MEASURING ECONOMIC IMPACT OF A DISASTER WITHOUT DOUBLE COUNTING: A THEORETICAL ANALYSIS

MEASURING ECONOMIC IMPACT OF A DISASTER WITHOUT DOUBLE COUNTING: A THEORETICAL ANALYSIS ABSTRACT : H. Tatano and K. Nakano Profssor, Disastr Prvntion Rsarch Institut, Kyoto Univrsity, Uji. Japan Ph.D Candidat, Graduat School of Informatics, Kyoto Univrsity, Kyoto, Japan Email: tatano@imdr.dpri.kyoto-u.ac.jp, nakano@imdr.dpri.kyoto-u.ac.jp This papr proposs a mthodology for consistnt masurmnt of conomic losss causd by natural hazards. A two-sctor conomic growth modl is usd to dscrib rcovry procss aftr a disastr. It is shown that sum of th diffrncs of discountd cash-flow in industrial sctors is consistnt masur as total conomic loss. It consists of Forgon Nt Rvnu and Rstoration Invstmnt. This papr also proposs how to masur th cascad ffct, which is th spillovr ffct of conomic impacts inducd by mutual rlation ships btwn industrial sctors A numrical simulation dmonstrats how much cascad ffct nlargs th total conomic loss and th incidnts KEYWORDS: doubl counting of conomic loss, cascad ffct, conomic growth modl. INTRODUCTION As to avoid doubl counting conomic losss causd by disastrs (Ros, 4, it is important to find th way to masur total conomic losss which includs conomic ffcts of lost valus of stocks and rcovry invstmnt in a consistnt way. In th past studis (Tatano t al., Nakano t al 7, authors showd that nt prsnt valu of th diffrnc of th cash flows btwn with and without a disastr is a consistnt masur of th total conomic losss ovr industrial sctors. This papr invstigats th masur of th total conomic loss in an ntir conomy which includs industrial sctors, housholds and outsid of th rgion by us of a conomic growth modl. Th mthodology can tak into account cascad ffct, which is th spillovr ffct of conomic impacts inducd by mutual rlation ships btwn industrial sctors. Th papr is structurd as follows: Th conomic growth modl with two industrial sctors is formulatd and conomic growth path bfor a disastr is invstigatd in sction. An opn conomic modl with two industrial sctors is dvlopd by Turnovsky and Sn (995. Th modl dvlopd in this papr dals with intrmdiat good, which hav not bn obsrvd in any prvious rsarchs. In sction 3, a disastr is assumd as unxpctd shock and its impacts to th conomy ar invstigatd. By comparison of th objctiv function btwn cass with and without disastr, a mthodology for consistnt masurmnt of conomic loss is invstigatd in sction 3. How to masur cascad ffct sparatly is invstigatd and a numrical simulation dmonstrats how much cascad ffct can nlarg th total conomic loss in sction 4.. FORMULATION OF AN ECONOMIC GROWTH MODELS WITH TWO SECTORS.. Prliminary Sttings It is assumd that an conomy has two industrial sctors and a houshold sctor. Th first industrial sctor producs an intrmdiat good (th intrmdiat goods sctor and th scond producs an final consumption good (th final goods sctor. Th final good sctor inputs intrmdiat good from th domstic markt. Intrmdiat good markt is assumd to b in prfct comptition and closd in th country. It is assumd that final consumption good markt is opn to th comptitiv intrnational markt and its production shar of th conomy is small. Th assumption mans that th pric of th final consumption good is tradd at fixd pric p dtrmind in th comptitiv intrnational markt. Th capital markt is opn and th intrst rat r is dtrmind by quilibrium in th intrnational markt. Th amount of th capital of th conomy is much smallr than that of th whol world so that its chang cannot affct th world intrst rat. Labor is assumd to b immobil btwn countris... Formulation of th Intrmdiat Good Sctor

