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We are delighted to present this book honoring Professor Jean-Michel Grandmont. Over his illustrious career, Jean-Michel has made a number of highly important contributions to economic theory, in particular on non-linear dynamics and sunspots. These topics have been the core of the International Conference on Instability and public policies in a globalized world: Conference in honor of Jean-Michel Grandmont, organized at Aix-Marseille School of Economics and GREQAM on June 6–8, 2013. This book presents the state-of-art in non-linear dynamics and sunspots by his colleagues, students and friends that have been influenced by and are admirers of his work.

Jean-Michel already stood out as a brilliant figure when he graduated at Polytechnique in 1962 among the very best, 10th out of more than 300, and he entered one of the most prestigious Corps, the Corps des Ponts et Chaussées. Fortunately, Jean-Michel went to research in economics although getting a business position is the standard career of french engineers from prestigious schools.

Jean-Michel obtained his Ph.D. in Berkeley in 1971 where he worked “On the Temporary Competitive Equilibrium” under the supervision of Gérard Debreu, Nobel Prize in Economics in 1983. Over the last 40 years, Jean-Michel contributed to diverse areas of economics ranging from the general equilibrium theory to monetary theory, learning, aggregation, non-linear dynamics and sunspots. On every count, his work has been characterized by insight, originality and technical clarity as well as rigor.

Jean-Michel came back to France in 1970 and joined the CEPREMAP in Paris where he stayed until 1996. There, he published pioneering papers on general equilibrium with money, temporary equilibria, learning, aggregation, and of course bifurcations and chaos in economic dynamics which strongly improved our understanding of the interrelationships between business cycles and economic growth. In 1996, he moved to the Centre de Recherche en Economie et Statistique (CREST), where he has remained ever since. He has been Associate Professor from 1977 to 1992 and Professor of Economics from 1992 to 2004 at Ecole Polytechnique. He is also affiliated to the Department of Economics of Ca’ Foscari University of Venice since 2004 and has been nominated a fellow of the Research Institute for Economics and Business Administration at Kobe University in 2008.

From 1996 to 2000, Jean-Michel served as the chairman of the Department of Economics at Ecole Polytechnique. During this period he was responsible for recruiting new faculty members and building an excellent undergraduate program, tasks in which he showed initiative and executive ability. As a result, the quality of the teaching at Ecole Polytechnique has been significantly improved and the number of students that the program attracted has been strongly increased.

Jean-Michel also got involved in promoting research in quantitative economics, mainly through the Econometric Society. He has been nominated Chairman in 1977, Vice President in 1988 and then President in 1990 of the Econometric Society.

He has received the Honorary Degree from Keio University in 2007 for his contributions to economic theory. He was also selected a foreign member of the American Academy of Arts and Sciences (AAAS) in 1992. He received the Palmes Académiques from the French Ministry of Education in 1995.

In addition, he has served on the editorial boards of many top journals in economics. In particular, he became an associate editor of Econometrica from 1976 to 1983 and of the Journal of Economic Theory from 1973 to 2015. Since 1995, he is a co-Editor of the International Journal of Economic Theory.

His major contributions can be classified into five groups of topics that remain however strongly linked through a general and comprehensive view of Economics.

  1. 1.

    General equilibrium with money. The 1983 book Money and Value provides a thorough and rigorous treatment of topics as various as the quantity theory of money, the classical dichotomy, short run and long run effects of monetary policy and expectations. See also [4, 5, 6, 7, 8, 12].

  2. 2.

    Temporary equilibrium. The papers [1, 2, 11] provide all the basic necessary tools to discuss the role of expectations in the determination of equilibrium. A number of applications followed: temporary competitive equilibria, temporary Keynesian equilibria, monetary policy in the short and in the long run [9, 10, 13, 14, 15, 16, 18, 26].

  3. 3.

    Learning. The rational expectations assumption naturally finds its justification into its relationships to the learning processes that lead the agents to revise their beliefs until they hold these expectations. Sufficient conditions on these processes, either for convergence or divergence, in non linear deterministic models, appear in [23, 28, 29, 30, 36].

  4. 4.

    Non linear cycles and sunspots. This important field was initiated in [19], and developed in [20, 22, 24, 25, 31, 34, 37]. It is still quite active today: non linear dynamics and chaos make most of the program of this book.

  5. 5.

