Microeconomia

1. U (X,Y) = 5X 0.5 Y0.5
U = 5X 0.5 Y0.5
U = Y0.5
5X 0.5

U = Y0.5(1/0.5)
5X 0.5

0.5
U = Y
5X 0.5

£ =5X0.55Y0.5 + λ (m - PxX – PyY)

∂£ = 2.5X-0.5 Y0.5 - λ Px = 0
∂X

∂£ = 2.5X0.5 Y-0.5 - λ Py = 0
∂Y

∂£ = m – PxX - PyY = 0
∂λ

∂U 5X0.5 Y0.5
∂X= Px
∂U 5X0.5 Y0.5 Py
∂Y

2.5 Y = Px
2.5 X Py

Y = Px
X Py

Y = PxX
Py

m = PxX + PyY
m = PxX +Py(PxX)
Py

m = PxX + PxX
m = 2PxX

2. U (X,Y) = 3X 2/5 Y 3/5
U = 3X 2/5 Y 3/5
5/3
U = Y 3/5 . 5/3
3X 2/5

U 5/3 = Y
(3X2/5) 5/3

£ = 3X2/5 Y3/5 + λ (m - PxX – PyY)

∂£ = 3 . 2/5 X-3/5 Y3/5 - λ Px = 0
∂X
6/5 X-3/5 Y3/5 = λ Px
∂£ = 9/5X2/5 Y -2/5 - λ Py = 0
∂Y
9/5 X2/5 Y -2/5-= λ Py
∂£ = m – PxX - PyY = 0
∂λ

∂U 3X2/5 Y3/5
∂X
= Px
∂U 3X2/5 Y3/5 Py
∂Y

6/5X-2/5 Y3/5 = Px
9/5X2/5 Y-2/5 Py

30Y3/5 Y2/5 = Px
45 X2/5 X3/5 Py

2Y = Px
5X Py

2Y = Px5X
Py

Y = Px5X
2Py

m = PxX + PyY
m = PxX +Py(Px5X)
2Py
m = PxX + Px5X
2

3. U (X,Y) = AX Yβ

U = AX Yβ
1/β
U = Y β . 1/ β
AX

( U )1/β = Y
( AX ) 1/β

£ = AX Y β + λ (m - PxX – PyY)

∂£ = AX-1 Yβ - λ Px = 0
∂X
AX-1 Yβ = λ Px

∂£ = βAX Y β-1 - λ Py = 0
∂Y
βAX Y β-1 =λ Py

∂£ = m – PxX - PyY = 0
∂λ

∂U AX Y β
∂X
= Px
∂U AX Y β Py
∂Y

AX-1 Yβ = Px
βAX Y β-1 Py

Yβ-β+1 =...