Population Distribution Dynamics: Analyzing the Time to Equality and Stationary State in a Social Mobility Markov Chain

Abstract: This study investigates the temporal evolution of population proportions between a "backward class" and a "higher caste" using a discrete-time Markov chain model. Starting with an initial distribution of 69% backward class and 31% higher caste, the objective is to determine the time required for these proportions to approximately equalize (50% each), and to understand the system's long-term behavior. A transition matrix derived from specified annual inter-class movements (7% from backward to higher, 1% from higher to backward) was used. Analysis reveals that the system does not converge to an equal distribution; instead, it stabilizes at a stationary distribution of approximately 12.5% backward class and 87.5% higher caste due to asymmetric transition rates. However, transient analysis shows that the population proportions are closest to equality (B_t ≈ 0.5, H_t ≈ 0.5) at approximately 5 years. Full convergence to the stationary distribution (within a 1% tolerance) is estimated to take around 49 years. This highlights a critical distinction between transient approximate equality and true long-term equilibrium in dynamic systems.