Th maximization problm of th intrmdiat good sctor is to choos capital ( K and labor ( to L maximiz th nt prsnt valu of th cash flows. Lt Y dnot th production of th intrmdiat good sctor and F ( K, L dnot its production function which is assumd to b constant rturn-to-scal. Adjustmnt cost (Barro and Sala-i-Martin, 4 is assumd to b ncssary for invstmnt. Th maximization problm for th intrmdiat good sctor is th following: max ( qf( K, L I( + T ( I / K wl dt, (. s.t. K & = I, (. K ( = givn, (.3 I, (.4 whr q, w and r ar th pric of th intrmdiat good and wdg rat of labor, and I dnots invstmnt for th sctor. Th adjustmnt cost is formulatd in th Eqn.. as th trm I T ( I / K. According to Hayashi (98, adjustmnt cost I T ( I / K is assumd to b homognous of dgr on for invstmnt I and capital K. Eqn.. is th capital accumulation quation. In th quation, w do not dal with dprciation of capital bcaus th rstoration assumd in this papr taks no mor than svral months and has no significant ffcts on th rstoration procss vn if dprciation is nglctd. Eqn..3 is th initial valu condition. Eqn..4 dscribs th assumption that a capital onc installd cannot b sold in th capital markt; that is th assumption of irrvrsibility invstmnt. L and w ar fixd as th valus at quilibrium. Th first ordr conditions of th optimality ar th following: q F / K + & μ + ( I / K T '( I / K = μr, (.5 + T ( I / K ( I / K T '( I / K + μ, (.6 = lim K μ =. (.7 t K whr μ is th shadow pric of capital. To driv th first ordr conditions, th irrvrsibility condition is ignord. It will b takn into account in th invstigation of th transitional dynamics aftr a disastr. Eqn..5 implis that marginal rvnu for capital accumulation is coincidnt to th intrst rat. Th first trm of th lft hand sid in th Eqn..5 is marginal product valu, th scond is capital gain and th third is incrasd rvnu du to rduction of th adjustmnt cost. Eqn..6 implis that marginal cost is coincidnt to th shadow pric. Eqn..7 is th transvrsality condition..3 Formulation of Final Good Sctor Th final goods sctor chooss capital K, intrmdiat input Z, and th invstmnt I to maximiz its nt prsnt valu of th cash flow. Not thaty and L ar th production and labor mployd in th final goods sctor. Production function of th final good sctor is formulatd as H ( K, Z, L, which is assumd constant rturn-to-scal. Maximization problm of th final good sctor is formulatd as, max ( py qz I wl dt, (.8 s.t. K & = I, (.9 K ( = givn, (. I. (. Th first ordr conditions of th maximization ar drivd in th sam mannr with th indrmdiat good sctor: p H / K = r, (. p H / Z = q, (.3 lim K λ =. (.4 K t Whr λ is th shadow pric of capital. Eqn.. mans that marginal rvnu for capital accumulation is qual to th intrst rat. Eqn..3 implis that marginal product valu of intrmdiat input is qual to q. Eqn..4 is th transvrsality condition.

.4 Houshold A rprsntativ houshold is assumd. Th houshold choos th consumption C and saving in forign bond As to maximiz th intrtmporal utility. Th houshold arns from th labor, forign bond, and th dividnd of th firm. Lt ρ b th tim prfrnc rat, π b th dividnd of th firm of th intrmdiat goods sctor and π b th dividnd of th firm of th final goods sctor. Th intrtmporal utility maximization problm of th houshold is formulatd as: max s.t. u( C dt, (.5 As& = wl + ras + π + π pc, (.6 As ( = givn, (.7 lim As ( t =.. (.8 t Eqn..8 is a Non Ponzi Gam (NPG condition, which is th condition that limits th houshold from xpanding th utility infinitly by borrowing. In ordr to avoid xtrm bhavior of th modl du to th assumption of an xognous intrst rat, w st ρ = r as in prvious rsarchs; s Turnovsky (997. Th first ordr condition is: u "( C C& / u( C = r ρ. (.9 Th outlin of th modl is summarizd in Figur. Dmand Dmand Capital Markt (opn Saving (Supply Intrmdiat good sctor Final good sctor Supply Final good Markt (opn Dmand Houshold Dmand Supply Dmand Intrmdiat good markt (closd, non-tradabl Dmand Labor Markt (closd Supply Figur Outlin of th modl.5. Economic Growth Path bfor a Disastr.5. Stady stat Th production function and adjustmnt cost function ar assumd as: α α β β η η F ( K, L = A K L, H ( K, Z, L = A K ( Z L, T ( I / K = ( γi /(K. (. Th first ordr conditions of th industrial sctors consist of Eqn..5-.7,.