What is an Auction?

I found a nice introductory description of auction from the viewpoint of economics or game theory in the following textbook on auction theory:

The below is the (partial) quotation from the chapter 2.3.1, titled "What is an Auction?":

[A]uctions Have become an effective tool to implement public policy. Their use now ranges from the allocation of radio spectrum necessary for mobile communication, to spot markets trading electricity and pollution permits, as well as being widely used in government procurement. 

We can now define an auction by one of its central properties: as a market clearing mechanism, to equate demand and supply. Other market mechanisms include fixed price sales (as in a supermarket) or bargaining (as in the negotiated sale of a house or a used car). Within the class of market mechanisms which allocate scarce resources, one particular characteristic of the auction is that the price formation process is explicit. That is, the rules that determine the final price are usually well-understood by all parties involved. 

Auctions are often used in the sale of goods for which there is no established market. Auctions were instrumental in the mass privatization in Eastern Europe given the absence of a price system that could guide the valuation process for firms being privatized. Rare or unique objects are typically sold in auctions as the markets for these objects are likely to be very thin. However, auctions are also used to sell Treasury bills and the markets for these assets are very thick. The reason is that only governments can legally produce such bonds and therefore the sale in an auction is an exercise in revenue maximization. 

Auctions are more flexible than a fixed price sale and perhaps less time-consuming than negotiating a price. Auctions are used to sell hundreds of goods, such as bales of wool or used cars, in a few hours. One can imagine how many hours it would take to sell 100 used cars through negotiated sales.

Lucas' View on Free Trade

Continued from the previous post, I would like to quote a couple of paragraphs from the Lucas' book on economic growth. In the introductory chapter, the author mentions the connection between international trade and economic growth, which illustrates his (and perhaps most economists') view on free trade. While this part is written as an introduction to Chapter 3 ("Making a Miracle"), his evocative illustration provides better economic understanding and insight  of free trade in general.  This could also contribute to the debate on free trade (especially, on TPP issues in Japan).

The most spectacular growth successes of the postwar world have been associated with growth in international trade. This is the single empirical generalization that strikes everyone who is trying to understand economic growth in the last 50 years. Countries like Japan, South Korea, Taiwan, Hong Kong, and Singapore began producing goods they had never made before and exporting them to the United States, successfully competing with American and European producers who had the advantages of decades of experience. At the other extreme, the Communist countries that cut themselves off from trade with the West stagnated, as did India and many Latin American economies that used tariff walls to protect inefficient domestic producers from outside competition. These observations seem to provide further confirmation of the usual economic arguments in favor of free trade, arguments that seem to me as true and as relevant now as they were when Hume an Smith first articulated them. 

But classis trade theory does not really help in understanding the connections between trade and growth that we see in the postwar period. One problem is that while some of the Asian successes - in Taiwan and Honk Kong - were associated with liberal trade policy, others - Japan, Korea, and Singapore - occurred in heavily managed environments, under policies that Smith would certainly have criticized as mercantilist. (I agree with Smith that the mercantilist economies would have hared even better without managed trade, but this view is obviously not a straightforward statement of the facts.) A second, more important, barrier to the application of the theory of gains-from-trade to postwar growth is that quantitative versions of the theory do not yield estimated benefits of tariff reduction that are of the right order of magnitude to account for the growth miracles. (...) These models support a compelling case for the importance of free trade. What they do not provide, though, is a theoretical link between free trade and economic growth that is both rapid and sustained.

Romer vs. Uzawa-Lucas

As I illustrated in the previous post, the most cited paper of Robert Lucas is written about (endogenous) economic growth. Surprisingly, its citation is even greater than those of Paul Romer (1986a, 1986b), the pioneering papers in this field (according to Google Scholar).
To understand the essence of these models and their differences, I have checked the Lucas' book on economic growth (this is actually a volume of collected papers), and found insightful exposition.



In the Introduction of the book, the author first provides a nice summary of Paul Romer's pioneering works.

