Let C be an n × n consumption matrix whose column sums are less than 1. Let x be the production vector that satisfies a final demand d , and let Δ x be a production vector that satisfies a different final demand Δ d .

a. Show that if the final demand changes from d to d + Δ d , then the new production level must be x + Δ x. Thus Δ x gives the amounts by which production must change in order to accommodate the change Δ d in demand.

b. Let Δd be the vector in Rn with 1 as the first entry and 0’s elsewhere. Explain why the corresponding production Δx 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.