1 – Introduction

1The study of what we refer today as ‘perfectly competitive markets’ has been the focus of economic theory throughout its history. Today economists have come to identify perfectly competitive markets with the so called ‘price taking’ behavior of participants in them, a term that signifies the inability (or lack of interest) of market participants to manipulate prices and thus influence transactions to their benefit. Of course, this is not the only ingredient of perfectly competitive markets, but it is certainly a crucial one. The framework developed by Walras [1874] naturally became the point of reference for the study of perfectly competitive markets and the Arrow-Debreu [1954] model of an exchange economy provided a widely accepted context of Walrasian ideas. It was soon recognized that there was a need for a justification of the price taking hypothesis.

2It is widely accepted among economists that in a system of markets where individual participants are small relative to the market size, individuals have a negligible effect on the determination of market outcomes, so they may be thought of as exhibiting a price taking behavior. This is the so called ‘price taking hypothesis.’ Of course, in order to make sense of this statement one has to attach a meaning to ‘a small individual relative to market size’. In this way it would be possible to distinguish when price taking is a reasonable assumption and when it is not. The significance of the price taking hypothesis in economics calls for a formal clarification of this point -a ‘theory of competition’ so to speak.

3The search for a justification of the price taking hypothesis has taken two paths. Aumann [1964] took the point of view that an atomless economy, that is an economy populated by an atomless measure space of individuals is the natural mathematical tool to model the idea of ‘negligibility’ of market participants. The interpretation of an atomless economy is that it constitutes an idealization of a ‘large finite economy’: an idealization of an Arrow-Debreu economy with a large number of individuals. Another more traditional approach of economic theory is the asymptotic study of equilibrium outcomes of finite economies, when the number of individuals increases without limit, i.e., the study of large finite economies themselves, rather than their idealization. This methodology starts by proposing a reasonable equilibrium notion when individuals are not price takers. The idea is that if we can identify conditions under which equilibrium outcomes of finite economies converge asymptotically (in some sense) to Walrasian ones, then we would have a context where individuals have negligible effect on market outcomes and hence when ‘price taking’ can be justified as a reasonable hypothesis.

4Of course those two methods to address the price taking hypothesis are not unrelated because the asymptotic study may be viewed as a link between ‘large finite’ and ‘atomless’ economies. In atomless economies ‘negligibility’ is built in the non atomicity of the measure of the space of agents. If a large finite economy is to be thought as a reasonable substitute of the idealized continuum model, it should be the case that equilibrium outcomes of a large finite economy are close to those of the atomless limit, i.e., the equilibrium outcomes of the former should asymptotically converge (in some sense) to those of the latter as the number of individuals increases.

5Asymptotic studies have been performed for a variety of equilibrium notions which have been developed in finite economies. For the most part asymptotic studies focus on the limits of the core [2] and Nash equilibria, mainly because those notions are associated with the traditional theories of Edgeworth [1881] and Cournot [1838], which are prevalent in economic theory.

6Here we study the asymptotic limits of Nash equilibria of strategic market games. Our interest in this is that these games feature an explicit physical description of the price formation and trade process. In this way, the ability of individuals to influence prices and trade volumes is explicit. This issue has been addressed by several authors, Dubey and Shubik [1978], Mas-Colell [1982], Peck and Shell [1989], Sahi and Yao [1989], Amir et al. [1990] among many others, albeit in the fragile context of sequences of economies obtained through replication. [3] Besides the particularity of this type of sequences (finite number of types of individual characteristics), the above results are shown only for the subset of ‘type symmetric’ Nash equilibria. Note that in replica sequences type symmetry is a property of the core (known as ‘equal treatment’), but not of Nash equilibria.

7By contrast, our results apply to more general sequences of economies with characteristics drawn from compact sets and do not depend on type symmetry. One of our results provides also a rate of convergence. In this way we address the issue of asymptotic convergence of Nash equilibria, at the same level of generality as some known core convergence results. Our approach is based on the idea developed in Koutsougeras [2009] of measuring individuals’ departure from price taking, via the wedge between the hyperplanes defined by the price vector and the supporting hyperplane of the indifference surface through the equilibrium bundle. We then demonstrate that under suitable assumptions on the distribution of individual characteristics [4] this wedge becomes arbitrarily small as the number of individuals converges to infinity.

