Echenique (2003)

Echenique, F. (2003), The Equilibrium Set of a Two Player Game with Complementarities is a Sublattice, Economic Theory, 22: 903-905.  Link 

Summary  I prove that the equilibrium set in a two-player game with complementarities, and totally ordered strategy spaces, is a sublattice of the joint strategy space.

It is widely known that in games with strategic complementarities (GSC), i.e., best reply correspondings are monotone increasing for all players, the set of pure-strategy Nash equilibria forms a non-empty complete lattice. This result implies the existence of smallest and largest Nash equilibria.

The current paper looks further into the structure of the equilibrium set of GSC, and shows that under certain restrictions the equilibrium set becomes not just a complete lattice but also a sublattice, as is written down in the Summary above.

The practical importance of this result, according to the author, is:

If the equilibrium set of a game is a sublattice, then we can find new equilibria from knowing that two profiles are equilibria, and by taking the componentwise join and meet of players’ strategies.

Note that we cannot obtain such strong property by a complete lattice structure alone. In this sense, a sublattice is indeed critical. The proof is extremely simple, which makes use of the observation that (a) when there are only 2 players and (b) their strategy spaces are completely ordered, (c) the other player's strategy must be completely ordered. The property (c) does not hold either (a) or (b) is not satisfied. Once (c) is verified, the rest of the proof is just to follow the definition of GSC.

My random thought: A sublattice property also arises in one-to-one two-sided matching markets (but neither in one-to-many nor many-to-many markets). I'm wondering if the idea of this paper can be somehow connected to the sublattice property of the set of stable matchings.

A final remark: A nicely written but a bit uninformative note.

Sotomayor (1996)

Sotomayor, M. (1996), A non constructive elementary proof of the existence of stable marriages, Games and Economic Behavior, 13: 135–7.  Link 

Abstract  Gale and Shapley showed in their well known paper of 1962 (Amer. Math. Monthly 69, 9–14) that stable matchings always exist for the marriage market. Their proof was constructed by means of an algorithm. Except for the existence of stable matchings, all the results for the marriage market which were proved by making use of the Gale and Shapley algorithm could also be proved without the algorithm. The purpose of this note is to fill out this case. We present here a nonconstructive proof of the existence of a stable matching for the marriage market, which is quite short and simple and applies directly to both cases of preferences: strict and nonstrict. 

In this short note, the author establishes the existence of stable matchings for the marriage market, i.e., one-to-one matching market, without employing any algorithms. Her key idea is a new concept that she calls a simple matching, which is defined as follows:

A matching is simple if, in the case a blocking pair (m, w) exists, w is single.

Since the set of simple matchings is nonempty and finite, there must exist a specific matching that I call here (for notational convenience) an M-simple matching:

A simple matching is M-simple if it cannot be (weakly) Pareto dominated for men by any other simple matching.

Then, she shows that such M-simple matching is stable. The basic logic is that the existence of blocking pair necessarily induces Pareto improvement for men (since w is single), which contradicts to the Pareto efficiency of M-simple matching for men.

As the author claims, the proof is very shout and simple, yet quite novel I think. To explain her proof strategy in a different way, she

1. focuses on the subset of stable matchings, 
2. characterize it by "M-simple matching," and
3. shows that the set is non-empty. 

It is immediate that M-simple matching is unique and coincides with M-optimal stable matching when all the preferences are strict. In this way, we can see that her idea is somewhat related to the existence of one-sided optimal stable matchings, a widely known result in the literature.

A minor remark: The definition of matching in the paper incorporates individual rationality, and consequently that of of stable matchings pays attention only on blocking pairs. This looks a bit non-standard while nothing is lost in her analysis.

A final remark: I like this paper very much :)

How did GS algorithm come out?

Roth, A. (2008), Deferred acceptance algorithms: history, theory, practice, and open questions, International Journal of Game Theory, 36: 537-569.

In this survey article on the Gale-Shapley's deferred acceptance algorithm, Al Roth mentions (in footnote 3) an amazing story about how GS discovered the algorithm (which of course established the theory of two-sided matchings):

At his birthday celebration in Stony Brook on 12 July 2007, David Gale related the story of his collaboration with Shapley to produce GS by saying that he (Gale) had proposed the model and definition of stability, and had sent to a number of colleagues the conjecture that a stable matching always existed. By return mail, Shapley proposed the deferred acceptance algorithm and the corresponding proof. 

IO Seminar (Rysman)

Original article (link) posted: 04/11/2005

Gowrisankaran and Rysman "Dynamics of Consumer Demand for New Durable Consumer Goods"

The paper proposes a dynamic model of consumer preferences for new consumer durable goods and estimates it. Consumers in their model are heterogeneous. They are assumed to choose between purchasing a current product and waiting for future products, making rational forecasts about the future distribution of prices and qualities. Two actual industries, DVD players and digital cameras are estimated.

