Mathematical Economics of Compound Rates of Interest: A Four-Thousand Year Overview Part I
复利的数学经济学：四千年的回顾 （一）
January 24, 2004
“Whoever enters here must know mathematics.” That was the motto of Plato’s Academy. Emphasizing such abstract ratios as the Pythagorean proportions of musical temperament and the calendrical regularities of the sun, moon and planets, its philosophy used the mathematics of nature to reveal an underlying harmony and order in the universe and hence, in an ideal society. But there was little quantitative analysis of economic relations. Although the Greek and Roman economies were increasingly wracked by debt, there was no measurement of this phenomenon, or of overall production, distribution and other macroeconomic measures.
“未通晓数学者禁入。”这是柏拉图学园的铭文。像毕达哥拉斯的音调协调，像日月星辰按历律列张于苍穹，它的哲学强调了这种抽象的“比例”，使用自然界的数学，揭示了宇宙所蕴含的和谐秩序，因此，理想的社会，亦复如是。然而，在这里，却几乎找不到什么经济学关系的量化分析。虽然希腊和罗马的经济颠覆于债务，却找不到对这种现象进行测量的计算方法，也找不到总产量、分配、其它宏观经济手段的测算方法。
The education of modern economists still consists largely of higher mathematics whose use remains more metaphysical than empirically measuring the most important trends. Over a century ago John Shield Nicholson (1893:122) remarked that “The traditional method of English political economy was more recently attacked, or rather warped,” by pushing the hypothetical or deductive side . . . to an extreme by the adoption of mathematical devices. . . . less able mathematicians have had less restraint and less insight; they have mistaken form for substance, and the expansion of a series of hypotheses for the linking together of a series of facts. This appears to me to be especially true of the mathematical theory of utility. I venture to think that a large part of it will have to be abandoned. It savors too much of the domestic hearth and the desert island.
然而，现代经济学家的教育包含了很大部份的高等数学，它的使用更多地是以抽象推理，而不是凭借经验来估算最重要的趋势。一个世纪以前，John Shield Nicholson评论道：“近来，英国政治经济学的传统方法受到更多攻击，更确切地说，是扭曲”，由于采纳了数学手段，将假设和演绎的方法推向了极致，有才干的数学家无不缺乏洞鉴烛明之能事，误把形式当本质，一堆堆假说联系起一堆堆事实。摆在我眼前的，似乎数学理论功莫大焉。不过，我斗胆地认为，大部份的它们将不得不被抛弃。它，太不食人间烟火了。
To contemporary economists, mathematics has become the badge of scientific method. But are the right things being mathematized? Do today’s models correlate the appropriate phenomena, or do they confuse cause and effect while omitting key dynamics? Is the use of mathematics scientific ipso facto, regardless of how it is applied?
对于当代的经济学家，数学已经成为科学方法的标志工具。然而，正确地做法是数学化吗？现代的模型与现象互为表里吗？或者，遗漏了关键机制，这些模型混淆了原因和结果？应用数学，而不考虑如何应用，其本身是科学的吗？
The preferred method of mathematical economics is general equilibrium analysis in an environment in which only small marginal disturbances are envisioned, not major structural problems or legal changes in the economic environment. Marginal analysis avoids dealing with quantum leaps. It selects and correlates a rather narrow set of phenomena (supply, labor and materials costs, the interest rate, income and the pattern of demand) to produce models that show how economies might settle at an equilibrium point if left free from outside political interference.
数学经济学所偏爱的办法是在某个环境条件下的一般等式分析。在这个经济环境中，仅预见到少量的边际干扰，并没有看到重要的结构问题或法律发生了改变。边际分析避免了讨论量的跃变。它选择了一系列非常狭窄的现象（供给、劳动力、原材料成本、利率、收入和需求模式）逐个加以对应，所导出的模型显示了经济是如何可能在一个点上找到它的平衡，对于政治的外来干扰，则不加考量。
许多经济学家接受过微积分和高等数学的训练，却并不感到有必要用数量等式来检测自己的理论。他们倾向于更少地将数学作为一门经验的计算工具，而更多地是将其作为一门解释性的语言，或者仅仅是一项科学装饰，打扮一下他们的“政策处方”。对实际运转中的财富和收入的内在的极化趋势，数学经济学极少进行统计分析。
In fact, the mathematical “badge of science” has distracted attention from the tendency for economies to veer out of balance.[1] The problem is that to achieve a single determinate, stable solution to any given problem (always posed as a “disturbance” to a pre-existing balance), general equilibrium theorists are driven to assume diminishing returns and diminishing marginal utility in order to “close the system.” A narrow set of variables is selected that all but ignore the economy’s growing debt overhead relative to its assets, and the associated flow of interest.
事实上，经济体都有失去平衡的趋势，而数学作为“科学的标志”却转移了对这一方面的关注。问题的关键是，为了获得任何给定问题（总是以对原先平衡产生“干扰的因素”的面目出现）的唯一恒定解，一般均衡理论家有着假定递减回报和递减边际效用的倾向，以使得“体系首尾相衔”。一套狭义的变量被选择，完全忽略了经济体里相对于资产不断增长的债务经费，以及相关的利息流。
Economies change their shape as they grow. This shape is distorted by the inherent tendency for financial claims – bonds, bank loans and other financial securities – to grow more rapidly than the economy’s ability to carry them, much less to pay them off. The volume of such claims tends to grow by purely mathematical principles of self-expansion, independently from underlying economic trends in wealth and income, and hence from the ability of debtors to pay.
经济一边发展，一边改变着自己的形态。形态的扭曲是因为内在的金融债权——债券、银行贷款和其它金融有价证券——的发展趋势，它们增长得太快，超过了经济的承受能力，更不用说对它们的清偿能力。这些证券的数量倾向于以纯数学的自我膨胀的原理来增长，脱离了财富和收入的基本经济走势，因此，也脱离了债务人的支付能力。
The task of economic regulation is reduced to setting an appropriate interest rate to keep all the economy’s moving parts in equilibrium. This interest rate is supposed to be controlled by the money supply. An array of measures is selected from the overall credit supply (or what is the same thing, debt securities) to represent “money.” This measure then is correlated with changes in goods and service prices, but not with prices for capital assets – bonds, stocks and real estate. Indeed, no adequate statistics presently exist to trace the value of land and other real estate.
经济管理和调节的任务被演绎为：设置一个适宜的利率就可以保持经济中所有活的部件在平衡中运转。利率被认为由货币供给来控制。一系列计量标准从总信用供给（或者那些同样的事物：债务证券）中挑选出来代表“货币”，然后，这种方法又与商品服务的价格联系起来，却与资本资产——债务、股票和房地产——的价格脱离开来。实际上，要查考土地和房地产的价值，目前仍没有充足的统计数据存在。
The resulting economic models foster an illusion that economies can carry any given volume of debt without having to change their structure, e.g., their pattern of wealth ownership. Self-equilibrating shifts in incomes and prices are assumed to enable a debt overhead of any given size to be paid. This approach reduces the debt problem to one of the degree to which taxes must be raised to carry the national debt, and to which businesses and consumers must cut back their investment and consumption to service their own debts and to pay these taxes.