2 – The model

8Let H, indexed by h = 1,2,…, N, be a finite set of agents. There are L, indexed by i = 1,2,…, L commodity types in the economy and the consumption set of each agent is identified with ℜ^L₊. Each individual h ∈ H is characterized by a preference relation and an initial endowment e_h ∈ ℜℜ^L₊{0}. We use the following assumption:

9Assumption 1 Preferences are continuous, convex, and strictly monotone.

10Denote by P_cm the set of preferences that satisfy (1) endowed with the topology of closed convergence. Let T ⊂ P_cm × ℜℜ^L₊. An economy is defined as a mapping E : H → T.

11We now turn to describe a strategic market game, which proposes an explicit model of how exchange in the economy takes place.

2.1 – Trade using inside money

12We will develop our results for the strategic market game version appearing in Postlewaite and Schmeidler [1978] and in Peck et al. [1992] which is described below.

13Trade in the economy is organized via a system of trading posts where individuals offer commodities for sale and place bids for purchases of commodities. Bids are placed in terms of a unit of account. The strategy set of each agent is . Given a strategy profile (b, q) ∈ ∏_h∈HS_h let Bⁱ = ∑_h∈Hbⁱ_h and Qⁱ = ∑_h∈Hqⁱ_h denote aggregate bids and offers for each i = l, 2, …, L. Also for each agent h denote Bⁱ_−h = ∑_k≠hbⁱ_k, Qⁱ_−h = ∑_k≠hqⁱ_k. For a given a strategy profile, the consumption of consumer h ∈ H is determined by x_h = e_h + z_h(b, q), where for i = 1, 2, …, L :

14

15and it is postulated that whenever the term 0/0 appears in the expressions above it is defined to equal zero. When BⁱQⁱ ≠ 0, the fraction has a natural interpretation as the (average) market clearing ‘price’. The relation is a ‘bookkeeping’ restriction which ensures that units of account remain at zero net supply (inside money). The interpretation of this allocation mechanism is that commodities (money) are distributed among non-bankrupt consumers in proportion to their bids (offers), while the purchases of bankrupt consumers are confiscated.

16An equilibrium is defined as a strategy profile (b, q) ∈ ∏_h∈HS_h that forms a Nash equilibrium in the ensuing game with strategic outcome function given by (1). Let N(E) ⊂ ∏_h∈HS_h denote the set of Nash equilibrium strategy profiles of the strategic market game and N(E) ⊂ ℜ^LN₊ the set of consumption allocations corresponding to the elements of N(E).

17The following notation and familiar facts will be useful in the sequel. Fix (b_−h, q_−h) ∈ ∏_k≠hS_k and let [5] . The set of allocations which an individual h ∈ H can achieve via the strategic outcome function is given by the convex set

18

19Conversely, . Thus, due to the bankruptcy rule, at an equilibrium with nonzero bids and offers we have: if and only if:

20

21We say that is fully active if, for the corresponding , we have , i.e., there is trade in all commodities. In the sequel we will focus on such equilibria. [6]

2.2 – Strategic vs. price taking behavior

22The methodology that we will follow uses the fact that price taking is characterized by a ‘tangency condition’: the Walrasian price vector defines a hyperplane that supports the set of bundles which are at least as good as the Walrasian one for each individual and separates that set from the budget set. Given a Nash equilibrium, the idea is to find a ‘surrogate’ vector that separates the ‘budget set’ from the set of bundles which are at least as good as the Nash equilibrium one for each individual and compare that to the strategic market game price vector. The asymptotic exercise becomes then to study under what conditions the difference between the two vectors becomes arbitrarily small as the number of individuals becomes arbitrarily large.

23Formally, let us fix a fully active corresponding to a strategy profile . Consider one h ∈ H and denote .

