Question

Let \(C\) be a consumptions matrix such that \({C^m} \to {\bf{0}}\) as \(m \to \infty \), and for \(m = {\bf{1}},{\bf{2}},......\) let \[{D_m} = I + C + .... + {C^m}\]. Find a difference equation that relates \({D_m}\) and \({D_{m + {\bf{1}}}}\) and thereby obtain an iterative procedure for computing formula (8) for \({\left( {I - C} \right)^{ - {\bf{1}}}}\).

Short answer

\({D_{m + 1}} = I + C{D_m}\)

Step-by-step solution

Find the expression for \({D_{m + {\bf{1}}}}\)

\(\begin{array}{c}{D_{m + 1}} = I + C + {C^2} + .... + {C^{m + 1}}\\ = I + C\left( {I + C + .... + {C^m}} \right)\end{array}\)

Reduce \({D_{m + {\bf{1}}}}\) in the form of \({D_m}\)

Substitute the value of \({D_m}\) in theequation \({D_{m + 1}} = I + C\left( {I + C + .... + {C^m}} \right)\).

\(\begin{array}{c}{D_{m + 1}} = I + C\left( {I + C + .... + {C^m}} \right)\\ = I + C{D_m}\end{array}\)

So, the relation between \({D_{m + 1}}\) and \({D_m}\) is \({D_{m + 1}} = I + C{D_m}\).