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  # Systems of Linear Equations
  
  Markov Chains and
  Dynamic Models
  
  The refinery and economic supply-demand models of Section 1.2 were static
  in the sense that we solved them once and that was it. There was one set of
  production levels required, not a sequence of levels that would be needed
  to describe an economy changing over time. A model that tries to predict
  the behavior of a system over a period of time is called a dynamic model.
  In this section we examine two dynamic linear models.
  
  The first dynamic model we consider involves probability. This model,
  called a Markov chain, will arise over and over in this book, so it is important
  to understand the model well. The concepts of probability we need for this
  model are simple and intuitive.
  
  A Markov chain is a probabilistic model that describes the random
  movement over time of some activity. At each period of time, the activity
  is in one of several possible states. States might be amounts won in gam-
  
  yield
  
  heating oil = 520,
  diesel oil = 890,
  gasoline = 1040
  
  This is getting quite close to our production goals. The overproduction
  of 20 to 40 gallons in each product might be a reasonable safety margin
  that is actually desirable.
  
  To get closer, we should decrease x2 and x3 a little.
  
  yield
  
  heating oil = 504,
  diesel oil = 852,
  gasoline = 1006
  
  This is an excellent fit. We have been a bit lucky. To do better, we
  would probably have to use fractional values. (In the Exercises the
  reader may need more tries to get this close.)
  
  We next consider a slightly more complicated supply-demand model.
  This model has the balancing advantage that trial-and-error calculations to
  estimate a solution are easier. The reader is warned that it takes a little while
  to get a feel for all the numbers in this model.
  
  Example 2. A Model of General Economic
  Supply-Demand
  
  We present a linear model due to W. Leontief, a Nobel Prize-winning
  economist. The model seeks to balance supply and demand throughout
  a whole economy. For each industry, there will be one supply-demand
  equation. In practical applications, Leontief economic models can have
  hundreds or thousands of specific industries. We consider an example
  with four industries.
  
  The left-hand side of each equation is the supply, the amount
  produced by the ith industry. Call this quantity xi; it is measured in
  dollars. On the right-hand side, we have the demand for the product
  of the ith industry. There are two parts to the demand. The first part
  is demand for the output by other industries (to create other products
  requires some of this product as input). The second part is consumer
  demand for the product.
  
  For a concrete instance, let us consider an economy of four general
  industries: energy, construction, transportation, and steel. Suppose
  that the supply-demand equations are
  
   Industrial Demands Supply Transport. Energy Construct. Steel Consumer Demand Energy: x1= Construct.: x2= Transport. : x3= Steel: x4=
  
  (6)
  
  The first equation, for energy, has the supply of energy x1 on the
  left. The terms on the right of this equation are the various demands
  that this supply must meet. The first term on the right, 4x1 is the
  input of energy required to produce our x1dollars of energy (A units
  of energy input for one unit of energy output). Also, the second term
  of .2x2 is the input of energy needed to make x2 dollars of construction.
  Similarly, terms .2x3 and .2x4 are energy inputs required for transportation
  and steel production. The final term of 100 is the fixed consumer
  demand.
  
  Each column gives the set of input demands of an industry. For
  example, the third column tells us that to produce the x3 dollars of
  transportation requires as input .2x3 dollars of energy, .2x3 dollars of
  construction, and .1x3 dollars of steel. In the previous refinery model,
  the demand for each product was a single constant quantity. In the
  Leontief model, there are many unknown demands that each industry's
  output must satisfy. There is an ultimate consumer demand for each
  output, but to meet this demand industries generate input demands on
  each other. Thus the demands are highly interrelated: Demand for
  energy depends on the production levels of other industries. and these
  production levels depend in turn on the demand for their outputs by
  other industries, and so on.
  
  When the levels of industrial output satisfy these supply-demand
  equations, economists say that the economy is in equilibrium.
  
  As in the refinery model, let us try to solve this system of equations
  by trial-and-error. As a first guess, let us set the production levels
  at twice the consumer demand (the doubling tries to account for the
  interindustry demands). So , and
  x4 = 0; these are our supplies. Given these production levels, we can
  compute the demands from (6).
  
   Supply Demand Energy: 200 Construct.: 100 Transport.: 200 Steel: 0
  
  (7)
  
  For our next approximations, let us try supply levels halfway
  between the supply and demand values in (7). That is,
  , and similarly, , and
  
   Supply Demand Energy: 220 Construct.: 140 Steel: 165 Steel: 15
  
  (8)
  
  The second approximation is only moderately better. The interaction
  effects between different industries are hard to predict. Adjusting production
  levels was much easier in the refinery problem, where the
  demand for each product was constant.
  
  Let us stop trying to be clever and just use the simple-minded
  approach of setting production levels (i.e., supply levels) equal to the
  previous demand levels. So from (8), we try
  
   Supply Demand Energy: 252 Construct. : 192 Transport. : 139 Steel: 30
  
  (9)
  
  The demand values here have been rounded to whole numbers. The
  supplies and demands are getting a little closer together in (9).Repeating
  the process of setting the new supply levels equal to the previous
  demand levels (i.e., the demands on the right side in (9)) yields
  
   Supply Demand Energy: 273 Construct. : 214 Transport. : 150 Steel: 33
  
  (10)
  
  Repeating this process again, we have
  
   Supply Demand Energy: 289 Construct. : 229 Transport. : 155 Steel: 36
  
  (11)
  
  Observe that in successive rounds (9), (10), (11), supplies are
  rising. This is because as we produce more, we need more input which
  requires us to produce still more, and so on. It may be that this iteration
  will go on forever, and no equilibrium exists. On the other hand, the
  gap between supplies and demands is decreasing.
  
  Leontief proposed a constraint on the input costs that we shall
  show (in Section 3.4) guarantees that an equilibrium exists. The constraint
  is
  
  Input Constraint. Every industry is profitable: Every industry must
  require less than \$1 of inputs to produce \$1 of output.
  
  In mathematical terms, this means that the sum of the coefficients
  in each column must be less than I. Our data in (6) satisfy this constraint,
  so an equilibrium does exist for this four-industry economy.
  Moreover, the iteration process of repeatedly setting production levels
  equal to the previous demands will converge to this equilibrium. The
  reader should check that the following numbers are equilibrium values
  (rounded to the nearest integer).
  
  Equilibrium: energy = 325, construction = 265,
  transportation = 168, steel = 43
  
  Note that any system of linear equations can be rewritten in the form
  of supply-demand equations with xi appearing alone on the left side of the
  ith equation, as in the Leontief supply-demand model (6). It is standard
  practice to solve large systems of linear equations by some sort of iterative
  method. The nature of the supply-demand equations suggested the iterative
  scheme we used here, letting the demands from one round be the production
  levels of the next round .
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