امروزه علم اقتصاد با گسترش و رشد قابل توجه به صورت یک موضوع ریاضی تبدیل شدهاست. ریاضیات موجود در نوشتههای اقتصادی ۵۰ سال گذشته که به عنوان ریاضیات پیشرفته تلقی شده بودند، اکنون از آن به عنوان زبان معمولی تشریح مباحث اقتصادی یاد میشود. ریاضیات در تمام شاخههای مختلف علم اقتصاد و سایر علوم اجتماعی نقش مهمی را ایفا میکند. امروزه کمتر اقتصاددانی وجود دارد که بتواند خود را از کاربرد ریاضیات در تشریح مباحث و مسائل اقتصادی و به خصوص موضوعات نظری اقتصاد که در حقیقت پایهٔ بررسیهای تجربی اقتصاد سنجی در این رشته را تشکیل میدهند، بی نیاز بداند. بنابراین اقتصاد ریاضی را نمیتوان مانند اقتصاد بخش عمومی ویا اقتصاد بینالملل به عنوان شاخهٔ مستقلی از علم اقتصاد تلقی نمود. بلکه باید آن را به عنوان ابزاری برای تحلیل مسائل و پدیدههای اقتصادی بهشمار آورد.
اهمیت اقتصاد ریاضی[ویرایش]
از دیرباز، دانش ریاضیات امکانات مناسبی را به منظور ارائهٔ تحلیلهای دقیق، توصیف روابط بین پدیدهها و نیز کاهش خطای پیشبینی در اختیار علوم مختلف قرار دادهاست. ماهیت کمی بیشتر متغیرهای اقتصادی در کنار عواملی مانند لزوم برنامهریزی و... سبب توسعهٔ کاربرد ریاضیات در اقتصاد گردیدهاست. سادهسازی و رعایت ایجاز و اختصار در ارائهٔ نظریههای اقتصادی و به خصوص دقت بالای این ابزار موجب شدهاست که اقتصاددانان از ریاضیات به عنوان ابزاری به منظور ارائهٔ نظریههایشان استفاده نمایند. بنابراین اقتصاد ریاضی، مانند اقتصادسنجی شاخهای مستقل در دانش اقتصاد مانند اقتصاد خرد یا اقتصاد کلان محسوب نمیشود بلکه اقتصاد ریاضی یک ابزار تحقیق و یک زبان برای ارائهٔ نظریههای اقتصادی بهشمار میآید.
برای بیان اهمیت نقش ریاضیات در اقتصاد، میتوان به موارد زیر اشاره کرد:
کاربرد ریاضیات محض در نظریههای اقتصادی[ویرایش]
در این ارتباط، کافی است به این مطلب اشاره شود که بیشتر اقتصاددانان مهم و برجستهٔ صد سال گذشته، ریاضی دان بودهاند. آلفرد مارشال، ویلیام استانی جونز، نات ویکسل و جان مینارد کینز از جملهٔ این اقتصاددانان هستند. در حالی که اقتصاددانان دیگری نیز وجود دارند که علم اقتصاد را در تحلیلهای خود با منطق آمیختهاند. از جمله میتوان به جان استوارت میل و ری هارود اشاره کرد. در رابطه با این که چرا زبان ریاضی بر زبان معمول در اقتصاد میتواند مزیت داشته باشد، باید به دلایل زیر اشاره نمود:
الف. اصطلاحات و عبارات ریاضی دقیق بوده و کمتر گمراهکننده اند[ویرایش]
در زبان معمول (نوشتاری)، یک واژه ممکن است معانی و تفاسیر مختلفی داشته باشد که هر یک از این تفاسیر مختصر تفاوتی نیز با دیگری داشته باشد، اما نه آن تفاوتی که بتواند در یک مبحث معین، معنی دقیق کلمه را مشخص نماید. بنابراین علاوه بر ابهامی که در معانی واژهها وجود دارد، باید گفت که هر واژه اغلب اوقات حتی هنگامی که به صورت تخصصی از آن استفاده میشود خالی از بار ارزشی نیست. برای مثال کاربرد واژهٔ تعادل در اقتصاد را در نظر بگیرید. این نکته که اقتصاد در حالت تعادل قرار دار، برای بیشتر مردم این مفهوم را خواهد داشت که اقتصاد در یک شرایط مطلوب قرار دارد. منظور از شرایط مطلوب در حقیقت از نظر کارایی اقتصادی و غیرهاست. در مقابل، «عدم تعادل»، نشان دهندهٔ وضعیت نامطلوب است، در حالی که در عمل چنین نیست. همان طوریکه کینز اشاره کردهاست، این امکان وجود دارد که ابهام کاربرد کلمهٔ «تعادل» در اقتصاد به خاطر این است که این کلمه در سطح وسیع، به صورت ناصحیح به کار برده شدهاست. به عنوان مثال در رابطه با بازارها، این کلمه ممکن است بیانگر یکی از وقوع زیر باشد:
۱. طرحها، برنامهها و اهداف خریداران و فروشندگان هر دو به وقوع بپیوندند.
