Losses from Fluvial Floods in Poland over the 21st Century – Estimation Using the Productivity Costs Method

This paper aims at the estimation of the impact of climate change on future losses caused by fluvial floods in Poland over the twenty-first century at the local level with the productivity costs valuation method. The daily data on river discharges published by (Piniewski et al., Hydrol Process 31:2210–2225, 2017), map of flood risk and value added generated in each county are used to estimate of the impact of climate change on the fluvial flood damage at the county level. This study supplements the findings of (Koks et al., Environ Res Lett 14:084042, 2019), (Alfieri et al., Clim Change 136:507–521, 2016) and (Feyen et al., Clim Change 112:47–62, 2012) with estimates of future flood losses in Poland using the productivity costs valuation that considers also the costs of disruptions of production chains and lost production at the finer level of spatial disaggregation. This method shows the overall increase in losses caused by fluvial floods in Poland due to climate change in comparison to the reference period of 1974–2000 by 47% in RCP4.5 and 83% in RCP8.5 scenario in 2024–2050 and by 32% in RCP4.5 and 51% in the RCP8.5 in 2074–2100.


Introduction
An increase in losses caused by floods is one of the most significant economic effects of climate change in Eastern Europe.There are already several estimates on the impact of climate change on losses caused by floods in Europe.First such estimates were published by Feyen et al. (2012) based on the spatial model and country-specific flood depth-damage functions.Alfieri et al. (2016) adopted a similar approach.While they use high-resolution hydraulic model and depth-damage functions, the economic potential of the lost capital is ignored.The value of damage to the flooded building is equal to the size multiplied by the value of the depth-damage function.Even though these functions are specific to the country, land use and economic sector (see e.g.Huizinga et al. (2017)), their level of detail is too scarce to fully reflect the different valuations of assets based on reconstruction cost.Further study by Koks et al. (2019) includes the calculation of economic losses at the regional disaggregation, which is sufficient to allow for international comparisons but fails to represent the local losses.This paper proposes a different methodology that values the damage based on the economic potential and applies this to the future losses caused by fluvial floods in Poland.To do this, the percentage of the area inundated in a given county based on flood risk maps and the value-added produced in this county is used to calculate vulnerability.In line with previous studies (e.g.Koks et al. 2019), the maximum annual river discharge published by Piniewski et al. (2017) is used to fit the Gumbel distribution to annual discharge maxima and to show how the probability of today's 100-year event will change over the twenty-first century.In the next step, the map of 100-year flood inundation is used to assess the vulnerability of a given county to the flood event and finally, the exposure is calculated based on the estimated production executed at the given area.These data allow for the costing of changes in flood damage over the twenty-first century using the productivity costs valuation method that was not previously used for Poland.In the other words, in contrast to other research, in this study, vulnerability is defined through the implicit production in the given area, not through the sheer volume of the building stock.Additionally, finer, county-level disaggregation allows for a more accurate reflection of flood losses.
The structure of this paper is as follows.In the next section, the methods used for the valuating of flood losses are described and the existing research on the impact of climate change on flood losses is summarized.Then, the data on past flood losses in Poland are presented.In the fourth section, the adopted model and approach are described.The fifth section shows the results in the four blocks, the first devoted to changes in hazard, the second describing the evolution of exposure, the third concentrating on future damage and the fourth showing the sensitivity of losses to different assumptions on vulnerability distribution.Section six presents the discussion and the seventh concludes.

Direct Losses
Direct losses are the first category of losses that are considered while valuing the costs of natural disasters.They are the immediate consequences of inundation.There are several valuation methods described in the literature that were used to value damages caused by floods.They convey different philosophies in the categories of costs.
In the case of the contingent valuation method, people are asked how much they are willing to pay to avoid a certain event.This method is the most comprehensive as it theoretically (assuming people provide accurate values) considers almost all the costs of floods, including the valuation of the costs of inconveniences related to movement, opportunity costs of time lost and other intangible damage.Such a method was quite commonly used to value past events, e.g. by Markantonis et al. (2013) to value the 2006 Evros river flood in Greece, Islam et al. (2016) for floods in Bangladesh or Navrud et al. (2012) for floods in Vietnam.Kellens et al. (2013) present a comprehensive review of the empirical studies related to floods.Nevertheless, as Botzen and van den Bergh (2012) show, the willingness to pay for flood insurance is heavily dependent on the perception of flood risk and people tend to underestimate small risks (e.g.Baillon et al 2022), so this method is very subjective and prone to the measurement errors.
In the productivity costs method, the damage from the flood (or any other event) is equal to the loss of production within industrial areas that were caused by the event.Such a valuation method is based on the stream of income or profits lost by companies due to the event (Penning-Rowsell et al. 2005).Examples include the valuation of damage caused by the flood in the industrial complex in the Netherlands by Booysen et al. (1999) or by the event in Germany in June 2013 (Sultana et al. 2018).To calculate flood losses using such a valuation method, either firm-level data can be used (Sultana et al. 2018) or region or even country-level macroeconomic aggregates (e.g.Jonkman et al. 2008).
Replacement cost is by far the easiest and the most popular method of valuation of losses caused by an extreme event.In this method, the costs of restoring the lost property are calculated.There are several variants of this method -for instance, Merz et al. (2010) argue that depreciated values should be used as replacement costs overestimate the damage and are inconsistent with national accounts.Usually, it is also supplemented by the additional valuation of impacts on agriculture (e.g.Morris andBrewin 2014 or Qiu et al. 2010) or by the valuation of life lost, injuries, or evacuation (e.g.Penning-Rowsell et al. (2014), Bockarjova et al. (2012)).McGrath et al. (2015) highlight the uncertainty related to the choice of the depth-damage function.Nevertheless, this is the most intuitive method of valuation of losses from extreme events as it aims at the valuation of the costs of restoration of the situation to the previous state.
Generally, for most ex-ante studies of flood impact in the changing climate, the replacement costs approach is used and widely accepted -such method was used e.g. in Rojas et al. (2013), Alfieri et al. (2015b) or Dottori et al. (2020).

