Sequences -- Exponential Models with Immigration -- Stretch Your Understanding

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

1. This first question is purely mathematical. Consider the family of models of the form

   p(1) = a

   p(n ) = m p(n - 1) + b

   These models are called linear dynamical systems because the function

   f(p) = m p + b

   on the right hand side of the change equation

   p(n) = f(p(n - 1))

   is a linear function. Do a series of experiments with models of this form with different values of the constants m and b and with different initial conditions a. Be sure to try the following values of m

   • m = 0
   • 0 < m < 1
   • m = 1
   • 1 < m
   • -1 < m < 0
   • m = -1
   • m < -1

   and be sure to try different values of b and of the initial condition, a. Describe the results of your experiments.

2. Economists often talk about the Law of Supply and Demand as one mechanism by which prices are determined in a free market economy. Discrete dynamical systems are often used to model this mechanism for items like farm products that have distinct "generations."

   The Law of Supply and Demand is based on two observations about people.

   • In general people buy more of a product when prices are low and buy less when prices are high. Economists and businessment talk about a demand function -- for example a function like

     D(p) =1000(2 - p)

     that describes the total demand D(p) for a product at the price p.

   • In general people make more of a product when the price is high than when it is low. This behavior is described by a supply function -- for example, a function like

     S(p) = 500(p - 0.50)

     that describes the total supply S(p) of a product at a price p. That is, S(p) is the total amount of the product that will be produced by producers when the price is p.

   

   At a given price p we can compute the excess demand -- the difference between the demand and the supply.

   Excess Demand = D(p) - S(p)

   If the excess demand is positive then there is a shortage of this particular product. In this situation producers are able to raise their prices. If the excess demand is negative then there is an oversupply of this particular product. In this situation prices tend to fall.

   The change in price is often modeled by a change equation of the form

   p(n) = k [D(p(n - 1)) - S(p(n - 1))]

   where k is a positive constant. Notice if the excess demand is positive this equation predicts that prices will rise and if the excess demand is negative it predicts that prices will fall. The size of the constant k affects how quickly prices change in response to the excess demand.

   Experiment with this example -- that is, with the supply and demand functions given above -- to see how these models behave and what they predict about prices. Start with k = .0001 amd then try other (positive) values of k. Be sure to try several different initial prices. Describe the results of your experiments.