    Aggregation. Aggregation is a basic question raised by the microeconomic foundations of macroeconomics. It is a difficult subject that Jean Michel has studied in different ways. The first paper [17] on this topic shows the boundaries of the Arrow theorem and provides extremely useful conditions for the applicability of the median voter theorem. Later studies deal with aggregate demand [21, 32], learning [33], and more recently finance [38].

Jean-Michel Grandmont also chose to devote a large fraction of his time to teaching and thesis management. Many students, in several institutions (CORE, University of Aix-Marseille, Bonn, Lisbon, Strasbourg, Paris, Venice, Yale, École Polytechnique), have benefited of Jean Michel’s advices and encouragements. A number of them figure prominently among the contributors to this volume.

One of the co-editors of this book has clear recollections on his first meeting with Jean-Michel. He was at the very end of his year of military service and, he believed, at the very end of his Ph.D. Thesis. He had been advised by one of his co-advisor, the late Louis-André Gérard-Varet, to meet Jean-Michel in order to get advices on his work. So he obtained very easily an appointment in Paris at CEPREMAP. When arriving in his office, it was quite obvious that Jean-Michel did not have time to look at the material of the thesis before the appointment. But then, after an extremely quick and highly efficient reading of the main parts of the chapters, Jean-Michel clearly explained that much more additional work was necessary in order to get an acceptable document. He provided a large number of remarks to improve many results and asked the disappointed young co-editor to come back after having revised his document. While strongly depressed, he decided to follow Jean-Michel’s advices and a few months later the thesis had indeed significantly improved! This is just one anecdote but it could almost literally apply to many former students from CORE, Lisbon, Paris, Marseille, Strasbourg or Yale who are contributors to this book.

1.1 Overview of the Papers

The 16 contributions presented in this book are grouped into four different topics on which Jean-Michel has made a large number of contributions, as presented previously.

1.1.1 Self-fulfilling Expectations and Sunspots

The opening paper by Jean-Paul Barinci and Jean-Pierre Drugeon, “Assessing the Local Stability Properties of Discrete Three-Dimensional Dynamical Systems: a Geometrical Approach with Triangles and Planes and an Application with some Cones”, provides a general analysis of the determinacy properties of three-dimensional discrete-time dynamical systems. The authors introduce a new geometrical argument which brings about a complete typology of the eigenvalues moduli and then provide a new apparatus for assessing from a geometrical standpoint the emergence of local bifurcations for parameterized economies. This general methodology is illustrated through the extensive characterization of the stability properties of a standard overlapping-generations (OLG) model with endogenous labor.

In the second paper, “From Sunspots to Black Holes: Singular Dynamics in Macroeconomic Models”, Paulo Brito, Luis Costa and Huw Dixon present conditions for the emergence of singularities in dynamic general equilibrium (DGE) models. The concept of impasse singularity is introduced to exhibit new types of DGE dynamics, in particular temporary determinacy/indeterminacy. These results are illustrated through two simple models: the Benhabib and Farmer (1994) one-sector model with aggregate externalities and one with a cyclical fiscal policy rule.

Brito, Costa and Dixon’s paper is followed by “Sunspot Fluctuations in Two-Sector Models with Variable Income Effects”. Here, Frédéric Dufourt, Kazuo Nishimura, Carine Nourry and Alain Venditti analyse a version of the Benhabib and Farmer (1996) two-sector model with sector-specific externalities in which they consider a class of utility functions inspired from the one considered in Jaimovich and Rebelo (2009) which is flexible enough to encompass varying degrees of income effect. They first show that local indeterminacy and sunspot fluctuations occur under plausible configurations regarding all structural parameters—in particular regarding the intensity of income effects. Second, they prove that, for any given size of income effect, there is a non-empty range of values for the Frisch elasticity of labor and the elasticity of intertemporal substitution in consumption such that indeterminacy occurs.

In the fourth paper of this group, “A Survey on Self-Fulfilling Mistakes”, Cars Hommes links some of his own work on expectations, learning and bounded rationality to the inspiring ideas of Jean-Michel Grandmont. In particular, his work on consistent expectations and behavioral learning equilibria may be seen as formalizations of Jean-Michel’s ideas of self-fulfilling mistakes contained in [36]. Some of his learning-to-forecast laboratory experiments with human subjects have also been strongly influenced by Jean-Michel’s ideas. Key features of self-fulfilling mistakes are multiple equilibria, excess volatility and persistence amplification.

