Homework 1bFinite Markov ChainsMATH 56A: STOCHASTIC PROCESSESHOMEWORKFrom the syllabus: There will be weekly homework. Students are encouraged to work ontheir homework in groups and to access all forms of aid including expert advice, internetand other resources. The work you hand in should, however, be in your own words and inyour own handwriting. Every student hands in his own homework. There will be a penaltyfor late homework.Homework 1bFinite Markov ChainsThis homework project is due next Thursday, Feb 14:Your assignment is to analyze the Leontief economic model and work out one example.Don’t do the calculations by hand.In the Leontief model, there are “factories” which require the output of other factories tomake their product. For each dollar of output, Factory i requires qijdollars worth of outputof factory j. The total amount that factory i needs to spend for each dollar of output isqi1+ qi2+ · · · + qir≤ 1.We always assume the sum is ≤ 1. (But it is allowed to be equal to 1.)Work out the following example and answer the questions (in complete sentences so thatyour kid brother can read it!)Example: We have 4 factories:S = SteelW = WaterE = Coal/Gas/OilP = PlasticTo produce $1 worth of steel, the steel factory needs 50c| worth of energy and 25c| worthof water (and no plastic). This goes into the matrix Q in the first line.Q =0 .25 .5 0.1 0 .4 .2.2 .1 .3 .1.1 .2 .2 .1Each factory keeps a stockpile of material, say 10$ worth of each item. When it get an orderfor goods, the factory uses its inventory and then orders replacements. So, the steel factory,after filling out an order for 1$ worth of steel will order 25c| worth of water and 50c| worth ofenergy.(*) Write in words: What do the numbers in the fourth row of the matrix mean?1The consumer wants 1$ of steel, 2$ of water, 10$ of energy and 2$ of plastic.(a) How much does each factory need to make?(b) Follow the money: Where do the 15$ go after 4 rounds?(c) How long does it take for all factories to regain 99% of their original inventory assumingthat they keep 10$ worth of each commodity in stock.(d) Follow the energy. Take the total amount of energy (your answer to part (a)) that isneeded. Where does it go?Theory: Assume we have a more general matrix Q representing the requirements of eachindustry in the Leontief model.(e) The rows of Q may not add up to less than 1. If row i adds up to 1, what does itmean about state i? Give an example.(f) Suppose that every row of Q adds up to at most p = 0.9. Then prove that each rowof Qnadds up topn= (0.9)nor less.Why does this imply that the sequenceI + Q + Q2+ Q3+ · · ·converges? [Hint: a series (infinite sum) of matrices converges if and only if, for each i andj the sum of the (i, j) entries converges. Use the comparison test, comparing these entriesto a geometric series to show that the series
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