(***********************************************************************
Mathematica-Compatible Notebook
This notebook can be used on any computer system with Mathematica 4.0,
MathReader 4.0, or any compatible application. The data for the notebook
starts with the line containing stars above.
To get the notebook into a Mathematica-compatible application, do one of
the following:
* Save the data starting with the line of stars above into a file
with a name ending in .nb, then open the file inside the application;
* Copy the data starting with the line of stars above to the
clipboard, then use the Paste menu command inside the application.
Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode. Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).
NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing the
word CacheID, otherwise Mathematica-compatible applications may try to
use invalid cache data.
For more information on notebooks and Mathematica-compatible
applications, contact Wolfram Research:
web: http://www.wolfram.com
email: info@wolfram.com
phone: +1-217-398-0700 (U.S.)
Notebook reader applications are available free of charge from
Wolfram Research.
***********************************************************************)
(*CacheID: 232*)
(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[ 8171, 221]*)
(*NotebookOutlinePosition[ 9025, 251]*)
(* CellTagsIndexPosition[ 8981, 247]*)
(*WindowFrame->Normal*)
Notebook[{
Cell[CellGroupData[{
Cell[TextData[StyleBox["Leontiev Input-Output Model",
FontFamily->"Times",
FontSize->24]], "Input",
TextAlignment->Center,
ImageRegion->{{-0, 1}, {0, 1}}],
Cell[BoxData[{
\(Author : \ Thomas\ Shores\), "\[IndentingNewLine]",
\(University\ of\ Nebraska\), "\[IndentingNewLine]",
\(Send\ comments\ \(to : \ tshores@math . unl . edu\)\)}], "Input",
FontFamily->"Helvetica",
FontWeight->"Plain",
FontVariations->{"CompatibilityType"->0}],
Cell[CellGroupData[{
Cell["Introduction", "Subsection",
CellGroupingRules->{"InputGrouping", 40},
ImageRegion->{{-0, 1}, {0, 1}}],
Cell[TextData[{
"Most administrative systems in industry or government produce two kinds \
of goods: those intended for external consumption, such as a salable \
product, public regulation, customer support, etc., and those intended for \
internal consumption, such as payroll accounting, worker training, facilities \
maintenance,etc. With respect to the latter goods, the system can be likened \
to a closed economy. That is, everything produced by the system is consumed \
by the system. Here is a simplified model of such a situation: suppose an \
administrative unit has four divisions serving the internal needs of the \
unit, which we label as (A)ccounting, (M)aintenance, (S)upplies and \
(T)raining. Each unit produces the ``commodity'' its name suggests, and \
charges the other divisions for its services. A study shows that the \
fraction of commodities consumed by each division is given by the following \
table, called an ",
StyleBox["input-output matrix",
FontSlant->"Italic"],
":\n\n Produced by: A M S T \n \
A 0.2 0.1 0.4 0.4 \n Consumed by: M 0.3 0.4 0.2 \
0.1 \n S 0.3 0.4 0.2 0.3 \n \
T 0.2 0.1 0.2 0.2 "
}], "Text",
CellGroupingRules->"InputGrouping",
ImageRegion->{{-0, 1}, {0, 1}}],
Cell[CellGroupData[{
Cell[TextData[{
"Note that the sum of entries in each column is one. This reflects the \
fact that the system is closed, i.e., everything produced by a given division \
is consumed by the divisions of the unit. If the sum were less than one, \
there would be a surplus of the commodity represented by the \ncolumn -- if \
greater than one, there would be a shortage of it!\n\nHere is the problem: \
what price should each division charge for its commodity so that each \
division earns exactly as much as it spends? Such a pricing scheme is called \
an ",
StyleBox["equilibrium price structure ",
FontSlant->"Italic"],
" in the parlance of economics; it assures that no division will earn too \
little to do its job. Let x , y , z and w be the price per unit \
commodity charged by A, M, S and T, respectively. The requirement of \
expenditures equalling earnings for each division results in the four \
equations following. Click inside the next cell and press to \
activate it....\n "
}], "Text",
CellGroupingRules->"InputGrouping",
ImageRegion->{{-0, 1}, {0, 1}}],
Cell["\<\
{ 0.2x + 0.1y + 0.4z + 0.4w == x,
0.3x + 0.4y + 0.2z + 0.1w == y,
0.3x + 0.4y + 0.2z + 0.3w == z,
0.2x + 0.1y + 0.2z + 0.2w == w }\
\>", "Input",
ImageRegion->{{-0, 1}, {0, 1}}]
}, Open ]],
Cell[CellGroupData[{
Cell["\<\
We can put this system into simpler form by multiplying each \
equation by -10 and bringing all terms to the left hand side to obtain the \
following. Click inside the next cell and press to activate \
it.\
\>", "Text",
CellGroupingRules->"InputGrouping",
ImageRegion->{{-0, 1}, {0, 1}}],
Cell["\<\
equations = {8x - y - 4z - 4w == 0,
-3x + 6y - 2z - 1w == 0,
-3x - 4y + 8z - 3w == 0,
-2x - 1y - 2z + 8w == 0 }\
\>", "Input",
ImageRegion->{{-0, 1}, {0, 1}}]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[{
"Clearly, a trivial solution to the system is x = y = z = w = 0 . This \
solution is not informative. Nonzero solutions give information about the \
relative value of each service. Are there any nontrivial solutions? Let's \
ask ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" for some help. Activate the next cell."
