General equilibrium with negligible private information

GENERAL EQUILIBRIUM WITH NEGLIGIBLE PRIVATE INFORMATION WU LEI (B.Sc., Shanghai Jiao Tong University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2009 Acknowledgement Many people have played an important role in the journey of my life. They held me up when I was down and set the path straight for me in difficult times. This thesis would not have been possible without their love and help. Among them, some deserve special mention. I am particularly indebted to my supervisor, Professor Yeneng Sun. In the past five years, he has showed great kindness and patience to me. He guided me through each step of my research. Professor Sun’s help is not limited to research. He always offers valuable suggestions and advice on matters beyond academics. I have benefited greatly from Professor Nicholas C. Yannelis with whom I coauthored two papers. I appreciate the hospitalities from Professor Yannelis and his wife Professor Anne P. Villamil during my visit to the University of Illinois at Urbana Champaign in 2007. I would like to thank my family for their unconditional love. My wife Sheryl has always been there to be my support. She helped correct and improve my English in this thesis. My parents and parents-in-law always have confidence in me and have shared their life iii iv Acknowledgement experience and wisdom to me. Special thanks go to my grandmother who brought me up. It is their love that keeps me moving on. Some friends have also contributed to my research in an indirect way, including but not limited to, Li Lu, Shen Demin and Xu Yuhong. Mr. Xu Yuhong provided me with the Tex template for this thesis, which was originally created by Mr. David Chew. Wu Lei July 2009 Contents Acknowledgement iii Summary ix Preliminaries 1.1 Large deterministic pure exchange economy . . . . . . . . . . . . . . . . . 1.1.1 An introduction to pure exchange economy . . . . . . . . . . . . . 1.1.2 Large deterministic pure exchange economy . . . . . . . . . . . . . 1.1.3 Competitive equilibrium, core, bargaining set and efficiency . . . . Modeling of uncertainty and private information . . . . . . . . . . . . . . 17 1.2.1 Macro state of nature and private information . . . . . . . . . . . 18 1.2.2 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Private information economy . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4 Fubini extension and the exact law of large numbers . . . . . . . . . . . . 24 1.2 v vi Contents On the Existence of Incentive Compatible and Efficient Rational Expectations Equilibrium 29 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Rational expectations equilibrium, incentive compatibility and efficiency 32 2.3 Assumptions and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4 Rational Expectations Equilibrium with Aggregate Signals 51 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Rational expectations equilibrium with aggregate signals . . . . . . . . . . 55 3.2.1 Empirical signal distribution . . . . . . . . . . . . . . . . . . . . . 55 3.2.2 Rational expectations equilibrium with aggregate signals, incentive 3.3 3.4 compatibility and efficiency . . . . . . . . . . . . . . . . . . . . . . 55 Assumptions and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Radner Equilibrium, Private Core and Insurance Equilibrium in Private Information Economy 71 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Economic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.1 Private information economy . . . . . . . . . . . . . . . . . . . . . 73 4.2.2 Induced large deterministic economy . . . . . . . . . . . . . . . . . 75 Contents vii 4.3 Radner equilibrium, private core and insurance equilibrium . . . . . . . . 79 4.4 Assumptions and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.5 Bargaining Set and Walrasian Equilibrium in Private Information Economy 95 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 Economic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2.1 Private information economy . . . . . . . . . . . . . . . . . . . . . 97 5.2.2 State contingent economy . . . . . . . . . . . . . . . . . . . . . . . 98 5.3 Bargaining set and Walrasian equilibrium . . . . . . . . . . . . . . . . . . 