This survey paper is followed by the contribution of Takashi Kamihigashi, “Regime-Switching Sunspot Equilibria in a One-Sector Growth Model with Aggregate Decreasing Returns and Small Externalities”. It is shown that regime-switching sunspot equilibria, in which labor supply is positive in one state while it is zero in the other, easily arise in a standard one-sector growth model with aggregate decreasing returns and arbitrarily small externalities. Regime-switching sunspot equilibria are explicitly constructed in the case where the utility function of consumption is linear. A stochastic optimal growth model whose optimal process appears to be a regime-switching sunspot equilibrium of the original economy without capital externality is also constructed.

The sixth paper of this group is provided by Antoine Le Riche and Francesco Magris, “Equilibrium Dynamics in a Two-Sector OLG Model with Liquidity Constraint”. They study a two-sector OLG economy in which a share of old age consumption expenditures must be paid out of money balances. It is first shown that the competitive equilibrium is dynamically efficient if and only if the share of capital on total income is large enough while a steady state capital per capita above its Golden Rule level is not consistent with a binding liquidity constraint. Assuming gross substitutability in consumption, they show that the dynamic efficiency property ensures the local determinacy of equilibrium and, as a consequence, rule out sunspot fluctuations. In addition, they provide a detailed analysis of the dynamic properties of the equilibrium focusing on flip and Hopf bifurcations under different sectoral capital intensity configurations.

The last contribution of this group is “Sunspots and Homoclinic Bifurcations in Continuous-Time Endogenous Growth Models”. Here, Hiromi Murakami, Kazuo Ni-shimura and Tadashi Shigoka consider a three-dimensional continuous-time general stationary model that includes one predetermined variable and two non-predetermined variables. Assuming that the model has a two-dimensional invariant manifold and that the manifold includes a one-dimensional closed curve that could be either a homoclinic orbit or a closed orbit, sunspot equilibria are constructed. These results are then applied to two-sector endogenous growth models that are variants of Lucas (1988) model.

1.1.2 Bubbles and Stabilizing Policy

In “Rational Land and Housing Bubbles in Infinite-Horizon Economies”, Stefano Bosi, Cuong Le Van and Ngoc-Sang Pham consider rational land and housing bubbles in an infinite-horizon general equilibrium model. Land is an input to produce while the house is a (durable) good to consume. It is shown that dividends on both these long-lived assets are endogenous and their sequences are computed. Then different concepts of bubbles, including individual and strong bubbles, are introduced and studied.

This paper is followed by “The Stabilizing Virtues of Monetary Policy on Endogenous Bubble Fluctuations” where Lise Clain-Chamosset-Yvrard and Thomas Seegmuller explore the stabilizing role of monetary policy on the existence of endogenous fluctuations when the economy experiences a rational bubble. Considering an OLG model, expectation-driven fluctuations are based on the co-existence of portfolio choices between three assets (capital, bonds and money), credit market imperfections and a collateral effect, and are shown to occur under a positive bubble on bonds. Then, the stabilizing role of a monetary policy managed by a (standard) Taylor rule is studied.

The last paper of this set is by Teresa Lloyd-Braga and Leonor Modesto, “Can Consumption Taxes Stabilize the Economy in the Presence of Consumption Externalities?”. The stabilization role of consumption taxes under a balanced-budget rule and in the presence of consumption externalities of the “keeping up with the Joneses” type is discussed in a finance constrained economy. Departing from a situation where sufficiently strong externalities make the steady state indeterminate, it is shown that sufficiently procyclical consumption tax rates are able to ensure local saddle path stability. However, government intervention with stabilization purposes may not be successful as this procyclicality leads to the appearance of another possibly indeterminate steady state with lower levels of output.

1.1.3 Growth

In “Uncertainty and Sentiment-Driven Equilibria”, Jess Benhabib, Pengfei Wang and Yi Wen construct a simple neoclassical model to capture the Keynesian idea that equilibrium aggregate supply is determined by aggregate demand and thus influenced by consumer sentiments about aggregate income. They show that when firms’ production and employment decisions must be based on expectations of aggregate demand and that realized demand follows from firms’ production and employment decisions through market-clearing mechanisms, rational expectations about aggregate demand can lead to stochastic sentiment-driven equilibrium despite the absence of production externalities, incomplete financial markets, strategic complementarity or any non-convexities in the model.

