The price of a product is assumed to be a function of the inventory high inventory results in a lower price. If the inventory is small, then fewer products are sold and the price is relatively high. However, the small inventory and the corresponding high price promise future profits and therefore lead to investment activities and a higher production level The higher production level then fills up the inventory leading to lower prices and a drop in production. Thus it is intuitively clear that the process may generate oscillatory behaviour, called the production cycle. To model the process, suppose the production level (units/time) changes according to where is the investment rate, is the inventory goal and the price is proportional to . Also suppose the inventory (units) satisfies where the sales rate depends on the buying reaction to price (elasticity) , the normal rate of consumption c and an input function that can representthe relative effect of advertising, the weather etc.
(a) If is constant, find the equilibrium point(s).