-.4. Ths conditions and th markt claring condition for th intrmdiat goods giv th Eqn..,.9, transvrsality conditions( Eqn..7,.4, and following diffrntial quations: I / K = ( + μ /γ (. β α ( β η pb β K K Θ = r (. β α ( β η pbα( β η K K Θ + ( + μ / γ μr = & μ (.3 β η (α ( β η Not that B= A A and Θ = L L. By stting K & = and μ& =, capital, intrmdiat good pric and othr variabls in th stady stat ar obtaind. Shadow pric of th capital in th intrmdiat sctor is qual to in th stady stat. It is assumd that mployd labor and wag rat of ach sctor is constant at th

valu in th stady stat. In quilibrium of this labor markt, wag rat is qual in th both sctors and marginal product valu is qual to th wag rat. In th prvious study, authors (8 dmonstratd th stady stat and saddl path xists in this modl..5. Consumption and Savings Th assumption ρ = r implis that th optimum consumption path is constant consumption. Intgrating th budgt constraint and NPG condition givs th consumption lvlc as: r = + C As( ( py IT ( I / K I dt. (.7 p Saving is drivd as follows by using budgt constraint and Eqn..7: A& s = wl + π + π r ( wl + π + π dt. (.8 This implis that th houshold savs whn incom xcds consumption, dtrmind by th prsnt valu of futur incom and drain upon it othrwis..5.3 Invstmnt Invstmnt of intrmdiat goods sctor is dtrmind by th Eqn... Invstmnt of final goods sctor is dtrmind by th Eqn.. according to th valu of. As usual rsult of opn conomic growth modls, Invstmnt is dtrmind indpndntly from saving. Not that irrvrsibility of invstmnt is assumd..5.4 Transitional Dynamics Onc initial capital K (, K ( is givn, μ is dtrmind by saddl point path. Thn invstmnt of intrmdiat goods sctor is dtrmind by th Eqn... In th cas of K ( < K, for xampl, capital is accumulatd by invstmnt and th production incrass. This lads to th dclin of μ and rach th stady stat finally. K 3. CONSISTENT MEASUREMENT OF ECONOMIC LOSSES CAUSED BY NATURAL HAZARDS 3.. Economic Impact of a Disastr A disastr vnt is assumd to occur whn th conomy is in its stady stat. Disastr is assumd as unxpctd shock. Capital stock is assumd to b dcrasd in th intrmdiat sctor in a discrt way by a disastr vnt. Aftr th disastr, houshold and firms rplan th optimal path of consumption, invstmnt and capital. If th irrvrsibility condition is ignord, first ordr conditions and th markt claring condition of th intrmdiat goods markt givs Eqn..,.9, transvrsality conditions (Eqn..7,.4, and Eqn..-3. Rstoration procss is dtrmind by ths conditions and initial valus. Howvr, in fact, th production of th intrmdiat sctor can not rcovr immdiatly du to th adjustmnt cost. Bcaus th capital of th final good sctor is not damagd undr our scnario, th amount of capital of th final good sctor is in xcss of th optimum lvl dtrmind by th Eqn.. and its marginal productivity dclins. This implis that th shadow pric of th capital of th final goods sctor falls blow. In this cas, th optimal dcision is to sll th xcss capital; that is ngativ invstmnt. If th adjustmnt cost for ngativ invstmnt is zro, th final goods sctor slls th xcss capital to adjust th amount of capital to th optimal valu. Howvr, onc capital is installd, it is difficult to sll at th sam pric. This papr assums that a