۲. نیازی به این نیست که برنامهها و اهداف خریداران و فروشندگان تجدید نظر شود.
۳. هیچ نیرویی وجود ندارد که باعث شود قیمت و مقادیر مبادله شده را تغییر دهد.
۴. قیمتها و مقادیر مبادله در طول زمان ثابت هستند.
در حالی که باید گفت معنی تعادل در هر یک از حالات فوق به هیچ وجه بیانگر وضعیت تعادل در حالات دیگر نیست. در زبان ریاضیات، وضعیت تعادل را میتوان توسط یک معادلهای که رابطهٔ دقیق بین مجموعهای از متغیرها را نشان میدهد، بیان نمود و این چنین معادلهای دیگر دارای ابهام نبوده و از هر گونه بار ارزشی نیز مصون است.
ب. زبان ریاضی درک و بیان مباحث پیچیده را آسان میسازد[ویرایش]
در دنیای واقعی، معمولاً هر متغیر اقتصادی در ارتباط یک طرفه یا دو طرفه با یک یا چند متغیر دیگر است. برای مثال سطح قیمتها، در نظریههای اقتصاد کلان تحت تأثیر متغیرهایی مانند مخارج دولت، حجم پول، نرخ ارز و... قرار دارد. بدیهی است تجزیه و تحلیل این روابط، به زبان توصیفی و هم چنین ترسیمی با محدودیتهای زیادی روبرو است. معادلات و الگوهای ریاضی، این امکان را فراهم میکند تا نظریه پرداز یا محقق اقتصاد، روابط بین متغیرها را به نحو ساده و در عین حال دقیق تر بیان نموده یا مطالعه نماید.
ج. بسیاری از متغیرهای اقتصاد ماهیت کمی دارند[ویرایش]
بخش قابل توجهی از متغیرهای اقتصادی، ماهیت کمی دارند. برای مثال متغیرهایی مانند سود، درآمد، هزینهٔ کل، هزینهٔ متوسط و... در اقتصاد خرد یا متغیرهایی مانند درآمد ملی، صادرات، واردات، نرخ تورم در اقتصاد کلان، کمی و قابل اندازهگیری هستند. به همین دلیل، ماهیت کمی این متغیرها، محاسبه و تعیین آنها را ایجاب میکند.
د. یک انسان اقتصادی یک ریاضیدان است[ویرایش]
بیشتر نظریههای اقتصادی بر این فرض بنا شدهاند که افراد و کارگزاران اقتصادی بهطور «عقلایی» رفتار میکنند. انتخاب یک عمل توسط یک فردی که عقلایی رفتار میکند، در حقیقت یک تمرین منطق عملی است. هنگامی که این انتخاب با کمیات و ارقام سر و کار داشته باشد، در این صورت میتوان گفت که این رفتار عقلایی در حقیقت یک تمرین ریاضی است. برای مثال، یک مصرفکننده که به صورت عقلایی رفتار میکند؛ به دنبال حداکثر نمودن رضایت خاطر خود با توجه به درآمد معین و قیمت کالاها است. بهطوریکه او میکوشد تا مقادیری از کالاها را خریداری کند که با توجه به درآمد معین و قیمتهای داده شده، بیشترین مطلوبیت را به دست آورد. بنابراین به علت این که منظور از رفتار یک فرد اقتصادی در حقیقت رفتار فردی است که هم ریاضیدان بوده و هم آشنایی کامل به علم منطق داشته باشد، در این صورت جای تعجب و شگفتی نیست که بیشتر قوانین اقتصادی که بر پایهٔ فرض «عقلایی» بنا شدهاند، اغلب از تفکرات کسانی که در منطق و در ریاضیات تعلیم دیدهاند، سرچشمه گرفتهاست.
