2. The basics of differential equations 1 AGEC 642 Fall 2018

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1 This documen was generaed a :30 PM, 0/9/8 Copyrigh 08 Richard T Woodward The basics of differenial equaions AGEC 64 Fall 08 I Wha is a differenial equaion? A differenial equaion is an equaion ha involves a derivaive of a funcion In our applicaions, ypically he equaion will define a funcion ha is equal o he derivaive wih respec o ime, eg, ( ) = f (, z, ) The LHS of his equaion will frequenly be wrien ( ), or jus Noe ha differenial equaions are used when ime is measured coninuously The erm difference equaion is used for he discree-ime analog Par of he difficuly (ie, hassle) of opimal conrol is ha he firs order condiions yield differenial equaions, which we have o inegrae o obain a closed form soluion Hence, o solve opimal conrol problems we have o undersand differenial equaions and be able o solve hem Noaion: We will frequenly wrie (), or jus, all meaning he same hing Mosly we'll use for compleeness and concision The correc meaning should be undersandable in cone; if no, ask Eample : Suppose is he disance raveled up o ime The rae of change in disance wih respec o ime is, and he unis of his measuremens are disance per uni of ime, such as miles per hour or cenimeers per second If you can choose your speed a any insan, hen your speed is a choice variable, which we will refer o as z Hence, he differenial equaion describing his relaionship is = z and he disance you ve raveled k afer k hours is k 0 = zd If your speed is consan, ie z=z, hen we can easily solve 0 k he differenial equaion k = zd + 0 = k z + 0 If 0 =0, hen k=k z So if k=05 hours 0 and z=50 miles per hour, hen k =5 miles Anoher opion would be o ravel 50 miles per hour for 5 minues and 70 miles per hour for 5 minues, which would give us = 50d + 70d = I is imporan o recognize ha when inegraing over ime, he inegrand (wha is inside he inegral, z in he case above) is always a rae per uni of ime as defined by how we measure So, for eample, if we changed our unis from miles per hour o miles per minue, hen he value of z would change from, say z o z'=z/60, and we would inegrae no from 0 o k bu from 0 o 60 k If insead of puing a speed in he inegral we These noes are based primarily on chaper of Léonard and Van Long

2 - pu a uiliy funcion, hen he correc inerpreaion of he uiliy funcion is also a rae a which uiliy is being creaed, ie, uiliy enjoyed per uni of ime Eample : Suppose an invesmen of 0 dollars a ime =0 is pu in an accoun so ha he r balance grows coninuously a he rae of ineres r, ie, = 0e In his case he differenial equaion is r 0e r = = r0e = r or d = r Ofen, however, we are given he differenial equaion iself, = r, which we need o solve An easy way o inegrae his epression is o use he fac ha we can wrie his as = r We wan o solve for as a funcion of oher suff, so we sar by inegraing boh sides, d = rd () We know ha ( ) ln = = ɺ, so he LHS can be easily inegraed d = ln ( ) plus a consan of inegraion The RHS is jus r So () can be rewrien ln = r + K, where K is he consan of inegraion Taking he ep of boh sides, we obain r + K K r r K = e = e e = Ae, where A = e has an unknown value ( ) If we know he value of a some poin in ime, eg if 0 = (ie we have dollars in he bank a ime =0) hen, subsiuing for =0, we can solve for A, r 0 = Ae = A Hence, he specific soluion is r = e This eample demonsraes he sandard process of solving differenial equaions: firs find he general soluion, hen use prior informaion abou value(s) a poin in ime o ge he specific soluion (Wha would he answer look like if insead of knowing, 0 we knew ha s = K for s>0?) ɺ An ineresing side noe: The psychological lieraure has found evidence ha people do no seek o maimize he inegral of uiliy over ime The bes-selling book Thinking Fas and Slow by Daniel Kahneman (Nobel Prize in Economics) alks a lo abou how people seem o pay more aenion he peak and end poins in an eperience If his is righ, hen he discouned uiliy model of economiss may be a very poor descripive model of behavior Noneheless, i is wha economiss normally use and here are good economic and heoreical reasons why discouned uiliy is a raional objecive