24The monotonicity of preferences implies that , i.e., lies on the boundary of the convex set c_h, which is C². Since preferences are also convex, by the separating hyperplane theorem there is a p_h ∈ ℜ^L₊, specifically , where Dg_h (·) denotes the gradient of g_h(·), such that

25

26Using the definition of c_h we have

27

28Now observe that if for some λ_h > 0, then the behavior of such an individual would be identical to price taking at the market clearing prices . To see this notice that because ( is active) there is a cheaper point, i.e., w ∈ ℜ^L₊ with . Since furthermore preferences are continuous and convex, the first part of (4) implies . Finally, .

29Therefore, the measurement

30

31serves as an indicator of ‘how far’ the strategic behavior of individual h falls from price taking. [7] Clearly, for each agent h we have and is Walrasian if (and only if) for each agent h. Therefore, a sequence of market game price-allocation pairs tends to become a price taking one, if (and only if) the above indicator tends to zero (in an appropriate sense) for all individuals.

32We are ready now to proceed with the results of this paper.

3 – Results

33For the results that follow we consider a sequence {E_n}_n∈N of economies E_n : H_n → P_cm × [0, r]^L, where #H_n → ∞, and associated x_n ∈ N(E_n), for each n ∈ N which are fully active. Let (b_n, q_n) ∈ N(E_n) corresponding strategies, and z_n,h = x_n,h − e_h the corresponding net trades for each h ∈ H.

34The following result is shown in Koutsougeras [2007] and its proof applies unchanged here.

35Theorem 1. For each ε > 0, there is an n_ε ∈ N so that for all n > n_ε

36

37Or, equivalently

38

39The above theorem asserts some kind of convergence (in measure) for our indicator. Its strength is that it requires no assumptions, so it applies to all sequences of active Nash equilibria. It is noteworthy that some sort of convergence already obtains only as a consequence of the number of individuals without any assumptions on the distribution of their characteristics. This fact reveals that a large number of market participants goes already a long way towards price taking.

40On the other hand it hardly fits the bill: we still need to show some convergence of the Nash equilibrium allocations themselves. Furthermore, we need to ensure that ‘most’ of the commodities are consumed by ‘most’ of the individuals who exhibit ‘almost’ a price taking behavior, as the above theorem asserts. We proceed with two lemmas which will be useful to us in pursuing this end. We skip the proofs which can be found in Koutsougeras [2009].

41Lemma 1. Let r be an upper bound for the endowments in the economy. Define where . Then .

42The above lemma asserts that the proportion of individuals whose net trades remain bounded above along a sequence is uniformly bounded below.

43Lemma 2. Suppose that . There is a subsequence (still indexed by n) and ε > 0, so that for each i = 1, 2, …, L, we have: .

44This lemma claims that the proportion of individuals whose consumption remains bounded away from zero along a sequence is uniformly bounded below.

45We now turn to develop an asymptotic convergence theorem, by introducing appropriate assumptions on the distribution of characteristics along a sequence of economies. In particular, consider a sequence of economies E_n : H_n → T where T ⊂ P_cm × [0, r]^L is compact. For such sequences the set of Nash equilibrium allocations is uniformly bounded as the following result shows.

46Proposition 1. Let {E_n}_n∈N be a sequence of economies, E_n : H_n → T, where #H_n → ∞ and let x_n ∈ N(E_n) for each n ∈ N be fully active. There is B ⊂ ℜ^L₊, which is bounded and depends only on T, such that for all n ∈ N, x_n,h ∈ B for each h ∈ H_n, i.e., the set of Nash equilibrium allocations remains uniformly bounded along a sequence of economies with characteristics drawn from T.

47We can now prove the following result.

48Theorem 2. Consider a sequence of economies{E_n}_n∈N, where E_n : H_n → T, #H_n → ∞, and T ⊂ P_cm × [0, s]^L is compact. Let(b_n, q_n) ∈ N(E_n) and suppose that for some β > 0, for n large enough, so that the corresponding x_n ∈ N(E_n) is fully active. Then given any ε > 0, there is N so that if # H_n > N, then δ_h(x_n) < ε, ∀h ∈ H_n.

