The Arrow-Debreu model: How math can hide a fatal conceptual error

I argue that math, like love, can cover a multitude of sins, and I use the neoclassical object of adoration, the Arrow-Debreu model, as the case in point. It is commonplace that the Arrow-Debreu (AD) model of general equilibrium does not describe the real world, but it is equally commonplace to accept it as representing the pure logic of the competitive capitalist economy in an idealized world free of transactions costs. I show that the AD model fails even as an idealized model; it actually mistakes the logic of pure capitalism. Unlike McKenzie’s model of idealized general equilibrium under constant returns to scale, Arrow and Debreu claim to have shown the existence of competitive equilibrium under decreasing returns to scale and positive pure profits. The AD model (again unlike the McKinzie model) needs to assign the profits to individuals and this is done using the notion of “ownership of the production set.” But this notion suffers from a fatal ambiguity. If Arrow and Debreu interpret it to mean “ownership of a corporation” then a simple argument in the form “labor can hire capital or capital can hire labor” defeats the alleged necessity of assigning residual claimancy to the corporation. A given corporation may or may not end up exploiting a set of production opportunities (represented by a production set) depending on whether it hires in labor and undertakes production or hires out its capital to others (all by assumption at the parametrically given prices). In the latter case, residual claimancy is elsewhere. There is no such property right as “ownership of a production set” in a private property market economy. The legal party which purchases or already owns all the inputs used up in production has the defensible legal claim on the outputs: there is no need to also “purchase the production set.” At any set of prices that allow positive pure profits, anyone in the idealized AD model could bid up the price of the inputs and thus try to reap a smaller but still positive profit. Therefore,pace Arrow and Debreu, there could be no equilibrium with positive pure profits. In the Appendix, the property rights fallacy that afflicts the AD model is shown to also afflict orthodox capital theory and corporate finance theory.     

Equilibrium with arbitrary market structure

Summary. Fifty years ago Arrow [1] introduced contingent commodities and Debreu [4] observed that this reinterpretation of a commodity was enough to apply the existing general equilibrium theory to uncertainty and time. This interpretation of general equilibrium theory is the Arrow-Debreu model. The complete market predicted by this theory is clearly unrealistic, and Radner [10] formulated and proved existence of equilibrium in a multiperiod model with incomplete markets.In this paper the Radner result is extended. Radner assumed a specific structure of markets, independence of preferences, indifference of preferences, and total and transitive preferences. All of these assumptions are dropped here. We - like Radner - keep assumptions implying compactness.Received: 17 April 2003, Revised: 26 March 2004, JEL Classification Numbers: D52, D40.     

Non-computability of competitive equilibrium

Summary. We provide a “computable counterexample” to the Arrow-Debreu competitive equilibrium existence theorem [2]. In particular, we find an exchange economy in which all components are (Turing) computable, but in which no competitive equilibrium is computable. This result can be interpreted as an impossibility result in both computability-bounded rationality (cf. Binmore [5], Richter and Wong [35]) and computational economics (cf. Scarf [39]). To prove the theorem, we establish a “computable counterexample” to Brouwer's Fixed Point Theorem (similar to Orevkov [32]) and a computable analogue of a characterization of excess demand functions (cf. Mas-Colell [26], Geanakoplos [16], Wong [50]). Received: September 9, 1997; revised version: December 17, 1997     

Economics,Democracy, and the Distribution of Capital Ownership

This article holds that widespread, practical access to capital acquisition is essential for sustainable widespread economic prosperity and democracy. The founders of the U.S.A. agreed that sustainable democracy required widespread ownership of land to provide a viable earning capacity sufficient to support robust participation in democratic government. The importance of widespread land ownership to individual prosperity and sustainable democracy was supported not only by the prevailing philosophical views of property, it was also apparent to the common man and woman. Compared to Europe, America offered widespread access to land ownership, higher wages, better work conditions, cheaper staples and greater individual freedom, equal opportunity, prosperity, and political participation. This conviction that widespread access to ownership is a necessary condition for widespread prosperity and sustainable democracy continued throughout most of the nineteenth century, but today public discourse reveals virtually no trace of this once universally held opinion. This article suggests that the disappearance of this conviction can be traced to an erroneous view shaped by neoclassical economics and Keynesian economics. According to this view, (1) the disappearance of the American frontier and industrialization made the goal of widespread capital ownership either impractical or of little or no economic significance and (2) by way of technological advance, sufficient earning capacity and consumer demand to promote growth and sustain democracy can be achieved, without widespread ownership, primarily through jobs and welfare. Although differing in many respects, both mainstream schools, along with Adam Smith’s classical economics, share one common but unstated economic assumption: the broader distribution of capital acquisition (in itself) has no fundamental relationship to the fuller employment of people and capital, the broader distribution of greater individual earning capacity, and growth. Contemporary thinking, shaped by these economic schools, also tacitly assumes that widespread capital ownership is not essential for the sustainable individual earning capacity needed to support robust democracy. This erroneous “ownership-neutrality assumption” (1) contradicts both the views of America’s founders and the colonial experience, and (2) provides theoretical justification for structuring capital markets and capital acquisition transactions to unfairly and dysfunctionally favor existing owners at the expense of broader ownership distribution, more widely shared prosperity, greater efficiency, ecologically friendly growth, and a vital democracy. America’s conscientious founders would be shocked by the diminished importance of the distribution of ownership in the mainstream analysis of prices, efficiency, production, growth, and democracy. Rather than enhancing democracy, they would view the “ownership-neutrality assumption” of mainstream economics as contributing to its deterioration and corruption. They would openly search for economic analysis built on an alternate assumption more consistent with their understanding of the requisite conditions for sustainable democracy. This article advances an economic analysis that suspends the ownership-neutrality assumption, replaces it with a “broader-ownership-growth assumption,” and suggests a voluntary market strategy for substantially broadening capital ownership, enhancing individual earning capacity, and providing the widespread economic prosperity needed for robust democracy.     

