Information Economies with Taste Diversity and Bounded Attention Spans 1

We consider an economy with a countably infinite number of both consumers, and pure public information goods. Each of these differentiated products is produced by a single monopoly firm that joins the economy if it can cover production costs. Thus, the product space is endogenous. We assume that the population of agents has diverse tastes, and also bounded attention spans for content. We show that this implies both that agents consume a finite number of public goods, and that each of these goods is consumed by a finite number of agents, at all Pareto efficient allocations. In effect, these two taste assumptions turn pure public goods into what amount to club goods, despite the lack of rivalry in consumption, or crowding of any type. Unfortunately, the equilibrium outcomes of Tiebout like competition between public good providers do not satisfy the First Welfare Theorem. Even nonanonymous, Lindahlian, price systems are not sufficient to signal all profit opportunities to firms. We conclude that information markets are likely to be inefficient, and there will always remain opportunities for economic profits in an information economy.


Introduction
Modern communication technologies have expanded the addressable market for creative works, e-commerce, social media platforms, and other information services immeasurably.The number and variety of information goods has proliferated, and have become increasingly central in ordinary consumption bundles.
Information goods and services of all kinds typically have high first copy costs.Historically, the marginal costs of producing and distributing copies of physical books, records, and movies was relatively low by comparison, although not completely trivial.
Electric distribution has lowered marginal production and distribution costs dramatically.Electronic information goods can be consumed by any number of agents without rivalry, and without significant cost to the producers.In effect, they are public goods.One might speculate that producing a relatively small number of such goods, each of which is consumed by a relatively large number of consumers would be efficient.Doing so takes advantage of the lack of rivalry in consumption, and spreads the first copy costs widely.
Instead, we see a proliferation of news, entertainment, literature, music, and videos, delivered through the web, as well as platforms, electronic games, and other information services.Important as this sector has become, it clearly does not fit the traditional model of competitive firms producing a fixed number of homogeneous products at market prices.How should we model a market in which each firm is a monopoly producer of a differentiated product, where the product space in endogenous, and prices, or consumer's willingness to pay, may not be known in advance?We offer the following observations which we think describe several key features of the information goods and markets that any such model should reflect.
1. Information goods are differentiated.Each Jessica Simpson song is different (however subtly).
We do not see economists or other scholars writing the same paper over and over again (Really!There are differences if you look hard enough).Each blog entry, poem, patent, movie, and web page is a unique product, although it may be similar to many other offerings.
2. Each of these differentiated products is provided in clear, finite amounts.Most pop songs last about three and a half minutes.Even Wagner operas come to an end eventually.Authors of popular books or important articles do not respond to high demand by adding pages to any great extent.Of course, they may write new contributions that are slightly differentiated and in the same spirit, but these are newly created goods rather than additional quantities of the original ones.
3. Agents across the economy have widely diverse tastes, however, tastes across individual agents are correlated to some degree.If the market for content is thick, most agents will be able to chose among information goods that are close substitutes in their estimation, and will share this taste with other agents, at least locally.For example, if two agents like one Katy Perry song, they are both likely to also enjoy another one almost as much.This implies nothing, however, about how either agent feels about Nicki Minaj.
4. As an economy gets larger, the number of intellectual products is likely to increase.In the limit there may be an infinity of intellectual goods on offer.Despite this, it need not be the case that the number of agents who choose to consume any given piece of content becomes large.If too many agents consume one public good, it creates a market opportunity to produce a slightly differentiated product that peels off some of the user-base and still makes a profit.(Lil Wayne in Spanish, Lil Wayne sings the Don Ho song book?) 5. Most critically, not only is it the case that agents do not consume an infinite amount of any given type of content, but they also do not consume an infinite number of types of content.Agents have finite attention spans (or endowments of time).A person might listen to an hour of radio, study two articles, and read a chapter of a book, but is unlikely to listen to one second of 3600 songs or read the first sentence of 1,000 books.Each time he starts to consume a new information good, he incurs an initiation, or switching cost.This may be due to the effort involved in clicking a mouse, retrieving a book from a shelf, or simply refocusing his attention.Eventually this cost becomes prohibitive, and this, in turn, implies that it can only be optimal to consume at most a finite set of information goods.
From a formal standpoint, we describe an economy with pure public information goods that are produced by monopolistic firms.The goods space in endogenous in the sense that firms enter and produce their goods only if there exists sufficient willingness to pay across consumers to cover costs.
We consider the limiting case of an economy with a countably infinite number of information goods and consumers.One problem with such a model is that an infinite number of agents might choose to consume any given public good.This would mean either that the average price paid by consumers would be zero, or that total revenue of the firm would be infinite.Neither fits observations, nor could be expressed as a meaningful optimization problem.
Our solution is to impose a taste diversity condition on agent's preference.As we mention above, this prevents any single information good from attracting too may consumers because it creates a profitable entry opportunity for similar information goods to enter the market.This has the interesting effect of turning pure public goods, into what amount to club goods, despite the lack of rivalry in consumption, or crowding of any type.
Using this approach to public goods, we show that the set of approximately Pareto optimal allocations is not empty.More importantly, we show that at any Pareto optimal (or approximately Pareto optimal) allocation, agents consume a finite number of public goods, and also that each of these good is consumed by a finite number of agents.This is consistent with the five desiderata outlined above.
We then explore two possible price systems in an attempt to decentralize these Pareto optimal allocations.The first price system assigns one (anonymous) price to each information good.These prices necessarily do not reflect the differing marginal willingness to pay for public goods across agents, and not surprisingly fail to generate the standard welfare theorems for the same reasons they do in finite public goods economies.We then proceed more directly along the lines of Lindahl (1919) by allowing personalized prices.Despite assuming that firms are fully informed about agent's preferences, and are able to impose first degree price discrimination, the First Welfare Theorem continues to fail.
The inability of price systems, even ones with extreme informational requirements, to deliver a First Welfare Theorem suggests that this environment has inescapable complexities that prevent market forces from removing arbitrage opportunities.The information economy may be fundamentally entrepreneurial and creative in nature.Production, in particular, cannot be decentralized by any reasonable price system.Thus, economic profits are to be expected.There will be five dollar bills on the ground in the information age, and very little content production will be mediated under the more familiar industrial age economic system of a fixed commodity set, and known, competitive prices.
The paper proceeds as follows: Sections 2 gives a brief review of some of the related literature.We describe our model in detail in Section 3, and characterize optimal allocations in Section 4. Section 5 confirms the failure of anonymous prices to decentralized efficient allocations.In Section 6 we define a notion called Entry-Free Project Equilibrium (EFPE) with nonanonymous prices.EFPE decentralizes consumer choices over goods, and the production choices of incumbent firms.It is also proof against entry from all potential firms even when we allow them the power of first degree price discrimination.Unfortunately, we show that the First Welfare Theorem continues to fail.We discuss the technical contribution of the paper in Section 7, and Section 8 concludes.

