7 Interdependenciesbetweencriticalinfrastructuresandtheeconomyamplifytheeffectsofdamagecaused 8 by disasters. The growing interest in impacts beyond physical damage has spurred a surge in literature on 9 economic modeling methodologies for estimating indirect economic impacts of disasters. In this review, 10 we propose a mathematical framework for categorizing modeling approaches that assess indirect economic 11 impactsacrossnaturalhazardsandanthropogenicdisasterssuchascyberattacks. Weconductacomparative 12 analysisofmacroeconomicmodels,focusingontheapproachescapturingsectoralinter-dependencies. These 13 include the Leontief Input-Output (I/O) model, the Inoperability Input-Output Model (IIM), the Dynamic 14 Inoperability Input-Output Model (DIIM), the Adaptive Regional Input-Output (ARIO) model, and the 15 ComputableGeneralEquilibrium(CGE)modelanditsextensions. Weevaluatetheirapplicabilitytodisaster 16 scenarios based on input data availability, the compatibility of model assumptions, and output capabilities. 17 We also reveal the functional relationships of input data and output metrics across economic modeling 18 approaches for inter-sectoral impacts. Furthermore, we examine how the damage mechanisms posed by 19 different types of disasters translate into model inputs and impact modeling processes. 20 INTRODUCTION 21 The growing interdependencies of modern infrastructures, coupled with the increasing frequencies of 22 natural hazards (NOAA, 2022) and cyber attacks (Statista, 2021), increase the potential vulnerabilities and 23 consequencestophysicalandcyberthreats(FEMA,2024)andchallengeconventionalresilienceframeworks 24 (OkuyamaandRose,2019). Understandingtherippleeffectsofdisastrouseventsonmoderninfrastructures 25 and the economy is crucial for guiding resilience interventions and mitigating future economic downturns. 26 1 Zhu et al., September 11, 2024 Historically, disaster loss estimation focused on quantifying physical damage and was primarily led by 27 engineers (Rose, 2009). In recent decades, the rising interest in the impacts of disasters beyond physical 28 damage has led to an evolving understanding and definition of disaster economic consequences. From 29 the engineering perspective, the National Research Council (1999) classifies losses into three categories: 30 primarydirect,secondarydirect,andindirect. Primarydirectlossesareattributedtotheimmediatephysical 31 destructionresultingfromtheeventitself. Secondarydirectlossesencompassadditionalimpactsstemming 32 fromfollow-onphysicaldamage,suchasfiredamagefromgaspipebreakageduringanearthquake. Indirect 33 lossesareactivity-relatedconsequencesofthephysicaldestructioncausedbytheevent,whichcanbefurther 34 subdivided into short-term and long-term indirect impacts. Short-term indirect impacts encompass factors 35 like reductions in spending, input-output losses for firms, and changes in future production, employment, 36 and income. Long-term indirect impacts involve more enduring effects such as migration patterns, shifts in 37 housing values, and government expenditures. 38 On the economic side, distinctions are drawn between stocks and flows (Rose, 2004). Stocks represent 39 quantities at a specific time, while flows are time series data that represent the outputs of stocks over time. 40 Direct stock losses result from property damage and direct flow losses stem from business interruptions 41 by the hazard itself, such as factory shutdowns following an earthquake (Brookshire et al., 1997; Rose, 42 2004). Indirect effects are economic impacts beyond direct damages (Rose, 2009). Indirect stock losses are 43 secondary direct losses per the National Research Council (1999) definition, while the indirect flow losses 44 arise from disruptions in successive rounds of suppliers and customers, such as reduced supplier orders or 45 consumer spending. Rose(2004) further introduced the termhigher-order effects to describeall flow losses 46 of interdependent activities, beyond direct and indirect flow losses of the focus activity. The term offers 47 a broader perspective by capturing both the ripple effects, due to economic interdependencies, and other 48 market responses, such as changes in income and spending. 