On Intersectoral allocations,factors substitutability and multiple long-run growth paths

Summary. This article considers a two-sector model of economic growth with “labour-augmenting” intersectoral external effects stemming from the aggregate capital stock. It is shown that equilibrium balanced growth paths with a non-trivial labour allocation scheme become available. A set of sufficient conditions for the existence of multiple equilibrium growth rays is provided and their determinacy properties are then characterised. Finally, examination of a parameterised C.E.S. economy illustrates the central role of non-unitary values for the elasticity of substitution in the multiplicity issue. Received: October 31, 2000; revised version: September 25, 2001     

Nash and Walras equilibrium via Brouwer

Summary. The existence of Nash and Walras equilibrium is proved via Brouwer's Fixed Point Theorem, without recourse to Kakutani's Fixed Point Theorem for correspondences. The domain of the Walras fixed point map is confined to the price simplex, even when there is production and weakly quasi-convex preferences. The key idea is to replace optimization with “satisficing improvement,” i.e., to replace the Maximum Principle with the “Satisficing Principle.” Received: July 9, 2001; revised version: February 25, 2002 RID="*" ID="*" I wish to thank Ken Arrow, Don Brown, and Andreu Mas-Colell for helpful comments. I first thought about using Brouwer's theorem without Kakutani's extension when I heard Herb Scarf's lectures on mathematical economics as an undergraduate in 1974, and then again when I read Tim Kehoe's 1980 Ph.D dissertation under Herb Scarf, but I did not resolve my confusion until I had to discuss Kehoe's presentation at the celebration for Herb Scarf's 65th birthday in September, 1995. RID="*" ID="*"Correspondence to: C. D. Aliprantis     

General equilibrium in economies with adverse selection

We model economies of adverse selection as Arrow–Debreu economies. In the spirit of Prescott and Townsend (Econometrica 52(1), 21–45, 1984a), we identify the consumption set of the individuals with the set of lotteries over net transfers. Thus, prices are linear in lotteries, but they may be non linear in commodity bundles. First, we study a weak equilibrium notion by viewing the economy of adverse selection as a pure exchange economy. The weak equilibrium set is non empty, but some of the allocations may be inefficient, and the equilibria indeterminate. Second, following Prescott and Townsend (Econometrica 52(1), 21–45, 1984a), we introduce an intermediary (firm) supplying feasible and incentive compatible measures. Equilibria are constrained efficient, but the equilibrium set is empty for an open set of economies containing the Rothschild and Stiglitz insurance economies. The research of A. Rustichini was supported by the NSF grant NSF/SES-0136556.     

Risky procurement with an insider bidder

Procurement auctions carry substantial risk when the value of the project is highly uncertain and known only to insiders. This paper reports the results from a series of experiments comparing the performance of three auction formats in such complex and risky settings. In the experiment, every bidder knows the private value for the project but only a single insider bidder knows the common-value part. In addition to the standard second-price and English auctions we test the “qualifying auction,” a two-stage format commonly used in the sale of complex and risky assets. The qualifying auction has a fully “revealing” equilibrium that implements the revenue-maximizing outcome but it also has an uninformative “babbling” equilibrium in which bidders place arbitrarily high bids in the first stage. In the experiments, the latter equilibrium has more drawing power, which causes the qualifying auction to perform worse than the English auction and only slightly better than a sealed-bid second-price auction. Compared to the two other formats, the English auction is roughly 40% more efficient, yields 50% more revenues, avoids windfall profits for the insider, while protecting uninformed bidders from losses.     