这样的经济模型孕育出了一种幻觉，即经济能够承担任何数量的债务，而不会改变自身的结构——比如说，它的财富拥有模式（财富所有制形式）。收入和价格的自我平衡被认为能令任何大小的债务经费得到支付。这种方法把债务问题演绎为程度问题：税赋必须以何种程度征收，以运转国家债务，商业和消费者必须以何种程度来削减投资和消费，以偿付自己的债务和支付这些税赋。
Excluded from the analysis is the finding that many debts are not repayable except by transferring ownership to creditors. This transfer changes the shape of the economy’s legal and political environment, as creditors act as rentiers to subordinate labor and capital to the economy’s financial dynamics.
被排除于这种分析的是这样的结论：除了将所有权转移给债权人，许多债务是无法支付的。这种转移改变了经济的法律和政治环境，而债权人作为食利者使得劳动力和资本依附于这种经济的金融机制。
Rent-seeking exploitation and the proverbial free lunch are all but ignored, yet real-world economics is all about obtaining a free lunch. That is why one seeks to become a political insider, after all, yet such considerations are deemed to transcend the narrow boundaries of economics. These boundaries have been narrowed precisely so as to limit the recognized “problems” only that limited part of economic life that can be mathematized, and indeed, mathematized without involving any changes in social structure (“the environment”).
寻租式的剥削和免费午餐这玩意儿可以完全被置之不理了，而真实世界的经济学充斥着如何获得一份免费的午餐。这就是为什么人们努力要成为政策内幕知情者，然而，毕竟人们会认为这种想法超出了经济学的狭窄边界。这些边界被狭隘，恰恰是因为要将那些公认的“问题”仅仅界定为能被数学化的经济生活的有限部份，实际上，数学化，却毫不涉及社会结构（“环境”）发生的任何改变。
A particular kind of mathematical methodology thus has come to determine what is selected for study, recognizing only problems that have a single determinate mathematical solution reached by or what systems analysts call negative feedback. As noted above, such entropic behavior is based on the assumption of a falling marginal utility of income: The more one earns, the less one feels a need to earn more. This is fortunate, because most models also assume diminishing returns to capital, which is assumed to be invested at falling profit rates. Income and wealth thus are portrayed as tapering off, not as soaring and polarizing until a financial collapse point, ecological limit or other kind of crisis is reached.
于是，一种特别的数学方法开始决定挑选什么来研究，它仅认识到有单一、确定的数学式解答的问题，它们的解答，由体系分析者叫做负反馈的玩意儿来获得。如上所述，这种回归的行为是建立在下降的收入边际效用的假定上，即一个人挣得越多，它就越不想去再挣。真是幸运，因为大多数模型也假定资本的递减回报，即它们被投资于下降的利润率。所以，收入和财富被描述为锥体般逐渐减少，而不是高涨并极化，直到金融崩溃、生态的极限和其它种类的危机时刻降临。
A model acknowledging that positive feedback occurs when the rich get richer at the expense of the poorer, and when the “real” economy is dominated by an expanding overhead of financial capital, will depict an economic polarization that has an indeterminate number of possible resolutions. The economic problem becomes essentially political in the sense that conflicting trends will intersect, forcing something to give. This is how the real world operates. But to analyze it would drive economists out of their hypothetical entropic universe into an unstable one in which the future is up for grabs. Such a body of study is deemed unscientific (or at least, uneconomic) precisely because it cannot be mathematized without becoming political.
如果有一种模型承认，当富人以穷者更穷为代价变得更富，当“真实”经济由一种膨胀的金融资本开支来主导，正反馈会发生；那么，这种模型将描绘出经济的极化有着一些尚不确定的解决方法。就不相容的趋势会交叉在一起并迫使一方屈服这个意义上来说，经济问题本质上是政治性的。这就是真实经济的运行。但要分析它，就会把经济学家从他们的假说的回归的世界中驱赶出来，进入到一个不稳定的世界中，在这种世界中，未来是无法餍足的攫取。这样一类的研究本身被认为是不科学的（或至少是，非经济的），恰恰是因为在数学化之前，它必须变得具有政治性。
The mathematical universe of modern economics was not created without some degree of protest. A generation ago F. J. Dyson (1964: 132f .) commented that “Mathematical intuition is more often conservative than revolutionary, more often hampering than liberating.” Citing Ernst Mach’s observation that “The power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations,” he worried that too much real-world complexity might be discarded. Nowhere is this problem more pronounced than in the treatment (or lack of recognition) of the economic role played by money and debt.
现代经济学的数学小宇宙是被创造出来之前，会有一定程度的抗议。30多年前，F. J. Dyson评论说，“数学的直觉知识常常是更保守的，而不是更创新进取的，常常是更束缚性的，而不是更解放思想的。”他引证了Ernst Mach的评论：“数学的力量在于它回避了所有多余的想法，在于它极圆满地节约了心智的活动”，他担忧真实世界太多的复杂性可能会被置之不理。最为清楚地揭示这个问题，还是要到货币和债务所扮演的经济角色的讨论——或认识的短缺——中寻找。
债务的指数增长
剧作家和小说家要比现代的经济学家更为关注债务问题。狄更斯、巴尔扎克和他们同时代的人，以及更早时期的英国戏剧，都充满了债务的形象描述。事实上，纳皮尔的对数（1500年发展起来）的最早应用是与莎翁的《一个冬天的故事》一起出现的。故事里的人物Polyxenes华丽地表达了他对受到的9个月好客款待的感激之情，打趣地使用了永远无法偿清的缠身债务这个比喻。要花时间感谢东道主，就会消磨掉主人更多的时间，用掉了更多的好客之情，又欠下了更多的感激之情，引发了永不停歇的人情债。
Nine changes of the watery star [the moon] hath been
The shepherd’s note since we have left our throne
Without a burden: time as long again
Would be fill’d up, my brother, with our thanks;
And yet we should for perpetuity,
Go hence in debt: and therefore like a cipher,
yet standing in rich place, I multiply
With one we-thank-you many thousands more
That go before it.
“Without a burden” means without debt. The “cipher . . . standing in a rich place” is the language used in the early 1600s to indicate the logarithmic exponential. The imagery was that of a debt mounting up unpaid at compound interest, multiplying to the point where it engulfs the kingdom.
“没有负担”意思是“没有债务”，“无足轻重的人……站立在富有之地”是1600时期使用的语言，表示对数函数。这里形象化表达的是一份未偿清的债务以复利堆积起来，繁殖到它能够吞下整个王国。
Economic writers in earlier times were more ready than their modern counterparts to confront the problem of debts growing so large as to be unpayable. In The Wealth of Nations (V, iii), Adam Smith observed that “Bankruptcy is always the end of great accumulation of debt. The liberation of the public revenue, if it has ever been brought about at all, has always been brought about by a bankruptcy; sometimes by an avowed one, but always by a real one, though frequently by a pretended payment.”
