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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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