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{SECT 0 {PARA 260 "" 0 "" {TEXT -1 33 "Leontief Input-Output Open Mode
ls" }}{PARA 0 "" 0 "" {TEXT -1 125 "Example: A town has three main ind
ustries: a coal-mining operation, an eletric power generating plant, a
nd a local railroad. " }}{PARA 0 "" 0 "" {TEXT -1 147 "To mine $1 of c
oal, the mining operation must purchase $.25 of electricity to run its
 equipment and $.25 of transportation for its shipping needs. " }}
{PARA 0 "" 0 "" {TEXT -1 166 "To produce $1 of eletricity, the generat
ing plant requires $.65 of coal for fuel, $.05 of its own electricity \+
to run auxilliary equipment, and $.05 of transportation." }}{PARA 0 "
" 0 "" {TEXT -1 131 "To provide $1 of transportation, the railroad req
uires $.55 of coal for fuel, and $.10 of elctricity for its auxilliary
 equipment. " }}{PARA 0 "" 0 "" {TEXT -1 348 "In a certain week, the c
oal-mining operation receives orders for $50,000 of coal from outside \+
the town, and the gnerating plant receive orders for $25,000 of electr
icity from outside. There is no outside demand for the local railroad.
 How much of the three industries produce in that week to exactly sati
sfy their own demand and the outside demand?" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 74 "Accordin
g to the information, we set up the following input-output matrix." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "A := matrix([[0,.65,.55],[.2
5,.05,.10],[.25,.05,0]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-
%'matrixG6#7%7%\"\"!$\"#l!\"#$\"#bF-7%$\"#DF-$\"\"&F-$\"#5F-7%F1F3F*" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 294 "We are solving the system X=AX+E, where X represents the
 outputs matrix for the three industiries and E represents the externa
l demands. Notice that we may write the system as X-AX=E or (I-A)X=E. \+
In this case, we are solving a nonhogenous system, BX=E where, B=I-A a
nd E is the external matrix." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "B := evalm(diag(1$3)-A);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"BG-%'matrixG6#7%7%\"\"\"$!#l!\"#$!#bF-7%$!#DF-$\"#&*F-$!#5F-7%F1$!\"
&F-F*" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "E := matrix(3,1,[50000,25000,0]);\n\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"EG-%'matrixG6#7%7#\"&++&7#\"&+]#7#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rref(augment(B,E));\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7&\"\"\"\"\"!F)$\"+_Z(3-
\"!\"%7&F)F(F)$\"+(=-jh&!\"&7&F)F)F($\"+))>+LGF0" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "Thus
 the total output of the coal-mining operation, the power generating p
lan, and the output of the railroad  would be about $102,087, $56,163 \+
and $28,330 repectively." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "2
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