Economic Consequences of the Floods
The most common way to assess the economic consequences and indirect impact of catastrophic events on the economy is to apply replacement cost for assessing the direct damages and then to extend the analysis with the input-output based models.Examples of such work are Jonkman et al (2008), where input-output analysis is applied to calculate the indirect impacts of floods in the Netherlands, and Koks and Thissen (2016) where input-output analysis is extended with supply-side constraints to accommodate production losses.Koks et al. (2019) used the latter model to estimate the indirect losses caused by floods in the future climate in Europe.Here, the damage to the production capacity of each of the three analyzed sectors (agriculture, industrial, commercial) is calculated based on the physical model and compared to the total value of assets accrued in the given region.Consequently, the labor supply to each sector and the possibility of substitution are not considered.In the next step, the multiregional impact assessment (MRIA) model based on the input-output tables for 270 European regions is applied to assess indirect losses.The particular strength of this approach lies in the spatial resolution (NUTS -2, which is equal to regions) that allows for modeling not only the interactions between the countries but also the interdependencies between the regions.In addition, recovery curves are added to account for the temporal dimension and the positive effects of the destruction of the production capacities in the affected region on the production in other regions (through substitution of the ceased activities).Consequently, in this approach, it is possible to assess the impact of the flood on the economy in a given region even though it is not directly affected by the event.Rose and Wei (2013) highlight the importance of resilience using the input-output model to study the economic effects of port activities disruptions.They conclude that including resilience (based on ad-hoc adjustment and expert judgment) reduces losses from such events by 70 percent.Consequently, the issue of adaptation is the significant challenge while estimating the effects of natural disasters on the economy.
Another strand of research uses computable general equilibrium analysis (CGE) to account for the indirect consequences of natural disasters.This approach allows for nonlinear effects of losses caused by extreme events and captures the changes in relative prices caused by the event.Such a model was used to estimate the economic costs of sea-level rise in Europe by Bosello et al. (2012), Pauw et al. (2011) to estimate the macroeconomic effects of drought in Malawi or Carrera et al. (2015) to assess the economic consequences of 2000 Po river flood.Zhou and Chen (2021) present a comprehensive review of the studies that use this type of model for the long-term assessment of the economic consequences of natural disasters and conclude that the final influence of disasters on the economy is highly dependent not only on the resilience of the economy but also on the specific assumptions on the particular model.
Another general class of models that are used for assessing the indirect effects of natural disasters are so-called integrated assessment models (IAMs), which allow for the feedback loops between the economy, greenhouse gas emission and climate change.For instance, Narita et al. (2010) used this kind of model to assess the economic impact of increased storminess due to the climate change and Zhang et al. (2021) to calculate the interdependencies between climate change and the Chinese economy.Generally, these models are criticized for using too simplistic climate damage functions or improper discounting rates that lead to the underestimation of the impact of climate change on the economy (Tol 2018).

Direct Losses
Flood losses are the biggest group of economic costs, which will be caused by climate change in Poland in the future.Costs of heat and cold waves as well as costs of wildfires will be far lower than the change in the costs of floods (MEnv 2013).The global distribution of the flood losses as a percentage of GDP based on the EM-DAT disaster database is shown in Fig. 1.Losses caused by a single flood have log-normal distribution with the median value of 0.016 and a mean of 0.25 percent of GDP.Moreover, this distribution is robust to floods in well-developed countries or floods within the last one or two decades.
The figure shows the four big floods in Poland between 1990 and 2018.All were on the right-hand side of the spectrum, which means they were more severe than the world's median.It may suggest that the distribution of flood losses in Poland is different than in the other parts of the world, but four events are far too few to either confirm or reject this hypothesis.

