APPLICATIONS OF DISCRETE DYNAMICAL SYSTEMS

1. Applications to finance

Consider the general situation when we have an account that contains dollars after compounding periods. Suppose that account is collecting 100 I per cent annual interest, compounded times per year. Assume a constant amount is added to the account at the end of each compounding period (or taken from the account if ). Let be the initial amount in the account, that is

You should see that the dynamical system

describes the relationship between the amount in the account at the end of compounding periods and the amount after compounding periods. This simple model applies to a wide range of financial applications.

In the general case one has

When then

2) Applications to economics

Let us consider supply and demand as it relates to a product that takes one unit of time to produce. Let be the supply of our product, the demand for our product and the price of one unit of our product, all in year

To develop a model we need certain assumptions. A reasonable set of assumptions is the following:

1. The supply of the product in any year depends positively on the price of the product the previous year;
2. The demand for the product in any year depends negatively on the present price of the product;
3. Each year, the price of the product is adjusted so that the demand equals the supply.

Let and represent the supply demand (the amount the consumers buy) and price per bushel of potatoes, all in year

Assume, at first, that the supply equation is presented as

Thus, if the price is 6 dollars per bushel this year, the producers will grow 4.8 units of potatoes next year. However, if the price is 12 dollars the production will be 9.6 units next year.

Assume that the demand equation is

In this case, if the price is 6 dollars per bushel, the consumers are willing to buy 12.8 units of potatoes, while if the price is 12 dollars, the consumers are only willing to buy 5.6 units of potatoes.

The third assumption states that the price next year will be adjusted so that the supply equals the demand, that is

Since

and

substitution gives

Thus

or

From the second assumption, demand in year depends on the price in year Assuming that the relationship is linear, the demand equation for the next year is

whereas the supply equation is

Now one can obtain that

where is the equilibrium value of this dynamical system