THE UNIVERSITY OF MARYLAND COLLEGE PARK, MARYLAND Economcs 600: August, 007 Dynamc Part: Problm St 5 Problms on Dffrntal Equatons and Contnuous Tm Optmzaton Quston Solv th followng two dffrntal quatons. ) y ( + y( = + t, y(0) = ) 3y ( + 6y( = -t, y(0) = Quston Rduc th n-th ordr lnar dffrntal quaton n n a x ( a x ( ax ( a0 x( 0 n n to a systm of frst-ordr dffrntal quatons. Quston 3 Rconsdr th Solow-Swan growth modl that w hav dscussd n th lcturs, and rlax two of th assumptons w mad, namly, that th captal stock dos not dprcat, and that thr s no tchncal progrss. To allow for dprcaton, spcfy th law of moton for th captal stock as K ( I( K(, (0, ) To allow for tchncal progrss, suppos that th producton functon s gvn by Y ( F[ K(, A( ] [ K( ] [xp( ], 0, (0, ) Drv th mpld law of moton for th ffctv captal-labor rato, ~ k ( K( /[ A( ], comput th (nontrval) stady stat for th ffctv captallabor rato, and analyz ts stablty proprts.
Quston 4 Consdr th xpctaton augmntd Phllps curv ( f ( u( ) (, f ' 0, (0,] whr s th obsrvd nflaton rat, u s th unmploymnt rat, and s th xpctd nflaton rat. Assum that xpctatons ar gradually corrctd accordng to ( ( ( ( ), 0 ) Obtan th (rducd form) dffrntal quaton as a functon of obsrvd quantts, and charactrz t. ) Now assum that th govrnmnt wshs to mantan th unmploymnt rat at a crtan targt u -. In th short run t s known that <, but n th long run =. Commnt th fasblty of ths polcy takng nto account th stablty proprts of th dffrntal quaton. Quston 5 Consdr th followng dmand and supply qulbrum modl. Th dmand cur s gvn by: D( = a + bp(, whr D s th quantty dmandd and p s th unt prc. Th supply curv s dfnd as S( = c + dp (, whr S s th quantty suppld and p s th xpctd prc. Expctatons ar formd accordng to an adaptv xpctatons mchansm, p ( = [p(-p (], >0. Show that n qulbrum, p = p = p -. Examn th stablty proprts of th paths of p( and p (. Quston 6 (a) Gv th gnral soluton for th lnar dffrntal quaton systm for x ( = Ax( () A 8 5 3 (b) Now augmnt th lnar dffrntal quaton systm you consdrd n (a) by a boundary conon. What s th rsultng soluton? Us th followng boundary conon:
x(t = 0) = x 0 = (,, ) Quston 7 (a) Gv th gnral soluton for th lnar dffrntal quaton systm x ( = Ax( () for 3 A 0 0 0 (b) Now augmnt th lnar dffrntal quaton systm you consdrd n (a) by a boundary conon. What s th rsultng soluton? Us th followng boundary conon: Quston 8 x(t = 0) = x 0 = (,, ) Consdr an conomy wth two commos, and. Th markt dmand for good ( =, ) s a functon of normalzd prcs (allowd to b ngatv) of both goods (th prcs ar dnotd by p( = p, p ( ) ): ( t D [p(] = a p ( bp (, j whr j dnots not, namly j = 3 -. Lkws, th supply of good ( =, ) s wrttn as: S [p(] = c p (. W assum that th a s and c s ar strctly postv. Suppos now that w hav th followng smpl adjustmnt procss for prcs: p ( = D [p(] - S [p(], =,. Dtrmn th stablty of ths adjustmnt procss. (Not: Thr s no nd to solv th prvous quaton xplctly.)
Quston 9. Consdr th followng nonlnar dffrntal quaton systm: y ( [ y(, y ( ], 0, 0 y y y ( [ y(, y ( ], 0, 0 y y Prformng a lnar approxmaton, w obtan whr / y j Quston 0 ~ y ( ) ~ ( ) ~ t A y t A y (, ~ y ( A ~ y ( A ~ y ( ), t ~ y ( y ( y, =,. Show that gvn th assumptons mad on th sgns of,, j =,, th stady stat s a saddl. Th fundamntal quatons for th contnuous-tm dtrmnstc optmal growth modl (th Cass-Koopmans modl) ar gvn by th followng dffrntal quaton systm n consumpton pr workr, c(, and th captal-labor rato, k(: dk( f [ k( ] c( nk(, du[ c( ]/ n f [ k( ], u[ c( ] () whr u C dnots th utlty functon, f C th producton functon, n s th rat of labor nput growth, and th subjctv dscount rat. Assum that u and f ar strctly concav functons, and that f satsfs th Inada conons f ( 0) 0, f (0), f ( ) 0 Also assum that k(0) > 0, and mpos th constrants k ( 0, c( 0, for all t. (a) Drv xprssons for th stady stats of k( and c(. (b) Draw th phas plan and plot th saddl path. (Hnt: Carfully drv th proprts of th phas path for th captal-labor rato.) (c) Lnarz th systm gvn by (), and confrm th systm s stablty proprts n th nghborhood of th stady stat.
Quston Th dmand curv n an ndustry at tm t s gvn by q( = a - bp(, whr a, b > 0, and p( and q( dnot prc and ndustry output at tm t, rspctvly. Thr s on larg frm n th ndustry that sts th prc, and a frng of small frms that accpt ths prc and sll thr ntr output at that prc. Nw frng frms ntr f th larg frm chargs a prc gratr than p*. Dnot th output of th frng frms by x(. Suppos that th ntal valu of x(0) s gvn, and that x( satsfs th dffrntal quaton x ( = k [p( p*]. Th larg frm s sals ar thn gvn by q( - x(. Suppos that th avrag cost of th larg frm s constant and qual to c. Thrfor, th prsnt dscountd valu of th larg frm s profts s gvn by 0 t [ p( c][ a x( bp( ] whr s th dscount rat. Assum that p* > c. (a) Apply th Maxmum Prncpl to ths problm, takng x as th stat varabl, and p as th control varabl. (b) Usng th frst-ordr conons, lmnat th Lagrangan multplr to obtan a lnar dffrntal quaton systm n x( and p(. Construct th phas plan n (x,p) spac, and assss th stablty proprts of th stady stat.