相比于现代，早些时期的经济学作家更敏感于债务增殖太大而无法支付的问题。在《国富论》中，斯密论述道：“破产总是大规模债务累积的结果。财政收入要得到解放，如果它曾发生过的话，总是由破产潮产生的，有时是通过公开宣布的破产，虽然时常是由名义支付【1】，但总是由真实的破产所产生。”
【1】通胀后，支付的数额不变，但实际价值却大大降低了。——章二
The tendency of debts to accumulate at compound rates of interest explains why Smith’s axiom applies so universally. The principle was described graphically by one of Smith’s contemporaries, the dissenting Anglican minister and actuarial mathematician Richard Price. It was Price who first popularized the distinction between compound and simple interest that later came to be associated mainly with Malthusian population theory.
债务以复利累积的趋势解释了为什么斯密的格言应用得如此之广。斯密的同时代人，政见不合的英国教会的教长、精算数学师Richard Price详细地解释了其间的原理。正是Price首先推广了复利和简单利率的区别，后来，这种区别主要与马尔萨斯的人口理论联系在了一起。
In the 1770s when Price and Smith wrote, Britain ’s war in America had forced the nation deeply into debt. It was largely out of their opposition to such debt that they urged Britain to grant freedom to its colonies. As for the debts that already had mounted up, Price proposed an idea that had been anticipated a half century earlier by Nathaniel Gould, a director of the Bank of England. Parliament would pay off the national debt by setting aside a million pounds sterling in a sinking fund, to accumulate at compound interest by reinvesting the dividends annually until the fund grew large enough to pay off the entire debt. Price’s 1772 Appeal to the Public on the Subject of the National Debt described the seeming magic of how money could grow at compound interest:
在十六世纪70年代，Price和斯密正在写作的年代，英国在美洲的战争迫使这人国家深深地陷入了债务。正是由于对这些债务的抗议和反对，两人敦促英国认可殖民地的自由。关于已经累积如山的债务，Price提出了一个半世纪前就被Nathaniel Gould（英格兰银行的董事）期盼的观点。议会可以将100万英镑留存为沉积基金（又意译为“偿债基金”），每年的股息再投资进去，以复利来积累，直到这笔基金增长到足以付清全部债务。Price的《关于国家债务向公众的呼吁》描述了钱以复利增长的魔力。
Money bearing compound interest increases at first slowly. But, the rate of increase being continually accelerated, it becomes in some time so rapid, as to mock all the powers of the imagination. One penny, put out at our Saviour’s birth at 5% compound interest, would, before this time, have increased to a greater sum than would be obtained in a 150 millions of Earths, all solid gold. But if put out to simple interest, it would, in the same time, have amounted to no more than 7 shillings 4½d.
带有复利的货币的增长，首先是缓慢的。不过，因为增长的速率是持续增加的，在某时，它就会增加得如此迅速，以至于所有的人类想像力都变得相形见拙。一便士，在救世主的生日生产出来，以5%的复利来计算，则在此时之前，就已经增长到从1亿5千万个地球上都不能获得的金额——以黄金实体来说。然而，如果以简单利率来计算，在同样的时间里，它将相当于差不多7先令4½便士。
Price elaborated this idea in his Observations on Reversionary Payments, first published in 1769 and running through six editions by 1803. “A shilling put out at 6% compound interest at our Saviour’s birth would . . . have increased to a greater sum than the whole solar system could hold, supposing it a sphere equal in diameter to the diameter of Saturn’s orbit.” He concluded that “A state need never, therefore, be under any difficulties, for, with the smallest savings, it may, in as little time as its interest can require, pay off the largest debts.”
Price在他的《继承支付的观察》中详尽说明了这个概念。这本书最先在1769年出版，到1803年，一共经历了六个版本。“一先令，在救世主的生日以复利6%发行出来……将增长到整个太阳系都不可能盛得下的金额，姑且假定它是一个直径等于土星轨道的球体。”他推论道：“所以，一个国家永远不必置于麻烦之中，因为，只需最小的储蓄，它就可以在利息所需要的时间内，偿清最巨大的债务。”
What Price had discovered was the exponential growth of money invested at interest, multiplying the original principal by plowing back the dividends into new saving. What he failed to appreciate was that never in history has any economy been able to turn a penny or any other sum into a surplus large enough to pay creditors a solid sphere of gold reaching out to Saturn’s orbit. Marx accordingly poked fun at Price’s calculations in his Grundrisse notebooks (1973: 842f .) on the ground that no society’s productive powers are able to support such compound rates of growth in interest claims. “The good Price was simply dazzled by the enormous quantities resulting from geometrical progression of numbers. . . . he regards capital as a self‑acting thing, without any regard to the conditions of reproduction of labour, as a mere self‑increasing number,” subject to the growth formula
Surplus = Capital (1 + interest rate)n
Economists estimate that during the two thousand years since the birth of Christ the European economy has grown at a compound annual rate of 0.2 percent, far less than the level at which interest rates have stood. No wonder Adam Smith found that no nation in history had paid off its public debt. As he observed in The Wealth of Nations (loc. cit.) England ’s tax revenues had become “a fund for paying, not the capital, but the interest only, of the money which had been borrowed . . .” Sinking funds established ostensibly to pay off this debt were not effective, as governments invariably reborrowed the money. Smith concluded that the availability of such funds merely “facilitates the contraction of new debts. It is a subsidiary fund always at hand to be mortgaged in aid of any other doubtful fund, upon which money is proposed to be raised in any exigency of the state.”
Price所揭示的就是以利息投资的货币的指数增长，将红利再投资为新的储蓄，不断增加了本金量。他没有认清的是，在历史上，从来没有任何经济体还债是这样把一便士变成土星轨道大小的黄金实体球。所以，马克思打趣Price《Grundrisse》笔记中的计算，认为没有一个社会的生产力能够维持利息债权的这种复利增长。“善良的普赖斯只不过被几何级数所得出的庞大数量弄得眼花目眩……他根本不考虑再生产和劳动条件的限制，把资本看作是自我运行的东西，仅仅只是一个自我增长的数字，”主观地看待增长公式：
盈余=资本×（1+利息率）n
经济学家曾估算过，基督出生后的两千年里，欧洲经济以每年2%的复利增长，远小于利率曾有的水平。难怪斯密在历史上没有发现一个国家偿清了它的公共债务。在《国富论》里，他提到，英国的税收已经成为一项支付资金，不是（偿付）曾经借来的钱的资金，而是（偿付）利息……。沉积资金的建立，表面上是偿清债务，但它却并不奏效，因为政府总是又再借入这笔钱。斯密断定，这种基金的用处只不过是“减轻新债务的收缩这个麻烦。它是一种随时方便取用的补充基金，为其它未定的——国家在紧急关头筹钱时所需要的——资金提供帮助。”
The contrast between geometric and arithmetic rates of growth has long been established in the popular mind not by reference to the interest rate at which savings and debts double and redouble, but by Thomas Robert Malthus with regard to the growth of population relative to the food supply. Malthus first published his theory in 1798, a generation after Price put forth his argument for a sinking fund. Picking up his fellow minister’s mathematical imagery, Malthus asserted that population tended to grow “geometrically” at compound rates, but was held back by the fact that the means of subsistence could only grow at an “arithmetic” rate, that is, at simple interest.
