From the Ideas of Edgeworth and Pareto in Exchange Economy to Multi-Objective Mathematical Programming

Transcription

1 SCIRE Journal of Mathematics htt://scireaorg/journal/mathematics ril Volume 2 Issue 2 ril 207 From the Ideas of Edgeorth and Pareto in Echange Econom to Multi-Objective Mathematical Programming Zdravko Dimitrov Slavov Christina Slavova Evans 2 Varna Free niversit Varna ulgaria 2 The George Washington niversit Washington DC S bstract In this aer e consider the first general theories of multi-objective mathematical rogramming The stem from otimization techniques in economics and are attributed to the economists Francis Edgeorth and Vilfredo Pareto We ill focus our attention on these ideas from a mathematical oint of vie Keords: otimization multi-objective mathematical rogramming Edgeorth-bo Pareto-otimal equilibrium Introduction The ke concet of ever otimization roblem is to seek the best solution that maimizes or minimizes one or more objective functions During otimization one often needs to consider several objective functions simultaneousl When more than one objective functions are associated the otimization roblem becomes multi-objective in hich case the usual otimization techniques for a scalar function cannot be used This te of roblem is knon 5

2 as either multi-criteria or vector otimization Mathematicall it is knon as multi-objective mathematical rogramming (MOMP and the folloing records eist in 200 Mathematics Subject Classification: 90C - Mathematical rogramming 90C29 - Multi-objective and goal rogramming 65K - Mathematical rogramming otimization and variational techniques 65K05 - Mathematical rogramming methods 49M - Numerical methods and 49M37 - Methods of nonlinear rogramming te The standard form of a MOMP roblem is to find a variable ( 2 m R m so as to m maimize (or minimize f f f f ( 2 n subject to g ( 0 i J i h ( 0 i i J q [ a b ] i J i i i m here f : i J } are given objective functions n 2 g : i J } are given inequalit { i n { i constraint functions 0 h : i J } are given equalit constraint functions q 0 { i q and b i are the loer and uer bounds for i Let the feasible domain (or constraint set be denoted b a b and J m 2 m is the inde set gi( 0 i J m { ( 2 m R : hi ( 0 i Jq } ai i bi i Jm s usual let us assume that set is nonemt but the inde sets emt or nonemt i i a i J and J q can be either We ill discuss onl maimization roblems Hoever each minimization roblem can be transformed to a maimization roblem ver easil b considering the negative of the objective function values Since the objective functions f : i J } ma conflict ith each other it is usuall difficult { i n to obtain the global maimum for each objective function at the same time So if no single oint maimizes all of the several objective functions at once then n rg ma( f i i is emt Therefore the target of the maimization roblem is to obtain a set of solutions that are Pareto-otimal 6

3 2 Notions and Definitions Edgeorth and Pareto are often called the fathers of multi-objective mathematical rogramming The notion of Edgeorth-Pareto otimalit as introduced b the Irish economist Francis Edgeorth ( in 88 Hoever this concet is usuall attributed to the French-Italian economist Vilfredo Pareto ( ho in 896 develoed it further see Figure Edgeorth and Pareto ere studing the natural ordering in the image sace in mathematical models of a simle echange market [2] The first mathematical consideration of this roblem as done b Kuhn and Tucker in 95 [4] [0] [] Since that time multi-objective otimization has became an active research field Figure : The fathers of multi-objective mathematical rogramming In [0] Harold Kuhn and lbert Tucker cite the 975 Nobel Laureate in Economics Tjalling Koomans [8] hen the talk about "vector maimum of Koomans' efficient oint te for several functions" Kuhn and Tucker also cite the 972 Nobel Laureate in Economics Kenneth rro [] ho contributed to the foundations of multi-objective otimization The origins of the mathematical foundations of multi-objective otimization can be traced back to the eriod from 895 to 906 During that eriod Georg Cantor and Feli Hausdorff laid the foundations of infinite dimensional ordered saces [2] [4] [3] In countless books and research aers on multi-objective mathematical rogramming one can find a mathematical definition essentiall as folloing: 7

