Mathematical Economics of Compound Rates of Interest: A Four-Thousand Year Overview Part I
復利的數學經濟學：四千年的回顧 （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.
“未通曉數學者禁入。”這是柏拉圖學園的銘文。像畢達哥拉斯的音調協調，像日月星辰按歷律列張于蒼穹，它的哲學強調了這種抽象的“比例”，使用自然界的數學，揭示了宇宙所蘊含的和諧秩序，因此，理想的社會，亦復如是。然而，在這里，卻幾乎找不到什么經濟學關系的量化分析。雖然希臘和羅馬的經濟顛覆于債務，卻找不到對這種現象進行測量的計算方法，也找不到總產量、分配、其它宏觀經濟手段的測算方法。
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，現代經濟學家的教育包含了很大部份的高等數學，它的使用更多地是以抽象推理，而不是憑借經驗來估算最重要的趨勢。一個世紀以前，John Shield Nicholson評論道：“近來，英國政治經濟學的傳統方法受到更多攻擊，更確切地說，是扭曲”，由于采納了數學手段，將假設和演繹的方法推向了極致，有才干的數學家無不缺乏洞鑒燭明之能事，誤把形式當本質，一堆堆假說聯系起一堆堆事實。擺在我眼前的，似乎數學理論功莫大焉。不過，我斗膽地認為，大部份的它們將不得不被拋棄。它，太不食人間煙火了。
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?
對于當代的經濟學家，數學已經成為科學方法的標志工具。然而，正確地做法是數學化嗎？現代的模型與現象互為表里嗎？或者，遺漏了關鍵機制，這些模型混淆了原因和結果？應用數學，而不考慮如何應用，其本身是科學的嗎？
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.
數學經濟學所偏愛的辦法是在某個環境條件下的一般等式分析。在這個經濟環境中，僅預見到少量的邊際干擾，并沒有看到重要的結構問題或法律發生了改變。邊際分析避免了討論量的躍變。它選擇了一系列非常狹窄的現象（供給、勞動力、原材料成本、利率、收入和需求模式）逐個加以對應，所導出的模型顯示了經濟是如何可能在一個點上找到它的平衡，對于政治的外來干擾，則不加考量。
許多經濟學家接受過微積分和高等數學的訓練，卻并不感到有必要用數量等式來檢測自己的理論。他們傾向于更少地將數學作為一門經驗的計算工具，而更多地是將其作為一門解釋性的語言，或者僅僅是一項科學裝飾，打扮一下他們的“政策處方”。對實際運轉中的財富和收入的內在的極化趨勢，數學經濟學極少進行統計分析。
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.
事實上，經濟體都有失去平衡的趨勢，而數學作為“科學的標志”卻轉移了對這一方面的關注。問題的關鍵是，為了獲得任何給定問題（總是以對原先平衡產生“干擾的因素”的面目出現）的唯一恒定解，一般均衡理論家有著假定遞減回報和遞減邊際效用的傾向，以使得“體系首尾相銜”。一套狹義的變量被選擇，完全忽略了經濟體里相對于資產不斷增長的債務經費，以及相關的利息流。
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.
經濟一邊發展，一邊改變著自己的形態。形態的扭曲是因為內在的金融債權——債券、銀行貸款和其它金融有價證券——的發展趨勢，它們增長得太快，超過了經濟的承受能力，更不用說對它們的清償能力。這些證券的數量傾向于以純數學的自我膨脹的原理來增長，脫離了財富和收入的基本經濟走勢，因此，也脫離了債務人的支付能力。
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.
經濟管理和調節的任務被演繹為：設置一個適宜的利率就可以保持經濟中所有活的部件在平衡中運轉。利率被認為由貨幣供給來控制。一系列計量標準從總信用供給（或者那些同樣的事物：債務證券）中挑選出來代表“貨幣”，然后，這種方法又與商品服務的價格聯系起來，卻與資本資產——債務、股票和房地產——的價格脫離開來。實際上，要查考土地和房地產的價值，目前仍沒有充足的統計數據存在。
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.
