Income tax deductions for charitable donations is one of the primary policies that the federal government uses to incentivize charitable giving. Any income that is donated to charity is exempt from the income tax. In this problem, assume that income spent on all other goods and services is subject to the income tax.

Old Saint Scrooge is a devout man who follows the practice of tithing hispre-tax income(tithing is the practice of donating 10 percent of income to charity). However, he grumbles every day that he has to donate such a large share of his daily $1,000 income. It is enough, he says, that he already pays an income tax rate of 50 percent.

1. (a)Letx1denote the amount Old Saint Scrooge donates to charity andx2denote his consumption of other goods and services. Old Saint Scrooge has Cobb-Douglas preferences with utility function
2. U(x1, x2) =x1^x2^
3. where >0 and >0. Given Scrooge's piety, what are valid values forandin Scrooge's utility
4. function?
5. (b)Old Saint Scrooge has become more kind-hearted. He still has Cobb-Douglass preferences, but with= 0.2 and= 0.8. Find the first-order necessary conditions for Scrooge's utility maximization problem.
6. (c)Find the optimal consumption bundle. What share of his income does Scrooge donate to charity?
7. (d)How much tax does the government collect from Scrooge at his optimal consumption bundle? How much tax does the government forgo by not taxing his charitable contributions?
8. (e)Suppose that instead of exempting charitable donations from tax, the government reduces the tax rate to 40 percent. This tax rate applies to all income, including income donated to charity. What is Old Saint Scrooge's new budget constraint?
9. (f)What is Scrooge's new optimal consumption bundle?
10. (g)Does Scrooge prefer the new government policy? Justify your answer.
11. (h)How much tax revenue does the government collect under the new policy? How does this revenue compare to the revenue collected under the old policy?