Indirect Losses
The literature on the past indirect losses from floods in Poland is scarce.There is no general equilibrium assessment of the long-term consequences of any of the four floods indicated above.Also, GDP growth in 1997, 2001, 2009, and 2010 was equal to 6.4, 1.3, 2.8, and 2.9 percent respectively and does not deviate from long-term equilibrium.1Some insights on the relationship between indirect and direct losses can be provided based on the estimates for other floods.For example, Carrera et al. (2015) assessed the economic impact of the 2000 Po river flood in Italy using the Computable General Equilibrium model atop of the direct estimates using depth-damage functions and concludes that while the total direct loss was between 3.3 and 8.8 billion Euro, it should be supplemented with 0.6-2 billion Euro indirect losses.Oosterhaven and Többen (2017) used the nonlinear programming method and suggest that for Germany the multipliers for floods are below 1.2 (what means that indirect loss is lower than 20 per cent of the direct loss), but the results are heavily dependent on the assumption on the flexibility of the production functions -in the standard input-output approach, the indirect damage is comparable in magnitude to the direct damage -so the multiplier is closer to 2. Similarly, Koks et al. (2019) show that in the future, the ratio between expected annual output losses, which can be considered as indirect and direct damage respectively is around 20 per cent.The model used in their study is based on the multiregional input-output tables.Nevertheless, the results are very sensitive to the assumptions on the flexibility of the production structure.
Overall, the impact assessment of the indirect losses from natural disasters is very difficult and depends on the assumptions of the model.So far, no impact assessment for the indirect losses for past floods in Poland was conducted.The Central Statistical Office estimates losses from 1997 flood in Poland to 3.6-3.7 billion USD, which was then equal to 2.3 per cent of GDP.According to Siwiec (2015), losses from the 2010 flood were equal to about 5 billion USD, equivalent to about 1.1 per cent of GDP.Nevertheless, these data do not include any indirect losses resulting from the interruptions in economic activity, break of supply chains, etc.Based on the ratios cited above, its magnitude can be roughly estimated to around 0.5 and 0.2 per cent of GDP in the case of 1997 and 2010 events, respectively, but the detailed study of these events was never fully conducted.

General Framework
To estimate the changes in the flood risk, one should know what projected changes in hazard, exposure, and vulnerability are.The frequent practice in defining flood risk is to create risk index of the following form: where R is risk index, E is exposure, and V is vulnerability (De Roo et al. 2007).Changes in hazard are assessed based on the river flows simulations by Piniewski et al. (2017)., while exposure is the percentage of area of the given county that will be flooded following the today's 100-year flood based on data and maps, published by Alfieri et al. (2014).Vulnerability reflects the extent of harm that is expected to be experienced once the flood hit.In the literature, there are several approaches to that issue.For example, De Roo et al. ( 2007) assumed that the vulnerability is inversely related to GDP per capita, as poorer countries are not so well prepared to the natural disasters and exposure is based on population density.On the other hand, in countries that produce more, the disruptions in production and breakages of value chains are costlier and in the absolute terms the capital accumulated in the given is more vulnerable (though it should be and usually is less vulnerable in the relative terms as more expensive buildings and equipment should be better protected).In the ideal circumstances, the information on vulnerability at the county level should be sourced from the information of severity of losses during the past events.However, no comprehensive data on past flood losses at the county level is available.There are attempts to create such a database, such as IOŚ ( 2019), but no data are publicly available so far.Therefore, for this study, the simplified method is used -vulnerability is proxied by the value of capital accumulated in a given area.It is calculated using the by the level of value added produced in a given county -therefore implicit assumption that the share of capital in value added over the whole country is constant is adopted.

