Pure exchange competitive equilibrium under uncertainty

Transcription

1 J Ambient Intell Human Comut 7) 8: DOI.7/s x ORIGINAL RESEARCH Pure exchange cometitive equilibrium under uncertainty Qiqiong Chen Yuanguo Zhu Received: 7 February 7 / Acceted: 4 Aril 7 / Published online: 3 May 7 Sringer-Verlag Berlin Heidelberg 7 Abstract We investigate in this aer a version of ure exchange cometitive equilibrium under uncertain circumstances. Those uncertain factors are embedded in each agent s reference, which is characterized by the uncertain utility function. By maximizing the exected utility of each agent, we formulate this kind of ure exchange cometitive equilibrium roblem into a quasi-variational inequality roblem. This idea is alied in a ure exchange economy which consists of two agents and two goods. And we find the cometitive equilibrium of this economy with each agent s reference being an uncertain variable. Keywords Cometitive equilibrium Walras law Quasivariational inequality Uncertain variable Introduction The theory of cometitive equilibrium was set u by Walras 874). He built a system of simultaneous equations to describe an economy and then showed that the system could be solved to give the equilibrium rices and quantities of commodities. However, the solutions did not reach until Wald 936) gave the first rigorous result on the National Natural Science Foundation of China Grant No. 6673) and China Scholarshi Council Grant No ). * Yuanguo Zhu ygzhu@njust.edu.cn Qiqiong Chen qiqiongchen@63.com Nanjing University of Science and Technology, Nanjing, China existence of the equilibrium in 936. Subsequently, with the develoments of other mathematical branches like linear rogramming, nonlinear analysis and game theory, many researchers obtained numerous existence results and develoed algorithms for calculating the equilibrium by using fixed-oint theory, for examle Arrow and Debreu 954), McKenzie 954), Gale 955) and Nikaidô 968), etc. Starting from 98s, with the develoment of the variational inequality theory, several researchers began to turn their attention to an alternative aroach to study general equilibrium by using a suitable equivalent variational inequalities formulation. It revealed that every equilibrium roblem may be formulated as a variational inequality roblem, see Dafermos 98, 99), Dafermos and Nagurney 984), Border 985) and Gabay and Moulin 98). Those studies indeed showed that variational inequality aroach reresented an innovative and owerful methodology. It rovided a more sohisticated tool not only to study the existence of the economic equilibrium, the sensitivity and stability analysis but to develo efficient comutational rocesses for finding the solutions. Our work is mainly based on the study of Anello et al. ), Donato et al. 8a, b, 6) and Milasi 3) and Liu s uncertainty theory 7). The latter is a new branch of axiomatic mathematics which is used to characterize the belief degrees of human beings rationally. As a result of the inefficient information and the comlexity of the economic environment as well as the sychological factors of human nature, there exist many uncertainties in human s references characterized by utility) in ractice. In our ure exchange model, we deal with maximizing the exected utility of each agents uncertain utility in the cometitive equilibrium roblem. Vol.: ) 3

2 76 Q. Chen, Y. Zhu Finally, the lan of the aer is as follows. For readers convenience, Sect. rovides the background knowledge of the cometitive equilibrium of the ure exchange economy with n agents and k goods in the determinate setting. Section 3 first stresses why do we introduce the uncertain utility in Sect. 3., then rocedes to resent the relimimaries of uncertainty theory in Sect. 3. and focuses on giving our model in Sect We get down to resent the main work in Sect. 4, which handle the ure exchange cometitive equilibrium of an economy consists of two agents and two goods with uncertain references. Finally, Sect. 5 is dedicated to the conclusion. n