3 - 3 II Normal differenial equaions The equaion = r is a firs-order differenial equaion (FODE), firs order in he sense ha is he firs derivaive of wih respec o In general, an ordinary m h order differenial equaion is an equaion of he form m ( m) ( m ) ( m ) ( ) = g(,,,,,, ) m For eample, a FODE would have on he LHS and and on he RHS A second order differenial equaion would have ɺ = on he LHS could have and, and on he RHS If you have he m h order equaion, in principle you can ge back o an equaion for as a funcion of by inegraing m imes Solving he m h order differenial equaion will lead o an m h order differenial equaion, and solving ha will lead o an m h order differenial equaion, and so on However, each ime you inegrae, you end up wih a new consan of inegraion like K above Hence, o reach a paricular soluion o an m h order equaions you will need m boundary condiions (ie he eac value of,, ɺɺ, or some oher ( m ) derivaive up o a some Despie heir name, hese values do no need o be known a any boundary; he value a any poin in ime will do III The unis and meaning of a differenial equaion A he risk of being redundan, le s look again a he real-world meaning of a differenial equaions If is a sae variable, hen i mus be measurable For eample, migh be ons, gallons, dollars, miles raveled, ec Similarly, he unis for he ime sep,, mus also be clearly defined, i could be seconds, hours, ec For eample, if you measure ime in hours, and is a disance measured in miles, hen he unis for =ɺ are miles per hour Noe ha his is no he number of miles ha are raveled in one hour, because he vehicle s speed could change coninuously over he hour Raher, i is an insananeous measure of speed, he disance ha he vehicle would ravel in one hour if he speed were held consan for ha hour The inegral 0 would be he number of miles acually raveled in one hour There is also inuiive meaning in he second derivaive, = = ɺɺ ; his is he rae of change in, ie he rae of acceleraion Noe ha if we measured he ime sep is seconds or years he value of would change accordingly, even hough he speed of he vehicle has no changed I is imporan o have a clear undersanding of wha, and ɺɺ mean inuiively If is a posiive number (size of your invenory for eample) and >0, hen your invenory is growing and if <0, hen your invenory is falling If >0 bu ɺɺ < 0, hen he invenory is sill growing, bu he speed a which i is growing is slowing down If <0 bu ɺɺ > 0, hen he invenory is falling, bu he speed a which i is falling is slowing down The figure below should be helpful in hinking hrough hese meanings

4 - 4 IV Equilibrium An equilibrium in a dynamic sysem is a poin a which all he variables do no change over ime (Sudens ofen confuse equilibrium wih opimum; be sure you undersand he difference) = g and g( ) = 0 hen is an equilibrium value of If ( ) Wha's he equilibrium for he FODE ɺ = a + b? Wha's he equilibrium for he FODE ɺ = r? Wha's he equilibrium for he FODE ɺ = a + b + c? Wha's he equilibrium if = 3 and = 4 +? V Linear firs-order differenial equaions (FODE) Linear firs-order differenial equaions are he simples form of differenial equaions, and he one we will be using mos ofen A linear FODE is an equaion of he form ɺ = a + b I is insrucive o walk hrough one way o solve such equaions Firs we muliply boh sides by e -a and reorganize: a a a e e a = e b The LHS of his equaion is he ime-derivaive of e -a (using he produc rule) The RHS can easily be inegraed, e b d = e C a + Hence, inegraing he LHS and he RHS we obain a a a e = e b + C, or, canceling e, a a a b

