Aging in Europe: Reforms, international diversification and behavioral reactions. Technical appendix

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1 Agng n Europe: Reforms nernaonal dversfcaon and beavoral reacons Axel Börsc-Supan a Klaus Härl a and Alexander Ludwg ab a Munc Researc nsue for e Economcs of Agng (MEA) b CMR Unversy of Cologne Tecncal appendx Ts annex descrbes our dynamc open-economy macroeconomc model a allows us o analyze e effecs of e labor marke reforms on GDP and consumpon per capa n an agng Europe. Man feaure s e specfcaon of labor supply w an endogenous and an exogenous componen. Wle we rea e reforms and e resulng varaon n e maxmum labor supply as exogenous ouseolds n our model endogenously adus er ours worked up o a maxmum and may us counerac pars of e labor marke reforms.. Demograpy Tme n our model s dscree and exends from =0 T. Eac model perod reflecs a me nerval of 5 years. Our demograpc proecons owever are more dealed w an annual perodcy. Tese dealed demograpc proecons form e background of our analyss. Demograpy s aken as exogenous. represens one of e man drvng force of our smulaon model n addon o exogenous canges n labor supply resrcons and penson polcy canges. Houseolds n our model economes ener economc lfe a age 5 wc we denoe by =0. Te maxmum age s 00 years. Accordngly e maxmum economc age denoed by s 85. We assume a ouseolds gve br beween ages 0 f e age of menopause. Mgraon s assumed o be done by e age of 5 (=0) so a we can rea newborns and mmgrans n e economc model alke. Accordngly n eac counry e sze of populaon of age n perod s gven recursvely by

2 () for 0 and 0 f M 0 were ς denoes e age-specfc condonal survval rae f e age-specfc ferly rae and M 0 e ne mgraon no counry. We assume a e oal ferly rae s consan unl 2050 as n e consan ferly varan of Uned aons World Populaon Prospecs (Uned aons 202). Lfe expecancy s compued from lfe ables provded by e Human Moraly Base (202) and proeced no e fuure by applyng e Lee-Carer meod (Lee and Carer 992). Daa and assumpons on mgraon s agan aken from Uned aons (202). Table summarzes e nformaon on oal ferly raes and lfe expecances a br. f 0 TABLE : Toal ferly raes and lfe expecances Lfe expecancy France Germany aly Uned Saes Toal ferly rae France Germany aly Uned Saes oes: Daa for Germany are for Wes Germany. Source: Auors compuaons based on Uned aons (202) and Human Moraly Daabase (202). 2. Producon Te producon secor n eac counry consss of a represenave frm a uses a Cobb- Douglas producon funcon gven by Y F K L K L (2) were K denoes e capal sock and L s aggregae effecve labor supply of counry a me. s e capal sare and s e ecnology level of counry growng a e exogenous rae g. 2

3 Te frm s problem s sac suc a wages and neres raes are gven by (3) w k (4) r k were k s e capal sock per effcen un of labor and δ s e deprecaon rae of capal. 3. Houseolds A ouseold faces an exogenously gven lm l of on ours of labor supply. Te ouseold endogenously decdes on e fracon of ours wc are acually suppled w 0. Accordngly oal labor supply of a ouseold s gven by e produc of e wo componens l and oal lesure s -l. Te ouseold derves uly from consumpon c and lesure -l and e ouseold s per perod uly funcon s gven by u ( c l ) c l. Te maxmzaon problem of a coor born n perod a =0 s gven by (5) max uc l 0 were s e pure me dscoun facor. n addon o pure dscounng ouseolds dscoun fuure uly w er uncondonal survval probably s. Here s +kk denoes e probably o survve from perod +k age k o perod +k+ age k+ n counry w s 0 =. k 0 k k A feaure of our model s uncerany abou e me of dea expressed n e erm n equaon (5). We assume a accdenal bequess resulng from premaure dea are axed by e governmen a a confscaory rae and used for oerwse neural governmen consumpon. We do no nclude nended bequess n our model. An alernave assumpon would be o redsrbue accdenal bequess o e populaon accordng o some sceme. Te redsrbuon would owever no affec our resuls muc and we erefore oped for s smplfyng assumpon. 3

4 Labor producvy canges over e lfe-cycle accordng o age-specfc producvy parameers ε. Hence e age-specfc wage s w w. Denong oal asses by a maxmzaon of e ouseold s neremporal uly s subec o a dynamc budge consran gven by (6) a a r l w p c were = for =0 r and =0 for >r and r s e exogenous reremen age. τ s e conrbuon rae o a PAYG fnanced publc penson sysem and p s penson ncome see below. Furermore maxmzaon s subec o e consran a e fracon of e ouseolds maxmum possble work ours wc s acually suppled s posve and may no exceed one (7). 0 n e varan of our model w fully exogenous labor supply we replace e consran (7) w e consran a = for all. 4. Pensons and penson reform Te only purpose of e governmen n our model s o organze a prooypcal publc penson sysem a s pay-as-you-go fnanced. Penson paymens are earnngs relaed. Precsely penson paymens are gven by (8) p w (- ) s were ρ denoes e ne replacemen rae (generosy of e penson sysem) and e conrbuon rae of e penson sysem n. s s e penson sock capurng e earnngs relaed componen of e sysem - wc accumulaes over e lfe-cycle accordng o (9) s s l r were r r r l l. Accordngly ouseolds earn one earnngs pon f ey receve average wage ncome n a gven perod and earnng pons are normalzed by e leng of e workng perod r. n our model wou cross-seconal eerogeney dfferences n wage ncome are only nduced by 4