A Brief Literature Review
This paper touches on three main literatures.We discuss them briefly below.
The first is the local public goods and club goods literature shaped by Tiebout (1956) and Buchanan (1965).Both point out that many "public" goods are provided by cities, states, and clubs, and are neither purely rival, nor nonrival.When agents impose negative externalities on one another due to crowding or congestion effects, these must be balanced against the benefit sharing the cost of public good production more widely.Thus, a market is created in which coalitions offer different bundles of tax/admission prices and semirival goods.Since consumption of these excludable goods is conditioned on agents voluntarily agreeing to pay the admission price, agents reveal their willingness to pay as they chose their most preferred offering.This provides a solution to the free-riding and market failure seen in the case of pure public goods economies.
Buchanan had in mind a framework of competitive clubs.Crowding was direct, and was a consequence of too many people trying to use the public good the club provided at once.For example, swimming pools, golf courses, and private schools, yield less benefit to each consumer as use approaches their capacity.
In contrast, Tiebout, had in mind a framework of jurisdictions tied to physical locations.Crowding might be direct, but it also could be a result of the relative scarcity of land close to an amenity such as park, or a fire station, whose consumption value decreases with distance.Tying consumption to location also suggests that equilibrium should result in a partition of agents, each living in one, and only one, location, and providing different bundles of goods.In contrast, agents might join several Buchanan type clubs, each providing a different semi-rival good instead of a bundle.
A large literature studying semirival goods in different contexts has since developed.For example, crowding might be nonanonymous in that different types agents may crowd each other differently.These crowding characteristics may even be chosen in response to market signals (see Conley and Wooders 2001).Congestion might depend on the number of visits each member makes to a club.The relative location and size of jurisdictions, or limitations on tax instruments available to jurisdictions, might play a role in outcomes.See Sandler (2023) for a recent survey.On the theory side in particular, there are many explorations of the nature of the core, the competitive equilibrium allocations, and the required price systems.See Allouch, Conley, and Wooders (2009), or Chan and van den Nouweland (2023) for recent treatments and discussions of this literature.
The current paper departs from this literature in a number of ways.We consider a large pure public good economy without crowding, congestion, or location.We show how differences in tastes for pure public goods, combined with inherent limits on the number of public goods agents are capable of consuming, generates outcomes very similar to Tiebout's and Buchanan's.In particular, we show that Pareto efficiency requires that no single public good is consumed by a meaningful fraction agents, and that all agents choose to consume a tiny subset of the public goods that are produced.We show this in the context of an economy with a countably infinite set of agents, and countably infinite, but endogenous, set of pure public goods.
The second literature has to do with innovation.Arrow-Debreu-McKenzie economies, and their extensions to public goods and local public goods, generally embed the idea of a fixed commodity space.The function of prices is to govern efficient production and consumption of these commodities.A key point that quickly emerges when trying to model an information economy for content and other creative works is that any reasonable treatment must treat the commodity space as endogenous.Armstrong (2006) takes a step in this direction with his exploration of endogenous product differentiation of platforms, but does not extend it to the full product space. 3  Millions of websites, videos, blog posts, songs, movies, books, and so on, appear every day, and there is no pre-existing description of such commodities for creators to reference.Exactly what kind of market signals content creators are responding to is not clear.Certainly, it is not a standard set of Walrasian prices.More likely it relates to some estimate on the part of creator of the demand for the work or content they have in mind based on current and projected consumer behavior (dare we say, on marketing studies).
3 Although the product development process is not well understood, ours is certainly not the first general equilibrium model with an infinite number of commodities.Classic models of savings and consumption allow agents to choose infinite streams of future consumption levels.For example Mas-Colell (1975) and Jones (1983) model private goods as a vector of characteristics, so that consumers' consumption bundles are represented by measures over the set of characteristics.These models assume an infinite number of agents and no production.We will not attempt to survey this very large literature here, but see Podczecka and Yannelis (2007) for a recent treatment with references and Aliprantis et al. (1990) for an older, but more comprehensive treatment of the literature.As far as we are aware, none of the above models include public goods.
Of course, the process of new product development has been considered by many other contributors to the literature.The Schumpeterian view emphasizes the importance of entrepreneurial activity (Schumpeter 1947), and more recent empirical work demonstrates the importance of entrepreneurship in growing economies (McMillan and Woodruff 2002), Baumol (1968) argues that such models are unlikely to develop under the modern maximization paradigm, where managers are cold calculating machines with no capacity for the inspired creation of new ideas or products.Some theoretical work in this direction has been attempted (see Nelson and Winter 1974, 1977, or 1978, for example), but it centers around off-equilibrium dynamic selection rather than neoclassical equilibrium analysis and therefore is more difficult to analyze with standard economic tools.Other models explore the self-selection of entrepreneurs based on skill (Lucas 1978 or Calvo andWellisz 1980) or risk aversion (Kihlstrom and Laffont 1979), or analyze the decision to enter into the production of a new good (as in Aron and Lazear 1990), but these models do not predict what types of new goods one might expect to develop.
In general, this literature tends to abstract from pricing or other market signals.Policy approaches to optimal innovation seem to be more the norm.See Jullien and Sand-Zantman (2021), Jullien andPavan (2018), andMazzucato (2021) for recent discussions.Where prices are considered, it is more often from the standpoint of using artificial intelligence and other new tools to maximize profits and choose discriminatory prices for existing information goods.See Lerner (2020), Shiozawa (2020) and Steel et al. (2021), and reference therein.
Despite the proliferation of the internet and related research in fields such as computer science, surprisingly few theoretical papers in economics have examined the knowledge goods created by the internet.Crémer et al. (2000), Besen et al. (2001), andLaffont et al. (2001) study issues of pricing and bargaining between internet service providers and internet backbone providers, and Jackson and Rogers (2007) explore a simple model of network formation that captures observed regularities in the network of links between websites.None of these studies, however, focus on the actual provision of content by website operators.Perhaps the closest work to ours in this respect is by Harper et al. (2005), who model the incentives for users to contribute to an online movie review website, though their model is purely decision-theoretic and does not consider market or strategic interactions.
Finally, the third literature relates to large general equilibrium economies.Aumann (1964) was the first to present a model of a private goods economy with a continuum of agents.He argues that it is only in such an environment that the competitive assumption that agents are price takers is fully justified.The attraction of his approach is that it provides a very clean mathematical structure to gain insights about large finite economies.For example, he was able to show that Debreu and Scarf's (1963) result that the core converges to the competitive allocations as the economy gets large holds exactly in the sense that the core and equilibrium allocations are equivalent in a continuum economy.
Expanding on the Lindahl approach for pricing public goods, Mas-Colell (1980) defines a valuation equilibrium and a cost-share equilibrium for partial equilibrium settings with a single private good and a finite set of public projects with no mathematical structure.This was generalized by De Simone and Graziano (2004) who include production and a Riesz space of private goods.We will provide a more detailed look at the technical contribution of this paper, and the reasons for your modeling choices, in the conclusion.

