$($Comprehensive Problem$)$Suppose that you are in the fall of your senior year and are faced with the choice of either getting a job when you graduate or going to law school. Of$$ course, your choice is not purely financial. $$ However, to make an informed decision you would like to know the financial implications of the two alternatives. $$ Let's assume that your alternatives are as$$ follows:$$

If you take the$$ "get a$$ job" route you expect to start off with a salary of $$$$35$ comma $000$ per year. There is no way to predict what will happen in the$$ future, your best guess is that your salary will grow at $3$ percent per year until you retire in $44$ years. As a law$$ student, you will be paying $$$$30$ comma $000$ per year tuition for each of the $3$ years you are in graduate school. $$ However, you can then expect a job with a starting salary of $$$$70$ comma $000$ per year. $$ Moreover, you expect your salary to grow by $9$ percent per year until you retire $39$ years later.$$

$$Clearly$,$ your total expected lifetime salary will be higher if you become a lawyer. $$ However, the additional future salary is not free. You will be paying $$$$30$ comma $000$ in tuition at the beginning of each of the $3$ years of law school. In$$ addition, you will be giving up a little more than $$$$108$ comma $000$ in lost income over the three years of law$$ school:$$$$35$ comma $000$ the first$$ year, $ $36$ comma $050$ the second$$ year, and $$$$37$ comma $132$ the third year.

a$.$To start your analysis of whether to go to law$$ school, calculate the present value of the future earnings that you will realize by going directly to$$ work, assume a discount rate of $12$ percent.

b$.$What is the present value today of your future earnings if you decide to attend law$$ school, assuming a discount rate of $12$percent$?$ Remember that you will be in law school for $3$ years before you start to work as a lawyer. $($Hint: assume that you are paid at the end of each year so that your first salary payment if you decide to go to law school occurs $4$ years from$$ now.$)$

c$.$If you pay your law school tuition at the beginning of each$$ year, what is the present value of your$$ tuition, assuming a discount rate of $12$percent$?$