Technology adoption and accumulation in a vintage-capital model

We present a model of capital accumulation and technology adoption in a vintage-capital framework. The model is an infinite-horizon/infinite-dimensional optimal control model: the firm employs a continuum of technologies (a continuum of heterogeneous capital goods). Capital goods are technology specific, their technology is related to vintage and technology progress. The entrepreneur maximizes the profits obtained by employing a continuum of technologies under the assumption of constant returns to scale and bearing adjustment costs for gross investments. The diffusion of a new technology is established by allowing the entrepreneur to invest in vintage capital goods.     

Comparing finite mechanisms

Summary. This paper obtains finite analogues to propositions that a previous literature obtained about the informational efficiency of mechanisms whose possible messages form a continuum. Upon reaching an equilibrium message, to which all persons “agree”, a mechanism obtains an action appropriate to the organization's environment. Each person's privately observed characteristic (a part of the organization's environment) enters her agreement rule. An example is the Walrasian mechanism in an exchange economy. There a message specifies a proposed trade vector for each trader as well as a price for each non-numeraire commodity. A trader agrees if the price of each non-numeraire commodity equals her marginal utility for that commodity (at the proposed trades) divided by her marginal utility for the numeraire. At an equilibrium message, the mechanism's action consists of the trades specified in that message, and (for classic economies) those trades are Pareto-optimal and individually rational. Even though the space of environments (characteristics) is a continuum, mechanisms with a continuum of possible messages are unrealistic, since transmitting every point of a continuum is impossible. In reality, messages have to be rounded off and the number of possible messages has to be finite. Moreover, reaching a continuum mechanism's equilibrium message typically requires infinite time and that difficulty is absent if the number of possible messages is finite. The question therefore arises whether results about continuum mechanisms have finite counterparts. If we measure a continuum mechanism's communication cost by its message-space dimension, then our corresponding cost measure for a finite mechanism is the (finite) number of possible equilibrium messages. We find that if two continuum mechanisms yield the same action but the first has higher message-space dimension, then a sufficiently fine finite approximation of the first has larger error than an approximation of the second if the cost of the first approximation is no higher than the cost of the second approximation. An approximation's “error” is the largest distance between the continuum mechanism's action and the approximation's action. We obtain bounds on error. We also study the performance of Direct Revelation (DR) mechanisms relative to “indirect” mechanisms, both yielding the same action, when the environment set grows. We find that as the environment-set dimension goes to infinity, so does the extra cost of the DR approximation, if the error of the DR approximation is at least as small as the error of the indirect approximation. While the paper deals with information-processing costs and not incentives, it is related to the incentive literature, since the Revelation Principle is central to much of that literature and one of our main results is the informational inefficiency of finite Direct Revelation mechanisms. Received: May 21, 2001; revised version: December 14, 2001 RID="*" ID="*" Earlier versions of this paper were presented at the Decentralization Conference, Washington University, St Louis, April 2000 and at the Eighth World Congress of the Econometric Society, August 2000. We are grateful for comments received on those occasions. The second author gratefully acknowledges support from National Science Foundation grant #IIS9712131. Correspondence to: T. Marschak     

The exact law of large numbers via Fubini extension and characterization of insurable risks

A Fubini extension is formally introduced as a probability space that extends the usual product probability space and retains the Fubini property. Simple measure-theoretic methods are applied to this framework to obtain various versions of the exact law of large numbers and their converses for a continuum of random variables or stochastic processes. A model for a large economy with individual risks is developed; and insurable risks are characterized by essential pairwise independence. The usual continuum product based on the Kolmogorov construction together with the Lebesgue measure as well as the usual finitely additive measure-theoretic framework is shown further to be not suitable for modeling individual risks. Measurable processes with essentially pairwise independent random variables that have any given variety of distributions exist in a rich product probability space that can also be constructed by extending the usual continuum product.     

Weak measurability and characterizations of risk

Summary. In the context of a continuum of random variables, arising, for example, as rates of return in financial markets with a continuum of assets, or as individual responses in games with a continuum of players, an important economic issue is to show how idiosyncratic risk can be removed through some device of aggregation or diversification when such risk is explicitly introduced into the model. In this paper, we use recent work of Al-Najjar [1] as a general backdrop to provide a review of the basic issues involved when the continuum is formulated as the Lebesgue interval. We present two examples to argue that the fundamental problem of the non-measurability of sample functions, originally identified by Doob, and further elaborated by Feldman, Gilles and Judd in the economic literature, simply cannot be bypassed by reinterpretations of standard results. We also provide an equivalence result in the spirit of Al-Najjar's efforts; but argue that this elementary result does not go beyond the standard law of large numbers for a sequence of real-valued iid random variables, and as such, is incapable of yielding anything of substantive economic interest beyond this law. Received: April 23, 1998; revised version: April 28, 1998     

Optimal Taxation and Monopsonistic Labor Market: Does Monopsony Justify the Minimum Wage?

Does monopsony on the labor market in itself justify the implementation of a minimum wage when it would not be used in a competitive economy? This issue is studied in a model of optimal taxation. We find that there is no room for the minimum wage when there are a continuum of skills with no isolated mass point at the bottom of the wage distribution. Accordingly, in the empirically relevant situation, where there is a continuum of wages at the bottom of the distribution, the minimum wage is not helpful.     

The nonatomic assignment model

Summary We formulate a model with a continuum of individuals to be assigned to a continuum of different positions which is an extension of the finite housing market version due to Shapley and Shubik. We show that optimal solutions to such a model exist and have properties similar to those established for finite models, namely, an equivalence among the following: (i) optimal solutions to the linear programming problem (and its dual) associated with the assignment model; (ii) the core of the associated market game; (iii) the Walrasian equilibria of the associated market economy.     