Agents and k goods cometitive equilibrium roblem of ure exchange economy In this section, we confine our scoe into n agents and k goods cometitive equilibrium roblem of the ure exchange economy and its variational formulation aroach in a determinate setting. For general definitions and roerties corresonding to consumers, rice, commodities, utility functions and economic equilibria we refer readers to Nagurney 993). A ure exchange economy is an economy without roductions. Every agents has an initial endowments like goods). The only issue is how those goods should be distributed and consumed to maximize each agent s utility. In this aer, we consider a ure exchange economy with n consumers indexed by i I ={,,, n} and k different goods indexed by j J ={,, k}. To each consumer i I, we assume that he/she has an initial endowment of the k commodities denoted as e i =e i, e i,, e ik ), where e ij is the quantity of good j. For each commodity j J is associated a rice j such that j. The vector =,,, k ) reresents market rices for those goods. To survive the market, we assume that everyone is endowed with at least one commodity with the ositive rice, that is for i I, there exists at least one j J such that e ij > and j >. also see Assumtion A). Furthermore each agent i has the budget e i, = e i >, where e i, is the inner roduct of the rice vector and quantity vector. In addition each agent i I is associated a consumtion set X i ={x i =x i, x i,, x ik ):x ij, j J} R k +. It reresents the amount of various goods that the customer can consume or trade with other individuals. Note that x ij is the amount of good j that individual i holds for j J. In the market, each consumer i chooses a consumtion bundle x i X i, and the allocation x =x, x, x n ) T R n k is the + total consumtion of the market. The cometitive equilibrium in a ure exchange economy, when ut it simly and informally, lies in two asects. From the ersective of individuals, they have to achieve their otimal choices within their budget constraints after the transaction. To some extent, it means all agents are better-off after the trade than before the trade. On the viewoint of goods market, the total consumtion cannot exceed the available commodity in the market. That is market clears. For individuals, a customary way which results in an otimal choice whether real or imagined) on the basis of the degree of hainess, satisfaction, gratification, enjoyment is using a reference, from the oint of economics. A reference is a sychological term used in relation to deciding what to choose between alternatives. In other words, it can be used to comare and rank various goods available in the economy. To measure references, economists brought utility functions into economics. Generally seaking, the utility function relevant to agent i, is exressed by u i x i ):R k + R for i I, where x i is the consumtion of agent i. In a ure exchange economy, the mechanism is each consumer oerates in the market to find his/her otimal consumtion bundles which maximize his/her utility after transacting with others. Of course, choices have to be subject to a natural budget constraint: the value of consumtion bundles of consumer i at the current rice, i.e. x i, cannot exceed his/her initial wealth e i. Hence this leads to the following maximization roblem. Given the current rice vector of all goods, for all i I, find x i B i ) such that u i x i )= max x i B i ) u i x i ) where B i ) ={x i R k + :x i e i } reresents the budget constraint. Note that B i ) is convex and closed and is unbounded when j = for some j. Since B i ) is homogenous of degree zero in rices, i.e. B i ) =B i λ) for all λ>, we restrict in the rice simlex P ={ R k + : j, j J j =, j J}. For the goods market, it should be cleared after each individual taking his/her action. That is to say for all goods, i I x ij e ij ), which means the total consumtions cannot exceed the total initial endowments. Thus these two asects give a mathematical descrition of cometitive equilibrium of a ure exchange economy as the following form. Find P and x = x, x,, x n ) T B) = i I B i ) such that u i x i )=max xi B i ) u i x i ) for all i I, and i I x ij e ij ) for all j J. As usual in the classical theory of economics, we need some assumtions under the economic setting. ) ) 3

