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File: Welfare Economics Pdf 126410 | Negishiwelfareweights
welfare economics and existence of an equilibrium for a competitive economy zyxwvutsrqponmlkjihgfedcbazyxwvutsrqponmlkjihgfedcba by takashi negishi tokyo i zyxwvutsrqponmlkjihgfedcbazyxwvutsrqponmlkjihgfedcba the proof of the existence of an equilibrium for a com petitive ...

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                                                                                                                                                                                                                   WELFARE ECONOMICS AND EXISTENCE OF AN 
                                                                                                                                                                                                                          EQUILIBRIUM  FOR A COMPETITIVE ECONOMY zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
                                                                                                                                                                                                                                                                                                                                                                                                                                      by Takashi Negishi, Tokyo. 
                                                                                                                                                                                                                   I. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA-      The proof  of  the existence  of  an  equilibrium  for  a  com- 
                                                                                                                                                     petitive economy is given by Arrow and Debreu [I] and many others 
                                                                                                                                                     such  as  Gale zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA[4],                                                                                                                                                                                            Kuhn [6],  McKenzie [8], zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA[g], and  Nikaido zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA[IO]. 
                                                                                                                                                       In this note, we shall give another proof of  the existence of  an equi- 
                                                                                                                                                     librium, putting emphasis on the welfare aspect of  the competitive 
                                                                                                                                                     equilibrium (1). 
                                                                                                                                                                                                                  As  is well known, an equilibrium point of  an economic system 
                                                                                                                                                     under perfect  competition  is  an efficient  state in Pareto's sense  in 
                                                                                                                                                      which  we  cannot  make anyone better off  without  making someone 
                                                                                                                                                      worse off.  In other words, it can be said that a competitive equi- 
                                                                                                                                                     librium is a maximum point of  some properly defined social welfare 
                                                                                                                                                      function subject to the resource and technological constraints. 
                                                                                                                                                                                                                  In the following, we  shall show that a competitive equilibrium 
                                                                                                                                                       is  a  maximum point  of  a social  welfare  function  which  is a linear 
                                                                                                                                                       combination of  utility functions  of  consumers, with  the weights  in 
                                                                                                                                                        the Combination  in  inverse  proportion  to the marginal  utilities  of 
                                                                                                                                                       income.  Then, the existence of  an equilibrium is equivalent to the 
                                                                                                                                                       existence  of  a  maximum  point  of  this  special  welfare  function. 
                                                                                                                                                       Therefore,  we  can prove  the former by showing the latter. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
                                                                                                                                                                                                                    2. - Let  us  construct  our  economic  model,  the  existence  of 
                                                                                                                                                        whose equilibrium we  shall prove, as follows.  Let there be m goods, 
                                                                                                                                                        n consumers, and I  firms.  Let zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAx, be  a  consumption  vector  (whose 
                                                                                                                                                       element  is  xtt2 o), xi be  an initial  holding  vector  (whose element 
                                                                                                                                                        is ;,,                                                                 3> o), and Uc (x,) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAbe  the  utility  (function) of                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         the ith consumer. 
                                                                                                                                                        Let yk be a production  vector of  the kth firm whose element ykr > o 
                                                                                                                                                           (< 0) is the output (input) of  the ith good, and Yk be  the possible 
                                                                                                                                                         set of  yk, i. e.,  the set  of  yk which  satisfies the restriction  on  pro- 
                                                                                                                                                         duction  Fk (ye) >_ 0. Let zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAP                                                                                                                                                                                                                                                                                                                                               (whose  element  P,2 0) be  the price 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          Let hik be the proportion  of 
                                                                                                                                                         vector.  For a non-free  good, P, > 0. 
                                                                                                                                                         profit  oi the kth firm  distributed  to the ith consumer. 
                                                                                                                                                                                                                      We define an equilibrium point under perfect competition: 
                                                                                                                                                                                                                       Definition I.  The following are the conditions of  an equilibrizcm 
                                                                                                                                                         point  (xi j  yk, P): 
                                                                                                                                                                                                                (l) .The author wishes to express his gratitude to Prof. K. J. Arrow and 
                                                                                                                                                          Mr.  H. Uzawa, both of  Stanford University,  for their valuable suggestions. 
