8 TheAdaptiveRegionalInput-Output(ARIO)modelispopularforquantifyingindirecteconomiclosses, 9 whichstemfrombusinessandsupplychaininterruption. However,refiningthismodeltostudynewcontexts 10 is challenging in its basic form due to low-resolution modeling of behavioral parameters and temporally 11 staticreconstructionrates. ThispaperpresentsarefinedARIO,orR-ARIOmodelthatincorporatesdynamic 12 reconstructionrates,sector-levelmodelingofbehavioralparameters,andexplicitmodelingofhousinglosses 13 separatelyfromproductivecapitallosses. Weperformaglobalvariance-basedsensitivityanalysistoidentify 14 themostinfluentialparametersonpredictedindirectlossfromtheR-ARIOmodel. Acasestudyapplication 15 to the 2016 Kumamoto Earthquake Sequence isolates trends in housing and economic recovery, capturing 16 temporal differences in reconstruction demand and uncertainty across economic indicators. 17 INTRODUCTION 18 Indirect losses stemming from the disruptions in production and supply chains make up a substantial 19 portion of post-disaster loss. The 1994 Northridge Earthquake, 2008 Wenchuan Earthquake, and 2011 20 Tohoku Earthquake generated 7.3, 124, and 211 billion U.S. dollars of indirect loss, respectively (Petak 21 and Elahi 2000; Wu et al. 2012; MacKenzie et al. 2012). These losses, amounting to 17%, 35%, and 37% 22 of the events’ total post-disaster losses, illustrate that exclusive prediction of direct losses can significantly 23 underestimate post-disaster impacts. 24 Severalmacroeconomicmodelingtoolshavebeendevelopedtoquantifypost-disasterindirectlossacross 25 sectorsinaregionaleconomy. Input-output(I-O)modelshavebeentraditionallyusedtomodelsuchimpacts. 26 1 Issa, March 20, 2024 These models leverage tables that characterize production inputs and outputs of each sector. However, a 27 limitationofbasicI-Omodelsisthattheycannotquantifysupply-sideshocksorcapturebusinessadaptation 28 suchassubstitutionorconservationofinputsbydefault(Koksetal.2016;GalbuseraandGiannopoulos2018). 29 Asaresult,I-Omodelstendtooverestimatedisasterimpacts(GalbuseraandGiannopoulos2018;Hallegatte 30 2014). Asecondapproachistousecomputablegeneralequilibrium(CGE)modelstosimulatepost-disaster 31 impacts by estimating how shocks to the supply and demand of goods and services map to changes across 32 economic indicators (e.g., Rose and Liao 2005). Unlike basic I-O models, they account for business 33 adaptation behaviors, economies of scale, and nonlinear impact functions. However, without refinement, 34 CGE models tend to overestimate resilient response by economic sectors, leading to underestimation of 35 disaster impacts (Rose and Liao 2005; Botzen et al. 2019). 36 The Adaptive Regional Input-Output model (ARIO) extends basic I-O models by incorporating some 37 CGEcharacteristicstoconsiderchangesinproductivecapacityduetoproductivecapitallossesandadaptive 38 behaviors by individual sectors (Hallegatte 2008). Examples of adaptive behaviors include overproduction, 39 which can be achieved through production recapture (e.g., overtime or extra shifts to compensate for lost 40 production)orresourceisolation(e.g.,modifyingoperationstorunwithouttypicalinputs). Bothtacticshave 41 been demonstrated to be highly effective in various post-disaster contexts (Wein and Rose 2011; Haywired 42 2019; Wei et al. 2020). The ARIO model has since been improved in Hallegatte (2014) to explicitly model 43 inventories and production bottlenecking. The ARIO model was first used to predict economic recovery 44 following Hurricane Katrina, and others have used it to evaluate climate change, earthquake, and flood 45 impacts (Ranger et al. 2011; Zhang et al. 2017; Liu et al. 2023). 46 While the ARIO model and its applications enable the evaluation of post-disaster indirect loss, several 47 barriers limit its use in new regions and contexts. First, the ARIO model assumes a constant, temporally 48 static reconstruction rate for each sector, leading to identical reconstruction demands across all time steps 49 until all productive capital has been reconstructed. While this rate can be modified, it cannot capture 50 differences across sectors or time. Second, it does not have an explicit mechanism for handling housing 51 losses, which often comprise a substantial portion of the total direct loss. Previous studies have assigned 52 all housing reconstruction demand to the real estate sector (Hallegatte 2014; Markhvida and Baker 2023). 53 This workaround captures housing-related reconstruction demands but distorts economic recovery for the 54 realestatesectorbecauseitimpliesthathousingispartoftheproductivecapitalofthatsector. Theextentof 55 distortion will vary depending on the amount of damaged housing and productive capital in the real estate 56 2 Issa, March 20, 2024 sector. 57 Finally, ARIO model behavioral parameters are modeled at an economy-level resolution. These 58 parameters—which characterize post-disaster adaptability, inventory, overproduction, and heterogeneity— 59 significantly influence the predicted indirect loss. One consequence of low-resolution modeling is that a 60 single change to one parameter must be applied to all sectors, regardless of inter-sector differences. Mod- 61 elers wishing to select these parameters for new study regions must use a "one-size-fits-all" approach. As a 62 result, several studies (e.g. Ranger et al. 2011; Zhang et al. 2017; Wang et al. 2018; Markhvida and Baker 63 2023) simply adopt the parameters used to model post-Katrina recovery introduced in Hallegatte (2008) or 64 Hallegatte(2014),despitetransferringthemtoanon-Katrinacontext. Furthermore,low-resolutionmodeling 65 makes it more difficult to refine specific parameters since empirical evidence often exists at the sector or 66 sector-category level, and it is not clear from sensitivity analyses which sectors contribute the most to the 67 variance in the predicted ARIO output. Due to these challenges, past studies have not been able to perform 68 sector-level sensitivity analyses or uncertainty quantification. 69 To address the abovementioned issues, this paper proposes the R-ARIO model to simulate post-disaster 70 economicrecovery. TheR-ARIOmodelimprovesonpreviousiterationsoftheARIOmodelbyintroducing 71 (i) dynamic reconstruction rates based on sector-specific reconstruction time curves, (ii) explicit modeling 72 of housing losses separate from productive capital losses, and (iii) sector-level modeling and uncertainty 73 quantification of behavioral parameters. In addition, we propose the use of global sensitivity analyses 74 to identify the most important behavioral parameters for further refinement. The R-ARIO model and the 75 accompanyingsensitivityanalysisapproacharedemonstratedinacasestudythatexploreseconomicrecovery 76 followingthe2016KumamotoEarthquake. Aspartofthecasestudy,weexploretheinfluenceofeachmodel 77 enhancement on the predicted indirect loss and illustrate how global sensitivity analyses can be used to 78 prioritize future refinement of behavioral parameters. 79 THE R-ARIO MODEL 80 In this section, we provide an overview of the R-ARIO model and describe its inputs, outputs, and 81 architecture. Subsections describe each model improvement in greater detail. 82 The R-ARIO model extends the work by Hallegatte (2014) in three ways: 83 1. Reconstructiondemandismodeleddynamicallythroughouttherecoveryperiodusingtime-dependent, 84 sector-specific reconstruction rates. 85 3 Issa, March 20, 2024 2. Housing losses are incorporated into the model explicitly and separately from productive capital 86 losses. 87 3. Behavioral parameter modeling is performed at the individual sector level to enable parameter 88 refinement that accounts for inter-sector differences and enables uncertainty quantification. We 89 propose a set of updated parameters that reflect these differences based on documented cases of 90 business adaptation. 