Difference Equations, Part 1

Difference Equations

Part 1: Examples and Explorations

In this module our principal object of study is the sequence, that is, a list of numbers of the form

y₀, y₁, y₂, y₃, ... ,

which goes on forever. We abbreviate such a sequence as {y_k}, with the understanding that the index k takes all nonnegative integer values, 0, 1, 2, and so on.

We will focus on sequences defined by difference equations -- for which we give two examples in this section, leading to careful definitions in subsequent sections. But first we examine how each sequence can be thought of as a "vector" in a "space of sequences."

Let S be the set of all sequences of real numbers. Explain why S is a vector space -- that is, why S is closed under addition and scalar multiplication.

Example 1: Loan Repayment. Can you afford to buy the car of your dreams? Suppose that car costs $20,000, and you can afford to pay $450 per month. Your credit union is offering new car loans at 6.9% interest. How many months would it take to pay off this loan?

Let's write y₀ = 20000 for the initial balance of the loan, and then calculate each successive monthly balance y₁, y₂, and so on. When the balance reaches 0, the loan is paid off. Since we are unlikely to land right on 0, the last payment may be less than $450.

To find y₁, we have to add the interest accrued in the first month, and then subtract the first payment. The monthly interest rate is one-twelfth the yearly rate, so

y₁ = 20000 + (0.069/12) 20000 - 450,

or $19,665. Then we calculate y₂ in the same way:

y₂ = 19665 + (0.069/12) 19665 - 450,

or $19,328.07. Note that each month's balance is computed from the preceding month's balance.

Find a formula for y_k+1 in terms of y_k.

Use your formula to answer the question about how many months it would take to pay off the loan. How big would the last payment be?

Plot the sequence {y_k} as a function of k, and confirm visually your answer for when the last payment would be made.

Describe in words what you observe about the shape of the graph of the sequence.

Example 2: A National Economy. A simplified model of a national economy has the form

y_k+2 - a(1 + b) y_k+1 + ab y_k = 1,

where y_k is the total national income during year k, a is the marginal propensity to consume (a constant < 1), and b is a constant of adjustment, which describes how the rate of private investment is affected by changes in consumer spending. (The original model was formulated by the economist Paul A. Samuelson.)

Note that there isn't any obvious starting place for the sequence of national incomes -- any year could be "year 0," and any amount could be the starting income. We have a hint about reasonable sizes for numbers from the constant term on the right: The left-hand side is a weighted sum of incomes, and it has to add up to 1. Thus, it is reasonable to assume that the monetary unit has been scaled so that the order of magnitude of an annual income is 1. Think of one unit representing a trillion dollars, say.

Note also that the formula allows us to compute y_k+2 in terms of y_k+1 and y_k. In order to generate a sequence from this formula, we have to know both y₀ and y₁. That is, the formula gives income for a current year in terms of income the year before and the year before that.

Start with a = 0.9, b = 0.5, y₀ = 1, and y₁ = 1.1. Calculate national incomes for 30 years, and plot the sequence as a function of k. What do you observe?

Experiment with starting amounts. What happens if the income drops to 0 in year 1? What happens if it is 0 in year 0, but positive in year 1? Try other combinations as well. How much influence do these starting amounts have on the eventual state of the economy?

Set the starting amounts to give whichever plot you found most interesting in the preceding step. Now experiment with the adjustment constant b, both raising it and lowering it (but keep it positive). How much influence does b have on the eventual state of the economy?

Finally, return b to its first value of 0.5, and experiment with a, trying a range of values between 0 and 1. How much influence does a have on the eventual state of the economy?

Can you quantify your observation in step 9 in a statement that relates "eventual state" to the actual value of a? If not, just go on -- we will return to this question in Part 3.

Our first two examples happen to be about economics -- micro and macro, respectively. However, sequences also play important roles in engineering, science, and other areas of application of mathematics, as do the formulas that generate the sequences and analysis of those formulas by linear algebra.

modules at math.duke.edu