Question 4. Arrow-DebreuEconomyConsider a world in which there are only two dates: 0 and 1. At date 1 there are threepossible states of nature: a good weather state (G), a fair weather state (F), and a badweather state (B). Denote S₁ as the set of these states, i.e., 8₁ € $₁ = {G, F, B}. Thestate at date zero is known. Denote probabilities of the three states as π = (0.4, 0.3,0.3).There is one non-storable consumption good, apple. There are three consumers inthis economy. Their preferences over apples are exactly the same and are given by thefollowing expected utility function& + β Σ Tsu (c),$1 ES1where subscript k = 1, 2, 3 denotes each consumer. In period 0, the three consumers havea linear utility and, in period 1, the three consumers have the same instantaneous utilityfunction:c²-yu (c)=where y = 0.2 (the coefficient of relative risk aversion). The consumers' time discountfactor, ß, is 0.98.The consumers differ in their endowments, which are given in the table below:Endowmentst=0t = 1SoG F B0.43.2 1.8 0.9Consumer 1Consumer 21.21.6 1.2 0.4Consumer 3 2.0 1.2 0.6 0.2Assume that atomic (Arrow-Debreu) securities are traded in this economy. One unitof 'G security' sells at time 0 at a price qc and pays one unit of consumption at time 1 ifstate 'G' occurs and nothing otherwise. One unit of 'F security' sells at time 0 at a priceqF and pays one unit of consumption at time 1 if state 'F' occurs and othing otherwise.One unit of 'B security' sells at time 0 at a price qв and pays one unit of consumption instate 'B' only.1. Write down the consumer's budget constraint for all times and states, and define aMarket Equilibrium in this economy. Is there any trade of atomic (Arrow-Debreu)securities possible in this economy?2. Write down the Lagrangian for the consumer's optimisation problem, find the firstorder necessary conditions, and characterise the equilibrium (i.e., compute the op-timal allocations and prices defined in the equilibrium). (3. At the equilibrium, calculate the forward price and risk premium for each atomicsecurity. What do your results suggest about the consumers' preference? ( Suppose that instead of atomic (Arrow-Debreu) securities there are three linearlyindependent securities, a riskless bond, a stock, and a one-period put option on this stockavailable for trade in this economy. The riskless bond pays 1 apple in every state, thestock pays 2, 1 and 0 apples in G, F and B, respectively. The put option has a strikeprice of 1.4. Write down the budget constraint for each consumer using the newly availablesecurities.5. Write down the Lagrangian for the consumer's optimisation problem, find the firstorder necessary conditions, and characterise the equilibrium (i.e., compute equilib-rium allocations and prices of the newly available securities).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.

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1. Write down the consumer's budget constraint for all times and states, and define a Market Equilibrium in this economy. Is there any trade of atomic (Arrow-Debreu) securities possible in this economy?