Socio-economic optimum of urban drainage expansion for climate change adaptation

Socio-economic optimization of public activities has been around for many years, perhaps most frequently in relation to infrastructure constructions and the health sector. It has, however, also been used in the water sector as described by Merreth (1997) as part of a comprehensive book on the economics of water resources. The purpose is to ensure that society uses its resources in a way that in an economic sense is optimal for society. Basically, among different alternatives, the one which has the greatest net present value (NPV) of all benefits less costs during the lifetime of the activity should be selected, hereby ensuring that society gets the most value for money. The cost–benefit analysis (CBA) is a common method to arrive at the economically optimal decision. All costs and benefits during the lifetime of the construction are discounted to their present value (PV) to find the construction with the greatest NPV. However, inclusion of non-monetary outcomes typically creates problems and mistrust as described by Purshouse & McAlister (2013).

Botto et al. (2014) developed an uncertainty-compliant design flood method and found a similarity between standard design methods and a cost–benefit procedure using linear cost functions. The resulting flood estimates were systematically greater than those obtained by standard flood frequency approaches. Their method was based on Monte Carlo simulations of artificial data sets. Qi (2017) developed a non-stationary cost–benefit approach by integrating a non-stationary generalized extreme value distribution into the CBA, thereby quantifying the design flood and the influence of non-stationarity on expected total costs. Like Botto et al. (2014), linear cost and damage functions were assumed, and the study was based on Monte Carlo simulations as well.

Multicriteria CBA has been discussed and advocated by Munda (2019). Ouwayemi et al. (2020, 2021) reported on the retrofit of drainage systems, where intangible benefits also were included. In a case study on climate adaptation in Hamburg, Dehnhardt et al. (2022) discussed the limits for using the CBA in policy making and found that a participatory process is important for the usefulness of the CBA. Dennig (2018) argued that the use of the CBA in climate change issues has put new challenges to the method, e.g. caused by a distinction between inter- and intra-generational preferences, which have led to new discounting approaches. Xu et al. (2020) found that staged optimization of urban drainage systems might improve the decision-making process by providing richer decision-making information than traditional implement-once plans. The valuation of environmental impacts and the controversy in discounting the future impact of PVs were discussed in detail by O'Mahony (2021). He demonstrated that the time horizon can have a considerable impact on results, even more so than the future impacts of the discount rate. A comprehensive report on the use of the CBA in projects related to the environment has been issued by OECD (2018). It provides a timely update on recent developments in the theory and practice of the CBA taking into account important theoretical developments in relation to the economics of climate change and to the treatment of uncertainty and discounting in policy or project assessments.

In Denmark, the use of the CBA has recently become mandatory (Miljøstyrelsen 2022; Finansministeriet 2023) to get allowance for expansion of the drainage system in cities above a 5-year rainfall event in case of separate sewer systems, and above a 10-year event in case of combined sewer systems, to account for climate development. Designing for a 5- or 10-year event, depending on the type of sewer system, has been required for many years, although drainage systems of lower capacity still can be found in many places. It is stated that the NPV of all costs and benefits for expansion should be assessed taking 2–5 (in most cases three) different T-year events into account, and the alternative with the greatest NPV should then be selected following a CBA approach. This procedure can be criticized for being too crude and too dependent on pre-selecting a few return periods for the analysis.

The purpose of the present analysis is to develop a procedure that can calculate the PV and the NPV of using continuous damage cost and capital cost functions of the return period T to avoid the subjectiveness of pre-selecting a few events for the analysis. Contrary to Botto et al. (2014) and Qi (2017), no specific extreme value distribution is applied, only estimates of construction and damage costs as functions of the rainfall return period are required. Moreover, instead of a requirement for linear cost functions, the construction and damage costs are here assumed to be log-linear functions of the return period, which is found to be a robust assumption. Moreover, in the presented method, there is no need for heavy Monte Carlo simulations, as the method is purely analytical. In the approach, the T-year event that minimizes the PV of all incurred costs is determined, and it is subsequently shown that this corresponds uniquely to the T-year event that maximizes the NPV. A comprehensive sensitivity analysis is performed, and the effect of taking the climate factor's dependence on the return period is addressed. Moreover, the impact of refinancing in the case that the lifetime of the construction is shorter than the time horizon for the analysis is assessed, and the socio-economic costs of delaying climate adaptation have been calculated. Finally, the effect of a time-varying discount rate has been investigated.

The climate factor is estimated based on climate model runs in both current and future climate 100 years ahead (Christensen et al. 1998; Gregersen et al. 2013) and depends on several different factors, where the rainfall return period T is the most important one. The effect of including this dependence is addressed below in the optimization section.

Applying a time horizon for the current analysis L = 100 years (the lifetime of the construction) and using CF(L) = 1.4 (standard climate factor in Denmark for L = 100 years), we find θ = 0.004.

First, immediate climate adaptation will be considered, however, later be extended to take delayed adaptation into account.

