occur naturally in line with price signals through the standard mechanisms of the model. The kinds of adaptation investment covered here are ones undertaken by the public sector because the social gains of the investment exceed the social costs, and might include building a dyke around a community which would cost much less than each household providing individual protection, or hardening a roadway so that it did not wash out interrupting commerce between two cities. For additional literature regarding different types of adaptation see Appendix 3A.

# Introducing adaptation into PAKMod

To accurately model the economic consequences of adaptation we need to link spending on adaptation to the amount of damages avoided at a national level. Unfortunately there is little literature outlining the effectiveness of adaptation, indeed, we found only one source relating spending on adaptation to avoided damages (Multi-hazard Mitigation Council 2019).<sup>26</sup> The report refers only to investments in buildings in the United States, however.

Given the lack of direct data, a more speculative approach was taken constructing a protection function that linked spending on adaptation to damage avoided in a flexible fashion, such that parameters can be refined as more information becomes available. The approach here follows that of de Bruin, Dellink and Tol (2009) and De Cian _et al._ (2016). As with Lecocq and Shalizi (2007), de Bruin, Dellink and Tol (2009) Bosello, Carraro and De Cian (2010), and Millner and Dietz (2015), we assume that the product of adaptation investment is adaptation capital<sup>27</sup> that capital does not have a productive use. Adaptation capital works only to reduce damages. It is further assumed that a country that adapts, will implement the projects that offer the most protection per unit of expenditure first, and that subsequent adaptations will generate less protection (decreasing returns). As discussed earlier in PAKMod total public investment is split into adaptation investment ( ) and productive investment ( ) and the stock of adaptation ( ) and productive capital ( ) are tracked II<sup>VV</sup> separately. Adaptation capital protects the economy from climate-related damage to productive capital VV tt tt II<sup>GGPP</sup> KK KK (say flooding). The more adaptation capital an economy accumulates the less productive capital will be damaged by natural disasters. If a road is constructed in a resilient manner (perhaps larger and more frequent culverts to reduce the likelihood of washout), the cost of the road beyond the standard cost is not considered as productive and is treated as an investment in the adaptation capital stock. From a societal point of view adaptation investment makes sense if the productive capital saved (protected) by the investments exceeds the amount of investment forgone in order to make the adaptation investment. The level of adaptation capital accumulates in the same way as productive capital and is assumed to have the same rate of depreciation, VV as productive capital. tt KK . δδ, VV VV VV tt tt−1 tt KK = (1 −δδ)KK + II

> 26 (World Bank 2010) attempts calculate the costs of different adaptation investments, but there is limited information on how much protection (damage forgone) different investments yield. The (Multi-hazard Mitigation Council 2019) estimates the ratio between the cost of adapting buildings to natural disasters and the damages avoided with these investments. The organization finds that 11 dollars can be saved for 1 dollar invested in the case of applying international building codes. However, these data are estimated for the Unites States and apply to natural disasters generally, without explicit links to climate change. The results are thus not easily transferable to climate change adaptation in Pakistan.

> 27 (Bosello, Carraro, and De Cian 2010), by contrast, use a flow variable for adaptation albeit in a model framework where a period is 10 years long.

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