Signaling bargaining power: Strategic delay versus restricted offers

Summary. I study the first-round separating equilibrium of a buyer-seller bargaining game, extended to allow for asymmetric information, strategically delayed offers and offers restricted to a portion of the good. When bargaining is over a consumption good, in equilibrium the “strong” buyer uses a restricted offer if his optimal consumption path is conservative relative to the “weak” buyer. A pure restricted offer may even be a costless, efficient signal. When the good is durable, a pure strategic delay is involved in signaling a strong bargaining position if the discount factor is high. Received: June 24, 1998; revised version: May 30, 1999     

Different employment of capitals in vertically integrated sectors: Smith after the Austrians

The Austrian notion of stages of production and the related principle of the greater productivity of roundabout methods, plus the neo-Austrian notions of vertical integration and vertical division of labour, are utilized in this paper in an attempt to reconstruct Smith’s convoluted arguments on the different employment of capitals in chapter 5, book 2, of the Wealth of Nations. Smith’s arguments are first clarified in the light of the two concepts of capital (money capital and productive capital), of the two aspects of productive labour (living labour and dead labour) and of the two viewpoints (of an individual and of society) on which Smith’s theory is based. The results of this clarification are then used to prove that, independently of Smith’s own words but in consistency with his theory, the notions of “quantity” and “productivity” of productive labour have a “vertical”, as well as a “horizontal”, dimension so that they fit both the input–output scheme and the Austrian framework of time-consuming methods of production.     

Existence of competitive equilibrium when incomes may be zero

Summary In a pure exchange economy, there exists a price vector which is a quasi-equilibrium (Debreu 1962), but this may not be a competitive equilibrium if some individuals' demand functions are discontinuous because their incomes may be zero. We show nonetheless, in a pure exchange economy with free disposal, that there is asequence of prices approaching the quasi-equilibrium along which total excess demands tend to a non-positive limit.Comments on earlier drafts from K. J. Arrow, D. Duffie, H. E. Scarf, and D. Schmeidler are gratefully acknowledged.     

Debreu’s social equilibrium theorem with asymmetric information and a continuum of agents

We provide several different generalizations of Debreu’s social equilibrium theorem by allowing for asymmetric information and a continuum of agents. The results not only extend the ones in Kim and Yannelis (J Econ Theory 77:330–353, 1977), Yannelis and Rustichini (Stud Econ Theory 2:23–48, 1991), but also new theorems are obtained which allow for a convexifying effect on aggregation (non-concavity assumption on the utility functions) and non-convex strategy sets (pure strategies). This is achieved by imposing the assumption of “many more agents than strategies” (Rustichini and Yannelis in Stud Econ Theory 1:249–265, 1991; Tourky and Yannelis in J Econ Theory 101:189–221, 2001; Podczeck in Econ Theory 22:699–725, 2003). To the memory of Gerard Debreu. A preliminary draft was presented in Paris, in April of 2005. I have benefited from the discussion, comments and questions of Erik Balder, Jean-Marc Bonnisseu, Bernard Cornet and Filipe Martins Da-Rocha and Conny Podczeck. A careful and knowledgeable referee made several useful comments and rescued me from a mishap.     

Competitive and core allocations in large economies with differential information

Summary. We study the core and competitive allocations in exchange economies with a continuum of traders and differential information. We show that if the economy is “irreducible”, then a competitive equilibrium, in the sense of Radner (1968, 1982), exists. Moreover, the set of competitive equilibrium allocations coincides with the “private core” (Yannelis, 1991). We also show that the “weak fine core” of an economy coincides with the set of competitive allocations of an associated symmetric information economy in which the traders information is the joint information of all the traders in the original economy. Received March 22, 2000; revised version: May 1, 2000     

A bargaining theory of the firm

Suppose that a firm has several owners and that the future is uncertain in the sense that one out of many different states of nature will realize tomorrow. An owner’s time preference and risk attitude will determine the importance he places on payoffs in the different states. It is a well-known problem in the literature that under incomplete asset markets, a conflict about the firm’s objective function tends to arise among its owners. In this paper, we take a new approach to this problem, which is based on non-cooperative bargaining. The owners of the firm play a bargaining game in order to choose the firm’s production plan and a scheme of transfers which are payable before the uncertainty about the future state of nature is resolved. We analyze the resulting firm decision in the limit of subgame-perfect equilibria in stationary strategies. Given the distribution of bargaining power, we obtain a unique prediction for a production plan and a transfer scheme. When markets are complete, the production plan chosen corresponds to the profit-maximizing production plan as in the Arrow–Debreu model. Contrary to that model, owners typically do use transfers to redistribute profits. When markets are incomplete, the production plan chosen is almost always different from the one in a transfer-free Drèze (pseudo-)equilibrium and again owners use transfers to redistribute profits. Nevertheless, our results do support the Drèze criterion as the appropriate objective function of the firm.     