4 Definition (a oint is called an ideal Pareto-otimal solution if and onl if fi fi for all and all i Jn The set of ideal Pareto-otimal solutions of is denoted b f Pareto-otimal set IPO (the are the global maimum of f on and is called an ideal (b oint is called a Pareto-otimal solution if and onl if there does not eist a oint such that fi fi for all i Jn and f k f k for some k Jn The set of the Pareto-otimal solutions of is denoted b f PO and is called a Paretootimal set The image of a Pareto-otimal set under the objective function is called a Paretofront set ie ( PO f PF ( f (c oint eist a oint f is called a strictl Pareto-otimal solution if and onl if there does not such that fi fi for all i Jn and The set of strictl Pareto-otimal solutions of is denoted b f SPO and is called a strictl Paretootimal set (d oint eist a oint is called a eakl Pareto-otimal solution if and onl if there does not such that fi fi for all i Jn The set of eakl Pareto-otimal solutions of is denoted b f WPO and is called a eakl Pareto-otimal set The image of a eakl Pareto-otimal set under the objective function is called a eakl Paretofront set ie ( WPO f WPF ( f f The above definition qualifies Pareto-otimal solutions in the global sense In literature the term Pareto-otimal is frequentl used snonmousl ith Edgeorth-Pareto otimal efficient non-inferior and non-dominated In this aer let the given functions f : i J } g : i J } and h : i J } be all { i n { i { i q continuous on In this case it is eas to rove that the feasible domain is comact Remark In our otimization roblem it can be shon that: PO f and f nonemt but IPO f n rg ma( f i i and f WPO are SPO ma be emt or nonemt; IPO f PO f SPO f PO f PO f WPO f PF f WPF f f ( ; WPO f and f and [3] and WPF are comact see also [7] [9] 8

5 Note that strictl Pareto-otimal solutions are the multi-objective analogue of unique otimal solutions in scalar otimization Remark 2 It is ell-knon that PO f IPO f if and onl if f nonemt [3] If the ideal Pareto-otimal set f n rg ma( f i i IPO is IPO is emt then e ill accet the elements into the Pareto-otimal strictl Pareto-otimal and eak Pareto-otimal sets as three different solutions of our otimization roblem ( suall a Pareto-otimal solution is not necessaril uniquel determined instead there are several Pareto-otimal solutions 3 Edgeorth-bo Diagram Echange and Equilibrium The general idea of the Edgeorth-bo diagram (or geometric interretation of echange is that there are to consumers: consumer and consumer and to goods: good and good We ill assume for mathematical simlicit that ever good is erfect homogeneous and erfectl divisible The divisibilit assumtion is ver convenient in economic analsis since it allos us to use continuit arguments [6] The Edgeorth-bo diagram gives us a grah of different distributions of goods beteen consumers No suose that consumers and have utilit functions and (strictl increasing concave and continuous resectivel consumer has indifference curves 2 and 3 and consumer has indifference curves 2 and 3 locus of allocations of goods for hich the utilit function is constant is called an indifference curve In order to create an Edgeorth-bo diagram for consumer and consumer e must lace them ithin the same barter echange market To do this e siml take one of the consumers for eamle consumer reflect their indifference curves about the and aes and lace them on the same grah as the other consumer ie consumer see Figure 2 9

6 Figure 2: Edgeorth-bo diagram and Pareto-otimal set 3 arter Echange No e have to achieve otimal distributions of the to goods beteen the to consumers In this case e get an otimization roblem: maimize the utilit functions and of both consumers and subject to a limited amount of goods and nalticall for fied 0 and 0 e obtain a multi-objective otimization roblem as follos: maimize (2 subject to Observe resectivel and resectivel and reresent the quantit of good roert of consumers and and reresent the quantit of good roert of consumers and From the roerties of utilit functions to roblem (3 ritten as: maimize and it follos that roblem (2 is equivalent (3 20

7 subject to The basic idea is as follos: The indifference curves join all the oints that give consumers the same level of utilit connecting all oints of tangenc beteen the indifference curves of both consumers e construct the contract curve and it reresents all Pareto-otimal allocations ie curve O O is a geometrical image of the Pareto-otimal set Thus the Pareto-otimal allocation is the allocation here it is not ossible to make one consumer better off ithout making another orse off see also Figure 2 From a mathematical oint of vie the Pareto-otimal allocations are solutions of multi-objective otimization roblem (3 Here it is true that there are several Pareto-otimal solutions The Pareto-otimal set does not deend on the initial distribution of goods among consumers but it deends on the initial qualities of the various goods see also [2] It can also be seen that the Pareto-otimal allocations are not related to a rices sstem This means that e analzed a barter echange ie mathematical model of echange ithout rices 32 Cometitive Equilibrium In addition let goods and have rices and resectivel and let consumers and have an initial allocation and of goods and resectivel From an economic oint of vie an echange econom is characterized b initial allocation of the goods rices sstem and utilit functions of the consumers In this case e also have to achieve the otimal distribution of the to goods beteen the to consumers Hence e get a ne otimization roblem nalticall for fied and 0 such that 0 and 0 and to ositive arameters ( and ( e obtain a multiobjective otimization roblem ritten as: maimize (4 subject to 2

