exercise:84096bfca6: Difference between revisions
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(Created page with "<div class="d-none"><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></div> In the Leontief economic model,<ref group="Notes" >W. W. Leontief, ''Input-Output Economics'' (Oxford: Oxford University Press, 1966).</ref> there are <math>n</math> industries 1, 2, \ldots, <math>n</math>. The <math>i</math>th industry requires...") |
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\newcommand{\mathds}{\mathbb}</math></div> In the Leontief economic model,<ref group="Notes" >W. W. Leontief, | \newcommand{\mathds}{\mathbb}</math></div> In the Leontief economic model,<ref group="Notes" >W. W. Leontief, ''Input-Output Economics'' (Oxford: Oxford University Press, 1966).</ref> there are <math>n</math> industries 1, 2, ..., <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>. | ||
''Input-Output Economics'' (Oxford: Oxford University Press, 1966).</ref> there | <ul style="list-style-type:lower-alpha"><li> Show that if the industries produce total amounts given by the vector | ||
are | |||
<math>n</math> industries 1, 2, | |||
\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>. | |||
<ul><li> 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 | <math>\mat{x} = (x_1,x_2,\ldots,x_n)</math> then the amounts of goods of each type that | ||
the | the | ||
Latest revision as of 23:01, 15 June 2024
[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,[Notes 1] there are [math]n[/math] industries 1, 2, ..., [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