Economic Models for Exchange of Goods

Suppose that on a remote island, the members of a tribe are engaged in three occupations:
1) farming, 2) manufacturing of tools and utensils, and 3) the weaving and sewing of clothing.
Assume that initially the tribe has no monetary system and that all goods and services are
bartered.

The tribe divides the goods and services up as follows:

Farmers:

The farmers keep half of their produce and give one quarter of their produce to the manufacturers and on quarter to the clothing producers.

Manufacturers:

The manufacturers divide the goods evenly among three groups, one third goes to each (including themselves).

Clothing weavers:

The group producing clothes gives half of the clothes too the farmers and divides the other
half evenly between themselves and the manufacturers.

This is represented by a table:

The first column of the table indicates the distribution of the goods produced by the farmers,
the second column indicates the distribution of the manufactured goods, and the third
column indicates the distribution of the clothing.

As the size of the tribe grows, the system of bartering becomes too difficult and, the island decides to institute a monetary system of exchange. For this simple economic system, assume that the prices for each of the three types of goods will reflect the values of the existing bartering
system.

Your assignment is to develop a system of monetary exchange.  Then assign values to the three types of goods that fairly represent the current bartering system.

<>This problem can be turned into a linear system using a model that was originally developed
by the Nobel prize winning economist Wassily Leontief.

As you work through this problem,  let x1 be the monetary value of the goods produced by the farmers, x2 be the value of the manufactured goods, and x3 be the value of the clothing produced.

According to the first row of the table, the value of the goods received by the farmers amounts to half the value of the farm goods produced, plus one third the value of
the manufactured products, and half the value of the clothing goods. Thus the total value of
the goods received by the farmers is 1/2(x1)+1/3(x2)+1/2(x3). If the system is fair, the total value of goods received by the farmers should equal x1, the total value of the farm goods produced.

Now open up a new document in javabars.

Draw three different bars, make them three different colors and label the bars x1, x2, x3.

Draw three equal sized mats, label these as farming, manufacturing , and clothing.

farming=x1

manufacturing=x2

<>clothing=x3

Divide the bars up according to how much goes to each of the three occupations. (i.e. farmers give 1/2 to themselves, 1/4 to manufacturing, and 1/4 to clothing.)

Place the proper proportions in each mat as given by the rows in the table.


Answer these questions on a separate piece of paper:

1) Write an equation for the farming mat, showing where each piece on the mat came from.  (Remember what x1, x2, and x3 stand for. )

2) What does this mat represent (in total)?

3) What should this mat be equal to?

4) Self Check: Now reread the blue paragraph above.  Does your answer match?

Answer the first three questions for the other mats: manufacturing and clothing.  Remember to show in your equation where each piece on the mat came from.  Take a moment to think about what each mat is representing and what each mat should equal.

 

The three equations you should now see from what we just did are:

Switch now to Graphing calculator.

Input these three equations we got from javabars substituting x1=x, x2=y, and x3=z, and graph these equations.

Answer these questions on a separate sheet of paper:

1) Do the planes intersect at a point or a line?

2) What is the significance of this?   How many solutions will this system of equations have?

3) How can you manipulate these three equations to solve them? (hint matrix form)

4) Once the system is in matrix form, what do you have to do in order to find the ratio between x1, x2, and x3?

5) Using your findings, develop a currency system for the island.  Assign prices to clothing, manufacturing, and farming. 

 

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