长久以来，几何和数学增长率的鲜明对比已在人们的心目中牢固树立，但并不是通过储蓄和债务在利率上翻倍再翻倍，而是通过马尔萨斯关于人口相对于食物供给的增长。马尔萨斯在1798年首次发表了他的理论，距离Price提出沉积基金论点时，已过了30年。拾起同侪教长的数学的形象化表述，马尔萨斯断言人口倾向于以复利方式“几何级数”增长，但会被生活资料仅能以“算术”率——即简单利率——增长这个实事所遏制。
Malthus did not foresee that fertility rates historically have tapered off as incomes have risen. Over time, breakthroughs in agricultural and mining technology have increased productivity in these sectors even more rapidly than has occurred in manufacturing, so that food and other consumption goods have increased even more rapidly than has population.
马尔萨斯并没有预见到：当收入上升时，历史上的生育率却是逐渐变小的。随着时间的推移，农业和采矿业技术突破提高了这些部门的生产率，甚至比制造业所发生的改变更为迅速，这样，食物和其它消费品的增加比人口的增加更为迅速。
Although Malthus’s demographic ideas have been disproved, the financial analysis drawn by Price remains apt. What indeed grow geometrically are the economy’s financial claims – its bonded debt, mortgages and bank loans, whose interest charges have tended throughout history to accrue in excess of society’s wealth-creating powers.
虽然马尔萨斯的人口论一直受到驳斥，但Price所进行的金融的分析却仍然完好如初。真正几何级数增长的是经济的金融债权——公债借款、抵押和银行贷款，它们的利息费用在整个历史上的累积，倾向于超过社会财富创造力。
It was inevitable that private individuals would attempt to make use of the compound interest principle by leaving savings to accumulate over a protracted span of time. In 1800 a Mr. Thelluson set up a trust that was to accumulate its income for a hundred years. At the expiration of that time the trust was to be divided among his descendants. His estate of £600,000 was estimated to yield £4500 per year (at 7 ½ percent), producing a final value of £19,000,000, some thirty times the original legacy.
不可避免，人们会想法利用复利原理，将储蓄在一段较长时间内累积。1800年，一个叫Thelluson的先生大人就建立了一家信托基金，将它的收入储蓄100年。到期时，这份信托基金将由子孙后辈间分享。据估算，他的60万英镑资产每年将产出4500镑（7.5%），最终创造出1900万的价值，即大约原遗产的30倍。
As matters turned out, Thelluson’s will was contested in a litigation that lasted some 62 years, from his death in 1797 through 1859. By the time the lawyers were paid, “the property was found to be so much encroached on by legal expenses that the actual sum inherited was not much beyond the amount originally bequeathed by the testator.”[2] Meanwhile, under the leadership of William Pitt, the government calculated that at four per cent compound interest the trust would own the entire public debt by the time a century had elapsed. Some legislation known at the time as Thelluson’s Act was speedily passed, limiting such trusts to twenty‑one years’ duration.
事实却不如人所料，Thelluson的遗嘱在一场持续62年的官司（从他1797年死亡到1859年）中受到质疑。到付清律师费后，“才发现这份财产让高昂的法律费用蚕食殆尽，以至于实际的继承金额并不比遗嘱人最初的遗赠数目高出多少。”与此同时，在William Pitt的领导下，政府计算出，以4%的复利，一个世纪过去，这家信托基金将把国家的全部债务纳为己有。一些法律，如那时的Thelluson法令，迅速地举手通过，将这种信托基金限制在21年的时期之内。
Orthodox academic models rarely acknowledge the problems posed by the exponential growth of debt overhead. Such models typically make government policies appear unnecessary to cope with this problem, by focusing on the kind of world that might exist if the financial overgrowth of savings and debts did not double every decade or so, having multiplied again and again over the past century. It thus has been left mainly to non-mainstream writers to address the structural problems created by an accumulation of interest-bearing debt. The socialist Proudhon (What is Property, quoted in Flürscheim 1902:326), for instance, observed that “If men, united in equality, gave to one of their number the exclusive right of property, and if this single proprietor placed with humanity a sum of 100 francs at compound interest, repayable to his successors of the twenty-fourth generation after the lapse of 600 years – this sum of 100 francs would, if invested at 5 per cent., amount to the sum of 107,854,010,777,600 francs, a sum 2,696 times as large as the capital of France, estimated at 4,000 millions (50 years ago), or 20 times as large as the value of the whole globe with all movable and unmovable wealth.”
正统的学院派极少承认由债务支出的指数增长引起的问题。正是这些学院派使得政府的政策不再将处理这个问题作为必须，只是专注于，如果储蓄和债务的金融过度生长不会每隔十年左右就翻倍，这样的世界就会生存下来。虽然它们在过去一个世纪已经反反复复的翻倍增殖，于是，负息债务的累积所引起的结构性问题，就留给了非主流的作家们去探讨。例如，社会主义者Proudhon（《财产是什么》）提到：“如果人们平等地联合起来，将财产的专有权给予他们中的一个成员，如果这个业主大发慈悲，将100法郎的一笔钱置于复利之下，600年后，偿还给他的第24位继承人——这笔100法郎的金额，如果利率为5%，将会变成107,854,010,777,600法郎，这是法国资本的2696倍，估计为40亿法郎（50年前），或者全球所有流动和固定资产总价值的20倍。”
Marx’s analysis of the financial dynamics of debt
The most sophisticated analysis of financial capital in the 19th century was that of Marx. In Volume III of Capital (ch. xxx) and Vol. III of Theories of Surplus Value (both published posthumously) he referred to high finance as being based on “imaginary” or “fictitious” capital. It was fictitious because it consisted of claims on tangible wealth rather than constituting the direct means of production. These financial claims took the form of bonds, mortgages, bank loans and commercial paper. Marx (Capital, III.461) called these financial claims “a void form of capital” inasmuch as they represented a financial overhead whose interest charges ate into industrial profits. Such profits were earned actively by employing labor to produce goods and services for sale – a process Marx summarized by the formula M-C-M’, spending money to produce commodities that would sell for yet more money. But the growth of interest-bearing financial capital was characterized by the disembodied M-M’, making money simply from money itself, i.e. in an essentially sterile way.
