[5] Solving Multiple Linear Equations A system of m linear equations and n unknown variables:
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1 [5] Solving Muliple Liner Equions A syse of liner equions nd n unknown vribles: nn = b = b n n : nn = b n n A= b, where A =, : : : n : : : : n = : n A = = = ( ) where, n j = ( ); = : j j ii i i in j n n b b nd b = : bn, Quesion: Le = (,, ) n be soluion vecor Soluion eiss? Is he soluion unique? When? Why? Liner Models-3
2 Definiion: Vecor Any ri is clled colun vecor Any n ri is clled row vecor Vecors re norlly denoed by lower cses (eg,, y,, b) Noe: When people lk bou vecors, hey re usully colun vecors Definiion: Orhogonliy Two vecors u= ( u,, u ) nd v= ( v,, v ) re sid o be orhogonl iff uv= uv + uv + + u v = 0 EX: = Second enry of vecor - ( ) ( ) u v - Firs enry of vecor Liner Models-3
3 Definiion: Nor (Lengh) of Vecor Nor (lengh) of v = ( v,, v ) v = v + + v EX: v = (, ) v = ( ) = + (by Pyhgors Theore) = + EX: v = (,3, 4) v = = 6 Definiion: Disnce beween Two Vecors For u = ( u,, u ) nd v = ( v,, v ), duv (, ) = u v = ( u v) + ( u v) + + ( u v) Liner Models-33
4 Definiion: -diensionl Eucliden Spce: = {( h, h,, h ) h,, h } Definiion: Liner Cobinion Le b,,,, n Suppose h,,, n b= n n cobinion of,,, n Then, b is sid o be liner Noe: = = A =,,, n = n n : n ( ) b= n n = A Sying h b is liner cobinion of,, n is equivlen o sying h A = b hs soluion Liner Models-34
5 Definiion: Liner Independence The r vecors,, r re sid o be linerly independen iff = = = r = 0 whenever r r = 0 Noe: This ens h none of,, r re liner cobinions of he ohers Suppose 0 (so he condiion for liner independen violed) Then, = r r r = r Theore: Miu Nuber of Linerly Independen Vecors ) The iu nuber of linerly independen vecors in is ) The vecors,,,, re linerly independen, iff ny b is liner cobinion of,,, (For his cse, we sy h,, spn ) Definiion: Rnk of Mri Le A n= (,,, n) Suppose h r ( n) is he iu nuber of he linerly independen coluns of A Then, rnk(a) = r Liner Models-35
6 Liner Models-36
7 Theore: For A n, rnk(a) nd rnk(a) n Proof: I is obvious h rnk( A) re in in n Observe h ll of he coluns in A Bu he iu nuber of linerly independen vecors is Thus, rnk( A) Definiion: Echelon For The echelon for of ri A is obined by pplying eleenry row nd/or colun operions o A o reduce A o ri B = [b ij ] such h b = 0 for ll i > j ij EX: , Theore: The rnk of ri A equls o he nuber of non-zero rows in is echelon for Liner Models-37
8 0 3 4 EX : rnk =, rnk = EX : A = r+ r r r r+ r EX 3: A = r+ r, /8 r3+ r r+ r3, r+ r r r Liner Models-38
9 Theore: ) For ny ri B n, ( ) ( rnk B = rnk B B) = rnk( BB ) ) A squre ri A n nis inverible iff rnk( A) = n Theore: Consider syse of liner siulneous equions, A nn = b, where nd n y be differen ( equions nd n unknowns) Assue h rnk( A) = Then, he followings hold for given b: ) There is les one soluion for ) If rnk( A) = < n, hen, here re infiniely ny soluions for 3) If rnk( A) = = n, hen, here is one unique soluion for Proof: ) Wihou loss of generliy, ssue h,, ( n ) re linerly independen Since,, spn,,, b= Se = + = = n = 0 Then, b = n n Thus, b is liner cobinion of he coluns of A Liner Models-39
10 ),, z z = z + + z Choose n rbirry c nd define + = cz ; ; = cz ; = c; Then, Thus, + + = = n = n n = ( cz) + + ( cz) + c + = c( + z, z ) = b = (,, ) n is lso soluion Since c is n rbirry nuber, here re infiniely ny soluions 3) A is now squre nd inverible Since rnk( A) =, les one soluion eis Suppose = (,, n) nd = (,, n) re wo soluions Then, A b A A = = A = A A = The unique soluion is: A b A = A = A b = A b Noe: If rnk( A) <, he syse y hve no soluion (inconsisency) or infiniely ny or unique Liner Models-40
