LM Equation Research Articles

Applications of degenerate kernels to potential flow across circular, elliptical cylinders and a thin airfoil

In this paper, we employ the boundary integral equation (BIE) to analytically derive the solution of potential flow across a circular cylinder, an elliptical cylinder and a thin airfoil. It is found that the symmetric or antisymmetric potential flow problem can be solved by using the UT or the LM equation alone. Both analytical and numerical approaches were considered. The key tool is using the degenerate kernel instead of the closed-form fundamental solution. A closed-form fundamental solution is expressed in terms of degenerate kernel by using polar and elliptical coordinates for a circular cylinder and an elliptical cylinder, respectively. The role of the dual BEM is also examined by showing the singular vectors of influence matrix. Besides, either the singular or the hypersingular BIE are respectively employed alone to solve the problem of symmetric and anti-symmetric flow field. Analytical derivation as well as the BEM is demonstrated. The orientation of the ellipse, as well as the angle of attack for the thin airfoil, is considered. The result shows that the degenerate kernel can be an alternative tool for solving some BVPs, although the complex variable is always employed. Once the degenerate kernel for the closed-form fundamental solution is available, the BIE is nothing more than the linear algebra and an analytical study is possible. The extension to 3-D object is promising once the degenerate kernel is available.

A stylized macro-model with interacting real, monetary and stock markets

We propose a model economy consisting of interdependent real, monetary and stock markets. The money market is influenced by the real one through a standard LM equation. Private expenditures depend on stock prices, which in turn are affected by interest rates and real profits, as these contribute to determine the participation level in the stock market. An evolutionary mechanism regulates agents’ participation in the stock market on the basis of a fitness measure that depends on the comparison between the stock return and the interest rate. Relying on analytical investigations complemented by numerical simulations, we study the economically relevant static and dynamic properties of the equilibrium, identifying the possible sources of instabilities and the channels through which they spread across markets. We aim at understanding what micro- and macro-factors affect the dynamics and, at the same time, how the dynamics of asset prices, which are ultimately influenced by the money market, behave over the business cycle. Starting from isolated markets, we show the effect of increasing the market interdependence on the national income, the stock price and the share of agents that participate in the stock market at the equilibrium. Moreover, we investigate the stabilizing/destabilizing role of market integration and the possible emergence of out-of-equilibrium dynamics.

‘The Provocative Joan Robinson’ Was Completely Ignorant of Keynes ’S Chapter 21 Is-Lm Model on Pp. 298–299 of the General Theory Even Though She Supposedly Read the Final 1935 Draft of the General Theory: The Exchanges of September to November, 1936 Between Keynes and Robinson Show Her Complete Ignorance

‘The Provocative Joan Robinson’ Was Completely Ignorant of Keynes ’S Chapter 21 Is-Lm Model on Pp. 298–299 of the General Theory Even Though She Supposedly Read the Final 1935 Draft of the General Theory: The Exchanges of September to November, 1936 Between Keynes and Robinson Show Her Complete Ignorance

ANALISIS KESEIMBANGAN TINGKAT SUKU BUNGA DAN GDP DI INDONESIA : TINJAUAN INTERAKSI KEBIJAKAN FISKAL DAN MONETER (1998 – 2011)

The aim of this study is to see how the equlibrium (internal balance) of interest rate and GDP created by the goods market (IS curve) and money market (LM). Then from the earlier internal balance, the study want to see how the fiscal and monetary policy affects the economy, particularly in influencing the internal balance when the two policies were interacted. The study was conducted with Two Stage Least Square (TSLS) method to finding IS – LM equation using time series data (1998 – 2011). Then do the trial and error to find an internal balance. After that the authors try to find the shift of IS – LM when there were policy intervention. The result showed that the earlier of internal balance is 7,79% for interest rate, and Rp. 438.011 billion for GDP. Based on estimation of time series data, the study concluded that, there is no equilbrium for internal balance when fiscal and monetary policy were interacted. When there is fiscal policy intervention, the result show that the policy effectively affect economic growth. While when there is monetary policy intervention, the result show that the economy not responsive to the policy. So when the fiscal and monetary policy were interacted, it can not be created the new equlibrium (internal balance).