100 5.4 Assumptions and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5 5.4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 A Basic Notations and Definitions 113 A.0.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 A.0.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Bibliography 115 Summary The general equilibrium model has gained its popularity in economics since it was introduced by Arrow-Debreu-Mckenzie in 1950s. This model has been extended in several directions in the last several decades. For example, Aumann [8] extended the general equilibrium model to large games with a continuum space of agents. Radner [45] introduced private information in the model to reflect the heterogeneity in information among agents. Sun and Yannelis (see [53] and [54]) built a private information economy model in which each agent has negligible information. This thesis will adopt Sun and Yannelis’ model and analyze various concepts of solutions for the model. It consists of four chapters. In Chapter One, we present all preliminaries for the succeeding chapters. It covers a large deterministic pure exchange economy model, a private information economy model, a framework for the modeling of uncertainty, and the mathematical background on Fubini extension product space introduced in Sun [49]. In Chapter Two, we formulate Radner’s rational expectations equilibrium (REE) for the private information economy model. We show that in the new formulation, rational expectations equilibrium exists. Furthermore, the resulting price in the equilibrium may ix x Summary depend only on the macro state of nature or even degenerate to a constant under certain conditions. In Chapter Three, we introduce a new notion of solution called rational expectations equilibrium with aggregate signals based on Radner’s definition of REE. In a REE with aggregate signals, agents make inference of information from not only commodity prices as they in REE, but also from the aggregate signal distributions announced by a centralized agent. Two theorems on the existence of equilibrium are proven. One important topic in general equilibrium analysis is the equivalence between various concepts of solution. The last two chapters endeavor this task. In Chapter Four, the equivalence between Radner equilibrium, private core and insurance equilibrium is established. And in the last chapter, we continue to prove the equivalence between bargaining set, Walrasian equilibrium and core in the ex ante sense. For the reader’s convenience, we list some notations and mathematical definitions in Appendix A. The reader may consult that part of the thesis when encountering unfamiliar notations or mathematical definitions. 5.4 Assumptions and results 107 1. for i ∈ I, Ui (xpi ) ≤ uπi (xπi ); 2. if Ax p (i, t)dλ = A e(i)dλ for P -almost all t ∈ T , then Ax π (i)dλ = Ae π (i)dλ for a coalition A ∈ I. In particular, if xp is feasible, then xπ is also feasible. In contrast to Φ, Ψ does not retain utility. In fact, it increases the utility of an agent as statement (1) indicates. Statement (2) shows that if an allocation xp in E is A-feasible, then the corresponding allocation xπ is also A-feasible. If A is taken to be the grand coalition I, then the statement (2) says that the feasibility of xp implies that of xπ . The following lemma gives a sufficient condition under which the mapping Ψ retains utility. Lemma 5.4.4. Let xp be an allocation for the private information economy E. Define xπ = Ψ(xp ). If xp is efficient, then Ui (xpi ) = uπi (xπi ) for i ∈ I. If we start with an efficient allocation xp for the private information economy E, we can obtain an allocation xπ = Ψ(xp ). By the above lemma, we know that agents have the same utility with these two allocation schemes. For the allocation xπ , we can construct an allocation y p for the economy E as y p = Φ(xπ ). Lemma 5.4.2 and 5.4.4 tell us that agents are indifferent to the allocations xp and y p . Moreover, for any coalition A, Ax p (i, t)dλ = Ay p (i, t)dλ for P -almost all t ∈ T . Hence, xp and y p are identical to agents in some sense. The following lemma (corresponds to Lemma in [52]) shows that the mapping Φ carries a core allocation for the state contingent economy E π to a core allocation for the private information economy E. As a matter of fact, the mapping Ψ also