Arrow-Debreu Economy Consider a world in which there are only two dates: 0 and 1. At date 1 there are three possible states of nature: a good weather state (G), a fair weather state (F), and a bad weather state (13). Denote S; as the set of these states, i.e., 31 E S; = {G, F, B}. The state at date zero is known. Denote probabilities of the three states as 11' = (0.4. 0.3,0.3). There is one non-storable consumption good, apple. There are three consumers in this economy. Their preferences over apples are exactly the same and are given by the following expected utility function c5+ Z 1f..u(6f,). 81681 where subscript k = 1. 2,3 denotes each consumer. In period 0. the three consumers have a linear utility and, in period 1, the three consumers have the same instantaneous utility function: where 1 = 0.2 (the coefcient of relative risk aversion). The consumers' time discount factor, 15', is 0.98. The consumers differ in their endowments. which are given in the table below: Endowments t = 0 t = 1 so G F B Consumer 1 0.4 3.2 1.8 0.9 Consumer 2 1.2 1.6 1.2 0.4 Consumer 3 2.0 1.2 0.6 0.2 Assume that atomic (Arrow-Debreu) securities are traded in this economy. One unit of 'G security' sells at time 0 at a price qc and pays one unit of consumption at time 1 if state 'G' occurs and nothing otherwise. One unit of 'F security' sells at time 0 at a price qr and pays one unit of consumption at time 1 if state 'F' occurs and nothing otherwise. One unit of '3 security' sells at time 0 at a price 93 and pays one unit of consumption in state '3' only. 1. Write down the consumer's budget constraint for all times and states, and dene a Market Equilibrium in this economy. Is there any trade of atomic (Arrow-Debreu) securities pomible in this economy? (1 mark) 2. Write down the Lagrangian for the consumer's optimisation problem, nd the rst order necessary conditions, and characterise the equilibrium (i.e., compute the op- timal allocations and prices defined in the equilibrium). (2 marks) 3. At the equilibrium, calculate the forward price and risk premium for each atomic security. What do your results suggest about the consumers' preference? (1 mark) Suppose that instead of atomic (Arrow-Debreu) securities there are three linearly independent securities, a riskless bond. a stock, and a onehperiod put option on this stock available for trade in this economy. The riskless bond pays 1 apple in every state, the stock pays 2, 1 and 0 apples in G, F and B, respectively. The put option has a strike price of 1. 4. Write down the budget constraint for each consumer using the newly available securities. (1 mark) 5. Write down the Lagrangian for the consumer's optimisation problem, find the rst order umary conditions, and characterise the equilibrium (i.e., compute equilib- rium allocations and prices of the newly available securities). (1 mark) 6. Now, price the newly available securities using the atomic prices from part 2. Com- ment on your results in light of the arbitrage-free markets. (1 mark)