for the net output ratio, or output net of climate damages as a fraction of output in the absence of climate change; and *T* for global average temperature increase in °C above 1900. The DICE damage function is

Or equivalently,

The DICE net output ratio can be viewed as combining two separate estimates: first, for low temperatures, William Nordhaus, the creator of DICE, estimates that damages are 1.8 percent of output at 2.5°C (Nordhaus 2007); second, at high temperatures, it is assumed by default that the quadratic relationship of damages to temperature in (1) or (2) continues to apply. Separate research addresses the low-temperature and high-temperature estimates, suggesting alternatives to each.

The DICE low-temperature damage estimate is based on an evaluation of several categories of climate damages at 2.5°C (Nordhaus 2008; Nordhaus and Boyer 2000). In a review and critique of the Nordhaus estimates as applied to the United States, Michael Hanemann develops alternative estimates for damages at 2.5°C, which are, in total, almost exactly four times the Nordhaus value (Hanemann 2008). If the same relationship applies worldwide, then a reasonable alternative at low temperatures is to keep the form of equation (1) or (2), but recalibrate damages to 7.1 percent of output at 2.5°C. This yields the equation

Neither the Nordhaus nor the Hanemann 2.5°C estimate provides a basis for projecting damages at much higher temperatures.^{8} It has become conventional to extrapolate the same quadratic relationship to higher temperatures, but there is no economic or scientific basis for that convention. The extrapolation implies that damages grow at a leisurely pace, especially in the Nordhaus version: from equations (2) and (3), it is easy to see that half of world output is not lost to climate damages until temperatures reach18.8°C according to DICE, or 9.1°C in the Hanemann variant.

In a discussion of damage functions and catastrophic risks, Martin Weitzman argues that even if the Nordhaus estimate is appropriate for low-temperature damages, the increasingly ominous scientific evidence about climate risks implies much greater losses at higher temperatures (Weitzman 2010). He suggests that damages should be modeled at 50 percent of output at 6°C and 99 percent at 12°C as better representations of the current understanding of climate risks; the latter temperature can be taken as representing the end of modern economic life, if not human life in general. In support of this disastrous projection for 12°C of warming, Weitzman cites recent research showing that at that temperature, areas where half the world’s population now lives would experience conditions, at least once a year, that human physiology cannot tolerate – resulting in death from heat stroke within a few hours (Sherwood and Huber 2010).

Weitzman creates a damage function that matches the DICE estimate at low temperatures, but rises to his suggested values at 6°C and 12°C. He modifies (2) by adding a higher power of *T* to the denominator:^{9}

When *T* is small, the quadratic term in (4) is more important, providing a close match to the original DICE damage function; when *T* is large, the higher-power term is more important, allowing the damage function to match Weitzman’s values for higher temperatures.

The same method can be applied to the Hanemann low-temperature estimate in (3); calibrating to Hanemann’s value at 2.5°C, and Weitzman’s values at 6°C and 12°C, we obtain Equations (2), (3), (4), and (5) incorporate all combinations of two low-temperature alternatives (Nordhaus and Hanemann), and two high-temperature alternatives (Nordhaus and Weitzman). Using their initials, these can be labeled as the N-N, H-N, NW, and H-W damages functions, respectively. They are displayed in Figure 3 (the graph presents damages as a share of GDP, not *R*), with large dots indicating the points used for calibration. Below 3°C, the low-temperature alternatives are dominant, and the high-temperature alternatives make no visible difference; at 6°C and above, the high-temperature alternatives determine the shape of the damage function. In particular, the two damage functions with the Weitzman high-temperature assumption are nearly identical above 6°C.^{10}

Source: Authors’calculations

^{8}Nordhaus presents some numerical estimates of damages at 6°C, suggesting they are between 8 percent and 11 percent of output (Nordhaus 2007); these estimates are not well documented, and do not appear to be used in the calibration of DICE.

^{9}This equation follows Weitzman’s method but differs slightly from his numerical estimates. He appears to have taken the DICE coefficient in (1) to be .00239 rather than .002839. Our equations (4) and (5) were fitted to minimize the sum of squared deviations from the Nordhaus and Hanemann damage estimates, respectively, at 2.5°C, and the Weitzman point estimates at 6°C and 12°C.

^{10} A small anomaly is that between 6°C and 12°C the N-W damage function, despite its lower low-temperature damages, is slightly higher than H-W; the gap is greatest at 6.9°C, where N-W damages are 3.8 percent above H-W. This anomaly, which is an artifact of our curve-fitting procedure, may explain one aspect of the results presented below: under conditions where high-temperature damages are likely to be important, the SCC can be greater with the N-W than with the H-W damage function. See, in particular, the upper estimates in Figure 5.