On Some Possible Explanations for the Continuing Denial of the Existence of Keynes’s IS-LM Model in Section 4 of Chapter 21 of the General Theory: Propaganda (from Joan Robinson), Ignorance (of the Keynes-Harrod Correspondence in Volume 13 Of the CWJMK in Late 1935), Anchoring (on Keynes’s Mention of Robinson’s Name in the Preface to the General Theory), Groupthink, and Cognitive

On Some Possible Explanations for the Continuing Denial of the Existence of Keynes’s IS-LM Model in Section 4 of Chapter 21 of the General Theory: Propaganda (from Joan Robinson), Ignorance (of the Keynes-Harrod Correspondence in Volume 13 Of the CWJMK in Late 1935), Anchoring (on Keynes’s Mention of Robinson’s Name in the Preface to the General Theory), Groupthink, and Cognitive

J. M Keynes Was Never a ‘Chapter 12’ Keynesian: The Claim That He Was a ‘Chapter 12’ Keynesian Was Manufactured by Joan Robinson and G. L. S. Shackle After His Death by Changing Keynes’s Definition of Uncertainty to Radical Uncertainty

The claim that Keynes regarded himself as a “Chapter Keynesian is inaccurate and misleading. Keynes’s chapter 12 discussion and definition of uncertainty in the General Theory is simply a footnote to his much more general theoretical discussion about uncertainty made in chapter 26 of the A Treatise on Probability in 1921 pages 309-312 which concentrated on economics specifically. This was demonstrated in the 1937-38 Keynes-Townshend correspondence, where there is no discussion of the 1937 QJE article or radical uncertainty. Keynes had two interrelated and interconnected models in the General Theory. They were the D-Z model, which dealt with expected aggregate demand, D, and the IS-LM(LP) model, which dealt with actual or realized aggregate demand, Y. Keynes incorporated uncertainty and expectations into his D-Z model of chapters 20 and 21 after having provided readers of the General Theory with a very brief introduction to the Theory of Effective Demand in Chapter 3 of the General Theory. Keynes’s use of his IS-LM(LP) model in chapter 15 (pp.199-209) provided a brief introduction to the final, complete presentation of his IS-LM(LP) model in Chapter 21 where Keynes brought all of the elements that made up the IS-LM(LP) model together in Section Four of Chapter 21. This analysis incorporated the LM(LP) equation from the classical and neoclassical model of the rate of interest (r) that Keynes had demonstrated to Harrod, in his letter of August 27th,1935, was only a single downward sloping curve in (r,Y) space that intersected nothing, so that the existence of a quantitative determinate equilibrium in classical and neoclassical theory was an impossibility in (r,Y) space. Harrod capitulated completely in his letter to Keynes of August 30th,1935 and acknowledged that Keynes had indeed come up with a very important equation and had made a major advance in economic theory. Keynes incorporated his August 27th analysis, using a version of Harrod’s original diagram, in chapter 14 of the General Theory on pages 179-182. Keynes showed diagrammatically that the classical and neoclassical theory of the rate of interest amounted to a single IS curve in (r,Y) space that intersected nothing. Data (in his February, 1937 QJE article, he reverts to a missing equation” characterization) was that could only be provided by the LM(LP) curve in order to have a quantitative, determinate, unique equilibrium as stated explicitly by Keynes on page 299 of the General Theory. These interrelated and interconnected models work in harmony together with each other. It is quite impossible to use one without the other and expect to fully grasp Keynes’s mathematical, technical analysis. The D-Z model incorporated expectations and uncertainty in order to analyze the anticipated or expected amount of Effective Demand, D. Based on this analysis, a specific actual or realized amount of Effective Demand, Y, results. Keynes then combines Y with r to show why monetary policy can be completely neutralized if liquidity preference becomes absolute in the highly elastic range of the LM(LP) curve. Keynes decided to present this result on pp.207-208 of the General Theory, as well as in his reply to Viner in the February, 1937 QJE article. Hicks’s version of Keynes’s IS-LM analysis has no foundation in any type of D-Z model and hence is incomplete, as it provided no microfoundations in the theory of the firm, aggregate production function, labor market or necessary and sufficient first and second order conditions for optimality as provided by Keynes in chapter 20 explicitly.

