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Alpha cares about his daily consumption c and leisure time t. His leisure time t =...

Alpha cares about his daily consumption c and leisure time t. His leisure time t = 24 − ℓ, where ℓ is the number of hours worked in a day. If he works ℓ hours in a day, he receives $wℓ income, which he consumes by the end of the day. Alpha’s daily utility function is u(c, t) = ct.

1. Write down Alpha’s budget equation which gives a relation between his daily consumption and leisure time.

2. Find the consumption and leisure time that maximizes Alpha’s daily utility.

3. Suppose instead of optimizing his daily utility, Alpha is satisfied as long as he obtains a daily utility of at least 12. Find all consumption and leisure choices that satisfy Alpha. For what value of w does he have a unique satisficing choice?

4. Now suppose that instead of optimizing his daily utility, Alpha chooses to work until his daily income is $48. Assuming w ≥ 3, find Alpha’s consumption and leisure choice in this case. Plot Alpha’s daily labor supply as a function of w (assuming w ≥ 3).

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