Question: Consider an economy consisting of one consumer with utility function \(u(x, y, z)=\) \(x^{\alpha} y^{\beta} z^{\gamma}\) and endowment \(\omega=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)\), where \((\alpha, \beta, \gamma) \gg 0\)

1. Let \(\omega=\left(\omega_{1}, \omega_{2}, \omega_{3}\right) \gg 0\). Answer the following questions:

1. Find all the feasible allocation(s).
2. Find all the Pareto efficient allocation(s).
3. Find all the equilibria.

(b) Let \(\omega_{1}=0\) and \(\left(\omega_{2}, \omega_{3}\right) \gg 0\). Answer the following questions:

1. Find all the feasible allocation(s).
2. Find all the Pareto efficient allocation(s).
3. Find all the equilibria.

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