On Keynes’s August 27th and 30th, 1935 Exchanges with Harrod, Who Acknowledged That Keynes Had Discovered the Missing LM Equation Needed to Complete the Classical –Neo Classical Theory of the Rate of Interest: The Ignorance of This Exchange, in Volume 13 of the CWJMK, Explains Why Economists, Who Have Written on Keynes, Have Overlooked His IS-LM Model in Section Four

Economists who have been writing on Keynes’s General Theory since 1936 have been unable to understand, grasp or comprehend Keynes’s clear statements in chapter 21 of the General Theory that ”… if we have all the facts before us, we shall have enough simultaneous equations to give us a determinate result” on p.299 of the General Theory and/or Keynes’s statement that “the quantitative effect could be derived from the three elements” on page 298 of the General Theory. This failure to grasp that Keynes is analyzing his IS-LM(LP) model in chapter 21,as he said he would do in greater detail than he did in section 4 of Chapter 15 in a footnote on page 209, has led to the false conclusion for the last 81 years that there was no IS-LM(LP) analysis provided by Keynes in the General Theory, when, in fact, this analysis, which is central to the entire General Theory, is provided in explicit fashion. The question of why no economist in the 20th or 21st century ,who has written on Keynes’s General Theory, has demonstrated any inkling about what Keynes was doing in chapter 21 of the General Theory, requires an analysis involving cognitive dissonance, group think, and the Benthamite Utilitarian orientation of the economics profession. Harrod’s clear admission, in his letter to Keynes of August 30th, 1935, that Keynes had incorporated his LM equation, missing from classical and neoclassical theories of the rate of interest, as specified on page 199 of the General Theory, to create a complete theory of the rate of interest, requires a complete revision of the history of economic thought regarding Keynes and the General Theory. Keynes’s discovery was a major theoretical achievement because Keynes had demonstrated to Harrod in his letter of August 27th that the classical and Neoclassical theory was simply a single downward sloping IS curve in (Y,R) space that intersected nothing. Keynes, exhibiting great patience and care with Harrod in their correspondence of 1935, gradually walked Harrod through the construction of Keynes’s IS-LM(LP) model .It is in Keynes’s letter of August 27th, 1935, where Keynes demonstrated to Harrod in a post script that the classical and neoclassical theory of the rate of interest was indeterminate because there was only a single , downward sloping IS curve in Keynes’s (r,Y) space. Harrod showed that he had finally understood the IS-LM model in his letter to Keynes of August 30th,1935, when he acknowledged that Keynes had made a major contribution and had reconstituted the theory of the rate of interest by having discovered the missing LM equation that made the previous theory determinate: “…has to be determined by some new equation not provided for in the classical system. Thus the way is clear for a radical reconstruction. Your new equation is the liquidity preference schedule…”. (Harrod, 1973, CWJMK, Vol. 13,p.550). The point made by Keynes to Harrod in his letter of August 27th, 1935 was carefully incorporated by Keynes in chapter 14 of the General Theory on pages 179-182 concerning the” missing data” provided by the LM equation, in Keynes‘s 1937 QJE article in his reply to Viner about the need to emphasize the completely elastic range of the LM curve, and in his two point summary on pp.222-223 of his conclusion of the 1937 QJE article, in which he identified what was missing from the neoclassical theory of the rate of interest that dealt only with the IS(mec) curve –the LM curve. It is quite impossible for Hicks to have first developed the IS-LM model in his April,1937 Econometrica article, as asserted by Krugman. This erroneous claim also represents the major error in Colin Champion’s recent 2019 work on the General Theory, which is partially based on a mistaken assessment by B. DeLong of the 1937 QJE article that overlooks the first point made in Keynes’s summary on pp.222-223, which was that the fundamental error of classical and neoclassical economics was a mistaken theory of the rate of interest, which was caused by their having overlooked the existence of uncertainty, as opposed to risk, in decision making.

Comparing the Original 1933, IS-LM (LP) Model of Keynes With the Advanced Version of February, 1936 Contained in Chapter 21 of the General Theory

Keynes’s original IS-LM (LP) model of 1933, although a major conceptual breakthrough, was technically and mathematically flawed because Keynes incorporated realized and expected variables in the same set of equations. Keynes had solved this problem by early 1936 by formulating a completely new model that David Champernowne and Brian Reddaway had never seen or heard of before in Keynes’s lectures between 1933 and 1935 – the D-Z model of chapters 20 and 21. Keynes provided a brief introduction, outline and summary of the D-Z model on pages 24-32 of chapter 3 of the General Theory. The D-Z model focused exclusively on expectations and uncertainty. The IS-LM (LP) model focused exclusively on the realized, actual outcomes taking place in the economy in the Goods and Money sectors. The labor market, production function, marginal optimizing conditions and theory of the firm, with uncertainty and profit expectations integrated into the formal model, was done within the framework of the D-Z model in chapters 20 and 21. Both Kahn and Joan Robinson had no inkling about what was transpiring in chapters 20 and 21 of the General Theory. Keynes’s final IS-LM(LP) model represents a major improvement over Keynes’s first exposition of his IS-LM model in December, 1933. Keynes’s consumption function, C, has an explicit marginal propensity to consume incorporated and investment multiplier defined. The variable, Y, is explicitly incorporated into the LM (LP) equation,M=L, as M=L(Y,r).

