Our task The optimization principle Available choices Preferences and utility functions Describing the representative consumer preferences

X Y U U(x, y), x X y Y (x, y ) U(x, y ) (x, y )

(x, y ) (x, y ) (x, y ) (x, y ) U(x, y ) > U(x, y ) (x, y ) (x, y ) U(x, y ) < U(x, y ) U(x, y ) = U(x, y ) X Y U x y U x > x 0 U(x, y) U(x 0, y), y y > y 0 U(x, y ) U(x, y 0 ), x U x U x > 0, U y U y > 0 X Y U(x 0, y 0 ) = U(x, y ) U(αx 0 + ( α)x, αy 0 + ( α)y ) > U(x 0, y 0 ) 0 < α < U

x y U(x, y) U(, ) U(x, y)

number of oranges 4 5 4 4 0 0 4 5 6 7 8 9 number of apples U = U x x + U y y U = 0 y x = U x du=0 X Y X,Y X Y X,Y U y (x, y)

number of oranges x y p x M M x y 4 p x x + y M 0 0 4 5 6 7 8 9 number of apples y y = M p x x px X Y X Y Y X = p X Y M ( x, ȳ) px + y p x + ȳ

x y U(x, y) p x x + y M L(x, y, λ) = U(x, y) + λ(m p x x y) U x (x, y) λp x = 0 U y (x, y) λ = 0 } U x U y = p x λ 0, M p x x y 0, λ(m p x x y) = 0 p x x + y < M

λ = 0 U x = U y = 0 λ > 0 p x x + y = M x, y U x (x, y ) U y (x, y ) = p x p x x + y = M x = x (p x,, M) y = y (p x,, M) X Y λ = U x(x, y ) p x = U y(x, y ) X Y X p x Ux p x Y Uy Uy Ux p x U X Y Y X x, y

number of oranges 4 5 4 4 0 0 4 5 6 7 8 9 number of apples U(x, y) (x, y ) V (p x,, M) = x,y {U(x, y) p xx + y = M} = U (x (p x,, M), y (p x,, M)) L = U(x, y ) + λ (M p x x y ) L M = λ + (U x λ p x ) x M + (U y λ ) y M L = λ x + (U x λ p x ) x + (U y λ ) y p x p x p x L = λ y + (U x λ p x ) x + (U y λ ) y = λ = λ x = λ y x y

λ = Ux p x = Uy V (p x,, M) M = U (x, y ) M x = U x M + U y y M = λ x p x M + y λ M y = λ ( p x x M + M ) = λ λ x = V p x V M V V M 0 y = f f ( ) > 0 V V M W (x, y) f(u(x, y)) W x W y = f (U)U x f (U)U y = U x U y

x y U(x, y) p x x + y = M x 0 y 0 L(x, y, λ) = U(x, y) + λ(m p x x y) + µ x + µ y U x (x, y) λp x + µ = 0 U y (x, y) λ + µ = 0 p x x + y = M µ 0 x 0 µ x = 0 µ 0 y 0 µ y = 0 µ µ U x (x, y) λp x = µ U y (x, y) λ = µ U x (x, y) λp x 0 x 0 x (U x (x, y) λp x ) = 0 U y (x, y) λ 0 y 0 y (U y (x, y) λ ) = 0 p x x + y = M x y M > 0 x = 0 U x (x, y) λp x < x x U x(x, y) = x 0 U x(x, y) = x > 0 U x (x, y) λp x = 0 x 0 U y(x, y) = y > 0 U y (x, y) λ = 0 y 0

x y U(x, y) p x x+ y M x 0 y 0 U(x, y) = x θ y θ 0 < θ < x y x > 0 y > 0 U ( y ) θ U x = θ x y = ( θ) ( ) θ x y U x = x 0 U y = U x > 0 U y > 0 y 0 x > 0 y > 0 p x x + y = M θ ( ) y θ x ( ( θ) x y ) θ = p x p x x + y = M x (p x,, M) = θm p x y (p x,, M) = ( θ)m ( ) θ ( ) θ θ θ V (p x,, M) = M p x

U(x, y) = x + y U x = x > 0 U y = > 0 y = p x = p L = x + y + λ(m px y) x λp = 0 λ = 0 px + y = M λ = px = y = M x (p, M) = p y (p, M) = M y M < V (p, M) = M p x y U x = x x 0 x > 0 U y = y L = x + y + λ(m px y) + µy x λp = 0 λ 0 y 0 ( λ) y = 0 px + y = M

λ = y = 0 λ = x λ = px px = y = M x = p y 0 M y = 0 px = M y = 0 x = M p λ 0 λ = px = M M 0 M x = {M, }, y = {M, 0} p V (p, M) = {M, } + {M, } p { M p, M = M p, V M = n + p i x i X i i = 0,,..., n α i > 0 i ( n U(x 0, x,..., x n ) = α i x ρ i ) ρ = (α 0 x ρ 0 + α x ρ + + α nx ρ n) ρ

X 0 p 0 = α 0 = n p i x i = p 0 x 0 + p x + + p n x n = M ( n L = α i x ρ i ) ( ρ + λ M ) n p i x i f(u) = U ρ ) n n L = α i x ρ i (M + λ p i x i L = n (α i x ρ i λp ix i ) + λm ρα i x ρ i λp i = 0 λ = ρα ix ρ i p i i = 0,..., n X 0 i =,..., n : ρα i x ρ i p i = ρα 0x ρ 0 x i = αi σ p σ i x 0 σ = p ρ 0 M = n p i x i = x 0 + n i= αi σ p σ i x 0 x M 0 = + n i= ασ i p σ i p k x k = αk σp σ k n M ασ i p σ i k = 0,..., n P P ( n α σ i p σ i ) σ

x k = ασ k p σ k P σ M = ( αk p k P ) σ M P k = 0,..., n M P α k σ p x /P ( n α i x ρ i α k x ρ k x k = αkp σ σ σ k MP ) ρ = α+σρ k = αkp σ σ k ( n = V (p 0,..., p n, M) = MP σ ( n p σρ k (MP σ ) ρ (MP σ ) ρ αkp σ σ k (MP σ ) ρ = MP σ P σ ρ V α σ kp σ k = M P ) ρ ) ρ ρ = 0 σ = U(x 0, x,..., x n ) = n p i x i = x αi i = x α0 0 xα xαn n α i n α i M i = 0,..., n