Comment
The model seems to be attractive because it examines dynamic aspects and heterogeneity of consumers that have not been done successfully in the literature, although everyone agrees the importance. I am bit wondering if incorporating the uncertain of the future markets has serious effects to the estimated results or not. In their model, consumers fully know when and what kind of products will appear. However, if the future market is uncertain in the sense that consumers only know the probability distribution of future products, there might be an additional waiting value. That is, consumers may be better off by postponing their purchase to resolve uncertainty. This option value of waiting is known as "Real Option" in finance. In real world, the arrival of new products typically depends on the result of R&D investments which is presumably stochastic. So, even firms cannot exactly know the schedule of future products in many cases. Therefore, to incorporate future uncertainty is an important extension I think.

Review on "The New Geography of Jobs" (by Enrico Moretti)

I found a nice review on the following book written by Aaron M. Renn in his blog:

The New Geography of Jobs by Enrico Moretti

Aaron spells out quite a few suggestions and cautions on the materials, yet his last paragraph below clearly indicates appreciation of what this new book achieved:

Despite a few things I thought could have been more developed or stronger, I think that, as a book making the case for the innovation economy and what it means, The New Geography of Jobs is a strong one, and I again would suggest it to people in cities that have not yet fully found their place in the new century.

In Memory of Prof. Hayami

I found an interesting article (through this tweet by Prof. Sawada) about Prof. Yujiro HAYAMI, a leading agricultural/development economist who passed away last December:
"Death of a layman's economist" (by Prof. Abdul Bayes at Financial Express)

The article says:

Leaving aside for a moment his hundreds of technical articles published in reputed international journals, I shall take the privilege of citing from his famous book on development economics that I have mentioned before. Using the concept of Prisoner's Dilemma - when two persons convicted of a murder are kept in separate cells without one knowing what the other person is saying to the investigating officer - he illustrates how much loss the inability among people to establish cooperative relationship due to lack of communication and trust could generate for the society. This loss can happen in all economic transactions. "For example, in the transaction of a commodity, a buyer may try to reduce payment to a seller on the false charge of quality deficiency in delivered commodities. Then, the seller will deliver low-quality commodities thereafter. As their mutual distrust is heightened, they will stop transactions and thereby close off a mutually profitable business opportunity".

Prof. Hayami's famous book mentioned above is the following.

It is very interesting to know that Hayami has applied game theoretical ideas to his work on developmental studies. I should have tried to talk with him (Actually, Prof. Hayami was my colleague in GRIPS...)

Hayami cites another example regarding the costs of non-cooperation: "If employment is so insecure that employees may be discharged any moment, they would make little effort to acquire specific knowledge and skill for efficient work in his organisation. Their employer would then be inclined to discharge these employees for their lack of effort. In this way, cooperative relationship will not be established with the little accumulation of skill and knowledge needed for efficient functioning of this organisation". 

According to Hayami hypothesis, if mutual trust between particular individuals is thus elevated to a moral code in the society, huge savings would be made in transaction costs. If such cooperative negotiations could be guaranteed, business plans could be promoted ex-post much more flexibly and efficiently than by rigid ex-ante specification of contingencies, especially in long-term transactions subject to high risk and uncertainty. By and large, trust is a social capital that needs to be nurtured, if necessary, by structuring the cooperative relations into a hierarchical organisation. If mutual trust between workers and management ceases to work, the cost of monitoring and enforcing the contract would be large.

(made bold by yyasuda)

Uzawa-Equivalence Theorem

I found an interesting section in a nice intermediate textbook on General Equilibrium Theory by Ross M. Starr:




In Section 18.4 titled "Uzawa-Equivalence Theorem" shows the equivalence of two existence theorems:

1. The existence of equilibrium in an economy characterized by a continuous excess demand function fulfilling Walras' Law 
2. The Brouwer Fixed-Point Theorem

Interestingly, the two apparently distinct results are mathematically equivalent, which is originally shown by Hirofumi Uzawa (1962) in his short note: "Walras' Existence Theorem and Brouwer's Fixed Point Theorem." Economic Studies Quarterly, 8: 59-62.

The importance of the theorem is stressed by Prof. Starr as follows:

What are we to make of the Uzawa Equivalence Theorem? It says that use of the Brouwer Fixed-Point Theorem is not merely one way to prove the existence of equilibrium.  In a fundamental sense, it is the only way. Any alternative proof of existence will include, inter alia, an implicit proof of the Brouwer Theorem. Hence, this mathematical method is essential; one cannot pursue this branch of economics without the Brouwer Theorem. If Walras (1874) provided an incomplete proof of existence of equilibrium, it was in part because the necessary mathematics was not yet available.

The paper is included in this volume of Uzawa's collected papers. (I thank Prof. Kawamura for the information.)

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.)