On Hicks's Own, Explicit Acknowledgment, Made in His 1937 Econometrica Article, That It Was J M Keynes Who Had First Presented an IS-LM Model in the General Theory: Hick's Model Was Only a Slight Improvement

Hicks was quite clear in his 1937 Econometrica paper that his model of IS-LM was based directly on Keynes’s IS-LM model of the General Theory with minor embellishments. In his closing paragraph, Hicks is very clear that it is Keynes, not Hicks, who proposed the first IS-LM model: “These, then, are a few of the things we can get out of our skeleton apparatus. But even if it may claim to be a slight extension of Mr. Keynes’ similar skeleton...” (Hicks, 1937, p. 158). Hicks had, of course, realized that Keynes had already considered, and rejected, his “really General” Theory because it removed both expectations and uncertainty from the original model. Keynes was quite explicit in his reply to Hicks: “At One Time I Tried the Equations, as You Have Done, with I in All of Them”. In fact, Hicks’s “equations”, which are actually not equations at all ,but mere functional relationships expressed between dependent and independent variables of the form y=f(x) and z=h(q), are identical to those presented by Keynes in chapter 21 of the General Theory on pages 298-299.The only actual equations appear in the General Theory on pages 115, 199, and 208-209. It is shown that Hicks’s figures one and two on page 153 are simply graphical translations of Keynes’s specific descriptions contained on pages 200-202, 207 of the General Theory and Keynes’s analysis of the missing LM equation from the neoclassical theory contained on pages 179-182 of the General Theory. Hicks’s discussion of an horizontal IS curve, in relation to Keynes’s discussion of Wicksell on page 242 of the General Theory, is the only “slight extension” offered by Hicks. It is demonstrated that the economics profession has been completely bamboozled for 82 years by the canards, fictions, fables, myths, tall tales, and stories told about Keynes by a mathematically innumerate, illiterate, inept, and confused economist, named Joan Robinson, based on her claims that in the General Theory, Keynes only used the Marshallian, partial equilibrium modeling approach. Keynes supposedly refused to use a system of mathematical, simultaneous equations because, as a fanatic disciple of Marshall, he would always burn his mathematical analysis and replace it with a strictly literary summary after checking to make sure that the mathematical analysis supported the literary analysis. Robinson had no idea about how to specify or analyze a system of simultaneous, mathematical equations because she had never taken an introduction to pre-algebra course. The entire history of macroeconomics needs to be completely rewritten by economists. They have integrated the faulty claims introduced by Joan Robinson, a self admitted mathematical illiterate, into their histories. In retrospect, one can ask the following question: How could a mathematical illiterate, like Joan Robinson, have been taken seriously as the major source upon which to base the foundation for an assessment of Keynes’s GT? It is simply a catastrophic, egregious error that has ramifications for questions about the scientific respectability of the economics profession. Hicks admitted in 1937 that he had made only a slight improvement over Keynes ‘s model. Keynes established that absolute liquidity preference, which was represented by the “flattened” horizontal portion of his LP equation in (r,Y) space on page 207 of the GT, which was called a Liquidity “Trap” by others, would occur once the long run rate of interest fell below a rate of 2% or 2.5%. This is identical to a short run rate approaching or equaling 0%, once the various premiums are subtracted from the long run rate or added to the short run rate. Krugman has constantly maintained this position since the early 1990’s in the face of spurious attacks based on the baseless assertion that there was no IS-LM model in the GT. Hicks’s own admission in 1937, never repeated or clarified afterwards by Hicks, that his model represented only a slight improvement over Keynes’s, should end any support for the baseless claims of Joan Robinson. The most severe errors currently embedded in histories of macro economics concerning the impact of the GT can all be traced directly to Joan Robinson. The reader is asked to consult the many papers I have available on SSRN chronicling her many, myriad errors, errors which are embedded in heterodox and orthodox assessments of Keynes’s GT.