The Model
Consider an economy with a countably infinite set of Agents, i    ℕ + where ℕ + {  1, , } is the set of positive natural numbers.We will use the convention that capi  ∞ tal letters represent subsets.Thus, i  I ( ) means that agent    i is in the coalition I which is an element of the power set of the set of all agents.
The economy has one, infinitely divisible, Private Good, The initial allocation of private good over agents is given by an Endowment Map, where ( Ω i) is the endowment of agent i.Note that we assume that this is bounded from above and below.
A Private Good Allocation Map is denoted, where X(i) = x i is the private good allocation of agent i, and XMAP denotes the set of all possible private good allocation maps.
The economy also includes a countably infinite set of potential Pure Public Goods, which we will refer to as Web Projects or just, Projects, below, where w  W  () means that project w is in the set of web projects, W = {w, , w,  w,  }, which is an element of the power set of all web projects, and may be finite, or countably infinite.Note that we do not impose any Euclidean structure on the project/public good space. 4he cost of producing a project in terms of private good is given by a bounded Cost Function, where The production of projects is paid for through a Tax Plan that collects enough private good from agents for each project to cover its costs, where TaxPlan(i, w) = t iw means that agent i contributes t iw ≥ 0 private good to the production of web project w and, TAXPLAN denotes the set of all possible tax plans.Note that a tax plan may require agents to contribute to projects to which they do not subscribe.
Given this, an Allocation is a triple, and is a Feasible Allocation given the endowment map if, Ω where UserMap is consistent with SubMap.