這樣的經濟模型孕育出了一種幻覺，即經濟能夠承擔任何數量的債務，而不會改變自身的結構——比如說，它的財富擁有模式（財富所有制形式）。收入和價格的自我平衡被認為能令任何大小的債務經費得到支付。這種方法把債務問題演繹為程度問題：稅賦必須以何種程度征收，以運轉國家債務，商業和消費者必須以何種程度來削減投資和消費，以償付自己的債務和支付這些稅賦。
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.
被排除于這種分析的是這樣的結論：除了將所有權轉移給債權人，許多債務是無法支付的。這種轉移改變了經濟的法律和政治環境，而債權人作為食利者使得勞動力和資本依附于這種經濟的金融機制。
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”).
尋租式的剝削和免費午餐這玩意兒可以完全被置之不理了，而真實世界的經濟學充斥著如何獲得一份免費的午餐。這就是為什么人們努力要成為政策內幕知情者，然而，畢竟人們會認為這種想法超出了經濟學的狹窄邊界。這些邊界被狹隘，恰恰是因為要將那些公認的“問題”僅僅界定為能被數學化的經濟生活的有限部份，實際上，數學化，卻毫不涉及社會結構（“環境”）發生的任何改變。
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，一種特別的數學方法開始決定挑選什么來研究，它僅認識到有單一、確定的數學式解答的問題，它們的解答，由體系分析者叫做負反饋的玩意兒來獲得。如上所述，這種回歸的行為是建立在下降的收入邊際效用的假定上，即一個人掙得越多，它就越不想去再掙。真是幸運，因為大多數模型也假定資本的遞減回報，即它們被投資于下降的利潤率。所以，收入和財富被描述為錐體般逐漸減少，而不是高漲并極化，直到金融崩潰、生態的極限和其它種類的危機時刻降臨。
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.
如果有一種模型承認，當富人以窮者更窮為代價變得更富，當“真實”經濟由一種膨脹的金融資本開支來主導，正反饋會發生；那么，這種模型將描繪出經濟的極化有著一些尚不確定的解決方法。就不相容的趨勢會交叉在一起并迫使一方屈服這個意義上來說，經濟問題本質上是政治性的。這就是真實經濟的運行。但要分析它，就會把經濟學家從他們的假說的回歸的世界中驅趕出來，進入到一個不穩定的世界中，在這種世界中，未來是無法饜足的攫取。這樣一類的研究本身被認為是不科學的（或至少是，非經濟的），恰恰是因為在數學化之前，它必須變得具有政治性。
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.
現代經濟學的數學小宇宙是被創造出來之前，會有一定程度的抗議。30多年前，F. J. Dyson評論說，“數學的直覺知識常常是更保守的，而不是更創新進取的，常常是更束縛性的，而不是更解放思想的。”他引證了Ernst Mach的評論：“數學的力量在于它回避了所有多余的想法，在于它極圓滿地節約了心智的活動”，他擔憂真實世界太多的復雜性可能會被置之不理。最為清楚地揭示這個問題，還是要到貨幣和債務所扮演的經濟角色的討論——或認識的短缺——中尋找。
債務的指數增長
劇作家和小說家要比現代的經濟學家更為關注債務問題。狄更斯、巴爾扎克和他們同時代的人，以及更早時期的英國戲劇，都充滿了債務的形象描述。事實上，納皮爾的對數（1500年發展起來）的最早應用是與莎翁的《一個冬天的故事》一起出現的。故事里的人物Polyxenes華麗地表達了他對受到的9個月好客款待的感激之情，打趣地使用了永遠無法償清的纏身債務這個比喻。要花時間感謝東道主，就會消磨掉主人更多的時間，用掉了更多的好客之情，又欠下了更多的感激之情，引發了永不停歇的人情債。
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.
“沒有負擔”意思是“沒有債務”，“無足輕重的人……站立在富有之地”是1600時期使用的語言，表示對數函數。這里形象化表達的是一份未償清的債務以復利堆積起來，繁殖到它能夠吞下整個王國。
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.”