On the Reason Why Keynes Brought Together All the Elements of His IS-LP(LM) Model in Chapter 21: He Had to Cover the Foundation Supplied by the D-Z Model of Chapter 20 First

Keynes carefully supplied a series of building blocks in the construction of his General Theory. Keynes’s Theory of Effective Demand consisted of two interrelated models. They are, first, the IS-LP(LM) model that determines the rate of interest and second, the D-Z model of expected aggregate demand (D) and expected aggregate supply (Z). Y, realized or actual aggregate demand, will be one of the set of all possible, expected D-Z (D=Z) outcomes that was introduced briefly in chapter 3 of the General Theory, where Keynes provided a brief outline of the theory he would fully develop in later chapters. The IS-LP(LM) model is based on the Liquidity Preference Function, derived on page 199 of chapter 15 after being introduced in chapter 13 and 14, and the Consumption function and Investment Multiplier functions, derived from chapter 10 following his introductory analysis related to the Consumption function in chapters 8 and 9. The Investment function is analyzed as a function of rate of interest and expectations regarding expected future prices and profits in chapters 11 and 12. Chapters 16 and 17 generalize the Liquidity Preference analysis of chapter 15 to all assets. Keynes does not present his mathematical model of D and Z, which he briefly introduced in chapter 3, until chapter 20. This chapter 20 model supplies the aggregate production function, theory of the firm (purely competitive), analysis of expected prices and profits, multiple equilibria, and necessary and sufficient optimality conditions needed to obtain the crucial condition that the expected real wage equaled the marginal product of labor. Chapter 20 deals with the output and labor markets. Keynes is now in position to bring together all of the elements of the IS-LP(LM) model in sections 4 and 5 of chapter 21 and then connect his results to chapter 20 by formally incorporating his money market analysis mathematically in section 6 of chapter 21. The Y variable has now explicitly been derived and supported with clear cut microeconomic foundations. Y appears in both the IS equation and LM equation. Both determine the rate of interest. The specific IS-LP(LM) equilibrium in (r, Y) space, of which there are many, since Keynes’s version of IS-LP(LM) incorporated expectations and uncertainty, will be in one to one correspondence with exactly one of the set of all possible D=Z outcomes specified by the Aggregate Supply Curve. The set or locus of all expected D=Z outcomes was derived in chapter 20 and called the Aggregate Supply Curve. D and Z were introduced in chapter 3. Keynes is now finished with the presentation of the General Theory. He spends four additional pages of Section Four of chapter 21 discussing the possible deficiencies in his IS-LP(LM) model that are due to possible interactions with other relevant variables that were not formally specified in his IS-LP(LM) model, but which are included in the D-Z model. Chapters 4-7 are correctly described by Keynes as digressions from his structured approach that add nothing to the formal development of his general theory. Chapters 22-24 examine how the General theory can be implemented. The most important part of these chapters is on pages 377-379, where Keynes puts forth his proposal for a Keynesian – Neoclassical synthesis. This was ignored by all economists of the 1930’s-1940’s. Chapter 18 is a literary summary of the IS-LP(LM) model only that does not incorporate the technical details provided in chapter 20 and 21. The appendix to chapter 19, where Keynes compared his chapter 20 and 21 models to Pigou’s, makes it clear that Pigou, and Modigliani, had only one equilibrium in the output market. Chapters 2 and 19 deal with Pigou and the money wage issue and are not part of Keynes’s formal models. The piece meal approach that economists take to reading Keynes’s GT is exemplified by Coddington’s 1976 Journal of Economic Literature article which provides a division of Keynesian readers into three different, conflicting categories, depending on the particular chapters picked from the GT by these readers which is being emphasized as providing the one, true message of the General Theory. Each category claims that Keynes’s “true” message can be found in the particular chapters that they have highlighted. The three Keynesian categories are Hydraulic, Fundamentalist, and Reconstituted Reductionist. In reality, Keynes, in the GT, is using all three of these characterizations of his work. For example, the labor market is in disequilibrium as determined by the output market, which is characterized as one of multiple equilibria. Keynes’s holistic approach can’t be broken up into different, disconnected parts if a reader is interested in understanding what Keynes really meant to say. What Keynes really meant to say is contained in all of the chapters in the General Theory.