Paul Romer (1986a, 1986b) worked out an explicit model of a growing economy that reconciled the opposing forces of increasing an diminishing returns, and did so in a way that generated sustained production growth and was at the same time consistent with market equilibrium of many, competing producers. The economics of Romer's model are closely related to the ideas of Allyn Young (1928), but his development of these ideas is entirely new. In the theory, goods are produced with a single kind of capital - Romer called it "knowledge capital" - and each producer's output depends both on his own stock of this capital and on the stock held by other firms. Aggregating over producers, production in the economy as a whole is subject to increasing returns: Every 10 percent increase in the total stock of knowledge capital leads to an output increase of more than 10 percent. But an individual producer, who has no control over the economy's total stock of capital, faces diminishing returns to increases in his own capital. Thus the fact of increasing inequality among the economies of the world is reconciled with the absence of a tendency to monopolization within each economy.

Then, the author relates Romer's idea with his own (Lucas, 1988).

Section 4 of my "On the Mechanics of Economic Development" constructs a model designed to deal with the problem posed by diminishing returns along the lines proposed by Romer. In doing this, I found it more convenient to make use of a model of Uzawa (1965) in which there is both physical and human capital but returns, private and social, depend only on the ratio of these two stocks. The theory replaces the increasing returns assumed by Romer with a kind of constant returns, yielding a system which is easier to analyze than Romer's but which circumvents the problems of diminishing returns in a similar way.
The human capital model I used involves an external effect of human capital, patterned on the external effect of knowledge capital that Romer introduced. But in my analysis, this external effect is not needed to ensure the existence of a competitive equilibrium the way it is in Romer's model. If this effect is removed, the model continues to be internally consistent and is in fact even easier to analyze. [footnote] 

[footnote]: Rebelo (1991) stripped the model down to its simplest one-capital-good "Ak" form. Caballe and Santos (1993) provide an elegant analysis of the off-balanced-path dynamics of an Uzawa model without a production externality.

References
Caballe, Jordi, and Manuel S. Santos (1993) "On Endogenous Growth with Physical and Human Capital." Journal of Political Economy, 101: 1042-1067. [313]
Lucas, Robert E., Jr (1988) "On the mechanics Economic Development." Journal of Monetary Economics, 22: 3-42. [13791]
Rebelo, Sergio (1991) "Long Run Policy Analysis and Long Run Growth." Journal of Political Economy, 99: 500-521. [2748]
Romer, Paul M. (1986a) "Increasing Returns and Long-Run Growth." Journal of Political Economy, 94: 1002-1037. [12099]
Romer, Paul M. (1986b) "Cake Eating, Chattering, and Jumps: Existence Results for Variational Problems." Econometrica, 54: 897-908. [58]
Uzawa, Hirofumi (1965) "Optimum Technical Change in an Aggregative Model of Economic Growth." International Economic Review, 6: 18-31. [1142]
Young, Allyn A. (1928) "Increasing Returns and Economic Progress." Economic Journal, 38: 527-542. [2014]

(Number in [ ] shows the citation in Google Scholar.)

Growth "higher" than Rational Expectations

Professor Robert E. Lucas Jr. at Chicago Univ. is perhaps the most famous macroeconomist in the (at least academic) world. He is especially well-known to his series of works on rational expectations.  In fact, he received the Nobel Prize in 1995 due to this contribution:

The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1995 was awarded to Robert E. Lucas Jr. "for having developed and applied the hypothesis of rational expectations, and thereby having transformed macroeconomic analysis and deepened our understanding of economic policy" (from the official website of the Nobel Prize).

However, somewhat surprisingly, I just noticed that Professor Lucas' most cited paper is NOT about rational expectations. According to Google Scholar (search result is here), his most cited paper is "On the mechanics of economic development" (Journal of Monetary Economics, 1988), which is cited more than three times as much as the second one, "Econometric policy evaluation: A critique." The citation of the former exceeds 13,000, which is amazingly high in the field of Economics (maybe in other fields, too).

