Cobb-Douglas Utility (Two Goods)

def. Two-goods Cobb-Douglas Utility function:

u (x_{1}, x_{2}) = x_{1}^{α} x_{2}^{1 - α}

let the income:

I (p_{1}, p_{2}) = p_{1} x_{1} + p_{2} x_{2}

Utility Maximization

max_{x_{1}, x_{2}} u = x_{1}^{α} x_{2}^{1 - α} such that I = p_{1} x_{1} + p_{2} x_{2}

⎩ ⎨ ⎧ x_{1} (p_{1}, p_{2}, I) = \frac{α I}{p _{1}} x_{2} (p_{1}, p_{2}, I) = \frac{( 1 - α ) I}{p _{2}}

v (p_{1}, p_{2}, I) : = u (x_{1} (p_{1}, p_{2}, I), x_{2} (p_{1}, p_{2}, I)) = \frac{( 1 - α ) ^{1 - α} α ^{α}}{p _{1}^{α} p _{2}^{1 - α}} \cdot I

min_{x_{1}, x_{2}} I = p_{1} x_{2} + p_{2} x_{2} such that u = x_{1}^{α} x_{2}^{1 - α}

⎩ ⎨ ⎧ h_{1} (p_{1}, p_{2}, \overset{u}{ˉ}) = (\frac{p _{1}}{p _{2}} \frac{1 - α}{α})^{α} \overset{u}{ˉ} h_{2} (p_{1}, p_{2}, \overset{u}{ˉ}) = (\frac{p _{2}}{p _{1}} \frac{α}{1 - α})^{1 - α} \overset{u}{ˉ}

E (p_{1}, p_{2}, \overset{u}{ˉ}) : = p_{1} h_{1} + p_{2} h_{2} = \frac{p _{1}^{α} p _{2}^{1 - α}}{( 1 - α ) ^{1 - α} α ^{α}} \cdot \overset{u}{ˉ}