91Ó°ÊÓ

11\. In a simple model of a national economy, \(I=I-\alpha C, C=\) \(\beta(I-C-G)\), where \(I\) is the national income, \(C\) is the rate of consumer spending and \(G\) is the rate of government expenditure. The model is restricted to its natural domain \(I \geqslant 0, C \geqslant 0, G \geqslant 0\) and the constants \(\alpha\) and \(\beta\) satisfy \(1<\alpha<\infty, 1 \leqslant \beta<\infty\) (a) Show that if the rate of government expenditure \(G=G_{0}\), a constant, then there is an equilibrium state. Classify the equilibrium state when \(\beta=1\) and show that then the economy oscillates. (b) Assume government expenditure is related to the national income by the rule \(G=G_{0}+k l\), where \(k>0\). Find the upper bound \(A\) on \(k\) for which an equilibrium state exists in the natural domain of \(t\) model. Describe both the position and the behaviour of this state \(\beta>1\), as \(k\) tends to the critical value \(A\).