a) We understand that as long as income is less than $40 per day, one earns $20 per hour with no tax. This means one can work only two hours of untaxed work(2 hrs. x $20/hr = $40) before hitting the level at which there is an income tax. If the worker works more than two hours, she pays 20% of her wages, or $4/hr. as a tax. So, if you work more than two hours a day, you are instead making $16/hr. instead of the original $20. We can represent this on a graph that shows daily income on the y-axis, and hours worked per day on the x-axis. As we begin to plot our points, we can see that someone who works 1 hour would make $20. Someone who works 2 hours would make $40. Up until x=2 our line has a slope of 20, which comes from the hourly pay. But then, at all points where x>2, or situations in which more than two hours are worked, we instead see a line with a slope of 16, which comes from the after-tax income for everyone earning over $40.
b) In this case, no matter what the worker does, the government is going to give him or her $10. The tax, however, remains at income above $40, it just takes less hours of work to reach that point, because you begin with the $10 the government has already given you. So, one can only work 1.5 hours at the full rate of $20 per hour before hitting an income of $40. beyond x=1.5 hours of work, after-tax income is back to the $16/hr rate. In part B as opposed to part A, it takes 30 minutes of work less to earn $40; so if $40 per day is a liveable income, this new guaranteed $10 per day is likely to inspire people to work fewer hours, because a slightly higher portion of their hours are at a reduced income than before.
c) In this situation, the switch occurs at the same time, because the worker is still guaranteed $10, and she still starts out making $20 per hour. In this case, because the tax is lower, the worker still earns $18 per hour as opposed to $16, meaning the line continues from x=1.5 at a slope of 18.