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The set of all possible feasible allocations for a given an endowment map is denoted,

FEASIBLE-ALLOCATION.
This definition says that: (1) the difference between any agent's endowment and private good allocation equals the limit of the partial sums of his assigned tax contribution to web projects, and (2) for all web projects that are produced under a given subscription map (that is, have at least one subscriber), the cost of production equals the limit of the partial sums of the assigned tax contribution to the web project over agents.Note that his allows for the possibility that an agent might subscribe to an infinite number of projects, and that a project might have an infinite number of subscribers.
Each agent i   has quasilinear Utility Function of the following form, where, x i : private good consumption of agent i.
V i (W i ): Transferable Utility value to agent i of subscribing to all the projects in W i .
TC i (W i ): Transaction Cost to agent i of subscribing to all the projects in W i .
The motivation for including a transaction cost for consuming a set of projects is that there are search, attention, and other fixed costs, associated with the act of calling up any given set of web pages, getting a set of books off the shelf, putting on a set of CDs, starting a game, or otherwise accessing content.This cost is independent of the time and effort required to consume and enjoy the content.Ultimately, it is this friction that prevents us from consuming an unlimited amount of content. 5Note that we do not impose monotonicity or convexity conditions on V(W) or TC(W).
We assume the following, 5 It might be interesting to generalize this by allowing all agents to receive both an increased flow of utility and an increased attention cost as they put more time or effort into consuming a given public project.This would allow us to consider the often neglected, but important, fact that one has only 24 hours a day to allocate not only to work and leisure, but also to consumption activities of all kinds.Consuming a leisurely dinner is different from wolfing one down.It may be more pleasurable to do the former, even with the same food in front of you, but it means that you must curtail how long you plays Grand Theft Auto later, or not be able to edit your Instagram page as much as you might wish.We will leave these considerations for future research.
∃ w and  where (W j  w / w) is the set of web projects W j with project w added, and project w deleted.
Assumption 1 is just a normalization that says if agents do not subscribe to any projects, they receive no consumption benefits, and pay no attention/transactions costs.
Assumption 2 says that there is an upper limit on the utility that agents can get from any set of subscriptions.To allow otherwise would be to imagine that agents can achieve Nirvana while consuming a finite set of information goods, or perhaps approach it as they consume projects without bound.While it certainly is conceivable that agents might attain enlightenment by reading a book that contains the universal truth, or that listening to the Dark Side of the Moon enough times brings one close to a bliss point, the economic problem disappears in these cases.Note that Assumption 2 implies that V i (W i ) − TC i (W i ) is bounded above.In addition, transaction costs are always non-negative.
Assumption 3 says that at some point, the attention cost of consuming one more web project exceeds any possible gain.We call this the "go to bed" constraint.
Assumption 4 is a weak way of capturing the idea of the existence of close substitutes for any web project.Specifically, the assumption says the following: There exist a strictly positive ε and a strictly positive proportion, δ (  0, 1] such that, a. for every user map, UserMap USERMAP,  and for every project w that has a positive   number of users, UserMap(w) = I w , under this map, there exists, i. a subcoalition I  I w , containing at least δ × ‖ I w ‖ − 1, agents, and ii.an alternative project w   for which no agent in subcoalition I  I w , is currently a user, such that, b. all agents in I prefer an allocation in which project w has been exchanged for by w at least ε private good.
Assumption 4 requires both a diversity of tastes, and the existence of commonly agreed upon substitutes.It captures the idea that while people are different, we tend to differentiate is similar ways.
For example, some people think that Beethoven's fifth symphony is one of the greatest orchestral pieces ever written, while others prefer the works of Mozart, or Bach.People are unlikely to agree on what the best music, work of art, or web platform over all might be.However, it is likely that significant fraction of the fans of the fifth symphony will agree that Beethoven's third is even better.It would be somewhat surprising, on the other hand, if a similar fraction rated Götterdämmerung more highly.Preferences of individual are not random and tend to rank commodities with similar characteristics similarly.
Note that Assumption 4 requires that subcoalition I must contain at least δ × ‖ I w ‖ − 1 agents.Since δ can be very small, this will only bite, in general, if a very large number of agents are consuming a given project w.For example, if δ = .0001and ‖ I w ‖ = 9,000, the size of subcoalition I would have to be at least .9 1 − = .1 − agents, that is, zero agents.

Optimal Allocations
A feasible allocation, {X, TaxPlan, SubMap}  FEASIBLE-ALLOCATION is Pareto Optimal if there does not exist another feasible allocation, {X, TaxPlan, SubMap}  FEASIBLE-ALLOCATION such that for all i  , Note that we use the strict inequality in this definition because of the quasilinear structure of the utility functions.If a weak Pareto improvement exists, then it is possible to transfer small amounts of utility from agents who strongly prefer a weakly Pareto improving allocation, to agents who are indifferent, so that all strongly prefer the new allocation.
A feasible allocation, {X, TaxPlan, SubMap}  FEASIBLE-ALLOCATION is -ε Pareto Optimal if there does not exist another feasible allocation, {X, TaxPlan, SubMap}  FEASIBLE-ALLOCATION such that for all i  , The first Lemma shows that in any Pareto optimal allocation, agents will choose to consume at most a finite number of projects.
Lemma 1.There exists an upper bound, n  ℕ + , on the number of project subscriptions any agent is assigned at any Pareto optimal allocation {X, TaxPlan, SubMap}: where W i = SubMap(i).
Proof: See appendix. The next Lemma shows that in any Pareto optimal allocation, all public projects will have a finite number of subscribers.
Lemma 2. There exists an upper bound, n  ℕ + , on the number of agents that are assigned to any project at any Pareto optimal allocation, {X, TaxPlan, SubMap}: ∀ w  , ‖ I w ‖ < n.
where I w = UserMap(w), and UserMap is consistent with SubMap.
Proof: See appendix. Put together, Lemmas 1 and 2 say that all Pareto optimal allocations in an economy that satisfies Assumptions 1 through 4 will have the properties outlined in the introduction.In particular, all agents subscribe to a finite number of projects, and all projects are subscribed to by a finite number of agents, at all Pareto optimal allocations.
Our first Theorem shows that there will always exist a Pareto optimal allocation, at least in an ε sense.
Proof: See appendix.

