Cheat Sheet on Linearization

Transcription

1 Cheat Sheet on Linearization Econ 504 September 18, 2008 A number of people have asked for help with linearization tricks. This is reallyamathissuethatyoushouldreviewonyourown,butherearesomeuseful hints. 1. Scalar Case Supposethatyouhaveanequationoftheform: Z t =f(x t ) whichyouwanttolinearizearoundapoint( Z, X)suchthat Z=f( X) Thepoint( Z, X)isoftentakentobethenonstochasticsteadystate. Assume that Z t and X t are scalars, and f is a continuously differentiable function. The starting fact is Taylor s theorem, which says that Z t =f(x t )=f( X)+f ( X)(X t X) wherethelastequalityisonlyuptoasecondorderresidual,whichweignorein what follows. If Z and X arebothnonzero,thepreviousexpressioncanbewritten(prove this!) as: z t =η f x t (1) 1

2 wherelowercasevariables denote percentage deviations (i.e. x t =(X t X)/ X) and η f = Xf ( X) f( X) istheelasticity off at X.Inthemodelswehavelookedat,thismeansthatthe percentagedeviationfromsteadystateofafunctionofavariablex t isequalto the elasticity of that function, evaluated at the steady state, times the percentage deviationofx t fromitsownsteadystate. Notethatthisisalmostthedefinition ofelasticityoff (=thepercentagechangeinthevalueofthefunctionduetoa one percent change in its argument.) Power trick. An important special case is the power function: Z t =X ω t forwhichη f =ω,independentlyof X (showthis),andhence z t =ωx t Youmaycallthisthe"powertrick."Whatisimportantisthatthe"trick"is justaspecialcaseofthemoregeneralrule(1). 2. Functions of Many Variables The argument generalizes to functions of more than one variable: if, for instance, Z t =f(x t,y t ) andwewanttoapproximateitsbehavioraroundapointatwhich Z=f( X,Ȳ) youshouldbeabletoshowthat where z t =η x f x t+η y f y t η x f = Xf 1 ( X,Ȳ) f( X,Ȳ),ηy f =Ȳf 2( X,Ȳ) f( X,Ȳ) arethepartial elasticitiesoff withrespecttox andy respectively.

3 then Youshouldnowshowvariousother"tricks"nowfollowfromtheabove: Sumtrick: if Z t =X t +Y t z t = X X+Ȳ x t+ Ȳ X+Ȳ y t Inwords,thepercentagedeviationfromsteadystateofasumistheweighted average of the percentage deviations of the members of the sum. Product trick: if Z t =X t Y t then Ratio trick. z t =x t +y t Z t =X t /Y t ==>z t =x t y t (assuming,ofcourse,thatȳ isnonzero). 3. Composite Functions Finally, suppose that so The Chain Rule gives: Z t =f(y t ),Y t =g(x t ) Z t =f(g(x t )) h(x t ) h (X t )=f (g(x t ))g (X t )=f (Y t )g (X t ) sothetaylorapproximationaroundapoint( Z, X,Ȳ)suchthat Z=f(Ȳ),Ȳ = g( X)is Z t = Z+h ( X)(X t X t )= Z+f (Ȳ)g ( X)(X t X) You should show the composite function trick: z t =η y f ηx gx t where η y f (Ȳ) =Ȳf f(ȳ),ηx g = Xg ( X) g( X)

4 aretheelasticitiesoff andgatthesteadystate. Note that this tells that z t =η y f (%deviationofy tfromitssteadystate) (2) As an example, let us linearize Y t =(ax θ t +(1 a)zθ t )γ/θ from Farmer s book, p. 39. Using lowercase letters for percentage deviations, y t =(γ/θ) thepercentagedeviation(p.d.)ofax θ t +(1 a)z θ t using the power trick; but: p.d.ofax θ t +(1 a)z θ t = using the sum trick; finally, a X θ a X θ +(1 a) Z (p.d.of θ axθ t) (1 a) Z θ + a X θ +(1 a) Z (p.d.of θ (1 a)zθ t ) p.d.of ax θ t =θx t p.d.of (1 a)z θ t =θz t using power trick again. Collecting all this: a X θ (1 a) Z θ y t = (γ/θ)[ a X θ +(1 a) Z θθx t+ a X θ +(1 a) Z θθz t] = γ[a X θ x t +(1 a) Z θ z t ]/Ȳθ/γ Thelastlineissimplyarearrangement. Notealsothatwehaveused(2)ateach step. 4. Loglinearization vs percentage deviations Consider the Taylor expansion of Z t =logx t =f(x t )

5 around Z=log( X), logx t =f( X)+f ( X)(X t X)=log X+x t wherex t =(X t X)/ X,asbefore. So,uptofirstorder, x t =logx t log X (3) In words, percentage deviations from steady state are the same as log-deviations (differences in logs), up to first order. For most practical purposes in our course, therefore, there is no difference between the two. (Often you will see the term "log linearizing" instead of linear approximation in percentage terms.) This equivalence(again, up to first order) may help linearizing quickly some expressions that become linear in logs. For example, if the production function is Cobb Douglas, Y t =A t K α t L1 α t taking logs you get logy t =loga t +αlogk t +(1 α)logl t Thesteadystateversionofthisis logȳ =logā+αlog K+(1 α)log L Substract the two and using(3), y t =a t +αk t +(1 α)l t 5. Caveats These derivations become generally invalid if some variable is zero in the nonstochastic steady state, since then you will have to make sure that you have not divided by zero in taking percentage approximations, etc. Also, the derivations are only accurate up to first order, as I have emphazised. For some questions(e.g. welfare evaluation) a first order approximationisnotgoodenough, andmoreaccuratemethodsneedtobeused. Inourcourse,however,thiswillmostprobablynotbeaconcern.