The economic crisis of 2008/2009 had confronted the mainstream economic theory with an unpalatable task of revisiting the notions and perils of the ideas which dominated the course of economic theory in the last few decades. In 2003, delivering a speech to the American Economic Association, Robert Lucas famously noted that the central problem of depression prevention had been solved by mainstream macroeconomic theory which was built by combining the rational expectation hypothesis with New Keynesian macroeconomics. Although one should not obscure the achivements of new classical macroeconomics and new Keynesian macroeconomics, the criticism of contemporary macroeconomic theory is not uniform. It stems from the unrecognized role of systemic shocks in the financial sector and the spillovers from Wall Street to the Main Street. In contemplating the the linkages of over-leveraging and biased financial deregulation, it should not come as a surprise that early warnings of the financial crisis, mainly leveraged borrowing in the U.S subprime mortgage market, were earmarked in the mainstream economic theory.
In fact, in 1970, George Akerlof's influential paper on the issue of adverse selection in the market for lemons, was a landmark achievment in the economic theory since it demonstrated the falacies of perfectly competitive market mechanism when the information on quality of various commodities is distributed unevenly. In addition, a series of papers in 1970s by Joseph Stiglitz on screening theory and asymetric information, has dealt exactly with the central origins of the 2008/2009 financial crisis. Subprime loans and highly-complex derivate schemes which enabled the exponential growth of overleveraging of the banking sector were most likely to be used by the least sophisticated and accordingly the most risky borrowers. The only difference is that in normal circumstance, banks would recognize adverse selection by rationing credit to risky borrowers but the continuous obsession with home-ownership and the reluctance of the Federal Reserve to "remove the bowl of punch when the party started" - to use the analogy of Preston Martin, former Vice President of the FED - added to the turbulence of overleverage that turned into the most disastrous financial meltdown after the Great Depression.
The fact is that contemporary macroeconomics had little to offer to predict the subsequent financial meltdown although Robert Shiller of Yale University has repeatedly warned against unstable stock market fundamentals, particular notorious price-earnings ratios after the dot-com bubble came to burst. However, the central element of the critic of mainstream economic theory should revisit the notorious paradigm of supply-side economics whose intelectual melange of fervent belief in tax cuts and a dangerous preoccupation with deregulation as the cure of the malaise which led to stagflation in early 1970s, have proved how dangerous the conclusions could become.
First, the rise of the supply-side economics in the political economy began in early 1980s. But the intelectual influence of the supply-side economics should not be confined to the theoretical paradigm itself. The field of the political economy of taxation manifested itself as the intelectual triumph of supply-side economics. The original idea of the Laffer curve, the relationship between tax rate and tax revenues, was not disputable after all. In fact, if tax rates reached predatory levels, decreases in total tax burden would yield considerable gains, not only in total tax revenue but also in terms of higher level of productivity. However, when average and marginal tax rates were at moderate levels, it would be foolish to believe immense revenue gains would ensue by reducing the rates of taxation to bottom-levels, arguing for significant gains in terms of employment growth, productivity boost and total tax revenues. Even though cross-country empirical evidence does suggest an increase in tax revenues amid the decline in average tax rate, the pattern is confined to the episodes where average and marginal tax rates were very high, exceeding 70 percent threshold. Once tax rates were reduced, there is no evidence of higher revenue gains.
The major peril of supply-side economics is the claim that tax reduction would boost the aggregate supply and stimulate productivity growth. On the other hand, the valuable contribution of supply-side economics is the notion that additional tax increases do not generate much higher revenue. One should not feel reluctant to recall the 1964 Kennedy-Johnson tax cut which decreased marginal tax rates substantially. Although supply-side economics has repeatedly blasted the intelectual heritage of Keynesian macroeconomics, the 1964 tax reform was itself a Keynesian prescription for the U.S recession in the years prior to Vietnam war. Back in early 1960s, Paul Samuelson wrote that "Congress could legislate, for example, a cut of three or four percentage points in the tax applicable to every income class, to take effect immediately under our withholding system in March or April, and to continue to the end of the year." (link). Therefore, Samuelson's mindful observation that additional spending would not automatically counteract the recession unless complemented by tax reductions, probably would not come due in the framework of supply-side economics. Moreover, what distinguished the supply-side economics from the framework of sound economic analysis taught in microeconomic and macroeconomic textbooks, was adverse propensity to enforce tax cuts for the rich while leaving the middle class and low-income households no pie from tax reductions. The striking features of income inequality in the U.S. suggest that from 1970s, median household income stagnated (link) while top 5 percent of households have received disproportionately windfall gains from tax reductions up the point where more than 85 percent of total income was earned by top 5 percent of households (link). Moreover, one should distinguish between patterns of good and bad inequality as Gary Becker recently suggested (link). It is envitable that income inequality has some great value in the society when market outcomes lead to better overall health, less stress and higher standard of living and the evidence is yet inconclusive whether the narrowing of income inequality would return health improvements for the poor - since poor health outcomes of low-income households are mainly attributed to deteriorating dietary habits and dangerous lifestyle.