This contribution is followed by “Technological Progress, Employment and the Lifetime of Capital”, in which Raouf Boucekkine, Natali Hritonenko and Yuri Yatsenko study the impact of technological progress on the level of employment in a vintage capital model where capital and labor are gross complement, labor supply is endogenous and indivisible, there is full employment, and the rate of labor-saving technological progress is endogenous. The stationary distributions of vintage capital goods and the corresponding equilibrium values for employment and capital lifetime are characterized. It is shown that both variables are non-monotonic functions of technological progress indicators.

The last paper of this group is “Nonbalanced Growth in a Neoclassical Two-sector Optimal Growth Model”. Here, Harutaka Takahashi considers a neoclassical two-sector optimal growth model with Cobb-Douglas technologies and sector specific technological progress. Non-balanced growth, i.e. the fact that the two sectors are characterized by different long-run balanced growth rates, is characterized. This result refers to the recent literature showing that Kaldor and Kuznets facts are compatible in standard growth theory.

1.1.4 General Equilibrium

In “An Argument for Positive Nominal Interest”, Gaetano Bloise and Herakles Polemarchakis consider a dynamic OLG economy in which money provides liquidity as a medium of exchange. A central bank, that sets the nominal rate of interest and distributes its profit to shareholders as dividends is characterized by shares that are traded in the asset market. It is shown that nominal rates of interest that tend to zero, but do not vanish, eliminate equilibrium allocations that do not converge to a Pareto optimal allocation.

The second paper of this set is “Winners and Losers from Price-Level Volatility: Money Taxation and Information Frictions” by Guido Cozzi, Aditya Goenka, Minwook Kang and Karl Shell which analyzes an economy with taxes and transfers denominated in dollars and an information friction that allows for volatility in equilibrium prices and allocations. When the price level is expected to be stable, the competitive equilibrium allocation is Pareto optimal. When the price level is volatile, it is not Pareto optimal, but the stable equilibrium allocations do not necessarily dominate the volatile ones, and there can be winners and losers from volatility.

The final paper of this book, “A Note on Information, Trade and Common Knowledge”, by Leonidas Koutsougeras and Nicholas C. Yannelis, is dealing with the well known no trade result in Milgrom-Stokey (1982) using the appropriate definition of efficiency among several available in the asymmetric information framework. It is shown that if Pareto efficiency is understood in the private information sense, i.e., allocations and possible reallocations are adapted to the private information of each individual, then the no trade result is valid.

All of us who have contributed to this book consider it a privilege to be able to honor Jean-Michel in this way. As colleagues, former students and friends, we have benefited from Jean-Michel’s insights, encouragement and perseverance. His inspiration, his friendship, his generosity, and his contributions to economics are distinguished and invaluable. We are delighted to contribute our papers to this book in his honor and we look forward to our continuing relationships with Jean-Michel in the years to come.

Writing for the friends and colleagues of Jean-Michel

Kazuo Nishimura, Alain Venditti and Nicholas Yannelis

1.2 Main Scientific Writings of Jean-Michel Grandmont

Book

Money and Value, Econometric Society Monographs, Cambridge University Press, 1983. French version, Economica, 1986.

Articles

  1. 1.

    “Continuity Properties of a von Neumann-Morgenstern Utility”, Journal of Economic Theory, 1972.

  2. 2.

    “On the Short Run Equilibrium in a Monetary Economy”, in J. Drze (Ed.), Allocation under Uncertainty, Equilibrium and Optimality, McMillan, 1974.

  3. 3.

    “A Technical Note on Classical Gains from Trade”, Journal of International Economics, 1972, with D. McFadden.

  4. 4.

    “On the Role of Money and the Existence of a Monetary Equilibrium”, Review of Economic Studies, 1972, with Y. Younès.

  5. 5.

    “On the Efficiency of a Monetary Equilibrium”, Review of Economic Studies, 1973, with Y. Younès.

  6. 6.

    “Sur la demande de monnaie de court terme et de long terme”, Annales de l’INSEE, 1972. English version “On the Short Run and Long Run Demand for Money”, European Economic Review, 1973.

  7. 7.

    “Sur les taux d’intérêt en France”, Revue Économique, 1973, with G. Neel.

  8. 8.

