Dynamical Systems
Continuous and Discrete Models

We are particularly interested in how things change over time. In this module we look at two different kinds of models used to study the way things change over time -- continuous models and discrete models. Both are examples of Dynamical Systems -- systems that change over time.

We illustrate the ideas involved by looking at several models of the way in which the price of a particular product might change over time. These models involve three related quantities.

The price of the product.
The supply for the product -- that is, how many units of the product are being made.
The demand for the product -- that is, how many units of the product are being bought.

These three quantities are related in several ways. The graph below shows how supply and demand might be related to the price.

Notice the following.

When the price is low the demand is high and as the price rises the demand falls. This is the most common situation. People usually buy more of something when it is cheaper. BUT this is not always true. Some people want to buy the best and they equate the best with the most expensive. Thus, they are more likely to buy something when it is more expensive than when it is cheaper. Salespeople are very adept at playing on this fact. Manufacturers often make a deluxe model that really isn't much better than the standard model knowing that some people will buy the deluxe model just because it is more expensive.
There is a price below which the supply is zero. This is very common. Manufacturing a product usually requires material and time and no one will make it unless the price is high enough to make some profit.
The supply is very low when the price is low and the supply rises as the price rises. The more profit that can be made from manufacturing a product the more people will want to manufacture it.
The particular functions shown in the graph above are linear. We often work with linear functions because they are easy to work with. But, in practice, the supply function relating supply to price is rarely linear and the demand function relating demand to price is rarely linear.

The study of supply and demand and price changes is a wonderful opportunity to discuss functions and graphs in the classroom. Many students equate the idea of a function with the idea of a formula. When asked what a function is, they will reply -- "something like y = sin x." This concept of a function is very narrow. It is true that some functions can be described by formulas but this is the exception rather than the rule.

The linear supply and demand functions in the graph above are very unrealistic. In particular, the demand for a product is often constant over a wide range of prices because there is a limited need for the product. For example, even if Corn Flakes were free there are only so many Corn Flakes a person can eat. Draw the graph of a demand function that is constant for low prices and then drops off for higher prices.
There are often some fairly wealthy people who are willing to pay whatever is necessary to buy a particular product. These people are said to be price insensitive. Draw the graph of a demand function for a product some of whose customers are price insensitive.
Make up some problems like the ones above that you might ask your students to help them learn how to make connections between the way in which supply and demand depend on price in the real world and graphs of these functions.

Ideally supply and demand are in balance -- manufacturers are making exactly the number of units that customers want to buy. The price at which this occurs is called the equilibrium price. Graphically, this is the price at which the supply function and the demand function intersect.

Many economists believe that the "correct" price for a product is the equilibrium price and that the marketplace will "find" this price. We will discuss several models for how this might happen later. For now we want to examine some consequences of this idea.

We begin by working with a very simple model in which both the supply function and the demand function are linear.


               S(p) = 1000 p - 400      when p > $0.40
               D(p) = 1000 - 500 p

What is the equilibrium price for this product?
We can often learn something from a little algebra. In this case we can rewrite the supply function as
```
          S(p) = 1000 p - 400      
               = 1000 (p - 0.40)
```
In this form we see that the supply really depends on the quantity
(p - 0.40).
This function is shown in the graph below. Price is on the x-axis, which runs from $0.00 to $5.00 and supply is on the y-axis, which runs from 0 to 5,000.
This is exactly what you would expect if the manufacturing cost of the product was $0.40 per unit. Then the profit per unit would be (p - 0.40). Since profit is the money actually earned by producers, the amount they produce depends more directly on their profit than on the selling price.
This gives us an idea for a whole family of supply functions that can help us investigate questions about how the effects of inflation percolate through an economy. These supply function are of the form
S(p) = 1000 (p - c)
The parameter c represents the unit manufacturing cost for this product. In our first example c was $0.40. The graph above is a live Java applet. You can change the value of c by clicking along the x-axis or by changing the value in the box and clicking the move button.
What happens when the manufacturing cost goes up by $0.10? Notice this is a somewhat open-ended question. There are many different things you can say -- for example, what happens to the equilibrium price? How much of the rise in the unit manufacturing cost was passed on to the consumer? How much was absorbed by the manufacturer? What happened to the total number of units sold as a result of the rise in manufacturing cost? What happened to the total profit for the manufacturer? What happened to the total amount of money spent by consumers on this product? How many people lost their jobs as a consequence of the rise in manufacturing cost?
What happens when the manufacturing cost goes up by $0.20?
What happens when the manufacturing cost goes up by R dollars?
These kinds of questions are good examples of the kinds of questions we can examine if we have a good model. This model is not very realistic but it does illustrate the power of modeling.
Make up some problems like the ones above that you might use in your classes. The easiest problems involve linear functions -- that is,
D(p) = a p + b
S(p) = r (p - c)
where a, b, r, and c are constants. These problems let students see some of the power of mathematics while they are learning to work with linear functions.
Students may notice some interesting patterns when both the supply function and the demand function are linear. They might ask whether these same patterns work when either or both of the supply and demand functions are not linear. Make up some problems with nonlinear supply and demand functions. Your problems should be motivated by the patterns that you notice with linear supply and demand functions to see whether these same patterns appear for nonlinear supply and demand functions as well. You can make up problems that provide some interesting insights into prices and the percolation of inflation and at the same time exercise very specific algebraic and computer or graphing calculator skills. For example, if you were working with quadratic functions you might work with demand functions like
This basic model can be a nice thread winding its way through the curriculum demonstrating how as we learn more mathematics we are able to learn more about our world.

The models we looked at above were static. They did not involve changes over time. Now we want to consider dynamic models or dynamical systems in which we look at how prices change over time.

We begin by thinking about how supply, demand, and prices interact. Consider the figure below.