While bad inequality, especially rents from non-market outcomes, have precipitated the decline in good inequality in the last two decades, there is an overwhelming evidence that stagnation of median household income (despite moderate productivity improvements) caused a somehow lower quality of the U.S. labor force and a widening gap in educational achievments of American children. The drawbacks of widening inequality were largely ignored by supply-side economics or justified on the hands-off approach to the issues of the poor. It should not be forgotten that negative income tax, which favored low-income families, was suggested by Milton Friedman, whom supply-siders have taken for the intelectual father without a detailed knowledge of his precious contribution to economics.
Second, supply-side economics has been perhaps known for favoring the deregulation as the cure for social ills and staggering income growth. Despite substantial euphoria caused by the pioneers of deregulation of banking and financial sector, the regulatory framework eventually jeopardized sound regulation that could prevent hazardous outcomes as shown in the seminal work of George Akerlof and Joseph Stiglitz. In fact, deregulation of the banking sector, hailed by supply-side economics as the triumph of its own ideology, laid the basis for rigorous financial innovation by special investment vehicles (SIV) and shadow banking institutions.
In fact, deregulation of the banking and financial sector was not the central issue per se. The main systemic flaw was rather the adoption of unsound regulation that did not predict the perils of over-leveraged banking sector and especially the system-wide spillovers during the financial crisis. Moreover, the loosening of the monetary policy and the series of fiscal stimulus have notified two main drawbacks in the macroeconomic outlook. The first is the invariant postponement of taxation fuelled by the mountain of government debt. And the second is the hidden explosive potential for inflation following the flood of money supply in the balance sheet of the banking sector.
Generally speaking, the intelectual adventure of supply-side economics has overlooked the possibility of pitfalls brought up by rigorous tax cuts to the wealthy and deregulation of banking and financial sector. It would not come due to label mainstream economic theory as a cataclysm which the financial crisis proved accordingly. It would be either insensible to tarnish the useful contribution of supply-side economics. In fact, tax cuts do generate systemic incentives, particularly in the response of the labor supply to tax reductions. However, the elusive quest for higher growth and job creation after reducing tax rates for the wealthy, is an important lesson we should learned from the unfortunate turn of supply-side economics in favoring deregulation without acknowledging the possibility of systemic banking collapse and the consequences carried over by society at large.
Showing posts with label Great Minds in Economics. Show all posts
Showing posts with label Great Minds in Economics. Show all posts
Friday, July 08, 2011
Friday, December 25, 2009
THE MOST INFLUENTIAL GLOBAL THINKERS
Foreign Policy composed the rank of the most influential global thinkers (link).
Monday, December 14, 2009
IN MEMORIAM: PAUL A. SAMUELSON
On Sunday, December 13th, Paul A. Samuelson has died at the age of 94 (link).
He was on the economic giants of the 20th century. His ideas reshaped the economic science and revolutionized the mode of economic thinking around the world. With the mathematical rigour and analytical mastermind, his groundbreaking approach to economic analysis transfored the economic science into a dynamic problem-solving tool. In this short essay, I will present my reflections on the life and contributions of Paul A. Samuelson to the economic science.
I first came across Paul Samuelson in the year before I entered the university. In the first year of the undergraduate class, Samuelson and Nordhaus's Economics was the assigned reading for the Introductory Macroeconomics. I read the textbook back and forth and I liked it; not because of its simplicity in introducing the analytical framework of economics but rather because of the clarity, intuition and incentives to undertake a rigorous pursuit of analytical economics at the theoretical and empirical level. In addition, Samuelson penetrated the application of linear programming to economic problem-solving.