The "mechanics" paper is a seminal pioneering work in endogenous growth theory and is built on the idea of Uzawa ("Optimum Technical Change in An Aggregative Model of Economic Growth", International Economic Review, 1965); because of this, the model is often called Uzawa-Lucas model. Professor Paolo Mattana, the author of "The Uzawa-Lucas Endogenous Growth Model" explain the model as follows:

R. Lucas, in the late 1980s, writes a path-breaking paper: by taking some initial intuitions of Uzawa (1965) a step further, he proposes a two-sector capital accumulation growth model where human capital plays the role of the key variable through which ongoing growth can be generated. Human capital is understood to refer, in Becker's tradition, to the skills and knowledge intensity of the labor force and is accumulated in the learning (or educational) sector via a linear constant-returns to scale technology, only requiring older vintages of human capital. The Uzawa-Lucas economy differs in a fundamental way from the standard neoclassical model; since a lower bound to the return of accumulation is implicitly imposed, the long-run growth rate basically reflects an endogenous equilibrium where only the "primitives" of a specific economy (endowments, technology and preferences) are relevant. Other factors, such as increasing population or exogenous technical progress, crucial in the traditional theory, have, conversely, no critical influence.

Dream Comes True: Japan edge USA!

Congratulations! Many many many thanks to Nadeshiko JAPAN!! We all are very proud of you :)

Japan are FIFA Women’s World Cup™ champions for the first time after a penalty shootout victory over USA, following a drama-charged 2-2 draw in Frankfurt. (Link to FIFA official site)

What a wonderful moment! They really became world champion!!

Two Papers on Repeated Games

Original article (link) posted: 31/10/2005

Sorin (1986) “On Repeated Games with Complete Information” Math. Of Operations Research, 11-1

Several properties of the sets of feasible payoffs for repeated games are shown. Particularly, the condition that the set of feasible payoffs are convex hull of the feasible payoffs in pure strategies is given. Namely, it is necessary and sufficient that a discount factor is larger than or equal to 1-1/N, where N is the number of the players.

Dal-Bo (2001) “Tacit Collusion under Interest Rate Fluctuations” Job Market Paper, UCLA

The paper examines the optimal tacit collusion equilibrium when the discount factor changes over time. It is shown that collusive prices and profits depend not only on the level of the discount factor but also on its volatility; they increase with a higher discount factor level and decrease with its volatility. The model is a variant of Rotemberg-Saloner model, where, instead of demand fluctuation, the discount factor is assumed to fluctuate.

Independence and Implementation

Continued from the previous post on bargaining theory, I am going to quote other parts from the Introduction. This time, my focus is on editor's (Prof. Thomson) view on a controversial axiom of Nash solution, contraction independence(*), as well as his brief survey on implementation of bargaining solutions.

(*) Contraction independence is often called the axiom of independence of irrelevant alternatives (IIA), which is defined as follows:

Contraction independence If, keeping the disagreement point constant, the feasible set contracts but the alternative chosen as solution outcome remains feasible, then it should remain the solution outcome.

On contraction independence
Contraction independence has been the object of the sharpest criticisms. Nash himself expressed some misgivings about it and Luce and Raiffa's (1957) objections are well known. In evaluating a bargaining situation, it is unavoidable and probably desirable that it be simplified and summarized, that it be reduced to its essential features. The issue is how much and what information should be discarded in the process, and one can make a convincing case that contraction independence ignores too much. Indeed, it covers contractions that affect the shape of the feasible in ways that seem relevant, for instance, the elimination of the only alternatives at which a particular agent's payoff is higher than at the initial solution outcome and the other agent's payoff lower than at the initial solution outcome; contraction independence prevents solutions from responding to such eliminations.
The main counterargument to this criticism was made by Nash himself: in practice, information is usually lacking about which alternatives are truly available, and a compromise under evaluation only competes against others that are not too different from itself. Modeling this lack or information explicitly is what an investigator should perhaps do, but there are also advantages to keeping the model simple; contraction independence is a formal way to express the idea.

