Chapter 2.6, Problem 8E

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 Δx be a production vector that satisfies a different final demand Δd.

1. a. Show that if die final demand changes from d to d + Δd, then die new production level must be x + Δx. Thus Δx gives die amounts by which production must change in order to accommodate the change Δd in demand.
2. b. Let Δd be the vector in ℝⁿ with 1 as die first entry and 0’s elsewhere. Explain why the corresponding production Δx is die first column of (I − C)⁻¹. This shows that the first column of (I − C)^−l gives the amounts the various sectors must produce to satisfy an increase of 1 unit in the final demand for output from sector 1.