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1 UADPhilEc, Dp. f Ecmics,, Uivsi f Ahss Lcu: Nichlas J. hcaakis Dcmb 2 Ec Advacd Maccmic h I: Mdul : Gwh G ad Ccls Basic wh mah im vaiabls. 2. Disc vaiabls Scks (a a pi f im,.. labu fc) ad Flws ( i a pid f im,.. GDP) im: =,,2,3,.., Disc vaiabl:,, 2 Ciuus vaiablss f ciuus fuci Ra f wh Oh smbls iclud ˆ,. Disc vaiabls,, N ha, Hc,, ,2 3,3 4,4 5,,,,,,,2,,,,2 sic,,,,2 ad,3,3,4 Ava a f wh such ha Hc
2 N ha, 2. Ciuus vaiabls Isad f Δ/ w us h fis divaiv f h fuci f Rcall ha fis divaiv f a fuci f is dfid as d f f f lim d h a f wh f ciuus im vaiabls is dfid as f d f d W us a d v a ciuus im vaiabl d h fis divaiv wih spc (w..) im d f d I ads d h a f wh f a ciuus im vaiabl is dfid as ˆ I ads ha h fllwi ais a quival: d f ˆ d f Laihmic ad xpial fucis Rcall ha h aual laihm f a psiiv umb, l(), is ha umb which wh usd as a xp f h ascdal umb = ivs us. ha is, l hus ( ) xp( ) is a xpial fuci w... his fuci is a mmb f a m al class f xpial fucis f h ( ) d fm A ( ) ( ) h fis divaiv f which is A A ( ) d d I h simpl cas wh ( ) ad A= w hav d ha is, h fis divaiv f h simpl xpial fuci is h fuci islf. I is vid ha all hih divaivs a h sam. 2
3 N ha sic fllws:, h aph f h fuci is as Ecmic siificac f Assum a sum Α ha ilds is wih a auall a. Af a a his will b Α(+ +) Af as i willl b Α(+) Nw assum ha h bak cmpuds h is v qua. Af a a his sum will b Α(+/4) 4 Af as w hav Α(+/4) 4 I al if w cmpud is ims a a: Af a a w will hav Α( (+/) Af as w will hav ΑΑ (+/ ) L us s h impac f cmpudi f A= ad a =% % I h fllwi abl w hav i h fis clum, ims f cmpudi p a ad i h scd clum h amu w will hav a h d f h a f h sum f u wih a aual is a f % 3
4 (+/ ) Wh ds ifii w hav ciuus cmpudi. Af a wih is a % wih ciuus cmpudi will iv us which is qual. his divs fm h v dfiii f which is: lim I is as pv ha wih a aual is a, wih ciuus cmpudi af as will b qual Pf Fis w pv f =: W sk h limi f h squc Dfi m=/. Hc wh ds ifii m m m lim lim m m m m lim m m m 4
5 h w pv f a : Rcall ha i al w hav: lim xm lim xm Hc m m lim lim lim m hus a sum f Α wih ciuus cmpudi wih a aual is a will bcm af as qual Α W ca us his fmula cmpu ps valus. h ps valu f a sum Β a pi wh h us a csa aual is a wih ciuus cmpudi is qual PV B. Cfim ha PV B. h quival ps valu fmula wih aual cmpudi is: B ( ) Nw assum a icm sam () fm im = im Τ. h ps valu f his sam is iv b h fmula PV d h quival fmula f a disc vaiabl wih aual cmpudi is PV ( ) I al, a ciuus im vaiabl () ha ws wih a csa wh a a pi f im i will b qual qual ad iiial valu Cfim ha 5
6 Exampl: Hw l wuld i ak f a cm wih a wh a dubl is GDP? 2 If h GDP is dubld i as af h iiial a i wuld b 2. aki laihms f bh sids w hav l 2 l l 2 l 2 Sic l f a wh a f %, h GDP will dubl i l ha is, i abu 7 as. N h fllwi ppis f h laihms If xl l xl If x l l xl If x l l x d If f xl x dx x d d dx dx x If f x l x xˆ d dx d x d x aki fis laihms ad h divaivs w hav dl dl x dl xl l xl d d d d dx d x ˆ xˆ ˆ d x d d x dl dl x dl x l l xl d d d d dx d x ˆ xˆ ˆ d x d d x dl dl x x l l x d d d dx x ˆ xˆ d x d x Exampl: Cbb-Dulas pduci fuci aki ls w hav h w ak divaivs a Y AK L a ly l A al K a l L 6
7 dly dl K dl L a a d d d Y K L a a Yˆ akˆ alˆ Y K L Rais ad wh as If h ai f w vaiabls x ad is csa h h bh w a h sam a Pf: x ˆ xˆ ˆ If h a is csa h i mais uchad hc is wh a is : ˆ xˆ ˆ xˆ ˆ F xampl, if h GDP p capia mais uchad h h GDP mus w a h sam a as h ppulai Ah xampl K K If k, k A L L ad w kw ha h wh as f k, A ad L a spcivl ˆ k, Aˆ, Lˆ. Wha a h wh as f K ad k? K ˆ k k Kˆ Aˆ Lˆ Kˆ Kˆ AL K k kˆ Kˆ Lˆ kˆ L 7
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