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The significance of this Theorem is to show that the attempt to prove Welfare Theorems for this economy is not vacuous.Pareto optimal allocations exist, at least in an sense.Discovering ε whether they can be decentralized by markets is therefore a worthwhile exercise.

Anonymous Prices
A Price System is a mapping from each agent and project to the nonnegative real numbers, plus the empty set, Price: ×   ⇒ ℝ +  ∅ where Price(i, w) = p iw means that if agent i wishes to subscribe to project w, he must pay p iw in private good to firm w, and, PRICE denotes the set of all possible price systems.
Firms, in this economy, are each monopoly producers of a single potential public project.Given a price system, each firm w must decide whether to produce their project, w.More formally, a Production Plan for an economy is denoted, where ProdPlan(w) = 1 means that firm w chooses to produce project w while ProdPlan(w) = 0 means that it chooses not to produce w, and, PRODPLAN denotes the set of all possible production plans.Projects that are not produced, are also not priced, The most natural approach to decentralization would be to use anonymous prices.Otherwise, we would need to assume that firms can figure out each consumer's willingness to pay for projects, and are also able to impose them on individuals.Although firms are each monopoly producers, identification, and the prevention of resale, are strong assumptions it would be preferable to avoid.
A price system, Price  PRICE, is said to be an Anonymous Price System if ∀ w  , and ∀ i, j  , it holds that, Price(i, w) = Price(j, w).
Unfortunately, anonymous decentralization is not generally possible, as the next Theorem demonstrates.6 Theorem 2. It is not possible to decentralize every Pareto optimal allocation with an anonymous price system.
Proof: Consider the following example.Suppose for each pair of adjacent agents, i and i+1, where i is an odd number, That is, each agent in these adjacent pairs, i and i+1, enjoys positive utility only if he subscribes to the single web project w = i.Also suppose, Given this, all Pareto optimal allocations would produce each of the odd numbered projects, w  ,  and assign subscriptions to agents i = w and i = w+1.Doing so generates a net transferable utility surplus of (11 − 1) + (21 1 − ) − 25 =5 from each adjacent pair of agents after production costs are paid, and transaction costs deducted.The associated tax system that completes the feasible allocation could be anything that divides the production cost of 25 between these pairs of agents, for example.Even numbered projects would not be produced.
Unfortunately, there is no anonymous price system which can decentralize this.To see this consider any odd numbered project w  .
• If Price(i, w) = Price(i+1, w) > 20, then no agent chooses to subscribe to w, and so revenue equals zero.
Obviously, no agent would choose to subscribe to any even numbered project at any positive price.Thus, none of the anonymous price systems that induce agents to choose the Pareto optimal set of subscriptions (Price(i, w) = Price(i+1, w) ≤ 10) collect enough revenue to cover the project cost of 25 which is necessary to incentivize firm w to choose to produce the project.We conclude that is not possible to decentralize every Pareto optimal allocation with an anonymous price system.

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Theorem 2 tells us that the Second Welfare Theorem fails for anonymous equilibrium.This should not be surprising given that projects are nonrival in consumption, and we would therefore expect that personalized Lindahlian prices should be required for decentralization.
It is easy to see to use the same example to construct a failure of the First Welfare Theorem.Suppose, ∀ w  ,  Cost(w) = 15.
Then Price(i, w) = Price(i+1, w) =15 results in agent i+1 choosing to subscribe to project w = i and receiving a surplus of 5. Agent i, on the other hand, chooses not to subscribe to anything.If he did so at this price, he would receive negative 5 units of utility.Since the all Pareto optimal allocations require that both agents to subscribe to project w = i, we see that anonymous prices can decentralize Pareto dominated allocations.Thus, both the First and Second Welfare Theorems fail for anonymous prices.

