Final answer:
The consumer's daily budget constraint is derived from having 24 hours to split between work and leisure, earning $20 per hour of work, receiving $100 daily from parents, and the cost of $8 per unit of the composite good c. To maximize utility with the utility function U(r, c) = rc, the consumer needs to select the optimal bundle of c and r subject to their budget constraint.
Step-by-step explanation:
To solve for the consumer's optimal bundle given a utility function U(r, c) = rc, we first need to outline the consumer's daily budget constraint. With 24 hours in a day, the consumer can choose to work or enjoy leisure. The wage offered is $20 per hour for labor, and the consumer also receives $100 daily from their parents. The price of the good c is $8 per unit. The time budget constraint indicates that the sum of working hours (l) and leisure hours (r) must equal 24. Thus, we can represent this as r + l = 24. Since each working hour yields $20, the income from working alone would be 20l.
Adding the parental support, the consumer's total daily income equates to 20l + 100. This income can be spent on goods or saved for leisure. The cost of one unit of the composite good c being $8, means the consumer can afford c = (20l + 100)/8 units. Substituting l with 24 - r (since working hours plus leisure hours equals 24), we have c = [(20(24 - r) + 100)]/8. Simplifying further, we get c = (480 - 20r + 100)/8 or c = 60 - (20/8)r. The consumer maximizes utility by choosing the bundle of c and r that provides the highest utility subjected to this budget constraint.