What is Wrong with the Following Statement? 'Just See What Hicks and Hansen Did with Keynes's Elaborate Analysis. Keynes Needed an Entire Book to Get His Point Across. Hicks and Hansen Needed Only a Few Pages and Got it All in One Diagram.' Arjo Klamer (2007). Hicks and Hansen Missed a Lot from 'Keynes's Elaborate Analysis'

While Hicks and Hansen got some of elaborate down correctly, they also got some of the analysis wrong. Hicks and Hansen needed a lot more than only a few pages and got it all in one diagram. for them to have been able to grasp Keynes's complete IS-LP(LM) model, as presented, summarized, critiqued, and mathematically developed by Keynes on pp. 298-306 of the General Theory. While Hicks did consider and mention a few of the possible problems with IS-LM, his discussion of problems pales into insignificance when compared to Keynes's pages of criticism in chapter 21 of the General Theory of his own IS-LP(LM) model. Hicks (Hansen) diagram was already completely described mathematically by Keynes on p. 207 of the General Theory. Any competent economist in 1936 could have drawn an upward sloping curve in Y,r space and then have taken into account Keynes's statement that The most striking examples of a complete breakdown of stability in the rate of interest, due to the liquidity function flattening out in one direction or the other, have occurred in very abnormal circumstances.(Keynes,1936; bold face added) by simply adding a vertical segment and a horizontal segment to the upward sloping part of the curve if they had actually read page 207 of the General Theory. More importantly, Hicks, like Modigliani in 1944, totally overlooked Keynes's mathematical development of his D-Z aggregate demand-aggregate supply model in chapter 20 of the General Theory. Hansen did a little bit better, but overlooked Keynes's aggregate production function and labor supply analysis. Hicks and Hansen thus had no underpinnings in microeconomics, in terms of aggregate production functions, theory of the firm, marginal analysis, and expectations of profit and price changes, for their version of IS-LM. Finally, both Hicks and Hansen removed Keynes's integration of uncertainty as a shift parameter from both the IS and LM curves. Hicks and Hansen got some parts right and got some parts wrong. Keynes's three elements, are, of course, the two IS equations and one LM(LP) equation explicitly derived on p.115 and p.199 of the GT along with the aggregated Y=C I version. All of these equations are superior in construction to the comparable equations that Keynes gave to his students in his December, 1933 lecture. Keynes expected an alert, competent, mathematically trained economist to be able to recognize that the equations could be solved to yield a quantitative, determinate, equilibrium solution in (Y,r) space that would be …valuable in introducing order and method into our enquiry… Keynes was mistaken. Keynes already had figured out that his UNCERTAINTY VARIABLE WAS GOING TO BE DROPPED from his GT analysis by all readers of the GT because they had no idea about what the weight of the evidence concept was. Economists still lack a basic understanding 81 years later. Nevertheless, Keynes was satisfied that both Harrod's and Hicks's versions of his original IS-LM(LP) work represented a major improvement in the manner that unemployment analysis was handled in the mid 1930's. That was why Keynes offered both Harrod and Hicks the possibility of creating and constructing a Keynes–Neoclassical Synthesis 19 years before Samuelson created his Keynes-Neoclassical synthesis. Unfortunately, both Hicks and Harrod failed to accept Keynes's offer.

J M Keynes, 'The State of the News (The Change in the Weight of the Evidence)' and the Weight of the Evidence: From Keynes's December, 1933 Student Lectures to Chapter 12 of the General Theory in February, 1936