马克思：债务的金融机制的分析
19世纪对金融资本最深刻的分析属于马克思。在《资本论》第三卷和《剩余价值理论》第三卷（它们都是马克思逝世后出版）中，马克思指出高金融是建立在“想像的”和“虚构的”资本基础上。它是虚构的，因为它由有形资本的债权组成，而不是形成直接的生产资料。鉴于这些金融债权代表一种金融开支，其利息费用侵蚀了工业利润，马克思（资本论III，461）称它们为“资本的空无形式”。工业利润是通过雇佣工人生产待出售的商品和提供服务来积极地挣得——马克思用M-C-M’这个公式来概括，消费货币以生产商品，商品出售获取更多的货币。但是，负息金融资本的增长，其特征是脱离实体的公式M-M’，仅仅是从钱本身来赚钱，也就是说，是一种阉割了实体的方式。【1】
【1】这关系到马克思的价值论。在生产过程中，工人加入了自己的劳动价值，产品增值了。从整个社会的宏观经济来看，商品的使用价值增大了，增多了。在M-M’中，货币本身不会增值，因为没有人的劳动价值加入其中。为什么M能变成M’，唯一的结论是：作为一种（金融）体系或制度，或者说社会和经济的结构，通过侵蚀（eat into）进入真实经济的利润来获取，它们是“无酬劳动”。由此可见，在学习经济学时，社会的结构、制度和体系是十分重要的因素。马克思说过：“理解一个时代的历史秘密必须到那个时代的经济学中寻找答案”。同时我们会进一步问：它们是如何形成的？或者具体的说，每一种结构、制度和体系是如何必然产生它的极化现象？这就把令人眼花缭乱的经济学化为了简单的问题。赫德森的高明之处，就是抓住的经济历史分析的任督二脉：债务和极化（贫富悬殊）。
Marx spelled out how financial capital tended to assert its domination over tangible capital above all in monetary crises and the foreclosures that followed in their wake. To illustrate how the inexorable force of such usury capital and its stipulated debt service tended to exceed debtors’ ability to pay, Marx (Capital, III: 463) cited Martin Luther’s imagery likening it to the beast Cacus, and Dr. Price’s calculations about the power of compound interest. The volume of financial claims grew willy-nilly as a result of this mathematical principle, inexorably surpassing the ability of the economy’s tangible productive powers to keep pace.
马克思阐明了金融资本是如何倾向于维护它对有形资本的统治，尤其是在货币危机和紧随其后的取消抵押品赎回（潮）中。要说明这些高利贷资本的无情的威力，以及它厘定的偿债服务中是如何趋向于超过债务人的支付能力，马克思（资本论III：463）引用了马丁·路德，路德将其形象地类比为怪兽Cacus，也引用了Price复利魔力的计算。不管人们愿不愿意，金融债权的数量都以数学原理的方式增长，漠然将经济的实际生产力远远抛在身后。
Yet after analyzing finance capital’s tendency to grow geometrically at compound interest, Marx abruptly dropped the subject. He believed that finance capital was becoming thoroughly subordinated to the dynamics of industrial capital. “In the course of its evolution, industrial capital must therefore subjugate these forms and transform them into derived or special functions of itself,” he speculated (1971:468). “It encounters these older forms in the epoch of its formation and development. It encounters them as antecedents, but not as antecedents established by itself, not as forms of its own life-process.” It was the destiny of industrial capitalism to mobilize finance capital to fund its own economic expansion. “Where capitalist production has developed all its manifold forms and has become the dominant mode of production,” Marx concluded, “interest-bearing capital is dominated by industrial capital, and commercial capital becomes merely a form of industrial capital, derived from the circulation process.”
不过，马克思在分析金融资本以复利的几何级数增长的趋势后，立即放下了这个主题。他相信金融资本正在完全的附属于工业资本机制。他推演说：“在工业资本产生过程中，它必须使这些形式从属于自己，并把它们转化为自己派生的或特殊的功能。工业资本在它形成和发展的时期碰到了这些更古老的形式。它遭逢它们时，恰如我们遇到我们的先辈……而不是我们遇到儿时的我们。”工业资本的使命是调集金融资本以为自己的经济扩张提供资金。马克思推断说：“在资本主义生产方式发展成熟，并成为生产的主导模式，生息资本便由工业资本主宰，商业资本仅仅是工业资本的一种形式，在流通过程派生而出。”
If this were true, the economy’s means of payment would be able to keep up with the exponential growth of interest-bearing debt claims. Alternatively, the overhang of financial capital – savings and their counterpart debts – would be wiped out by monetary crises. More than any other economist, Marx drew attention to this problem, but he nonetheless viewed finance capital as playing a subordinate role. He thus shared the 19th-century optimism of the French utopian socialist Fourier and of the St. Simonians that industrial progress might solve the debt problem by mobilizing savings more productively than ever before had been the case.
果真如此，那么，经济的支付方式将能够跟得上负息债权的指数增长。同时，过剩的金融资本——储蓄及其对应的债务——会通过货币危机来抹除。马克思超越了其它的经济学家，对这个问题给予了重视，但是，他将金融资本视为附属角色。因此，他分享了19世纪法国乌托邦社会主义者傅立叶的乐观主义，分享了圣·西蒙主义者——工业的进步将更有效率地调动储蓄，可以解决债务问题——的乐观主义。
German historical economists such as Roscher pointed to the fact that interest rates tended to fall steadily with the progress of civilization; at least, rates had been falling since medieval times. Credit laws were becoming more humanitarian, and debtors’ prisons were being phased out throughout Europe as more lenient bankruptcy laws were freeing debtors to start afresh with clean slates. Most important, European and North American public debts were on their way to being paid off during the relatively war-free century 1815-1914. Lending was mobilized to fund heavy transport, industry, mining and construction. For awhile, the economy’s debt burden seemed likely to become self-amortizing. The broad consensus was that the debt problem was curing itself by being co-opted into a more socially productive credit system.
德国的历史学派，如Roscher等人指出，随着文明的进展，利率趋向于稳定的下降；至少自从中世纪以来，一直在下降。信贷法正在变得更人性化，债务人监狱从整个欧洲淡出，而更为仁慈的破产法使得债务人随着债务清算而可以重新起步。最重要的是，欧洲和北美的公共债务是在相对没有战争的1815-1914年之间一步一步被偿清的。贷款用于重型运输、工业、采矿业和建筑业的融资。一时之间，经济的债务负担似乎很可能够自动被偿清。广泛的共识是，债务问题通过指定的社会生产性信用体系而自我愈合。
Modern observers can trace how these salutary trends gave way to the overgrowth of debt experienced in recent decades. The drives of finance capital have tended to overshadow those of industrial development. Indeed, finance capital was absorbing industrial capital even in the Victorian era as emperors of finance surpassed captains of industry.
细心的观察者可以踪迹到这些有利的趋势是如何让步给最近几十年所发生的债务的过度增长。金融资本内在动力的趋势胜过了工业发展的内在动力。实际上，甚至在维多利亚时代，当金融君主超越工业首脑，金融资本是在吞并工业资本。
Flürscheim的金融批评
马克思和他的社会主义追随者将抨击主要集中在工业及其利润上，而不是那些拿走利息的金融家。Henry George吸引了许多追随者，他将焦点集中在食利者身上，土地拥有者收割地租、拿走食利收入，其致富是以社会的损失为代价，可是，他们的租的所得收入却显示了经济的进步。George的一个欧洲追随者，Michael Flürscheim，写了几本书，其中一本主要关注负息债务引起的问题。（在《经济迷宫管窥》中）他写道：“确实，雇主是海绵，吮吸了利润，劳工产品的大部份的价值（马克思称它为剩余价值Mehrwerth）竟然全让渡给了租和利息的主子，还有中间商，这些人一起从劳工身上尽可能迅速地挤出利润，留给劳工的只是一点点血汗钱。更糟的是，他们拿走了劳工充分发挥生产潜力的能力。”于是，他建议劳工和资本应该联合起来，对“真正的敌人”，食利者，发起进攻。
To do this in an informed way, finance capital had to be distinguished from physical capital. Pointing out (p. 347) that “Tribute Claim Capital constitutes the Bulk of the World’s Capital,” Flürscheim explained that “When an orator or writer has to reply to a socialist’s attack upon capital as the oppressor of labor, he points to what orthodox economy calls capital, and speaks of our wonderful progress due to our improved means of production and distribution, whereas his antagonist thinks of Government bonds, of land monopoly, of mining rights, of all kinds of tribute claims selling at Exchange for certain amounts, and not at all falling under the orthodox definition of capital, though representing that capital which people principally have in view when they use the term.” But he was almost alone in focusing on this target.