8 It is imortant to note that rices and of the goods is an additional result in the rocess to solving the multi-objective otimization roblem (4 ie the rices sstem is derivative and a ver imortant roduct of maimization of the utilit functions of the consumers in an echange econom gain from the roerties of utilit functions equivalent to roblem (5 ritten as: and maimize it follos that roblem (4 is (5 subject to It is knon that roblem (3 has a solution but obtaining a solution is difficult for roblem (5 Let us assume that there eist arameters 0 and 0 such that roblem (5 has a unique solution ie roblem (5 has an ideal Pareto-otimal solution In other ords e assume that e have a cometitive equilibrium allocation this is at oint E in Figure 3 and the equilibrium rices are and 22

9 23 Figure 3: Cometitive equilibrium allocation - oint E Remark 3 Note that the sloe in absolute value at oint E of the contract curves and 3 is equal to k see Figure 3 Let us assume that utilit functions and have first artial derivatives therefore e have that ( ( ( grad grad For : ( - marginal rate of substitution and : e obtain the tangenc beteen indifference curves is the oint here (for roblem (3 and if this oint is a cometitive equilibrium allocation then k (for roblem (5 [5] [6] Consider the linear sstem of constraints in roblem (5 ritten as: It is eas to sho mathematicall that the above sstem is equivalent to the folloing sstem:

10 Obviousl e obtain roblem (5 is equivalent to roblem (6 that is: maimize (6 subject to Remark 4 Note that each solution to roblem (6 is also a solution to roblem (3: therefore each cometitive equilibrium allocation is Pareto-otimal Remark 5 lgebraicall consumer trades q for q and similarl consumer trades q for q ; therefore q q It is eas to rove that the folloing statements are equivalent: (a (b ; ; (c (d ; In all above cases the consumers do not reall trade ie q 0 and q 0 If consumers trade goods then ( ( 0 and ( ( 0 Of course e can also consider these ideas in a ure echange econom ith n 2 consumers and m 2 goods 33 Numerical Eamle Here e consider an illustrative eamle to demonstrate the concets of otimalit and equilibrium in a simle echange econom ith to consumers and to goods 24

11 Let the consumers have Cobb-Douglas utilit functions and 0 0 and given b here (0 and 0 No e ill find the Pareto-otimal set and the cometitive equilibrium 0 0 ased on Remark 3 it is eas to sho that roblem (3 is equivalent to the folloing sstem: First e comute and ; therefore ie here 0 ( s a result e obtain that the Pareto-otimal set is equal to 4 {( R : ( 0 ( } and OO : ( infinite number of Pareto-otimal allocations Here there are three cases: ( If then and O O is a straight line (2 If then and O O is a conve curve (3 If then and O O is a concave curve see Figure 2 In fact there is an In this eamle all of the Pareto-otimal allocations are strictl Pareto-otimal and eakl Pareto-otimal and the ideal Pareto-otimal set is emt Clearl roblem (6 is equivalent to the folloing sstem: k k k k We solve this sstem and the result is as follos: 25

12 26 ( ; ( ; ; ; k If the rices lie on a unit simle ie then the rices are ( and ( ( In the general case e obtain s ( and s ( ( here R s or s and s ( here R s ; therefore ( and ( References [] K rro Social Choice and Individual Values Coles Commission for Research in Economics Monograh N:2 John Wile and Sons 95 [2] M Ehrgott Gandibleu Multi-criteria Otimization: State of the rt nnotated ibliograhic Surves Kluer cademic Press 2002 [3] M Ehrgott Multi-criteria Otimization Sringer 2005 [4] M Ehrgott Vilfredo Pareto and multi-objective otimization Documenta Mathematica Etra volume: Otimization Stories [5] Ellickson Cometitive Equilibrium: Theor and lications Cambridge niversit Press 997 [6] Feldman R Serrano Welfare Economics and Social Choice Theor Sringer 2006 [7] J Jahn Vector Otimization: Theor lications and Etensions Sringer 2004

13 [8] T Koomans nalsis of Production as an efficient Combination of ctivities in T Koomans editor Coles Commission for Research in Economics Monograh N:3 John Wile and Sons [9] D Luc Theor of Vector Otimization Sringer 989 [0] H Kuhn Tucker Nonlinear rogramming in J Neman editor Proceedings of the 2nd erkele Smosium on Mathematical Statistics and Probabilit niversit of California Press erkele C [] M Lutacik Mathematical Otimization and Economic nalsis Sringer 200 [2] Z Slavov Structure of the Pareto otimalit set ith fied resources and consumtion sets lied Mathematics and Comutation 54 ( [3] R Steuer Multile Criteria Otimization: Theor Comutation and lication John Wile and Sons