J M Keynes’s “the State of the News” variable, W, which he integrated into his formal, simultaneous, four equation IS-LP( LM) model in his December 1933 student lectures, represented the change, either of an increase in the amount of relevant Knowledge, K, (evidence, data, information) in the Weight of the Evidence, w, or an increase in the amount of relevant ignorance,I, in the Weight of the Evidence, w. Keynes formalized his w variable in chapter 26 of the A Treatise on Probability on page 315 for insertion into his c coefficient model, the first decision weight rule constructed in the history of decision science, w=K/(K plus I). The completeness of the relevant evidence, w, was defined as being an index between 0 and 1, so that the completeness of the relevant evidence could be graded as 0 ≤ w ≤ 1. Keynes’s W is a technical innovation that represents the change in w, which Keynes defined in the first paragraph of chapter 6 as a quantitative measure that had to be normalized, as probability had to be normalized, on the unit interval [0,1], as w= K/(K plus I) or 0 ≤ K/(K plus I)≤ 1. Thus, W=∆w =∆K/(∆K plus ∆I). As in the A Treatise on Probability, an increase in K, where Keynes used information, data, and evidence as synonyms for K, always leads to an increase in w. On the other hand, an increase in I, Ignorance, will always lead to a decrease in w. Keynes’s W allows a decision maker to figure out the change in w over time so as to deal with changing circumstances by adjusting his level of confidence and liquidity preference. Keynes’s innovation refutes the claim of Stephen Stigler, made in 2002 , that Keynes made no more contributions in the area of probability and statistics after 1921. It is not surprising that Keynes chose capital W in December, 1933 to represent the change in w, given that Keynes had used w to define the weight of the evidence in his c coefficient model in chapter 26 of A Treatise on Probability on p. 315. Keynes’s IS -LP(LM) model of chapters 15 and 21 of the General Theory is built to integrate Keynes’s weight of the evidence into both the IS and LM equations in his formal model. W deals with the change in w over time Keynes changed his emphasis from changes in w, as given by W, to w itself in the General Theory when he defined uncertainty to be an inverse function of weight on p.148 of chapter 12 of the General Theory. A large and erroneous literature had grown since 1990 among heterodox economists that is based on a confused and confusing article published by J Runde in 1990 that claims that Keynes gave three conflicting definitions of weight. Bertrand Russell, F.Y. Edgeworth, C. D. Broad, and William Ernest Johnson make no mention of any such inconsistencies or conflicts in Keynes’s analysis in their book reviews of the A Treatise on Probability. That is because there are no inconsistencies or conflicts in Keynes’s analysis. Needless to say, Runde has never cited any of these reviews in any of his published work.

Firms’ money demand and monetary policy

Standard New Keynesian models for monetary policy analysis are cashless. When the nominal interest rate is the central bank's operating instrument, the LM equation is endogenous and, it is argued, can be ignored. The modern theoretical and quantitative debate on the importance of money for the conduct of monetary policy, however, overlooks firms' money demand. Working in an otherwise baseline New Keynesian setup, this paper shows that the monetary policy transmission mechanism is critically affected by the firms' money demand choice. Specifically, we prove that equilibrium determinacy may require either an active interest-rate policy (i.e., overreacting to inflation) or a passive interest-rate policy (i.e., underreacting to inflation), depending on the elasticity of production with respect to real money balances. We then calibrate the model to U.S. quarterly data and develop a sensitivity analysis in order to investigate the quantitative implications of our theoretical results. We find that macroeconomic stability is more likely to be guaranteed under an active, although not overly aggressive, monetary-policy stance.

Accuracy Assessment of Five Equations Used for Estimating the Glomerular Filtration Rate in Korean Adults.

BackgroundWe aimed to assess the performance of the five creatinine-based equations commonly used for estimates of the glomerular filtration rate (eGFR), namely, the creatinine-based Chronic Kidney Disease Epidemiology Collaboration (CKD-EPIcr), Asian CKD-EPI, revised Lund–Malmö (revised LM), full age spectrum (FAS), and Korean FAS equations, in the Korean population.MethodsA total of 1,312 patients, aged 20 yr and above who underwent 51Cr-EDTA GFR measurements (mGFR), were enrolled. The bias (eGFR–mGFR) and precision (root mean square error [RMSE]) were calculated. The accuracy (P30) of four eGFR equations was compared to that of the CKD-EPIcr equation. P30 was defined as the percentage of patients whose eGFR was within±30% of the mGFR.ResultsThe mean bias (mL·min-1·1.73 m-2) of the five eGFR equation was as follows: CKD-EPIcr, -0.6; Asian CKD-EPI, 2.7; revised LM, -6.5; FAS, -2.5; and Korean FAS, -0.2. The bias of the Asian CKD-EPI, revised LM, and FAS equations showed a significant difference from zero (P<0.001). The RMSE values were as follows: CKD-EPIcr, 15.6; Asian CKD-EPI, 15.6; revised LM, 17.9; FAS, 16.3; and Korean FAS, 15.8. There were no significant differences in the P30 except for the Asian CKD-EPI equation: CKD-EPIcr, 76.6%; Asian CKD-EPI, 74.7%; revised LM, 75.8%; FAS, 76.0%; and Korean FAS, 75.8%.ConclusionsThe CKD-EPIcr and Korean FAS equations showed equivalent analytical and clinical performances in the Korean adult population.

