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A low-income family has a daily budget of M dollars to spend on an essential starch food (S) and a non-essential protein food (PF). The family's preferences are equal to U(PF, S) = PF + S^0.5. The protein food costs eight times the starch food. In a diagram, measure units of starch food along the horizontal and draw the family's budget constraint when M = $12 and ps = $1.
1. Write the family's Lagrangian function.
2. Write the first order conditions of the maximization problem.
3. Find the family's optimal bundle.
4. In the diagram, add an indifference curve and illustrate the family's optimal bundle.
5. How much money should this family receive in aid to decide to buy one pound of protein food?
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