}], "Text",
CellGroupingRules->"InputGrouping",
ImageRegion->{{-0, 1}, {0, 1}}],
Cell["rules = Solve[equations, {x,y,z,w}]", "Input",
ImageRegion->{{-0, 1}, {0, 1}}]
}, Open ]],
Cell[CellGroupData[{
Cell["\<\
Very interesting ... what do you make of this ? Try the next \
cell...\
\>", "Text",
CellGroupingRules->"InputGrouping",
ImageRegion->{{-0, 1}, {0, 1}}],
Cell["equations /. rules", "Input",
ImageRegion->{{-0, 1}, {0, 1}}]
}, Open ]]
}, Closed]],
Cell[CellGroupData[{
Cell["Exercise", "Subsection",
CellGroupingRules->{"InputGrouping", 40},
ImageRegion->{{-0, 1}, {0, 1}}],
Cell["\<\
Suppose there are exactly five companies (Ace, Brown, Collegiate, \
Drake, and Excalibur) that publish textbooks, and that universities have the \
option of buying any book from any publisher. Also suppose that a constant \
proportion of universities that buy books from a given company (say, Ace) \
will purchase books from another company (Drake) the next year, and that \
the empirical statistics for this are:
current year: Ace Brown Collegiate Drake \
Excalibur
next year: Ace 0.30 0.20 0.15 \
0.13 0.12
Brown 0.15 0.35 0.30 \
0.10 0.22
Collegiate 0.25 0.05 0.40 \
0.20 0.25
Drake 0.10 0.12 0.02 \
0.33 0.10
Excalibur 0.20 0.28 0.13 \
0.24 0.31
Here 12% of the universities that bought from Brown in 1991 bought from \
Drake in 1992, for example. Let a,b,c,d,e be the proportion of universities \
buying books from Ace, Brown,
Collegiate, Drake, and Excalibur, respectively.
QUESTION: Does this situation have an equilibrium state? If so, find it. \
(Note that in order to represent the state as proportions of the market, we \
also need the equation a + b + c + d + e = 1. ) \
\>", "Text",
CellGroupingRules->"InputGrouping",
ImageRegion->{{-0, 1}, {0, 1}}]
}, Closed]]
}, Closed]]
},
FrontEndVersion->"4.0 for X",
ScreenRectangle->{{0, 1152}, {0, 864}},
WindowToolbars->{},
CellGrouping->Manual,
WindowSize->{520, 600},
WindowMargins->{{Automatic, 308}, {104, Automatic}},
PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}},
ShowCellLabel->True,
ShowCellTags->False,
RenderingOptions->{"ObjectDithering"->True,
"RasterDithering"->False}
]
(***********************************************************************
Cached data follows. If you edit this Notebook file directly, not using
Mathematica, you must remove the line containing CacheID at the top of
the file. The cache data will then be recreated when you save this file
from within Mathematica.
***********************************************************************)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[1739, 51, 162, 4, 40, "Input"],
Cell[1904, 57, 297, 6, 64, "Input"],
Cell[CellGroupData[{
Cell[2226, 67, 112, 2, 45, "Subsection",
CellGroupingRules->{"InputGrouping", 40}],
Cell[2341, 71, 1375, 22, 70, "Text",
CellGroupingRules->"InputGrouping"],
Cell[CellGroupData[{
Cell[3741, 97, 1106, 19, 70, "Text",
CellGroupingRules->"InputGrouping"],
Cell[4850, 118, 189, 6, 70, "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell[5076, 129, 312, 7, 70, "Text",
CellGroupingRules->"InputGrouping"],
Cell[5391, 138, 170, 6, 70, "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell[5598, 149, 436, 10, 70, "Text",
CellGroupingRules->"InputGrouping"],
Cell[6037, 161, 86, 1, 70, "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell[6160, 167, 167, 5, 70, "Text",
CellGroupingRules->"InputGrouping"],
Cell[6330, 174, 69, 1, 70, "Input"]
}, Open ]]
}, Closed]],
Cell[CellGroupData[{
Cell[6448, 181, 108, 2, 29, "Subsection",
CellGroupingRules->{"InputGrouping", 40}],
Cell[6559, 185, 1584, 32, 70, "Text",
CellGroupingRules->"InputGrouping"]
}, Closed]]
}, Closed]]
}
]
*)
(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)