The Lund–Malmö creatinine‐based glomerular filtration rate prediction equation for adults also performs well in children

Objective. To evaluate the clinical performance in a paediatric population of the Lund–Malmö creatinine‐based glomerular filtration rate (GFR) prediction equations, primarily developed for adults. Material and methods. Iohexol clearance was used as the gold standard in 85 paediatric Caucasian patients (0.3–17 years; 37 F/48 M). One Lund–Malmö equation was based on age and gender (LM) and one included lean body mass (LM‐LBM). Comparisons focused on correlation (adjusted R2), bias (median percent error) and accuracy (proportions of predicted GFR differing ⩽30 % from measured GFR) (mL/min/1.73 m2). The performances were compared with those of the Modification of Diet in Renal Disease (MDRD) Study equation, a dedicated paediatric creatinine equation, Counahan–Barratt (CB) and a cystatin C‐based equation. Results. The MDRD equation performed poorly with a median bias of 96 %. Of the remaining equations, only the LM‐LBM produced significant bias (+10 % in median) according to line of identity regression analysis. The LM equation yielded marginally higher accuracy (76 %) than the LM‐LBM equation (74 %) and the CB (73 %), but lower than the cystatin C‐based equation (82 %). However, the estimated accuracy figures for these four equations were generally imprecise and none of the differences compared with the LM equation was statistically significant. Conclusion. In contrast to most creatinine‐based GFR prediction equations, the LM equation performs adequately for both children and adults. This may be due to the unique model‐building principles used when the LM equation was established. Further validation in a larger paediatric population is necessary.

Analytical derivations for one-dimensional eigenproblems using dual boundary element method and multiple reciprocity method

In this paper, MRM (multiple reciprocity method) and DBEM (dual boundary element method) are combined to solve one-dimensional eigenproblems. It is found that the hypersingular equation (LM method) for MRM provides an additional constraint to determine the eigenfunctions in case of failure using the singular equation (UT) method only. Also, augmented eigenvalues can be deleted by employing dual equations. The hypersingular equation of DBEM (LM equation) plays an important role in determining the eigenvalues and eigenmodes. Four numerical examples are given to verify the validity of the present formulation.

Monetary, fiscal and exchange intervention policy in a two-country intertemporal disequilibrium model

A symmetric two-country, two-period model, with flexible exchange rates and rigid first-period money wages is constructed. There is a single world output and perfect capital mobility. Agents have perfect foresight and optimise intertemporally. Aoki's method of ‘sums’ and ‘differences’ is used to analyse the model and to relate it to closed and small open versions. A key structural feature is that the intertemporal budget contraint, not the LM equation, determines a country's output. International shocks are transmitted through the real interest rate. Anticipated future spending increases are contractionary abroad, so a permanent increase may be, too. This modifies Mundell's findings.

On the missing equation: Comment

Canto and Miles (CM) present and analyze graphically “a third, alternative method for closing the IS-LM system of equations”—the ‘wedge model’. According to CM, “the result is a more general set of results” and “the two traditional alternatives are only special cases.” The CM model, however, is logically inconsistent. Because CM implicitly assume simultaneous capital market equilibrium and disequilibrium, CM's assumptions are mutually inconsistent. Thus, none of the reported results logically follow. Furthermore, the CM model is not very general. This paper proves analytically that GNP is determined without the IS and LM equations both in the CM model and in the obvious logical patchups of the CM model. Therefore, for example, in the CM model exogenous changes in business-fixed investment or consumption have no predicted effect on national income.

Keynes’s Labor Market: A Reply

Professor Tuchscherer (1984) offers a number of comments on my interpretation of Keynes's labor market, but the main points of his criticism concern three issues. First is the labor supply function. Second is the meaning of involuntary unemployment. Third is the determination of money wages. If differences on these three points can be reduced, other comments loosely scattered through his text and footnotes will not be major barriers to understanding, so I shall confine my response to these issues. Unlike Tuchscherer, I do not see all of these issues as resolvable within the labor sub-system, so I reproduce the principal equations used in my interpretation of Keynes's general equilibrium theory. The output market and money market equations are summarized in two equilibrium relations, the IS and LM equations. These equations are, as I have said, as close to conventional as I can make them.