Note that the columns in this table all add up to 1 Explain

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

Now, let p_F, p_T, p_C, p_CM, p_SB 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, p_F 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 p_F, p_T, p_C, p_CM, andp_SB subject to the following conditions:

0.25p_F + 0.15p_T + 0.25p_C + 0.18p_CM + 0.20p_SB = p_F
0.15p_F + 0.28p_T + 0.18p_C + 0.17p_CM + 0.05p_SB = p_T
0.22p_F + 0.19p_T + 0.22p_C + 0.22p_CM + 0.10p_SB = p_C
0.20p_F + 0.15p_T + 0.20p_C + 0.28p_CM + 0.15p_SB = p_CM
0.18p_F + 0.23p_T + 0.15p_C + 0.15p_CM + 0.50p_SB = p_SB

Solution

Add all the 5 equations. You will get 0=0. All the pF, pT etc. on the LHS add up to a total of 1 because each occurs only once in the 5 equations with a value of 1 on the RHS. This is necassary to get a consistent solution ( here there are 5 unknows in 5 equations, easily solvable). Each row depicts the income of one occupation\'s employee, as a weighted sum of others\' incomes, to make the city self sufficient.