91 Figure 1 illustrates the R-ARIO model workflow, which consists of three main steps. 92 Assemble inputs Simulated or observed disaster over region economic recovery realizationsfrom time of shock to full recoveryRun R-ARIO model Local input-output tableValue addedLocal final demandFixed assetsExportsImportsPre-disaster, sector-level economic data Sector-level direct lossesSector-levelreconstruction time curvesBuilding-level direct lossesBuilding-level reconstruction timesaggregateaggregate Sector-levelbehavioral parameters012345020406080100 012345-40-2002040 012345-40-2002040 012345-40-2002040value added (% change)time after earthquakeSimulate demand Update supply and economic metricsStage I Stage IIIFor each timestep:R-ARIO Simulate productionStage II Fig. 1.General overview of the R-ARIO model workflow. First,astudyregionanddisasteraredefinedfortheanalysis. Disastersarebasedonobservedpastevents 93 orhypotheticalevents(e.g.,usingsimulation-basedscenarios). Thedisasterisusedtodeterminethespatial 94 extent of the study, and the amount of capital loss per sector resulting from the damage. 95 Next, input data specific to the regional economy (comprised of 𝑁𝑠sectors) is assembled. The required 96 data falls into four categories: pre-disaster economic activity (e.g., value added, exports), monetary losses 97 due to direct damage, reconstruction time curves, and behavioral parameters. These are detailed in Table 1. 98 Sector-level data encompasses a series of "baseline" inputs used to quantify steady-state economic 99 4 Issa, March 20, 2024 activity. The first of these inputs is the local input-output (I-O) table, a matrix representing the flow of 100 goods and services exchanged between sectors in the defined economy, indicating how the outputs of one 101 sectorbecomeaninputforothers. Thistableisusedtoderiveinput-outputratiosthatcontroltheamountof 102 inputs necessary to fulfill productive tasks. Next, value added and fixed assets are provided for each sector. 103 Value added represents each sector’s net contribution to the defined economy’s output. It is calculated as 104 the difference between a given sector’s total output and the value of intermediate inputs it consumes from 105 supplyingsectors. Fixedassetsrepresentthevalueofproductivecapitalleveragedbyeachsectortoproduce 106 goodsandservices. Fixedassetsareassumedtoequalthetotalreplacementvalueofbuildingswithinagiven 107 sector. Finally, exports, imports, and local demand are provided. Exports represent the value of goods and 108 servicesproducedbyeachsectorwithinthedefinedeconomythataresoldoutsidethestudyregion. Imports 109 represent the value of goods and services produced by each sector outside the study region that are brought 110 into the defined economy. Local final demand refers to the total demand for goods and services by final 111 consumers. 112 Sector-level reconstruction time curves are time-dependent functions representing the reconstruction 113 trajectory of damaged buildings within a specific sector. These inputs are used to determine sector-specific 114 reconstruction rates of productive capital as part of the first enhancement in this study. User-provided 115 reconstruction time curves only account for the time it takes to reconstruct buildings and do not include 116 indirect delays that impede the start of repairs or slow work due to a lack of needed inputs. We describe the 117 dynamic reconstruction rates below. 118 Sector-level direct losses are monetary losses directly attributed to damage from the disaster. These 119 inputscontrolthelossofproductivecapitalandtheapproximatedropinproductivecapacity,attheonsetof 120 the disaster for each sector. 121 Finally, sector-level behavioral parameters characterize adaptation and inventory mechanics of each 122 sectorfollowingthedisaster. LiketheARIOmodel,theR-ARIOmodelconsidersfivebehavioralparameters 123 selected by the user to characterize the regional economy. These are discussed in greater detail later. 124 Running the R-ARIO model 125 At the beginning of each simulation, behavioral parameters are sampled from user-defined distributions 126 for uncertainty quantification purposes. Each simulation tracks economic recovery at discrete time steps 127 over a user-defined period. At each time step, a three-stage calculation (as shown in Figure 1) is performed 128 5 Issa, March 20, 2024 TABLE1. InputstotheR-ARIOmodel. Here, 𝑁𝑠representsthenumberofsectorsinthedefinedeconomy, and(𝑁𝑠+1)represents the number of sectors, plus housing. 𝑁steprefers to the length of the time domain used as the x-axis of each recovery curve. Input Category Size Units Differences in treat- ment (R-ARIO versus ARIO) Local input-output tableSector-level economic data𝑁𝑠×𝑁𝑠 Monetary value None Value added Sector-level economic data𝑁𝑠×1 Monetary value None Exports Sector-level economic data𝑁𝑠×1 Monetary value None Imports Sector-level economic data𝑁𝑠×1 Monetary value None Local final demand Sector-level economic data𝑁𝑠×1 Monetary value None Direct losses Sector-level direct losses(𝑁𝑠+1)×1 Monetary value None Reconstruction time curvesSector-level reconstruction time curves(𝑁𝑠+1)×𝑁stepUnitless (Fraction of damaged capital reconstructed)Used to determine time-dependant recon- struction time rates in R-ARIO. Not used in the ARIO model. Behavioral parame- tersSector-level behavioral parameters𝑁𝑠×5 Varies by parameterModeled at the sector- level resolution in R- ARIO. Modeled at the economy-level in ARIO. to estimate key economic metrics at the sector level. 129 Stage I: Simulate demand 130 Demand𝐷𝑖(𝑡)for each sector 𝑖is computed at each time step ( 𝑡) as the sum of inventory orders, local 131 final demand, reconstruction demand due to damaged productive capital, and exports: 132 𝐷𝑖(𝑡)=∑︁ all𝑗𝑂𝑗,𝑖(𝑡)+𝐶𝑖(𝑡)+𝑅𝑖(𝑡)+𝐸𝑖(𝑡) (1) 133 6 Issa, March 20, 2024 where: 134 𝑂𝑗,𝑖(𝑡)= Orders (intermediate consumption) from sector 𝑗to sector𝑖at time𝑡 𝐶𝑖(𝑡)= Local final demand to sector 𝑖at time𝑡 𝑅𝑖(𝑡)= Reconstruction demand for sector 𝑖at time𝑡 𝐸𝑖(𝑡)= Exports of sector 𝑖at time𝑡135 At each time step, demand 𝐷𝑖(𝑡)is satisfied by two sources: production and imports. If these two 136 sources cannot fulfill demand, sectors begin proportionally rationing (using the ratio between production 137 and demand at the current time step) across 𝑂𝑗,𝑖(𝑡),𝐶𝑖(𝑡),𝑅𝑖(𝑡), and𝐸𝑖(𝑡). 138 A key component in the calculation of demand 𝐷𝑖(𝑡)is reconstruction demand 𝑅𝑖(𝑡). Fulfillment 139 of reconstruction demand drives the restoration of productive capital over time. 𝑅𝑖(𝑡)is estimated using 140 Equation 2: 141 𝑅𝑖(𝑡)=∑︁ all𝑗(𝑅𝐷𝑀 𝑗,𝑖(𝑡)×rate 𝑗(𝑡)) (2) 142 where: 143 𝑅𝐷𝑀 𝑖, 𝑗(𝑡)= Reconstruction demand from sector 𝑗to sector𝑖at time𝑡; rate 𝑗(𝑡)= Rate of reconstruction of sector 𝑗’s productive assets at time 𝑡144 Totakeintoaccountthetime-dependentnatureofreconstruction,theR-ARIOmodelintroducestheterm 145 rate 𝑗(𝑡), which represents the rate of reconstruction at the current time step, described further below. 146 Stage II: Simulate production 147 In the absence of supply-side constraints, the production of sector 𝑖would equal the demand for sector 𝑖 148 at each time step. However, the R-ARIO model constrains the estimated production in two ways. It is first 149 constrainedbyproductioncapacity, 𝑃𝑐𝑎𝑝 𝑖(𝑡),whenproductivecapitalisinsufficienttomeetdemand(e.g.,in 150 caseswithsignificantdirectdamagetoasector). Productionisalsoconstrainedbyinventories. Itisassumed 151 that if inventories are lower than their required levels, then production is reduced. 152 Thefinalvalueofproduction, 𝑃𝑎 𝑖(𝑡),accountsforboththeproductioncapacityandinventoryconstraints. 