Bank's capital structure under non-diversifiable risk

Summary. The aim of this paper is to study the design of optimal capital structure of a “large” intermediary when the intermediary faces a non-diversifiable risk, within the standard costly-state-verification (CSV) model. I demonstrate that, under weaker conditions, a “large” intermediary realizes more efficient allocation by issuing both debt and equity than by issuing only debt. Unlike Diamond (1984) and Williamson (1986), the set of optimal contracts involves ex ante monitoring made by shareholders of the intermediary. Changes in parameters, such as the variance of the aggregate risk or the cost of monitoring, affect bankruptcy costs and the capital structure. Received: October 12, 1998; revised version: March 20, 2001     

Economic Growth and (Re-)Distributive Policies in a Non-cooperative World

Many models show that redistribution is bad for growth. This paper argues that in a non-cooperative world optimizing, redistributing (“left-wing”) governments mimic non-redistributing (“right-wing”) policies for fear of capital loss if capital markets become highly integrated and the countries are technologically similar. “Left-right” competition leads to more redistribution and lower GDP growth than “left-left” competition. Efficiency differences allow for higher GDP growth and more redistribution than one's opponent. Irrespective of efficiency differences, however, “left-wing” governments have higher GDP growth when competing with other “left-wing” governments. The results may explain why one observes a positive correlation between redistribution and growth across countries, and why capital inflows and current account deficits may be good for relatively high growth.     

Evenly rotating economy: A new modeling technique for an old equilibrium construct

In the middle of the twentieth century, just five years before Arrow and Debreu proved the existence of an equilibrium for a competitive economy in the Walrasian system, Ludwig von Mises introduced the English-speaking world to his alternative equilibrium construct: the evenly rotating economy. In contrast to Arrow and Debreu, which characterizes equilibrium as a unique vector of prices and quantities, Mises depicts equilibrium as a pattern of behavior. After reviewing the Misesian conception of equilibrium and its failure to take hold in the profession, I turn to the modern literature. I contend that the evenly rotating economy is a special case of the now-prevalent class of search-theoretic exchange models. As such, I argue that this class of models is particularly well suited for applications considered by economists working in the Austrian tradition.     

General equilibrium and the new neoclassical synthesis

We present a general equilibrium model of the new neoclassical synthesis that has the same level of generality as the Arrow–Debreu model. This involves a stochastic multi-period economy with a monetary sector and sticky commodity prices. We formulate the notion of a sticky price equilibrium where all agents form rational expectations on prices for commodities and assets, interest rates, and rationing. We present a general result showing that monetary policy imposes no restrictions whatsoever on nominal equilibrium price levels and that the set of sticky price equilibria has a dimension equal to the number of terminal date-events. Stickiness of prices implies that this indeterminacy is real.     

EDGAR on the internet: The welfare effects of wider information distribution in an experimental market for risky assets

Policies such as the SEC’s Fair Disclosure Rule, and technologies such as SEC EDGAR, aim to disseminate corporate disclosures to a wider audience of investors in risky assets. In this study, we adopt an experimental approach to measure whether this wider disclosure is beneficial to these investors. Price-clearing equilibrium models based on utility maximization and non-revealing and fully-revealing prices predict that in a pure exchange economy, an arbitrary trader would prefer that no investors are informed rather than all are informed; non-revealing theory further predicts that an arbitrary trader would prefer a situation in which all traders are informed rather than half the traders are informed. These predictions can be summarized as “None > All > Half”. A laboratory study was conducted to test these predictions. Where previous studies have largely focused on information dissemination and its effects on equilibrium price and insider profits, we focus instead on traders’ expected utility, as measured by their preferences for markets in which none, half, or all traders are informed. Our experimental result contradicts the prediction and indicates “Half > None > All”, i.e. subjects favor a situation where a random half is informed. The implication is that in addition to testing predictions of price equilibrium, experiments should also be used to verify analytical welfare predictions of expected utility under different policy choices. JEL Classification D82, D53, G14, L86 This work was largely completed while this author was at The Hong Kong University of Science and Technology.     

The Influence of Socioeconomic Factors on Economic Growth

The main goal of this paper is to analyse the relationship between social capital and economic growth taking into account the role of fiscal policy from theoretical and empirical points of view. To achieve this goal, “Human Capital and Public Capital Effects on Economic Growth” is focused on the effects of two traditional factors: human capital and public capital effects on economic growth. “Social Capital Effects on Economic Growth” considers qualitative variables introducing some socioeconomic effects on economic growth process analysis. In this case, social capital the main variable will be considered. “Empirical Analysis,” an empirical analysis is developed considering the case of European countries prior to the EU enlargement. Finally, in Conclusions,” the main conclusions will be resumed.