Matching and bargaining models of markets: approximating small markets by large markets

Summary. We show that the equilibrium of a matching and bargaining model of a market in which there is a finite number of agents at each date need not be near the equilibrium of a market with a continuum of agents, although matching probabilities are the same in both markets. Holding the matching process fixed, as the finite market becomes large its equilibrium approaches the equilibrium of its continuum limit.Received: January 22, 1996; revised version: September 24, 1996This revised version was published online in February 2005 with corrections to the cover date.     

Does commodity money eliminate the indeterminacy of equilibrium?

Previous studies have shown that a random-matching model with divisible fiat money and without constraint on agents’ money inventories possesses a continuum of stationary single-price equilibria. Wallace (J. Econom. Theory 81 (1998) 223) conjectures that the indeterminacy can be eliminated by the use of commodity money. Instead, I find that in a similar random-matching model with dividend-yielding commodity money, a continuum of stationary single-price equilibria exists when the utility of dividend is not too high. This result casts doubt on the conventional belief that the indeterminacy of monetary equilibrium is be caused only by the nominal nature of money.     

To Each According to . . . ? Markets, Tournaments, and the Matching Problem with Borrowing Constraints

We compare the performance of markets and tournaments as allocative mechanisms in an economy with borrowing constraints. The economy consists of a continuum of individuals who differ in their initial wealth and ability level. These must be assigned to a continuum of investment opportunities or inputs of different productivity. With perfect capital markets matching is efficient under both mechanisms. Markets, however, generate higher aggregate consumption because of the waste associated with the production of signals under tournaments. When borrowing constraints are present, tournaments dominate markets in terms of matching efficiency and, for sufficiently powerful signalling technologies, also in terms of aggregate consumption.     

Core and value for a public-goods economy: An example

An example of a public-goods economy with a continuum of agents is presented in which the Shapley value does not lie in the core.     

Indeterminacy of intertemporal equilibria under imperfect competition

Summary. It is known that overlapping generations models with imperfectly competitive firms may exhibit a continuum of stationary equilibria. The reason of this indeterminacy is that different price expectation functions of consumers lead to different objective demand functions against which firms maximize. All these expectation functions fulfill perfect foresight in the equilibrium, but they can be arbitrary off the equilibrium. In this paper it is shown that it is not this arbitrariness which is responsible for the indeterminacy, but that the continuum of stationary equilibria emerges even if expectation functions are rational. Received: March 25, 1999; revised version: February 16, 2000     

Nash equilibria equal competitive equilibria

With strictly monotone preferences and a continuum of traders, there is a game in which the set of Nash equilibria is exactly the same as the set of competitive equilibria.     

Essential equilibria of large generalized games

We characterize the essential stability of games with a continuum of players, where strategy profiles may affect objective functions and admissible strategies. Taking into account the perturbations defined by a continuous mapping from a complete metric space of parameters to the space of continuous games, we prove that essential stability is a generic property and every game has a stable subset of equilibria. These results are extended to discontinuous large generalized games assuming that only payoff functions are subject to perturbations. We apply our results in an electoral game with a continuum of Cournot-Nash equilibria, where the unique essential equilibrium is that only politically engaged players participate in the electoral process. In addition, employing our results for discontinuous games, we determine the stability properties of competitive prices in large economies.     

Stationary monetary equilibria with strictly increasing value functions and non-discrete money holdings distributions: An indeterminacy result

In this paper, we present a search model with divisible money in which there exists a continuum of monetary equilibria with strictly increasing continuous value functions and with non-discrete money holdings distributions.     

Individual risk and Lebesgue extension without aggregate uncertainty

Many economic models include random shocks imposed on a large number (continuum) of economic agents with individual risk. In this context, an exact law of large numbers and its converse is presented in [Y.N. Sun, The exact law of large numbers via Fubini extension and characterization of insurable risks, J. Econ. Theory 126 (2006) 31-69] to characterize the cancellation of individual risk via aggregation. However, it is well known that the Lebesgue unit interval is not suitable for modeling a continuum of agents in the particular setting. The purpose of this paper is to show that an extension of the Lebesgue unit interval does work well as an agent space with various desirable properties associated with individual risk.     

Indeterminacy in incomplete market economies

Summary It is shown that in a two-period economy with a continuum of states and real assets, the following holds: (1) if the asset structure is complete, then generically the number of equilibria is finite; (2) if there are a finite number of real assets (this can approximate completeness arbitrarily close) then, for a nonempty open set of economies, there are a continuum of distinct equilibria. Asymptotic versions (on the number of states and on the number of assets) of the result are also given. It is argued, therefore, that incompleteness, by itself, may be a leading source of indeterminacy.I want to thank R. A. Dara, J. Geanakoplos and the audience of a larger number of presentations (at Brown, Stanford, Columbia, Harvard, Paris,...) for useful comments.     

Robustness to strategic uncertainty

We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.     

Local envy-freeness and equal-income Walrasian allocations

This paper introduces a local version of envy-freeness and investigates its implications in a continuum agent economy with connected preferences. We show that the set of locally envy-free and Pareto efficient allocations coincides with the set of equal-income Walrasian allocations.     

Continua of Hahn-unsatisfactory equilibria: Some examples

Quantity-constrained fix-price equilibria may occur even at Walrasian equilibrium prices. Structurally stable examples are given to support the claim that there may be a continuum of such equilibria, even in a neighborhood of the Walrasian one.     

Strategic Complementarity in the Dynamic Private Provision of a Discrete Public Good

Individual voluntary contributions to a discrete public good are shown to be strategic complements in a dynamic private provision game. This is in contrast to a public good that can take on any value in a continuum where they are strategic substitutes.