Consider an open production model having n industries with . In order to produce $1 of

Chapter 10, Problem T2

(choose chapter or problem)

Consider an open production model having n industries with . In order to produce $1 of its own output, the jth industry must spend for the output of the ith industry (for all ), but the jth industry (for all ) spends nothing for its own output. Construct the consumption matrix , show that it is productive, and determine an expression for . In determining an expression for , use a computer to study the cases when , 3, 4, and 5; then make a conjecture and prove your conjecture to be true. [Hint: If (i.e., the matrix with every entry equal to 1), first show that and then express your value of in terms of n, , and .]