کاربرد ریاضیات عملی در تصمیم گیریهای اقتصادی[ویرایش]
روش ریاضیات عملی، اغلب به عنوان روشی به منظور تحلیل روابط بین مقادیر متغیرها و ارزشهای عددی آنها مورد استفاده قرار میگیرد. بیشتر مفاهیم و متغیرهایی که مورد توجه اقتصاددانان است از جمله قیمت، تولید، اشتغال، درآمد، موازنهٔ تجاری و... را میتوان با اعداد اندازهگیری نمود. به همین خاطر است اقتصاددانانی که به مطالعهٔ روابط بین مقادیر اقتصادی میپردازند از ریاضیات عملی استفاده میکنند.
یکی از مزایایی که در رابطه با مطالعهٔ روابطی که بر اساس مقادیر کمی نوشته شدهاند وجود دارد، این است که به هنگامی که رابطهٔ دقیق بین مقادیر مشخص شد، در این صورت میتوان با فرض این که این رابطه در طول زمان ثابت بماند، وضعیت آیندهٔ اقتصاد را نیز پیشبینی نمود. بر اساس همین پیشبینی است که میتوان اهداف اقتصادی در آینده را مورد توجه قرار داده و سپس با ترکیب سیاستهای مختلف، به این اهداف نائل آمد.
برای مثال اگر سیاست اقتصادی و یکی از اهداف اقتصادی این باشد که اشتغال کامل و ثبات در موازنهٔ تجاری هماهنگ با هم وجود داشته باشد، در این صورت رسیدن به این هدف میسر نخواهد بود، مگر آن که اطلاعی از روابط بین متغیرهای مهم اقتصادی با یکدیگر در اختیار داشته باشیم. به عبارت دیگر میبایستی روابط دقیق واردات و صادرات را داشته و علاوه بر آن لازم است سطح درآمدی که در آن اشتغال کامل به وجود میآید، را نیز برآورد کرد. از ترکیب این اطلاعات است که قادر خواهیم بود تا چگونگی رسیدن به هدف فوق را بررسی نماییم. در این مورد به خصوص، اگر فرض کنیم که واردات بهطور مستقیم تابعی از درآمد ملی بوده و صادرات متغیری مستقل است، در این صورت میتوان موازنهٔ تجاری را به صورت روابط زیر بیان نمود: واردات: (Z=f(y صادرات: X=Xavg موازنهٔ تجاری: (X-Z=Xavg-f(y حال اگر فرض کنیم که موازنهٔ تجاری با کسری روبرو باشد و اقتصاد نیز در حالت رکود و با بیکاری مواجه باشد، در این صورت میتوان به این نتیجه رسید که چنانچه بخواهیم به اشتغال کامل برسیم، لازم است تولید ناخالص ملی را افزایش داد تا به تدریج بتوان به هدف اشتغال کامل رسید. اما با افزایش تولید ناخالص ملی از طرف دیگر مشاهده میشود که وضعیت تراز تجاری، با فرض ثابت بودن صادرات؛ بدتر خواهد شد. بنابراین مشاهده میشود که رسیدن به اشتغال کامل ممکن است تحت شرایط فوق، عدم تعادل در تراز تجاری را تشدید نماید و این جاست که هماهنگی سیاستهای مختلف با توجه به پیشبینی متغیرهای مهم برای رسیدن به اهداف اقتصادی با اهمیت است. از این مثال میتوان نتیجه گرفت که با بهکارگیری ریاضیات، به خوبی میتوان مسائل اقتصادی را مورد تحلیل قرار داد.
کاربرد ریاضیات در تحلیل آمارهای اقتصادی[ویرایش]
استفاده از ریاضیات مقدماتی میتواند به صورت قابل توجهی برای خلاصه کردن و تفسیرهای آماری، اقتصادی و اجتماعی سودمند باشد. ریاضیات پیشرفته نیز در نظریههای اقتصادسنجی که هدف آنها تخمین روابط بین متغیرهای مهم اقتصادی است، کاربرد زیادی دارد. برای مثال بیشتر نتایج سرشماری و بررسیهای آماری در ارتباط با تولید، توزیع و مصرف در اقتصاد که در حقیقت وضعیت اقتصادی، مالی و اجتماعی کشور را توجیه مینماید، را میتوان با استفاده از مفاهیم سادهٔ ریاضی، تجزیه و تحلیل نمود. به عبارت دیگر، انجام محاسباتی از قبیل درصد گیری، میانگین و واریانس و آشنایی با ریاضیات مقدماتی میتواند در تحلیل آمارها مفید باشد.