Together with Milton Friedman, Paul A. Samuelson is the economic giant of the 20th century. Hardly any economist could take the same place in the scope of influence as an economic thinker. He conducted the Neoclassical synthesis. As an interested reader can verify in his Nobel prize lecture (link), Samuelson's synthesis combined a Keynesian macroeconomics with a rigorous Marshallian microeconomics. In microeconomics, Samuelson extended the Marshallian analysis of partial equilibrium with strong mathematical articulation of demand and supply curves, cost curves and deadweight loss. On the abstract level, together with Abram Bergson, he constructed social welfare functions based on three marginal conditions and extracted from earlier work of Kaldor-Hicks-Scitovsky analysis (link). In macroeconomics, Samuelson further affirmed the dominance of Keynesian macroeconomics with a strong emphasis on the role of fiscal policy in stimulating full-employment output. In addition, he invented the term multiplier and the acceleration, the former relating to the effect of change in exogenous macro variables on endogenous variable (notably, output) and the latter referring to the partial adjustment of aggregate investment to the capital stock. Samuelson-Hansen multiplier-accelerator principles spurred the theoretical foundations of Keynesian economic policy. He also popularized Overlapping Generations Model which later became the corner stone of innovations in the modeling of aging population. In macroeconomics, Samuelson also proposed the so-called "Samuelson-Mishi condition for the efficient provision of public goods. When the condition is satisfied, it implies that further substitution of private goods provision for public goods will result in a diminishing social utility.
Assuming Pareto efficiency, Samuelson-Mishi condition satisfied the criteria for Lindahl equlibrium. The equilibrium states that when individuals are willing to pay for the provision of public goods according to marginal benefits, it will be Pareto efficient. However, such condition is not compatible with the incentive mechanism since it requires the complete knowledge of individual demand functions for particular public good which could result in the asymmetric distribution of benefits in response to relevation-principled taxation.
As a student of international economics, I came across the influential theoretical work of Paul Samuelson. Modern international economics is a combination of mathematical economics, advanced microeconomics, game theory and international finance. One of the most interesting and penetrating areas of international economics are theorems in international trade. Under particular assumptions theorem postulate axiomatic explanations based on previous statements. Back in 1941, he proposed Stolper-Samuelson theorem together with Wolfgang Stolper. The theorem quickly became a source of academic debate. In its simplest form, the theorem states the following: assuming constant returns to scale and perfect competition, a rise in the relative price of good will lead to higher return on the factor which is used more intensively in the production of the good and to the fall in the return to the other factor. Stolper and Samuelson wrote:
"Second only in political appeal to the argument that tariffs increase employment is the popular notion that the standard of living of the American worker must be protected against the ruinous competition of cheap foreign labor… In other words, whatever will happen to wages in wage good (labor intensive) industry will happen to labor as a whole. And this answer is independent of whether the wage good will be exported or imported."
The theorem showed that the international trade between two countries could lead to the opposition of international trade since the relative price of labor-abundant good in the high-wage country will be higher than the world price of that good, reflecting the relative abundance of capital or human capital. The theorem quickly became the main theoretical weapon of opponents to free trade. Even today, Stolper-Samuelson is the best explanation of why labor unions in high-wage countries oppose free trade agreements and further economic integration with low-wage countries.
Another important contribution of professor Samuelson is the so called Ballasa-Samuelson effect which states that higher growth productivity growth rate in tradable goods relative to non-tradables will lead to the real exchange rate appreciation. Balassa-Samuelson effect also went through numerous time-series regression. The effect has been tested 60 times in 98 countries. Cross-section regression studies of Ballasa-Samuelson effect were analyzed in 142 countries. In a vast majority, the empirical evidence of Ballasa-Samuelson hypothesis was supported. The main empirical findings emphasize that productivity differential between tradeable and non-tradable sector is positively correlated with differences in relative prices. The empirical evidence also supported Samuelson's initial proposition that productivity differentials translate into higher purchasing power parity through real exchange rate appreciation.
