Problems

According to the Institute of Economic Modelling Sciences, the Slakan economy has alternate years of growth and decline, as in the following model. The number \(V\) of vloskan (the unit of currency) in the Slakan Treasury is assumed to behave as a continuous variable, as follows. In a year of growth it increases continuously at an annual rate \(aV_{0}\left(1+(V/V_{0})\right)^{2}.\) During a year of decline, as long as there is still money in the Treasury, the amount decreases continuously at an annual rate \(bV_{0}\left(1+(V/V_{0})\right)^{2};\) but if \(V\) becomes zero, it remains zero until the end of the year. Here \(a,b\) and \(V_{0}\) are positive constants. A year of growth has just begun and there are \(k_{0}V_{0}\) vloskan in the Treasury, where \(0\leqslant k_{0} < a^{-1}-1\). Explain the significance of these inequalities for the model to be remotely sensible. If \(k_{0}\) is as above and at the end of one year there are \(k_{1}V_{0}\) vloskan in the Treasury, where \(k_{1} > 0\), find the condition involving \(b\) which \(k_{1}\) must satisfy so that there will be some vloskan left after a further year. Under what condition (involving \(a,b\) and \(k_{0}\)) does the model predict that unlimited growth will take place in the third year (but not before)?