要根据事实依据做这事儿，就得区分金融资本与有形资本。Flürscheim指出（p. 347）“贡赋权资本构成了世界资本的主要部份。”他解释说：“当演讲者和不得不回答社会主义者对资本的抨击，即资本是劳工的压迫者时，他就会指出正统经济学叫做资本的事物，说起我们的非凡进步是归因于生产资料和分配方式的提高和改善，然而，他的对手想起的却是政府债券、土地垄断权、采矿权、票据交换所里各种各样的收益（贡赋）权……根本不在正统的资本定义中，可是，它们代表着人们使用资本这个词时脑海浮现的东西。” 关注这个对象的Flürscheim几乎是孤独的。
Flürscheim elaborated that “All exertions, all improvements in the methods and tools of labor, the strictest economy, the severest self-denial, are all powerless to compete with the rapidity of self-increase possessed by capital placed at compound interest, and they cannot keep up with its demands.” To illustrate this point he composed the following allegory to illustrate the dynamic at work. Many ages after man was driven from Paradise and told by the Lord “to earn his bread by the sweat of his brow, mercy began to prevail. A loving angel was sent down by the Great Master, charged with the task of lightening the burden. The angel’s name was Spirit of Invention. He began his work by teaching man to make useful tools,” to tame animals, mobilize water power, air and wind power, fire and steam power to drive machinery.
Flürscheim详细地解释道：“劳动工具和方式的所有运用和改善、最严谨的经济、最苛刻的自我牺牲，都无力与置于复利之上的资本所拥有的自我增殖能力相匹敌。它们根本无法满足它的要求。”为了形象地说明这一点，他创作了下面的寓言，说明了实际运转的机制。人类被赶出伊甸园许多年代后，上帝告诉他们“要用辛勤的汗水来挣得面包。于是派了一个充满爱心的天使，天使承担着（为人类）减轻负担的任务。它的名字叫创造精灵。他从教人们如何制造有用的工具开始了他的工作”，然后教人们驯养动物、利用水力和风力、火和蒸气动力来运转机器。
“It seemed that at last the golden era had come of which men had dreamed for ages past, without ever hoping to attain it. Without trouble, with almost no exertion, except that of wealth for the satisfaction of wants which, in former times, even the richest did not know or dream of.” But “that envious spirit, that fallen angel, Satan, who once before, in the shape of the serpent, had driven man from Paradise by seducing him to sin,” was jealous and angry that his own empire would soon be over for ever.
“人类许多年代的梦想，从来不曾希望成真的黄金时代似乎最终降临了。不再有苦恼，不再有千辛万苦，只有财富运用于满足人们的需要，在过去，这是最富有的人都不曾梦想过的。”然而，“忌妒的幽灵，那个堕落的天使、曾以蛇的模样出现，唆使人类犯下罪恶，驱除于伊甸园的撒旦”眼看着他的王国将从此告终，不禁妒火中烧，恼羞成怒。
Among the follies of man, one little imp, called Interest, managed to attract Satan’s attention. “‘What is the matter with you, Interest?’ he asked the saucy imp. ‘You don’t seem to be so dejected as your comrades are?’”
在人类的愚蠢中，有个小捣蛋鬼，叫利息，设法吸引撒旦的注意。“利息，你怎么啦？”撒旦问这个活泼鬼，“你似乎不像你的同伴那样一蹶不振啊？”
“‘Why should I be dejected, master?’ replied the spirit, ‘Am I not one of your favorite soldiers? Haven’t I always been victorious under your august guidance?’”
“为什么我要像他们，我的主子？”这个精灵回答说，“我不是你最赏识的战士吗？凭借你的威仪，我不是一直节节胜利吗？”
But Satan answered sadly, “Alas, You are no match for the Spirit of Invention.” The imp, however, volunteered to demonstrate his prowess in a duel.
撒旦面带愁容，回答道：“哎呀，你不是创新精灵的对手啊。”然而，这个小捣蛋鬼自告奋勇地要在决斗中展示自己的勇气和技艺。
“‘You little imp! Fight the powerful angel who is defeating all my army?’ laughed Satan.”
“你这个调皮鬼！竟然要与打得我溃不成军的强大天使战斗！”撒旦嘲笑道。
“‘Yes, I alone; provided, of course, you allow my son, Compound Interest, to help me.’” He explained with regard to the goblins of technology, that “Instead of their being a source of blessing to mankind, I shall make them the producers of untold misery – worse than ever man suffered from thy hands.” So “Satan let him have his way. The battle of giants began.”
“是的，我去单挑，当然，请你允许我带上我的儿子，复利，帮我的忙。”关于技术这个小妖精，他解释道：“它们是人类的福祉之源，不，我会让它们成为不可言状的痛苦的缔造者——比出自你手的痛苦还要痛苦”。于是“撒旦让它自行其是。巨人的战争爆发了。”
In the beginning the angel laughed, for, though twenty squares were passed, no noticeable diminution of his forces was perceptible. Demon Interest said nothing, but attended to business, quietly doubling his army on every succeeding square. At the thirtieth square the angel ceased to laugh, and soon saw he was lost.
战斗伊始，天使大笑，因为，虽然20个方格过去了，但他的军队并没有明显、能够察觉的减少。魔鬼利率一声不吭，专心关照自己的事务，很快在以后每一次获取的方格中，让军队翻了倍。到了第三十个方时，天使的笑嘎然而止，刹时明白：他失败了。
‘I despised you, little fellow,’ he signed despairingly, ‘and I am punished for my vanity. I see there is no use fighting against you. Demon Interest is more powerful than the Spirit of Invention. I am your slave. Command your servant!’
“我看不起你，小家伙，”他沮丧地叹息，“我因为虚荣自大而受到处罚。再和你斗，已毫无用处。魔鬼利息比创新精灵更强大。我是你的奴隶了，我的陛下，你发令吧。”
‘I am the only servant of my great master,’ dryly replied the demon. ‘Here I see him coming. He will give you his orders.’
“我只是我伟大的主人的仆人”，这个魔鬼冷冰冰地回答。“他很快就来，你静候他的命令吧。”
And Satan gave his orders. He commanded that the angel was to continue in his work with all his troops, which were to be increased with all possible exertion, so that humanity – which did not know the nature of the antagonist it had to fight against – would always keep in fresh hope of final success when the new troops were forthcoming. But as fast as they appeared, Demon Interest was to send forth a larger army to capture the new forces, to enslave them, and – instead of their benefiting man – make them increase the slave-chains which weigh him down.