153 Computing the value of 𝑃𝑎 𝑖(𝑡)follows three principal calculations. First, each sector’s required inventory 154 levels𝑆𝑟 𝑗,𝑖(𝑡)are computed. 𝑆𝑟 𝑗,𝑖(𝑡)represents the amount of input 𝑗necessary to meet the local production 155 level of sector 𝑖over the duration of inventory 𝑗, calculated as: 156 7 Issa, March 20, 2024 𝑆𝑟 𝑗,𝑖(𝑡)=  𝑛𝑗×𝑃𝑐𝑎𝑝 𝑖(𝑡)−𝐼𝑖(𝑡)×𝐴𝑗,𝑖(𝑡) if𝐷𝑖(𝑡)>𝑃𝑐𝑎𝑝 𝑖(𝑡) 𝑛𝑗×𝐷𝑖(𝑡)×𝑃𝑐𝑎𝑝 𝑖(𝑡)−𝐼𝑖(𝑡) 𝑃𝑐𝑎𝑝 𝑖(𝑡)×𝐴𝑗,𝑖(𝑡)if𝐷𝑖(𝑡)≤𝑃𝑐𝑎𝑝 𝑖(𝑡)(3) 157 where: 158 𝑛𝑗= Target inventory level of supplying sector 𝑗in days of demand 𝑃𝑐𝑎𝑝 𝑖(𝑡)= Production capacity of sector 𝑖at time𝑡 𝐼𝑖(𝑡)= Imports of sector 𝑖at time𝑡 𝐴𝑗,𝑖= I-O table coefficients (required units of input from sector 𝑗to produce unit of sector 𝑖)159 Next,themaximumpossibleproductionofsector 𝑖,𝑃𝑚𝑎𝑥 𝑗,𝑖(𝑡),dependsupontheactualinventorylevelof 160 input𝑗. If required inventory 𝑆𝑟 𝑗,𝑖(𝑡)is not met, then the maximum possible production 𝑃𝑚𝑎𝑥 𝑗,𝑖(𝑡)is reduced 161 proportionally, taking into consideration inventory substitution effects (heterogeneity): 162 𝑃𝑚𝑎𝑥 𝑗,𝑖(𝑡)=  𝑃𝑐𝑎𝑝 𝑖(𝑡)−𝐼𝑖(𝑡)×min 1,𝑆𝑗,𝑖(𝑡) 𝜓𝑗×𝑆𝑗,𝑖(𝑡) +𝐼𝑖(𝑡) if𝐷𝑖(𝑡)>𝑃𝑐𝑎𝑝 𝑖(𝑡) min 𝐷𝑖(𝑡),𝐷𝑖(𝑡)×𝑃𝑐𝑎𝑝 𝑖(𝑡)−𝐼𝑖(𝑡) 𝑃𝑐𝑎𝑝 𝑖(𝑡)×min 1,𝑆𝑗,𝑖(𝑡) 𝜓𝑗×𝑆𝑗,𝑖(𝑡) +𝐼𝑖(𝑡)! if𝐷𝑖(𝑡)≤𝑃𝑐𝑎𝑝 𝑖(𝑡)(4) 163 where: 164 𝑃𝑚𝑎𝑥 𝑗,𝑖(𝑡)= Maximum production of sector 𝑗to sector𝑖 𝑆𝑗,𝑖(𝑡)= Actual inventory of input 𝑗for sector𝑖at time𝑡 𝑆𝑟(𝑗,𝑖)(𝑡)= Required inventory of sector 𝑗to sector𝑖at time𝑡 𝜓𝑗= Production reduction parameter (heterogeneity) of sector 𝑗165 Foreachinputsector 𝑗,ifthecurrentinventoryisgreaterthanorequalto 𝑆𝑟 𝑗,𝑖(𝑡),nosectoralconstraints 166 are applied. On the other hand, if a given sector cannot meet the required inventory, then its production is 167 reduced using a ratio that considers heterogeneity in disaster losses and impacts. The top and bottom terms 168 in Equation 4 account for the case in which the production capacity of sector 𝑖is insufficient and sufficient 169 to fulfill demand, respectively. 170 8 Issa, March 20, 2024 Finally, the actual production 𝑃𝑎 𝑖(𝑡)is taken as the minimum of all sectoral constraints: 171 𝑃𝑎 𝑖(𝑡)=min 𝑃𝑚𝑎𝑥 𝑗,𝑖(𝑡),for all𝑗 (5) 172 Stage III: Update supply and key economic metrics 173 Demandandproductionarethenusedtoupdatesupplyandcalculatekeyeconomicmetricssuchasvalue 174 added, which is calculated as production minus intermediate production and imports: 175 𝑉𝐴𝑖(𝑡)=𝑃𝑎 𝑖(𝑡)−𝐼𝑖(𝑡)−∑︁ all𝑗𝐴𝑗,𝑖×𝑃𝑎 𝑖(𝑡) (6) 176 where: 177 𝑉𝐴𝑖(𝑡)= Value added of sector 𝑖at time𝑡 𝐼𝑖(𝑡)= Imports to sector 𝑖at time𝑡 𝐴𝑗,𝑖= Coefficients of the I-O table178 By repeating the calculations in Stages I through III across all sectors, value added, production, and 179 unsatisfied demand can be tracked over time to produce economic recovery curves. Uncertainty across 180 different economic metrics can be captured by rerunning the R-ARIO model using different behavioral 181 parametersamplings. Theresultingrecoverycurveensemblesforeachsector,whichconsideruncertaintyin 182 the assumed behavioral parameters, are the final output of the R-ARIO model. 183 ThefollowingthreesectionscovertheimplementationofthreeR-ARIOenhancementsingreaterdetail. 184 Dynamic reconstruction rate 185 Reconstruction demand plays a significant role in the economic recovery process, since it is a critical 186 component of sector-specific demand 𝐷𝑖(𝑡), and drives the restoration of productive capital (and hence, 187 production capacity) over time. Equation 2 indicates that the reconstruction demand is driven by the 188 assumed rate of reconstruction, rate 𝑗(𝑡). 189 Past iterations of the ARIO model (Hallegatte 2008; Hallegatte 2014) assume a constant value of 190 rate 𝑗(𝑡)for all timesteps. The original ARIO model assumed a constant half-year reconstruction time for 191 all sectors, and hence, a rate of1 0.5throughout the recovery process. Markhvida and Baker (2023) extended 192 this assumption to account for differences in reconstruction speed across sectors by using the time to 95% 193 9 Issa, March 20, 2024 reconstruction for sector 𝑗(i.e.,𝜏𝑗,95), based on sector-specific reconstruction times. While this allows for 194 differing reconstruction processes for each sector, it still employs a constant reconstruction rate equivalent 195 to1 𝜏𝑗,95across the recovery period for each sector. 196 Toaccountfortemporalvariationsinthisrate,theR-ARIOmodelleveragesa"dynamic"reconstruction 197 rate for each sector 𝑗that is updated throughout a simulation based on reconstruction progress (Figure 2b). 198 00.511.500.20.40.60.81 00.20.40.60.81Recovery Progress0123Slope of Recovery Curvefraction of damaged productive capital reconstructedTime (years) 𝑟𝑎𝑡𝑒!𝑡(fraction / year)fraction of damaged productive capital reconstructed(a)(b) Fig.2.IntheR-ARIOmodel,themappingbetweenrate 𝑗(𝑡)andreconstructionprogress(right)isdeveloped using the user-provided reconstruction time curve for sector 𝑗(left). For a given sector, the mapping between rate 𝑗(𝑡)and the fraction of damaged productive capital recon- 199 structed (Figure 2b) is derived using user-provided reconstruction time curves (Figure 2a). This is done by 200 takingthederivativeofthereconstructiontimecurvewithrespecttotimeandthenmappingitdirectlytothe 201 fraction of damaged productive capital reconstructed. 202 Explicit consideration of housing losses 203 Housingdamagecanproduceasignificantportionofpost-disasterlossandreconstructiondemandinan 204 impacted region. Previous applications of the ARIO model typically assigned this reconstruction demand 205 to the real estate sector (Hallegatte 2014; Markhvida and Baker 2023). While this approach accounts for 206 housing loss in the analysis, it treats their replacement costs as productive capital. Since the ARIO model 207 usestheratiooflosstoproductivecapitaltoestimatetheinitialdropinproduction,thisapproachcandistort 208 economic recovery for the real estate sector and cause unintended upstream and downstream ripple effects. 209 Ratherthanassigninghousinglossesandproductivecapitaltoanindividualsector,theimprovedR-ARIO 210 approach assigns housing losses to a distinct housing "sector." This sector only generates reconstruction 211 10 Issa, March 20, 2024 demands, and is assumed to hold no productive capital. The housing sector does not contribute to any 212 macroeconomiccalculationsofinputs,outputs,orproduction. Withthistreatment,itisaccountedforinthe 213 analysis without influencing the initial drop in production for non-housing sectors. 214 Sector-level behavioral parameter modeling 215 Finally, the R-ARIO model utilitzes sector-level behavioral parameter modeling to enable more gran- 216 ular refinement, uncertainty quantification, and global sensitivity analysis. Behavioral parameters, which 217 characterizesector-leveladaptationandinventorymechanics,significantlyinfluencethepredictedeconomic 218 recovery and indirect loss. For example, past sensitivity analyses have shown that the choice of inventory 219 parameters𝑛𝑠and𝜏𝑠canmovepredictedchangesinvalueaddedshortlyafterthedisasterfrommoderate(< 220 20%) to economic collapse (100%) (Hallegatte 2014). 221 TheR-ARIOmodelconsidersthesamefiveARIObehavioralparametersthatcontroleconomicrecovery 222 dynamics. Time to maximum overproduction ( 𝜏𝛼) introduces a temporal dimension, defining the duration 223 needed for the production system to adjust and reach peak overproduction capacity. Time of inventory 224 restoration ( 𝜏𝑠) quantifies the duration required to restore inventory levels to the predefined target after 225 a disruption. The maximum overproduction parameter ( 𝛼𝑚𝑎𝑥) defines the upper limit of overproduction 226 capacity in response to increased demand. Target inventory level ( 𝑛𝑗) represents the temporal dimension 227 of inventory management, specifying the duration for which available inventory can support production. 