# Consider a world in which there are only two dates: 0 and 1. At date 1 there are three possible…

Question 3, Arrow-Debreu Economy (8 marks) 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 (GW), a fair weather state (FW), and a bad weather state (BW). Denote S1 as the set of these states, i.e., s1 ? S1 = {GW, FW, BW}. The state at date zero is known: call it s0. Denote probability of the three states as p (·), p (S1) = (0.4, 0.3, 0.3). There is one non-storable consumption good – say apples. There are three consumers in this economy. Their preferences over apples are exactly the same and are given by the following expected utility function c i 0 (s0) + ß X s1?S1 p (s1) u c i 1 (s1) , where subscript i = 1, 2, 3 denotes consumers. In period 0, all agents have a linear utility while in period 1, the three consumers have the same CRRA instantaneous utility function: u (c) = c 1-? 1-? , where the coefficient of RRA is ? = 0.2. The consumers’ time discount factor, ß, is 0.98. The consumers differ in their endowments which are given in the table below: Table 2 Endowments t = 0 t = 1 s0 GW FW BW 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 Arrow-Debreu securities are traded in this economy. One unit of ’GW security’ sells at time 0 at a price q(GW) and pays one unit of consumption at time 1 if state ’GW’ occurs and nothing otherwise. One unit of ’FW security’ sells at time 0 at a price q(FW) and pays one unit of consumption at time 1 if state ’FW’ occurs and nothing otherwise. One unit of ’BW security’ sells at time 0 at a price q(BW) and pays one unit of consumption in state ’BW’ only. 1. Write down the consumer’s budget constraint for all times and states, and define a Sequential Market Equilibrium in this economy. Is there any trade of Arrow-Debreu securities possible in this economy? (1 mark) 2. Write down the Lagrangian for the consumer’s optimisation problem, find the first order necessary conditions, and characterise the equilibrium (i.e., find the allocation and price defined in the equilibrium). (1 mark) 3. At the equilibrium, calculate the forward price and risk premium for each atomic security. What do your results suggest about the consumers’ preference? Prove your intuition, and comment on your results. (1 mark) c Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material. (ECON3107 only) Suppose that instead of Arrow-Debreu securities there are two securities, a (risky) bond, a riskless bond, and a stock, available for trade in this economy. The risky bond pays 1 apple in GW and FW, but 0 in BW, the riskless bond pays 1 apple in every state, while the stock pays 2, 1 and 0 apples in GW, FW and BW, respectively. Each trader is able to commit to their claim. 4. Write down the budget constraint for each consumer. (1 mark) 5. Write down the Lagrangian for the consumer’s optimisation problem, find the first order necessary conditions, and characterise the equilibrium (i.e., find the allocation and price defined in the equilibrium). (2 marks) 6. What is the price for risky bond, riskless bond and stock? What are the risk premia/discount for these securities? (1 mark) 7. Compare your results with part 3. and comment on your results in light of the arbitrage-free markets.(1 mark)

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