Productivity Cost Method
In the usual setting (such as this adopted by Feyen et al. (2012) or Alfieri et al. (2016)), the replacement costs method is used.In the other words, a physical model is applied to calculate the depth of future floods, and then depth-damage functions paired with the land coverage maps are used to determine the future losses.In such setting, the vulnerability is implicitly defined as the value of building, estimated based on its age, size, usage etc. and does not take into account the economic potential of the buildings.Given these problems, Koks et al. (2019) add an additional layer through the modeling the input-output flows and trade between the European regions with the input-output model to assess the indirect impact of future flooding.In this paper, a different approach is adopted -instead of estimating indirect losses at the regional level, the production at the county level is used.Therefore, this method includes the first-order indirect effects of floods, but ignores the long-run economic consequences, such as the shifts in production patterns and substitution effects.
The adopted procedure to estimate future flood losses based on the productivity cost method consists of a few steps.In the first step, daily discharges database prepared by Piniewski et al. (2017) is used to estimate the parameters of the Gumbel distribution of annual maximum of daily flow for each of the nine models in model ensemble, three scenarios (reference, RCP4.5 and RCP8.5) and the three time periods (current, near future (2024-2050) and distant future (2074-2100)).The estimates of parameters are averaged out over the model ensemble and used to obtain the changes in the frequency of what is now considered a 10-year, 50-year, 100-year, and 500-year flood.The effectiveness of this method in projecting future flood frequency is confirmed by Yoshimura et al. (2008) and Roudier et al. (2016).These distributions are calculated at the level of the subbasins.
There are several potential distributions of the annual maxima that can be fitted.The most used are the Generalized Extreme Value (GEV) distribution (McFadden 1977) and Gumbel distribution (Gumbel 1941).In this paper the Gumbel distribution is applied with the PDF of the form: Gumbel distribution is used primarily because due to the data limitations and the nature of estimations (distribution parameters were derived based on 30 observed values), the estimation of the full shape of the Generalized Extreme Value (GEV) distribution would not be possible.Papalexiou and Koutsoyiannis (2013) argue that the length of record strongly influences the estimate of the GEV shape parameter.While estimating the GEV distribution over the grid points, this translates into unreasonably different parameters across different models in the ensemble and neighboring grid points.Furthermore, Mirosław-Świątek et al. (2020) show that the Gumbel distribution is the best fit for modeling the river discharges, volume of flood waves, and residence time -even if for some grid points, Weibull and GEV distributions provided better fits, the differences were negligible.
In the second step, exposure is assessed based on data by Alfieri et al. (2014).To do that, the map of 100-year flood inundation is overlapped with the map with county borders, and for each county, the percentage of area (regardless of depth), covered by water is calculated.Consequently, the implicit assumption that the capital at risk is uniformly distributed across the county area is adopted.
Furthermore, vulnerability is formally defined as the level of economic activity in a given county.As the direct data on the value added in a given area is not available, in this study, it is approximated with the percentage of national PIT and CIT tax revenues multiplied by the national value added.This indicator is used as a proxy for economic activity generated in a given area.This measure has many drawbacks (e.g. it underestimates capital income, it does not consider agricultural losses etc.), but at the county level, there is no better indicator of economic activity.On the other hand, coarser region aggregation (e.g., using NUTS-3 regions) would not allow for adequate picture of the changes in the frequency of floods as well as local vulnerability.The data on personal income tax should reflect the part of value added that is attributed to labor, while the data on corporate income tax should be proportional to the capitalrelated part of value added.While this proxy is imperfect due to the tax avoidance, underreporting etc., it should accurately reflect the chunk of value added produced in ( 2) The vulnerability is then scaled, such that the expected value of flood losses is in line with past observations, depending on the chosen source of information.Due to this scaling, the percentage of capital (or productive capacity) that is destroyed on average during current floods matches observations.While technically, the total economic losses should be used (i.e.including the indirect damage), due to the unavailability of such information, the direct damage is applied.Therefore, the results may be underestimated, but assuming that the relationship between direct and indirect damage remains constant overtime, they would be underestimated by the same factor throughout the projection period and trends still hold.
Future changes to exposure are not considered in this paper.There are several reasons for that -in general, reduction of exposure is described in the literature simply as a percentage of avoided losses without any specific analysis, and the scale of resilience is virtually impossible to project and depends on the local politics.For the same reason, no flood protection estimates of potential losses are presented.

Changes of Flood Hazard
The description of the change in high flows (95 th percentile of daily flows) directly from Piniewski et al. (2017) is a useful starting point.As Fig. 2 shows, an increase in high (95 th percentile) flows is expected in almost all the Vistula and Odra basins, especially in the central part of Poland.The change is greater in the RCP8.5 scenario than in the case of the RCP4.5 scenario and in the distant future (2074-2100) than in the near future .Moreover, all nine different models are relatively consistent regarding the direction of change and the spatial distribution of changes in flows.Furthermore, Piniewski et al. (2017) also argue that their projections are consistent with those made by Roudier et al. (2016) and Alfieri et al. (2015a), although the changes projected in the latter study seemed for them too large.
Nevertheless, even though the change in the 95 th percentile flow should be correlated with flood risk, the change in 100-year maximum flow can be vastly different due to the properties of the Gumbel distribution.That is indeed the case for the Vistula basin, especially in the eastern part.
Regardless of the scenario, data show that both the mean, and variance of the annual maximum daily flow will decrease in the southern part of Poland and increase in the western part (see Fig. 3 for mean, figure with variance was relegated to appendix).These changes are, however, more significant in the RCP8.5 scenario and in the distant future (2074-2100), than in the nearest future .Also, it is worth to note, that annual maxima of daily flows follow different pattern than indicator of high flows, as described by Piniewski et al. (2017) The changes in mean and variance of the annual maximum daily flows depict the interplay between two main trends that will shape the frequency of floods this century.The first one is increased autumn and winter precipitation, which will drive a surge in the expected value of annual maximum flows in the western part of Poland.As this shift is driven by severe weather events, the variance of the annual maximum daily flow will increase.On the other hand, the warmer climate and longer dry spells will lead to the higher evapotranspiration and lower flows in the Vistula basin.These factors will play a greater role in the more distant future, where the climate change, especially in the RCP8.5 scenario, will become severe.Therefore, in the nearest future (2024-2050), the mean annual maximum flow in Poland will increase even in the Vistula basin, but when the severe changes in weather patterns will hit Poland in the second part of the century, increased evapotranspiration will lead to slow drying of the rivers and consequent fall in the risk of flood in eastern part of Poland Table 1.
The changes in the risk of floods are presented in Figs. 4 and 5, showing the return periods for today's 10-year and 100-year floods.The results for the 2024-2050 are, in general, consistent with the maps on the mean and variance of the maximum flow, as yielded by the Gumbel parameter estimates-in the near future (2024-2050), the median return periods of all levels of flood will decrease and the change will be more severe in the RCP8.5 scenario than in the RCP4.5.However, in the second half of the twenty-first century, the evapotranspiration will start to intensify leading to increase in the length of return periods for floods in the eastern part of Poland.Moreover, even though in 2100 the median return periods will be lower than in the reference period, the mean increases, what suggests, that the changes in flood regimes in the eastern part of Poland, are more substantial than the increase in flood frequency in the western part.However, as less production activity is concentrated in areas with a decrease in flood frequency, the benefits from the fall in flood frequency will be smaller than the change in losses in the western part of country, and the overall impact on the expected losses is positive.The evolution of vulnerability and the impact of climate change on the expected losses are described in Sections 5.2 and 5.3, respectively.

