[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

In the Leontief economic model, there

are [math]n[/math] industries 1, 2, \ldots, [math]n[/math]. The [math]i[/math]th industry requires an amount [math]0 \leq q_{ij} \leq 1[/math] of goods (in dollar value) from company [math]j[/math] to produce 1 dollar's worth of goods. The outside demand on the industries, in dollar value, is given by the vector [math]\mat{d} = (d_1,d_2,\ldots,d_n)[/math]. Let [math]\mat{Q}[/math] be the matrix with entries [math]q_{ij}[/math].

• Show that if the industries produce total amounts given by the vector [math]\mat{x} = (x_1,x_2,\ldots,x_n)[/math] then the amounts of goods of each type that the industries will need just to meet their internal demands is given by the vector [math]\mat{x} \mat{Q}[/math].
• Show that in order to meet the outside demand [math]\mat{d}[/math] and the internal demands the industries must produce total amounts given by a vector [math]\mat{x} = (x_1,x_2,\ldots,x_n)[/math] which satisfies the equation [math]\mat{x} = \mat{x} \mat{Q} + \mat{d}[/math].
• Show that if [math]\mat{Q}[/math] is the [math]\mat{Q}[/math]-matrix for an absorbing Markov chain, then it is possible to meet any outside demand [math]\mat{d}[/math].
• Assume that the row sums of [math]\mat{Q}[/math] are less than or equal to 1. Give an economic interpretation of this condition. Form a Markov chain by taking the states to be the industries and the transition probabilites to be the [math]q_{ij}[/math]. Add one absorbing state 0. Define
  [[math]] q_{i0} = 1 - \sum_j q_{ij}\ . [[/math]]
  Show that this chain will be absorbing if every company is either making a profit or ultimately depends upon a profit-making company.
• Define [math]\mat{x} \mat{c}[/math] to be the gross national product. Find an expression for the gross national product in terms of the demand vector [math]\mat{d}[/math] and the vector [math]\mat{t}[/math] giving the expected time to absorption.

Notes