Suppose in a far away land of Eigenbazistan, in a small country town called Matr
ID: 3693011 • Letter: S
Question
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.
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
1) Note that the columns in this table all add up to 1. Explain why this happens.
2) 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. Use Matlab to find pF, pT, pC, pCM, pSB.
I know you need to use the commands eye(n,m) and rref(A) but I'm not sure how to implement them.
Explanation / Answer
C = [0.25 0.15 0.25 0.18 0.20;
0.15 0.28 0.18 0.17 0.05;
0.22 0.19 0.22 0.22 0.10;
0.20 0.15 0.20 0.28 0.15;
0.18 0.23 0.15 0.15 0.50]
I = eye(5)
d = [0;0;0;0;0]
now augmenting the matrix [C - I | d] and writting down the general solution to (C - I)p = 0.
M=rref([(C - I )d])
Output: ans = 1.0000 0 0 0 -0.8025 0
0 1.0000 0 0 -0.5919 0
0 0 1.0000 0 -0.7108 0
0 0 0 1.0000 -0.7520 0
0 0 0 0 0 0