                                                                                                                                                            This work  is  supported  by the Office  of  Naval  Research, U.S.A. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     - 93 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             a) Equalities zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAof                                                                                                                                                                                                                                                                                                                                                                                                                                                    demand and supply for non-free  goods: 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              ._                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               - 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               xi, 5 0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA#                                                                                                                                                                                                                                                                                                                                         Pj zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA(Xi xi5                                                                                                                                                                                                                                         - Ck ykj -  Xij) = 0 > 
                                                                                                                                                                                                                                                                                                                                                                  xi %if - ck zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAyk$ - 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   for zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAj                                                                                                       = I, ... , m. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   b) The equilibrium of  consumers: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAxi is  a  maximum  point  of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
                                                                                                                                                                                                                                                                                                                                                             Ui (xi) .subject to 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          zj P# xi, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA5 cj pj Xij + max [o , C, iki C, P, yk,]  = Mj , 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               for  i = I, ... , n. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    c)  The equilibrium  of  firms:  y,  is  a  maximum  point  of  Cj 
                                                                                                                                                                                                                                                                                                                                                       P, ykf subject to 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 for k  = 1 I  ... I t'. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Fk (yi) 2 0  (yk E YL) , 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Next, we  define a  welfare maximum  point  as follows: 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Definition  2.  Consider  the  weighted  sum  of  utility  functions 
                                                                                                                                                                                                                                                                                                                                                                   Xi cc,  Ui (xi)  with  weights  air 0, i = I , ... , n , I;(  ui = I , as a 
                                                                                                                                                                                                                                                                                                                                                       social welfare  function.  We  call a point  (xi, y,), which  maximizes 
                                                                                                                                                                                                                                                                                                                                                        it, subject  to the condition  of  r10  excess  of  demand  over supply, 
                                                                                                                                                                                                                                                                                                                                                                   xi xi 5                                                                                                                                                                                                                                                                                                                                                                                                     + Ck yk , and production subject to the restriction  on 
                                                                                                                                                                                                                                                                                                                                                        Fk (yk) 2 o , k..=  I , ... , r , a  welfare maximum point. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                3.  - The  assumptions  on  utility  functions  and  production 
                                                                                                                                                                                                                                                                                                                                                       restrictions  are as follows: 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Assumption I.  U, (xi) is  continuous, increasing,  and concave; 
                                                                                                                                                                                                                                                                                                                                                        more precisely, we  can make it concave by a  strictly positive  mo- 
                                                                                                                                                                                                                                                                                                                                                          notone transformation.  See Fenchel [3]. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Roughly speaking, this assumption  implies that, among utility 
                                                                                                                                                                                                                                                                                                                                                          functions which satisfy the same indifference map, there is a utility 
                                                                                                                                                                                                                                                                                                                                                        function with non-increasing  marginal utility. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Assumption  2.  Fk (yk)  is  continuous  and  concave,  and 
                                                                                                                                                                                                                                                                                                                                                        Fk (y*k) > o for some y'k  such that  I;, Y'k  < xi Xi (2).  Furthermore, 
                                                                                                                                                                                                                                                                                                                                                          the sets Yk and their vector sum Y sati'sfy the followvhg conditions: 
                                                                                                                                                                                                                                                                                                                                                          o E  Y,, Y n B = o (B isaclosedpositive orthant), Y n (-  3')                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    = 0. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             The  concavity  of  F,  implies  non-increasing  returns.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          The 
