Deriving demand from quadratic utility function

Question

How do you derive the demand for utility $u(x_1, x_2) = x_1^2 + x_2^2 $ and initial endowment is $\omega = (2,2) $?

I believe this demand has three cases.

Thank you

Answer 1

The demand function is derived based on the prices $p_{1}, p_{2}$. Since $x_{1}, x_{2}$ contribute to utility equally and this is a linear utility function, we favor one over the other based on which we can obtain more of.

Our initial wealth is: $w = 2p_{1} + 2p_{2}$, based on the valuation of the initial endowment.

If $p_{1} < p_{2}$, then $x_{1}(p_{1}, p_{2}) = \frac{2p_{1} + 2p_{2}}{p_{1}}$ and $x_{2}(p_{1}, p_{2}) = 0$. The case is symmetric for when $p_{1} > p_{2}$.

Now when $p_{1} = p_{2}$, we demand $\{ (x_{1}, x_{2}) : p_{1}x_{1} + p_{2}x_{2} = 2p_{1} + 2p_{2}\}$.