Equilibrium and Efficiency
In our construction, projects which are not produced, are not priced (ProdPlan(w) = 0 ⇒ ∀ i  , Price(i, w) = ∅).This means that our price system embeds incomplete markets, and so, perhaps, it should hardly be surprising that markets fail.Exactly how to complete markets, however is not immediate.
The economy defined here contemplates a countable infinity of projects that might, or might not, be produced in equilibrium.A complete price system, therefore, is tasked not only with allocating consumers to projects, but also with decentralizing innovation in the sense that it optimally incentivizes the entry of new commodities to the goods space.A complete price system would seem to require posting a marginal willingness to pay, for infinite number of agents, for every conceivable variation of a web project.
It seems unlikely that anyone would bother to investigate the economic viability of such obviously undesirable potential projects as: David Hasselhoff sings Mongolian throat music, on ice!, or a 37 volume work on the life and times of a hamster that Che Guevara owned as child.Some projects are not even worth considering from the standpoint of the economy as whole, much less deriving, and posting, a set of personalized prices for every agent in the economy.
A less exhaustive approach would be to imagine producers as entrepreneurs who conduct marketing studies to determine if there is enough willingness to pay in an economy to cover the costs of the potential project that they are capable of producing.In effect, each firm searches for a set of non-anonymous prices that would justify its entry into the market.These prices, however, do not become part of the equilibrium price system unless the firm actually enters.This approach defines a type of "entry-free" equilibrium notion.Each firm holds constant the prices of incumbent firms, and the subscription choices of all agents, but tries to discover if there are prices that would induce enough agents to add its potential project to their existing set of subscriptions, while collecting enough revenue to cover cost.Thus, incumbent firms are price-takers, but potential entrants are price-makers.
As price-takers, incumbent firms assume all agents assigned positive price will subscribe to their project in equilibrium, Firms therefore decide to produce their project if the sum of these prices covers costs, In general, firms may make profits, and so these would need to be returned to agents through a system of ownership shares.This would complicate the definition of equilibrium significantly.Since agents' utility functions are quasilinear, transfer of profits to agents would have no effect on either the equilibrium subscription choices, or the set of Pareto optimal allocations.We therefore simplify by constraining the equilibrium price system to sum to the costs of each of the projects that are produced in equilibrium in the definition below.
An Entry-Free Project Equilibrium (EFPE) with respect to an endowment map consists of Ω a triple, ∑ w=1 w Price(i, w), and and i ∀   such that Price(i, w) > 0, Condition 1 says that, for all agents, the equilibrium subscription plan and private good allocation are affordable.Note that this implies that both, lim w ⇒ ∞ ∑ w=1 w Price(i, w), and lim w ⇒ ∞ ∑ w=1 w TaxPlan(i, w), must be finite even if the number of projects that an agent subscribes to, ‖ W i , ‖ is not bounded.
Condition 2 says that, for all agents, the equilibrium subscription plan is optimal given prices.In particular, for each agent i  , there does not exist an alternative subscription map, SubMap, that leaves him better off under price system Price, than the equilibrium subscription map, SubMap.
Condition 3 says that firms produce all projects for which the sum of prices over agents equals the cost of the project.Projects for which prices sum to less than cost are not produced.Note that incumbent firms that produce in equilibrium are price-takers.
Condition 4 says no firm that does not produce in equilibrium can find a set of prices for agents that would cause agents to add the firm's potential project to their subscription map, and which would cover the cost of producing the project.Thus, the equilibrium is "entry-free" in the sense that even if all non-incumbent firms were price-makers capable of perfect first-degree price discrimination, profitable entry by any potential firm is infeasible.
The next Theorem shows that even with non-anonymous prices and first degree price discrimination, it is still possible support Pareto dominated allocations as equilibria.
Theorem 3. EFPE allocations may or may not be Pareto optimal.
Proof: Consider the following example.For each n  ℕ + , construct a coalition, I n , consisting of ten adjacent agents as follows, That is, for each coalition of ten adjacent agents, each agent in the coalition gets 20 units of utility if he subscribes to all of the first five adjacent "low" projects starting from w = 10×(n − 1)+1, 25 units of utility if he subscribes to all of the second five adjacent "high" projects starting from w = 10×(n 1)+6 − , and zero utility otherwise.Also suppose that, Consider two feasible allocations.The "low" plan is defined as follows: In contrast, the "high" plan is defined as follows: Observe that both plans are feasible since for each ten agent coalition, a total of five projects are subscribed to.The sum of prices for each project is 10 × 2 = 20, which equals the cost of each project.Since each agent pays a total of 10 in taxes and has an endowment of 15, they are left with 5 units of private good to consume.Also observe that the "low" allocation leaves each agent i  I n a utility of: while the "high" allocation differs only in that V i (W n,high ) = 25, and so leaves agents with a utility of 25.It is easy to see that, not only that the high plan Pareto dominates the low plan, but that the high plan is, in fact, Pareto optimal.
We claim the low allocation is an EFPE for the following price system Price low  PRICE, and To prove this, we show each of the four equilibrium conditions is satisfied.
(2) Following SubMap low gives each agent a utility 20.The alternative of choosing to subscribe any other set of projects offered in equilibrium besides W n,low gives agents net negative utility.
Only the "low" projects are priced and produced, and Even though for agents i  I n , Price(i, w) low = 0 for all projects w  W n,low such that n  n, when transaction costs are subtracted, net utility is negative.Thus, all agents chose to follow the subscription map in equilibrium. ( and so only projects for which prices exactly cover costs are produced. (4) Finally, consider any w  s  uch that ProdPlan(w) = 0.Only "high" projects are not produced, and adding any single high project, w W n,high to a "low" set of subscriptions drops the utility an agent receives from subscriptions to zero: V i (W n,low  w) = 0.The marginal willingness to pay for a subscription to a single high project, therefore, is less than zero, since adding it results in a loss for 21 units of utility (accounting for higher transaction costs).Thus, no extension of the price system to a non-produced project, Price(i, w), would yield enough revenue to cover its production costs.

We conclude that
Price low ProdPlan low {X low , TaxPlan low , SubMap low } is an EFPE that supports a Pareto dominated feasible allocation.
On the other hand, it is easy to check that ProdPlan high (w) = 0 if ∄ n  ℕ + such that w  W n,high and {X high , TaxPlan high , SubMap high } is also an EFPE, and that it supports a Pareto optimal allocation. Theorem 3 tells us that the First Welfare Theorem fails regardless of whether prices are anonymous or non-anonymous.Given this, we do not try to prove a Second Welfare Theorem.Supporting Pareto optimal allocations as equilibria, when in general, equilibria may not be Pareto optimal, is of limited value.
We see that despite our equilibrium concept incorporating first-degree price discrimination, an infinite set of prices for an infinite set of goods, and price-making potential entrants for every possible new project, markets fail profoundly.We discuss the implications of this in the concluding Section.

