"ECONOMICS2Exercise 3.1. Benevolent Dictators? Benevolent means the desire to help people out charitable. Benevolent dictatorship is atheoretical form of a person given the authority to exercise their power over the group leastbut does in a relative manner that benefits the group. Dan being chosen as the benevolentdictator he has to carry out his powers in a friendly manner. For the benefit of Ann and Bob,he has to choose a possible option doesn’t favor any of the two. It means the utility has to bealmost equal on the choice. Therefore, Dan cannot choose Cinema; U 1 ? U 4, Football; U 3 ? U 1; and Television U 2.5 ? U 0.5. A B A B A B ? Social welfare function (SWF) is a function that takes individual utility and outputs a realnumber out of it, and that represents the social well- being of society. To calculate utilitarianof social welfare, you get the sum of all individual utilities. 1 1 ? 2? = ( ) Yi W = ( ) 5 + 4 + 3 + 3.5 = 7.75? =1 ? =1 ? 2 ? Where: W is the society’s welfare; U is the utility of individual i; N is the total number ofpeople. Welfare maximization occur at a Pareto optimal assignment of inputs. In that case,the maximum total utility is 7.75. Therefore, Dan should choose options that provide a closeutility satisfaction to 7.75. That is Cinema and Football. Utilitarian criterion changes in aperson’s choice arise from choice in social welfare.? In Rawlsian social welfare, the measure of society welfare is equal to that of the individualwho is worst off.In this case, it means maximizing societal welfare would requiremaximizing the utility of the one with a lower choice option. UA = 1+3+2.5+1.5 =8, UB = 4+1+0.5+2 = 7.5.ECONOMICS3It means maximizing Bob’s utility. Therefore, the option to be considered is Bob’s choices. ? Bergsonian social welfare was developed to state the magnitude of decisions needed forderivation of maximum welfare conditions. It states that in a 2-person society the graphicalillustration of welfare maximization is for consumer theory of an individual of varying use ofgoods. According to Bergson, he noted that a status advancementarise from improving someperson’s social position at the expense of others (Hall, 2005). This is described as equitydimension, and individual shows it with the highest utility satisfaction. UA = 1+3+2.5+1.5 =8, UB = 4+1+0.5+2 = 7.5. It means welfare can be maximized by choosing Ann’s option since she enjoys high utility.? Among the options excluded, the one that represents the Pareto efficient is the Bergsoniansocial welfare function. A choice or allocation is considered to be Pareto efficient if in theallocation there is none of other individuals are better off, and some are worst off. It meansthere is no relationship between Pareto efficient and equity. The outcomes in Pareto efficientmay be so inequitable such that the option with the best output is theater. ? For Bob and Ann to choose theater, utility for all of them has to be equal. UA = UB.Mostsuitable assignment uses utility of the households who are least-well-off in society. In amatter of facts, the Rawlsian criterion is widely recognized for maintaining the social welfarefunction. This can be done through the following function: W = min (Y1, Y2… Yn). W = 0.5 (4, 1, 2) = 2, 0.5, 1. Contract curves Contract curve is used to represent final allocations of commodities between two individualsthat could occur as a result of beneficial trading or choices. The set of points in this locus arePareto efficient allocation. The utility function of Robinson is UR = F1 + C1, and that of Friday ECONOMICS4is min (F2, C2), where Fi (resp. Ci) is the amount of F (resp. C) consumed by individuals I =Robinson & Friday. It then means that Robinson has 4 pieces of fish and 3 pieces’ coconuts, andFriday has 6 pieces of fish and 2 pieces of coconut. Let ? denotes the endowment allocation. Thecontract curve will be as follows; let x represent Robinson and y to represent Friday. Y AX 0 B BARf 2 R – Robinson, F -FridayA RC 1 B FC 1Contract curveBFf 2 0X A AY B This graph indicates a reflection of Friday’s indifference curve over that of Robinsons’.a. According to Pareto’s efficiency, an allocation is only preferred if it improves thestandards of one individual while not having adverse effects on others. An indicationof a region of mutual advantage has to be shown when plotting on a graph. This is thearea where all Pareto preferred endowment point occurs. In the graph below the yellow ECONOMICS5marked region is the region of mutual advantage. And E2 is Pareto preferred to E1; E3is Not preferred to E1 (Friday is worse off). b. Within limits, it is possible to say that the distribution of fish is optimal. Pareto optimalconsiders an allocation optimal if it cannot make someone better off without makingthe other one worse off (Hall, 2005). Analysis of the graph shows that the point is notpreferred to E1 because at that point Friday is worse off.1010 E 1Rc 1 EE 2 3Fc 1 E 10210SOCIAL WELFARE FUNCTIONSWF is a function that is used to rank societal states regarding less desirable, more desirable, orindifference on every possible set of social states. Hypothetically consider a society consisting oftwo individuals, A &B, considering constant timeline. There is an existence of one good (X), theconsumption of which is the only source of utility. Let total utility enjoyed byA B U & U for utility enjoyed by B. so we have: ECONOMICS6A A A U ? U (X ) B B B U ? U (X )This implies that; utilitarianism does not carry any particular implication as such for the wayoutput should be distributed. This gives it a kink point on its curve at the point where the twoindividuals have the same satisfaction (as shown in the graph). Point f is the kink point. At this point, the utility gained is the same for all individuals regardlessof the product. "