228 Finally, the production reduction (or heterogeneity) parameter ( 𝜓) captures the response of businesses to 229 disaster impacts, influencing the extent to which production is reduced when inventories are insufficient. 230 Past studies (e.g. Ranger et al. 2011; Zhang et al. 2017; Wang et al. 2018; Liu et al. 2023; Markhvida 231 andBaker2023)assignidenticalparametervaluesforeachsectorintheeconomy,typicallyusingthevalues 232 proposedbyHallegatte(2014)andindicatedinFigure3. Anexceptiontothisisforsectorswithnon-stockable 233 goods—inthosecases,thetargetinventorylevel 𝑛𝑗issetto3daystoaccountforthefactthatmanysectors 234 cannot store long-lasting inventories (e.g., utilities). 235 The R-ARIO model includes updated behavioral parameters split across seven major sector categories 236 (Figure3). Thisproposedsetretainssomepriorvaluesandmakesamendmentswhereevidenceisavailable. 237 Furthermore, we maintain treatment for "non-stockable" goods for utilities sectors (Hallegatte 2014), as 238 described earlier in this section. We use these parameters later as part of the case study. 239 For each category-parameter pair, we also define lower and upper bounds (denoted by the blue bars in 240 11 Issa, March 20, 2024 Figure 3) for use in uncertainty quantification and sensitivity studies. Hence, the parameter values selected 241 serve as the central values of their corresponding sampling distribution. 01020Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicestime to overproduce 02040Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicestime to restock 11.21.4Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicesmax overproduction 050100150Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicesinventory (days) 0.60.81Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicesHeterogenity01020Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicestime to overproduce 02040Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicestime to restock 11.21.4Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicesmax overproduction 050100150Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicesinventory (days) 0.60.81Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicesHeterogenity01020Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicestime to overproduce 02040Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicestime to restock 11.21.4Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicesmax overproduction 050100150Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicesinventory (days) 0.60.81Agriculture, forestry + fisheryMiningBeverages + FoodsManufacturingConstructionUtilitiesServicesHeterogenityLegendDeterministic parameter value, Hallegatte (2014)Mean value of parameter distributionDistribution range, [lb, ub]𝜏!(months) 𝛼"#$𝜏%(days) 𝑛%(days) 𝜓(a)(b) (c)(d) (e)N.A N.A Fig. 3.Proposed distributions of behavioral parameters for each of the defined sector categories, compared against the default values in Hallegatte 2014. 242 Overproduction parameters 243 We refine overproduction parameters 𝜏𝛼and𝛼𝑚𝑎𝑥(Figure 3a, Figure 3c). First, we reduce 𝜏𝛼from 12 244 monthsto6monthsforManufacturingandBeverages+Foodssectors,drawinginsightsfromtheaccelerated 245 deployment of production recapture strategies following events like the 2016 Kumamoto Earthquake (S&P 246 Global 2016a; S&P Global 2016b; Maruya et al. 2017). We reduce 𝜏𝛼from 12 months to 1 month for 247 Services sectors. Services typically carry little inventory compared to other sectors, and can adapt rapidly 248 12 Issa, March 20, 2024 due to high teleworking potential compared to non-services industries (OECD 2021). For Utilities sectors, 249 where responsiveness to consumer demand is critical, we reduce 𝜏𝛼to 3 days to reflect agility in adjusting 250 production continuously to match demand, many times throughout the day (e.g., energy utilities). Finally, 251 we lower𝛼𝑚𝑎𝑥from 125% to 110% across all sectors to better align with empirical industrial productivity 252 indices(IIPs)followingthe2011TohokuEarthquake(Kajitanietal.2013;MinistryofEconomy,Tradeand 253 Industry (METI) 2018). 254 Inventory parameters 255 In most cases, we maintain the default values of inventory parameters 𝜏𝑠and𝑛𝑠(Figure 3b, Figure 3d), 256 exceptforUtilitiesandServicessectors,wherewereduce 𝜏𝑠from30daysto3days. ThisimpliesthatUtility 257 sectors can rapidly replenish their inventories when productive capital remains undamaged. Similarly, for 258 Services sectors, we reduce 𝜏𝑠from 30 days to 14 days to reflect rapid adaptability. Finally, we reduce 𝑛𝑠259 from 90 days to 30 days for Services sectors, because Services sectors carry little inventory compared to 260 other sectors. 261 Heterogeneity parameter 262 Finally, we reduce the heterogeneity parameter 𝜓from 0.8 to 0.7 for all sectors to reflect recent studies 263 of post-shock substitutional elasticity (Figure 3e). Fujiy et al. (2022) suggests that elasticities (the degree 264 to which consumers or producers can switch between different goods or services in response to changes in 265 pricesoravailability)areslightlyhigher(between0.38and0.41)thanpreviouslyreportedinAtalay(2017). 266 Suchanincreaseimpliesthatproductionreductions(thatarisewheninventoriesareinsufficient)aresoftened, 267 due to increased flexibility. As a result, 𝜓should decrease to reflect increased input substitutability. 268 R-ARIO BEHAVIORAL PARAMETER SENSITIVITY ANALYSIS 269 In this section, we describe a Sobol sensitivity analysis (Saltelli et al. 2010) to quantify the influence of 270 R-ARIO behavioral parameters on the predicted indirect loss. We apply this approach for the parameters 271 assigned to the 𝑁𝑐𝑎𝑡=7sector categories defined in Figure 3. 272 First, we select sampling bounds for the behavioral parameters based on the upper and lower bounds in 273 Figure 3. Sobol sampling is used to efficiently cover the sample space. 274 Next, we generate samples of the behavioral parameters. Each sample of parameters X(𝑘)is a𝑁𝑐𝑎𝑡×5 275 matrix, accounting for the five types of behavioral parameters: 276 13 Issa, March 20, 2024 X(𝑘)=𝜓(𝑘) 1𝑛(𝑘) 𝑠,1𝜏(𝑘) 𝑠,1𝛼(𝑘) 𝑚𝑎𝑥, 1𝜏(𝑘) 𝛼,1 𝜓(𝑘) 2𝑛(𝑘) 𝑠,2𝜏(𝑘) 𝑠,2𝛼(𝑘) 𝑚𝑎𝑥, 2𝜏(𝑘) 𝛼,2 ............... 𝜓(𝑘) 𝑁𝑐𝑎𝑡𝑛(𝑘) 𝑠,𝑁𝑐𝑎𝑡𝜏(𝑘) 𝑠,𝑁𝑐𝑎𝑡𝛼(𝑘) 𝑚𝑎𝑥,𝑁 𝑐𝑎𝑡𝜏(𝑘) 𝛼,𝑁 𝑐𝑎𝑡(7) 277 where𝑘is an index indicating the sample number, ranging from 1 to 𝑁𝑠𝑖𝑚. 278 When using the proposed behavioral parameters introduced in the previous section, 𝑁𝑐𝑎𝑡=7, resulting 279 inatotalof35variablesinthesensitivityanalysis. Forsample 𝑘,weruntheR-ARIOmodelandrecordand 280 the𝐿𝑜𝑠𝑠(total indirect loss across the economy) . 281 Usingtheensembleof 𝑁𝑠𝑖𝑚samplesalongwiththerecordedoutput,weestimateSobolindices 𝑆1,𝑖and 282 𝑆𝑇,𝑖for each of the 𝑁𝑐𝑎𝑡×5variables. Each index uses an 𝑖subscript to denote one of the 35 variables in 283 the analysis (e.g., 𝛼𝑚𝑎𝑥for the Manufacturing category). 284 𝑆1,𝑖,thefirst-orderSobolindex,measuresthecontributioninoutputvarianceassociatedwithmodifying 285 a variable in isolation: 286 𝑆1,𝑖=V𝑎𝑟[EX∼𝑖[𝐿𝑜𝑠𝑠|𝑋𝑖] V𝑎𝑟[𝐿𝑜𝑠𝑠](8) 287 where𝑋𝑖isR-ARIObehavioralparametervariable 𝑖(associatedwithaspecificparameter-categorypair) 288 andX∼𝑖denotes the set of all variables except 𝑋𝑖. 289 𝑆𝑇,𝑖, the total-order Sobol index, measures a variable’s first- and higher-order influence on predicting 290 the model output. Unlike 𝑆1,𝑖,𝑆𝑇,𝑖measures higher (or total-order) influence that accounts for all levels of 291 interaction: 292 𝑆𝑇,𝑖=EX∼𝑖[V𝑎𝑟𝑋𝑖[𝐿𝑜𝑠𝑠|X∼𝑖]] V𝑎𝑟[𝐿𝑜𝑠𝑠](9) 293 The inequality 0≤𝑆1,𝑖≤𝑆𝑇,𝑖≤1must hold for all cases, in addition to: 294 ∑︁ 𝑖𝑆1,𝑖<1 (10) 295 Finally,weusevaluesof 𝑆1,𝑖and𝑆𝑇,𝑖torankeachvariable. Anyvariablesthatheavilyinfluenceindirect 296 losseswillhavehighindexvalues. Suchvariablesshouldbeprioritizedforsubsequentbehavioralparameter 297 refinement efforts. 