Countably Infinite Public Goods Economies
Aumann 's (1964) pathbreaking approach provided many useful insights precisely because the continuum economy he considered was clearly an economically meaningful limit case of a large finite economy.Without this clear connection, his results might simply have been artifacts of the modeling approach and so would give misleading or even incorrect intuitions about large finite economies.
Unfortunately, the connection between finite and continuum cases is much less clear in the case of pure public goods economies as they are usually written.See Muench (1972), for the canonical treatment and Berliant and Rothstein (2000) for a more recent treatment that discusses some issues outlined below.
The central problem with the continuum approach to public goods economies is that private goods and public goods are measured in different ways that make them fundamentally incomparable.Consider an economy with a fixed, finite set of public and private goods.Note that the level of a public good is determined by the integral over agents' contributions.
Suppose that the average contribution of agents is strictly positive.Then the ratio of the public to private good is infinite for all consumers.(For example, if we each donate a dollar to public radio and there are infinitely many of us, then public radio would have an infinite number of dollars to spend.)How would one compare two such allocations?Is it meaningful to say that one infinity of public goods is preferred to another?Would we be better or worse off if we halved our contributions given that we would still be funding all public goods at infinite levels?
On the other hand, suppose that the average agents' contribution is zero.Would their Lindahl taxes be infinitesimal in this case?How would agents express demand for public goods when faced with infinitesimal Lindahl prices?In any event, how would we distinguish between different allocations in which the public goods level is unmeasurable, or almost zero?For such allocations, the ratio of public to private goods consumption for each consumer could be bounded or undefined.Differentiating these two cases is important, but is not possible in a standard continuum economy.
We conclude that the direct extension of the Aumann approach to public goods creates significant mathematical and economic difficulties that make it hard to understand it as a limit of a large finite economy.
In this paper, we propose an approach to large public goods economies that is departs from the existing literature in several ways: 1.The economy has a countably infinite number of consumers instead of a continuum.We are unaware of any other work in public goods using countable infinities.
2. The economy has a countably infinite number of public goods.
3. The public goods that are produced are endogenously determined in equilibrium, 4. We assume taste diversity, and limited attention span.We show that this implies that each agent subscribes to, and pays for, a finite number of projects, and that each public project is subscribed to by a finite number of agents.
Our approach solves the measure incompatibility problem described above.At no point is it necessary to contemplate infinitesimals or infinities, something we believe to be outside the abilities of consumers.We also believe that this provides a more reasonable limiting case of a large public goods economy, and generates outcomes that match many features of the information economy in which we actually live.