5 age producvy. Te sceme approxmaes well e curren legslaon n Germany and France w er earnngs pon sysems and aly w e new enrans noonal defned conrbuon sysem. Houseolds fully undersand e lnkage beween conrbuons o e penson sysem and penson paymens n old age. Terefore relave o a Beverdgan penson sysem n wc penson paymens are lump-sum labor supply dsorons are smaller n suc a Bsmarckan sceme because of e earnngs relaed lnkage. Tey are no zero owever as ofen clamed n e publc debae because e rae of reurn on e capal marke exceeds e mplc reurn of e penson sysem n a dynamcally effcen economy 2 o e effec a labor s dsored. Due o compounded neres s effec s sronger for younger ouseolds and monooncally decreases (bu never becomes zero) w age. We assume a e budge of e penson sysem s balanced n all suc a (0) p w. r r Te man polcy parameer s e ne replacemen rae ρ; e conrbuon rae responds passvely o balance e penson sysem s budge. f ρ s large publc pensons crowd ou prvae savng roug e ouseolds consumpon/savng decson gven by (5) and (6). f ρ = 0 all old age provson wll be prvae savngs. Ts represens e exbook lfe-cycle model n wc neremporal consumpon smoong over e lfe cycle provdes e reremen ncome roug savng n young age and dssavng afer reremen. 5. Equlbrum We defne equlbrum n s economy sequenally. Gven nal capal socks K 0 a compeve equlbrum of e economy s defned as sequences of dsaggregae varables for e ouseolds l a sequences of aggregae varables L K prces for c labor as well as conrbuon raes o e penson sysem common world neres rae r suc a w C n eac counry and a 2 We calbrae our model suc a e economy s dynamcally effcen. 5

6 6. Gven prces and nal condons ouseolds maxmze lfe-me uly n (5) subec o e consrans n (6) o (9). 2. Facor prces equal er margnal producves as gven n equaons (3) and (4). 3. Governmen polces sasfy (8) n every perod and ) ( a G were G s (oerwse neural) governmen consumpon (fnanced by axng accdenal bequess a a confscaory rae). 4. All markes clear n all. l L 0 for all a K 0 K L K G K c 0 6. umercal mplemenaon Our me lne as four perods: a pase-n perod a calbraon perod a proecon perod and a pase-ou perod. Frs we sar calculaons 0 years before e calbraon perod begns w e assumpon of an arfcal nal seady sae n 850. Te me perod beween 960 and 200 s en used as calbraon perod n order o deermne e srucural parameers of e model. Our proecons run from 200 roug 200. Resuls are dsplayed unl year 2050 o sow e man perod of populaon agng. Te pase-ou perod afer 200 as wo pars: a ranson o a seady-sae populaon n 2200 and an addonal 00-year perod unl e macroeconomc model reaces a fnal seady sae n For eac sequence of ouer loop varables (see below) we solve e ouseold model by loopng backward n age applyng frs-order meods and appropraely andlng e consrans. We can caracerze soluons n closed form condonal on guesses for fnal asse oldngs

7 of ouseolds (respecvely fnal consumpon) a for eac perod and counry. To avod cosly nner loops for soluon of e ouseold model we updae ese guesses only w oer ouer loop varables as descrbed nex. We deermne e equlbrum pa of e overlappng generaons model by usng e Gauss- Sedel-Quas-ewon meod developed by Ludwg (2007). Te algorm searces for equlbrum pas (ouer loop varables) of capal o oupu raos penson conrbuon raes (or dependng on e penson sysem scenaro replacemen raes) average earnngs and fnal perod ouseold asses n eac counry. 7. Calbraon We solve e model a a 5 year frequency. We calbrae a subse of srucural model parameers as frs-sage parameers wou usng e model. Accordngly we se e capal sare o 0.40 based on esmaes of e wage sare. We se e long-run grow rae of producvy g o 0.05 based on esmaes of real TFP grow n OECD counres. Te deprecaon rae s 5% wc gven our calbraon arge of a capal oupu rao of 2.65 (based on esmaes of e sock of fxed asses o oupu) and our coce of e capal sare mples an nvesmen o oupu rao of abou 20%. Te ner-emporal elascy of subsuon (ES /) s se o 0.5 by reference o oer sudes see e.g. e revew provded n Bansal and Yaron (2004). Te dscoun facor as well as e consumpon weg n e uly funcon are calbraed by usng e model suc a produces a capal oupu rao of abou 2.65 and a sare of me devoed o labor of abou one rd. Ts requres seng e dscoun facor close o one (0.99).e. pure me dscounng s weak. Recall owever a ouseolds dscoun fuure uly w e probably of survvng no e fuure. Te value of e consumpon sare n uly φ s

8 TABLE 2: Srucural model parameers : capal sare n producon 0.40 g: grow rae of labor producvy 0.05 : deprecaon rae of capal 0.05 : ecnology level : dscoun facor 0.99 : nverse ner-emporal subsaon elascy 2.0 φ: consumpon sare parameer Bblograpy Bansal R. and A. Yaron (2004). Rsks for e long run: A poenal resoluon of asse prcng puzzles. Te ournal of Fnance 59 (4) Human Moraly Daabase (202). Unversy of Calforna Berkeley (USA) and Max Planck nsue for Demograpc Researc (Germany). Lee R. D. and L. Carer (992). Modelng and Forecasng U.S. Moraly. ournal of e Amercan Sascal Assocaon Ludwg A. (2007). Te Gauss Sedel Quas-ewon meod: A ybrd algorm for solvng dynamc economc models. ournal of Economc Dynamcs & Conrol Uned aons (202). World Populaon Prospecs: Te 202 Revson. Uned aons Populaon Dvson. ew York. 8