298 14 Issa, March 20, 2024 It is important to note that variables at the sector category level are used here and in the case study to 299 illustratethismethod. Sector-levelanalysescanbeobtainedbyexpanding X(𝑘)toconsider𝑁𝑠×5variables, 300 where𝑁𝑠is the number of sectors. 301 CASE STUDY: 2016 KUMAMOTO EARTHQUAKE 302 Ananalysisofthe2016KumamotoEarthquakesinJapanisusedheretodemonstratetheR-ARIOmodel, 303 identify key drivers of indirect loss, and compare predicted recovery times obtained from variants of the 304 model. Webeginwithanoverviewofthestudyregionanddisaster,followedbyasummaryofmodelinputs, 305 implementation, and analysis results. Finally, we discuss the application of a variance-based sensitivity 306 analysis on the selected behavioral parameters. 307 Overview of study region and disaster 308 LocatedinthesouthernislandofKyushuinJapan,Kumamotoisoneofthecountry’s47prefecturesand 309 home to 1.3% of its population. The capital, Kumamoto City, is home to over 40% of the prefecture’s 1.7 310 million population (Figure 4), and serves as a key economic hub. Kumamoto’s 2016 GDP was roughly 6 311 trillion yen, just over 1% of Japan’s GDP. Latitude LongitudeM7.2M6.2Mashiki TownKumamoto City Population in100m x 100m grid cell 50250km Fig. 4.Epicenters of major earthquakes (indicated by circles) and population density in the Kumamoto Prefecture (indicated by shading), along with fault traces for the Futagawa-Hinagu fault zone. 312 The Kumamoto Earthquake sequence occurred along the Futgawa-Hinahgu fault in the Kumamoto 313 prefecture, beginning with a 𝑀𝑤6.2 foreshock on April 14th, followed by a 𝑀𝑤7.0 mainshock two days 314 15 Issa, March 20, 2024 later. Multiple significant aftershocks in the following days caused additional destruction. Damage and 315 fatalitiesfromtheforeshockwereconcentratedinMashikiTown,asmallsuburbnorthofthefaultzone. The 316 mainshock exacerbated the damage in Mashiki and extended the radius of influence to nearby Kumamoto 317 City. Twohundredseventy-threeconfirmedcasualtieshavebeenreported(KumamotoCabinetOffice2016). 318 Over 198,000 homes in the prefecture experienced some form of damage, with over 20% experiencing 319 collapse(KumamotoPrefecture2022). Asaresult,over60%oflossesreportedbytheOfficeoftheCabinet 320 stemmed from housing. Commerce and industrial assets sustained significant damages, causing cascading 321 supply chain disruption across Japan. Damage to buildings and infrastructure in the prefecture produced 322 losses of roughly 3.79 trillion yen as of September 14th, 2016 (Kumamoto Cabinet Office 2016). 323 Assemble inputs 324 Next, we describe the assembly of the necessary input data for the R-ARIO model (Table 1). This data 325 falls into four categories: pre-disaster economic data, monetary losses due to direct damage, repair time 326 curves, and ARIO behavioral parameters. 327 Pre-disaster economic data 328 We assemble the input-output table and key economic metrics (e.g., value added, total final demand, 329 exports, imports, and local demand) for Kumamoto from official prefectural data (Kumamoto Prefecture 330 2020) for the 2015 fiscal year. The input-output table, consisting of 37 productive sectors, is visually 331 representedintheelectronicsupplement,FigureS1. Wecomputesector-levelfixedassetsusingreplacement 332 costs provided by Sompo Inc. (Table S1). 333 Direct losses 334 Wederivesector-specificdirectlossesbyaggregatingbuilding-levelclaimsdatasuppliedbySompoInc 335 (TableS2). Totalbuildingdamagelossesacrossallsectorsare1.76trillionYen,andfurtherdetailsregarding 336 the treatment of losses can be found in the electronic supplement. We assume that 75% of reconstruction 337 demand from these losses are distributed to the construction sector, and the remaining 25% are distributed 338 to manufacturing sectors, consistent with past studies (e.g., Hallegatte 2008; Markhvida and Baker 2023) 339 Reconstruction time curves 340 Sector-specific reconstruction time curves are used to determine the rate of reconstruction at each time 341 step in the R-ARIO model. We develop each curve using building-level reconstruction times, which are 342 estimated using a proprietary model by Sompo Inc. These times are strictly limited to repairs and do not 343 16 Issa, March 20, 2024 include impediments to reconstruction progress. Upon converting building-level reconstruction data into 344 sector-level reconstruction trajectories, we observe that the time to 95% reconstruction is achieved within 345 six months. Trajectories for each sector are illustrated in the electronic supplement, Figure S2. 346 Behavioral parameters 347 The proposed set of behavioral parameters in Figure 3 is employed for the case study. We tabulate 348 assigned sector categories for each of the 37 productive sectors in the electronic supplement, Table S3. 349 The impact of the refined set, relative to the default ARIO parameters introduced in Hallegatte (2014), is 350 described in the next section. 351 R-ARIO model results 352 In this section, we describe the R-ARIO-predicted post-earthquake indirect loss, value added dynamics, 353 andreconstructionovertimeatvariousresolutions. Next,weexploretheinfluenceoftheR-ARIOmodelon 354 recovery time and quantify the influence of the proposed behavioral parameters on predicted indirect loss. 355 Post-earthquake economic loss and recovery at the economy-level 356 ToexaminetheimpactoftheR-ARIOmodelrefinements,wesimulateregionaleconomicrecoveryusing 357 several variants of the model, as listed below: 358 •DynamicReconstruction+explicitHousinglosses+BehavioralParameterRefinement(DR+H+BPR) : 359 this is the complete R-ARIO model proposed in this study. 360 •DynamicReconstruction+explicitHousinglosses(DR+H) :thisistheR-ARIOmodelintroduced 361 inthispaper,butitusestheHallegatte(2014)behavioralparametersratherthantheproposedrefined 362 sector-level behavioral parameters. 363 •Baseline: this is equivalent to the original ARIO model. 364 Indirect losses predicted by the DR+H+BPR model over the first 30 days amount to roughly 88 billion, 365 whichiswithinreportedestimatesbytheCabinetOfficeduringthesameperiod(81to113billionyen). The 366 DR+H and Baseline model predictions are also within this range, at 91 and 101 billion yen, respectively. 367 Indirect loss estimates over an analysis period of five years following the earthquake, aggregated at the 368 economylevel,areshowninFigures5c-eforeachcase,alongwiththeassociatedpost-earthquakedynamics 369 in value added (Figure 5b), and productive capital recovery trajectory (Figure 5a). 370 17 Issa, March 20, 2024 0123-25-20-15-10-5051005001000150000.10.2 05001000150000.10.2 05001000150000.10.20123020406080100 median581 billion Yenmedian102 billion Yenmedian257 billion Yen Dynamic reconstruction + Explicit housing losses + Behavioral parameter refinement (DR+H+BPR)BaselineDynamic reconstruction + Explicit housing losses (DR+H)Time after earthquake (years)Time after earthquake (years)Indirect loss (billion Yen)ProbabilityProbabilityProbability% of damaged capital reconstructed % change to pre-EQ value added(a)(b)(c)(d)(e)Fig. 5.Results for each of the three models explored in this case study, based on an ensemble of 1000 simulationsandarecoveryperiodof5years. Panels(a)and(b)representtherecoveryofproductivecapital andvalueadded,respectively,eachtakenatthe50thpercentile. Thehistogramsin(c),(d),and(e)illustrate the indirect losses across the ensemble for different variants of the R-ARIO model. Dashed lines indicate the 50th percentile value in each case. Figure5a showsthepredictedcapital recovery,accountingfor economicconstraintsthatimpede repairs 371 of productive capital. Recovery is rapid during the first year following the disaster. The DR+H and 372 DR+H+BPR models — which both incorporate dynamic reconstruction — follow very similar trajectories 373 and exhibit higher rates of recovery, particularly in the first few months. Across all three models, more 374 than 50% of damaged capital recovers within the first six months of the initial shock, and 95% recovers 375 within 2 years. Both the DR and DR+BPR models predict a shorter time to 95% recovery of 1.25 years, 376 comparedto1.60yearsfortheBaselinemodel. Thedynamicreconstructionassumptioncanaccountforthe 377 swift progress made during the first month of recovery (reconstruction time curves for Kumamoto sectors 378 generally exhibited rapid reconstruction initially), while the baseline model is forced to leverage a constant 379 reconstruction rate that cannot capture this progress. 