ECON 4230/35 Fourth seminar
Problem 3: General equlibrium analysis
Two consumers with identical utility functions
0,2
uA (x1 , x2 ) = uB (x1 , x2 ) = x0,6
1 x2
(a) Marshallian demand functions
0,2
max x0,6
1 x2
s.t.
p 1 x1 + p 2 x2 = m
Simplify by logarithmic transformaton. The Lagrangian becomes:
L(x1 , x2 ) = 0, 6lnx1 + 0, 2lnx2 − λ(p1 x1 + p2 x2 − m)
0, 6
− λp1 = 0
x1
⇒ p1 X1 =
0, 6
λ
(FOC1)
0, 2
0, 2
(FOC2)
− λp2 = 0
⇒ p2 X2 =
x2
λ
Insert into the budget constraint and calculate the Marshallian demand:
x1A (p1 , p2 , mA ) =
3mA
4p1
x2A (p1 , p2 , mA ) =
mA
4p2
x1B (p1 , p2 , mB ) =
3mB
4p1
x2B (p1 , p2 , mB ) =
mB
4p2
(b) Market clearing prices and equilibrium allocation
ωA = (12, 0)
ωB = (0, 12)
mA = 12p1
mB = 12p2
x1A = 9
x2A =
x1B =
3p1
p2
9p2
P1
x2B = 3
1
Since x1A + x1B = 12
⇒9+
9p2
= 12
p1
x1A = 9
⇒3
x1B = 3
p2
=1
p1
⇒
p2
=3
p1
x2A = 9
x2B = 3
(c) Pareto efficient allocations
0,2
max x0,6
1 x2
(12 − x1 )0,6 (12 − x2 )0,2 = c
s.t.
Simplify by logarithmic transformaton of both the maximand and the
constraint. The Lagrangian becomes:
L(x1 , x2 ) = 0, 6lnx1 + 0, 2lnx2 − λ(0, 6ln(12 − X1 ) + 0, 2ln(12 − X2 ) − lnc)
Sovle to find
x1 = x2 = x
(d) Welfare maximizing allocation
W = (uA , uB ) = uA + uB
max(x1A )0,6 (x2A )0,2 + (x1B )0,6 (x2B )0,2
s.t.
x1A + x1B = 12 = x2A + x2B
Or
max(x1A )0,6 (x2A )0,2 + (12 − x1A )0,6 (12 − x2A )0,2
0, 6(x1A )−0,4 (x2A )0,2 − 0, 6(12 − x1A )−0,4 (12 − x2A )0,2 = 0
(FOC1)
0, 2(x1A )0,6 (x2A )−0,8 − 0, 2(12 − x1A )0,6 (12 − x2A )−0,8 = 0
(FOC2)
Solve to find
x1A = x1B = 6
x2A = x2B = 6
2
(e) Welfare maximizing allocation given endowment
ωA = (8, 0)
ωB = (4, 12)
Then we know the income of the two consumers and we can find expressions for the marshallian demand for both goods. These expressions makes
us calculate the price ratio:
p1
=3
p2
And then we end up with the welfare maximizing allocation:
x1A = 6
x1B = 6
x2A = 6
3
x2B = 6