Notice that when the price is below the equilibrium price the demand is higher than the supply. We call this situation excess demand. This usually happens when the price is below the equilibrium price. Under these circumstances prices will tend to rise because buyers will be competing with each other and sellers will realize that they can charge a bit more.

On the other hand when the price is above the equilibrium price the supply is greater than the demand. We call this situation excess supply. In this case price will tend to fall as sellers see the product sitting on their shelves and try to move it with sales and special promotions.

The key quantity is

D(p) - S(p)

This is called the excess demand function. Notice that it is positive when demand is greater than supply and negative when supply is great than demand. Thus, we expect prices to rise when the excess demand is positive and prices to fall when the excess demand is negative. The greater the absolute value of the excess demand the stronger the pressures causing prices to change.

Now we need to reflect a bit about the kind of product whose price we are investigating. We want to make a distinction between two kinds of products that require two different kinds of models.

Products like farm products that are best described by a discrete dynamical system in which time is measured in fairly large steps. Farmers typically plant their crops in the spring and harvest them in the fall. They determine how much they will plant in the spring based on the price they received the preceding year and they try to charge the same price in the fall. In the fall the price may rise or fall depending on supply and demand. Thus, it makes sense to describe the price by a sequence of numbers
p₁, p₂, ... p_n, ...
with p₁ representing the price for the first year, p₂ representing the price for the second year, and so forth.
Because farm products often spoil and storage is expensive, each year's crop is distinct from the preceding and following year's crops. Farmers make their production decisions once a year and, thus, we think of time in one year increments. Many products behave in similar ways. For example, airlines need to make decisions about buying airplanes well in advance of when they plan to use them and airline seats are even more perishable than tomatoes. When an airplane takes off with an empty seat the revenue from that seat is lost forever. This perishability is one reason that farm prices and air fares vary so much.
In this situation we will look at a change equation of the form
p_{n + 1} = p_n + k ( D(p_n) - S(p_n) )
where k is a positive constant. This simple model captures the ideas we discussed above. It tells us how to compute each year's price based on the supply and demand situation at the preceding year's price. When the demand is greater than the supply the price will rise and when the demand is lower than the supply the price will fall. The size of the constant k and the absolute value of the excess demand determine how much the price will rise or fall each year. The value of k is determined by the behavior of producers and consumers. In some markets people react very strongly and in others they react more slowly. Family farmers, for example, face very strong pressures -- they often borrow money in the spring to finance planting and expect to pay it back in the fall when they harvest their crops. If buyers aren't buying their crops they face very strong pressures to lower their prices and make at least enough money to pay off their debts.
Many products are produced continuously and their prices are continually being adjusted. In this case we represent prices by a continuous function
p(t)
and the way that prices change by a differential equation. Differential equations are sometimes called continuous dynamical systems. We will look at the differential equation
p'(t) = k (D(p(t)) - S(p(t)))
where k is a positive constant. The size of the constant k and the absolute value of the excess demand determine how fast the price will rise or fall. The value of k is determined by the behavior of producers and consumers.

All the problems in this set of problems look at the same supply and demand functions.

S(p) = 1000 p - 400
D(p) = 1000 - 500 p

Use your computer algebra system window to help do these problems. In general, you will be expected to use your computer algebra system and window whenever it is appropriate without being specifically reminded.

In your CAS window you will work with numerical approximations for the solutions to initial value problems. If you know how to find the exact solutions, you should do so.

Look at the discrete dynamical system
p_{n + 1} = p_n + k ( D(p_n) - S(p_n) )
with k = 0.0002 and the initial condition p₁ = 0.50 and describe what happens.
answer
Look at the same discrete dynamical system with k = 0.0002 and the initial condition p₁ = 1.50 and describe what happens.
Look at the same discrete dynamical system with k = 0.0004 and the initial condition p₁ = 0.50 and describe what happens.
Look at the same discrete dynamical system with k = 0.0006 and the initial condition p₁ = 0.50 and describe what happens.
Look at the same discrete dynamical system with k = 0.0008 and the initial condition p₁ = 0.50 and describe what happens.
Look at the same discrete dynamical system with k = 0.0010 and the initial condition p₁ = 0.50 and describe what happens.
Look at the same discrete dynamical system with k = 0.0012 and the initial condition p₁ = 0.50 and describe what happens.
Look at the same discrete dynamical system with k = 0.0014 and the initial condition p₁ = 0.50 and describe what happens.
Relate your results above to the real-world meaning of the constant k.
Look at the continuous dynamical system or initial value problem
p'(t) = k (D(p(t)) - S(p(t))
with k = 0.0002 and the initial condition p(0) = 0.50 and describe what happens.
answer
Look at the same continuous dynamical system with k = 0.0002 and the initial condition p(0) = 1.50 and describe what happens.
Look at the same continuous dynamical system with k = 0.0004 and the initial condition p(0) = 0.50 and describe what happens.
Look at the same continuous dynamical system with k = 0.0006 and the initial condition p(0) = 0.50 and describe what happens.
Look at the same continuous dynamical system with k = 0.0008 and the initial condition p(0) = 0.50 and describe what happens.
Look at the same continuous dynamical system with k = 0.0010 and the initial condition p(0) = 0.50 and describe what happens.
Look at the same continuous dynamical system with k = 0.0012 and the initial condition p(0) = 0.50 and describe what happens.
Look at the same continuous dynamical system with k = 0.0014 and the initial condition p(0) = 0.50 and describe what happens.
Relate your results above to the real-world meaning of the constant k.
Discuss the differences between the discrete dynamical systems and the corresponding continuous dynamical systems (initial value problems).

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Dynamical Systems Continuous and Discrete Models

Dynamical Systems
Continuous and Discrete Models