380 Figure5bshowstherecoveryofvalueadded. Eachmodelpredictsanequalinitial19%dropinprefectural 381 value added. Value added recovers rapidly in the first year and returns to pre-disaster values at 0.7, 1.0, and 382 1.5 years for the DR+H+BPR, DR+H, and Baseline models, respectively. Due to assumed overproduction, 383 value added continues to increase, with median trajectories peaking at 2.03, 2.00, and 1.59 percent of pre- 384 earthquake value added for the DR+H+BPR, Baseline and DR+H models, respectively. Beyond the peak, 385 18 Issa, March 20, 2024 value added descends and eventually converges to pre-disaster values. As observed with recovery times, 386 DR+H+BPRpeaksthequickest,followedbyDR+H,andBaseline. Ofthethreemodels,theBaselinemodel 387 is the only one to exhibit non-monotonic recovery due to supply-side constraints that impede recovery at 388 around six months. 389 Figures 5c-e show the total predicted indirect loss over the analysis period. Each realization of value 390 addedcanbeintegratedacrosstimetoobtainacorrespondingrealizationoftotalindirectlossattheeconomy 391 level. Of the three models, DR+H+BPR predicts the lowest median indirect loss over the entire recovery 392 period,atroughly102billionyen. Withoutbehavioralparameterrefinement,theDR+HandBaselinemodel 393 predict substantially higher median indirect losses of 257 billion yen and 581 billion yen, respectively. 394 The significantly higher losses predicted by the DR+H and Baseline are due in part to longer 𝜏𝛼values 395 associated with the default ARIO behavioral parameter settings. Notably, the significant median indirect 396 losspredictedbytheBaselinemodelisduetoitsmuchlongerperiodofnon-recoverycomparedtotheother 397 two models. Overall, including behavioral parameter refinement reduces the predicted median indirect loss 398 overtherecoveryperiodby155billionyenwhencomparedtotheDR+Hmodel(i.e.,thedifferencebetween 399 median losses in 5c and d). 400 Sector-level economic recovery 401 The results in Figure 5 can be disaggregated by sector to reveal recovery attributes not visible at the 402 aggregate economy level. For each sector, we extract building recovery, production capacity, production, 403 demand,andvalueaddedovertime. Figure6showsConstructionsectorresultstodemonstraterelationships 404 between demand, production capacity, and value added over time. As part of this example, we examine the 405 single realization associated with the median indirect loss shown in Figure 5. 406 Figure 6a shows that capital is nearly reconstructed within two years of the disaster. This resembles the 407 economy-level productive capital recovery trajectory in Figure 5a. 408 Figure 6b shows trends in production and demand. Immediately following the disaster, demand for the 409 Construction sector dramatically increases. This sharp increase in demand is expected because 75% of all 410 reconstructiondemandisassignedtotheConstructionsector. Duringthissameperiod,thesectorlosesover 411 20% of its pre-disaster production capacity, constraining production, and hence, ability to fulfill demand. 412 Within 0.7 years, production capacity is restored to its pre-disaster level but is still unable to meet demand 413 and begins moving into overproduction. An extra capacity equivalent to 20% of pre-disaster production is 414 19 Issa, March 20, 2024 012345020406080100 012345-40-2002040 012345-40-2002040 012345-40-2002040012345020406080100 012345-40-2002040 012345-40-2002040 012345-40-2002040012345020406080100 012345-40-2002040 012345-40-2002040 012345-40-2002040012345020406080100 012345-40-2002040 012345-40-2002040 012345-40-2002040(b) (c)(d)(a)damaged capital reconstructed(as % of pre-EQ fixed capital)Change in unsatisfied demand (as % of pre-EQ production)Time after earthquake (years) Time after earthquake (years)Time after earthquake (years)Time after earthquake (years)Change in value added (as % of pre-EQ value added)Change in production and demand(as % of pre-EQ production) 012345-15-10-505101550th percentile demand50th percentile production50th percentile production capacityDemandProductionProduction capacity 012345-15-10-505101550th percentile demand50th percentile production50th percentile production capacityUnsatisfied demandValue addedReconstructed capitalFig. 6.Results for a single ARIO realization (associated with the 50th percentile indirect loss in Figure 5c) for the Construction sector. graduallyaddedtothesectorroughly1.6yearsafterthedisaster. Interestingly,whilecapacitygrowstomeet 415 elevated demand, actual production (light blue curve) cannot keep up and production plateaus at roughly 416 5% over baseline production due to supply-side constraints. During this plateaued period of production, 417 demand is rapidly decaying and the Construction sector’s actual production can fulfill all demand by year 2 418 (the orange and light blue curves merge). Demand returns to pre-earthquake levels (along with production) 419 shortly after. Production capacity, by comparison, is significantly slower to return to baseline. 420 Figure 6c shows the amount of unsatisfied demand (the difference between the demand and production 421 curvesinFigure6b)overtheentirerecoveryperiod. Bytheendoftheyear2,thedemandunsatisfiedreturns 422 to 0, indicating that all demand can be fulfilled by production beyond this point. 423 Finally, the value added trajectory (Figure 6d) quantifies the changes in value added over time, and is 424 20 Issa, March 20, 2024 used to calculate the indirect losses of the sector (through integration of trajectory over the entire recovery 425 period). Value added takes the sharpest loss immediately following the disaster, decreasing by an amount 426 equivalent to over 20% of its pre-disaster value. Due to the rapid initiation of overproduction, value added 427 is restored to its pre-earthquake value added within 0.7 years of the initial drop. The time to recover value 428 addedtakesroughly35%ofthetimetorecoverallphysicalcapitalshowninFigure6a. Thistrend,whereby 429 sectors achieve quicker recovery of lost value added compared to recovery of damaged productive capital, 430 is observed for nearly all sectors in the Kumamoto economy. Value added peaks at roughly 0.8 years after 431 the earthquake, plateaus for an additional year, then gradually decreases to its pre-earthquake value before 432 year three. Other sector-specific trajectories, including uncertainty bounds, can be found in the electronic 433 supplement, Figures S4-S9. 434 Sector-level losses 435 We integrate sector-specific value added curves for each of the 37 sectors to quantify the absolute total 436 loss, and the losses as a fraction of the pre-earthquake value added, broken down by direct and indirect 437 sources. Results for each sector can be found in the electronic supplement, Figure S10. To simplify 438 presentation,wesumthesector-level50thpercentileindirectlossesforeachofthesevencategoriesproposed 439 in Figure 3. 