MATH 56A SPRING 2008 STOCHASTIC PROCESSES 511.6.3. Leontief model. The final example is the Leontief economic model.In this model, the numbers in a substochastic matrix are interpretedas being the input requirements of several industries or factories. Istarted with the substochastic matrix:Q =!0.3 0.50.4 0.2"=!q11q12q21q22"Q being substochastic means(1) qij≥ 0,(2) q11+ q12≤ 1 and(3) q21+ q22≤ 1.The states are factories A and B plus one more C (the bank). qijis the amount of output of factory j which factory i needs for eachdollar’s worth of output. For example, the numbers in the first rowmean that for each dollar of product A, factory A needs30c| worth of product A50c| worth of product Bwhich leaves:20c| −→ profit! −→ C(bank)Similarly, Factory B makes 40c| worth of profit for each dollar of output.Putting in the third state C which is recurrent. (Assume the factoriesuse the same bank. Or think of C as representing all banks put to-gether.) Then we get the following probability transition function:P =C A BCAB10 0.2 .3 .5.4 .4 .2The rows add to 1. So, this gives a Markov chain. The question is:What are we measuring the probability of ?This matrix is keeping track of the money as it is being passed backand forth between the factories and the bank. Since the bank is re-current, all of the money eventually ends up in the bank. To make itrandom, we think of the money as a pile of one dollar bills. ThenXn= location of one random dollar at time n.For example, suppose you get 100$.You put 50$ in the bank CYou buy 30$ of product AYou buy 20$ of product B.52 FINITE MARKOV CHAINSThenX0= (.5, .3, .2)because, if you “mark” one of the dollars, the probability that themarked dollar will go to the bank is 0.5, the probability of that markeddollar going to Factory A is 0.3 and for Factory B it is 0.2.Suppose that each factory keeps a stockpile of supplies. After fillingyour order, each factory will have used up a certain amount of itsinventory. It will order supplies to replenish its stockpile and the moneywill move:X1= X0P = (.64, .17, .19)represents the location of the money after one day. For example,Factory A always puts 1/5 of its income into the bank. So, it puts30/5 = 6$ in the bank. Factory B puts 40% of its income into thebank. So, it puts 20 ∗ 4/10 = 8$ into the bank. So50 + 6 + 8 = 64$will be in the bank after one day. This is the first coordinate of X1times 100$.Xn= X0Pngives the distribution of the money you put into the system after ndays. Notice that, in this model, the total amount of money neverchanges! But the amount of goods produced can be very large.Using what we know about Markov chains we can answer questionsabout the output of the factories.Question: How much does Factory A need to produce in total?The answer is the A coordinate of the vector100$ · (X0+ X0P + X0P2+ · · · )We can ignore the first (C) coordinate since the money in the bankjust sits there and doesn’t do anything. So, the answer is equal to(30, 20)(I + Q + Q2+ · · · )1where the ( )1means 1st coordinate. The vector (30, 20) represents100X0. Using the formula that the series I + Q + Q2· · · converges to(I − Q)−1we get(30, 20)(I − Q)−11= ($88.888..., $80.555...)1= $8889Students figured out that what I wrote on the board did not makesense because the numbers must be greater than 30 and 20. Here, Iused Excel to calculate the matrix inverse and product more accurately.(I used “= index(minverse(...), i,
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