撒旦下了命令。他命令天使带领其部队继续工作；运用了最大的努力，部队不断地强大，于是，人类——可怜他们并不知道他们曾反抗的敌人的天性——眼看着新的部队不断集结，始终怀有最后成功的希望。然而，它们刚一集结，魔鬼利息立马就生长出更多的军队，俘虏了这些新生力量，并奴役他们，同时，并不让它们裨益于人类，而是将它们变成更牢靠的奴役锁链，沉沉地套在人类的脖梁上。
To the surprise of the king, this series of doublings “produced an amount larger than the treasures of his whole kingdom could buy. It is this kind of chess-game which capital is continually playing with labor.” The remarkable growth of compound interest would “soon swallow products, capital, the earth and even the workers.”
使国外惊奇的是，这种连续翻倍“产生出的数量远远超过了整个王国的宝藏。资本一直用于玩弄劳动者的，正是这种棋盘游戏。“复利的非凡增长会很快吞掉产品、资本、土地、甚至劳动者。”
Flürscheim concluded (1902: 333f .) by asking, “What is compound interest? Is it anything else than the fresh investment of earnings of capital?” He added the story that “Napoleon Bonaparte, when shown an interest table, said, after some reflection: ‘The deadly facts herein lead me to wonder that this monster Interest has not devoured the whole human race.’ It would have done so long ago if bankruptcy and revolution had not been counter-poisons.”
Flürscheim的推断如是质问：“复利是什么？除了资本收益的再投资，还是别的什么东西呢？”对这个故事，他补充说，“有人给拿破仑看一张利息表，他沉思片刻，说道：‘此中关窍令我纳闷，这个利息怪物居然还没吞掉全人类。’如果破产和革命不曾作为抗毒剂，恐怕早已如此了。”
This problem has a pedigree dating back some four thousand years. What is surprising is the clarity with which ancient economies dealt with it in a more straightforward way than is favored today.
这个问题自有其渊源，可以追溯到四千年前。令人吃惊的是，古代经济体处理这个问题所具有的清晰思辩色彩，它们所采用的处理方式比今天所宠爱的方式更直截坦率。
What the Babylonians recognized that modern economists don’t
At past Heilbronn symposia I have discussed the importance of tracing civilization’s economic trends back to Sumer and Babylonia.[3] It was in this epoch, over two thousand years prior to classical antiquity, that the basic elements of modern economic relations first appear. These elements included interest-bearing debt and ways of coping with the problems caused by its spread.
巴比伦人认识到现代经济学家未曾认识到的
在Heilbronn评论集中，我曾讨论过将文明的经济走向追溯至苏美尔和巴比伦。正是在古典时期的两千年前的这个纪元中，现代经济关系的基本元素第一次出现。这些元素包括负息债务和采用什么方法来处理其蔓延时所引起的问题。
Mathematics played a major role in the training of scribes. This hardly is surprising, as cuneiform writing’s first application (c. 3200 BC) was to economic account keeping. Already in the 3rd millennium BC, scribes were trained in mathematical procedures such as manpower allocation problems (e.g., how many men were needed to dig canals of a given size or to produce a given amount of bricks), the surveying techniques needed to calculate surface measurements (including the geometric analysis of squares and circular shapes), astronomical computations and even quadratic equations. Scribes also were trained to calculate the expected growth of herds and the exponential growth of interest-bearing investments.[4]
数学在书吏的培训中有着重要的作用。这并不奇怪，因为最早的楔形文字书写用于经济的账目记录，是在公元前3500年。公元前3000年，书吏在数学方面的培训包括如人力分配问题（例如，需要多少人挖一条水渠或多少人生产一定数量的砖）、地面测算所需要的技巧（包括正方形和环形的几何分析）、天体的计算、二次方程等。同时，书吏也被培训于计算畜群的预期增长和有息投资的指数增长。
Rather than reflecting economic productivity, profitability or the general ability to pay, the accrual of interest was essentially a mathematical phenomenon. For ease of computation, the normal commercial rate had been built into the system of sexagesimal weights and measures. Interest accrued at the rate of one shekel per mina per month, that is, the “unit fraction,” 1/60th. This rate remained constant over many centuries (indeed, millennia), and worked out to 12/60ths per year, 60/60ths in five years. Compounding occurred quinquennially, once the initial principal had reproduced itself in five years.[5] Interest rates in Greece , Rome and the Byzantine Empire likewise were based on ease of computation in their local systems of weights, measures and arithmetic.[6]
利息的增长并没有反映出经济的生产率、利润率或一般支付能力，本质上，它是一种数学的现象。为了计算的方便，正常的商业利率融入了六十进制的度量衡体系之中。利息是在每迈纳每月1谢克尔的利率上增长的，也就是说，“单分数”1/60。许多世纪（实际上，是千年）过去了，这个利率始终不变，每年产生出12/60，五年则60/60。最初的本金在五年里繁殖出自体一次，则复合率每五年出现一次。在希腊、罗马和拜占庭，利率同样是建立在方便计算的基础之上，用本地的重量、尺度和数学的系统来进行计算。
The fact that these interest rates were not economically based or responsive to changing economic conditions made repayment problems inevitable. Debt problems also develop today, of course, but contemporary theory insists that economies can adjust to any given level of debt charges. The Babylonians made no such assumption. Their student exercises show that they recognized that herds, for instance, increased at a slower pace than did the growth of debts mounting up at 20 percent per year, to say nothing of agrarian rates typically around 33 1/3 percent.
这些利率并非以不断变化的经济条件为基础，或者说，并不反应出改变的经济条件，这样，出现支付问题就不可避免。当然，今天的债务问题也会不断发展，但是，当代的理论却坚持：经济能调节到任何债务费用的规定水平。巴比伦人则没有这样的臆断。他们的学生习题显示他们认识到：畜群，例如，增长的速率要比债务的增长（以每年20%累积，更不要说农业利率一般在33 1/3左右）更慢。
In light of these exercises I would like to make a suggestion that initially may seem outrageous. Mesopotamian economic thought c. 2000 BC rested on a more realistic mathematical foundation than does today’s orthodoxy. At least the Babylonians appear to have recognized that over time the debt overhead became more and more intrusive as it tended to exceed the ability to pay, culminating in a concentration of property ownership in the hands of creditors.