3 Pure exchange cometitive equilibrium under uncertainty Assumtions: A. Each consumtion set X i is a nonemty, convex and closed subset of R k +. A. For all consumer i I there exists j J such that j > and e ij >. A3. For all consumer i I, the utility function u i is continuously differentiable and strictly concave on R +. A4. lim u xi + i x i ) =, for all i I. x i B i ) A5. The market is regulated by Walras law, i.e., i I x i e i ) =, in other words, j J j x ij e ij )=, i I for any P. Thanks to the work of Anello et al. ) see Theorem ), the cometitive equilibrium roblem ) is equivalent to the quasi-variational inequality roblem below. Find P, x = x, x,, x n ) T B ) = i I B i ) such that i I u i x i ), x i x i + i I x i e i ),, for all, x) P B ), 3) where is the gradient oeration. And Donato et al. 8) gave the existence theorem of solutions to roblem 3) in Theorem 4. From 3), x, ) B ) P is a solution to the quasi-variational inequality 3) if and only if is the solution to i I x i e i ),, for all P, and for all i I, x i is a solution to u i x i ), x i x i, for all x i B i ). 5) Now we resent a rocess to find the solution of roblem 3) which is also a cometitive equilibrium of a ure exchange economy consisting of n agents and k goods. We coe with it through two rocedures. Firstly, let us fix P and for all i I, find the unique solution x i to the quasi-variational inequality 5). From Assumtions A3 and A4, Theorem -.3 and Remark -. in Chater of Oden and Kikuchi 98), roblem 5) has a unique solution for each P, denoted by x i ) x i for short) for all i I. Furthermore, with the assumtion of Walas law, the solution to the quasi-variational inequality 5) lies in the set ϝ i ) ={x i X i : j J jx ij e ij )=} for all i I. Thus, when we fix, we can solve 5) by cases directly. Case. Assume that all j >, we have some x ij = e ij j j j x ij e ij ) by 6). Then we substitute x ij in 5) with e ij j j j x ij e ij ), which reduces that solving 5) is equivalent to solving an equation with k 4) 6) 76 unknowns. If the k equation system has solutions in the remise that x ij >, then the solution x i is a solution to 5). Case. Assume that some j =, i.e j > for j j, which is a secial case of case. Then follow the line of case to find the solution to 5). Case 3. If the equation with k unknowns in case does not have a solution, then we find the solution to 5) in the boundary of ϝ i ). That is there exists a subset J of J who has k elements such that x ij = for all j J. Thus we have some x ij = e ij + j j j e ij which is the solution j to 5) under some conditions. And at the same time we need to rove the continuity of x i on P. Finally with x = x, x,, x n ) T at hand by the first rocedure, we look for the solution to the variational inequality concerning with as follows: i I x i e i ),, for all P. If we set Gx) = i I x i e i ), then Gx) is continuous on P. Since P is comact, 4) has a solution. For solving 4), corresonding to the cases of x i, we reduce it to solve a k equation system first since j J j =. If the equation system does not have a solution, we find the solution in the boundary of P. Those rocesses are similar to solve the quasi-variational inequality in the first rocedure. Thus the solution air, x) is the solution we want. 3 Exected utility under uncertainties For clearness, we first exlain why do we study the cometitive equilibrium of ure exchange economy with uncertainty in Sect. 3. exlicitly. Then we resent some concets and lemmas of uncertainty theory connected with the ure exchange economy we will study in Sect. 3.. Finally, we roose our model in Sect Uncertain factors in references As we said in Sect., a reference is a sychological terminology. In sychology, references could be conceived