Changes in Exposure
To assess the exposure of each county to the flood-related damages, 100-year flood hazard maps, produced and availed by Alfieri et al. (2014) are used.Therefore, to assess the exposure of a given county to a 100-year flood, the map of inundation with the borders of specific counties is intersected to calculate the percentage of an area covered by a 100-year flood.The results are presented in Fig. 6.Generally, the southern part of Poland is more vulnerable, especially areas along Vistula or Odra, including large cities of Wrocław, Kraków, and Opole.Areas in the western part of the country and in the southern catchments of Vistula and Odra, which will be affected by the climate change, are areas that are exposed to flooding.The decrease in flood frequency, which is expected in the second half of the twenty-first century will apply mostly to the northeastern counties, which are not vulnerable anyway.Therefore, even though the average return periods of a

Changes in Damage
The procedure of calculating the change in flood losses based on productivity costs out of the change in flood frequency requires specific assumptions.The current expected value of flood losses serves as a starting point for the analysis of their changes in the future.Unfortunately, there is no national and comprehensive database on the past losses from floods and their indirect consequences, although some efforts are being made to create such databases. 2There are some national papers showing such estimates at the local scale (see e.g.Borowska-Stefańska (2016) for the analysis of three cities in central Poland or Głosińska (2013) for two cities in the Zachodniopomorskie region) or for particular events (e.g.Kundzewicz et al. (1999) for 1997, Chojnacki (2003) for 2001, and Biedroń and Bogdańska-Warmuz (2012) for 2010 flood).Moreover, there is a paper by Siwiec (2015), who collected data on losses caused by extreme events in Poland between 2001 and 2011.Furthermore, there are international sources, which can be used-estimates by Alfieri et al. (2015b) or the EM-DAT database (EM-DAT 2008).Siwiec (2015) collected data on losses caused by extreme events in Poland through questionnaires, which were filled by 87 different institutions at the local, regional, and national level, who are responsible for crisis management.Even though much data is lacking, and it was imputed by different institutions, this is the first such a detailed approach to estimate flood losses in Poland.The result of this exercise was used in MEnv (2013) for estimating the inaction costs in different climate scenarios.Generally, losses reported based on questionnaires are higher than estimates based on physical damages described below-they amount to 0.19% of GDP, while the EM-DAT ( 2008) database shows less than 0.1%.However, once smaller events (happening every year and causing local losses equaling to 0.07-0.08% of GDP) are excluded, this estimate is equal to 0.11% of GDP, which seems much closer to assessments based on other sources.Another potential source of information on the flood losses in the past is the EM-DAT database (EM-DAT 2008), which shows the consequences (deaths, number of people affected and economic losses) of natural disasters over the years.Although the spatial and temporal coverage is impressive, only for four floods in Poland there is information on flood damage (1997, 2001, 2009 and 2010).Nevertheless, this source is very useful to inform on the average damage of flood once it occurs (even though the coverage for Poland is small, the overall number of events is large enough to provide robust estimate on the value of damages caused by average flood).Furthermore, the regularity of distribution shown in Fig. 1 and its closeness to the log-normal distribution suggest that there exists an universal distribution of flood damage that is valid for all countries in the world.The estimates of the expected value of losses, based on this source, are close to the numbers produced by Alfieri et al. (2015b) -to the protection variant if the world average is adopted, and to the no protection version if we use only the values for floods observed in Poland.This implies that the average flood in Poland in costlier than the average flood worldwide.
Table 2 shows the losses caused by floods in Poland, based on diverse sources and approaches described above.The expected value of losses from severe floods in Poland ranks from 664 million PLN (2015 prices), if the EM-DAT (2008) database is considered and the average value of flood losses in Poland amounts to 3.4 billion PLN, if all flood losses are considered based on MEnv (2013).The two lowest values seem unreasonable-as Poland seems more vulnerable to flooding than an average country in the world and flood protection as described by Alfieri et al. (2015b) is not present.Also, the highest value seems implausible, as it counts in flash floods, which are not considered in this paper.The remaining three values are quite similar-from 0.09% of GDP annually in Alfieri et al. (2015b) no protection estimate and in EM-DAT database to 0.11% based on MEnv (2013).The exact choice from these three values is difficult-MEnv (2013) figure will be used in further elaboration, because these values are comprehensive estimates based on national data.None of the studies above uses the productivity cost method; therefore, it is impossible to decide on baseline data based on this.However, the main goal of this paper is to show how losses will change in the future -with such approach baseline value of losses is of secondary importance.
