Linear Algebra/Topic: Input-Output Analysis/Solutions

Solutions Hint: these systems are easiest to solve on a computer. Problem 1 With the steel-auto system given above, estimate next year's total productions in these cases. <li> Next year's external demands are: up from this year for steel, and unchanged for autos. <li> Next year's external demands are: up for steel, and up for autos. <li> Next year's external demands are: up for steel, and up for autos. Answer These answers were given by Octave. <li> With the external use of steel as and the external use of autos as , we get , . <li> , <li> , Problem 2 In the steel-auto system, the ratio for the use of steel by the auto industry is , about . Imagine that a new process for making autos reduces this ratio to . <li> How will the predictions for next year's total productions change compared to the first example discussed above (i.e., taking next year's external demands to be for steel and for autos)? <li> Predict next year's totals if, in addition, the external demand for autos rises to be because the new cars are cheaper. Answer Octave gives these answers. <li> , <li> , Problem 3 This table gives the numbers for the auto-steel system from a different year, 1947 (see Leontief 1951). The units here are billions of 1947 dollars. <TR> <TD style="border-right:2px solid black; border-bottom:2px solid black;"> <TD style="border-bottom:2px solid black;">  used by    steel   <TD style="border-bottom:2px solid black;">  used by     auto   <TD style="border-bottom:2px solid black;">  used by    others   <TD style="border-bottom:2px solid black;">  total   <TR> <TD style="border-right:2px solid black;">   value of    steel   <TD>  6.90   <TD>  1.28   <TD> <TD>  18.69   <TR> <TD style="border-right:2px solid black;">  value of    auto   <TD>  0   <TD>  4.40   <TD> <TD>  14.27   <li> Solve for total output if next year's external demands are: steel's demand up 10% and auto's demand up 15%. <li> How do the ratios compare to those given above in the discussion for the 1958 economy? <li> Solve the 1947 equations with the 1958 external demands (note the difference in units; a 1947 dollar buys about what $1.30 in 1958 dollars buys). How far off are the predictions for total output? Answer <li> These are the equations. Octave gives and . <li> These are the ratios. <TR> <TD> <TR> <TD style="border-right:2px solid black;border-bottom:2px solid black;">  1947   <TD style="border-bottom:2px solid black;">  by steel   <TD style="border-bottom:2px solid black;">  by autos   <TR> <TD style="border-right:2px solid black;">  use of steel   <TD>  0.63   <TD>  0.09   <TR> <TD style="border-right:2px solid black;">  use of autos   <TD>  0.00   <TD>  0.69 <TD><TR> <TD style="border-right:2px solid black;border-bottom:2px solid black;">  1958   <TD style="border-bottom:2px solid black;">  by steel   <TD style="border-bottom:2px solid black;">  by autos   <TR> <TD style="border-right:2px solid black;">  use of steel   <TD>  0.79   <TD>  0.09   <TR> <TD style="border-right:2px solid black;">  use of autos   <TD>  0.00   <TD>  0.70    <li> Octave gives (in billions of 1947 dollars) and . In billions of 1958 dollars that is and . Problem 4 Predict next year's total productions of each of the three sectors of the hypothetical economy shown below <TR> <TD style="border-right:2px solid black; border-bottom:2px solid black;"> <TD style="border-bottom:2px solid black;">  used by    farm   <TD style="border-bottom:2px solid black;">  used by     rail   <TD style="border-bottom:2px solid black;">  used by     shipping   <TD style="border-bottom:2px solid black;">  used by    others   <TD style="border-bottom:2px solid black;">  total   <TR> <TD style="border-right:2px solid black;">   value of    farm   <TD>  25   <TD>  50   <TD>  100   <TD> <TD>  500   <TR> <TD style="border-right:2px solid black;">  value of    rail   <TD>  25   <TD>  50   <TD>  50   <TD> <TD>  300   <TR> <TD style="border-right:2px solid black;">  value of    shipping   <TD>  15   <TD>  10   <TD>  0   <TD> <TD>  500   if next year's external demands are as stated. <li> for farm, for rail, for shipping <li> for farm, for rail, for shipping Problem 5 This table gives the interrelationships among three segments of an economy (see Clark Coupe 1967). <TR> <TD style="border-right:2px solid black; border-bottom:2px solid black;"> <TD style="border-bottom:2px solid black;">  used by    food   <TD style="border-bottom:2px solid black;">  used by     wholesale   <TD style="border-bottom:2px solid black;">  used by     retail   <TD style="border-bottom:2px solid black;">  used by    others   <TD style="border-bottom:2px solid black;">  total   <TR> <TD style="border-right:2px solid black;">   value of    food   <TD>  0   <TD>  2 318   <TD>  4 679   <TD> <TD>  11 869   <TR> <TD style="border-right:2px solid black;">  value of    wholesale   <TD>  393   <TD>  1 089   <TD>  22 459   <TD> <TD>  122 242   <TR> <TD style="border-right:2px solid black;">  value of    retail   <TD>  3   <TD>  53   <TD>  75   <TD> <TD>  116 041   We will do an Input-Output analysis on this system. <li> Fill in the numbers for this year's external demands. <li> Set up the linear system, leaving next year's external demands blank. <li> Solve the system where next year's external demands are calculated by taking this year's external demands and inflating them 10%. Do all three sectors increase their total business by 10%? Do they all even increase at the same rate? <li> Solve the system where next year's external demands are calculated by taking this year's external demands and reducing them 7%. (The study from which these numbers are taken concluded that because of the closing of a local military facility, overall personal income in the area would fall 7%, so this might be a first guess at what would actually happen.) References Wassily W. Leontief Input-Output Economics Scientific American 185 4 Oct. 1951 15. David H. Clark John D. Coupe The Bangor Area Economy Its Present and Future Reprot to the City of Bangor, ME Mar. 1967.