v v \\ I Consider a world in which there are only two dates: 0 and 1. At date 1 there are three possible stat. of nature: a good weather state (G), a fair weather state (F), and a bad weather state (B). Denote S] as the set of these states, i.e., 31 E .81 = {G', F, B}. The state at date zero is known. Denote probabiliti. of the three states as 7r = (0.4, 0.3, 0.3). There is one non-storable consumption good, apple. There are three consumers in this economy. Their preferences over appl. are exactly the same and are given by the following expected utility function c+ Z 1r..u(c':1). 51631 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: 1? L: 1'1. (6) = 1 _ '71 where 7 = 0.2 (the coefcient of relative risk aversion). The consumers' time discount factor, ,8, is 0.98. The consumers differ in their endowments, which are given in the table below: Endowments t = 0 t = 1 50 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 '0: security' sells at time 0 at a price qg 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 (p: 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 13 and pays one unit of consumption in state 'B' only. 1. Write down the consumer's budget constraint for all tim- and stats, and dene a Market Equilibrium in this economy. Is there any trade of atomic (Arrow-Debreu) securities possible 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 dened 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 one-period 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, nd the rst order necessary 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)