possesses this property. That is, it maps a core allocation for E to a core allocation for E π . This is the content of Lemma in [52]. Since we not use that result, we will not include it here. Lemma 5.4.5. If xπ is a core allocation for the state contingent economy E π , then xp = Φ(xπ ) is a core allocation for the private information economy E. 108 Chapter 5. Bargaining Set and Walrasian Equilibrium in Private Information Economy As for deterministic economies, the famous core equivalence theorem can also be proven for the private information economy as shown in the Proposition of [52]. We state it below. Lemma 5.4.6. An allocation xπ for the private information economy E is a core allocation if and only if it is a Walrasian equilibrium allocation. The purpose of this chapter is to show the equivalence between the concept of bargaining set and Walrasian equilibrium in the private information economy. As a common approach in the study of private information economy, we often resort to results for the corresponding state contingent economy. Hence, it is important to investigate the properties of various concepts in these two economies. The following two lemmas establish the relationship between bargaining sets in a private information economy and its state contingent economy. Lemma 5.4.7. Let xp be a bargaining allocation for the private information economy E. Define xπ = Ψ(xp ). Then xπ is a bargaining allocation for the state contingent economy Eπ. Proof Let (W, y π ) be an objection to xπ , we shall show that such an objection is unjustified. To this end, we need to find a counterobjection to (W, y π ). Since (W, y π ) is an objection to xπ , we have (1) uπi (xπi ) < uπi (yiπ ) for λ-almost all i ∈ W . Define y p = Φ(y π ). By Lemma 5.4.2, we have 1. W y p (i, t)dλ = W e(i)dλ for P -almost all t ∈ T . 2. Ui (yip ) = uπi (yiπ ) for λ-almost all i ∈ W . It follows that 1. W y p (i, t)dλ = W e(i)dλ for P -almost all t ∈ T . W y π (i)dλ = W eπ (i)dλ and (2) 5.4 Assumptions and results 109 2. Ui (yip ) > uπi (xπi ) for λ-almost all i ∈ W . Since xp is assumed to be efficient according to the definition, by Lemma 5.4.4, Ui (xpi ) = uπi (xπi ). Hence, it follows from the above inequality that Ui (xpi ) < Ui (yip ) for λ-almost all i ∈ W . Together with the first equality above, we know that (W, y p ) is an objection to xp . Since xp is a bargaining allocation, there must exist a counterobjection to (W, y p ). Suppose (V, z p ) is such a counterobjection. By definition 1. V z p (i, t)dλ = V e(i)dλ for P -almost all t ∈ T . 2. Ui (zip ) > Ui (yip ) for λ-almost all i ∈ V ∩ W and Ui (zip ) > Ui (xpi ) for λ-almost all i ∈ V \ W. Let z π = Ψ(z p ), we want to show that (V, z π ) is a counterobjection to (W, y π ). By Lemma 5.4.3, uπi (ziπ ) ≥ Ui (zip ) and V z π (i)dλ = V eπ (i)dλ. Hence, z π is V -feasible. Also note that 1. for i ∈ V ∩ W , uπi (ziπ ) ≥ Ui (zip ) > Ui (yip ) = uπi (yiπ ), 2. for i ∈ V \ W , uπi (ziπ ) ≥ Ui (zip ) > Ui (xpi ) = uπi (xπi ), hence (V, z π ) is a counterobjection to (W, y π ). This indicates that there is no justified objection to xπ . Hence xπ is a bargaining allocation. Q.E.D. Now, we are ready to state the main theorem of this chapter. Theorem 5.4.8. Bargaining set and Walrasian equilibrium coincide in a private information economy E in the sense that W E(E) = B(E). Proof The proof that a Walrasian equilibrium is a bargaining allocation is standard. So we omit it. We prove the converse, i.e., a bargaining allocation is also a Walrasian equilibrium. 110 Chapter 5. Bargaining Set and Walrasian Equilibrium in Private Information Economy The proof is simple and we only need to weave together all the above mentioned lemmas. We sketch the outline. Let xp be a bargaining set for E. Define xπ = Ψ(xp ). By Lemma 5.4.7, xπ is a bargaining set for E π . By Mas-Colell’s equivalence theorem for bargaining set (see Theorem 1.1.16, Chapter 1), xπ is a Walrasian equilibrium allocation, which in turn is a core allocation by the standard Core Equivalence Theorem. By Lemma 5.4.5 and 5.4.6, xp is a Walrasian equilibrium allocation for E. This completes the proof. Q.E.D. Combining with Lemma 5.4.6, we have the following corollary. Corollary 5.4.9. Bargaining set, core and Walrasian equilibrium coincide in a private information economy E in the sense that W E(E) = B(E) = C(E). 