On implementation
Whether a solution is implementable depends on the type of game forms that are used, and on the behavioral assumptions made about how agents confronted with such games behave. For implementation by normal form games and when agents calculate best responses taking as given the choices made by the other agents, a critical property for what is then called Nash-implementability is Maskin-monotonicity (actually an invariance property with respect to enlargements of the lower contour set at the chosen alternative). Most solutions are not Maskin-monotonic and therefore not Nash-implementable by normal form games. However, an implementation of the Kalai-Smorodinsky solution by a sequential game is possible (Moulin, 1984). Later contributions delivered the Nash solution (Howard, 1992), the egalitarian solution (Bossert and Tan, 1995)./// For subgame perfect implementation, a general theorem is offered by Miyagawa (2002). It covers all solutions obtained, after normalizing problems so that the ideal point has equal coordinates, by maximizing a monotone and quasi-concave function of the agents' payoffs. Implementation is by means of a stage game and the equilibrium notion is subgame perfection.

References
Bossert and Tan (1995) "An arbitration game and the egalitarian bargaining solution", Social Choice and Welfare, Volume 12, Number 1, 29-41.
Luce and Raiffa (1957), Games and Decisions: Introduction and Critical Survey, Wiley.
Miyagawa (2002) "Subgame-perfect implementation of bargaining solutions", Games and Economic Behavior, Volume 41, Issue 2, 292-308.
Moulin (1984) "Implementing the Kalai-Smorodinsky bargaining solution", Journal of Economic Theory, Volume 33, Issue 1, 32-45.

Bargaining Solutions: Nash, Egalitarian and Kalai-Smorodinsky

A leading researcher in bargaining theory, Professor William Thomson, recently edited the notable collection of papers in axiomatic bargaining:

Bargaining and the Theory of Cooperative Games: John Nash and Beyond


The following is quoted from the Introduction written by the editor. The parts I refer below focus on three representative solutions of the bargaining problem(*) in the literature, while his Introduction covers much more materials. I strongly recommend those who are interested in bargaining theory to read this insightful survey article.

Almost sixty years ago, Nash (1950) published his seminal paper on what is now known as the "axiomatic theory of bargaining"./// He formulated a list of properties, or "axioms", that he thought a solution should satisfy, and he established the existence and the uniqueness of a solution satisfying all of the axioms; this solution is now called the "Nash solution".

Nash's model has been one of the most successful paradigms of game theory. His paper is the founding stone of a literature that now comprises several hundred theoretical papers. The Nash solution is presented in all game theory textbooks./// Together with the Shapley value (Shapley, 1953) and the core (Gillies, 1959), it constitutes the obligatory background on cooperative games in most economics graduate programs.

In spite of the large number of reasonable solutions that one can easily define, only three solutions and variants have consistently emerged: in addition to the Nash solution, they are egalitarian(**) and Kalai-Smorodinsky solutions(***)./// The dominance of these three solutions and these variants is a central conclusion to be drawn from the literature.

The Nash solution has come out somewhat more often than the other two, and the claim can perhaps be made that it is indeed special./// But the argument is a little dangerous. Earlier, we talked about the theorist's need to simplify and summarize in order to analyze, and in axiomatic analysis simplification often takes the form of independence and invariance axioms. The Nash solution satisfies many invariance conditions, thus it is not much of a surprise that it should have dome out often. On the other hand, the monotonicity axioms that have generally led  to the Kalai-Smorodinsky and egalitarian solutions are readily understood and endorsed by the man on the street.

It is mainly on the basis of monotonicity properties that the Kalai-Smorodinsky solution should e seen as an important challenger to the Nash solution, the egalitarian solution presenting another appealing choice. This latter solution enjoys even stronger monotonicity requirements and like the Nash solution, it satisfies strong independence conditions. Unlike both the Nash and Kalai-Smorodinsky solutions, it requires interpersonal comparisons of utility however, which, depending upon the context, may be seen as a desirable feature or a limitation.

(*) A bargaining problem consists of a pair (S, d) where S, the feasible set, is the subset of alternatives, and d, the disagreement point, is a point of S./// A bargaining solution defined on a class of problems is a function that associates with each problem (S, d) in the class a unique point of S, the solution outcome of (S, d).

(**) The egalitarian solution (Kalai, 1977a) selects the maximal point of S at which utility gains from d are equal. More generally, by making utility gains proportional to a fixed vector of weights, we obtain the weighted egalitarian solution relative to these weights (Kalai, 1977b)

(***) The Kalai-Smorodinsky solution (Kalai and Smorodinsky, 1975) selects the maximal point of S that is proportional to the profile of maximal payoffs that agents can separately reach among the points of S that dominate d.