相比于現代，早些時期的經濟學作家更敏感于債務增殖太大而無法支付的問題。在《國富論》中，斯密論述道：“破產總是大規模債務累積的結果。財政收入要得到解放，如果它曾發生過的話，總是由破產潮產生的，有時是通過公開宣布的破產，雖然時常是由名義支付【1】，但總是由真實的破產所產生。”
【1】通脹后，支付的數額不變，但實際價值卻大大降低了。——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.
債務以復利累積的趨勢解釋了為什么斯密的格言應用得如此之廣。斯密的同時代人，政見不合的英國教會的教長、精算數學師Richard Price詳細地解釋了其間的原理。正是Price首先推廣了復利和簡單利率的區別，后來，這種區別主要與馬爾薩斯的人口理論聯系在了一起。
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:
在十六世紀70年代，Price和斯密正在寫作的年代，英國在美洲的戰爭迫使這人國家深深地陷入了債務。正是由于對這些債務的抗議和反對，兩人敦促英國認可殖民地的自由。關于已經累積如山的債務，Price提出了一個半世紀前就被Nathaniel Gould（英格蘭銀行的董事）期盼的觀點。議會可以將100萬英鎊留存為沉積基金（又意譯為“償債基金”），每年的股息再投資進去，以復利來積累，直到這筆基金增長到足以付清全部債務。Price的《關于國家債務向公眾的呼吁》描述了錢以復利增長的魔力。
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.
帶有復利的貨幣的增長，首先是緩慢的。不過，因為增長的速率是持續增加的，在某時，它就會增加得如此迅速，以至于所有的人類想像力都變得相形見拙。一便士，在救世主的生日生產出來，以5%的復利來計算，則在此時之前，就已經增長到從1億5千萬個地球上都不能獲得的金額——以黃金實體來說。然而，如果以簡單利率來計算，在同樣的時間里，它將相當于差不多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《繼承支付的觀察》中詳盡說明了這個概念。這本書最先在1769年出版，到1803年，一共經歷了六個版本。“一先令，在救世主的生日以復利6%發行出來……將增長到整個太陽系都不可能盛得下的金額，姑且假定它是一個直徑等于土星軌道的球體。”他推論道：“所以，一個國家永遠不必置于麻煩之中，因為，只需最小的儲蓄，它就可以在利息所需要的時間內，償清最巨大的債務。”
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資的貨幣的指數增長，將紅利再投資為新的儲蓄，不斷增加了本金量。他沒有認清的是，在歷史上，從來沒有任何經濟體還債是這樣把一便士變成土星軌道大小的黃金實體球。所以，馬克思打趣Price《Grundrisse》筆記中的計算，認為沒有一個社會的生產力能夠維持利息債權的這種復利增長。“善良的普賴斯只不過被幾何級數所得出的龐大數量弄得眼花目眩……他根本不考慮再生產和勞動條件的限制，把資本看作是自我運行的東西，僅僅只是一個自我增長的數字，”主觀地看待增長公式：
盈余=資本×（1+利息率）n
經濟學家曾估算過，基督出生后的兩千年里，歐洲經濟以每年2%的復利增長，遠小于利率曾有的水平。難怪斯密在歷史上沒有發現一個國家償清了它的公共債務。在《國富論》里，他提到，英國的稅收已經成為一項支付資金，不是（償付）曾經借來的錢的資金，而是（償付）利息……。沉積資金的建立，表面上是償清債務，但它卻并不奏效，因為政府總是又再借入這筆錢。斯密斷定，這種基金的用處只不過是“減輕新債務的收縮這個麻煩。它是一種隨時方便取用的補充基金，為其它未定的——國家在緊急關頭籌錢時所需要的——資金提供幫助。”
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.
