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 yE Where a is a random variable governed by a two-state Markov chain with values 8: = 81 =0 or at = 82 = 1. The Markov chain has time-invariant transition probabil- ities denoted by " (s:+1 = #'|st = ) = * (s'|#), and the probability distribution over the initial state is no (#). The aggregate endowment at t is Y (a,) = y, + y Let d' denote the stochastic process of consumption for agent i. Household i orders consumption streams according to where a (s' ) is the probability of the history at = (80, #1, ..., #t) (a) Give a formula for a (s') (4 Marks) Let # c (0, 1) be a Pareto weight on household 1. Consider the planning problem el,c2 max {0In (c ) + (1 - 8) In (c)} Where the maximisation is subject to 4 ( # ! ) + 4 ( # )