Y2K* Stephanie Schmitt-Grohé. Rutgers Uni ersity, 75 Hamilton Street, New Brunswick, New Jersey

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1 Revew of Economc Dynamcs 2, Ž Arcle ID redy , avalable onlne a hp: on Y2K* Sephane Schm-Grohé Rugers Unersy, 75 Hamlon Sree, New Brunswc, New Jersey E-mal: grohe@econ.rugers.edu and Marın Urbe Unersy of Pennsylana, 3718 Locus Wal, Phladelpha, Pennsylana E-mal: urbe@econ.upenn.edu Receved Sepember 8, 1998; acceped February 17, 1999 Ths paper sudes, whn a general equlbrum model, he dynamcs of Year 2000 Ž Y2K. -ype shocs: ancpaed, permanen losses n oupu whose magnude can be lessened by nvesng resources n advance. The mpled dynamcs replcae hree observed characerscs of hose rggered by he Y2K bug. 1 Precauonary nvesmen: Invesmen n solvng he Y2K problem begns before he year Invesmen delay: Alhough economc agens have been aware of he Y2K problem snce he 1960s, nvesmen dd no begn unl recenly. 3 Invesmen acceleraon: As he new mllennum approaches, he amoun of resources allocaed o solvng he Y2K problem ncreases. In addon, he model predcs ha Y2K nvesmen peas a he end of Journal of Economc Leraure Classfcaon Numbers: E22, E Academc Press Key Words: Y2K problem; nvesmen dynamcs 1. INTRODUCTION A number of professonal economss have compared he expeced recesson assocaed wh he Year 2000 Ž Y2K. compuer dae problem wh ha caused by he ol prce shocn Ž Yarden, In ancpaon of hs problem, as he mllennum comes o an end, boh he governmen and he prvae secor have been allocang sgnfcan amouns * We han he edor, Boyan Jovanovc, an anonymous referee, and semnar parcpans a he Federal Reserve Board for commens $30.00 Copyrgh 1999 by Academc Press All rghs of reproducon n any form reserved. 850

2 Y2K 851 of resources o s remedaon. Federal Reserve Governor Edward W. Kelley, Jr., esmaes ha resources allocaed o solvng he Y2K problem wll cos he U.S. economy 110 of 1% of GDP n 1998 Žsee hs esmony before he U.S. Senae Commee on Commerce, Scence, and Transporaon on Aprl 28, Alhough a a slower pace, smlar effors are under way n he res of he world. The Garner Group has esmaed ha worldwde he cos assocaed wh solvng he Y2K problem wll oal 300 o 600 bllon U.S. dollars. The macroeconomc dynamcs rggered by he Y2K problem are characerzed by he followng hree facs: Frs, he mllennum bug has nduced precauonary nesmen n he sense ha he allocaon of resources amed a solvng he Y2K problem began before he year Second, here has been nesmen delay. Alhough economc agens have been well aware of he Y2K problem snce he 1960s, he allocaon of real resources devoed o s soluon dd no begn unl he 1990s. Thrd, nvesmen n he Y2K problem has been accelerang, parcularly snced In hs paper, we embed he Y2K problem no a smple dynamc general equlbrum framewor. We model he Y2K problem as a suaon n whch before he year 2000 agens learn ha n he year 2000 oupu wll experence a permanen declne. Agens can lessen he oupu declne by nvesng resources n advance. The fac ha resources allocaed o solvng he Y2K problem become producve only n he year 2000 s he ey elemen drvng he dynamcs of he model. We sudy he Y2K problem whn he conex of a sandard opmzng growh model feaurng an ArrowKurz Ž A-K. echnology for he producon of goods. Ths echnology allows agens o shf resources across me a a consan rae of reurn. In spe of s smplcy, he model can accoun for he hree man facs assocaed wh he Y2K problem: precauonary nvesmen, nvesmen delay, and acceleraon. In addon, he model predcs ha Y2K nvesmen wll pea a he end of THE MODEL Consder an economy populaed by a large number of dencal, nfnely lved consumers wh preferences descrbed by he uly funcon H e u c d, 1 0 where c denoes consumpon n perod and 0 denoes he subjec- ve dscoun facor. The nsan uly funcon u s assumed o be wce connuously dfferenable, srcly ncreasng, and srcly concave. The represenave consumer s endowed wh an nal soc of capal 0.

