answer plss

1. Suppose an individual's "health preferences" are represented by the utility function u(5,e, h} = s[2 a] + h, whore s is sugar, e is exercise, and h is its general state of health. In turn, it is adversely (i.e., negatively) affected by s and positively aliected by e. Specically, suppose h[s,e} = lne {33,92}. Overall, a and e are subject to the respective constraints [I 5 s E 2 and IKE E e 5' 2. a. How much a and a would the individual choose to consume? Verify that this is indeed a maximum. Next, suppose that while the direct effects of s and e on u are certain, the effects of these on health are not. Rather, each of them may or may not have the purported (is, claimed] eect on health. For a: = s or e, let to, denote the probabilityr that factor I does have the purported health effect and {l 113,} denote the probability that a: does not, and assume it, and tr, are independent. In the event that s does aect health but s does not, then it is given by ids) = s:f2. Similarly, if e is effective but a is not, then his) = in e; and if neither is effective, then it = 0. b. For each case, determine the optimal choices of s and e if the individual knew whether . or not the purported health clients were true. c. Returning to the case in which the true effects on health are uncertain, write the decision problem facing the individual and characterize an interior solution assuming the individual meadmizes expected utility. 1.i'ilhet, if any, restrictions on '31", and w, ensure that an interior solution exists? d. Discuss the comparative static eects of it, and ire on the optimal s and e in part c. e. Suppose that regardless of whether or not the purported health eEects of s and e are true, health is subject to an exogenous adverse shock that occurs with probability we. In that event, Ms, as) = hie [32,!2} a. What eect would this have on the optimal values of s and e? f. Suppose the individual is currently at an interior solution as in part c. The government is considering a. proposal to ban sugar entirely (a E U}. Would this be benecial or detriniental to the individual? g. Finally, suppose that sugar cannot be obtained directly by the individual. However, in order to encourage people to exercise, insurers provide one unit of sugar for each unit f. Suppose the individual is currently at an interior solution as in part c. The government is considering a proposal to ban sugar entirely (s E 0). Would this be benecial or detriniental to the individual? g. Finally, suppose that sugar cannot be obtained directly by the individual. However, in order to encourage people to exercise, insurers provide one unit of sugar for each unit of c, i.c., s E c. How will this s'ect the choice of s and e at an interior solution? 2. In a pure exchange economy with two consumers and two goods, when consumer i (i = 1,2} consumes its 2 0 units of good if, consumer 1 gets utility 1:11 + Men} - dsfm and consumer 2 gets utility $13 + own], where :35' 3} ID, gt\" s: D, and :9 E {0,1,9'2}. Consumer 1 initially owns the entire endmvment e1 :- U of good 1 in the economy and consumer 2 owns the entire endowment e3 2: i] of good 2. The problems below that ask for \"conditions on the economy\" are asking for restrictions on what is given exogenously. 1 a. Describe the consumers' preferences in English. b. Characterize all the Pareto efcient [Pareto optimal} allocations in which both con" sumers consume positive amounts of both goods. Under what conditions on the economy is there a Pareto efcient allocation with $22 = t]? o. Characterize the competitive [Walrasian] equilihria in which consumer 1 treats the consumption of good 2 by consumer 2 [consumar 2's choice] as given, unaffected by actions of consumer 1. Can there be more than one competitive equilibrium allocation in this economy? Under what conditions on the economy is a competitive equilibrium allocation Pareto efficient when H = D? Explain your answer. Under what conditions on the economy is a competitive allocation Pareto efficient when H = 132? d. Consider a competitive equilibrium in which the price of good 2 is 1 and consumer 2 is charged a fee of 31:32 in addition to the purchase cost of good 2. The fee paid by consumer 2 is given as a lump sum transfer to consumer 1 in compensation for consumer 1's utility loss due to consumer 2's consumption. [Consumer 1 acts as if the fee con sumer 2 pays cannot be s'ectsd by anything consumer 1 does.) Characterize a resulting competitive equilibrium allocation. Is it necessarily Pareto eicient? e. Suppose, instead, that consumer 2 is free to bargain with consumer 1 without being charged the fee in part d. Consumer 2, knowing the utility function of consumer 1, chooses a consumption bundle [313,332] 1s 1], and offers consumer 1 two alternatives. Either {1) consumer 2 will give consumer 1 e2 s22 units of good 2 in return for receiving 2:12 3.. {1 units of good 1 from consumer I, or {2) consumer 1 will consume its endowment e1 of good 1 and get none of good 2. Con sumer 1 must choose one of the alternatives, {1) or {2}. Characterize a pure subgame perfect Nash equilibrium [SPE) in which the consumersI payoffs are their utilities. Which alternative does consumer I choose? Show that consumer 1 gets the same payo as if it chose the other alternative. Is a. SPE allocation necessarily Pareto eieient? f. Compare the quantity of good 2 consumed by consumer 2 in a competitive equilibrium allocation in part d with that in a SPE allocation in part c. How do the entire allocations in parts d and e dier Erom each other {if they do}? Explain why the dierences arise if they do. 3. Two roommates are considering buying an airrconditioner that costs $100. It is common knowledge that the utility function of roommate i is of the form a; (191;, q, ti} = the 5q + t, where o E {EL 1},31- E {0,613}. Here, 1; = 1 represents buying the air condiin'oner, q = [1' represents not doing so, and the utility is measured in dollars. The values of Sgt = 1, 2, are private. The roommates want to buy the air oonditioner if and only if their joint valuation of 3119; exoeeds the $1M oust. There is no external nancial help. An outcome can be expressed formally by a social choice function fhg} = [awhd-y},t1{91,92],tg(dl,dgj), such that Q{1,2) = I if and only if til + 192 :r 100 and 311011 that 31(31, 5'2} + tah 32) i: '3. a Consider a direct mechanism where each roommate i is asked for a report 1'9- of his {3; and where Q(I91,Eg} = 1 if and only if 6'1 + 192 3::- 10D.Deriye conditions that symmetric {g-[191, 92]}, 195 E {It}, Ell}, need to satisfy to truthfully implement the social choice function f in dominant strategies. [Symmetry means t1(or, ,5) = tg, o] for each or, ,6 E {l}? 60}.} Using the synnnetry assumption, express all conditions on {ii} in terms of t1{o,},o, E {0.61)} only and simplify the conditions. 3. Two roommates are considering buying an airrconditioner that costs $100. It is common knowledge that the utility function of roommate i is of the form a; (191;, q, ti} = the 5q + t, where o E {EL 1},31- E {0,613}. Here, 1; = 1 represents buying the air condiin'oner, q = [1' represents not doing so, and the utility is measured in dollars. The values of Sgt = 1, 2, are private. The roommates want to buy the air oonditioner if and only if their joint valuation of 3119; exoeeds the $1M oust. There is no external nancial help. An outcome can be expressed formally by a social choice function fhg} = [awhd-y},t1{91,92],tg(dl,dgj), such that Q{1,2) = I if and only if til + 192 :r 100 and 311011 that 31(31, 5'2} + tah 32) i: '3. a Consider a direct mechanism where each roommate i is asked for a report 1'9- of his {3; and where Q(I91,Eg} = 1 if and only if 6'1 + 192 3::- 10D.Deriye conditions that symmetric {g-[191, 92]}, 195 E {It}, Ell}, need to satisfy to truthfully implement the social choice function f in dominant strategies. [Symmetry means t1(or, ,5) = tg, o] for each or, ,6 E {l}? 60}.} Using the synnnetry assumption, express all conditions on {ii} in terms of t1{o,},o, E {0.61)} only and simplify the conditions