Question: Consider a consumer with preferences: \(U\left(x_{1}, x_{2}\right)=x_{1}^{a} x_{2}^{b}\) and income \(m\).

1. Formulate the consumer's choice problem and explain why it can be reduced to a single variable problem.
2. Can you assume instead that the consumer maximizes \(U\left(x_{1}, x_{2}\right)=x_{1}^{c} x_{2}^{1-c} ?\) what are the restrictions on \(c\) you need to make?
3. Write down the necessary conditions for an interior solution (first order conditions) for the consumer's choice problem.
4. Solve explicitly for the optimal bundle, \(\left(x_{1}^{*}, x_{2}^{*}\right)\), that satisfies the first order conditions. In general, which bundles (that do not satisfy the first order conditions) can also solve the consumer's choice problem? Given the preferences above, does the consumer prefers those bundles over \(\left(x_{1}^{*}, x_{2}^{*}\right)\) ?

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