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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: U_{1} = x_{1} y_{1}^{2}

The budget constraint is as follows: P_{x} x_{1} + P_{y} y_{1} = M

The utility maximization is as follows:

Max: x_{1} y_{1}^{2}

s.t

P_{x} x_{1} + P_{y} y_{1} = M

Using langrages equation: L = x_{1} y_{1}^{2} - λ(M - P_{x} x_{1} - P_{y} y_{1})

Now, taking the first order derivative:

∂L/(∂x_{1}) = y_{1}^{2} - λ(P_{x}) = 0

∂L/(∂y_{1}) = 2y_{1} x_{1 }- λ(P_{y}) = 0

∂L/∂λ = M-P_{x} x_{1} - P_{y} y_{1} =0

Solving the first two equations:

2y_{1}x_{1}/P_{y} = (y_{1}^{2})/P_{x}

y_{1} = 2x_{1}P_{x})/P_{y}

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

M-P_{x} x_{1} - P_{y} (2x_{1} P_{x})/P_{y} = 0

x_{1} = M/(3P_{x})

And

y_{1} = 2M/(3P_{y})

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**Consumer 2:**

The utility function is as follows: U_{2} = x_{2} y_{2}

The budget constraint is as follows: P_{x} x_{2} + P_{y} y_{2} = M

The utility maximization is as follows:

Max: x_{2} y_{2}

s.t

P_{x} x_{2} + P_{y} y_{2} = M

Using langrages equation: L = x_{2} y_{2} - λ(M - P_{x} x_{2} - P_{y} y_{2})

Now, taking the first order derivative:

∂L/(∂x_{2}) = y_{2}-λ(P_{x}) = 0

∂L/(∂y_{2}) = x_{2}-λ(P_{y}) = 0

∂L/∂λ = M-P_{x} x_{2} - P_{y} y_{2} = 0

Solving the first two equations:

x_{2}/P_{y} = y_{2}/P_{x}

y_{2} = (x_{2} P_{x})/P_{y}

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

M-P_{x} x_{2} - P_{y} (x= P_{x})/P_{y} = 0

x_{2} = M/(2P_{x})

And

y_{2} = M/(2P_{y})

Total demand

X = x_{1} + x_{2} = M/(3P_{x}) + M/(2P_{x}) = 5M/(6P_{x})

Y = y_{1} + y_{2} = 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} = 1X^{2} = X^{2}

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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