Eonomis 205A Fll 2008 K Kletzer A Primer on Continuous-time Eonomi Dnmis A Liner Differentil Eqution Sstems (i) Simplest se We egin with the simple liner first-orer ifferentil eqution The generl solution is ẋ (0) 0 0 e t n the initil onition is stisfie the prtiulr solution The growth rte of is given n 0 e t Net we solve the two-imensionl sstem given ẋ (0) 0 ẏ (0) 0 It is useful to write these in mtri form s ẋẏ 0 (0) 0 (0) 0 0 The solution for this two-imensionl sstem is simpl 0 e t 0 e t Eh of these prolems the one-imensionl n the two-imensionl re emples of initilvlue prolems This sstem possesses single ste stte 0 0n 0 0 A phse igrm plots out s funtion of for ifferent possile initil vlues of n The resulting lous relting to t eh time given n prtiulr initil vlues 0 n 0 epits trjetor long whih n move over time Arrows re usull rwn to epit the iretion of motion in the -plne s t inreses Consier three ses In the first the prmeters n re othpositive For ninitilpoint
( 0 0 ) tht is not the ste stte trjetor moves w from the origin This is the unstle se In the seon n re oth negtive n n trjetor onverges to the ste stte This is the stle se In this se trjetories smptotill pproh the ste stte prllel to the -is if <<0 n onversel In the thir se one root or is positive n the other is negtive For the emple in whih >0 > n trjetor suh tht 0 0onverges to the ste stte t the rte An trjetor suh tht 0 0moves w from the ste stte n onverges towrs the -is smptotill This is the sle-pth stle se Consier n emple of sle-pth nmis suh tht n The solution is given 0 et 0 e t As t grows without oun pprohes zero s grows without oun s 0e t (ii) Generl se Let z e n n-imensionl olumn vetor over the rels n ż Az where z (0) z 0 The mtri A isn n with onstnt rel oeffiients Impose the restrition tht A is nonsingulr For emple onsier the two-imensionl ifferentil eqution sstem ẋẏ (0) (0) We hnle this prolem hnging oorintes We wnt to fin hnge of sis from new sis suh tht the initil-vlue prolem eomes ŷ ẋ t â 0 0 whih hs the solution given in prt (i) ŷ ŷ ât 0 e t ŷ 0 e (0) ŷ (0) 0 0 0 This is one fining nonsingulr 2 2 mti M with onstnt oeffiients suh tht â 0 M n M M We n lws o this given the ssumptions me so fr The ifferentil eqution sstems re ientil ut epresse in ifferent oorintes ŷ 0 ŷ 0 2 to
3 The solution for the initil-vlue prolem ẋẏ (0) (0) 0 0 is given M ât 0 e So the onl issue left is fining M Strt oserving tht M les to where t ŷ 0 e â 0 M 0 This n e rewritten s two equtions m m 2 M where M â âm âm 2 m m 2 m 2 m 22 m m 2 â 0 ŷ 0 M m2 m22 0 0 M M m m 2 0 0 â 0 0 n m 2 m 22 m 2 m 22 These impl tht in the generl se we seek solutions to 0 0 m 2 m 22 A λi ν 0 where I is the ientit mtri of sme imension s A λ is slr n ν is olumn vetor For A nonsingulr this onl hs solutions suh tht et A λi 0 whih is n th orer polnomil You just solve for the n vlues of λ λ λ n n then sustitute to fin the orresponing n olumn vetors ν j M is the mtri me up of the n olumn vetors The roots λ λ n n e omple numers If root is omple then its omple onjugte is lso root (omple roots re isusse in susetion (iii)) For the two-imensionl se we hve tht â λ λ 2 ν The generl solutions for n re given α ν e λ t + α 2 ν 2 e λ2t m m 2 n ν 2 m 2 m 22
with the strighforwr etension to the n imensionl se Here α n α 2 re solutions to (0) (0) α ν + α 2 ν 2 We will use the onventionl terminolog: λ is lle n eigenvlue n ν is lle n eigenvetor et A λi 0is lle the hrteristi polnomil An emple: let 6 A 2 The eigenvlues n ssoite eigenvetors re λ 4 λ 2 ν ν 2 Note tht n non-zero multiple of these eigenvetors is lso n eigenvetor The generl solutions for n re α 2 2 e 4t + α2 3 e t The initil onitions re stisfie hoies of α n α 2 tht stisf + α2 whih implies (iii) Comple roots α 2 α 2 0 6 0 5 5 3 0 0 n α 2 3 0 5 + 6 0 5 The roots of the hrteristi polnomil for the n-imensionl liner initil-vlue prolem n e omple numers For emple onsier the sstem whih hs the hrteristi polnomil with the two roots ẋ ẏ 0 0 λ 2 +0 λ i n λ 2 i The eigenvetors ssoite with eh of these eigenvlues re i ν n ν 2 The generl solution is given the rel prt of α e it + α2 i i i e it 3 4