                                                                                                                                                                                                                                                                                                                                                            conditions  on  Yk and  Y  are  explained  in  Arrow  and Debreu  [I], 
                                                                                                                                                                                                                                                                                                                                                          p.  276. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               We  get  from  Assumption  2  and the conditions  of  no  excess 
                                                                                                                                                                                                                                                                                                                                                          of  demand  over. supply in  Definition  2, or  equalities  of  demand 
                                                                                                                                                                                                                                                                                                                                                            and supply for non-free goods in Definition  I, the following lemma: 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Lemma  I.  The domain of  xi and yk can be restricted as xi E Ti, 
                                                                                                                                                                                                                                                                                                                                                          yI E Tr , Ti , Tr being  suitably large  convex, compact sets, without 
                                                                                                                                                                                                                                                                                                                                                            causing  any change in the definitions of  a  welfare  maximum point 
                                                                                                                                                                                                                                                                                                                                                            and an equilibrium  point ("). 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Other lemmas we  shall use  in  this paper  are: 
                                                                                                                                                                                                                                                                                                                                                               ~- 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    (*)  This  condition  is  needed  for  the  application  of  the  Kuhn-Tucker 
                                                                                                                                                                                                                                                                                                                                                               Theorem.  See Lemnia  z  below. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (a)                                                                                  Arrow and Debreu  [rJ, pp.  276, 277.  279. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         - 94 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Lemma zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA2. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA(Kuhn-Tucker                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Theorem)  Let  f zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA(x) and                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 g  (x) = 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             = { g, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA(x) , ... , g, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA(x) }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  be concave in x 2 o and g (x) satisfy Slater's 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        condition that there is a vector x0 such that xO2 o and g (xo) > 0. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Then x maximizes  f  (x) subject  to the restrictions  that x2 o  and 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     g (x) 2 o if,  and only if,  there is  a  vector                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        such that zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA(G  , G)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                is  a 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     non-negative  saddle  point  of  the  Lagrangian  Q (x , u) = f  (x) + zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +u.g(x), i.e., 'p(x, G)<'p(X,<)icp(x,  tx)  for  all  x20 and 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             .u>  o .  See  Kuhn and Tucker [7]  and Arrow,  Hurwicz  and Uza- 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           wa [21,  PP- 32-37. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Lemma 3.  (Kakutani's  Fixed  Point  Theorem)  Let  K  be  a 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            1, 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           compact  convex set in n dimensional Euclidian, space Rn and f  (x) 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   be  a  point-to-set,  upper semi-continuous mapping from  K  into K, 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           whose image is non-void  and convex.  Then, there is  a fixed point 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         such that ;I:  = f  (i). See Kakutani [5]  and Nikaido [IO]. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    4. - We are now  in  the  position  to state  the  following  theo- 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   rems on a  welfare maximum  point. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            THEOREM I.  For  any set  of  weights  M, , there  is  a  welfare  ma- 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   ximztin point zrnder  Assi~m$tions I and  2. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         From Assumption I and Definition  z  the social  welfare 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Proof. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             function  is  continuous.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         From  Lemma  I  the domain  IS  compact. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        As a continuous function on the compact domain the social welfare 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             function has a maximum. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               THEOREM 2.  A  welfare  mnximiriit  point  is  a  saddle  point  of 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             'p  (XI r  yk  >  pf I  pk)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              xf 01,  us                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (xi) - xf  p, ( xs  xtz  - xk ykf - xf ;a,)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              + 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        + Ck pk F, (J'x) 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             where x.2 0, yk are  maximizing  variables  and  P,> o , pk>  0, are 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   The  necessary  and  szt ficient  condition  for  it 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        minimizing  variables. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        is  as follows (I): 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                a,U~~j-Pj>o, xTUi& -Pj 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       for  j  = I, ... , m. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    p& Fk (yl.) = 0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Fk  (yk) 2 0, 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  for  k  = I, ... , Y. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Proof.  By  putting  x, = 0, the  assumption  Fk (yk) > 0, 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              xk Y*k < x X, , guarantees  the satisfaction  of  Slater's  condition  in 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Lemma  2.  Then we  can  apply the Kuhn-Tucker  Theorem.  The 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   second half  of  the theorem follows from the definition of  the saddle 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   point. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               (4)  Here  UG!, stands  for  the  left-hand  derivative  of                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         U, with  respect 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              to x,j. 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                - 95 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
                                                                                                                                                                                                                                                                                       5.  -  Next,  we  shall prove  the following theorems on  a  compe- 
                                                                                                                                                                                                        titive equilibrium  point. 
