Suppose in a far away land of Eigenbazistan, in a small country town called Matrixville, there lived a Farmer, a Tailor, a Carpenter, a Coal Miner and Slacker Bob. The Farmer produced food; the Tailor, clothes; the Carpenter, housing; the Coal Miner supplied energy; and Slacker Bob made High Quality 100 Proof Moonshine, half of which he drank himself. Let us make the following assumptions:

Everyone buys from and sells to the central pool (i.e. there is no outside supply and demand)

Everything produced is consumed For these reasons this is called a closed exchange model.

Next we must specify what fraction of each of the goods is consumed by each person in our town. Here is a table containing this information:

	Food	Clothes	Housing	Energy	High Quality 100 Proof Moonshine
Farmer	0.25	0.15	0.25	0.18	0.20
Tailor	0.15	0.28	0.18	0.17	0.05
Carpenter	0.22	0.19	0.22	0.22	0.10
Coal Miner	0.20	0.15	0.20	0.28	0.15
Slacker Bob	0.18	0.23	0.15	0.15	0.50

So for example, the Carpenter consumes 22% of all food, 19% of all clothes, 22% of all housing, 22% of all energy and 10% of all High Quality 100 Proof Moonshine.

Now, let pF, pT, pC, pCM, pSB denote the incomes of the Farmer, Tailor, Carpenter, Coal Miner and Slacker Bob, respectively. Note that each of these quantities not only denotes the incomes of each of our esteemed citizens, but also the cost of the corresponding goods. So for example, pF is the Farmer's income as well as the cost of all the food. So if the Farmer produces $100 worth of food, then his income will also be $100 since all of this food is bought out and the profits go to the Farmer.

The idea is, of course, to be able to figure out how should we price the goods in order for the citizens of Matrixville to survive; i.e. we must find pF, pT, pC, pCM, and pSB subject to the following conditions:

0.25pF + 0.15pT + 0.25pC + 0.18pCM + 0.20pSB = pF

0.15pF + 0.28pT + 0.18pC + 0.17pCM + 0.05pSB = pT

0.22pF + 0.19pT + 0.22pC + 0.22pCM + 0.10pSB = pC

0.20pF + 0.15pT + 0.20pC + 0.28pCM + 0.15pSB = pCM

0.18pF + 0.23pT + 0.15pC + 0.15pCM + 0.50pSB = pSB

a) Note that the columns in this table all add up to 1. Explain why this happens.

b) Explain where this system of equations came fromand what it means (Think about what the left hand side and right hand side of each equation means?)

c) Do you think this system of linear equations will have a unique solution?

Give reason to your answer.