of as an individual s attitude towards a set of objects, tyically reflected in an exlicit decision-making rocess. Alternatively, one could interret the term reference to mean evaluative judgment in the sense of liking or disliking an object which is the most tyical definition emloyed in sychology. However, it does not mean that a reference is necessarily stable over time. Preference can be notably modified by decision-making rocesses, such as choices, even unconsciously. Consequently, reference can be affected by a erson s surroundings and ubringing in terms of geograhical location, cultural background, religious beliefs, and education. Hence 3

4 76 Q. Chen, Y. Zhu an individual s reference is full of uncertain factors. It does ose lots of difficulties to decision-making roblems. In general, how a erson values their consumtions in one state when comaring to another will rely on his/her judgment to what extent that the state in question will actually occur. Here the judgment is characterized by belief degree. Thus the references for consumtions in different states will deend heavily on the ersonal belief degree of how likely and what extent we conjecture those states are. Since the belief degree is a surjective concet, in order to rationally deal with it, Liu 7) ioneered the uncertainty theory in 7. From then on, a fruitful develoments have been witnessed in various branches of alied mathematics, for instance uncertain rogramming Ding and Zhu 4), uncertain otimal control Zhu ), uncertain differential game Yang and Gao 3), uncertain differential equations Ge and Zhu ; Yang and Yao 6, etc.) as well as other fields such as finance Chen and Gao 3), management Gao et al. 7) and economics Yang and Gao 6, 7). To describe references that have some uncertain factors mathematically, we introduce the concet: uncertain utility functions in this aer, which also aears in references Yao and Ji 4; Zhou et al. 7). We will denote the utility function ũx, ξ) instead of ux) as connecting to some uncertain factors, which is reflected by uncertain variable ξ as well as the consumtion level x. 3. Preliminaries on uncertainty theory In uncertainty theory, the most fundamental concet is uncertain measure which is a tye of set function satisfying the axioms of uncertainty theory. Then uncertainty sace and uncertain variable are set u on the basis of it. For details of uncertainty theory, the readers are referred to read books Liu 7, 5) and the references therein. Now we just flesh out the basic concets from uncertainty theory we will use in this aer without further demonstrations. Let Γ be a nonemty set universal set), and let be a σ -algebra over Γ. Then Γ, ) is called a measurable sace and each element Λ in is called a measurable set, which is also called event in uncertainty theory. A set function defined on the σ-algebra over Γ is called an uncertain measure if it satisfies the following three axioms: Axiom Normality) Γ )= for the universal set Γ. Axiom Duality Axiom) {Λ} + {Λ c }= for any event Λ. Axiom 3 Subadditivity Axiom) For every countable sequence of events Λ, Λ,, we have { } Λ i i= i= {Λ i }. Then the trilet Γ,, ) is called an uncertainty sace. An uncertain variable is a function ξ from an uncertainty sace Γ,, ) to the set of real numbers such that {ξ B} is an event for any Borel set B in R. In order to describe uncertain variables, Liu introduces the concet uncertainty distribution. Note that it is a carrier of incomlete information of the uncertain variable. Definition Liu 7) The uncertainty distribution Φt) of an uncertain variable ξ is defined by Φt) = {ξ t} for any real numbers. An uncertainty distribution Φt) is said to be regular if it is continuous and strictly increasing with resect to t at which < Φt) <, and lim t Φt) =, lim t + Φt) =. If the uncertain variable ξ has a regular uncertainty distribution Φx), then the inverse function Φ α) of Φx) is called the inverse uncertainty distribution of ξ. Note that a linear uncertain variable ξ is defined as it has a linear uncertainty distribution if, if t a; t a Φt) =, if a<t<b; b a, if t b. It is denoted as a, b), where a and b are real numbers with a < b. Definition Liu 7) Let ξ be an uncertain variable. Then the exected value of ξ is Eξ] = {ξ t}dt {ξ t}dt rovided at least one of the two integral is finite. Definition 3 Liu 9) The uncertain variables ξ, ξ,, ξ n are said to be indeendent if { n } n ξ i B i ) = {ξ i B i } i= i= for any Borel sets B, B,, B n. We also need the following lemma. 7) 3

