Question 1 (20 Marks) An economy consists of two consumers, named i = 1, 2. The economy exists in discrete time for periods t 2 0. There is one good in the economy, which is not storable and arrives in the form of an endowment stream owned by each consumer. The endowments to consumer i = 1, 2 are yt = St y? = 1 Where st is a random variable governed by a two-state Markov chain with values St = $1 = 0 or St = $2 = 1. The Markov chain has time-invariant transition probabil- ities denoted by 7 (st+1 = s'|st = $) = " (s'|s), and the probability distribution over the initial state is To (s). The aggregate endowment at t is Y (st) = yt + yt. Let c' denote the stochastic process of consumption for agent i. Household i orders consumption streams according to U (c') = E E Bt In [ci (st) ] i (st) t=0 st where 7 (s') is the probability of the history s' = (so, $1, ..., St) (a) Give a formula for 7 (st) (4 Marks)