長久以來，幾何和數學增長率的鮮明對比已在人們的心目中牢固樹立，但并不是通過儲蓄和債務在利率上翻倍再翻倍，而是通過馬爾薩斯關于人口相對于食物供給的增長。馬爾薩斯在1798年首次發表了他的理論，距離Price提出沉積基金論點時，已過了30年。拾起同儕教長的數學的形象化表述，馬爾薩斯斷言人口傾向于以復利方式“幾何級數”增長，但會被生活資料僅能以“算術”率——即簡單利率——增長這個實事所遏制。
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.
馬爾薩斯并沒有預見到：當收入上升時，歷史上的生育率卻是逐漸變小的。隨著時間的推移，農業和采礦業技術突破提高了這些部門的生產率，甚至比制造業所發生的改變更為迅速，這樣，食物和其它消費品的增加比人口的增加更為迅速。
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.
雖然馬爾薩斯的人口論一直受到駁斥，但Price所進行的金融的分析卻仍然完好如初。真正幾何級數增長的是經濟的金融債權——公債借款、抵押和銀行貸款，它們的利息費用在整個歷史上的累積，傾向于超過社會財富創造力。
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，人們會想法利用復利原理，將儲蓄在一段較長時間內累積。1800年，一個叫Thelluson的先生大人就建立了一家信托基金，將它的收入儲蓄100年。到期時，這份信托基金將由子孫后輩間分享。據估算，他的60萬英鎊資產每年將產出4500鎊（7.5%），最終創造出1900萬的價值，即大約原遺產的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.
事實卻不如人所料，Thelluson的遺囑在一場持續62年的官司（從他1797年死亡到1859年）中受到質疑。到付清律師費后，“才發現這份財產讓高昂的法律費用蠶食殆盡，以至于實際的繼承金額并不比遺囑人最初的遺贈數目高出多少。”與此同時，在William Pitt的領導下，政府計算出，以4%的復利，一個世紀過去，這家信托基金將把國家的全部債務納為己有。一些法律，如那時的Thelluson法令，迅速地舉手通過，將這種信托基金限制在21年的時期之內。
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.”
正統的學院派極少承認由債務支出的指數增長引起的問題。正是這些學院派使得政府的政策不再將處理這個問題作為必須，只是專注于，如果儲蓄和債務的金融過度生長不會每隔十年左右就翻倍，這樣的世界就會生存下來。雖然它們在過去一個世紀已經反反復復的翻倍增殖，于是，負息債務的累積所引起的結構性問題，就留給了非主流的作家們去探討。例如，社會主義者Proudhon（《財產是什么》）提到：“如果人們平等地聯合起來，將財產的專有權給予他們中的一個成員，如果這個業主大發慈悲，將100法郎的一筆錢置于復利之下，600年后，償還給他的第24位繼承人——這筆100法郎的金額，如果利率為5%，將會變成107,854,010,777,600法郎，這是法國資本的2696倍，估計為40億法郎（50年前），或者全球所有流動和固定資產總價值的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.
馬克思：債務的金融機制的分析
19世紀對金融資本最深刻的分析屬于馬克思。在《資本論》第三卷和《剩余價值理論》第三卷（它們都是馬克思逝世后出版）中，馬克思指出高金融是建立在“想像的”和“虛構的”資本基礎上。它是虛構的，因為它由有形資本的債權組成，而不是形成直接的生產資料。鑒于這些金融債權代表一種金融開支，其利息費用侵蝕了工業利潤，馬克思（資本論III，461）稱它們為“資本的空無形式”。工業利潤是通過雇傭工人生產待出售的商品和提供服務來積極地掙得——馬克思用M-C-M’這個公式來概括，消費貨幣以生產商品，商品出售獲取更多的貨幣。但是，負息金融資本的增長，其特征是脫離實體的公式M-M’，僅僅是從錢本身來賺錢，也就是說，是一種閹割了實體的方式。【1】
【1】這關系到馬克思的價值論。在生產過程中，工人加入了自己的勞動價值，產品增值了。從整個社會的宏觀經濟來看，商品的使用價值增大了，增多了。在M-M’中，貨幣本身不會增值，因為沒有人的勞動價值加入其中。為什么M能變成M’，唯一的結論是：作為一種（金融）體系或制度，或者說社會和經濟的結構，通過侵蝕（eat into）進入真實經濟的利潤來獲取，它們是“無酬勞動”。由此可見，在學習經濟學時，社會的結構、制度和體系是十分重要的因素。馬克思說過：“理解一個時代的歷史秘密必須到那個時代的經濟學中尋找答案”。同時我們會進一步問：它們是如何形成的？或者具體的說，每一種結構、制度和體系是如何必然產生它的極化現象？這就把令人眼花繚亂的經濟學化為了簡單的問題。赫德森的高明之處，就是抓住的經濟歷史分析的任督二脈：債務和極化（貧富懸殊）。
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.
