FREC 240 Assignment #6 -- Intro to Matrix Algebra
(1.)  If ...
A = 1 2   B = 2 3 4    C = 3 4    D = 4 5 6
    3 4       5 6 7        5 6        7 8 9
                           7 8        3 2 1
then calculate the following (show your work):

a.  B' + C
b.  C' - B		 c.  AB
d.  BC - A		 e.  ABC
f.  (ABC)-1		 g.  B'C' + D
h.  BD		 i.  BDC
j.  (BDC)-1
(2.)  Q(mxn) is a matrix of quantities of m different goods sold monthly in n different store locations.  P is a (1xm) row vector of goods prices (the stores all use the same price set).  A is a (nx1) column vector of  1's. 
  1. What information does PQ give you?
  2. What information does QA give you?
  3. What information does PQA give you?
(3.)  Suppose you have estimated the following demand system for Hops, Flumps and Tongs:
QH = 20 -2.0PH -0.5PF +0.3PT
QF = 10 -0.4PH -1.5PF +0.4PT
QT = 20 +0.2PH +0.3PF -1.0PT
or in matrix form
Q  =  C + AP
where Q is the column vector of demand quantities, C is the column vector of constant terms, A is the (3x3) matrix of slope coefficients, and P is the column vector of prices.
  1. If P' = [ $4.00  $3.00  $6.00 ], use matrix algebra to calculate the market equilibrium set of quantities Q.
  2. If actual supply is Q' = [ 10  5  15 ] use matrix algebra to calculate the market equilibrium price set P.
(4.)  An economy's production is split between intermediate demand (stuff bought and used by firms to make other stuff) and final demand (stuff bought and consumed by households).  Mathematically, AX + D = X, where A is the proportion of total output X satisfying intermediate demand, and D is final demand.  Solving for X yields X = (1-A)-1D.  So the total extra output necessary to satisfy a $1 increase in final demand will be $(1-A)-1.  For example, if A = 0.6 (60% of output goes intermediate demand, 40% goes to final demand), then it will take 1/(1-0.6) = $2.50 increase in total output to satisfy a $1 increase in final demand.  The final demand multiplier is 2.5. 

Using the same logic, we can disaggregate this economy into multiple industry sectors, and use matrix algebra to analyze the linkages between different sectors and calculate final demand multipliers for each sector.  Download the 8-sector Transactions Table for the U.S. economy (1999 data, in $millions) from the class website (www.udel.edu/johnmack/frec240/240hw6.html), or just type the data below into a spreadsheet:

U.S. Industry Transactions, 1999 ($millions)
  
  SECTOR                1         2          3          4          5          6          7          8
1 Agriculture     $67,179       $87     $6,486   $139,679       $191     $4,775    $13,464    $12,070
2 Mining             $348   $31,951     $7,560   $102,449    $58,919        $31         $5        $30
3 Construction     $3,232    $4,082       $846    $29,043    $53,881    $13,635    $65,611    $28,956
4 Manufacturing   $48,321   $15,400   $316,212 $1,410,342    $77,978   $109,898    $20,481   $321,771
5 Trans/Utils     $13,253   $12,204    $26,700   $189,446   $219,281    $76,342    $57,983   $119,980
6 Trade           $14,602    $3,682    $88,143   $243,338    $18,701    $41,944     $5,496    $70,772
7 Financial       $19,250   $37,582    $16,122    $74,092    $43,295   $119,886   $454,962   $247,877
8 Services         $9,826    $6,541   $111,242   $262,398   $162,462   $246,647   $211,945   $552,838
  Other/Imports      $245    $2,083     $1,122    $41,226    $25,678    $20,584    $34,216    $30,590
  Total Intermed $176,256  $113,611   $574,433 $2,492,012   $660,386   $633,742   $864,165 $1,384,883
  Value added    $103,000   $48,145   $491,941 $1,617,034   $672,056 $1,187,230 $1,821,347 $2,195,714
  Total Output   $279,256  $161,756 $1,066,373 $4,109,046 $1,332,441 $1,820,972 $2,685,512 $3,580,597

Each column in the upper 8x8 section of this table shows industry sector j's purchases of intermediate goods from each sector i.  Each row shows sector i's sales to sector j.  Total intermediate demands, value added and total outputs are shown in the final three rows. 
  1. Underneath this part of the spreadsheet, calculate the 8x8 technical coefficients matrix A showing each sector j's direct requirements for its own product and other sectors' products, expressed as a decimal percent of its total output.  (Divide each cell in the upper 8x8 section by the Total Output cell below it.)
  2. Below the technical coefficients table, create an 8x8 identity matrix I. 
  3. Below the identity matrix, calculate the technology matrix (I - A).  This table shows each sector's proportional net output of each product (the negative terms reflect net consumption).
  4. Below the technology matrix, use Excel's MINVERSE function to calculate the multiplier matrix (I - A)-1.  This table shows each sector j's direct and indirect requirements necessary to supply an extra dollar of its product to end users.  
  5. Sum each column of multipliers to obtain aggregate sector multipliers.  These show the total increases in economic output across all sectors that will be generated from an additional $1 of final demand for industry j's product.
  6. Which sectors have the highest aggregate multipliers?  Can you explain why?