49Remark 1. In the theorem above, the set of commodities is held fixed along the sequence. However, the number of commodities could vary as well. In that case, the solution is still valid as long as , i.e., the number of agents increases faster than the number of commodities. [8]

50The theorem above is the desired result, which asserts that along a sequence of economies with a distribution of characteristics drawn from a compact set, strategic behavior becomes asymptotically nearly price taking. The interpretation is that in an economy with a large number of individuals none of whom has exceptional characteristics, strategic behavior is approximately the same as price taking, the approximation becoming finer the larger the number of individuals. Notice that the proof suggests also a rate of convergence, although it cannot be argued that this rate is ‘tight’.

4 – Purely competitive sequences of economies

51The results of the previous section can become more transparent by considering sequences of economies converging to a limit. To this end in this section we will consider ‘purely competitive’ sequences of economies [Hildenbrand, 1974:138] which are defined as follows.

52Let T ⊂ P_cm × ℜ^L₊ be compact. Consider a sequence {E_n}_n∈N, where E_n : H_n → T such that:

1. #H_n → ∞.
2. The sequence of distributions of characteristics (μ_n) converges weakly on T.
3. If μ = lim μ_n, then ∫ e dμ_n → ∫ e dμ.
4. ∫ e dμ > 0.

53Denote now by τ_n the joint distribution of (E_n, x_n) : H_n → T × ℜ^L. The sequence (τ_n)_n∈N is tight since the sequences of its marginal distributions are tight, so we may assume, by passing to a subsequence if necessary, that τ_n → τ weakly. Hence, this sequence of economies and associated allocations admits a continuous representation [Hildenbrand, 1974, proposition 2: 139]: there is an atomless measure space (H, H, ν), (E, x) : H → T × ℜ^L and measurable functions a_n : H → H_n, so that in H and the respective distributions of and (E, x) are τ_n and τ respectively.

54Using this continuous representation, our indicator can be extended in a natural way on H, by . The meaning of theorem (2) can be made more transparent as follows:

55Lemme 3. in measure.

56The following proposition establishes that the allocation x is Walrasian for the economy E, provided that the associated sequence of strategic prices does not converge to the boundary of ℜ^L₊.

57Proposition 2. Let x_n ∈ N(E_n), for each n ∈ N be fully active and suppose that the sequence of associated strategic market game prices {π_n}_n∈N are such that no subsequence converges to the boundary of ℜ^L₊. Then δ_h (x) = 0, ae in H.

Concluding remarks

58Notice that the proof of theorem (2) provides a rate of convergence which depends on the set of characteristics T (the constants C and s), but it also depends on the sequence itself (the constant ξ, which in turn depends on β —the uniform lower bound on offers). This is sensible because in strategic market games there is no parameter β that works for all possible sequences: it is possible that some sequences of prices converge to the boundary of the simplex, irrespectively of the set of characteristics. In that case the corresponding sequence of active equilibria converges to one where some markets are inactive, which typically will not be Walrasian. For the same reason a similar qualification on the sequence of Nash equilibria was needed in proposition (2). Hence, the results in this paper must be understood as asserting that the limit of all sequences of Nash equilibria which remain active in the limit is Walrasian.

59We have established here a generalization of the idea that Nash equilibria converge to Walrasian ones as the number of individuals increases without limit. We showed that this conclusion holds for all Nash equilibria which are fully active in the limit and this conclusion holds for general sequences of economies with characteristics drawn from a compact set. In this way we have developed an asymptotic argument for the emergence of price taking behavior based on non-cooperative games, in the same level of generality as the analogous results based on cooperative games.

60By now the sufficient conditions for the emergence of price taking behavior are well understood. There are two directions where we believe that would be worthwhile to pursue. First, the extension of these results in the asymmetric information context. When information becomes one of the private characteristics of individuals a number of issues arise which make a straightforward extension of the existing results impossible. Second, we believe that it is worthwhile to explore the possibility of formulating necessary conditions for the emergence of price taking. It is unclear whether the sufficient conditions that we have developed for the emergence of competition are also necessary, i.e., whether or not competition may arise in a richer set of circumstances e.g. with small numbers of individuals.