49 Figure 1 presents a general framework for indirect economic impact modeling from the engineering 50 perspective. This framework highlights the critical interfaces that need to be considered in the analysis. A 51 disaster first directly impacts the built and social environment. These direct impacts cause disruptions to 52 one or more economic system components, triggering a ripple effect within the interdependent economic 53 system,andresultinginindirectimpacts. Whilemacroeconomicmodelingnavigatestheevolvingdynamics 54 within the economic system, understanding the interface between hazards, physical environments, and the 55 economic system is essential for the model setup. 56 2 Zhu et al., September 11, 2024 Macroeconomic modelingEconomic activitiesDisruption in economyBuilt environment Social environmentDisasterBuilding stocks Transportation Utilities Employees InvestorsConsumersCapital stock Labor Production capacity Demand Supply Import ExportProduction Supply AdaptationDemand Economic performance metricFig. 1.General framework for indirect economic impact modeling. The light gray box highlights the steps analyzed by macroeconomic models. Although several reviews have focused on macroeconomic modeling and evaluation of indirect impacts 57 of natural hazards and cyber attacks (e.g. Rose, 2004; Kelly, 2015; Koks et al., 2016; Galbusera and 58 Giannopoulos, 2018; Botzen et al., 2019), they are constrained in at least one of three ways. First, the 59 treatmentofthemathematicalformulationofeachmodelislimited. Mostreviewscomparethemathematical 60 formulationsatonlyahighlevel,ifatall,andfocusonthequalitativeimplicationsofeachmodelingapproach. 61 For example, Rose (2004) considers the appropriate model to study short-term versus long-term indirect 62 economic impacts of a disaster. Second, the reviews describe the differences in input data, mathematical 63 formulations,andoutputmetricsacrossmodelsatahighlevel,ifatall(e.g.Kelly,2015;Koksetal.,2016). 64 Third, the existing literature focuses on reviewing modeling approaches and theoretical assumptions (e.g. 65 Galbusera and Giannopoulos, 2018; Botzen et al., 2019), without a systematic review and categorization of 66 disaster scenarios based on their impact on modeling inputs. 67 This paper aims to advance the existing literature and provide a detailed mathematical framework 68 to categorize approaches that assess inter-sectoral impacts from natural hazards and cyber attacks. Our 69 contribution is three-fold. First, we conduct a comparative analysis of macroeconomic models for disaster 70 impacts, focusing on their assumptions and data requirements. We gauge the applicability of each model to 71 disaster scenarios based on input data availability, the compatibility of model assumptions, and the model 72 outputs’relevance. Second,basedondetailedmathematicaldescriptions,werevealtherelationshipsbetween 73 models’inputsandoutputs. Therelationshipsgiveanalystsanactionableguideformappingandcomparing 74 inputs and outputs across models. Third, we outline the different sets of damage mechanisms posed by 75 3 Zhu et al., September 11, 2024 various disasters, and categorize the attributes of disasters that influence model inputs or processes. 76 The remainder of the paper is organized as follows. First, we review the existing literature and provide 77 an overview of the three major economic modeling approaches in disaster analysis. We then build on the 78 literature review to provide detailed mathematical formulations for the classes of models that address the 79 sectoral interdependencies. Then, we review the literature on the economic impacts of natural hazards and 80 cyber attacks, and characterize how disaster impacts are translated into model inputs across models and 81 disaster types. Based on thereview ofmodeling approaches andeconomic modelingof naturalhazards and 82 cyber attacks, we identify gaps in current approaches and potential future research avenues. 