    “Money in the Pure Consumption Loan Model”, Journal of Economic Theory, 1973, with G. Laroque.

  9. 9.

    “Foreign Exchange Markets: A Temporary General Equilibrium Approach”, CORE DP, 1973, Catholic University of Louvain, with A.P. Kirman.

  10. 10.

    “Monnaie et Banque Centrale”, Annales de l’INSEE, 1973, with G. Laroque. English version: “On Money and Banking”, Review of Economic Studies, 1975.

  11. 11.

    “Stochastic Processes of Temporary Equilibria”, Journal of Mathematical Economics, 1974, with W. Hildenbrand.

  12. 12.

    “On the Liquidity Trap”, Econometrica, 1976, with G. Laroque.

  13. 13.

    “On Temporary Keynesian Equilibria”, Review of Economic Studies, 1976, with G. Laroque.

  14. 14.

    “Temporary General Equilibrium Theory”, Econometrica, 1977. Invited Lecture at the World Congress of the Econometric Society, Toronto, 1975. Shorter version in M. Intriligator (Ed.), Frontiers of Quantitative Analysis, North Holland, 1977. French version: “Théorie de l’équilibre temporaire”, Revue Économique, 1976.

  15. 15.

    “Equilibrium with Quantity Rationing and Recontracting”, Journal of Economic Theory, 1978, with G. Laroque and Y. Younès.

  16. 16.

    “The Logic of the Fix Price Method”, Scandinavian Journal of Economics, 1977. Reprinted in S. Ström and L. Werin (Eds), Topics in Disequilibrium Economics, McMillan, 1978.

  17. 17.

    “Intermediate Preferences and the Majority Rule”, Econometrica, 1978. Reprinted in Aggregation and Revelation of Preferences, J.J. Laffont (Ed.), North Holland, 1979.

  18. 18.

    “Classical and Keynesian Unemployment in the IS-LM Model”, in Monetary Theory and Institutions, M. de Cecco and J.P. Fitoussi (Eds), McMillan, 1986. Spanish version in Analysis Economico, 1984, Mexico.

  19. 19.

    “On Endogenous Competitive Business Cycles”, Walras-Bowley Lecture at the North American Summer meetings of the Econometric Society, Stanford, 1984. Published in Econometrica, 1985. Summary in H. Sonnenschein (Ed.), Models of Economic Dynamics, Springer-Verlag, 1986. French Version: “Cycles concurrentiels endognes”, Cahiers du Séminaire d’Econométrie, 1985.

  20. 20.

    “Periodic and Aperiodic Behaviour in Discrete Onedimensional Dynamic Systems”, in Contributions to Mathematical Economics, W. Hildenbrand and A. Mas-Colell (Eds), North Holland, 1986.

  21. 21.

    “Distributions of Preferences and the Law of Demand”, Econometrica, 1987.

  22. 22.

    “Stabilizing Competitive Business Cycles”, Journal of Economic Theory, 1986. Reprinted in Nonlinear Economic Dynamics, J.M. Grandmont (Ed.), Academic Press, 1987.

  23. 23.

    “Stability of Cycles and Expectations”, Journal of Economic Theory, 1986, with G. Laroque, Reprinted in Nonlinear Economic Dynamics, J.M. Grandmont (Ed.), Academic Press, 1987.

  24. 24.

    “Nonlinear Difference Equations, Bifurcations and Chaos: An Introduction”, CEPREMAP, June 1988, and Stanford Technical Reports, 1988.

  25. 25.

    “Local Bifurcations and Stationary Sunspots”, in W.A. Barnett, J. Geweke and K. Shell (Eds.), Economic Complexity: Chaos, Sunspots, Bubbles and Nonlinearity, Cambridge University Press, 1989.

  26. 26.

    “Keynesian Issues and Economic Theory”, Scandinavian Journal of Economics, June 1989.

  27. 27.

    “Report on M. Allais’ Scientific Work”, Scandinavian Journal of Economics, January 1989.

  28. 28.

    “Stability, Expectations and Predetermined Variables”, with G. Laroque, in P. Champsaur et alii (Eds), Essays in Honour of E. Malinvaud, Vol. I, MIT Press, 1990.

  29. 29.

    “Temporary Equilibrium: Money, Expectations and Dynamics”, in L. McKenzie and S. Zamagni (Eds.), Value and Capital. Fifty Years later, McMillan, 1991.