440 Total loss (trillion Yen)Total loss as fraction of pre-EQ value addedSector categories00.51Absolute losses (trillion Yen)MiningAgricultureUtilitiesManufacturingConstructionBeverages+FoodServicesHousing direct lossmedian indirect loss-0.4-0.200.20.4Losses as fraction of pre-EQ value addedMiningAgricultureUtilitiesManufacturingConstructionBeverages+FoodServicesHousing(a)(b) Fig.7.50thpercentiledirectandindirectlossesacrosstheseveneconomicsectorcategories(plushousing) in terms of absolute monetary value in trillion yen (left) and fraction of pre- disaster value added. Figure 7 shows the direct and indirect losses per sector category, in absolute values and as a fraction 441 of the total value added across all sectors within a category. The 50th percentile indirect loss across the 442 21 Issa, March 20, 2024 economy (102 billion yen) is small relative to total losses (1.7 trillion yen). In absolute monetary terms, 443 the Services category incurs the greatest indirect loss across all categories, followed by the Beverages + 444 Food and Utilities categories. Interpreting these results within the context of the economic recovery yields 445 a number of insights. 446 The Construction category’s indirect loss (both in absolute terms and as a fraction of its pre-earthquake 447 value added) is relatively low compared to other categories. This relatively small loss stems from the 448 substantial gains due to overproduction (e.g., that of the kind observed in Figure 6b). Such overproduction 449 is expected, since the demand for reconstruction following the disaster is substantial. When integrating the 450 value added trajectory to obtain the indirect loss, the initial shock is barely significant enough to counteract 451 large gains from overproduction. 452 TheManufacturingcategory(e.g.,iron+steel,productionmachinery,plasticproducts+rubberproducts 453 sectors)experiencesanetgaininvalueaddedduetostrongoverproductionacrossseveralsectors,as25%of 454 the reconstruction demand goes to manufacturing sectors. Across a handful of Manufacturing sectors, the 455 gain from overproduction counteracts the initial drop in value added. 456 The Mining category (which consists solely of the mining sector) also experiences a net gain in value 457 added. This result is attributed to a gentle initial drop in value added, and a notably extended duration of 458 overproduction. The mining sector is implicitly critical to the reconstruction of economic capital, since it 459 is the primary supplier to the electricity, gas, and heat sector (within the utilities category), which supplies 460 a significant number of manufacturing sectors (per the I-O table in the electronic supplement). The Mining 461 category is among the few sectors to incur zero direct damage in the analysis inputs, which influences the 462 observed net gain. 463 Influence of ARIO model on recovery time 464 Previous sections illustrated how ARIO-predicted recovery in value added, production, and productive 465 capital can be disaggregated at the individual sector level. These sector-level trajectories can be used to 466 extract time-to-recovery statistics, such as the time to restore lost value added. Such measures can then be 467 used to compare recovery performance across sectors. Figure 8 provides a comparison of time-to-recovery 468 metrics for the seven sector categories, plus housing. All reported metrics are based on 50th percentile 469 recoverytrajectoriesacrossanensembleof1000simulations. Timesrepresentedatthesectorcategorylevel 470 are estimated by averaging times across all sectors within a category. 471 22 Issa, March 20, 2024 Foreachsector,weextracttheR-ARIO-predictedtimestorecover95%ofthelostvalueadded,production, 472 and productive capital. Across most categories, the median value added is restored to pre-earthquake levels 473 within one year. The Manufacturing and Agriculture categories are the quickest and slowest to recover, at 474 0.5 and 1.2 years, respectively. Recovery of production is nearly identical to that of value added across 475 all categories. Times to recover value added and production for housing are set to zero in Figure 8, since 476 housing is not a productive sector. 477 00.511.52MiningAgricultureUtilitiesManufacturingConstructionBeverages+FoodServicesHousing50th percentile tprod,pre-EQ50th percentile tVA,pre-EQ50th percentile trecon,ARIO50th percentile trecon,SOMPOSector categories Time to recovery (years)95% production recovery timeLegend95% value added recovery time95% productive capital recovery time (R-ARIO)95% productive capital reconstruction time (SOMPO) 00.511.52ServicesBeverages+FoodConstructionManufacturingUtilitiesAgricultureMining50th percentile tProduction,pre-EQ50th percentile tVA,pre-EQ50th percentile trecon,ARIO50th percentile trecon,SOMPO Fig. 8.R-ARIO-predicted time to recover 95% of lost production, R-ARIO-predicted time to recover 95% of lost value added, R-ARIO-predicted time to recover 95% of damaged productive capital, and time to reconstruct 95% of damaged capital based on user-provided reconstruction time curves. In all four cases, sector category averages are shown. The R-ARIO-predicted time to recover 95% of damaged productive capital accounts for supply and 478 reconstructionconstraints. Acrossallsectorcategorieswithdamagedcapital(i.e,nottheMiningcategory), 479 Manufacturingsectorsexperiencetheswiftestrecoveryofproductiveassetsonaverage,atroughlyoneyear. 480 Thereconstructiontimedatausedtogeneratereconstructiontimecurves(andhence,thedynamicrecon- 481 structionratesusedintheR-ARIOmodelforindividualsectors)isgeneratedusingaproprietarycatastrophe 482 model by Sompo Inc. While these repair time curves are useful for enabling dynamic reconstruction rates, 483 they do not take into account supply chain disruptions, or the capacity of the construction sector to fulfill 484 post-disaster reconstruction demand. Past studies, such as Markhvida and Baker (2023), have demonstrated 485 thattheresultingsector-andcommunity-levelrepairtimesgeneratedbysimilarmodels(e.g.,HAZUS(Fed- 486 23 Issa, March 20, 2024 eral Emergency Management Agency (FEMA) 2020)) can be significantly lower than the recovery time 487 estimates made by the ARIO model. 488 Figure 8 compares R-ARIO-predicted productive capital recovery times and Sompo-provided recon- 489 struction times. The average time to repair 95% of damaged capital is 0.28 years, per Sompo-provided 490 reconstruction time data (which do not include supply or reconstruction delays). The R-ARIO model pre- 491 dicts significantly longer average recovery times, at 1.1 years. The longer estimate provided by the R-ARIO 492 model is consistent with documented reports of capital recovery, particularly housing. One year after the 493 KumamotoEarthquake,thousandsofhouseholdswerestillresidingintemporaryhousing(TakedaandInaba 494 2022). 495 ARIO model parameter sensitivity 496 Next, we perform a Sobol sensitivity analysis to understand the relative importance of sector-specific 497 behavioral parameters on the predicted indirect losses shown in Figure 5. We compute 𝑆1,𝑖and𝑆𝑇,𝑖using 498 equations 8 and 9, respectively. The results, displayed in Figure 9, show several features of the analysis. 499 Thebehavioralparametervariablesrelatingtoinventory(particularly 𝑛𝑠)significantlyinfluencetheindirect 500 losses. While this finding is consistent with the ARIO sensitivity studies in Markhvida and Baker (2023) 501 and Hallegatte (2014), our results unveil new sector-specific insights. 502 taustocktaualphapsinjalphaMiningAgricultureUtilitiesManufacturingConstructionBeverages+FoodServicesHousing 0.000.000.000.010.000.000.010.000.000.000.000.000.000.000.000.00-0.010.000.000.010.00-0.010.000.00-0.010.000.080.000.000.000.000.000.00-0.010.000.000.000.140.130.20 00.050.10.150.2 taustocktaualphapsinjalphaMiningAgricultureUtilitiesManufacturingConstructionBeverages+FoodServicesHousing 0.000.010.060.090.000.060.080.000.000.000.000.010.000.010.020.000.010.010.050.060.000.070.070.000.010.020.000.000.000.000.010.000.010.000.010.000.190.300.210.30 00.050.10.150.20.250.3S1First-order Sobol index)STTotal-order Sobol index) 𝜏!