References
Gillies (1959) "Solutions to general non-zero-sum games" in Contributions to the Theory of Games IV, Princeton University Press, 47-85.
Kalai (1977a) "Nonsymmetric Nash solutions and replications of 2-person bargaining", International Journal of Game Theory, Volume 6, Number 3, 129-133.
Kalai (1977b) "Proportional Solutions to Bargaining Situations: Interpersonal Utility Comparisons", Econometrica, Vol. 45, No. 7 (Oct., 1977), 1623-1630.
Kalai and Smorodinsky (1975) "Other Solutions to Nash's Bargaining Problem", Econometrica, Vol. 43, No. 3 (May, 1975), 513-518.
Nash (1950) "The Bargaining Problem", Econometrica, Vol. 18, No. 2 (Apr., 1950), 155-162.
Shapley (1953) "A Value for n-Person Games" in Contributions to the Theory of Games II, Princeton University Press, 307-317.

A Maskin's IO paper

Original article (link) posted: 30/10/2005

Maskin (1999) “Uncertainty and entry deterrence” Economic Theory, 14

A model where capacity installation by an incumbent firm serves to deter others from entering the industry is considered. The paper shows that uncertainty about demand or costs forces the incumbent to choose a higher capacity level than it would under certainty. The intuitive reason is explained in Introduction, which is stated as follows;

To deter entry, the incumbent must install enough capacity so that, if entry occurred, the entrant’s profit would be zero (or negative). Under certainty, the incumbent will install no more capacity than it would use were entry to occur. With uncertainty, when demand is high, an incumbent that has installed the certainty level of capacity still continues to produce at capacity; price simply rises to reflect the higher demand. But, when demand is low, the incumbent will wish to produce at less than full capacity. This means that the fall in price when demand is low is not so large as the rise in price when demand is high, and so if the entrant’s is zero under certainty, it is positive with certainty. To deter entry, therefore, the incumbent must increase capacity above the certainty level to ensure that when demand is high it produces enough to drive the entrant’s expected profit back down to zero.

Matching and Market

I found insightful comments on the relationship between matching theory and market economy (more specifically, general equilibrium) in the following paper:
Vincent Crawford (1991), "Comparative Statics in Matching Markets" Journal of Economic Theory, 54: 389-400.

Perhaps the most important advantage of the matching approach is its robustness to heterogeneity. A traditional competitive equilibrium cannot exist in general unless the goods traded in each market are homogeneous, because all goods in the same market must sell at the same price. A traditional model of a labor market with the degree of heterogeneity normally encountered therefore has the structure of a multi-market general equilibrium model. But because the markets in such a model are very thin, the usual arguments in support of price-taking are strained. The theory of matching markets replaces this collection of thin markets with a single market game, in which the terms of partnerships are determined endogenously, along with the matching, via negotiations between prospective partners. Gale and Shapley's notion of stability(*), suitable generalized, formalizes the idea of competition, and thereby makes it possible to evaluate the robustness of traditional competitive analysis to heterogeneity. (Stable outcomes in matching markets can in fact be viewed as traditional competitive equilibria when prices are allowed to reflect the differences between matches; see, for example, Shapley and Shubik, 1972(**))

The author, Vince Crawford, who is known as a leading researcher in game theory has written a few influential papers on matching theory. Especially, the following two are of great importance since they initiated the area of (many-to-one) matching with monetary transfers.

"Job Matching with Heterogeneous Firms and Workers"

with Elsie Marie Knoer, Econometrica, Vol. 49(2): 437-450, 1981.

"Job Matching, Coalition Formation, and Gross Substitutes"

with Alexander S. Kelso, Jr., Econometrica, Vol. 50(6): 1483-1504, 1982.

* Gale and Shapley (1962) "College Admissions and the Stability of Marriage" American Mathematics Monthly, 69: 9-15.

** Shapley and Shubik (1972) "The Assignment Game. 1. The Core" International Journal of Game Theory, 1: 111-130.