83 OVERVIEW OF MODELING APPROACHES 84 There are three broad classes of economic models for assessing indirect impacts: Input-Output (I/O) 85 models, Computable General Equilibrium (CGE) models, and econometric models (Rose, 2004; Galbusera 86 and Giannopoulos, 2018). Understanding the distinct strengths and limitations of each class of models is 87 crucial for selecting a suitable methodology. In this section, we summarize the key features and major 88 developments of each. 89 Input-Output Models 90 The I/O model is a widely-used framework that describes how an industry’s products are distributed 91 acrossallsectorsoftheeconomythroughtradeflowscapturedinthesurveyedInput-Outputtables(Leontief, 92 1944, 1986). The I/O formulation comprises a system of linear equations that quantitatively describe 93 inter-sectoral trade flows. The capability to assess demand-driven changes in sectoral output makes I/O 94 modelsapplicabletoeconomicassessmentstudiesofdisaster-inducedshocks(Kelly,2015). Oneofthekey 95 advantagesofI/Omodelsliesintheirabilitytoevaluatethepropagationofdisruptionsacrossdifferentsectors, 96 providingvaluableinsightsintoeconomy-wideinterdependencies(Rose,2004;Okuyama,2007). Moreover, 97 I/Omodelsofferacleardistinctionbetweendirectandindirectimpacts,enhancingtheunderstandingofthe 98 ripple effects of a shock to a specific sector (Kelly, 2015). 99 However, I/O models suffer from three main limitations (Rose, 2004; Okuyama, 2007; Botzen et al., 100 2019). Notably,theydonotaccountforpricefluctuations,whichcaninfluencethedemandforintermediate 101 and final goods, and potentially alter the economic landscape after disasters. Moreover, supply-side shocks, 102 technological advancements, and the potential for input and import substitution are not fully considered in 103 the model, which may not adequately capture adaptive behavior and other forms of economic resilience 104 4 Zhu et al., September 11, 2024 thatcanmanifestduringtherecoveryprocess. Additionally,I/Omodelsassumelinearitybetweeninputand 105 output. This can result in an oversimplified representation of the complex dynamics of the economy. 106 Several extensions have been developed based on the I/O framework, including the Leontief dynamic 107 model (Miller and Blair, 1985), the Inoperability Input Output Model (IIM) (Haimes and Jiang, 2001; 108 Haimes et al., 2005), the Dynamic Inoperability Input Output Model (DIIM) (Lian and Haimes, 2006), the 109 Multi-Regional Impact Assessment (MRIA) model (Koks and Thissen, 2016), and the Adaptive Regional 110 Input-Output (ARIO) model (Hallegatte, 2008; Hallegatte, 2014). The concept of inoperability of an 111 economicsectorisdefinedasthepercentageofoutputreducedfromtheidealoutputandisthemainfocusof 112 theanalysisinIIMmodels. TheLeontiefdynamicmodel,DIIM,andARIOmodelsincorporatetheadditional 113 timedimensionofeconomicactivities,whiletheMRIAmodelfurtherconsidersgeographicheterogeneities 114 in activities and impacts. The supply-driven variation of the I/O model or Ghosh model (Ghosh, 1958) 115 inverses the interpretation of the I/O framework to represent the relationship between primary inputs and 116 output. However,theGhoshmodelhasfacedcriticismregardingitsassumptionsofperfectdemandelasticity 117 and input substitutability (Oosterhaven, 1988; Gruver, 1989), leading to debates about its plausibility and 118 applicability in real-world scenarios (Oosterhaven, 1989; De Mesnard, 2009). 119 Computable General Equilibrium Models 120 A CGE model expands the traditional I/O framework to incorporate the response of all sectors and 121 consumers to changes in the prices of commodities in the economy. A CGE comprises cost-minimizing 122 firms,utility-maximizinghouseholds,andlump-sumpaymentsbythegovernment. AnequilibriumofaCGE 123 is a vector of prices and quantities of commodities that clear all markets (Shoven and Whalley, 1984). The 124 CGE framework accommodates substitution between inputs in production activities and goods consumed 125 by households; supply constraints; substitution between imports, exports, and domestic production; and 126 savings,investment,andborrowingdecisions(Hosoeetal.,2010;Lofgrenetal.,2002;Caprosetal.,2013). 