馬克思闡明了金融資本是如何傾向于維護它對有形資本的統治，尤其是在貨幣危機和緊隨其后的取消抵押品贖回（潮）中。要說明這些高利貸資本的無情的威力，以及它厘定的償債服務中是如何趨向于超過債務人的支付能力，馬克思（資本論III：463）引用了馬丁·路德，路德將其形象地類比為怪獸Cacus，也引用了Price復利魔力的計算。不管人們愿不愿意，金融債權的數量都以數學原理的方式增長，漠然將經濟的實際生產力遠遠拋在身后。
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.”
不過，馬克思在分析金融資本以復利的幾何級數增長的趨勢后，立即放下了這個主題。他相信金融資本正在完全的附屬于工業資本機制。他推演說：“在工業資本產生過程中，它必須使這些形式從屬于自己，并把它們轉化為自己派生的或特殊的功能。工業資本在它形成和發展的時期碰到了這些更古老的形式。它遭逢它們時，恰如我們遇到我們的先輩……而不是我們遇到兒時的我們。”工業資本的使命是調集金融資本以為自己的經濟擴張提供資金。馬克思推斷說：“在資本主義生產方式發展成熟，并成為生產的主導模式，生息資本便由工業資本主宰，商業資本僅僅是工業資本的一種形式，在流通過程派生而出。”
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，經濟的支付方式將能夠跟得上負息債權的指數增長。同時，過剩的金融資本——儲蓄及其對應的債務——會通過貨幣危機來抹除。馬克思超越了其它的經濟學家，對這個問題給予了重視，但是，他將金融資本視為附屬角色。因此，他分享了19世紀法國烏托邦社會主義者傅立葉的樂觀主義，分享了圣·西蒙主義者——工業的進步將更有效率地調動儲蓄，可以解決債務問題——的樂觀主義。
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.
德國的歷史學派，如Roscher等人指出，隨著文明的進展，利率趨向于穩定的下降；至少自從中世紀以來，一直在下降。信貸法正在變得更人性化，債務人監獄從整個歐洲淡出，而更為仁慈的破產法使得債務人隨著債務清算而可以重新起步。最重要的是，歐洲和北美的公共債務是在相對沒有戰爭的1815-1914年之間一步一步被償清的。貸款用于重型運輸、工業、采礦業和建筑業的融資。一時之間，經濟的債務負擔似乎很可能夠自動被償清。廣泛的共識是，債務問題通過指定的社會生產性信用體系而自我愈合。
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.
細心的觀察者可以蹤跡到這些有利的趨勢是如何讓步給最近幾十年所發生的債務的過度增長。金融資本內在動力的趨勢勝過了工業發展的內在動力。實際上，甚至在維多利亞時代，當金融君主超越工業首腦，金融資本是在吞并工業資本。
Flürscheim的金融批評
馬克思和他的社會主義追隨者將抨擊主要集中在工業及其利潤上，而不是那些拿走利息的金融家。Henry George吸引了許多追隨者，他將焦點集中在食利者身上，土地擁有者收割地租、拿走食利收入，其致富是以社會的損失為代價，可是，他們的租的所得收入卻顯示了經濟的進步。George的一個歐洲追隨者，Michael Flürscheim，寫了幾本書，其中一本主要關注負息債務引起的問題。（在《經濟迷宮管窺》中）他寫道：“確實，雇主是海綿，吮吸了利潤，勞工產品的大部份的價值（馬克思稱它為剩余價值Mehrwerth）竟然全讓渡給了租和利息的主子，還有中間商，這些人一起從勞工身上盡可能迅速地擠出利潤，留給勞工的只是一點點血汗錢。更糟的是，他們拿走了勞工充分發揮生產潛力的能力。”于是，他建議勞工和資本應該聯合起來，對“真正的敵人”，食利者，發起進攻。
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.