5 Pure exchange cometitive equilibrium under uncertainty Lemma Liu and Ha ) Let ξ, ξ,, ξ n be indeendent uncertain variables with regular uncertainty distributions Φ, Φ,, Φ n, resectively. If the function f z, z,, z n ) is strictly increasing with resect to z, z,, z m and strictly decreasing with resect to z m+,, z n, then ξ = f ξ, ξ,, ξ n ) has an exected value Eξ] = Φ n f Φ α), Φ α))dα, α),, Φ α), Φ α),, m m+ where Φ i is the inverse uncertainty distribution of ξ i for i =,,, n. 763 Find =, ) P ={ =, ):,, + = } and x B ) = i=, B i ) such that i x i )=max xi B i ) i x i ), where i x i ) 8) = Eũ i x i, ξ i )], for i =,, and e j x j )+e j x j ) for j =,. Then we will follow the line introduced in Sect. to study the cometitive equilibrium above. Now the key roblem is to calculate the exected utility of an uncertain utility. We give the theorem as follows. Theorem Let ξ, ξ,, ξ n be indeendent uncertain variables with regular uncertainty distributions Φ, Φ,, Φ n, resectively. If the function ũx, z, z,, z n ) is strictly increasing with resect to z, z,, z m and strictly decreasing with resect to z m+,, z n, then the exect utility of ũx, ξ) =ũx, ξ, ξ,, ξ n ) is 3.3 Exected utility model formulation In our model, we assume that the uncertainties are endowed in customers references through the uncertain variable, which is the uncertain utility function. It is a maing ũx, ξ):r k + R R, where x is the customer s consumtion bundle, ξ an uncertain variable defined in an uncertainty sace Γ,, ). Note that the uncertain utility function ũx, ξ) is also an uncertain variable. Since ξ is an uncertain variable defined in the uncertainty sace Γ,, ), it is a measurable function from Γ,, ) to the set of real numbers. Bear in mind that the function ũ is continuous on R k + R, thus the comosite function ũx, ξ) is also a measurable function from Γ,, ) to the set of real numbers. Hence ũx, ξ) is an uncertain variable. In the context of uncertainty theory, it is much more reasonable to describe an agent s consumtion references through its exected value Eux, ξ)] when he/she has the uncertain utility. In the ure exchange economy we will study, in order to describe some uncertain factors in every agent s reference, we assign each of them an uncertain utility function ũ i x i, ξ i ) for i =,. For convenience, we denote i x i )=Eũ i x i, ξ i )] the exected utility of the uncertain utility function ũ i x i, ξ i ) for i =,. To avoid confusion, we distinguish the uncertain ure exchange economy here as the classic ure exchange economy without uncertainty. In stead of maximizing the utility function of each customer in the classic ure exchange economy, we maximize the exected utility in the uncertain ure exchange economy. Hence according to the definition of exected utility, the cometitive equilibrium roblem ) in the case of two agents and two goods under uncertainty will assume the following form: Eũx, ξ)]= Proof We set f z, z,, z n )=ũx, z, z,, z n ). Then f z, z,, z n ) is strictly increasing with resect to z, z,, z m and strictly decreasing with resect to z m+,, z n. It follows from Lemma that That is Φ n ũx, Φ α))dα. Ef ξ, ξ,, ξ n )] = Eũx, ξ)] = Φ n ũx, Φ Remark Let ξ a, b), then α), Φ α),, Φ α), Φ α),, Φ m+ α))dα. f Φ As we said before, we can find the cometitive equilibrium of a ure exchange economy with uncertain references if we have the exected utility at hand. Then we could formulate the cometitive equilibrium of this kind to a quasi-variational inequality, and solve it directly by following the line of what we have said in Sect.. m α), Φ m+ α),, Φ m α), α),, Φ α))dα. Eũx, ξ)] = ũx, α)a + αb)dα. m n α), Φ α),, Φ α), Φ α),, m+ 3