3 852 SCHMITT-GROHE AND URIBE Oupu, y, s assumed o be pershable and o be produced wh he lnear echnology y A, where A, he margnal produc of capal ne of deprecaon, s assumed o be srcly posve. In perod zero, agens learn ha begnnng n perod T 0 hey wll experence a loss n ncome gven by g. Here T s mean o represen he year 2000 and g he Y2K problem. Agens can nves resources n advance o reduce he magnude of he Y2K problem. Specfcally, we assume ha agens can buld Y2K capal, whch we denoe by I, and ha he Y2K problem s a decreasng funcon of I, g gž I., where g: R R s wce connuously dfferenable and sasfes lm gž x. x 0. Le denoe Y2K nvesmen. Then he law of moon of he soc of Y2K capal s assumed o ae he form İ Ž., I0 0, Ž 2. where : R R s assumed o be wce connuously dfferenable and o sasfy 0 0 and lm Ž x. x 0. Y2K nvesmen s assumed o be rreversble; ha s, 0. Ž 3. To ensure ha agens wll always fnd opmal o nves n solvng he Y2K problem, we mpose he followng assumpon: Assumpon 1. 0 g 0 A. Ths assumpon says ha f he household chooses no o nves any resources n solvng he Y2K problem unl he year 2000 Ž 0 T., hen a he begnnng of he new mllennum, he rae of reurn on Y2K nvesmen, 0 g 0, exceeds he rae of reurn on physcal capal, A. Oupu can be allocaed o consumpon, nvesmen n physcal capal, or nvesmen n solvng he Y2K problem. Thus, he flow resource consran of he household s gven by ½ A T c, Ž 4. A gž I. T where denoes ne nvesmen n physcal capal a me. The house- hold chooses sequences, I, c, 4 o maxmze 1 subjec o 2, 3, and

4 Y2K 853 4, gven 0 and I 0. The frs-order condons of he consumer s problem are Ž Arrow and Kurz, , 3, 4, and už c. Ž 5. A Ž 8. ½ T gž I. T Ž 9. lm e 0 Ž 10. lm e 0 Ž 11. lm e 0 Ž 12. lm e I 0, Ž 13. where and are me-dfferenable Lagrange mulplers assocaed wh 2 and 4, respecvely. 3. Y2K INVESTMENT DYNAMICS Our frs resul s ha once Y2K nvesmen becomes posve, mus connue o be posve unl he year To see hs, suppose ha 0 for some T and ha 0 for some T. Then, by 7, Ž.. Because A 0, Eqs. 6 and 9 ogeher wh he assumed concavy of mply ha 0, whch volaes Condon 8. Furhermore, f nvesmen s posve a any me before he year 2000, hen nvesmen ncreases over me unl he begnnng of he mllennum. To see hs, noe frs ha f 0 for T, hen, as shown above, 0 T. Thus, by Eq. 7, Ž. T. Because and are dfferenable, hs equaon mples ha s also dfferen- able for T. From dfferenang hs expresson and usng Eqs. 6 and 9, follows ha A Ž. Ž. 0 T. We summarze hese resuls n he followng proposon.

5 854 SCHMITT-GROHE AND URIBE PROPOSITION 1 Ž Accelerang Y2K Invesmen.. If 0 for some T, hen 0 for all T. Furhermore, for all T. Nex, we esablsh ha he model exhbs precauonary nvesmen, n he sense ha agens fnd opmal o begn o allocae resources o solvng he Y2K problem before he arrval of he year PROPOSITION 2 Ž Precauonary Y2K Invesmen.. If Assumpon 1 s sasfed, hen here exss a T such ha 0. Proof. We esablsh he proposon n hree seps, by showng ha a f 0 T, hen 0; Ž b. T f 0 T, hen 0 T; and c 0 s mpossble. a Suppose ha 0 T and ha 0. Then from 7, follows ha Ž. T T T T. On he oher hand, 8 mples ha lm T lm T 0, or, because and are connuous, 0. Bu hs conradcs Ž. T T T T T because s srcly concave. Ž b. Suppose ha 0 T. ByŽ a,. T 0. Suppose ha 0 for some T. Le nf : 04. Then n any nerval around, and such ha 0 and 0. I follows from 8 ha 0. Equaons 6 and 9 mply ha and are connuous. I follows from Assumpon 1 ha. There- fore, 0. Thus, he concavy of and he fac ha 0 mply ha Ž., conradcng Ž 7.. Ž c. Suppose 0. Then for T, A and g 0. Thus, T Ž A. g 0 A e g 0 A e 0. If g 0 A T T T T T 0, hen a he rae, volang 8 because grows a he rae A. If T g 0 TA 0, hen e T g 0 TA, vola- Ž ng 11. Fnally, f g 0 A 0, hen. T T T T 0 AŽ Ž 0. gž 0... The rgh-hand sde of hs expresson s less han one by Assumpon 1; hus he lef-hand sde volaes 8. In he followng proposon, we show ha f a any me n he new mllennum Y2K nvesmen s posve, hen s rae of reurn mus be a leas as large as he rae of reurn on physcal capal. PROPOSITION 3. If 0 for T, hen Ž. gž I. A. Proof. Suppose ha 0. Then, because I I for, 9 m- ples ha gž I.. Usng Ž 6. and negrang hs expresson yeld gž I. gž I. Ž. Ž A.Ž. e e, A A