5 where e iθt os θt + i sin θt Sustituting we hve the generl solution α sin t os t + α2 os t sin t n the prtiulr solution sin t os t 0 + 0 os t sin t whih ou n hek setting t to zero in the generl solution For this emple ll trjetories re ounterlokwise irles roun the origin prts In the generl se in two imensions we might hve roots tht re omple with non-zero rel The generl solution will e the rel prt of whih equls λ γ + θi n λ 2 γ θi α ν e (γ+θi)t + α 2 ν 2 e (γ θi)t e γt α ν e θit + α 2 ν 2 e θit The prt insie the squre rkets is simpl irulr motion (lokwise or ounterlokwise epening on the oeffiients of the mtri A) The eponentil in front epns or ontrts these les epening on whether γ is positive or negtive B Non-liner nmil sstems In mn eonomis pplitions the nmis will not e represente liner ifferentil equtions In tpil ses we n onl prtill hrterize the nmis of our moel lthough in some speil instnes ou n integrte the sstem An emple of non-liner moel is the stnr one-setor optiml growth moel with iminishing returns to pitl The nmis re given k f (k) nk n ċ σ () (f (k) n ρ)
for σ () u () The initil onitions re etermine k u () 0 n the trnsverslit onition 6 lim u ( t e ρt t ) k t 0 We nnot integrte the eqution for ċ unless oth the intertemporl elstiit of sustitution n mrginl proutivit of pitl re onstnt One pproh to hrterizing the nmis of pitl umultion is to emine these ner ste stte We impose the In onitions so tht ste stte with positive k n eists First the sstem n e pproimte ner the ste stte tking first-orer Tlor series epnsion out the ste stte: n ċ f (k) σ ( ) k f (k) n(k k ) ( ) (k k )+(f (k ) n ρ) σ ( ) σ ( ) σ 2 ( ) ( ) We will el with the prtiulr solution tht stisfies the initil n trnsverslit onitions fter we hrterize the generl solutions to the two-imensionl linerize ifferentil eqution sstem n Notie tht this emple simplifies to k ρ (k k ) ( ) ċ f (k ) σ ( ) (k k ) when we impose the ste-stte onitions ċ 0n k 0 In mtri form the pproimte sstem is ċ k 0 f (k ) ρ σ( ) ( ) (k k ) where the oeffiients in the mtri A re ll onstnt The hrteristi polnomil is whih hs solutions In this se λ ± 2 λ (ρ λ) + f (k ) ( ρ ± σ ( ) 0 ρ 2 4 f (k ) σ ( ) λ+ >ρ n λ < 0 n the linerize sstem is sle-pth stle in neighorhoo of the ste stte ( k )There is n eigenvetor ssoite with eh of these eigenvlues The one ssoite with the negtive )
7 eigenvlue will give us trjetor for ( k k ) tht onverges to the ste stte This trjetor will e the onl one tht stisfies the trnsverslit onition n the itionl fesiilit onition tht k t 0 for ll t Wht is the slope of this eigenvetor? Our hnge of oorintes ove le us to the eqution Aν j λ j ν j 0 for eh eigenvlue n ssoite eigenvetor So we hve the equtions n where f (k ) ν 2 λ ν σ ( ) ν + ρν 2 λ ν 2 ν For λ λ these impl tht the slope of the stle sle pth ν /ν 2 ρ λ whihis positive n steeper thn the slope of the k 0lous t the ste stte in the phse igrm The slope of the unstle sle pth ρ λ + is negtive euse λ + >ρ It is importnt to know tht the solution to the linerize ifferentil eqution sstem pproimtes the nmis of the originl non-liner sstem ner the ste stte A si theorem in nmil sstems proves tht in neighorhoo of ste stte the trjetories for the generl solution of the lineriztion of the ifferentil eqution sstem out the ste stte re ritrril lose (in the epsilon-elt sense) to the trjetories of the non-liner sstem uner generl onitions For ontinuousl ifferentile funtions u () n f (k) the trjetories of the linerize n non-liner sstem re tngent to eh other t the ste stte This mens tht if ou fin the slope of the stle sle pth for the liner pproimtion (the slope of the stle root eigenvetor) ou hve foun the slope of the sle pth for the non-liner sstem t the ste stte It lso mens tht the growth rtes of ( ) n (k k ) re pproimtel equl to λ long the stle sle pth in neighorhoo of the ste stte The most importnt onition for this theorem to hol is tht the eigenvlues of the linerize sstem ll hve non-zero rel prts If some of the eigenvlues re omple with zero rel prt then the linerize sstem is not n pproimte sstem n its nmis m hve little in ommon with the nmis of the originl non-liner sstem This is lso n importnt theorem in nmil ν ν 2
8 sstems In generl non-liner sstem n hve mn ste sttes When ou n pproimte its nmis lineriztion of the ifferentil eqution sstem (the ommon se) ou will e oing so roun prtiulr ste stte The liner sstem will hve unique ste stte so ou will hve ifferent liner pproimtion for eh ste stte when there is more thn one When the linerize sstem is stle we ll the originl sstem loll stle When it is unstle we ll the non-liner sstem loll unstle n so forth for lol sle-pth stilit Sometimes we n s something or even lot out the nmis of non-liner sstem w from the ste stte ut we nnot o so using lol pproimtions