6 764 Q. Chen, Y. Zhu 4 Cometitive equilibrium in the case of two agents and two goods with uncertain utility The aim of this aer is to study the cometitive equilibrium of ure exchange economy in the case of two agents and two goods for both agents who want to maximize their exected utility under indeterminant setting. In this section, we are concerned with finding the cometitive equilibrium of the ure exchange economy with two agents and two goods with uncertain references. We consider a marketlace consisting of two different goods, namely, good and good and two agents, i.e. agent and agent. To make sure they can survive the market, both of them are endowed with initial goods vectors: e =, 4) and e =3, 6), resectively. It means that agent has units good and 4 units good while agent owns 3 units good and 6 units good. The consumtion vector related to agent and agent are x =x, x ) and x =x, x ), where x ij reresents the quantity that the agent i consumes the good j for i =, and j =,. We assume that the law of one rice is fulfilled, i.e. traders scoe out oortunities to the extent that each commodity is sold and urchased at only once rice. Good is associated a real number reresenting its rice while good is associated its rice. We also assume a cometitive behavior, i.e. both agents do not erceive that they can have any influence on these market rices. Moreover, the rice list, ) for good and good satisfies + =. Now we assume that agent has an uncertain utility function defined as u x, x ), ξ, ξ ))= x ξ 4 x ξ + 8ξ x ξ 5x ) with the indeendent uncertain variables ξ, ) and ξ, 3). Meanwhile, suose that agent s uncertain utility function is u x, x ), ξ, ξ ))= x ξ 6 x ξ + 4x +ξ + )x with ξ, ) and ξ, 5) are indeendent. In view of Theorem, together with the remark followed Theorem, agent s exected utility is x )=Ex, x ), ξ, ξ )) = x x + 4x + 5x. Similarly, we get agent s exected utility is x )=Ex, x ), ξ, ξ ))= x x + 4x + 7x We will find the cometitive equilibrium of this economy under the exected utility through the following stes: st Check assumtions on this economy. nd For fixed P and for i =,, find the unique solution x i to the quasi-variational inequality i x i ), x i x i, for all x i B i ). And rove the continuity of x i on P. 3rd Look for the solution =, ) to the variational inequality concerning with =, ) : z,, for all P, 9) where z =z, z ), z = x )+ x 3) is the aggregate excess demand for good. x is the agent s excess demand related good while x 3 is agent s. Similarly, z = x 4)+ x 6) is the aggregate excess demand for good. Now let us follow these stes one by one. First of all, for maings and, we just need to check assumtions A3 and A4 are satisfied since A and A are already satisfied, and A5 is the stiulation. Since both and are continuously differentiable and strictly concave on X and X R +, resectively, where X is agent s consumtion set whereas X is agent s. This meets the requirement of assumtion A3. Furthermore, the coerciveness condition of their exected utility functions matches assumtion A4. We now roceed to find the cometitive equilibrium of this economy. We handle agent s exected utility maximization roblem first. Obviously, x )=x 4, x 5). Then we fix P, and find x B ) ={x =x, x ): x )+ x 4) } such that x 4)x x )+ x 5)x x ), for all x B. ) Note that x ij is the function of, which is x ij ). But for the sake of simlicity, we write x ij instead of x ij ) for i, j =,. Since the oerator x ) is strongly monotone, there exists a unique solution x to the variational inequality ). Under the assumtion that the market is regulated by Walras law, the solution to ) lies in the set below: {x R + : x )+ x 4) =}. We calculate the solution to ) directly by cases. Case. Assume that neither = nor =, i.e.,. Then by ), we have got x = 4 x ) and x + 4. Furthermore, from ), we have ) 3