Conclusions
In this paper, we were motivated by two concerns.First, we wanted to provide a positive analysis of the properties of an information economy based as much as possible on the institutional details and constraints we observe in the real world.Second, we wanted to provide a model of a countably infinite public goods economy that was an economically, and mathematically, meaningful limit of a large finite public good economy.
We argued that it was unlikely that the levels of projects consumed by agents would grow without bound as an economy gets large.At some point, agents cease to be able to even contemplate the vastness of such consumption bundles.Certainly, agents would not be able to comprehend infinities of information goods.However, if the number of public goods is finite, this necessarily im -plies that the private good contribution levels to project production (Lindahl taxes, for example) would have to converge to zero.This clearly is not happening in any real world economy we see, no matter how large.
We proposed an alternative: as the economy gets large, the number of projects (which we think of as internet or information goods) also gets large.Thus, the commodity set is not fixed as it is in traditional Arrow-Debreu-MacKenzie/Samuelson economies.We showed that under fairly mild conditions, the Pareto optimal allocations of this economy involve an infinite number of public projects being produced, each of which is consumed by a finite number of agents.In addition, each agent consumes only a finite and bounded number of projects.
What drives these results are two basic assumptions: (1) agents have limited attention spans, and it costs an agent a certain amount of attention to begin to consume each public project and (2) there are likely to be close substitutes for any project that are at least slightly preferred, all else equal, by a fraction of any set of consumers.In effect, limited attention spans, and diversity of tastes, turn a pure public goods economy into a club good economy, even though there is no crowding.
Although the optimal allocations are exactly what we believe we see in the real world, things get tricky when the question of how to support these allocations as equilibrium outcomes is considered.It is immediately clear that efficient decentralization is impossible using anonymous prices.
We offer an alternative non-anonymous equilibrium concept called Entry-Free Project Equilibrium.EFPE has extremely high information requirements and assumes that firms are able to engage in first-degree price discrimination.Despite this, the First Welfare Theorem fails.Even when the equilibrium is Entry-Free, it is still possible to support Pareto dominated allocations as equilibria.Pareto improvements sometimes require coordinated, multilateral, entry, and significant shifts in demand behavior.Prices that correctly incentivize only unilateral entry of any potential firm turn out to be insufficient.While this might be viewed as a negative result, we think it is better viewed as a positive conclusion.Information goods are infinitely variable, and have the kind of discrete, non-metrizable structure we describe in the paper.There is no sense in which a Taylor Swift song can be compared to a Bach cantata.At least we know of no hedonic or other quantitative metric that could do so.They are simply different.
The market, therefore, has a hard time signaling that a certain new public project should be produced.Entrepreneurs take educated guesses about what will succeed, but there are no arbitrage opportunities implied by disparities in the cost-revenue signals from an equilibrium price system that are visible to all.Thus, we argue that the conclusion should be that we generally do not see first best outcomes in such an economy.It is possible to get rich (that is, make economic profits) if you happen to stumble on a public project that you can produce cheaply and which is in high demand.It is not at all surprising that no one beat you to it.There may indeed be five dollar bills lying on the ground in the new information economy.
Thus, both conditions of the feasibility are satisfied, and so, {X, TaxPlan, SubMap}  FEASIBLE-ALLOCATION.
Next, we show that {X, TaxPlan, SubMap} Pareto dominates {X, TaxPlan, SubMap}.Note that ∀ i  , and w  , TaxPlan(i, w) ≤ TaxPlan(i, w), since taxes at both allocations are identical except for those projects that happen not to be produced under SubMap, and for which all taxes drop to 0. This implies for ∀ i  , Since ∀ i  , i  j, SubMap(i, w) = SubMap(i, w), all agents besides agent j consume the same level of private good, and subscribe to the same set of projects, and so are indifferent between the two feasible allocations.
Agent j, consumes at least the same level of private good in both feasible allocations, but receives more transferable utility from SubMap(j, w) (which gives zero instead of negative utility).
We conclude that {X, TaxPlan, SubMap} weakly Pareto dominates {X, TaxPlan, SubMap}.Since utility is quasilinear, it is possible to reallocate private good away from agent j, and to all other agents, such that all are strictly better off, which completes the proof. Lemma 2. There exists an upper bound, n  ℕ + , on the number of agents that are assigned to any project at any Pareto optimal allocation {X, TaxPlan, SubMap}: ∀ w  , ‖ I w ‖ < n.
where I w = UserMap(w), and UserMap is consistent with SubMap.Together this implies that ∀ i ∉ I are exactly as well off, while ∀ j  I are strictly better off at {X, TaxPlan, SubMap}.Since utility is quasilinear, it is possible to reallocate private good away from agent j, and to all other agents, such that all are strictly better off, which completes the argument.
To see this, note that ∀ i  , and ∀ w  , w  w, TaxPlan(i, w) = TaxPlan(i, w).Thus, for all projects besides w, Thus, the TaxPlan exactly pays for all public projects.Part (3) requires ∀ i  , ∑ w=1 w TaxPlan(i, w), which balances the budget, and complete the proof that {X, TaxPlan, SubMap} is feasible. The Mean Utility of a feasible allocation is denoted, MeanUtility: FEASIBLE-ALLOCATION ⇒ ℝ + and defined as, ī .
By Assumption 2, the utility value of any set of subscriptions assigned by SubMap to any agent has an upper bound.Since endowments are also bounded above, MeanUtility of any feasible allocation is must also be bounded above.This implies that {U} is a bounded above sub-set of ℝ + , and it therefore must have a supremum, U*.In turn, there must exist a sequence of feasible plans, {X s , TaxPlan s , SubMap s }  FEASIBLE-ALLOCATION such that for any 0 ε  , ∃ s  ℕ + such that ∀ s  s, ε  U* − MeanUtility(X s , TaxPlan s , SubMap s ), Now suppose for any ε  0, that their existed a feasible allocation {X, TaxPlan, SubMap}  FEASIBLE-ALLOCATION such that ∀ s  s, ∀ i   U i (X(i), SubMap(i)) U  i (X s (i), SubMap s (i)) + .ε That is, ∀ s  s, {X s , TaxPlan s , SubMap s } is -Pareto dominated by ε {X, TaxPlan, SubMap}.This implies, MeanUtility(X, TaxPlan, SubMap))  MeanUtility(X s , TaxPlan s , SubMap s ) + .ε But, U* ≥ MeanUtility(X, TaxPlan, SubMap) which implies, U* MeanUtility(X s , TaxPlan s , SubMap s ) + .ε and so, U* − MeanUtility(X s , TaxPlan s , SubMap s ) ,  ε contradicting the construction of the supremum, U*. 

Proof:
Suppose instead no such upper-bound exists.Then for any n  ℕ + , ∃ w   such that, ‖ I w ‖ ℕ  + .But by Assumption 4, ∃ ε 0,  δ (0, 1],  a subcoalition I  I w  UserMap(w), and a project w   such that,‖ I ‖ ≥ δ × ‖ I w ‖ − 1, and ∀ j  I ,Recall, however, that I w can be chosen to contain an arbitrarily large number of users, and since,‖ I ‖ ≥ δ × ‖ I w ‖ − 1,and Cost(w) ≤ c, is bounded, ‖ I can also be made large enough so that ‖ ε > Cost(w) ÷ ‖ I ‖.
is the private good cost of producing project w  . Each agent is assigned, or chooses, a set of Subscriptions to projects which collectively define a w  ( )   means that project w is subscribed to by all agents i  I w , and, USERMAP denotes the set of all possible user mappings.A subscription mapping, SubMap  SUBMAP, and user mapping, UserMap  USERMAP, are said to be Consistent if ∀ i  , and ∀ w  ,