5.5 Discussion This chapter examines Mas-Colell’s type of bargaining set in a private information economy model. It establishes the classical equivalence theorem between bargaining set and Walrasian equilibrium for the private information economy model under the conditions of conditionally pairwise independent signals and non-trivial signal process. This result, coupled with Sun and Yannelis’ equivalence theorem for core and Walrasian equilibrium in [52], indicates that Walrasian equilibrium, core and bargaining set all coincide in a private information economy under the above-mentioned conditions. Although we follow Mas-Colell’s definition of bargaining set, it shall be noted that our definition deviates slightly from the original one. As Mas-Colell mentioned (without a proof) in [40], an allocation in the bargaining set is efficient in a large deterministic economy. We conjecture that this statement continues to hold in a private information economy model. However, we could not find a proof for it for the time being. Sun and Yannelis [52] show that when an allocation for the private information economy is efficient, the mapping Ψ retains the utility of each agent in the corresponding state 5.5 Discussion contingent economy. Hence, in order to use this property, we need to impose the efficiency condition explicitly in the definition. In the literature, it is found that Serrano and Vohra [48] also require a bargaining set allocation to be efficient. Our definition is in accordance with theirs. Einy, Moreno and Shitovitz [20] also provide an equivalence theorem between bargaining set and Walrasian equilibrium. The main difference is that in their definition, an allocation is related to private information (measurable with respect to a σ-algebra of private information). Hence, the bargaining set should be regarded as a refinement to the notion of private core (see [58]). For this reason, they call it private bargaining set in the paper. On the contrary, we define the bargaining set in an ex ante sense. In our model, an allocation is a function of the aggregate private information signals, i.e., signal profile. Nevertheless, both definitions are based on the same idea that a threat by a coalition to break the contracts is credible only if there is no valid counterobjection to it. 111 Appendix A Basic Notations and Definitions A.0.1 Notations 1. R+ : the set of all nonnegative real numbers, R+ = {x ∈ R : x ≥ 0}. 2. Rn : n-dimensional Euclidean space. 3. Rn+ : the positive cone of Rn , Rn+ = {(x1 , . . . , xn ) : xi ≥ 0}. 4. xy: inner product of two vectors x and y in Rn . If x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ), then xy = n i=1 xi yi . 5. x ≥ y (or x ≤ y): x, y ∈ Rn , xi ≥ yi (or xi ≤ yi ), i = 1, . . . , n, where xi and yi are the i-th component of x and y respectively. 6. x > y (or x < y): x, y ∈ Rn , xi ≥ yi (or xi ≤ yi ), i = 1, . . . , n, but x = y, where xi and yi are the i-th component of x and y respectively. 7. x >> y (or x yi (or xi < yi ), i = 1, . . . , n, where xi and yi are the i-th component of x and y respectively. 8. BRn+ : the Borel σ-algebra on Rn+ . 9. ∆n : the unit simplex in Rn+ , ∆n = {(x1 , . . . , xn ) ∈ Rn+ : n j=1 xj = 1}. 113 114 Chapter A. Basic Notations and Definitions A.0.2 Definitions Definition A.0.1. A measure space (I, I, λ) is atomless (or nonatomic) if for each A ∈ I, λ(A) > 0, there exists a subset B of A such that < λ(B) < λ(A). In particular, a singleton is of measure zero in an atomless space. For more details, see p. 31 of [12]. Definition A.0.2. Let A be a subset of X, the indicator function 1A on X is defined as 1A (x) =    if x ∈ A   otherwise. Definition A.0.3. 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GENERAL EQUILIBRIUM WITH NEGLIGIBLE PRIVATE INFORMATION WU LEI NATIONAL UNIVERSITY OF SINGAPORE 2009 GENERAL EQUILIBRIUM WITH NEGLIGIBLE PRIVATE INFORMATION WU LEI 2009 [...]