27. Each individual in a population of size N is, in each period, either active or inactive.

If an individual is active in a period then, independent of all else, that individual will be active in the next period with probability α. Similarly, if an individual is inactive in a period then, independent of all else, that individual will be inactive in the next period with probability β. Let Xn denote the number of individuals that are active in period n.

(a) Argue that Xn, n  0 is a Markov chain.

(b) Find E[Xn|X0 = i].

(c) Derive an expression for its transition probabilities.

(d) Find the long-run proportion of time that exactly j people are active.

Hint for (d): Consider first the case where N = 1.