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General Equilibrium Assignment - Walras Law Problem - Find Equating demand and supply of each good.
Solution - GENERAL EQUILIBRIUM
Consumer 1:
The utility function is as follows: U1 = x1 y12
The budget constraint is as follows: Px x1 + Py y1 = M
The utility maximization is as follows:
Max: x1 y12
s.t
Px x1 + Py y1 = M
Using langrages equation: L = x1 y12 - λ(M - Px x1 - Py y1)
Now, taking the first order derivative:
∂L/(∂x1) = y12 - λ(Px) = 0
∂L/(∂y1) = 2y1 x1 - λ(Py) = 0
∂L/∂λ = M-Px x1 - Py y1 =0
Solving the first two equations:
2y1x1/Py = (y12)/Px
y1 = 2x1Px)/Py
Putting this value in equation 3, we get the demand functions:
M-Px x1 - Py (2x1 Px)/Py = 0
x1 = M/(3Px)
And
y1 = 2M/(3Py)
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Consumer 2:
The utility function is as follows: U2 = x2 y2
The budget constraint is as follows: Px x2 + Py y2 = M
The utility maximization is as follows:
Max: x2 y2
s.t
Px x2 + Py y2 = M
Using langrages equation: L = x2 y2 - λ(M - Px x2 - Py y2)
Now, taking the first order derivative:
∂L/(∂x2) = y2-λ(Px) = 0
∂L/(∂y2) = x2-λ(Py) = 0
∂L/∂λ = M-Px x2 - Py y2 = 0
Solving the first two equations:
x2/Py = y2/Px
y2 = (x2 Px)/Py
Putting this value in equation 3, we get the demand functions:
M-Px x2 - Py (x= Px)/Py = 0
x2 = M/(2Px)
And
y2 = M/(2Py)
Total demand
X = x1 + x2 = M/(3Px) + M/(2Px) = 5M/(6Px)
Y = y1 + y2 = 2M/(3Py) + M/(2Py) = 7M/(6Py)
Firm 1
Production function is as follows:
Y= ½ly
Total cost is as follows:
C = w * ly = 1 * 2Y = 2Y
The marginal cost is the supply curve:
dTC/dY = 2
Firm 2
Production function is as follows: X = √(lx)
Total cost is as follows:
C = w * lx = 1X2 = X2
The marginal cost is the supply curve:
dTC/dX = 2X
Equating demand and supply of each good:
Good X
Demand = Supply
5M/(6Px) = 2X
Px = 5M/12X
Good Y
Demand = Supply
7M/(6Py) = 2
Py = 7M/12
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