  30. 30.

    “Economic Dynamics with Learning: Some Instability Examples”, with G. Laroque, in Barnett, W., Cornet, B. et alii (Eds.), Equilibrium Theory and Applications, Cambridge University Press, 1991.

  31. 31.

    “Expectations Driven Business Cycles”, European Economic Review, April 1991.

  32. 32.

    “Transformations of the Commodity Space, Behavioral Heterogeneity and the Aggregation Problem”, Journal of Economic Theory, June 1992.

  33. 33.

    “Aggregation, Learning and Rationality”, Proceedings of the 1992 World Congress of the International Economic Association (Moscow), McMillan, 1996.

  34. 34.

    “Expectations Driven Nonlinear Business Cycles”, Proceedings of the German Academy of Sciences, Wesdeutscher Verlag 1993, and in FIEF Studies on Business Cycles, Oxford University Press, 1994. Shorter version in MITA Journal of Economics (Tokyo), 1994.

  35. 35.

    “Behavioral Heterogeneity and Cournot Oligopoly Equilibrium”, Ricerche Economiche, 1993.

  36. 36.

    “Expectations Formation and Stability of Large Socioeconomic Systems”, Econometrica, 1998.

  37. 37.

    “Capital Labor Substitution and Nonlinear Endogenous Business Cycles”, Journal of Economic Theory, 1998, with P. Pintus and R. De Vilder.

  38. 38.

    “Heterogenous Probabilities in Complete Asset Markets”, Advances in Mathematical Economics, 1999 (Springer Verlag, S. Kusuoka and T. Maruyama, Eds), with L. Calvet and I. Lemaire. Japanese translation in Mita Journal of Economics, Tokyo, 1999.

1.3 List of Papers

Self-fulfilling Expectations and Sunspots

1. Jean-Paul Barinci and Jean-Pierre Drugeon: “Assessing the Local Stability Properties of Discrete Three-Dimensional Dynamical Systems: a Geometrical Approach with Triangles and Planes & an Application with some Cones”.

2. Paulo Brito, Luis Costa, and Huw Dixon: “From Sunspots to Black Holes: Singular Dynamics in Macroeconomic Models”.

3. Frédéric Dufourt, Kazuo Nishimura, Carine Nourry and Alain Venditti: “Sunspot Fluctuations in Two-Sector Models with Variable Income Effects”.

4. Cars Hommes: “From Self-Fullling Mistakes to Behavioral Learning Equilibria”.

5. Takashi Kamihigashi: “Regime-Switching Sunspot Equilibria in a One-Sector Growth Model with Aggregate Decreasing Returns and Small Externalities”.

6. Antoine Le Riche and Francesco Magris: “Equilibrium Dynamics in a Two-Sector OLG Model with Liquidity Constraint”.

7. Hiromi Murakami, Kazuo Nishimura and Tadashi Shigoka: “Homoclinic orbit and stationary sunspot equilibrium in a three-dimensional continuous-time model with a predetermined variable”.

Bubbles and Stabilizing Policy

8. Stefano Bosi Cuong Le Van and Ngoc-Sang Pham: “Rational Land and Housing Bubbles in Infinite-Horizon Economies”.

9. Lise Clain-Chamosset-Yvrard and Thomas Seegmuller: “The Stabilizing Virtues of Monetary Policy on Endogenous Bubble Fluctuations”.

10. Teresa Lloyd-Braga and Leonor Modesto: “Can Consumption Taxes Stabilize the Economy in the Presence of Consumption Externalities?”.

Growth

11. Jess Benhabib, Pengfei Wang and Yi Wen: “Uncertainty and Sentiment-Driven Equilibria”.

12. Raouf Boucekkine, Natali Hritonenko and Yuri Yatsenko: “Technological Progress, Employment and the Lifetime of Capital”.

13. Harutaka Takahashi: “Nonbalanced Growth in a Neoclassical Two-sector Optimal Growth Model”.

General Equilibrium

14. Gaetano Bloise and Herakles Polemarchakis: “An Argument for Positive Nominal Interest”.

15. Guido Cozzi, Aditya Goenka, Minwook Kang and Karl Shell: “Winners and Losers from Price-Level Volatility: Money Taxation and Information Frictions”.

16. Leonidas Koutsougeras and Nicholas C. Yannelis: “A Note on Information, Trade and Common Knowledge”.