It should be noted that PV(T) is independent of the parameter b.

The deviation between the curves is increasing for decreasing values of T, although it remains minor. Thus, it seems unnecessary explicitly to account for the return period dependence on the climate factor.

A great advantage of the analytical method for optimizing the adaptation level is a straightforward procedure for sensitivity analysis. Using Equation (10) and varying the different involved parameters ±20% results in a comprehensive sensitivity analysis, as shown in Table 1.

As seen from the table, the optimal values for the return period and the corresponding minimal PV of the costs are relatively stable for changes in the parameters. The most sensitive parameters are the slopes of the log-linear damage and cost curves followed by the discount rate and the maintenance fraction, which shows that the economic parameters are crucial for determining the optimum.

Including refinancing in year 70 in the present example results in T_opt = 78.3 years and PV_min = 77.1 × 10⁶ DKK. It is seen that both the optimal T-value and the corresponding PV are only moderately increased, which is ascribed to the influence of the discounting. Even if the capital investment is repeated at time 70, the optimal adaptation return period and the PV are only increased by, respectively, 8 years and 7 × 10⁶ DKK. The lack of correspondence between the usually applied economic time horizon and the common lifetime for construction is currently debated and may lead to a change of the recommended economic time horizon from 100 to 70 years.

In the CBA, the usual procedure consists of separately calculating the costs and benefits of different projects and then choosing the project that maximizes the NPV, i.e. the difference between the discounted benefits and costs.

Shortage of capital often leads to a delay in investments. Thus, the consequences in a socio-economic context are briefly investigated.

Note that the PV expression is independent of the parameter b.

It is seen that a delay in climate adaptation will imply a significant increase in the minimum PV of the socio-economic PV, primarily due to damages happening between the present and the time of adaptation.

A time-varying discount rate has recently been recommended in Denmark (Finansministeriet 2023) to obtain a better balance between macro-economic planning and socio-economic evaluations. The discount rate is set to r₁ = 0.035 from t = 0 to t₁ = 35 years, to r₂ = 0.025 from t₁ to t₂ = 70 years and to r₃ = 0.015 from t₂ to L = 100 years.

The advantage of the presented method for finding the optimal T-year event is that it takes a continuum of rainfall T-year events into account instead of pre-selecting a few events. Much of the subjectiveness and lack of precision is hereby avoided. It is interesting that, even if the log-linear approach to the damage function appears unprecise, the sensitivity analysis proved such lack of precision to be relatively unimportant. It may be argued, however, that the relatively low sensitivity to the model parameters may be due to a too-simplistic model approach. Nevertheless, the analysis reveals important information on the relative impact of the optimum of the different parameters. According to Koutsoyiannis (2023), the exponential tail assumption for extreme rainfall events is crude but considered acceptable here as uncertainties in the economic assumptions and parameters are dominant. It is an ongoing discussion whether a constant or a time-varying discount rate should be applied. The results here suggest that simplification by using a constant rate is not crucial. A decision on delay of capital investment for climate adaptation may require more elements than just the minimum of the PV of the involved socio-economic costs, e.g. capital availability, but the socio-economic minimum should clearly be considered. Although it is straightforward to optimize the adaptation return period using future return periods, it was decided here to optimize the return period in the current climate, as statements on future return periods are subject to additional uncertainty, see Rosbjerg (2024). It was considered desirable here to use the example presented in Miljøstyrelsen (2022), even though the curves around the optima of PV and NPV are relatively flat. Other cases are found to result in much more distinctive optima. Finally, it should be emphasized that intangible benefits are disregarded in the present analysis because the regulations set up by Miljøstyrelsen (2022) do not permit the inclusion of such benefits.

A procedure has been developed that makes it possible analytically to determine the optimal rainfall return period for climate adaptation based on the knowledge of, respectively, expected damage costs and the costs of climate adaptation as functions of the rainfall return period. Both functions can be approximated by log-linear relations, applicable also in the case of moderate deviations. The optimal return period is found by minimizing the PV of all incurred costs or, corresponding to a usual CBA, maximizing the NPV defined as the benefits less the costs. By using continuous discounting, a continuum of return periods can be considered, thus avoiding subjectivity by selecting a few return periods for comparison. A comprehensive sensitivity analysis shows that the method provides a robust solution as deviations from the basic data imply only moderate deviations from the optimal solution. Refinancing in the second half of the time horizon as well as applying a time-varying discount rate are shown to be of limited importance. The method has been extended to estimate the socio-economic costs of delaying climate adaptation, which is useful information for decision-making. The obtained results are found novel in the sense that a continuous calculation of the PV as well as the NPV as functions of the adaptation level has been obtained, and that an explicit formula for the optimal adaptation level has been determined. Further research should investigate to which extent the model assumptions limit the generality and applicability of the obtained results.

The comments of the three anonymous reviewers are greatly acknowledged.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.