A worker has two jobs, and he can choose to work any number of hours in a day on each job, up to the upper limit (if any), but can only work on one job at a time. The first job pays £12 an hour and has an upper limit of 7 hours a day. The second job pays £5 an hour and has no upper limit. The worker will always choose the first job if he can. Consider his budget constraint with the amount of free time on horizontal axis and his consumption expenditure (which equals his daily income) on the vertical axis. Based on this information, read the following statements and select the correct one(s).

Question

BlackTom AI · Accepted Answer

Total time endowment H = 24 hours (standard assumption). Hours worked = H − free time. The worker uses job 1 (£12/hr) up to 7 hours, then job 2 (£5/hr) for any additional working hours; marginal trade-offs between free time and consumption equal the wage of the job at the margin.

1) Relevant knowledge points and concepts
- Time budget: free time f plus hours worked w = H (here H = 24).
- Income as a piecewise linear function of hours worked because job 1 has a 7-hour cap.
- Budget constraint slope (dY/df) equals minus the wage of the marginal job (because increasing free time by 1 reduces income by the wage forgone).

2) Complete reasoning process step by step
Let f = free time, w = hours worked = 24 − f.
If w ≤ 7 (i.e., f ≥ 17): all work can be done on job 1 at £12/hr. Income Y = 12w = 12(24 − f) = 288 − 12f → slope = dY/df = −12.
If w > 7 (i.e., f < 17): first 7 hours at £12, remaining (w − 7) at £5. Income Y = 12·7 + 5(w − 7) = 84 + 5(24 − f − 7) = 84 + 5(17 − f) = 169 − 5f → slope = dY/df = −5.

3) Intermediate calculations shown
- For f ≥ 17: Y = 12(24 − f) = 288 − 12f.
- For f < 17: Y = 84 + 5(17 − f) = 169 − 5f.
- For example, f = 8 → w = 16 → Y = 169 − 5·8 = 169 − 40 = £129.

4) Why the correct answer is correct
When free time is small (f < 17) the worker is working more than 7 hours, so the marginal hour of free time displaces work on job 2 (wage £5) → slope = −5. When free time is large (f ≥ 17) the worker would only be working on job 1 if at all, so the marginal hour of free time displaces the £12 job → slope = −12. Thus the statement "The slope ... is −5 when the hours of free time is small, and −12 when the hours of free time is large" is correct.

5) Why the other options are incorrect
- "Kinked at 7 hours of free time": incorrect; the kink occurs where hours worked = 7, i.e. free time f = 24 − 7 = 17 hours, not 7.
- "The worker will never choose to consume exactly 17 hours of free time": incorrect/unsupported — 17 hours is feasible (it is the kink point) and could be chosen depending on preferences.
- "For 8 hours of free time, maximum expenditure is £96": incorrect calculation. With f = 8, income = 169 − 5·8 = £129 (not £96).

6) Conclusion
The correct statement is: The slope of the budget constraint is −5 when free time is small, and −12 when free time is large, which matches the provided answer.

类似问题

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