依据这些习题，我想提一个乍看之下似乎不合常规的建议。相比今天的经济思想，公元2000年前的美索不达米亚经济思想立足于一种更为现实的数学根基，至少巴比伦人认识到：随着时间的流逝，债务开支会变得越来越强人所难，超过了支付的能力，并以资产所有权集中在债权人手里而达到其顶点。
Scribal students (nearly all of whom were employed in temple and palace bookkeeping) were taught to calculate how rapidly investments doubled when lent out at interest. A model exercise appears in a Berlin cuneiform text (VAT 8528): How long does it take a mina of silver to double at the normal commercial rate of interest of 1/60th (that is, one shekel per mina) per month? (This often is expressed a 20 percent annual interest, inasmuch as 12/60ths = 1/5 = 20 percent.) The solution involves calculating powers of 2 (22 = 4, 23 = 8 and so forth).[7]
传授给将来要做书吏的学生（他们几乎都从事于庙堂和宫廷记账）的知识之一是：计算以某一利息借出的投资（资本）是以怎样的速率翻倍。一个例题出现在柏林的楔形文字原本中：在正常的商业利率为每月1/60（即每迈纳1谢克尔）时，1迈纳的银要花多少时间来翻倍？（这常常表达为20%的年利率，因为12/60=1/5=20%）解答方法涉及了2次幂的计算。（22=4，23=8，等等）
The answer is five years at simple interest, as compounding began only once the principal sum had entirely reproduced itself after 60 months had passed. At this rate a mina multiplies fourfold in 10 years, eightfold in 15 years, sixteenfold in 20 years, and so forth. A related problem (VAT 8525) asks how long it will take for one mina to become 64, that is, 26. The answer is 30 years, six times the basic five‑year doubling period (Illustration 1).
答案是，以简单利率来算，5年，因为，复利开始计算时，是在本金在60个月之后完全增殖了自己。以复利来算，那么10年内，1迈纳便翻了4倍，15年，8倍，20年，16倍，如此等等。另一个原文里的问题是：1迈纳花多长时间变为64，即26。答案是30年，6倍于基础翻倍时间（5年）。
The basic idea of interest-bearing debt is one of doubling times. An ancient Egyptian saying that “If wealth is placed where it bears interest, it comes back to you redoubled.”[8] Babylonian agricultural debts at the typical 33 1/3% rate doubled in three years. The Laws of Hammurapi appear to reflect the view that held that when creditors had received interest equal to their original principal – after three years of service – the debt should be deemed to be paid off and the debt bondservants freed.
负息债务的基本观念是倍增观念。一个古埃及人说：“如果财富置于它生息的地方，它翻着倍儿回到你手里。”巴比伦的农业债务利率一般为33 1/3%，三年即翻倍。汉默拉比法典显然反映了这样的观点，它认为，当债权人所收到的利息已等于其本金——三年的利息支付期满后——债务就应当视为偿清，债奴也就自由了。
Babylonians recognized that while debts grew exponentially, the rest of the economy (what today is called the “real” economy) grows less rapidly. Today’s economists have not come to terms with this problem with such clarity. Instead of a conceptual view that calls for a strong ruler or state to maintain equity and to restore economic balance when it is disturbed, today’s general equilibrium models reflect the play of supply and demand in debt-free economies that do not tend to polarize or to generate other structural problems.
巴比伦人认识到，当债务以指数增长，经济的其它部份（今天称为“真实”经济）增长得相对要慢。今天的经济学家从未清醒地认识到这一点。今天的一般均衡模式并不具有一种方案远景，当经济遭受扰乱时，召唤出一个强有力的统治者或政府来维持公正，重建经济的平衡，相反，这种模式反映的是没有极化或者引起结构问题的趋势的无债务经济体中供需的游戏。
Adam Smith grounded such ideology in a Deist religious view of the Lord as having started up the universe and then let it proceed harmoniously by its own laws of motion. But in Babylonia the earning capacity of subsistence rural producers hardly could be reconciled with creditor claims mounting up at the typical 33 1/3 percent rate of interest for agricultural loans (or even at the commercial 20 percent rate). Such charges were unsustainable for economies as a whole. At no time in history has agricultural output grown at sustained rates approaching these levels. In situations where the loan proceeds were used for basic consumption needs, interest charges ate into the cultivator’s modest resources and finally absorbed them in toto. Once the usury process got underway and debtors were called on to pay sums beyond their ability to produce, creditors were enabled to draw the land and other wealth into their own hands.
斯密将这样的思想体系建立在上帝的自然宗教观的基础上，是上帝创造了天地成了物，然后让它们以自己的运动定律和谐地运行着。然而，在巴比伦，靠劳动维系生计的乡村庄稼人几乎不可能达到债主债权的要求，这些乡村贷款以33 1/3%的利率累积（或者20%的商业利率）。这些费用对整个经济体来说，是无法持续的。历史上从来没有农业产出的增长达到这样的水平。在借入资金用于基本消费需要的情况下，利息费用就吃进了庄稼人的有限资财，最终将它们全部吞下。一旦高利贷的进程开始，一旦债务人被迫支付超过他们生产能力的数额，债权人就能够将土地和其它财富纳为己有。
Economic relations were put back in balance by Babylonian rulers acting from outside the economic system. They cancelled agrarian barley debts no less frequently than every thirty years, proclaiming clean slates on the occasion of their ascending to the throne, or as military or economic conditions dictated.[9] Modern economies would rely on income and price adjustments. But prices for most essentials, and most non-commercial incomes in Mesopotamia , were administered or set by custom. There was no idea that the economy by itself might automatically provide such balance.
巴比伦的统治者通过从经济体系外部来发挥作用，将经济关系重置回平衡之中。每隔三十年，或更短时间，他们在登基时，或面对军事和经济状况的客观需求时，宣布“凤凰涅槃”，取消农业上的大麦债务。现代经济学只会依赖收入和价格的调整。然而，在美索不达米亚，生活必需品的价格，以及大多数非商业的收入，是由惯例所支配或确定。在那里，找不到经济本身会自动产生平衡的观念。
Today’s economists have a problem analyzing the relationship between the debt overhead and the capacity to pay. Academic orthodoxy holds that economies can adjust to any volume of debt, given sufficient price and income flexibility to facilitate the transfer of revenue and assets to creditors. What is not recognized is that the resulting economic polarization reduces the economy’s ability to function well. In addition to missing this negative feedback (the proverbial vicious circle), modern economists tend to overlook the fact that interest-bearing debt grows according to its own exponential laws of increase. The economy rarely can keep up.
现代经济学家在分析债务开支和支付能力的关系上出了问题。学院正统学派坚持说，假如价格和收入具有充分的弹性，能够让收入和资产顺利移交债权人，经济就可以调整到任何债务水平。他们所没有认识到的是，因之而来的经济极化降低了经济的运转能力。除了忽略负反馈（众所周知的恶性循环），现代经济学家大抵都忽略了这样的事实：负息债务仅仅是根据它自己的指数增长率来增殖。经济如何跟得上？
If Sumerian and Babylonian students could learn the mathematics of compound interest and the associated exponential growth of debts, it should not be out of reach for modern economists to do so. But today’s economic ideology does not encourage mathematical models based on intersecting financial and physical trends. For it is at such points of intersection that something has to give, that is, a political solution must be imposed from outside the system.
如果苏美尔和巴比伦的学生都能明白复利的数学，明白相关的债务的指数增长，对于现代的经济学家，并非什么难事。然而，今天的经济学意识形态并不鼓励以金融和实体趋势相割断为基础的数学模式。因为，正是在这样的割断点上，必须做点什么，也就是说，必须从体系外部施加一种政治上的解决之道。
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