𝛼"#$𝜏%𝑛%𝜓𝜏!𝛼"#$𝜏%𝑛%𝜓(a)(b) Fig. 9.(a) First-, and (b) total-order Sobol indices across the seven sector categories and behavioral parameters, measured with respect to the R-ARIO-predicted indirect loss across the prefecture. Figure 9a shows that not all category-specific behavioral parameter variables for 𝑛𝑠exhibit the same 503 modeling importance. For example, the 𝑛𝑠parameter variables for Manufacturing, Services, and Beverages 504 24 Issa, March 20, 2024 + Food sectors are significantly more important than for those of other categories, and explain 20%, 14%, 505 and 13% of the total first-order variance of the indirect loss, respectively. 506 When higher order effects are considered (Figure 9b), the importance of 𝑛𝑠holds, and sector category 507 rankings based on 𝑆1,𝑖hold true for 𝑆𝑇,𝑖. Interestingly, inventory parameter 𝜏𝑠and heterogeneity parameter 508 𝜓, which have near-zero first-order indices, have more significant 𝑆𝑇,𝑖values. Similar to 𝑛𝑠, both𝜏𝑠and𝜓 509 parameter settings for Manufacturing, Services, and Beverages + Food sectors have high importance. The 510 𝜏𝑠parameters yield slightly higher 𝑆𝑇,𝑖values than𝜓in most cases. 511 Across 35 variables in this analysis, the top 3 variables when ranked using first- and total-order indices 512 are associated with the 𝑛𝑠behavioral parameter. Across all settings of 𝑛𝑠, the value assigned to Services 513 categoryisthemostsignificantinbothfirst-andtotal-ordercontexts. Whilethisresultmotivatedourcareful 514 scrutiny of the 𝑛𝑠parameter for the Services category, the results in this section suggest that priority for 515 future refinements should be considered for the Manufacturing and Beverages + Food categories as well. 516 CONCLUSIONS 517 This paper presented R-ARIO, a refined ARIO model to simulate post-disaster economic recovery. The 518 R-ARIOmodelincorporates(i)explicitmodelingofhousinglossesseparatefromproductivecapitallosses, 519 (ii) dynamic reconstruction rates based on sector-specific reconstruction time curves, and (iii) sector-level 520 modeling of behavioral parameters. The enhancements aim to improve indirect loss estimation, capture 521 temporal differences in reconstruction demand, and enable uncertainty quantification, sensitivity studies, 522 and refinement at the sector level. 523 We proposed a refined set of parameters across seven sector categories that address inter-sector dif- 524 ferences, in accordance with available empirical observation and recent studies on post-disaster business 525 adaptation. Theseparametersreflectinter-sectordifferencesandcanaccommodatecontext-specificchanges 526 based on new evidence. 527 Weusedaglobalsensitivityanalysistoevaluatetherelativeimportanceofspecificbehavioralparameters 528 at the sector category level and guide subsequent refinement of the most influential parameters. Parameter 529 bounds and a parameter sampling procedure are selected. An ensemble of simulated behavioral parameters 530 is then generated, along with R-ARIO model outputs for each sample. Finally, Sobol indices are generated 531 foreachcategory-parameterpairtoindicatethemostinfluentialsector-parameterpairsandruleoutvariables 532 with negligible influence. 533 25 Issa, March 20, 2024 WeappliedtheR-ARIOmodeltoexploreeconomicrecoveryfollowingthe2016KumamotoEarthquake 534 in Japan. Indirect losses predicted by the R-ARIO model over the first 30 days following the disaster align 535 closelywiththe81to113billionyenrangeestimatedbytheCabinetOffice. Overalongerfive-yearanalysis 536 period, the R-ARIO model predicts a median indirect loss of 102 billion yen. When sector-level behavioral 537 parameter modeling is omitted (and older default parameters are used in place of the proposed set) this loss 538 more than doubles to 257 billion yen. The dynamic reconstruction assumption is responsible for properly 539 modeling the high rate of recovery within the first month following the disaster, which previous, constant 540 reconstruction rate assumptions cannot capture. Furthermore, explicit and separate modeling of housing 541 losses prevents distortion of economic recovery caused by injecting significant direct damage into the real 542 estate sector. 543 UsingtheR-ARIOmodel,aggregateindirectlossesamountto5.4%ofthemediantotal(direct+indirect) 544 loss of 1.86 trillion yen. We evaluated sector-level indirect loss estimates to unveil trends across specific 545 sector categories. Overall, the Services category generated the largest portion of indirect losses in absolute 546 monetaryterms,followedbyBeverages+FoodandUtilitiescategories. TheConstructioncategoryseeslow 547 indirectloss(asafractionofpre-earthquakevalueadded)followingtheearthquakeduetothecompensating 548 effect of overproduction. On the other hand, the Manufacturing category and the Mining category both 549 experiencenetgainsinvalueadded,duetostrongoverproductiontosupportreconstructionthatcounteracts 550 initial sector-level shocks. 551 Next, we evaluated the economic and productive capital recovery times for each sector category. In 552 mostcases,thevalueaddedrecoverswithinayear,withtheManufacturingcategoryrecoveringthequickest 553 (0.5 years on average) and Agriculture the slowest (1.2 years on average). The average time to recover lost 554 production is nearly identical to the time to recover value added for all productive sector categories. When 555 comparingR-ARIO-predictedtimestorestorelostproductivecapitalwithuser-providedreconstructiontime 556 curves, we found that the R-ARIO model extends the average time to 95% recovery of productive capital 557 (acrossallsectorcategories)from3.5monthsto13.5months. ThelongerestimateprovidedbytheR-ARIO 558 model,whichincludessupplychainimpactsthatimpederepairs,ismoreconsistentwithdocumentedreports 559 of recovery, particularly housing. 560 Finally,weappliedtheproposedsensitivityanalysisapproachtotheKumamotocasestudy. Sobolindices 561 weregeneratedforthe35variables(representingbehavioralparameter-sectorcategorypairings)considered 562 in the analysis. The inventory parameter 𝑛𝑠for Manufacturing, Services and Beverages+Food categories 563 26 Issa, March 20, 2024 explains20%,14%,and13%ofthetotalfirst-ordervarianceinpredictedindirectlosses,respectively. These 564 trendsholdforthetotal-ordervarianceaswell. Whileinventoryparametersaremostimportantoverall,there 565 issignificantvariabilityinimportancebetweencategories. Therefore,effortstorefinebehavioralparameters 566 should focus on the subset of variables with significant influence on indirect loss. 567 DATA AVAILABILITY STATEMENT 568 Some or all data, models, or code that support the findings of this study are available from the corre- 569 sponding author upon reasonable request. Source code for the R-ARIO implementation used in the study is 570 available at github.com/0marissa/SR-ARIO. 571 ACKNOWLEDGEMENTS 572 WethankSompoHoldings,Inc.,includingMr. RyuSaito,Mr. ShinjiYamada,andMr. KatsuyoshiSekii, 573 for their support of this work, provision of data, and helpful feedback regarding our questions. This work 574 was additionally supported by the National Science Foundation under NSF grant number CMMI-2053014 575 and the Stanford Urban Resilience Initiative. Any opinions, findings, and conclusions or recommendations 576 expressedinthismaterialarethoseoftheauthorsanddonotnecessarilyreflecttheviewsofSompoHoldings, 577 Inc. or the National Science Foundation. 578 SUPPLEMENTAL MATERIALS 579 Figs. S1–10 and Tables S1-3, are available online as part of an electronic supplement in the ASCE 580 Library (ascelibrary.org). 581 27 Issa, March 20, 2024