In finance, Paul Samuelson penetrated the analytical aspects of lifetime portfolio selection. In 1972 he published The Mathematics of a Speculative Price which later became the ground of option pricing. Based on discoveries of Bachelier's pioneering work, he laid the foundations of stohastic price movements and random forecasting matches. His pioneering work in financial theory of speculation and random walk (stohastic) movements in stock prices became the underlying theoretical foundation in the emerging financial industry. In an article entitled Probability, Utility and the Independence Axiom (Econometrica, 1952), he discussed the role of probability models in measuring the overall utility. In this sense, he relied on Keynesian defence of subjective theory of probability and argued that thesubjective perception of probability does not inhibit the proper functioning of financial markets. In 1965, he published A Proof that Properly Anticipated Prices Fluctuate Randomly where he provided the foundation of the efficient market hypothesis that has been further developed by Eugene Fama and other scholars. For a detailed discussion of Paul Samuelson's contribution to financial economics, see Merton Miller's contribution in Britannica (link).
In addition to his theoretical and empirical work, he is the founding member of the Econometric Society and its president in 1951. In 1961, he was the president of American Economic Association. In the political sense, Paul A. Samuelson influenced the economic policy of the Kennedy Administration. In 1960, the U.S headed for the recession. President Kennedy, following Samuelson's advice, enacted tax cuts and a balanced budget. In 1964, when Kennedy tax cuts were enacted, top marginal tax rate was reduced from 91 percent to 70 percent. The economic reasoning behind tax reductions was firmly laid in the Keynesian multiplier (1/(1+c)(1+t)). Paul Samuelson and Walter Heller (Chairman of Council of Economic Advisers during Kennedy Administration) argued that lower tax rate would stimulate consumption spending and boosted output and employment. Throughout the 1960s, the U.S economy experienced one of the longest periods of stable economic growth, favorable employment outlook and balanced federal budget. Here is how JFK, following Samuelson's advice, supported the tax reduction (link). Also, David Greenberg's article on Kennedy tax reduction is a worthy source of further information on that topic (link).
On Sunday, the economic titan passed away. He not only revolutionized the field of economic science but also spurred the interest for economics and popularized it in a manner that turned dismal science into a problem-solving science based on theoretical foundations and empirical verification of theoretical postulates. His approach to economic analysis combined Marshallian microeconomics and Keynesian macroeconomics which he joined together after the WW2 in a Neoclassical synthesis. Compared to other economic thinkers, he knew how to formulate theoretical postulates in a manner that stimulates the research interest for further investigation.
He will be missed and remembered as the giant of the economic thought and a titan of economic theory.

He was on the economic giants of the 20th century. His ideas reshaped the economic science and revolutionized the mode of economic thinking around the world. With the mathematical rigour and analytical mastermind, his groundbreaking approach to economic analysis transfored the economic science into a dynamic problem-solving tool. In this short essay, I will present my reflections on the life and contributions of Paul A. Samuelson to the economic science.
I first came across Paul Samuelson in the year before I entered the university. In the first year of the undergraduate class, Samuelson and Nordhaus's Economics was the assigned reading for the Introductory Macroeconomics. I read the textbook back and forth and I liked it; not because of its simplicity in introducing the analytical framework of economics but rather because of the clarity, intuition and incentives to undertake a rigorous pursuit of analytical economics at the theoretical and empirical level. In addition, Samuelson penetrated the application of linear programming to economic problem-solving.
Together with Milton Friedman, Paul A. Samuelson is the economic giant of the 20th century. Hardly any economist could take the same place in the scope of influence as an economic thinker. He conducted the Neoclassical synthesis. As an interested reader can verify in his Nobel prize lecture (link), Samuelson's synthesis combined a Keynesian macroeconomics with a rigorous Marshallian microeconomics. In microeconomics, Samuelson extended the Marshallian analysis of partial equilibrium with strong mathematical articulation of demand and supply curves, cost curves and deadweight loss. On the abstract level, together with Abram Bergson, he constructed social welfare functions based on three marginal conditions and extracted from earlier work of Kaldor-Hicks-Scitovsky analysis (link). In macroeconomics, Samuelson further affirmed the dominance of Keynesian macroeconomics with a strong emphasis on the role of fiscal policy in stimulating full-employment output. In addition, he invented the term multiplier and the acceleration, the former relating to the effect of change in exogenous macro variables on endogenous variable (notably, output) and the latter referring to the partial adjustment of aggregate investment to the capital stock. Samuelson-Hansen multiplier-accelerator principles spurred the theoretical foundations of Keynesian economic policy. He also popularized Overlapping Generations Model which later became the corner stone of innovations in the modeling of aging population. In macroeconomics, Samuelson also proposed the so-called "Samuelson-Mishi condition for the efficient provision of public goods. When the condition is satisfied, it implies that further substitution of private goods provision for public goods will result in a diminishing social utility.