5 - 5 b a = + Ce () a I is always a good idea o check your inegraion In his case, aking he derivaive of () a wih respec o we obain = = ace a b Bu, rewriing () we know ha Ce = +, so we can wrie a b = a + a, or = a + b Anoher relaively simple FODE: Suppose we have a FODE ha can be wrien in he form = h g ( ) ( ) ɺ If we le f ( ) = h( ), hen our FODE can be rewrien f ( ) = g ( ) Then boh sides can be inegraed wr o obain f ( ) d = g ( ) d ( ) = ( ) f d g d If you have o deal wih more complicaed differenial equaions here are a number of good compuer programs ha can help Given he sophisicaed sofware available oday (eg, Malab, Maple, & Mahemaica), solving complicaed differenial equaions enirely by hand is almos like doing OLS wih a calculaor We will go over he use of such sofware in he compuer lab (and see he Malab uorial ha accompanies hese noes) VI Auonomous ODEs A differenial equaion is said o be auonomous if i does no depend on More formally, according o Weissein s MahWorld, For an auonomous ODE, he soluion is independen of he ime a which he iniial condiions are applied In economics, we frequenly seek o specify our problems o be auonomous since we ypically feel ha economic changes are a funcion of he sae of he sysem, he choices made, and random shocks, no he calendar dae For eample, he differenial equaion = a + b is no auonomous, since he rae of change in depends no only on he value of bu he ime, On he oher hand, he funcion = y + b is auonomous, a leas as long as y is no a funcion of ime VII Sysems of differenial equaions and phase diagrams Frequenly in OC we have o deal wih more han one differenial equaion a a ime In he simples OC problems, for eample, we have a differenial equaion for he sae variable,, and anoher for he co-sae variable, λ ɺ In oher cases we migh have wo

6 - 6 sae variables, eg, wo inerdependen fish socks or he marke shares of wo compeing firms Wihou solving eplicily for he enire ime pah of he wo variables, we can learn quie a lo abou he naure of a wo-variable sysem using wha is called a phase diagram A phase diagram presens he equilibria, sabiliy and dynamic evoluion of a sysem Phase diagrams are appropriae only if you have wo auonomous differenial equaions An eample of a phase diagram is shown below We will discuss he seps o developing a phase diagram oward he end of hese noes The ype of sysem porrayed here is known as a saddle poin or saddle pah, and is frequenly encounered in economic models The solid lines are called isoclines, indicaing ha along hese lines here is no direc pressure on one of he variables o change, i = 0 The equilibrium occurs where he isoclines cross where boh variables do no change The dashed lines heading oward he equilibrium are called separarices since hey separae he space in ha no rajecory ever crosses hese lines; if a pah reaches a separari i never leaves i The doed lines porray represenaive rajecories no on he separarices Noe ha when a rajecory crosses an isocline is slope is consisen wih he isocline For eample, he boom righ rajecory is horizonal a he poin where i crosses he ɺ = 0 isocline In a saddle poin sysem, only poins on he separarices will lead o

7 - 7 he equilibrium; if a saring poin is no on one of hese lines i will permanenly diverge from he equilibrium VIII Homogeneous and non-homogeneous sysems Consider firs a sysem of linear differenial equaions a a b = a a b ɺ or, using mari noaion, 3 = A b Using a phase diagram, he equilibrium of his sysem could easily be idenified as he poin where = 0 or A = b We can, herefore, find he equilibrium values,, by inversion, = A b I is also frequenly ineresing o know how variables behave around he equilibrium For eample, do and end oward he equilibrium, or away from i? I urns ou ha ecep in a special case (see L&VL p 0), he dynamics of he sysem in 3 will be idenical o he dynamics of he relaed homogeneous sysem in which he b is dropped: 4 = A, wih he equilibrium simply relocaed from A - b o he origin IX Analysis of he naure of he equilibria in sysems of differenial equaions The naure of he equilibrium of a sysem of differenial equaion can be deermined by looking a he Eigen values of he sysem The firs sep in idenifying he Eigen values is o guess ha he soluion o he homogenous differenial equaion sysem akes a form analogous o he scalar case discussed above 3 Tha is, we could guess ha he soluion will look somehing like 5 = a e λ where a is a vecor of consans, no all zero Taking he ime derivaive of his funcion we obain, 6 = λa e λ So, seing he RHSs of 4 and 6 equal, we ge λ λ λa e = A = Aae Canceling e λ, we ge λa=aa or [A λi]a=0 or a λ a a 0 a a λ = a 3 This secion, like almos all of his lecure, is based very closely on chaper of Leonard & Van Long A suden in he pas has quesioned his secion If you oo quesion his, I would welcome a clarificaion and/or correcion of he derivaion