11 Theore: Consider syse of liner siulneous equions, A nn = b Define A = ( Ab, ) = (,,,, b) Then, he following ( n+ ) n resuls hold: ) If rnk( A) < rnk( A ), hen, here is no soluion for (In his cse, we sy h he syse is inconsisen) If rnk( A) = rnk( A ), here is les one soluion ) If rnk( A) = rnk( A ) < n, hen, here re infiniely ny soluions for 3) If rnk( A) = rnk( A ) = n, hen, here is one unique soluion for Proof: ) Observe h rnk( A ) = rnk( A) ens h b is liner cobinion of he coluns of A Th ens h he syse hs les one soluion liner cobinion rnk A ( ) < rnk( A ) ens h b is no Liner Models-4
12 ) Wihou loss of generliy, ssue h he firs r (r < ) coluns of A,,,, r, re linerly independen Since b is liner cobinion of,,, r, here eis r rel nubers, z, z,, z r such h b= z + z + + zr r In ddiion, here eiss c, c,, cr such h = c + c + + c Se = r + j = zj dcj for j,, r, d is n rbirry rel nuber Then, r+ r r = d nd = r = + = n 0, where r r r+ r+ r+ r+ n n = z + + z + d( c c ) = b r r r+ r r Since d is n rbirry nuber, here re infiniely ny n vecors = (,, ) n h sisfy A= b 3) Since rnk( A ) = rnk( A ), here is les one soluion Observe h A A is n n n ri wih ( ) ( ) rnk A A = rnk A = n Thus, A A is inverible Le soluions Then, = (,, n) nd = (,, n) be wo A = b= A AA= AA ( AA) AA= ( AA) AA = Noe: The unique soluion for 3) is: ( ) ( ) ( A b AA = AA= AA Ab = AA) Ab Liner Models-4
13 EX : Check wheher he following hree vecors re linerly independen: v = (,, ) ; v = (6,4,) ; v = (9,,7) r+ r, r+ r3 7 r+ r rnk 4 = 7 Linerly dependen EX : Check wheher u = (4,,8) is liner cobinion of v nd v r+ r, r+ r3 8 r+ r rnk 4 = 3 8 The hree vecors re linerly independen u cnno be liner cobinion of nd v v Liner Models-43
14 EX 3: v = ; v = Do hey spn? v = v EX 4: How bou v = (,0) nd v = (,)? EX 5: + = + = Is here soluion? = A = b rnk( A) = ; rnk( A ) = rnk = 0 0 = rnk( A) < rnk( A ) No soluion Liner Models-44
15 EX 6: + = b + = b For wh vlues of nd b will he syse hve soluion(s)? b For les one soluion, i should be h hppens only if b = b Wih b = b, rnk A = n Thus, here re infiniely ny soluions b rnk = I b ( ) rnk( A ) = = < EX 7: + 3 = = = = 8 3 Soluion eiss? How ny? = A= b rnk( A) 3< 4= Liner Models-45
16 A = 3 r+ r ( r /) r+ r r r r+ 3 r + rnk( A) = < = 4 r+ r4 Cn show A = Since rnk( A) = rnk( A ) = < 3= n, here re infiniely ny soluions Finding soluions: A iplies h he soluions sisfy: = + = 3 3 Liner Models-46
17 Se 3 = λ (λ ) + 3 = λ + = λ 3 3 λ + 3 λ + 0 = = λ 3 0 λ 3 3 λ + λ λ = 0 = λ 3 λ 3 = 3 λ + λ = 3 + λ for ny λ 3 λ Cn show h hese soluions sisfy ll of he four originl equions Theore: Consider syse of liner equions, An nn = bn (n equions nd n unknowns) Define A Ab n b = (, ) = (,,,, ) Then, he following resuls hold: ) If rnk( A) = n, here is one unique soluion for given b: = A b ) If rnk( A) < n nd rnk( A) = rnk( A ), here re infiniely ny soluions for given b 3) If rnk( A) < rnk( A ), here is no soluion Liner Models-47
18 [6] Liner Econoic Models () A Siple Keynesin Model Assupions: No foreign rde: X (eper) = 0 nd M (ipor) = 0 No Firs invesens (I) nd governen spending (G) re fied (Fro now on, subscrip o ens fied vribles in given syse) The ggrege prive consupion ependiure (C) is liner funcion of ggreged incoe (Y) Model GDP ideniy: Y = C+ I + G Consupion: C = + by, > 0 nd 0< b < o o An econoic odel consiss of wo ypes of equions: definiionl equions which re rue by definiion (eg, GDP ideniy); behviorl equions which describe econoic gens econoic decisions (eg, consupion funcion) Liner Models-48
19 An econoic odel consiss of wo ypes of vribles nd consns: eogenous vribles whose vlues re no deerined by he equions ( I o nd G o ); endogenous vribles whose vlues re deerined by solving he equions (C nd Y ); consns which re fied inerceps or coefficiens The odel cn be wrien: Y C = I + G o by + C = o Y Io + Go b = C A b The ri A nd he vecor b conin consns nd eogenous vribles The vecor conins only endogenous vribles An econoic odel is clled coplee if he nuber of equions in i equls he nuber of endogenous vribles nd hs unique soluion The originl for of n econoic odel is clled srucurl for The soluion of he odel is clled reduced for: Srucurl for: A Reduced for: = b = A b Liner Models-49