Assuming Pareto efficiency, Samuelson-Mishi condition satisfied the criteria for Lindahl equlibrium. The equilibrium states that when individuals are willing to pay for the provision of public goods according to marginal benefits, it will be Pareto efficient. However, such condition is not compatible with the incentive mechanism since it requires the complete knowledge of individual demand functions for particular public good which could result in the asymmetric distribution of benefits in response to relevation-principled taxation.
As a student of international economics, I came across the influential theoretical work of Paul Samuelson. Modern international economics is a combination of mathematical economics, advanced microeconomics, game theory and international finance. One of the most interesting and penetrating areas of international economics are theorems in international trade. Under particular assumptions theorem postulate axiomatic explanations based on previous statements. Back in 1941, he proposed Stolper-Samuelson theorem together with Wolfgang Stolper. The theorem quickly became a source of academic debate. In its simplest form, the theorem states the following: assuming constant returns to scale and perfect competition, a rise in the relative price of good will lead to higher return on the factor which is used more intensively in the production of the good and to the fall in the return to the other factor. Stolper and Samuelson wrote:
"Second only in political appeal to the argument that tariffs increase employment is the popular notion that the standard of living of the American worker must be protected against the ruinous competition of cheap foreign labor… In other words, whatever will happen to wages in wage good (labor intensive) industry will happen to labor as a whole. And this answer is independent of whether the wage good will be exported or imported."
The theorem showed that the international trade between two countries could lead to the opposition of international trade since the relative price of labor-abundant good in the high-wage country will be higher than the world price of that good, reflecting the relative abundance of capital or human capital. The theorem quickly became the main theoretical weapon of opponents to free trade. Even today, Stolper-Samuelson is the best explanation of why labor unions in high-wage countries oppose free trade agreements and further economic integration with low-wage countries.
Another important contribution of professor Samuelson is the so called Ballasa-Samuelson effect which states that higher growth productivity growth rate in tradable goods relative to non-tradables will lead to the real exchange rate appreciation. Balassa-Samuelson effect also went through numerous time-series regression. The effect has been tested 60 times in 98 countries. Cross-section regression studies of Ballasa-Samuelson effect were analyzed in 142 countries. In a vast majority, the empirical evidence of Ballasa-Samuelson hypothesis was supported. The main empirical findings emphasize that productivity differential between tradeable and non-tradable sector is positively correlated with differences in relative prices. The empirical evidence also supported Samuelson's initial proposition that productivity differentials translate into higher purchasing power parity through real exchange rate appreciation.
In finance, Paul Samuelson penetrated the analytical aspects of lifetime portfolio selection. In 1972 he published The Mathematics of a Speculative Price which later became the ground of option pricing. Based on discoveries of Bachelier's pioneering work, he laid the foundations of stohastic price movements and random forecasting matches. His pioneering work in financial theory of speculation and random walk (stohastic) movements in stock prices became the underlying theoretical foundation in the emerging financial industry. In an article entitled Probability, Utility and the Independence Axiom (Econometrica, 1952), he discussed the role of probability models in measuring the overall utility. In this sense, he relied on Keynesian defence of subjective theory of probability and argued that thesubjective perception of probability does not inhibit the proper functioning of financial markets. In 1965, he published A Proof that Properly Anticipated Prices Fluctuate Randomly where he provided the foundation of the efficient market hypothesis that has been further developed by Eugene Fama and other scholars. For a detailed discussion of Paul Samuelson's contribution to financial economics, see Merton Miller's contribution in Britannica (link).
In addition to his theoretical and empirical work, he is the founding member of the Econometric Society and its president in 1951. In 1961, he was the president of American Economic Association. In the political sense, Paul A. Samuelson influenced the economic policy of the Kennedy Administration. In 1960, the U.S headed for the recession. President Kennedy, following Samuelson's advice, enacted tax cuts and a balanced budget. In 1964, when Kennedy tax cuts were enacted, top marginal tax rate was reduced from 91 percent to 70 percent. The economic reasoning behind tax reductions was firmly laid in the Keynesian multiplier (1/(1+c)(1+t)). Paul Samuelson and Walter Heller (Chairman of Council of Economic Advisers during Kennedy Administration) argued that lower tax rate would stimulate consumption spending and boosted output and employment. Throughout the 1960s, the U.S economy experienced one of the longest periods of stable economic growth, favorable employment outlook and balanced federal budget. Here is how JFK, following Samuelson's advice, supported the tax reduction (link). Also, David Greenberg's article on Kennedy tax reduction is a worthy source of further information on that topic (link).