capital onc installd can b sold at a pric much lowr than th pric at which it was purchasd. In this cas, th final goods sctor dos not sll th capital installd and kps it as it is; that is I =. Thn th first ordr conditions ar th transvrsality conditions (Eqn..7,.4, capital accumulation quations (Eqn..,.9, Eqn.., and th following quation: β α ( β η pbα( β η( K K Θ + ( + μ / γ μr = & μ. (3. Authors(8 invstigatd th transitional dynamics aftr a disastr with phas diagram. Du to th damag of capital of th intrmdiat goods sctor, productivity of capital of th sctor riss. It lads to th ris of th shadow pric of capital of th sctor and rstoration invstmnt is inducd. Capital and production rcovr by th invstmnt. Dlay of th rcovry of th intrmdiat sctor lads to not only th dcras of th production of th intrmdiat good sctor but also that of th final good sctor. Figur 3 shows a numrical simulation for a rstoration procss aftr a disastr. Th paramtrs ar st as α =.75, β =.65, η =.5, γ =, p =, L =, A =, A =. T h cass in which 99 % and % of capital of th intrmdiat good sctor is damagd ar invstigatd. Th Tim-Elimination Mthod (Mulligan and Sala-i-Martin, 993 is usd for th

calculation. Panl (a and (b in Figur.3 show capital rcovry procss in th intrmdiat good sctor and final good sctor, rspctivly. Th horizontal axis is tim aftr a disastr. Th valu is normalizd by th stady stat valu. Panl (b shows that capital in th final good sctor is not damagd in th scnario..8.6.4..8.6.4. K K Y Y 5 5 5 3 t (acapital rcovry procss in th Intrmdiat good sctor Y Y K K.5.5 H= GDP GDP L.9.8.7 5 5 5 3 t (bcapital rcovry procss in th Final good sctor 8 6 4 I 5 5 5 3 t (cproduction rcovry in th Intrmdiat good sctor 5 5 5 3 t (Rstoration invstmnt path of th Intrmdiat good sctor.6 q 5 45 4 35 3 5 5 5 5 3 t (dproduction rcovry procss of th Final good sctor (=GDP rcovry 5 5 5 3 t (fpric of th Intrmdiat good As - 4-4 -3 4 5 5 5 3 t CêC.9.8-4 4-5 4 (ghoushold savings in forign asst aftr th disastr.7.6 5 5 5 3 t (hconsumption aftr th disastr 99% damag in capital % damag in capital no damag in capital Figur 3 A numrical simulation for rcovry procss aftr a disastr

Panl (c and (d in Figur.3 show th production rcovry procss in th Intrmdiat good sctor and th final good sctor, rspctivly. Panl (d shows that production of th final good sctor, whos capital is not damagd, dclins aftr th disastr. Thrfor, it is dmonstratd that cascad ffct is inducd in this modl. At th sam tim, from th markt clar condition on th intrmdiat good, panl (d also shows th GDP rcovry procss aftr th disastr. Panl ( shows th rstoration invstmnt path in th intrmdiat good sctor. Panl ( shows that pak of th rstoration invstmnt is dlayd in th cas that capital is gratly damagd (99% damag in capital in Figur.3. This dlay rspons dpnds on th shap of th adjustmnt cost function. Panl (f shows that pric of th intrmdiat good ris aftr th disastr. It is inducd by marginal productivity of intrmdiat good in final good sctor rising. Panl (g shows that th houshold incras dbt for forign countris aftr th disastr. This is bcaus th houshold supplis th rcovry funds to th intrmdiat good sctor by borrowing. This lads to th dcras of th consumption blow th bfor lvl trnally. Panl (h shows that th houshold consumption dcras vn aftr th production rcovrs. 