Once the data on losses at the country level are assumed, the losses at the county level must be calculated.Here, the method to do that is simple-data on national GDP is broken down into counties, using the aggregate total revenues from corporate and personal income taxes.This creates an indicator of exposure measuring the intensity of production activity in a given county.Further to this, vulnerability estimates from the previous section are used.In line with Eq. 1, the product of vulnerability and exposure is used to estimate the expected value of flood losses at the county level (expected annual damage).These monetary damage is then rescaled to reflect the value of flood losses at the country level and related to local value added for the ease of interpretation.
Figure 7 shows the result of such procedure-baseline expected annual damage (EAD) at the county level.
Given the previous analysis of vulnerability and the value of flows, the presented baseline value of losses is not surprising-they are concentrated in the southern part of Poland along rivers with high flows-Vistula and Odra as well as the estuaries of both rivers.Furthermore, areas in the southern part of Poland with a bigger expected value of losses are also more densely populated and exhibit greater concentration of economic activity.As these areas are also those with the most significant growth in the future frequency of floods, an increase in losses there should be expected.
Figure 8 shows the spatial changes in the expected value of flood losses, considering flood hazard (as described in Section 5.1), vulnerability (as shown in Section 5.2) and exposure.The combination of significant baseline potential losses and increase in the flood frequency (especially in the first half of twenty-first century) yielded a surge in damage, especially in the southwestern part of Poland.In line with previous reasoning, the most significant increase will be observed in the RCP8.5 scenario in the first half of the century-the expected value of losses will increase in almost whole Poland, with the most significant impact along Odra, Warta, and Vistula below the mouth of San.The shift is also visible in the RCP4.5 scenario but is much less pronounced.
In the second half of twenty-first century, increased evapotranspiration will lead to fall in damage in large parts of eastern and central Poland.The expected losses along the southern Odra and Vistula will fall in comparison to 2050 but remain significantly above the baseline values of flood losses-so the protection will be still needed.After 2070, the differences between RCP4.5 and RCP8.5 fade out, and the impact of emission on the expected value of flood damage is similar.This is due to the fact, that in RCP8.5, both changes in precipitation and evapotranspiration are more pronounced and their additional impacts in comparison to RCP4.5 more or less balance each other.
As the expected value of losses will increase in better developed areas, one should expect that the damage summed over the whole area of the country shall also rise.Figure 9 shows the losses as a percentage of GDP for two scenarios and different assumptions on initial damage.As expected, until 2050, they will rise by 47% in the RCP4.5 and 83% in the RCP8.5 scenario.Therefore, even though the changes in flood frequency are not uniform across the country and there are even areas where it is expected to decrease, the spatial distribution of the future precipitation changes will cause flood damage across the country to rise quite significantly.
In the second half of twenty-first century, the impact of increased evapotranspiration on flood losses will be more profound and the damages will be reduced, though they will remain above the baseline level-by 32% in RCP4.5 and 51% in RCP8.5.These changes are visible even though the mean return period over the Odra and Vistula basin will increase (the median will fall slightly though).This shows, how important it is to consider the spatial distribution of the changes in flood frequency, as the increased frequency of fluvial floods will continue to exert pressure on the Polish economy, even though this influence will be milder after 2070.
At the country level, the annual expected value of flood losses will increase until 2050 by 2-3 bn PLN in the RCP4.5 and 3-4 bn PLN in the RCP8.5 scenario (in 2015 prices, see Fig. 10).Out of these numbers, climate change will cause 43 and 54 percent, respectively, while the remainder is due to the growth in GDP (and subsequent change in the value of assets).Therefore, by 2050, the Polish economy will lose on average 1-1.3 billion PLN annually due to the changes in flood frequency if measures to limit GHG emissions are adopted and enforced (RCP4.5 scenario) and 1.8-2.2PLN otherwise (in the RCP8.5 scenario).After 2050, these numbers, would be slightly lower (0.9-1.1bn PLN in RCP4.5 and 1.5-1.8bnPLN in RCP8.5), but the overall annual expected value of losses will increase due to the GDP growth.

