Documentos de trabajo
Tastes and singular economies
E.Accinelli, A.Piria & M.Puchet
Documento No. 13/01 Diciembre, 2001
Tastes and Singular Economies Elvio Accinelli
Alfredo Piriay
Martin Puchetz
November 2001
Abstract The object of this paper is to show some examples of economies in which singular equilibria occur as a consequence of utility functions and where this equilibria play a crucial rolle to understand the behavior of the economy as a system. An economy will be called singular if little changes in the tastes of the consumer imply big changes in the equilibria set.
Resumen El objetivo de este trabajo es el de mostrar algunos ejemplos de economas en las que las singularidades aparecen como consecuencia de las funciones de utilidad, algunas de las cuales al modi carse aunque sea de manera insigni cante producen grandes cambios en el comportamiento de la actividad economica.
Fac. de Ingeniera, IMERL, CC 30. Montevideo y Fac. de Ingeniera, IMERL, CC 30. Montevideo z UNAM, M exico D.F.
[email protected]
Uruguay. elvio@ ng.edu.uy. Uruguay. apiria@ ng.edu.uy.
1
1 Introduction An exchange economy is characterized by the set E = fi ; wi ; I g ; where I is a nite index set, one for each agent, i represents a preference relation for the agent indexed by i; endowments of the agent i are denoted by the symbol wi : As usual preferences are binary relations in the product space X X where X is the consumption space. In our work X = R+l that is we will deal only with economies with a nite number of agents and goods and we will assume that preferences can be represented by utility functions. The object of this paper is to show that in some kind of economies, little changes in tastes may imply big changes in the economic behavior of the economy considered as a system. It is well known that in a neighborhood of a singular economy in the traditional sense, there exist economies with di erent number of walrasian equilibria, see [Accinelli, E. (96)]. Singularity in this frame means that for xed preferences and given endowments there exist at least one set of social weights were the excess utility function vanished: 2 where e;w () = 0 and rankJe;w () < n ; 1; where is the vector of social weights, e;w is the excess utility function for endowments and preferences xed, and Je;w () denotes the jacobian of this function, see [Accinelli, E.(99)]. In this paper we will show cases, where singularities appear as a property of utilities, in the sense that changes in the number of equilibria appear in a neighborhood of certain kind of utility functions, the singular utilities. We will consider endowment as given and we will show cases in which in a neighborhood of a given utility function there exist economies with a di erent number of equilibria. This means that if the economy is singular in this sense, a little modi cation in the utilities of the consumers or little mistakes in the measure or appreciation of his tastes may give raise to an unforeseen behavior. The mathematical statement of this phenomenon is that in a neighborhood of a singular utility, the system is structurally unstable. Existence of singularities may be an answer to questions like: Why do crisis exist ? certainly this is an ambitious question. If this would be a real possibility to explain this topics, then a crisis would be the result of the structural conditions and not a result of exogenous movements in fundamentals. We think that this is the main argument to analyze the structural characteristics of di erent kind of singularities. As our object is to show that changes in utility functions may imply big changes in the behavior of the economy, we will follow the Negishi approach. In this approach the characterization of the walrasian equilibrium set is given by the excess utility function. This function play a fundamental role in our work because:
the utilities appear explicitly in the excess utility function, and zeroes of this function are in one to one correspondence with the set of walrasian equilibria. 1
Then, changes in the utility functions appear directly related with changes in the walrasian equilibrium set. In the following section we will characterize the Negishi approach and in the third section the space of utility functions, will be a metric space, next we will show some examples of economies with this kind of singular utilities, and we conclude with some comments about the economic meaning of this kind of singularities.
2 The Negishi approach and the excess utility function Consider the social welfare function: W : R+ln ! R de ned as:
W (x) =
n X i=1
i ui (xi ):
(1)
where ui is the utility function of the agent indexed with i; and
= (1 ; 2 ; :::; n ) 2 + =
(
2 Rn
++
:
n X i=1
)
i = 1 :
Each i represents the social weight of the agent i in the market. As it is well known if x 2 Rnl solves the maximization problem of W (x) subject to being a feasible allocation i.e., ( ) n n X X nl x 2 F = x 2 R+ : xi wi i=1
i=1
then x is a Pareto optimal allocation see [Mas-Colell, A. Whinston, M.]. Reciprocally it can be proved that if a feasible allocation is Pareto optimal then there exists a 2 (
= 2 R+n :
n X i=1
)
i = 1 ;
such that x ; maximizes W : If we will consider every Pareto optimal allocation we need to consider cases where j = 0 for some j 2 f1; 2; :::; ng. In these cases the agents indexed in this subset will be out of the market. As utilities are strictly increasing the maximization process implies that this agent will receive xj = 0: Since we consider that each agent has a non-null endowment this allocation can not be an equilibrium allocation. Then we can restrict ourselves, without loss of generality, to consider only cases where 2 + : Characterized the set of Pareto optimal allocations, our next step is to choose the elements x in the Pareto optimal set such that can be supported by a price p; satisfying px = pwi for all i = 1; 2; :::; n i.e., an equilibrium allocation. Let E = fui ; wi gni=1 be an exchange economy, to nd the walrasian equilibria we will de ne the excess utility function. 2
De nition 1 Let ei;ui ;wi : + ! R; i = 1; 2; :::; n be the function l X (x()) ;x () ; w ; ei;ui ;wi () = @ui@x ij ij ij
j =1
(2)
is the excess utility function for the agent i; ui is his utility function and wij is the endowment of this agent in the commodity j: The bundle set xi () is risen from x () that maximize W (x) s.t.: F , with i = 1; 2; :::; n and j = 1; 2; :::; l:
De nition 2 The excess utility function, eu1 ;u2;:::un;w : + ! Rn is the vector eu;w () = (e1;u1 ;w1 (); e2;u2 ;w2 (); :::; en;un ;wn ()) : Assuming conditions such that the solution of the maximization program involving the welfare social function will be attained in the interior of
