question 1 c thank you

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 yt = 1 Where st is a random variable governed by a two-state Markov chain with values $t = $1 = 0 or St = $2 = 1. The Markov chain has time-invariant transition probabil- ities denoted by 7 (st+1 = s'|st = s) = " (s'|s), and the probability distribution over the initial state is To (s). The aggregate endowment at t is Y (st) = y! + y?. Let c' denote the stochastic process of consumption for agent i. Household i orders consumption streams according to U (c') = EE s' In [ci (s') ] i (s') 1= 0 where 7 (s' ) is the probability of the history s' = (so, $1, ..., St) (a) Give a formula for 7 (s' ) (4 Marks) Let O E (0, 1) be a Pareto weight on household 1. Consider the planning problem max {0In (c' ) + (1 - 0) In (c2) } Where the maximisation is subject to ct (s' ) + 7 ( s') SY ( st) (b) Solve the Pareto problem, taking 0 as a parameter (4 Marks) (c) Define Arrow-Debreu competitive equilibria with history dependent Arrow-Debreu securities traded once and for all at time 0. Be careful to define all of the object that compose a competitive equilibrium (4 Marks) (d) Compute the competitive equilibrium price system (i.e., find the prices of all of the Arrow-Debreu securities) (4 Marks) (e) Tell the relationship between the solutions (indexed by 0) of the Pareto problem and the competitive equilibrium allocation. If you wish, refer to the two welfare theorems (4 Marks)