Suppose that an agent has Cobb-Douglas utility over R^{2}_{+ }, i.e. U(x_{1}, x_{2}) = x^{a}_{1} x_{2}^{1-a}, where 0 < α < 1.

The agent has wealth w > 0 and faces linear prices p_{1}, p_{2} > 0 for each good.

Can you please find the tangency points between the agent's utility function and the budget constraint.

What are the two potential corner solutions to the agent's utility maximization problem and can either of them be optimal?

Find the agent's optimal consumption bundle.

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