... example [49], [52], [54]) introduced a private information economy model in which agents have no direct knowledge of the real state of nature (i.e., uncertainty); they instead receive a noisy private information signal (i.e., asymmetric information) This section covers their model of uncertainty and private information 1.2.1 Macro state of nature and private information For simplicity, we assume that... a private information economy, agents have no complete information They may be uncertain about the state of nature, about their own utilities or about other agents’ status Each agent acquires the knowledge of truth through a piece of private information To facilitate the modeling of uncertainty and private information, a framework will be provided in the second section Based on this framework, a private. .. the initial endowment of agent i when her private information signal is q Initial endowments are the commodities an agent brings to the market for exchange In our model, initial endowment of an agent depends on her private information signal For each signal profile t ∈ T , we assume that e(i, ti ) is λ-integrable Private Information Economy Model A private information economy model consists of an underlying... made with some degree of uncertainty In this section, we will establish the private information economy model which is based on the large deterministic pure exchange economy model but has the feature of uncertainty In the private information economy model, agents’ characteristics are contingent on the underlying macro state of nature and signal profiles Each of them is informed of a noisy private information. .. treatment of private information economy For this reason, in the first section, we will introduce a large deterministic pure exchange economy model and discuss relevant results including the existence of Walrasian equilibrium, optimality of Walrasian equilibrium, core equivalence theorem and equivalence theorem of bargaining set 1 2 Chapter 1 Preliminaries The centerpiece of this thesis is private information. .. deterministic pure exchange economy model that will be frequently used in our treatment of private information economy; in the second section, we provide a framework to model uncertainty and private information in an economy; in the third section, we present the main economic model of this thesis, namely, private information economy model; in the last section, we discuss Fubini extension and the exact... the aggregate initial endowments Given these preparation, we can now define the notion of competitive equilibrium (or Walrasian equilibrium) for the large deterministic pure exchange economic E Definition 1.1.4 (Competitive Equilibrium or Walrasian Equilibrium) A competitive equilibrium (or Walrasian equilibrium) for the large deterministic pure exchange economic E = {(I, I, λ), u, e} is a pair of an... contrast to a deterministic economy model, utility functions and initial endowments depend on the underlying macro state of nature and private information signals This model is summarized in the following definition: Definition 1.3.1 (Private Information Economy (PIE)) A private information economy E = (I × Ω, I F, λ ˜ P ), u, e, (ti , i ∈ I), s consists of ˜ 1 a probability space (Ω, F, P ) of uncertainty,... signals, we do not preclude the possibility of an agent receiving a certain signal with probability zero 1.2 Modeling of uncertainty and private information We call the collective signals t of all agents a signal profile Alternatively, t can be viewed as a mapping from I to T 0 For each i ∈ I, t(i) is agent i’s private information signal t(i) is often written as ti 6 wherever no confusion will arise... operational inefficiency In short, information is noisy and it is difficult for market participants to fathom the exact happenings of a market For this reason, we do not presume agents know the real macro state of nature Instead, they are informed of a signal conveying limited information on the realization of macro state of nature This piece of information is called private information signal Signals are . GENERAL EQUILIBRIUM WITH NEGLIGIBLE PRIVATE INFORMATION WU LEI (B.Sc., Shanghai Jiao Tong University) A THESIS SUBMITTED FOR. extended the general equilibrium model to large games with a continuum space of agents. Radner [45] in- troduced private information in the model to reflect the heterogeneity in information among. piece of private information. To facilitate the modeling of uncertainty and private information, a framework will be provided in the second section. Based on this framework, a private information