8 - 8 For nonrivial soluions, ie, a 0, his requires ha [A λi] be singular, ie, A λi =0 A value λ ha saisfies his is called an Eigen value or a characerisic roo For a mari, solving he equaion where A λi is equal o a λ a λ a a = This is a nonlinear equaion, quadraic if λ =λ =λ For any ( )( ) 0 real mari A, he Eigen values form par of a mari B such ha here eiss a real mari T such ha T AT=B B can ake one of four forms, λ 0 λ 0 ( a) B =, ( b) 0 λ B = 0 λ λ 0 α β ( c) B =, ( d ) = λ B β α where λ and λ are disinc real roos, λ is a double roo and α±iβ are conjugae comple roos where i = Depending on he roos, he sabiliy of he sysem falls ino si caegories (see Léonard and van Long p 98) and hese deermine wheher he sysem is sable (converging owards he equilibrium) or unsable Some inuiion abou sabiliy of he homogenous sysem can be found by looking a 5 If λ is negaive, hen as increases will approach zero, he equilibrium of he homogeneous sysem If λ is posiive, hen will grow as increases If λ=[λ, λ ], and one is greaer han zero and he oher is less han zero, hings are likely o be more complicaed You can calculae hese Eigen values by hand by solving he equaion A λi =0, or you can use a sofware package o solve for he Eigen values The following sequence of Malab commands will calculae Eigen values of a mari EDU>> syms A B a b c d a b EDU>> A=[a,b;c,d] A = c d EDU>> B=eig(A) Wih he resul being B = [ /*a+/*d+/*(a^-*a*d+d^+4*b*c)^(/)] [ /*a+/*d-/*(a^-*a*d+d^+4*b*c)^(/)] As noed by L&VL (p 00): i Such a sysem has a sable equilibrium if and only if is characerisic roos have negaive real pars ii A saddle poin occurs if and only if he deerminan of A is negaive iii A sufficien condiion for insabiliy is ha he race of A>0 For all bu case d above, he deerminan of A, A =λ λ and r A=λ +λ, so condiions ii and iii can be evaluaed wih he roos, or wih he original A mari

9 - 9 Nonlinear sysems Of course, he above analysis is only direcly relevan o linear sysems However, i can be shown ha if a linear approimaion of he sysem is sable (unsable), hen he rue sysem is also sable (unsable) in he neighborhood of he equilibrium We presen a nonlinear eample below A sep-by-sep approach o analyzing sysems of differenial equaions Here are seps ha I use o analyze he dynamics of a sysem of wo differenial equaions There are a variey of approaches o drawing phase diagrams; his is a way ha I find quie inuiive and helps me avoid careless misakes Find a reduced form for he epressions and in erms of only and and eogenous parameers All oher variables mus be eliminaed from he equaions or assumed o be consan Solve for he inequaliies 0 and 0 This should leave you wih wo inequaliies in erms of and ha, if saisfied, mean ha and 0 3 Find he equilibria: he values of and such ha = = Graph he isoclines, ie he funcions = and 0 in he (, ) plane 0 = 5 Using he inequaliies found in, deermine he rajecories for and on eiher side of he isoclines Tha is, on which side of he isoclines is each variable is increasing ( > 0 and > 0 ) and where are hey decreasing ( < 0 and < 0 ) Hin: i is easies if you carry ou seps 4 and 5 separaely for each isocline firs before puing he wo ogeher 6 Take a linear approimaion of he sysem s dynamics in he neighborhood of each equilibrium and epress i as a mari of he form = A 7 Check o see if any of he hree condiions from L&VL p 00 (i iii above) are saisfied Then, if necessary, find he Eigen values of his linear sysem of equaions and, following Léonard and Van Long p 98, evaluae he sysem s sabiliy Eample Consider he following eample from Léonard and Van Long (p 0): is capial sock and is he sock of polluion Capial growh is assumed o be a consan fracion, s, of oupu, α wih α<, and depreciaes a he rae δ, so ha he rae of change in capial can be wrien α = s δ The sock of polluion,, grows as a funcion of capial β (β>) bu decays a he rae β γ<, = γ A fool-proof approach o creaing phase diagrams α Sep : Auomaic: = s δ and = β γ