127 Extensionsof thestandardCGE incorporatetechnological detailandthe responseofmultiple agentswithin 128 individual sectors (Böhringer and Rutherford, 2008; Böhringer and Rutherford, 2009). 129 Some assumptions underpinning CGEs may not hold when applied to disaster disruption. First, as- 130 suming perfectly competitive markets, cost-minimizing firms and utility-maximizing households, always at 131 equilibrium, do not necessarily hold under critical conditions (Rose, 2004). Disasters often damage infras- 132 tructures and cause supply shortages and service interruptions, causing the supply of goods and services to 133 5 Zhu et al., September 11, 2024 bevoluntaryordrivenbygovernmentinterventions. Firmsandhouseholdsmayalsopurchasecertaingoods 134 and services based on their immediate needs and not based on relative prices. Second, in the immediate 135 aftermath of a disaster, substitution between consumed goods and services may not be possible (Rose and 136 Guha, 2004). Understanding the post-disaster response of agents remains a challenge. Finally, the multiple 137 physical and economic inter-dependencies in a CGE can hinder the decomposition of total impacts into 138 indirect impacts, making it difficult to identify drivers of indirect impacts (Rose and Liao, 2002). 139 Econometric Models 140 IncontrasttoI/OandCGEmodelsthatarebasedoneconomictheory,econometricmodelsusehistorical 141 datatoestimatetheeconomicimpactsbasedonregressionsofregioncharacteristicsanddisasterindicators, 142 such as level of educational attainment, financial system development, number and types of disasters, direct 143 damages, and fatalities (Albala-Bertrand, 1993; Noy, 2009). These models provide correlations between 144 disasters and economic outcomes. Early studies (e.g. Skidmore and Toya, 2002) conduct cross-sectional 145 regressionsbyanalyzingabroadrangeofregionsandcontrollingforregionalcharacteristicssuchasbaseline 146 income,educationallevel,andtradeopenness. However,thesemodelscanbesusceptibletoomittedvariable 147 bias (Raddatz, 2009). Thus, more recent research has shifted to panel or longitudinal analyses, which can 148 incorporate location-fixed effects and evaluate the lagged effects of disasters through a dynamic assessment 149 of the disaster-economy relationship (Hsiang, 2010; Felbermayr and Gröschl, 2014). 150 Nevertheless, the focal point of econometric models is typically the effects of disasters on economy- 151 widegrowthratherthansectoraleconomicactivityandinter-sectoraleffects(Kelly,2015). Moreover,these 152 empirical analyses are often constrained by data availability (Botzen et al., 2019; Kelly, 2015). Given these 153 limitations, we omit them from furher review here. 154 MODEL COMPARISON 155 This section compares the I/O and CGE model classes on the dimensions of input data requirements, 156 theoretical assumptions, mathematical formulations, and output metrics. The comparison includes the 157 Leontief I/O model,IIM, DIIM, ARIO, the basicCGE model, and their extensions. Thesemodels and their 158 variationscomprisethevastmajorityofmodelsintheI/OandCGEclassesandarewidelydeployedindisaster 159 impact analysis. Moreover, we link the input data and output metrics of each model with the mathematical 160 formulation of each model, allowing for an end-to-end model comparison across all four dimensions. 161 We evaluate how assumptions within each dimension affect the capabilities of each model. Through the 162 6 Zhu et al., September 11, 2024 TABLE 1. Required inputs for Leontief I/O, IIM, DIIM, and CGE models for indirect economic impact analysis. . Model Inputs Leontief I/O IIM DIIM ARIO CGE I/O matrix ✓ ✓ ✓ ✓ ✓ Social Accounting Matrix (SAM) ✓ ✓ Baseline prices of commodities ✓ Elasticities of substitution (firms and households) ✓ Behavioral parameters ✓ Fixed assets ✓ Model Inputs for Model Extensions Trade matrix ✓ ✓ ✓ ✓ ✓ Export subsidies/import tariffs ✓ Elasticities of substitution (imports and exports) ✓ Investment matrix ✓ Taxes and subsidies ✓ Disaster Scenario Inputs Loss vector ✓ ✓ ✓ ✓ ✓ Perturbation vector ✓ ✓ ✓ ✓ Recovery rate ✓ ✓ ✓ Reconstruction demand distribution ✓ comparisons, we aim to guide model construction and cross-model translation of output metrics. 