Existence of Trajectories with Unbounded Consumption for a Model with Innovations

We consider the model with discrete innovations introduced in [1] and studied in [2–9]. In this model the state of the economy is determined by a set of operating technologies, a collection of funds corresponding to these technologies, and a set of known, but as yet not implemented technologies. To introduce a new technology, expenditures of the already utilized types of funds are required. As a result of these expenditures, the new technology at the next instant of time will be introduced into action with a certain initial reserve of new funds. We consider a single-product economy which deals with two production factors: labor L and fundsK . The time is assumed to be discrete and the amount of labor is constant and equal to unity. The state of the economy is determined by a set of operating technologies, a collection of funds corresponding to these technologies and a set of known, but as yet not implemented technologies. A technology is a pair ( f ,v) where f is a production function of two variables K , L and v ∈ [0,1). Possessing at time t funds K and labor resources L, the economy utilizing the technology ( f ,v) will produce during a unit time interval, a product in the amount of f (K ,L). Moreover, at time t + 1 the economy will still have in its possession the used old funds in the amount of vK . To introduce a new technology, expenditures of the already utilized types of funds are required. As a result of these expenditures, the new technology at the next instant of time will be introduced into action with a certain initial reserve of the new funds. Let I = {0,1, . . .} and {( f i,vi) : i∈ I} be the set of all technologies which can be utilized in the production process. At time t ∈ I the state of the economy is given in the form ( I 0,I t n, ( K t ,C i t ) ( i∈ I 0 )) , (1.1)