Stochastic Analysis of Number of Corporations in A Market Derived from Strategic Policies of Individual Corporations for Market Entry and Retreat

A stochastic model is developed for describing a market lifecycle expressed in terms of the number of corporations N in the market. Each corporation independently determines the probability of market entry if it is not in the market yet or the probability of market retreat if it is already in the market. These probabilities may depend on time t, the number of corporations in the market at time t and the number of corporations which have retreated from the market by time t. Of interest is the number of corporations in the market at time t, thereby enabling one to analyze the market lifecycle in terms of strategic actions of individual corporations. Rigorous analysis of this process becomes numerically intractable since the corresponding state space explodes as N increases. In order to overcome this difficulty, we propose temporally inhomogeneous marginal processes describing the states of individual corporations. The process of interest is then approximated as the independent sum of such marginal processes. An algorithmic procedure is developed for computing the probability distribution of the number of corporations in the market based on spectral analysis of the temporally inhomogeneous marginal processes combined with a bivariate generating function approach. Corporations are classified into three groups: RT (Risk-Taking), RN (Risk-Neutral), and RA (Risk-Aversive), where these groups are characterized by specifying the transition probabilities of the underlying marginal processes. It is numerically observed that any class alone is not sufficient to form a market and a typical market lifecycle emerges only through the presence of an appropriate combination of corporations from the three classes.