![{
𝑥1 units 𝑎 + 𝑥2units 𝑏 + 𝑥3units 𝑐 = 1unit 𝑎
𝑥1 units 𝑎 + 𝑥2units 𝑏 + 𝑥3units 𝑐 = 1unit 𝑏
𝑥1 units 𝑎 + 𝑥2units 𝑏 + 𝑥3units 𝑐 = 1unit 𝑐
. (2)
Taking the coefficients d . . . l to represent the constant values of the arbitrary goods or services
and x1, x2, x3 to represent said goods or services a, b, and c from the previous system of
equations, we have the following system of linear equations:
{
𝑑𝑥1 + 𝑒𝑥2 + 𝑓𝑥3 = 1𝑥1
𝑔𝑥1 + ℎ𝑥2 + 𝑖𝑥3 = 1𝑥2
𝑗𝑥1 + 𝑘𝑥2 + 𝑙𝑥3 = 1𝑥3
. (3)
The coefficients in the system of linear equations may be represented as the following
Consumption Matrix C:
C = [
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
𝑗 𝑘 𝑙
] . (4)
Furthermore, we must consider the outside demand of the local economy’s goods and services,
which can be represented by the following:
𝐝 = [
𝑥
𝑦
𝑧
] . (5)
Having established our arbitrary information into an algebraic form, we may employ
several algebraic techniques to discover what the local economy’s total demand for each good or
service is. To solve for the total demand, we must get the matrix into the form
C𝐱 = 𝐝 , ( 𝟔)
where x is the total demand. In other words, we can find the total demand by solving the matrix
in the form](https://image.slidesharecdn.com/ac740152-b86d-4568-8833-06bb7253d3ed-161209171646/85/LinearAssignmentGroupReview-2-320.jpg)
![(I – C) 𝐱 = 𝐝 , ( 𝟕)
where I is the Identity Matrix and C is the Consumption Matrix augmented with the outside
demand d. After using a computer algebra system, like SAGE, the reduction would be as
follows:
[I − C| 𝐝] = [
𝑑 𝑒 𝑓 𝑥
𝑔 ℎ 𝑖 𝑦
𝑗 𝑘 𝑙 𝑧
] ⟶ [
1 0 0 𝑥′
0 1 0 𝑦′
0 0 1 𝑧′
] , (8)
where the total demand for the economy is x and is
𝐱 = {
𝑥′
𝑦′
𝑧′
, (9)
where x’, y’, and z’ are all integers.
This method of analyzing economic demand is useful as it has the potential to describe
multiple economic phenomena, like total demand, equilibrium between bartering artisans, and
values of goods and services.
Model A (Anthony): This economic model represents a plumber, a painter, and an
electrician all trading services. They are also working for themselves in a portion of the trade.
Their total work time amounts to 10 hours per week. The three laborers consume different
quantities of each resource and are wondering how to price their services based on who
consumes which resource, and how much.
The analysis yields the following wages in a fair work-trade.
𝐝 = {
𝑥1 = 39
𝑥2 = 42
𝑥3 = 54
, (10)](https://image.slidesharecdn.com/ac740152-b86d-4568-8833-06bb7253d3ed-161209171646/85/LinearAssignmentGroupReview-3-320.jpg)
![𝐯 = [
2285.71
2250
2178.57
] . (12)
From this, the deduction of how much must be made in consulting was determined within the
model, given the profit target to be $2285.71for person x, $2250 for person y and $2178.57 for
person z. These values can be compared to those in part D because the target profits are identical
for all individuals within the model, only the amount of exchange within the group is different.
Model D (Westley): This economic model represented three consultants who together
are working on a multidisciplinary project. They distributed the work load by purchasing a
portion of each other’s services, however in this specific model it represented three consultants
that resorted to a lot of internal spending in regards to their own services. This model determined
the total dollar amount of consulting for outside orders, they are as follows:
𝐝 = {
𝑥 = $1208.33
𝑦 = $291.67
𝑧 = $541.67
. (13)
These amounts are directly related to the dollar amount of consulting each consultant
performed in this specific week. So, consultant x performed $1208.33 worth of services,
consultant y performed $291.67, and consultant z performed $541.67. In fact, by using this model
we can determine any dollar amount for any span out of the year that each consultant performs
just by altering the matrix and computing the model, via software such as SAGE.](https://image.slidesharecdn.com/ac740152-b86d-4568-8833-06bb7253d3ed-161209171646/85/LinearAssignmentGroupReview-5-320.jpg)
LinearAssignmentGroupReview
- 1. Westley Goebel, Anthony D’Andredi, Wilson Collins, Banks Osborne Dr. Michael Freeze MAT 335 30 November 2015 Group Economic Model Analysis In a theoretical, local economy, certain quantities of arbitrary goods or services are needed to produce certain quantities of other arbitrary goods or services. In many circumstances, the quantities of each good or service needed to produce the other goods or services remain in constant proportion to the total output of the produced commodity. This allows us to observe in a theoretical economy that produces only a few commodities certain points of equilibrium between supply and production. For example, to produce one unit of a good can be represented by the consumption equation 𝑥1 units 𝑎 + 𝑥2units 𝑏 + 𝑥3units 𝑐 = 1unit 𝑎, (1) where x1, x2, x3 are the amounts of inputs a, b, and c, respectively. We must balance the consumption equation, and others like it, to determine the proper proportion of inputs to be supplied to produce a single good. We will discuss in further detail the process and the other equations in a future paragraph. There are a couple of questions which may be explored using this model. One example of which seeks to find an equilibrium value of goods and services when given the hourly rates of artisans who trade services of differing values among themselves. Also, another model can be used to determine the total demand for an economy’s goods when given the normal supply and consumption rates and an outside order for more production in a local economy. Let us consider the consumption equations for some arbitrary goods or services in the local economy, where the term “units” is the equivalent of Dollars U.S.:
- 2. { 𝑥1 units 𝑎 + 𝑥2units 𝑏 + 𝑥3units 𝑐 = 1unit 𝑎 𝑥1 units 𝑎 + 𝑥2units 𝑏 + 𝑥3units 𝑐 = 1unit 𝑏 𝑥1 units 𝑎 + 𝑥2units 𝑏 + 𝑥3units 𝑐 = 1unit 𝑐 . (2) Taking the coefficients d . . . l to represent the constant values of the arbitrary goods or services and x1, x2, x3 to represent said goods or services a, b, and c from the previous system of equations, we have the following system of linear equations: { 𝑑𝑥1 + 𝑒𝑥2 + 𝑓𝑥3 = 1𝑥1 𝑔𝑥1 + ℎ𝑥2 + 𝑖𝑥3 = 1𝑥2 𝑗𝑥1 + 𝑘𝑥2 + 𝑙𝑥3 = 1𝑥3 . (3) The coefficients in the system of linear equations may be represented as the following Consumption Matrix C: C = [ 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 𝑗 𝑘 𝑙 ] . (4) Furthermore, we must consider the outside demand of the local economy’s goods and services, which can be represented by the following: 𝐝 = [ 𝑥 𝑦 𝑧 ] . (5) Having established our arbitrary information into an algebraic form, we may employ several algebraic techniques to discover what the local economy’s total demand for each good or service is. To solve for the total demand, we must get the matrix into the form C𝐱 = 𝐝 , ( 𝟔) where x is the total demand. In other words, we can find the total demand by solving the matrix in the form
- 3. (I – C) 𝐱 = 𝐝 , ( 𝟕) where I is the Identity Matrix and C is the Consumption Matrix augmented with the outside demand d. After using a computer algebra system, like SAGE, the reduction would be as follows: [I − C| 𝐝] = [ 𝑑 𝑒 𝑓 𝑥 𝑔 ℎ 𝑖 𝑦 𝑗 𝑘 𝑙 𝑧 ] ⟶ [ 1 0 0 𝑥′ 0 1 0 𝑦′ 0 0 1 𝑧′ ] , (8) where the total demand for the economy is x and is 𝐱 = { 𝑥′ 𝑦′ 𝑧′ , (9) where x’, y’, and z’ are all integers. This method of analyzing economic demand is useful as it has the potential to describe multiple economic phenomena, like total demand, equilibrium between bartering artisans, and values of goods and services. Model A (Anthony): This economic model represents a plumber, a painter, and an electrician all trading services. They are also working for themselves in a portion of the trade. Their total work time amounts to 10 hours per week. The three laborers consume different quantities of each resource and are wondering how to price their services based on who consumes which resource, and how much. The analysis yields the following wages in a fair work-trade. 𝐝 = { 𝑥1 = 39 𝑥2 = 42 𝑥3 = 54 , (10)
- 4. where x1, x2, and x3 are the electrician’s, plumber’s, and painter’s wages, respectively. These amounts are in dollars and represent how valuable the workers’ time is. The electrician supplies the most valuable time and therefore receives the highest wage. He is followed by the plumber and the painter supplies the least valuable time and has the lowest wage. In reality the wages could be any dollar amount as long as the respect the time value ratio, however it is specified that the rates must be whole numbers between thirty and sixty. Hence, the results must be as shown. Model B (Banks): As alluded to, the linear algebra model can describe demand for an economy’s goods and services. This model revealed that the total demand, which includes the normal internal demand and the excess external demand, of the local economy’s goods is 𝐝 = { 𝑐 = 102087 𝑒 = 56163 𝑡 = 28330 . (11) Consequently, we can conclude that the local economy needs to produce $102087 worth of coal, $56163 worth of electricity, and $28330 of transportation to adequately provide for both their internal and external demands. Model C (Wilson): This linear algebra model describes the basic sales in services, or possibly goods, required by an economic group to produce a specific amount of profit if there are known costs associated with the sails that occur do to exchange within the group. In this particular instance the model final equation represents a situation in which the exchange between individuals is relatively low and how that reduced exchange affects the relative sales required to meet a specified profit and the overall impact on the economy. The results were given as a vector,
- 5. 𝐯 = [ 2285.71 2250 2178.57 ] . (12) From this, the deduction of how much must be made in consulting was determined within the model, given the profit target to be $2285.71for person x, $2250 for person y and $2178.57 for person z. These values can be compared to those in part D because the target profits are identical for all individuals within the model, only the amount of exchange within the group is different. Model D (Westley): This economic model represented three consultants who together are working on a multidisciplinary project. They distributed the work load by purchasing a portion of each other’s services, however in this specific model it represented three consultants that resorted to a lot of internal spending in regards to their own services. This model determined the total dollar amount of consulting for outside orders, they are as follows: 𝐝 = { 𝑥 = $1208.33 𝑦 = $291.67 𝑧 = $541.67 . (13) These amounts are directly related to the dollar amount of consulting each consultant performed in this specific week. So, consultant x performed $1208.33 worth of services, consultant y performed $291.67, and consultant z performed $541.67. In fact, by using this model we can determine any dollar amount for any span out of the year that each consultant performs just by altering the matrix and computing the model, via software such as SAGE.