Answer:
-1(I - A) -1
Consider the following interindustry transactions matrix (X) and total outputs vector (z) for a two- sector economy.
a. What are the two elements in the final demand (fd) vector?
b. What are the two elements in the total inputs vector (q) and in the final-payments (fp) vector?
What does the complete transactions table look like now?
c. What is the algebraic representation of the A matrix, where aij = xij/qj?
A = X-1
or
Calculate the numeric values for this A matrix and for (I-A):
d. Using your hand-held calculator or a spreadsheet, what is the inverse of this Leontief matrix?
e. With this A matrix and the following new final-demand vector (now y), calculate the first 3 rounds of spending which would occur in seeking the new outputs which would satisfy the new final-demand.
What is the equation for this 'rounds of spending' solution to the model?
It is the matrix equation for the sum of an infinite series:
T = A0y + A1y + A2y + A3y + ... + Any,
Where y is the vector to be traced and A0 turns out to be the identity matrix.
Round 0:
Round 1:
Round 2:
f. How would the solution using the inverse compare with that using rounds of spending as above?
It would be identical. In this case, however, with such high coefficients, convergence would take over 20 rounds.
g = D(Bg + e + x - 
(Bg + e)) to
g = (I - D(I - )B)-1D((I - )e + x):
g = D(Bg + e + x - (Bg + e))
Expand, regroup, and manipulate to solution in terms of g:
g = DBg + D(e + x) - D (Bg + e))
g = DBg + De + Dx - DBg - De
g = DBg - DBg +De - De + Dx