Lagrange Maxima http://stavrakoudis.econ.uoi.gr18 20131 / 31 1. TheBeatlesCauseIdontcaretoomuchformoneyFormoneycantbuymelove2. Dunn,Gilbert&WilsonIfmoneydoesntmakeyouhappy,thenyouprobablyarentspendingitright3. MargaretThatcherNo-onewouldremembertheGoodSamaritanifhedonlyhadgoodintentions;hehadmoneyaswell.2 / 31 EconomicsofHappinesshttp://www.youtube.com/watch?v=YCV8IPlP-GEhttp://www.theeconomicsofhappiness.orgHappiness: ARevolutioninEconomicsBrunoS.Frey,AloisStutzer,MatthiasBenzhttp://books.google.gr/books?id=0tqGSAAACAAJHappinessEconomicsShariLape nahttp://books.google.gr/books?id=X7pnFd_uVPsC , ;3 / 31 U (x, y) = x1/2y1/24 / 31 U (x, y) = x1/2y1/25 / 31 U (x, y) = x1/2y1/2I = x + y= 86 / 31Utility-02.wxm U (x, y) = x1/3y2/37 / 31 . :U= x1/3y2/3= 2 I = x + 4 y= 12 y:y=232x y= x 124 :232x= x 124 x3/212 x1/2+ 272= 0 ;8 / 31 ndroot solve . , . . ndroot :ndroot(eq,x,min,max) eq x (min,max)1 eq : x^(3/2) - 12*x^(1/2) + 2^(7/2) = 0;2 find_root (eq, x, 1, 2);9 / 31Utility-03.wxm CobbDouglas:U(x, y) = 5 x3/5y2/5 , :Px= 5 , Py= 2 I =25x, y :I = Px x + Py y x, y U, Px x + Py y I ;10 / 31 1 2 3 y x;x y; x y;11 / 31 Lagrange :f (x, y) :g(x, y) = c :L(x, y, ) = f (x, y) + (g(x, y) c) :Lx= 0Ly= 0L= 012 / 31 Lagrange :U(x, y) = 5 x3/5y2/5 :5 x + 2 y= 25 :L(x, y, ) = 5 x3/5y2/5+ (2 x + 3 y+ 2 z 120) :Lx= 0Ly= 0L= 013 / 31 Lagrange1 2 3 Lagrange x, y, 4 3 Lagrange x, y, 5 3 3 x

, y

U, , .14 / 31Lagrange 1 .U(x, y) = 5 x3/5y2/51 U(x,y) := 5 * x^(3/5) * y^(2/5);15 / 31Lagrange 2 .Px= 5Py= 2I = 251 Px : 5;2 Py : 2;3 I : 25;16 / 31Lagrange 3 Lagrange:L = U(x, y) + C(x, y)1 L(x,y,lambda) := (U(x,y) + lambda*(I - Px*x - Py*y));17 / 31Lagrange 4 :Lx= 3 x25y25 5 = 0Ly= 2 x35y35 5 = 0L= 5 x 2 y+ 25 = 01 eq1 : diff(L(x,y,lambda), x) = 0;2 eq2 : diff(L(x,y,lambda), y) = 0;3 eq3 : diff(L(x,y,lambda), lambda) = 0;18 / 31Lagrange 5 :1 sol : solve([eq1, eq2, eq3], [x,y,lambda]);2 xmax : rhs(sol[1][1]);3 ymax : rhs(sol[1][2]);4 lmax : rhs(sol[1][3]); :xmax= 3ymax= 5max=

3535 0.73619 / 31Lagrange Maxima :1 U(x,y) := 5 * x^(3/5) * y^(2/5);2 Px : 5;3 Py : 2;4 I : 25;5 L(x,y,lambda) := (U(x,y) + lambda*(I - Px*x - Py*y));6 eq1 : diff(L(x,y,lambda), x) = 0;7 eq2 : diff(L(x,y,lambda), y) = 0;8 eq3 : diff(L(x,y,lambda), lambda) = 0;9 sol : solve([eq1, eq2, eq3], [x,y,lambda]);10 xmax : rhs(sol[1][1]);11 ymax : rhs(sol[1][2]);12 lmax : rhs(sol[1][3]);20 / 31Utility-04.wxm Lagrange .LI= = 0.736, 1 0.736 .21 / 31 Cobb-Douglas CobbDouglas :U(x, y) = Axay1a(1)A > 0 0 < a < 1. PxPy xy I , :I = Px x + Py y (2) :xmax= aIPxymax= (1 a)IPy : =(1 a) APy

PyPx

a

a1 a

a22 / 31 U= Axay1aA = 5a = 3/5Px= 5Py= 2xmax= aIPx=35255= 3ymax= (1 a)IPy=

1 35

252=25252= 5 =(1 a) APy

PyPx

a

a1 a

a=

3535 0.73623 / 31 1 U(x,y) := A * x^(a) * y^(1-a);2 L(x,y,lambda) := (U(x,y) + lambda*(I - Px*x - Py*y));3 eq1 : diff(L(x,y,lambda), x) = 0;4 eq2 : diff(L(x,y,lambda), y) = 0;5 eq3 : diff(L(x,y,lambda), lambda) = 0;6 sol : solve([eq1, eq2, eq3], [x,y,lambda]);7 xmax : rhs(sol[1][1]);8 ymax : rhs(sol[1][2]);9 lmax : rhs(sol[1][3]);24 / 31Utility-06.wxm :U (x, y, z) =

2 x2+ y+z2+ y zx, y, x - Px= 2,Py= 3 Pz= 2 . I = 120, :Px x + Py y+ Pz z= 120 xmax, ymax, zmax - .25 / 31Utility-08.wxm . .U(x, y) = x3/4y1/4I = 24 Px= 2, Py= 11 24 32 , xmax, ymax U; ;2 24 32, Px, Py xmax, ymax; ;26 / 31 27 / 31Utility-09.wxm : , .MUx=UxMUy=Uy .28 / 31MarginalUtility.wxm :1 ;2 ;3 ;4 ;5No-name ;6 ; 29 / 31 1 Cobb-DouglasU= Axay1a . .2 ;3 ;4 (cardinalutility);5 ;30 / 31 , 31 / 31