163 Weadoptthefollowingdefinitionsforthecommonterminologies: intermediateinputs refertogoodsand 164 services; production factors refer to capital, labor, and natural resources; and commodities refer to goods, 165 services, and production factors. 166 Input Data Requirements 167 Table 1 summarizes model inputs for the considered cases. The I/Omatrix is an essential input for all 168 models and comprises the intermediate sales between sectors and gross sectoral output (Miller and Blair, 169 1985). Open economy models incorporate trade with other regions and require the trade matrix in addition 170 totheI/Omatrix. The tradematrix recordsthetradeofmoneybetweeneachsectoroftheregionaleconomy 171 and the regions outside the focus region. The loss vector describes the direct losses, in monetary values, 172 in each sector, and is also a crucial disaster scenario input in all macroeconomic models. The Leontief I/O 173 model requires no additional information for analysis. 174 TheIIMfurtherrequiresthe perturbationvector ,whichisthedirectloss,normalizedbythegrossoutput 175 (Santos and Haimes, 2004). The dynamic extension of the IIM model, the DIIM, goes a step further by 176 incorporatingthetimerequiredforeachsectortorecoverfrominitialinoperabilitytoadesiredpost-recovery 177 state (Lian and Haimes, 2006). 178 7 Zhu et al., September 11, 2024 The ARIO model necessitates more extensive input data. In addition to the I/O matrix, and potentially 179 the trade matrix, ARIO also considers sector-specific inputs for value-added, total final demand, imports, 180 exports, and fixed assets. The additional inputs are available in the SAM 1. Moreover, the ARIO model 181 explicitly incorporates a set of user-defined behavioral parameters that characterize the adaptive capacities 182 ofeachsector. Theseparametersincludeoverproductioncapacity,timetoachievemaximumoverproduction, 183 target inventory levels, time for inventory restoration, and the production reduction parameter, also referred 184 to asheterogeneity (Hallegatte, 2014; Issa et al., 2024). 185 At its most basic form, the CGE model requires information from the SAM, the elasticities of substi- 186 tution between inputs in all activities and between commodities for all households, and baseline prices of 187 commodities and production factors. The SAM reports monetary values in a calendar year and extends the 188 I/O matrix to include all payments to production factors, such as labor and capital, as well as payments to 189 and from households and the government. By dividing the entries of the SAM with the respective baseline 190 prices, modelers derive the baseline quantities and stock of production factors. In addition to the SAM, 191 household consumption decisions are also based on the elasticity of substitution between consumed goods 192 andservices. Productiondecisionsbyfirmsalsorequiretheelasticitiesofsubstitutionbetweenintermediate 193 inputs and production factors. Dynamic CGE models require the investment matrix which quantifies the 194 allocation of savings to all sectors, in monetary values. A CGE of an open economy requires additional 195 information on inter-regional trade for all commodities, along with baseline import and export prices. The 196 derivation of traded quantities follows from the division of the entries of the trade matrix with the baseline 197 import and export prices. The baseline prices can include price markups due to taxes and subsidies. 