#### The car problem

**Situation 1: **

It is the spring of your 2nd class year, and you are faced with the prospect of purchasing an automobile. While your classmates are busy studying advertisements and sifting through reports by J. D. Power and Associates, you prudently decide to utilize your knowledge of discrete dynamical systems to aid in your decision. You have narrowed your choices to a Toyota Celica GT-S($\$17000$), an Alfa Romeo($\$20000$) and an Audi Quatro ($\$23000$). Based on an assessment of your financial situation, you realize that you should pay no more than $\$400$ per month. Furthermore, you have decided to try to pay off your loan in four years. The Marine Midland Bank of Highland Falls has agreed to give you a loan for the cost of the automobile at a very reasonable annual interest rate of $8\%$ compounded monthly. Assume that you make no down payment and that the cost of automobiles listed includes tax, title, and licensing.

1. How many months will it take you to pay off the loan for each of the three vehicles?

2. If you pay $\$400$ a month, what is the final monthly payment for each of the three cars?

3. Which car(s) can you afford under your restrictions?

4. Since you really had your heart set on the Audi Quatro, you decide to relax the $\$400$ restriction. Therefore, paying the loan off in 4 years, you want to find out how much the monthly payments would be for each of the three cars.

5. What is the equilibrium value of your dynamical system as a function of the monthly payment?

6. What is the significance of the equilibrium value? What if your monthly payments were $\$400$?

7. What about the stability of this equilibrium value? HINTS: Try some different payment values and some different starting values.

**Situation 2: **

You are rapidly approaching the time when you must decide which automobile you are going to purchase. Despite your detailed analysis of the problem, you seem to be no closer to a decision than you were when you started. Your decision becomes even more difficult when you receive word that your wealthy uncle has invested a sizeable sum of money in a money market account bearing your name. He will give you control of the account on your 21st birthday, which is days away. The only stipulation is that you do not close out the account until after you graduate. This fortuitous turn of events allows you to consider another possibility. You know that you can lease the Alfa Romeo for $\$375$ a month. You want to check out the feasibility of leasing the Alfa for the next 13 months and then using the money market account to purchase a brand new automobile after graduation. Your money market account earns $10\%$ annual interest compounded monthly.

8. Assume you pay $\$375$ a month out of your money market account after the interest has been added to your account. What do you need to know to solve this problem? Why?

9. How much money needs to be in the money market account to begin with for you to purchase each of the three car types?

**Situation 3: **

You become intrigued with the idea of leasing an Audi Quatro instead of purchasing a car. In fact, you consider leasing the Audi Quatro for three or four years while keeping the money market account intact. You have uncovered three leasing plans. (1) Pay $1.5\%$ of the purchase cost as down payment and then pay $\$400$ each month for the term of the lease. (2) Pay no down payment. The monthly payments are $\$50$, $\$65$, $\$80$, $\$95$, $\$110$, $\$125$, and so on. (3) Pay $\$50$ down. Your monthly payment is increased by $4\%$ each month. In addition, the leasing company charges you a monthly premium of $2\%$ of the total amount of money you have paid. So, for example the first month's payment would be $\$50+\$2+\$1=\$53$.

10. What are the merits and drawbacks of each of these three leasing options?

11. If you decide to lease for three years, which option is the least expensive? What about four years? Or five?

12. Do any of these plans have an equilibrium value? Why would that be important?

13. Any other questions you'd like to answer? Please, bring them up in class.