要根據事實依據做這事兒，就得區分金融資本與有形資本。Flürscheim指出（p. 347）“貢賦權資本構成了世界資本的主要部份。”他解釋說：“當演講者和作者不得不回答社會主義者對資本的抨擊，即資本是勞工的壓迫者時，他就會指出正統經濟學叫做資本的事物，說起我們的非凡進步是歸因于生產資料和分配方式的提高和改善，然而，他的對手想起的卻是政府債券、土地壟斷權、采礦權、票據交換所里各種各樣的收益（貢賦）權……根本不在正統的資本定義中，可是，它們代表著人們使用資本這個詞時腦海浮現的東西。” 關注這個對象的Flürscheim幾乎是孤獨的。
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詳細地解釋道：“勞動工具和方式的所有運用和改善、最嚴謹的經濟、最苛刻的自我犧牲，都無力與置于復利之上的資本所擁有的自我增殖能力相匹敵。它們根本無法滿足它的要求。”為了形象地說明這一點，他創作了下面的寓言，說明了實際運轉的機制。人類被趕出伊甸園許多年代后，上帝告訴他們“要用辛勤的汗水來掙得面包。于是派了一個充滿愛心的天使，天使承擔著（為人類）減輕負擔的任務。它的名字叫創造精靈。他從教人們如何制造有用的工具開始了他的工作”，然后教人們馴養動物、利用水力和風力、火和蒸氣動力來運轉機器。
“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.
“人類許多年代的夢想，從來不曾希望成真的黃金時代似乎最終降臨了。不再有苦惱，不再有千辛萬苦，只有財富運用于滿足人們的需要，在過去，這是最富有的人都不曾夢想過的。”然而，“忌妒的幽靈，那個墮落的天使、曾以蛇的模樣出現，唆使人類犯下罪惡，驅除于伊甸園的撒旦”眼看著他的王國將從此告終，不禁妒火中燒，惱羞成怒。
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?’”
在人類的愚蠢中，有個小搗蛋鬼，叫利息，設法吸引撒旦的注意。“利息，你怎么啦？”撒旦問這個活潑鬼，“你似乎不像你的同伴那樣一蹶不振啊？”
“‘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?’”
“為什么我要像他們，我的主子？”這個精靈回答說，“我不是你最賞識的戰士嗎？憑借你的威儀，我不是一直節節勝利嗎？”
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.
撒旦面帶愁容，回答道：“哎呀，你不是創新精靈的對手啊。”然而，這個小搗蛋鬼自告奮勇地要在決斗中展示自己的勇氣和技藝。
“‘You little imp! Fight the powerful angel who is defeating all my army?’ laughed Satan.”
“你這個調皮鬼！竟然要與打得我潰不成軍的強大天使戰斗！”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，我去單挑，當然，請你允許我帶上我的兒子，復利，幫我的忙。”關于技術這個小妖精，他解釋道：“它們是人類的福祉之源，不，我會讓它們成為不可言狀的痛苦的締造者——比出自你手的痛苦還要痛苦”。于是“撒旦讓它自行其是。巨人的戰爭爆發了。”
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.
戰斗伊始，天使大笑，因為，雖然20個方格過去了，但他的軍隊并沒有明顯、能夠察覺的減少。魔鬼利率一聲不吭，專心關照自己的事務，很快在以后每一次獲取的方格中，讓軍隊翻了倍。到了第三十個方時，天使的笑嘎然而止，剎時明白：他失敗了。
‘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，”他沮喪地嘆息，“我因為虛榮自大而受到處罰。再和你斗，已毫無用處。魔鬼利息比創新精靈更強大。我是你的奴隸了，我的陛下，你發令吧。”
‘I am the only servant of my great master,’ dryly replied the demon. ‘Here I see him coming. He will give you his orders.’