Sensitivity of Losses to Selected Vulnerability Indicators
A precise definition of the vulnerability and the spatial resolution is of the profound importance for the accurate estimation of the future flood losses, indicating the gains from the finer spatial disaggregation.Figure 11 shows the future value of flood losses for Poland as a percentage of GDP for different vulnerability indicators -proportional to total income, capital income, population, and area.If the vulnerability is defined with area (what is technically equivalent to the assumptions on equal spread of economic activity across country), the losses are expected to fall in both RCP4.5 and RCP8.5 scenarios, reflecting the lower share of area that will be inundated in the future climate scenarios.Once the population is considered, the losses are increasing in relative terms by almost 20 per cent in the RCP4.5 and by 54 per cent in the RCP8.5 scenario by 2050.Consequently, they are higher, reflecting the population density in areas that will be subject to more intense flooding in 21 st century.However, if total income is considered (sum of personal income tax and corporate income tax revenues), the damage increases even more reflecting higher productivity of areas in which the frequency of floods is projected to rise.It can surge even more if vulnerability is reflected by the share in capital income, but the difference is relatively minor as capital taxes revenues are proportional to labor incomes.Furthermore, the application of the total income is more justified as during floods not only capital is destroyed, but also labor is disrupted.
The difference between the future value of damage with different distribution of vulnerability assumptions highlights the importance of spatial details in modelling future flood losses.Economic activity is usually concentrated along rivers and therefore even though the total area subject to flooding may decrease, the actual losses may surge.In the case of Koks et al. (2019), the LISFLOOD hydrological model is set at 0.5° resolution and Gumbel distribution is modeled at this level, which in Poland is equivalent to about 50 km.Production and economic losses are modeled at the NUTS-2 level of spatial disaggregation.While such resolution is very useful for the cross-country comparisons and estimation of the order of magnitude of the future flood losses, more refined spatial resolution allows for better insights on the changes in local flood conditions and even affect spatial planning at the cost of a less sophisticated model, as input-output tables are not available at the county or municipality level and only recently it became available at the regional level.In this respect, this paper is complementary to previous research in this area.

Discussion
Even though instead of widely used reconstruction costs approach, in this paper the productivity costs valuation is used, it is useful to compare the above results with existing estimates.According to the data by Alfieri et al. (2015b), with no economic development, the damage caused by floods in Poland barely changes between 2020 and 2080.Therefore, the expected change in annual losses between these years is due to the changes in economic development.However, if changes between 1990 and 2080 are considered, an increase in average annual damage should be expected from 0.8‰ of GDP in 1990 to 2.2‰ in 2020, 1.7‰ in 2050 and 1.9‰ in 2080.However, the range of uncertainty (differences between model ensembles) is huge (0.5‰ to 1.3‰ in 1990 and 0.6‰ to 4.1‰ in 2080).Therefore, the magnitude of change is similar to those found by Feyen et al. (2012), but the numbers vary.These values are in line with the results presented above for the RCP8.5 scenario, even though in the estimates by Alfieri et al. (2015b), there is no fall after 2050 and the surge is observed already in 2020.Rojas et al. (2013) present the expected annual damage for different flood protection levels -for the laxest protection against a 50-year flood, the damage is expected to increase from €432 million (2006 prices) in the current climate to €4647 million in 2050 and €8282 million in 2080.The main reason for this discrepancy between the results presented above and those published by Rojas et al. (2013) is the quick pace of the economic growth in the Shared Socioeconomic Pathways scenario -expected annual damage as a percentage of GDP remains roughly constant between 2000 and 2080.The time pattern is analogous to that described above -in the first half of the century damage increases, but then they tend to fall.Rojas et al. (2013) present also spatial distribution of their results.In their findings only southern part of Poland will experience an increase in EAD, while the western part along the Odra River will enjoy a reduction in losses.Moreover, they report an increase in EAD until 2050 in the eastern part (including Podlaskie and Lubelskie regions).As these results were based on older and less detailed flood inundation models, these discrepancies are not surprising, but they highlight the importance of socioeconomic scenarios and modeling approach in the assessment of future flood losses.Dottori et al. (2020) show an updated version of the previous work by Alfieri (2018) with the use of newer climate model projections, digital elevation models, and depthdamage functions.The depth-damage functions were adopted from Huizinga et al. (2017), who included country-specific socioeconomic conditions in regressions to estimate the costs of construction.In this setting, Dottori et al. (2020) estimated the damage functions based on the regional administrative level (NUTS2).In no adaptation scenario, they show an increase in flood damage in Poland from 0.14% of GDP in the baseline scenario to 0.15% in the 1.5 °C increase scenario by 2050 and 0.19% in the 2 °C increase scenario.Although, these results can not be directly compared to the results presented in this paper due to the different approach to the scenarios, four observations are immediately visible.First, the baseline damage in this study is considerably lower.Second, the relative change in the losses due to the climate change is similar to the results presented in this paper.Third, this change in damage is relatively equally distributed across the country.Fourth, there is no effect of increased evapotranspiration after reaching a certain threshold (for 3 °C increase scenario damage increases to 0.29% of GDP).There are several reasons for the differences -baseline damage is higher because it is estimated based on depth-damage functions and not the actual data.In addition, the LISFLOOD model is built at a much coarser spatial resolution than the SWAT model used by Piniewski et al. (2017), what explains the different spatial distribution of changes in losses and the lack of detailed impact of increase in evapotranspiration.Koks et al. (2019) supplemented the earlier findings with indirect macroeconomic effects caused by the floods using the multiregional impact assessment model.In their case, the expected annual output losses in Poland (which is defined as additional change in output atop of the expected annual damages) is falling in the future climate, reflecting the share of the western part of Poland in reconstruction effects in western Europe.While this model allows to show the broader macroeconomic impacts of the changing patterns of future floods, it does not provide insights on changing future damage at the local level.
Summing up, the results presented in this study are broadly in line with other papers using the digital elevation modelling (DEM), depth-damage functions and the reconstruction costs approach, but it shows the benefits of using finer spatial disaggregation and the proxy of economic activity to model future losses.At the aggregate level, the future damage presented in this paper; increases more because economic activity is concentrated in area that will be affected by changes to flood patterns and is relatively less intense in areas where the frequency of floods will fall, what explains the discrepancy between the results presented above and previous papers where the economic activity was modeled at the regional level.In addition, the indisputable practical advantage of the method presented in this paper is its simplicity and applicability to other regions where the regional input-output table (as used in Koks et al. (2019)) are not available.Also, there is no need to use land-cover maps, which are unavailable or unreliable for some regions.Consequently, the method presented in this paper can be successfully used when the DEM modeling is unavailable or when there are no detailed land-use maps.In addition, it allows to consider the productivity losses caused by the extreme events and for the finer spatial disaggregation reflecting the mostly local character of flood damage in Central Europe.

