$$$ A Discrete Price Model $$$

Objective:

To study price evolution in an economy with discrete trading times.

We have studied continuous price evolution in the text (see Section 5.4); now we wish to consider a model for the evolution of prices in an economy with discrete trading times. Perhaps buyers and sellers only come to market once a week, for example.

As in the text, we assume that the trading times occur at regular time intervals of length Dt.

If k = 0, 1, 2, ... n indexes the trading times, then the demand and supply functions are:

Demand: D_k = a + bp_k

Supply: Q_k = c + dp_k-1

(There is a subscript of k-1 in the supply function because the suppliers base the amount to bring to the market on the price at the previous market.)

To model this problem using recursive formulas, we ...

Set the number of trading sessions:

Set the trading session counter:

We start at a price other than

the equilibrium price:

The market-clearing assumption D_k = Q_k

gives us the recursive formula

Define the demand function:

Define the supply function:

Note how the prices and associated supply values change as time goes by:

The equilibrium price (from Checkpoint 4b) is

Here are the graphs for price and supply; the parameters a, b, c, and d are defined

globally below the graphs.

Define the coefficients:

1. What happens to the price as k increases? What happens to the supply?

Describe as fully as possible.

2. Now experiment by increasing d slowly (say, in steps of 0.1) through the range from 1

to 2. Describe what happens. You may wish to change n to a larger value, if the graph

does not seem to be settling down soon enough.

At what value of d does the behavior of the graph start to change radically? Can you

give an intuitive explanation for what you have observed? What does an increase in the

parameter say about the attitudes of the suppliers? Think about what the parameters

b and d stand for.

3. Consider the recursive formula for p_k:

Let's simplify it a little by calling those fractions a and b:

p_k = a + b p_k-1

On a piece of paper, write down a formula for p₁ in terms of p₀, a and b.

Now find a formula for p₂ in terms of p₁, a and b.

Modify it to find a formula for p₂ which depends only on a, b, and p₀.

Express p₃ in terms of a, b, and p₀.

Do the same thing for p₁₀.

Find a formula for p_k which involves only a, b, and p₀.

Use your formula to explain what happened when you changed the value of d in part 2

above.

4. Now let's compare the continuous and discrete solutions to this problem.

We set the value of g to 0.17, as in the text:

Recall the continuous solution:

For the discrete problem, k plays the role of time; we have computed price, supply, and

demand at each time t_k. So if we graph the function P(k), we will see the solution to

the corresponding continuous problem.

Move the definitions of a, b, c, and d down near the graph below, to make it easier to change them:

Select three representative values of d for this example. (You want to choose 3 values

which make the solution do 3 very different things.) For each of them,

- graph the discrete solution

- graph the continuous solution

- describe below the graph how the discrete and continuous solutions differ

- print this page.

Turn in your worksheet with these three graphs.

As a group, write a maximum of one page discussing the topics covered in this lab. Explain what you have learned and how it fits in with the bigger picture of the calculus you've studied so far in this class. Hand in the writeup with your lab.