WDBIUEI' E WDI'lEl III WHICH. tEIE HIE til-111}F FWD [lat-E5: U and 1. H1] UBEE J. tEI'E HIE TJlI'EE possible state: of nature: a good weather state (G), a fair weather state [F], and a had wEathEr state {B}. Denote 31 as the set of these states, i.e., 31 E 5'1 = {G,F,B}. The state at date zero is known. Denote probabilities of the three states as \"II" = [0.4, {13, 0.3}. ThEre is one nonstorahle 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 at +13 2 a.u(c.}. FIE-5'1 where suhscript Fr. 2 1, 2, 3 denotes each consumer. In period l}, the three consumers have a linear utility and, in pariod 1, the thrEe consumErs have the same instantaneous utility flmction: cl\"if Elia}: 11', where y = [1.2 [the coeicient of relative risk aversion}. The consumers' time discount factor, ,3, is 0.93- The consumers diEer in their endowments, which are given in the table below: Endowments t = l} t = 1 5:} G F B ConsumEr 1 [1.4 3.2 1.3 0.9 Consumer 2 1.2 1.6 1.2 0.4 lUousumer 3 2.0 1.2 [LIE I12 Assume that atomic [ArrowDehrEu} securities are traded in this Economy. One unit of 'G security' sells at time {i at a price qr; and pays one unit of consumption at time 1 if state \"3' occurs and nothing otherwise. Due unit of \"F securityI sells at time II] at a price up and pays one unit of consumption at time 1 if state 'F' occurs and nothing othErwise. Clue unit of 'E seclu'ity' sells at time II} at a price 133 and pays one unit of consumption in state '3' only. 2. Write down the Lagrangian for the consumer's optimisation problem, find the first 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)