On Sunday, the economic titan passed away. He not only revolutionized the field of economic science but also spurred the interest for economics and popularized it in a manner that turned dismal science into a problem-solving science based on theoretical foundations and empirical verification of theoretical postulates. His approach to economic analysis combined Marshallian microeconomics and Keynesian macroeconomics which he joined together after the WW2 in a Neoclassical synthesis. Compared to other economic thinkers, he knew how to formulate theoretical postulates in a manner that stimulates the research interest for further investigation.
He will be missed and remembered as the giant of the economic thought and a titan of economic theory.
Friday, June 19, 2009
INTERVIEW WITH PAUL A. SAMUELSON
Wednesday, October 15, 2008
NOBEL PRIZE IN ECONOMICS 2008
The Nobel laureate in economics in 2008 is Paul Krugman (here):
"IT WAS widely expected that Paul Krugman, who won the the 2008 Nobel prize for economics on Monday October 13th, would claim the award one day. In 1991 he had received the John Bates Clark medal for the best young economist, which is widely seen as a stepping stone to a Nobel award. What is more of a surprise is that he was honoured rather sooner in his life than many other winners. Like most Nobel laureates in economics, Mr Krugman was recognised for research undertaken early in his career—in this case for his pioneering work on modelling trade between countries whose firms grow more profitable the bigger they become. At 55, he is only four years older than the youngest ever winner, Kenneth Arrow, who was 51 when he won in 1972. But he is a fresh-faced youngster in comparison with Leonid Hurwicz, one of last year’s winners, who was 90 when he shared the prize."
"IT WAS widely expected that Paul Krugman, who won the the 2008 Nobel prize for economics on Monday October 13th, would claim the award one day. In 1991 he had received the John Bates Clark medal for the best young economist, which is widely seen as a stepping stone to a Nobel award. What is more of a surprise is that he was honoured rather sooner in his life than many other winners. Like most Nobel laureates in economics, Mr Krugman was recognised for research undertaken early in his career—in this case for his pioneering work on modelling trade between countries whose firms grow more profitable the bigger they become. At 55, he is only four years older than the youngest ever winner, Kenneth Arrow, who was 51 when he won in 1972. But he is a fresh-faced youngster in comparison with Leonid Hurwicz, one of last year’s winners, who was 90 when he shared the prize."
Monday, January 07, 2008
THE 2008 YOUNG ECONOMIST AWARD
The 2008 Young Economist Award goes to Raj Chetty of Berkeley. Raj Chetty has done extensive research on taxation, unemployment, social insurance and risk preferences. His notable spectrum of research includes "Dividend Taxes and Corporate Behavior: Evidence from the 2003 Dividend Tax Cut". The work has been published in Quarterly Journal of Economics together with Emmanuel Saez. Raj Chetty is also a faculty member at the National Bureau of Economic Research.
Source: The American (link)
Source: The American (link)
Monday, December 10, 2007
INTERVIEW WITH JOHN NASH, THE 1994 NOBEL LAUREATE IN ECONOMICS
PBS has published an interview with dr. John Nash of Princeton University. John Nash received a Nobel prize in economics in 1994 for pioneering the equilibrium analysis of non-cooperative games. The interview can be read and seen as a video here.
On Discovering Math
"I was in grade school. I would be doing arithmetic, and I found myself working with larger numbers than other students would be using. I would have several digits, and they would have maybe two or three digits. I would do multiplication and basic operation, but with larger numbers. Later on in adolescence, I got some practice in using a calculator machine, where you could multiply and add, subtract and divide really large numbers like 10 digits."
John Nash
On Discovering Math
"I was in grade school. I would be doing arithmetic, and I found myself working with larger numbers than other students would be using. I would have several digits, and they would have maybe two or three digits. I would do multiplication and basic operation, but with larger numbers. Later on in adolescence, I got some practice in using a calculator machine, where you could multiply and add, subtract and divide really large numbers like 10 digits."
John Nash
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