$[16$ marks$]$ Robert$$s pre$-$tax income is $$100,000.$ According to the income tax system, the first $$40,000$ of income is taxed at $10\%,$ income between $$40,000$ and $$80,000$ is taxed at $20\%,$ and income above $$80,000$ is taxed at $40\%.$ a$.[4$ marks$]$ Draw a diagram accurately showing the average and marginal tax rates for incomes between $0$ and $$120,000.$ b$.[2$ marks$]$ Calculate what taxes Robert legally owes. c$.[2$ marks$]$ Assuming that Robert can hide some of his income, find and draw his marginal benefit of hiding income, with $$unreported income$($U $)$ on the horizontal axis$\mathrm{.}4$ marks$]$ There is a p $=10\%$ chance that the government will find out the amount of income Robert does not report. If he is caught, Robert faces a fine formula for unreported income, F $=25($U$/1000)2($Which means the marginal fine is M F $=0\mathrm{.}00005$U $).$ Using the fact that the M C $=$ pM F $,$ find Robert$$s choice of unreported income and show this on a diagram. How much he will owe the government if he is caught? e$.[4$ marks$]$ Now suppose Robert $$feels bad$$ when he does not report some of his income. Specifically, M C $=$ pM F $+$ h$,$ where h is the honesty$-$cost per dollar of U $.$ If h $=0\mathrm{.}15,$ how much will U be now? How high must h be for U $=0?$ Explain.