                                                                                                                                                                                                                                                                                       THEOREM 3. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAConditions  b), c)  of                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        an equilibrium #oint  in Defi- 
                                                                                                                                                                                                        nition I can be  written resfiectively in the following form: 
                                                                                                                                                                                                                                                                                                                                b') zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAx, is  a                                                                                                                                       saddle #oint  of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
                                                                                                                                                                                                                                                                                                                                                                                                                                                      'pz zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA(x. I zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA6,)                                                                       = u, (xs) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA- 6, (X, p, xi, - M,) 
                                                                                                                                                                                                         where x, 2 o  are  maximizing  variables  and  6,2 o  are  minimzzing 
                                                                                                                                                                                                          vaviables.                                                                                                                                                                                              The necessary  and  suscient  condition for  it is: 
                                                                                                                                                                                                                                                                                                                                ult, + ', pj>                                                                                                                                                                                                                                                                                   >                                                ui~, - 'j                                                                                                                                                                                                 5  J                                                                                                                                                                                                                    for  j  = I, .... , m 
                                                                                                                                                                                                                                                                                                                                         xj Pj x,j - M* = 0 
                                                                                                                                                                                                                                                                                                                                    c') yk is a  saddle  poznt  of 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   'pk (yk >  ph)                                                                                                                                                                                                                                                                                              p, yhl  - pk  Fk (yk) 
                                                                                                                                                                                                           where  yk  are  maximizing variables and  pa> 0 are  minimiziag varia- 
                                                                                                                                                                                                            bles.  The aecessary and  suficient  condition for it is: 
                                                                                                                                                                                                                                                                                                                                                                                        -  FkYE, -O                                                                                                                                                                      20, P, - pk Fl:kjI 0, for i = I, ... , m, 
                                                                                                                                                                                                                                                                                                                                  FA  (yi) 2 0 . 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               -                                                                                                                                                                                                                                                                                                                                                                                                                                             Then, putting x, = 0, 
                                                                                                                                                                                                                                                                                          Proof.  b) --* b').  x, > o implies A[, > o . 
                                                                                                                                                                                                             Slater's condition in Lemma 2 can be satisfied and the Kuhn-Tucker 
                                                                                                                                                                                                              Theorem  can be  applied. 
                                                                                                                                                                                                                                                                                                                                        c) --* c'). The  assumption  F, (y*J > o implies  Slater's condi- 
                                                                                                                                                                                                              tion  and the Kuhn-Tucker  Theorem  can  be  applied. 
                                                                                                                                                                                                                                                                                            THEOREM 4.  At any welfare maximum fioint, the conditzons a), 
                                                                                                                                                                                                              c), of  an equilibrium  in Definition zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAI are  satisfied. 
                                                                                                                                                                                                                                                                                            Proof.  Compare  Definitions  I  and 2  and  use  Theorem, 2  and 
                                                                                                                                                                                                              the second half  of  Theorem 3. 
                                                                                                                                                                                                                                                                                              6.  -  From  Theorem 4 we  know that if  condition  b) of  an equi- 
                                                                                                                                                                                                             librium  is  satisfied  at  a  welfare  maximum  point  for  some  set  of 
                                                                                                                                                                                                              weights  a,, then it is an equilibrium point.  We have to seek such 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                For this, we  construct the following mapping: 
                                                                                                                                                                                                              a  set  of  weights. 
                                                                                                                                                                                                                                                                                                                                     a) For any point a = (al, ... , am) on the n - I dimensional 
                                                                                                                                                                                                              simplex 3-1 we  get a welfare maximum point  (xot  yoI:, Poj, P"k) 
                                                                                                                                                                                                               and, by P'Q, = Poj[ Z,  Poj , we have (xoi , yor , P'O), where Pro E Sm-'. 
                                                                                                                                                                                                                                                                                                                                        b) By  a) and Lemma  I, it can be  considered  that all  (xi, 
                                                                                                                                                                                                               ya, P) are  contained  in  a  convex  compact  set  K = IIc Tt X TIr 
                                                                                                                                                                                                                 rk x s1-1.  We  can take a  positive  number A  such that. 
                                                                                                                                                                                                                                                                                                                                                                      ZdIMi - Xj P, zit[< A                                                                                                                                                                                                                                                                                                                                                                                                                         for any  (xi, yk  , P) E K. 
                                                                                                                                                                                                                For any a€ .W1, (x,, yk, P) € K, by 
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...Welfare economics and existence of an equilibrium for a competitive economy zyxwvutsrqponmlkjihgfedcbazyxwvutsrqponmlkjihgfedcba by takashi negishi tokyo i the proof com petitive is given arrow debreu many others such as gale kuhn mckenzie nikaido in this note we shall give another equi librium putting emphasis on aspect well known point economic system under perfect competition efficient state pareto s sense which cannot make anyone better off without making someone worse other words it can be said that maximum some properly defined social function subject to resource technological constraints following show linear combination utility functions consumers with weights inverse proportion marginal utilities income then equivalent special therefore prove former showing latter let us construct our model whose follows there m goods n firms zyxwvutsrqponmlkjihgfedcbazyxwvutsrqponmlkjihgfedcbax consumption vector element xtt o xi initial holding uc x zyxwvutsrqponmlkjihgfedcbazyxwvutsrqponmlk...

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