“我只是我偉大的主人的仆人”，這個魔鬼冷冰冰地回答。“他很快就來，你靜候他的命令吧。”
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。他命令天使帶領其部隊繼續工作；運用了最大的努力，部隊不斷地強大，于是，人類——可憐他們并不知道他們曾反抗的敵人的天性——眼看著新的部隊不斷集結，始終懷有最后成功的希望。然而，它們剛一集結，魔鬼利息立馬就生長出更多的軍隊，俘虜了這些新生力量，并奴役他們，同時，并不讓它們裨益于人類，而是將它們變成更牢靠的奴役鎖鏈，沉沉地套在人類的脖梁上。
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.”
使國外驚奇的是，這種連續翻倍“產生出的數量遠遠超過了整個王國的寶藏。資本一直用于玩弄勞動者的，正是這種棋盤游戲。“復利的非凡增長會很快吞掉產品、資本、土地、甚至勞動者。”
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的推斷如是質問：“復利是什么？除了資本收益的再投資，還是別的什么東西呢？”對這個故事，他補充說，“有人給拿破侖看一張利息表，他沉思片刻，說道：‘此中關竅令我納悶，這個利息怪物居然還沒吞掉全人類。’如果破產和革命不曾作為抗毒劑，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.
這個問題自有其淵源，可以追溯到四千年前。令人吃驚的是，古代經濟體處理這個問題所具有的清晰思辯色彩，它們所采用的處理方式比今天所寵愛的方式更直截坦率。
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.
巴比倫人認識到現代經濟學家未曾認識到的
在Heilbronn評論集中，我曾討論過將文明的經濟走向追溯至蘇美爾和巴比倫。正是在古典時期的兩千年前的這個紀元中，現代經濟關系的基本元素第一次出現。這些元素包括負息債務和采用什么方法來處理其蔓延時所引起的問題。
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]
數學在書吏的培訓中有著重要的作用。這并不奇怪，因為最早的楔形文字書寫用于經濟的賬目記錄，是在公元前3500年。公元前3000年，書吏在數學方面的培訓包括如人力分配問題（例如，需要多少人挖一條水渠或多少人生產一定數量的磚）、地面測算所需要的技巧（包括正方形和環形的幾何分析）、天體的計算、二次方程等。同時，書吏也被培訓于計算畜群的預期增長和有息投資的指數增長。
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]
利息的增長并沒有反映出經濟的生產率、利潤率或一般支付能力，本質上，它是一種數學的現象。為了計算的方便，正常的商業利率融入了六十進制的度量衡體系之中。利息是在每邁納每月1謝克爾的利率上增長的，也就是說，“單分數”1/60。許多世紀（實際上，是千年）過去了，這個利率始終不變，每年產生出12/60，五年則60/60。最初的本金在五年里繁殖出自體一次，則復合率每五年出現一次。在希臘、羅馬和拜占庭，利率同樣是建立在方便計算的基礎之上，用本地的重量、尺度和數學的系統來進行計算。
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.
這些利率并非以不斷變化的經濟條件為基礎，或者說，并不反應出改變的經濟條件，這樣，出現支付問題就不可避免。當然，今天的債務問題也會不斷發展，但是，當代的理論卻堅持：經濟能調節到任何債務費用的規定水平。巴比倫人則沒有這樣的臆斷。他們的學生習題顯示他們認識到：畜群，例如，增長的速率要比債務的增長（以每年20%累積，更不要說農業利率一般在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.