7 Pure exchange cometitive equilibrium under uncertainty + x + Then to solve ) is equivalent to solve the system as follows: + The solution to ) is under the condition that for, ) P, 4 Obviously, both x and x are continuous on P. Note that we will use x ij ) not x ij in case for clearness. Case. Either = i.e. = ) or = i.e. = ). Subcase.. For =, ), we have the budget constraint as Hence the solution to ) is ) 4] x + 4 = Subcase.. If =, ), then the budget constraint is Thus the solution to ) in this subcase is x x ). x + 4, x = 4 x ). x = + x = + 4 ) + 4, ) + 5. ) ] + 4, ) ] + 5, B, ) ={x R + :x 4 } ={x R + :x 4}. x, ), x, ))=4, 4). B, ) ={x R + :x } ={x R + :x } x, ), x, ))=, 5). ) 3) 4) It is easy to see that the solution in case is continuous on. Case 3. If the system ) does not have solutions, we then find the solution to ) on the boundary of the set 765 {x R + : x )+ x 4) =}, which is either x = or x =. The air { x =, x 5) = 4 +, is the solution to the variational inequality ) if and only if ) + 4 <, for, ) P. If 6) does not hold we reach that { x = + 4, x =, is the solution to the variational inequality ) if and only if 4 ) + 5 <, for, ) P. Note that the solutions under this case are continuous in P. Next we move to agent s exected utility maximization roblem. It easy to see that x )=x 4, x 7). Then fix P, we find x B ) such that x 4)x x )+ x 7)x x ), for x B ). 9) Similarly, we discuss the solutions to 9) by cases as we did for agent s and its counterarts are as follows. Case. Solving 9) is equivalent to solving + x + ] 3 4 x x ), which is further equivalent to solve the system + x + ) 3 4 =, x 3 + 6, x = 6 x 3). The solution to ) is x = + 3 x = + 6 ) ] + 4 ) ] ) 7) 8) ) ) 3

8 766 Q. Chen, Y. Zhu under the condition that 3 6 ) + 4, ) + 7. ) In consistence with the revious cases, we discuss the solution to 8) through three cases. Case. If conditions 4) and ) hold, the solutions to ) and 9) are 3) and ). Then solve { z z =, 9), =. That is Case. Either = or =. Subcase.. If =, then the solution to 9) is x, ), x, ))=4, 6). Subcase.. Otherwise, the solution to 9) is x, ), x, ))=3, 7). Case 3. If the system ) does not have solutions, we find the solution to 9) in the boundary of {x R + : x 3)+ x 6) =}, which is either x = or x =. The air { x =, x 3) = is the solution to 9) if and only if 3 ) + 4 <. 4) =. =, 3) So the solution to 3) is = 3 5, =, which is also the 5 solution to 7). Case. If conditions 6) and 4) hold, the solution the solutions to ) and 9) are 5) and 3). Then solve 8) is equivalent to solve the system { + =, 3) >, =. It does not admit any solutions. Thus we find the soluton to 8) in the boundary of P, i.e. =, ). It is the solution to 8) if and only if z z <. But from x, ), x, )) = 4, 4) and x, ), x, )) = 4, 6) we arrive at that If condition 4) does not hold, then we have { x = 3 + 6, x = is the solution to 9) if and only if 6 ) + 7 <. 5) 6) z z = x )+ x 3) x 4) x 6)] = 3 >, which contradicts with z z <. Hence =, ) is not the solution to 8). Case 3. If conditions 8) and 6) hold, the solution to ) and 9) are 7) and 5). Then to solve 8) is equivalent to solve the system { + =, 3), = >. Note that x and x of all cases are continuous on P. Now let us handle the variational inequality: find =, ) P such that z ) + z ) ), for all =, ) P, 7) where z = x )+ x 3) and z = x 4)+ x 6) are the aggregate excess demand functions of good and good, resectively. Since + =, from 7), we get z z ) ), for, ]. 8) It also does not have a solution on P. Then we turn our eyes to the boundary of P. That is =, ). From the variational inequality 8), it is solvable if and only if z z >. From x, ), x, )) =, 5) and x, ), x, )) = 3, 7), we have z z = <. It is a contradition. Thus =, ) is also not the solution to 8). All in all, = 3 5, = is the unique solution to 8). 5 Hence the cometitive economic equilibrium of our model is 3