198 Fromthecomparison,ARIOandCGEmodelsarerelativelymoredata-intensive. Althoughasubstantial 199 portionoftheinputdataissystematicallycollectedthroughperiodiccensusesandsurveys,somenecessitate 200 additional effort for acquisition. Specifically, the ARIO model requires a nuanced understanding of the 201 regionaleconomiccapacitiesandorganizationalstructuresgleanedfrompriordisasterresponsestodetermine 202 the behavioral parameters (Hallegatte, 2008). In the case of CGE models, the calibration of elasticities of 203 substitution can require additional panel data (Németh et al., 2011) and technological data (Balistreri and 204 Brown, 2023) among others. Moreover, when considering disaster scenario inputs, assumptions may be 205 1In the ARIO literature, these additional inputs are available in the I/O table. It is a matrix comprising annual inter-sectoral sales, final demand, and value added, which contains a subset of information in a SAM of CGE models. To reconcile between the different input matrices, in this paper we adopt the following convention: The ’I/O matrix’ records the dollar value of annual sales between sectors. The SAM refers to the matrix recording annual value added, final demand from households, the government, and for investment, and taxes, if applicable, in addition to inter-sectoral sales. 8 Zhu et al., September 11, 2024 imperative owing to the unavailability of empirical data, e.g.,forrecovery rates and the distribution of 206 reconstruction demand in ARIO (Markhvida and Baker, 2023) and substitution elasticities in a CGE (Rose 207 and Guha, 2004). 208 Theoretical Assumptions 209 The I/O and CGE model classes rely on different assumptions, which map to their formulation and 210 application. In the Leontief I/O model, disruptions are typically accounted for as changes in demand. The 211 model assesses the overall impact by analyzing the resulting changes in sectoral activity. The relationship 212 between the demand reduction and the changes in sectoral activity is linear, which is determined by the 213 technical coefficients that represent the interdependencies between different sectors. The technical coeffi- 214 cients for each sector are the normalized sales volumes between the focus sector and all other sectors in 215 the I/O matrix, normalized by the total output, in dollar values, of the focus sector. A key assumption in 216 the Leontief I/O model is that the technical coefficients remain fixed before and after the disaster, implying 217 that production technology does not adjust following an event during the modeled period. Additionally, the 218 modelassumesthatinputsremaininfixedproportionstooutput,whichindicatestheabsenceofsubstitution 219 between different intermediate inputs. 220 TheIIMandDIIMmodelsadoptthesameunderlyingassumptionsastheLeontiefI/Omodel. Tosimulate 221 thedynamicprocessoftheeconomicsectorrecovery,theDIIMmodelrequiresassumptionsforthetrajectory 222 of demand and the recovery rate–typically an exponential recovery curve between the initial inoperability 223 state and the final post-recovery inoperability state of each sector. This curve shape was originally chosen 224 by the High-altitude Electromagnetic Pulse (HEMP) Commission (Lian and Haimes, 2006). 225 The ARIO model shares the assumptions of fixed intermediate input ratios and no input substitutions 226 in sectoral production, but has a different approach for initial economic system disruptions and resulting 227 impacts. ARIOalsoconsidersconstraintsfromtheproductionside,suchasreductioninproductivecapacity 228 and lack of inventory of required input. Moreover, ARIO requires additional demand from the sectors 229 responsible for reconstruction. For instance, if direct losses result from damages to buildings and their 230 content,reconstructiondemandisallocatedtotheconstructionandmanufacturingsectors(Hallegatte,2008; 231 FederalEmergencyManagementAgency(FEMA),2022). Althoughsomeguidelinesorbenchmarksexistfor 232 thepost-disasterdistributionofdemandfromthereconstructionsectors(e.g.Hallegatte,2008;Markhvidaand 233 Baker,2023),thedemanddistributionsaredeterminedonacase-by-casebasis. Inaddition,ARIOaccounts 234 9 Zhu et al., September 11, 2024 fortheadaptationofeconomicactivities,buttheseadaptationsareassumedtobedrivenbyquantitiesrather 235 than prices. When imbalances in supply and demand occur, ARIO assumes constant adaptation rates to 236 achieve overproduction capacities and restore inventories during the recovery process. 237 CGE models are distinct from I/O models in that prices impact production activities and consumer 238 p