依據這些習題，我想提一個乍看之下似乎不合常規的建議。相比今天的經濟思想，公元2000年前的美索不達米亞經濟思想立足于一種更為現實的數學根基，至少巴比倫人認識到：隨著時間的流逝，債務開支會變得越來越強人所難，超過了支付的能力，并以資產所有權集中在債權人手里而達到其頂點。
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]
傳授給將來要做書吏的學生（他們幾乎都從事于廟堂和宮廷記賬）的知識之一是：計算以某一利息借出的投資（資本）是以怎樣的速率翻倍。一個例題出現在柏林的楔形文字原本中：在正常的商業利率為每月1/60（即每邁納1謝克爾）時，1邁納的銀要花多少時間來翻倍？（這常常表達為20%的年利率，因為12/60=1/5=20%）解答方法涉及了2次冪的計算。（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，以簡單利率來算，5年，因為，復利開始計算時，是在本金在60個月之后完全增殖了自己。以復利來算，那么10年內，1邁納便翻了4倍，15年，8倍，20年，16倍，如此等等。另一個原文里的問題是：1邁納花多長時間變為64，即26。答案是30年，6倍于基礎翻倍時間（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.
負息債務的基本觀念是倍增觀念。一個古埃及人說：“如果財富置于它生息的地方，它翻著倍兒回到你手里。”巴比倫的農業債務利率一般為33 1/3%，三年即翻倍。漢默拉比法典顯然反映了這樣的觀點，它認為，當債權人所收到的利息已等于其本金——三年的利息支付期滿后——債務就應當視為償清，債奴也就自由了。
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.
巴比倫人認識到，當債務以指數增長，經濟的其它部份（今天稱為“真實”經濟）增長得相對要慢。今天的經濟學家從未清醒地認識到這一點。今天的一般均衡模式并不具有一種方案遠景，當經濟遭受擾亂時，召喚出一個強有力的統治者或政府來維持公正，重建經濟的平衡，相反，這種模式反映的是沒有極化或者引起結構問題的趨勢的無債務經濟體中供需的游戲。
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.
斯密將這樣的思想體系建立在上帝的自然宗教觀的基礎上，是上帝創造了天地成了物，然后讓它們以自己的運動定律和諧地運行著。然而，在巴比倫，靠勞動維系生計的鄉村莊稼人幾乎不可能達到債主債權的要求，這些鄉村貸款以33 1/3%的利率累積（或者20%的商業利率）。這些費用對整個經濟體來說，是無法持續的。歷史上從來沒有農業產出的增長達到這樣的水平。在借入資金用于基本消費需要的情況下，利息費用就吃進了莊稼人的有限資財，最終將它們全部吞下。一旦高利貸的進程開始，一旦債務人被迫支付超過他們生產能力的數額，債權人就能夠將土地和其它財富納為己有。
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.
巴比倫的統治者通過從經濟體系外部來發揮作用，將經濟關系重置回平衡之中。每隔三十年，或更短時間，他們在登基時，或面對軍事和經濟狀況的客觀需求時，宣布“鳳凰涅槃”，取消農業上的大麥債務。現代經濟學只會依賴收入和價格的調整。然而，在美索不達米亞，生活必需品的價格，以及大多數非商業的收入，是由慣例所支配或確定。在那里，找不到經濟本身會自動產生平衡的觀念。
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.
現代經濟學家在分析債務開支和支付能力的關系上出了問題。學院正統學派堅持說，假如價格和收入具有充分的彈性，能夠讓收入和資產順利移交債權人，經濟就可以調整到任何債務水平。他們所沒有認識到的是，因之而來的經濟極化降低了經濟的運轉能力。除了忽略負反饋（眾所周知的惡性循環），現代經濟學家大抵都忽略了這樣的事實：負息債務僅僅是根據它自己的指數增長率來增殖。經濟如何跟得上？
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.
如果蘇美爾和巴比倫的學生都能明白復利的數學，明白相關的債務的指數增長，對于現代的經濟學家，并非什么難事。然而，今天的經濟學意識形態并不鼓勵以金融和實體趨勢相割斷為基礎的數學模式。因為，正是在這樣的割斷點上，必須做點什么，也就是說，必須從體系外部施加一種政治上的解決之道。
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