3.3 Consistnt Masurmnt of Economic Losss Losss of ach industrial sctor can b masurd by diffrnc of th discountd cash flows btwn th cass with and without a disastr. Lt V, V dnot th valu of th firm of intrmdiat goods sctor and final goods sctor valuatd at th tim disastr occurs. Lt th suffix of indicat th variabls in th cas with disastr and ^ indicat th variabls in th cas without disastr. Losss of ach industrial sctor can b dscribd as: V Vˆ = π π rt rt ˆ dt = q Y qy ˆ ˆ + Iˆ ( + T ( Iˆ / Kˆ dt, (3. ( ( ( π ˆ π rt dt ( py q ( Z ( pyˆ qz ˆ ˆ V Vˆ = = dt. (3.3 Tabl shows that losss of ach industrial sctor consist of Forgon Nt Rvnu and Rstoration Invstmnt. Nt rvnu mans rvnu minus costs of intrmdiat input. On th othr hand, th consumption of houshold dcrass as th dividnd incom dcrass, and it lads to th dclin of th houshold s utility. Consumption of th houshold dcrass vry priod by r π π + π π r ( t ( ˆ ˆ dt. (3.4 p Dclin of th houshold utility can b valuatd by Compnsating Variation (CV. To kp th sam utility aftr th disastr, it is ncssary and sufficint to compnsat th valu of th dcras of th consumption vry trm. Th prsnt valu of th loss of th houshold is coincidnt to th sum of th loss of th industrial sctors. This indicats that ffct of th dcras of capital in th intrmdiat sctor arrivs at th houshold sctor. Thus, it can b concludd that th total conomic loss causd by natural disastr can b masurd in industrial sctor, occurrnc sid, or houshold sctor, arrival sid. Tabl Consistnt masurmnt of conomic losss aftr a disastr Industry Houshold Intrmdiat goods sctor Dclin of Utility Forgon Nt Rvnu V q Y qyˆ Vˆ + V Vˆ ( ˆ dt Rstoration Invstmnt Iˆ ( + T ( Iˆ / Kˆ dt Final goods sctor Forgon Nt Rvnu ( py q Z ( pyˆ qz ˆ ˆ ˆ Total V V + V Vˆ dt Vˆ V Vˆ + If total conomic loss is masurd in industrial sctor, it is ncssary and sufficint to sum up th losss of th ach industrial sctor. It is important point to masur th forgon nt rvnu in th final good sctor, which V

is causd by cascad ffct, vn though it is not damagd physically. Tabl shows what should b masurd in ach sctor for consistnt masurmnt of total conomic loss in th whol conomy: 4. MEASURING CASCADE EFFECT SEPARATELY 4. Mchanism Inducing Cascad Effct Although th mthodology invstigatd abov can masur th total conomic loss including cascad ffct consistntly, it cannot masur th impact of th cascad ffct itslf. A mthodology to masur th cascad ffct sparatly is invstigatd blow. In th invstigation abov, cascad ffct occurs bcaus th intrmdiat good markt is closd and th intrmdiat good cannot b transportd altrnativly from outsid. If th intrmdiat good markt is opn, final good sctor can obtain th intrmdiat good from intrnational markt without any chang of its pric. In this cas, th intrmdiat good pric is dtrmind by th quilibrium in th intrnational markt. Th first ordr conditions ar Eqn.., transvrsality conditions (Eqn..7,.4, capital accumulation quations (Eqn..,.9 and following quations: α α q A αk L + ( + μ / γ μr = & μ, (3.5 β β γ η pa βk ( Z L = r, (3.6 β β γ η pa ( β η K ( Z L = q. (3.7 Eqn.3.7 implis that if final good sctor is not physically damagd and th capital is not dcrasd, its dmand for intrmdiat good is not changd from th stady stat valu, Z. Final good sctor can import th amount Z and its production dos not dcras. In this cas, cascad ffct is not inducd. This implis that it is Y important factor to cascad ffct whthr th intrmdiat good can b obtaind altrnativly from outsid. In othr words, substitutability is a factor inducing cascad ffct. 4.. How to Sparat th Cascad Effct Cascad ffct can b masurd by th diffrnc of loss btwn cass in which th intrmdiat good markt is closd and opn. Tabl illustrats th cascad ffct in ach sctor and total conomic loss. Not that th suffix ~ indicats th variabls in th cas that intrmdiat good markt is opn. Tabl : Tabl of loss masurd in ach sctor whn cascad ffct is sparatd Industry Houshold Intrmdiat goods sctor Dclin of Utility (Primary Effct Forgon Nt Rvnu (Primary Effct ~ V ~ q Y Y V ( dt Forgon Rvnu (Cascad Effct ~ q Y qy ( ˆ ˆ dt Rstoration Invstmnt (Primary Effct ~ ~ ~ I + T I K ( ( / dt Rstoration Invstmnt (Cascad Effct ( Iˆ ( + T ( Iˆ / Kˆ I ~ ( + T ( I ~ / K ~ Final goods sctor Forgon Nt Rvnu (Cascad Effct py q Z pyˆ qzˆ ( ˆ ( dt Total V ˆ Vˆ dt Dclin of Utility (Cascad Effct ~ V Vˆ + V Vˆ V + V V V + V ˆ Vˆ Tabl 3 shows a numrical calculation rsult whn th cascad ffct is masurd sparatly. Th valu of ach paramtrs is th sam as in th sction 3.. Th focusd scnario is th cas that 99 % of capital of th intrmdiat good sctor is damagd. From Tabl 3, it is found that th cascad ffct nlargs th total