10 - 0 Sep : Solve for 0 and 0 and idenify associaed spaces in he phase diagram 0 0 s δ 0 α s α α δ δ s ( ) ( α δ s ) γ 0 β γ β β Noe: since >0 by assumpion, he inequaliy does no flip when dividing by a sep 3, while since α <0, he inequaliy flips from 3 o 4 Sep 3: Idenify he equilibrium, and = β γ α α δ δ = 0 s δ = 0 = = s s ( ) s β α = γ Subsiuing in he value for yields ( δ ) ɺ ɺ ( ) ( α ) Sep 4: Graph he isoclines, = 0 and = 0 γ α = δ s and = β γ (see below) Sep 5: Idenify he regions where and are increasing and decreasing using he resuls from he firs sep: ( ) ( α 0 δ s ) ɺ and ɺ 0 β γ This means ha is increasing o he lef of is isocline, and is increasing below is isocline

11 - Puing he wo ogeher yields Sep 6: Find a linear approimaion of he dynamics of he sysem Le ˆ and ˆ be he equilibria idenified above In he neigborhood around his poin α ( )( ) β ( ) ( ) sˆ ˆ sˆ ˆ α δ + α δ ˆ ˆ ˆ ˆ ˆ β γ + β γ We know ha in mos cases in he neighborhood of he equilibrium, he dynamics will be he same as ha of he homogeneous sysem of equaions α ( α ˆ δ )( ˆ ) β β ˆ ( ˆ ) γ ( ˆ ) = s = α ɺ ( αsˆ δ ) or 0 ˆ = β ˆ β ˆ γ ɺ Sep 7: Solving for he Eigen values, yields λ = γ, and λ α sˆ α = δ, which a he equilibrium value of ( ) ( α = δ s ), simplifies o λ δ ( α ) ( ) = These are boh negaive, implying ha we fall in case b, (wih opposie arrows) so ha i is globally sable in he neighborhood he equilibrium Wha would happen if producion were adversely affeced by polluion, ie if oupu ook he form α /τ? Separarices As noed above, in some models here eiss an imporan line called a separari These are imporan economically for hey can help us undersand how sae variables will change over ime as hey approach an equilibrium or oherwise change over ime Karp (lecure

12 - noes) defines a separari as a line in he phase space ha rajecories never cross The reason his happens is ha he slope of a separari is he same as he slope of he rajecory Tha is, along he separari, in he, plane he separari is he se of poins along which = How do we find he separari? Recalling ha he homogeneous sysem is se so ha he equilibrium is a he origin, his means ha we're looking for a funcion of he form =K ɺ so ha / =K We hen solve he equaion = K by simply plugging in he wo sae equaions and solving In linear sysems here are wo separarices In nonlinear equaions he same basic principle would hold, bu he equaion would be nonlinear (and no doub more difficul) X Reading for ne lecure: Léonard & Van Long pp 7-5 XI References Weissein, Eric W Auonomous From MahWorld A Wolfram Web Resource hp://mahworldwolframcom/auonomoushml [Originally accessed May 7, 004]