Conclusions
According to the literature (e.g.MEnv ( 2013)), the increase in the frequency of fluvial floods will be the most severe consequence of the climate change in Poland.It is also quite extensively researched with estimates done previously by Feyen et al. (2012) or Alfieri et al. (2016).However, none of these studies so far considered the distribution of capital and the potential of given regions to create GDP, considering the productivity of capital accrued in given areas.This paper shows that losses caused by fluvial floods expressed as a percentage of GDP in Poland will increase by 47% in the RCP4.5 and 83% in the RCP8.5 scenario by 2050 if productivity costs valuation method is used.After that, they will slightly decrease due to the increased evapotranspiration, but they will remain above the baseline no-climate-change level by 32% and 51% on average in the RCP4.5 and the RCP8.5 scenarios in 2074-2100.Considering that the distribution of productive capacity across the country tends to amplify the expected changes in losses from floods in comparison to the existing estimates.
This study contributes to the assessment of the impact of climate change on the Polish economy, describing the consequences of the changing climate on flood losses.To fully describe how the changes in the frequency of floods caused by the climate change will shape the changes in the Polish economy, this direct influence should be plugged into a more general model of the economy in line with other assessments of the direct consequences of the climate change such as Gąska (2022) for windstorm losses.Especially interesting and promising research avenue could be model similar to the Poledna et al. (2018) calibrated at a local scale as local changes to losses from floods are available.Also, the standard general equilibrium model can be very useful in describing the impact of increased losses on the economy.Furthermore, the results of this exercise are sensitive to the assumptions on the climate models.As this area of research is developing very quickly, the reliability of the estimate hinges upon the regular update of both climate data and simulations of river flows like those published by Piniewski et al. (2017), so such study should be repeated every couple of years in line with advances in climate and physical river flows modeling.

Fig. 1
Fig. 1 The distribution of losses from floods as a percentage of GDP between 1962 and 2018.Source: EM-DAT (2008) a given county.Furthermore, the future vulnerability is scaled up in line with GDP growth scenario published by Organization of Economic Cooperation and Development (OECD) as Shared Socioeconomic Pathway 2: Middle of the Road (O'Neill et al. 2017) to reflect the impact of growth in the value of accumulated assets.

Fig. 2
Fig. 2 The multimodel ensemble mean change in the high flow indicator QH for the Vistula and Odra River basins, for the near future (NF) and far future (FF) under Representative Concentration Pathways (RCPs) 4.5 and 8.5.Source: Piniewski et al. (2017)

Fig. 3
Fig.3The multimodel ensemble mean change in the expected value of the annual daily flow maximum relative to the current climate.Source: Own calculations based on data byPiniewski et al. (2017)

Fig. 6
Fig.6The area endangered by 100-year flood as a percentage of area of each county.Source: Own calculations based onAlfieri et al. (2014)

Fig. 10
Fig. 10 Decomposition of annual losses from floods into GDP growth component and change in risk component based on Alfieri et al. (2015b) scenarios without protection (ANP) and with protection (AP) and severe events from MEnv (2013) -2015 prices

Fig. 11
Fig. 11 Future value of flood losses for different vulnerability indicators

Fig. 12 Fig. 14 Fig. 15
Fig.12The multimodel ensemble mean change in the mean of the annual daily flow maximum (data based).Source: Own calculations on the basis of data byPiniewski et al. (2017)

Table 1
Mean and median return periods for 10-, 20-, 100-and 500-years floods based on estimated Gumbel distributions for each subbasin

Table 2
Comparison of the expected annual value of flood damage based on various sources and methods (2015 prices)