conomic loss. Total of th primary ffct is 3.3 but th total conomic loss is 5.44 bcaus of th cascad ffct. For this cas, th cascad ffct nlargs th total conomic loss about.5 tims largr. In tabl 3, Cascad Effct on th Forgon Nt Rvnu in th intrmdiat good sctor is ngativ. Thr ar two rasons for this fact. Th first is that th rcovry spd of th production in th intrmdiat sctor is mad high du to th cascad ffct. Th scond is that th pric of intrmdiat good riss. Cascad Effct on th Rstoration Invstmnt is positiv bcaus largr amount of rstoration invstmnt is mad in th intrmdiat good sctor immdiatly. Forgon Nt Rvnu in th Final good sctor is positiv, and it givs th most significant impact on th xpanding of total conomic loss. Tabl 3 A numrical simulation rsult of masuring cascad ffct and total conomic loss Intrmdiat good sctor Forgon Nt Rvnu (Primary Effct 3. (Cascad Effct -.7 Rstoration Invstmnt (Primary Effct.5 (Cascad Effct.7 Final good sctor Forgon Nt Rvnu (Cascad Effct.87 Total 5.44 5. SUMMARY This papr proposd a mthodology for consistnt masurmnt of conomic losss causd by natural hazards. A two-sctor conomic growth modl was usd to dscrib rcovry procss aftr a disastr. It was shown that, in th cas that capital is damagd in industrial sctors, sum of th diffrncs of discountd cash-flow in industrial sctors can b a consistnt masur as total conomic loss. It consists of Forgon Nt Rvnu and Rstoration Invstmnt. This papr also proposs how to masur th cascad ffct, which is th spillovr ffct of conomic impacts inducd by mutual rlation ships btwn industrial sctors. Cascad ffct can b masurd sparatly by comparing two cass in which th intrmdiat good markt is closd in th country and opn to th world. A numrical calculation dmonstratd that cascad ffct can nlarg th total conomic loss about.5 tims. It indicats that, in masuring total conomic loss, it is important to tak into account cascad ffct in ordr to avoid undrstimation. REFERENCES Turnovsky (997. Intrnational Macroconomic Dynamics, MIT Prss. Sn, P. and Turnovsky, S.(995. Invstmnt in a Two-Sctor dpndnt conomy, Journal of th Japans and Intrnational Economics 9, 9-55. Tatano, H., Isob, W. and Okada, N.(. Economic Evaluation of Sismic Risks, Proc. of Joint Sminar on Urban Disastr Managmnt, 36-39, Bijing, China. Nakano.K., Tatano.H., Kajitani.Y., Fujimi.T., Tsuchiya.S.(7. Estimation of th Economic Loss in Industrial Sctors Causd by th 4 Mid-Niigata Earthquak, Rviw of Infrastructur planning, 4:, 89-98.(in Japans Nakano.K. and Tatano.H.(8 Economic Rstoration Procss aftr Natural Disastrs undr Mutual Rlationships btwn Industrial Sctors, Procdings of th IEEE Intrnational Confrnc on Systms, Man and Cybrntics, (accptd. Ros, A. (4 Economic Principls, Issus, and Rsarch Prioritis in Hazard Loss Estimation, in Okuyama, Y. and Chang, S.(ds Modling Spatial and Economic Impacts of Disastr, Advancs in Spatial Scinc, Springr,3-36. Barro S.J and Sala-i-Martin.X. (4. Economic Growth, MIT Prss. Mulligun.C.B and Sala-i-Martin, X.(993 Transitional dynamics in two-sctor modls of Endognous Growth, Th Quartrly Journal of Economics, 8:3, 739-773. Hayashi.F.(98 Tobin's Marginal and Avrag q: A Noclassical Intrprtation., Economtrica, 5, 3-4.