Let C be an n X n consumption matrix whose column sums are less than 1. Let x be the production vector that satisfies a final demand d, and let ?x be a production vector that satisfies a different final demand ?d.
a. Show that if the final demand changes from d to d + ?d, then the new production level must be x + ?x. Thus ?x gives the amounts by which production must change in order to accommodate the change ?d in demand.
b. Let ?d be the vector in Rn with 1 as the first entry and 0's elsewhere. Explain why the corresponding production ?x is the first column of (I - C)-1. This shows that the first column of (I - C)-1 gives the amounts the various sectors must produce to satisfy an increase of 1 unit in the final demand for output from sector 1.