In his current job, Smith can work as many hours per day as he chooses, and he will be paid 1/hr for the first 8 hours he works, and 2.50/hr for each hour over 8. Faced with this payment schedule, Smith chooses to work 12 hr/day. If Smith is offered a new job that pays 1.50/hr for as many hours as he chooses to work, will he take it? Explain.

the answer I have provided, but I have struggled to explain it, please explain it step by step, thanks !

Income = hw . w = (24 - h,) . w = (24 - h,) . 1 = 24 - hu where hw denote hours of work and hi hours of leisure. If he works more than 8 hours/day, the first 8 hours he is paid E1/hr, and the rest (16 - h,) hours he is paid E2.50/hr: Income = 8 . 1 + (16 - h,) . 2.5 = 48 - 2.5h. So his budget constraint has a kink (see the graph below, curve ABC): (48 - 2.5hu hi