6 Y2K 855 whch volaes 13 unless g I A 0. The resul follows mme- daely from he fac ha Ž. 0. We are now ready o show a ey predcon of he model, namely ha agens may opmally choose o delay Y2K nvesmen. PROPOSITION 4 Ž Y2K Invesmen Delay.. For T suffcenly large, here exss a Ž 0, T. such ha 0 for all. Proof. Le Ž 0, T. and suppose ha 0 T. By Proposon 1, 0 T. From 7, 9, and 6, hen follows ha Ž A.ŽT. ŽT. e e, whch, usng T 5 and smplfyng, be- comes Ž. e AŽT. Ž.. Snce 0, 0 Ž 0. T, and 0, hs expresson mples ha 0 e AŽT. Ž.. Thus, lm Ž. T T T 0, volang he condon Ž. gž I. A derved n Proposon 3. T In he case ha agens choose o delay Y2K nvesmen, he ranson from no nvesmen o posve nvesmen wll be smooh. In fac, as he nex proposon shows nvesmen s connuous everywhere. PROPOSITION 5 No Jumps n Y2K Invesmen. T s connuous. Proof. Suppose nvesmen s dsconnuous a. Then, because s connuous, 0 such ha for any 0 one can fnd a pon Ž. sasfyng Ž. and Ž. Ž. Ž.. Suppose 0. Then, by 7,. Because and are connuous, as 0, Ž. and Ž.. Thus, for suffcenly small, Ž., whch, by Ž. Ž. 7, mples ha Ž. 0. Bu f Ž. 0, hen Ž. Ž. Ž. 0; hus, for suffcenly small we have, whch volaes 8. If 0, hen Ž. Ž. Ž. Ž.. Also, Ž. mus be posve. Thus, he connuy of and and he fac ha Ž. Ž. Ž. mply ha for suffcenly small,, whch volaes Ž 8.. Ž. Ž. Ž. The presence of physcal capal s crucal n generang connuy n Y2K nvesmen. One can show ha n an endowmen economy Y2K nvesmen dsplays a dscree declne n he year 2000 ŽSchm-Grohe and Urbe, We complee he characerzaon of equlbrum by sudyng he dynamcs of Y2K nvesmen n he new mllennum. The followng proposon shows ha Y2K nvesmen peas wh he arrval of he new mllennum, s monooncally decreasng hereafer, and converges o zero n a connuous fashon. PROPOSITION 6 Ž Y2K Invesmen Dynamcs n he New Mllennum.. a Deceleraon: If 0 for T, hen 0 for all T. Ž b. Inesmen conerges o zero: lm 0. Ž c. Inesmen peas a he arral of he year 2000: T for all.

7 856 SCHMITT-GROHE AND URIBE Proof. a We frs prove ha f * 0 for some * T, hen 0 *. Suppose, conrary o he clam, ha 0 for some *. Le nf *: 04. Then n any nerval around and such ha 0 and 0. I follows from 8 ha 0.By Proposon 3 and Condons 8 and 9 s connuous and less han, whch s also connuous. Therefore, 0. Thus, he concavy of and 0 mply ha Ž., conradcng 7. We now show deceleraon. Suppose 0 for some T. I hen follows mmedaely from he above argumen ha 0 T. Therefore, T. Dfferenang hs expresson and usng 6 and 9 yeld Ž. Ž. Ž. gž I. A. From he facs ha s ncreasng and concave and ha he expresson n square braces s nonposve Ž Proposon 3. follows ha Y2K nvesmen s nonncreasng. Assume ha 0 for some. Then dfferenang 2 yelds Ž. Ž. g Ž I. 0, whch mples ha 0 for some, volang he fac ha nvesmen s nonncreasng. Ž b. I follows from a ha s nonncreasng for T. Thus, because s bounded below by zero, mus converge. Suppose converges o some posve consan *. Then I and Ž. gž I. 0, volang Proposon 3. c The resul follows drecly from a, b, and Proposons 1 and 2. REFERENCES Arrow, K. J., and Kurz, M. Ž Publc Inesmen, he Rae of Reurn, and Opmal Fscal Polcy. Balmore: The Johns Hopns Unversy Press. Schm-Grohe, S., and Urbe, M. Ž Y2K, Worng Paper 9832, Rugers Unversy. Yarden, E. Ž 1998., Year 2000 Recesson? mmeo, Cener for Cybereconomcs and Y2K.

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