9 Pure exchange cometitive equilibrium under uncertainty 8 49, x) = 3 5, 3 3,. 5) Before we interret our results from the oint of economics, we need to recall some requirements of the cometitive equilibrium, x).. Satisfaction: both agents weakly refer his/her bundle to any other affordable bundle as well as better-off after the trade than before the trade. That is i x i ) i e i ) for i =,.. Market clearance: the demand of the market equals the suly of the market. That is x + x = e + e. In our model, the equilibrium rice of good is = 3 5 while that of good is =. And the efficient allocation of goods called Pareto efficiency) for agent is 5 x = 8 3, x = 49. Similarly, agent s efficient allocation is x = 37 3, x = We can see from the Pareto efficiency of this market, the suly of good good ) equals the demand because = = 4 + 6). That is the market 3 clears. In addition, at the equilibrium rice-list, both agents can afford their allocation for given their initial endowment, i.e. ) ) = and 3 ) ) =. Furthermore, 8 3, 49 ) = >, 4) = reveals that agent weakly refers the consumtion bundle 8 3, 49 ) to the initial endowments, 4). 3 And 37 3, 8 ) = > 3, 6) = shows that agent a little strongly refers the bundle 37 3, 8 ) to the initial endowment 3, ). In other words, 3 both agents are better-off after the trade than before the trade. In other words, two goods are distributed efficiently between two agents. 5 Conclusion 767 We stressed the ure exchange cometitive equilibrium in the case that the economy consisted in two agents and two goods under uncertainty in this aer. The uncertainty was incororated into the consumers references through uncertain variables. We named it uncertain utility for each agent. Then we defined this version of cometitive equilibrium roblem by maximizing the exected utility of each agent s uncertain utility function under their budget constraints. Thus it was formulated into a quasi-variational inequality roblem through variational inequality aroach. Finally, we reached the cometitive equilibrium by solving the corresonding quasi-variational inequality directly. Acknowledgements The first author is also deely indebted to Shawnee State University of USA for the suort during her visiting duration. References Anello G, Donato M, Milasi M ) A quasi-variational aroach to a cometitive economic equilibrium roblem without strong monotonicity assumtion. J Glob Otim 48:79 87 Arrow K, Debreu G 954) Existence of an equilibrium for a cometitive economy. Econometrica :65 9 Border K 985) Fixed oint theorems with alications to economics and game theory. Cambrige University Press, Cambrige Chen X, Gao J 3) Uncertain term structure model of interest rate. Soft Comut 74): Dafermos S 98) Traffic equilibrium and variational inequalities. Trans Sci 4:4 54 Dafermos S 99) Exchange rice equilibrium and variational inequalities. Mathe Program 46:39 4 Dafermos S, Nagurney A 984) A network formulation of market equilibrium roblems and variational inequalities. Oer Res Lett 3:47 5 Ding C, Zhu Y 4) Two emirical uncertain models for roject scheduling roblem. J Oer Res Soc 66:47 48 Donato M, Milasi M, Vitanza C 8) Quasi-variational inequality aroach of a cometitive economic equilibrium roblem with utility function: existence of equilibrium. Math Mod Methods Al Sci 83): Donato M, Milasi M, Vitanza C 8) An existence result of a quasivariational inequality associated to an equilibrium roblem. J Glob Otim 4:87 97 Donato M, Milasi M, Vitanza C 6) On study of an economic equilibrium with variational inequality arguments. J Otim Al 68: Gabay D, Moulin H 98) On the uniqueness and stability of Nash equilibria in noncooerative games. In: Benoussan A, Kleindorfer PR, Taiero CS eds) Control alied stochastic, of econometrics and mangement science. Series: contributions to economic analysis, 3. North-Holland, Amsterdam, 7 93 Gale D 955) The law of suly and demand. Math Scand 3:55 69 Gao J, Yang X, Liu D 7) Uncertain shaley value of coalitional game with alication to suly chain alliance. Al Soft Comut. doi:.6/j.asoc Ge X, Zhu Y ) A necessary condition of otimality for uncertain otimal control roblem. Fuzzy Otim Decis Mak 64):

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