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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: P_x x₁ + P_y y₁ = M

The utility maximization is as follows:

Max: x₁ y₁²

s.t

P_x x₁ + P_y y₁ = M

Using langrages equation: L = x₁ y₁² - λ(M - P_x x₁ - P_y y₁)

Now, taking the first order derivative:

∂L/(∂x₁) = y₁² - λ(P_x) = 0

∂L/(∂y₁) = 2y₁ x₁ - λ(P_y) = 0

∂L/∂λ = M-P_x x₁ - P_y y₁ =0

Solving the first two equations:

2y₁x₁/P_y = (y₁²)/P_x

y₁ = 2x₁P_x)/P_y

Putting this value in equation 3, we get the demand functions:

M-P_x x₁ - P_y (2x₁ P_x)/P_y = 0

x₁ = M/(3P_x)

And

y₁ = 2M/(3P_y)

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Consumer 2:

The utility function is as follows: U₂ = x₂ y₂

The budget constraint is as follows: P_x x₂ + P_y y₂ = M

The utility maximization is as follows:

Max: x₂ y₂

s.t

P_x x₂ + P_y y₂ = M

Using langrages equation: L = x₂ y₂ - λ(M - P_x x₂ - P_y y₂)

Now, taking the first order derivative:

∂L/(∂x₂) = y₂-λ(P_x) = 0

∂L/(∂y₂) = x₂-λ(P_y) = 0

∂L/∂λ = M-P_x x₂ - P_y y₂ = 0

Solving the first two equations:

x₂/P_y = y₂/P_x

y₂ = (x₂ P_x)/P_y

Putting this value in equation 3, we get the demand functions:

M-P_x x₂ - P_y (x= P_x)/P_y = 0

x₂ = M/(2P_x)

And

y₂ = M/(2P_y)

Total demand

X = x₁ + x₂ = M/(3P_x) + M/(2P_x) = 5M/(6P_x)

Y = y₁ + y₂ = 2M/(3P_y) + M/(2P_y) = 7M/(6P_y)

Firm 1

Production function is as follows:

Y= ½l_y

Total cost is as follows:

C = w * l_y = 1 * 2Y = 2Y

The marginal cost is the supply curve:

dTC/dY = 2

Firm 2

Production function is as follows: X = √(l_x)

Total cost is as follows:

C = w * l_x = 1X² = X²

The marginal cost is the supply curve:

dTC/dX = 2X

Equating demand and supply of each good:

Good X

Demand = Supply

5M/(6P_x) = 2X

P_x = 5M/12X

Good Y

Demand = Supply

7M/(6P_y) = 2

P_y = 7M/12

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