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# Class Note for ECON 702 at UMass(2)

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Date Created: 02/06/15
Pareto optimality and the utility possibility frontier Definition 1 Suppose X is the set of all feasible social alternatives and N is the set of all individuals An alternative x x1 x e X is Pareto optimal or efficient if there is no alternative xv x139 x e X such that u1x139unx Z u1x1unxn with strict inequality for at least one component Note Similarly we can say a feasible utility vector U1 u is Pareto optimal if there is no feasible vector u139 un39 such that u139 u 2 u1 u with strict inequality for at least one component Definition 2 1 The utility possibility set is the set of all feasible utility vectors US ulun e Rquot u1un S u1x1un xn for some x1xn e X 2 The utility possibility frontier is de ned as u1un e US there is no xv x139x1e X such that u1x139un x1 Z u1un UPF w1th str1ct 1nequa11ty for at least one component Example 1 An affine utility possibility frontier Suppose a xed amount of money say 10 must be divided between two individuals John and George Let X1 and X2 denote the amount money that John and George receive thus X1X210 We assume that John s utility function is 111X1X1 and George s is 112X22X2 There are many different ways of dividing the money The following table shows several possibilities and the corresponding utility levels X1 John s utility George s utility X2 0 0 20 10 1 1 18 2 2 16 8 2 1 10 10 0 George39s ulilily Thus the graph of the utility possibility frontier is as follows Along the frontier it is not possible for one individual to consume more unless the other individual consumes less Thus the utility possibility frontier is downward sloping Any allocation corresponding to the inside of the frontier is not Pareto optimal because by moving towards the frontier from it we can improve the situation of both individuals 20 10 Johns utility On the other hand any allocation corresponding to the outside of the frontier is not feasible Note An algebraic expression of the utility possibility frontier can be easily obtained Note that u1x1 and uz2xz Thus using the fact that x1xz10 we have u1uz210 Note that the frontier expands if M is increased Example 2 A bowed out utility possibility frontier A utility possibility frontier does not have to be affine If a utility function is concave for at least one individual then the utility possibility frontier is also concave ie it bows out Suppose a fixed amount of money say 10 must be divided between two individuals John and George We now assume that John s utility function is linear as before but George s is concave that is u1x1x1 and M2 x2 le The followin table shows several possibilities and the corresponding utility levels X1 John s utility George s utility X2 0 0 316 10 l l 3 2 2 283 8 9 9 l l 10 10 0 0 The graph of the utility possibility frontier in this case is as follows the utility possibility frontier is bowed out George39s utility Again any allocation corresponding to the inside of the frontier is not Pareto optimal because by moving towards the frontier from it we can improve the situation of both individuals Allocations corresponding to the 316 frontier are Pareto optimal Note Again an algebraic expression of the utility possibility frontier can be easily obtained Note that u1x1 and M2 le Thus 10 using the fact that x1xz10 we have John39suility u1 u2 2 10 which is equivalent to M2 10 u1 Example 3 UPF for quasi linear utility functions Suppose there are two individuals and two goods x and y Utility functions are u lx1yl and the initial endowments of the two goods are 9 and 7 Because x1 x2 f and y1 y2 7 we obtain the utility possibility frontier in the following manner Step 1 Fix x1 Then x2 is also fixed because x1 x2 f Then add M1 and u2 to get u1 u2 1x1 2f x17 Note that 1x1 2f x17 is a constant This equation is called the utility possibility mini frontier for 1 This gives pairs of maximal utilities when 1 is fixed Step 2 Now choose x to maximize 1x1 2 f x1 7 Now the utility possibility mini frontier for x is the utility possibility frontier Because 1x 2 f x 7 is a constant the utility possibility frontier is a downward sloping straight line with slope 1 Note that the utility possibility frontier always takes the form of U IN where IN f1 f2 J7 is the maximum utility value of the entire society Indeed we can prove the following if the utility functions are quasilinear Theorem 1 11 an is Pareto optimal if and only if Zul VN ZEN Proof In the proof we will use the fact that statement p I3 q is equivalent to the stament of Qq I3 6p U Suppose 11 an is not Pareto optimal Then there is some feasible u139un39 such that u 2 u for all 139 e N with strict inequality for at least one 139 e N and L39 IN Thenbecause L39gt u wehave INgt L1Thus TN TN TN TN Zul vN is not true ZEN o vtN u D Suppose Zul VN is nottrue ie a u lt INLet e 151v iTv Clearly egt ONowwedef1ne LII u 9 Then gt u for all l N and Thus 11 an is not Pareto optimal QED ii N Example 4 UPF for Cobb Douglas utility functions Suppose there are two individuals and two goods X and y Utility functions are u a logq 1 a logyl and the initial endowments of the two goods are E and 7 Step 1 Fix x1 Then because u1 a1 logx1l a1logy1 implies y taxmw we have u a 10gf x 1 amogo exp 1 l z 1 0 1 This equation is called the utility possibility mini frontier for 1 Step 2 Now choose x to maximize all log x1 u2 a2 logo x1 1 a2 log7 expu1 1 Now the utility possibility mini 1 frontier for x is the utility possibility frontier 15 125 075 05 025 02 04 06 08 l

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