Question

Multinational companies in the Americas, Asia, and Europe have assets of \(\$ 4\) trillion. At the start, \(\$ 2\) trillion are in the Americas and \(\$ 2\) trillion are in Europe. Each year \(1 / 2\) of the Americas money stays home and \(1 / 4\) goes to each of Asia and Europe. For Asia and Europe, \(1 / 2\) stays home and \(1 / 2\) is sent to the Americas.
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a) Let \(C_{k}\) be the column vector with the assets of the Americas, Asia, and Europe at the beginning of year \(k\). Find the transition matrix \(T\) that gives the amount in year \(k+1: C_{k+1}=T C_{k}\)
b) Find the eigenvalues and eigenvectors of \(T\).
c) Find the limiting distribution of the \(\$ 4\) trillion as the world ends
d) Find the distribution of the \(\$ 4\) trillion at year \(k\).