20 The soluion vlues of endogenous vribles re clled equilibriu vlues de( A) = = b; b Io + Go de( A) = = + Io + Go; Io + Go de( A ) = = + b( Io + Go) b Y + Io + Go = C b + b( Io + Go) Coprive sic nlysis concerns how equilibriu endogenous vribles rec o chnges in eogenous vribles When G chnges by ΔY ΔG = ΔC b bδg ΔG Y / G = C/ G b b, how uch would Y chnge? Liner Models-50
21 () Anoher Siple Keynesin odel Assupions: No Epor (X) is eogenous Firs invesens (I) nd governen spending (G) re eogenous The ggrege prive consupion ependiure (C) is liner funcion of ggreged incoe (Y) Ipor is lso liner funcion of Y Model Y = C+ I + G + X M o o o C = + by, > 0 nd 0< b < M = c+ dy, c > 0 nd d > 0 The odel cn be wrien: Y C+ M = I + G + X by + C = dy + M = c o o o Y Io + Go + Xo b 0 C = d 0 M c A b Liner Models-5
22 The soluion vlues of endogenous vribles: de( A) = b 0 = b+ d; d 0 3 I + G + X o o o de( A) = 0 = I + G + X + c; c 0 I + G + X o o o de( A ) = b 0 = bc+ d + b( I + G + X ); d c I + G + X o o o o o o o o o de( A ) = b = c+ d bc+ d( I + G + X ) d 0 c o o o Y Io + Go + Xo + c C bc d b( Io Go o) = X b d M + c+ d bc+ d( Io + Go + o) X When G chnges by ΔG, how uch would Y chnge? ΔY ΔG C b G Δ = Δ b d + ΔM dδg Liner Models-5
23 (3) Effec of subsidy on equilibriu price nd quniy Assupion: Perfecly copeiive cell phone rke Eogenous subsidy (S) per cell phone Q = + bp, > 0 nd b < 0, d Q = c+ d( P+ S ), c 0 nd d > 0 Q s d = Q, bc d < 0 s o b Q d = P c ds + o A b de( A) = = b d; d b b de( A ) = = bc ( + dso) d; c+ ds d o de( A ) = = c+ dso c+ ds Q bc d + bdso = P b d c ds + o dδs dδs bδs Δ P = ; Δ ( P+ S) = +Δ S = b d b d b d o ΔQ bdδs = P b d dδs Δ Liner Models-53
24 (4) Wge Gps nd Inernionl Trde (Fro Klein, p 09) Trde wih developing counries will widen he wge gp beween skilled nd unskilled workers? Assupions: Two secors: Teile (T) nd Copuer (C) Two inpus: Skilled (S) nd unskilled workers (U) Technologies: = S / T nd = U / T re fied ST T UT = S / C nd = U / C re fied SC C UC nd > ST < SC UT UC T C The wo oupu rkes re perfecly copeiive in he long run so h long-run profis = 0 The wo lbor rkes re lso perfecly copeiive so h one wge (w S ) for skilled workers nd one wge (w U ) for unskilled workers: ws S T + wu U T = pt T ( ST / T) ws + ( UT / T) wu = p T ws S C + wu u C = pc C ( SC / C) ws + ( UC / C) wu = p C where p nd p re prices T C The oupu prices re eogenous o US econoy (becuse hey re deerined by perfecly copeiive world rkes) Liner Models-54
25 Zero Profis iply ws S T + wu U T = pt T nd ws S C + wu U C = pc C ST w T ST S UT + wu = p T nd T UT SC w C SC S UC + wu = p C C UC ST UT ws pt SC UC w = U p C A de( A) = STUC SCUT ; b pt UT de( A ) = UC pt UT pc p = ; C UC ST pt de( A ) = ST pc SC pt p = SC C ws UCpT w = U de( A) STpC UT SC p p C T If US rde wih developing counries, Δ < 0 (nd Δ 0): pt p C ΔwS U w U de( A) Δ C SC ΔpT Δp T Liner Models-55
26 (5) Ordinry Les Squres (OLS) Wish o eplin i3 = genderi y i = hwgei, using i = yrschooli, i = yrepi,, nd ec (i =,,, ) Regression odel: yi βo βi βi βn i, n = , where u i is n error er h is no reled o,,, For ll people ( > n), y y, n β0 u y, n β u = + : : : : : y, n βn u X β + u n n u i i i n Assue h ll coluns of X re linerly independen [This is clled he ssupion of no perfec ulicollineriy] rnk( X ) = rnk( X X ) = n : Idelly, wish o find β such h y = Xβ nd u = 0 Bu such β generlly does no eis unless ccidenlly he coluns of X spn y: Since rnk( X ) = n <, no ll y (in fc, very few) re liner cobinions of he colun of X Liner Models-56
27 If β eiss, i should be h β = ( XX) Xy β Since XXis n n n ri nd rnk( X X ) ˆ = n, i is inverible Ordinry Les Squres (OLS) esior: ˆ β = ( XX) Xy Vecor of residuls: û = y Xβˆ uˆ I y X( XX) = Xy= ( I X( XX) X) y Ny, which is no zero ri The ri N is syeric nd idepoen ri NX = [ I X ( X X ) X ] X = 0 Xuˆ = XNy = 0 All coluns of X re orhogonl o û Liner Models-57
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