اقتصاد ریاضی در مقایسه با اقتصاد سنجی[ویرایش]
همان طور که پیشتر اشاره شد، اقتصاد ریاضی یک زبان و یک ابزار برای بیان روابط دقیق بین متغیرهای اقتصادی است. به عبارت دیگر، اقتصاد ریاضی ابزاری است که به وسیلهٔ آن میتوان یک نظریهٔ اقتصادی را در قالب روابط ریاضی بیان نمود. اما باید توجه داشت که روابط بین متغیرها در اقتصاد ریاضی، یک رابطهٔ دقیق است. در یک رابطهٔ اقتصاد ریاضی، نوع روابط وعلامت مورد انتظار پارامترها تعیین میشود. اما برآورد مقدار عددی پارامترها در حوزهٔ مباحث اقتصادسنجی قرار میگیرد. به عبارت دیگر در رابطه با کاربرد ریاضیات در تحلیل آمارهای اقتصادی باید گفت که از کاربرد مفاهیم ریاضی با توجه به نظریههای اقتصادی میتوان بر اساس ارقام دورههای گذشته، روابط بین متغیرهای اقتصادی را برآورد نمود. بنابراین مدلهای اقتصادسنجی، بر اساس مدلهای ریاضی شکل میگیرند که مبنای آنها نظریههای اقتصادی هستند.
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. By convention, these applied methods are beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.
Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications.
Broad applications include:
Formal economic modeling began in the 19th century with the use of differential calculus to represent and explain economic behavior, such as utility maximization, an early economic application of mathematical optimization. Economics became more mathematical as a discipline throughout the first half of the 20th century, but introduction of new and generalized techniques in the period around the Second World War, as in game theory, would greatly broaden the use of mathematical formulations in economics.
This rapid systematizing of economics alarmed critics of the discipline as well as some noted economists. John Maynard Keynes, Robert Heilbroner, Friedrich Hayek and others have criticized the broad use of mathematical models for human behavior, arguing that some human choices are irreducible to mathematics.
The use of mathematics in the service of social and economic analysis dates back to the 17th century. Then, mainly in German universities, a style of instruction emerged which dealt specifically with detailed presentation of data as it related to public administration. Gottfried Achenwall lectured in this fashion, coining the term statistics. At the same time, a small group of professors in England established a method of "reasoning by figures upon things relating to government" and referred to this practice as Political Arithmetick. Sir William Petty wrote at length on issues that would later concern economists, such as taxation, Velocity of money and national income, but while his analysis was numerical, he rejected abstract mathematical methodology. Petty's use of detailed numerical data (along with John Graunt) would influence statisticians and economists for some time, even though Petty's works were largely ignored by English scholars.
The mathematization of economics began in earnest in the 19th century. Most of the economic analysis of the time was what would later be called classical economics. Subjects were discussed and dispensed with through algebraic means, but calculus was not used. More importantly, until Johann Heinrich von Thünen's The Isolated State in 1826, economists did not develop explicit and abstract models for behavior in order to apply the tools of mathematics. Thünen's model of farmland use represents the first example of marginal analysis. Thünen's work was largely theoretical, but he also mined empirical data in order to attempt to support his generalizations. In comparison to his contemporaries, Thünen built economic models and tools, rather than applying previous tools to new problems.
Meanwhile, a new cohort of scholars trained in the mathematical methods of the physical sciences gravitated to economics, advocating and applying those methods to their subject, and described today as moving from geometry to mechanics. These included W.S. Jevons who presented paper on a "general mathematical theory of political economy" in 1862, providing an outline for use of the theory of marginal utility in political economy. In 1871, he published The Principles of Political Economy, declaring that the subject as science "must be mathematical simply because it deals with quantities." Jevons expected that only collection of statistics for price and quantities would permit the subject as presented to become an exact science. Others preceded and followed in expanding mathematical representations of economic problems.
Marginalists and the roots of neoclassical economics
Augustin Cournot and Léon Walras built the tools of the discipline axiomatically around utility, arguing that individuals sought to maximize their utility across choices in a way that could be described mathematically. At the time, it was thought that utility was quantifiable, in units known as utils. Cournot, Walras and Francis Ysidro Edgeworth are considered the precursors to modern mathematical economics.
Cournot, a professor of mathematics, developed a mathematical treatment in 1838 for duopoly—a market condition defined by competition between two sellers. This treatment of competition, first published in Researches into the Mathematical Principles of Wealth, is referred to as Cournot duopoly. It is assumed that both sellers had equal access to the market and could produce their goods without cost. Further, it assumed that both goods were homogeneous. Each seller would vary her output based on the output of the other and the market price would be determined by the total quantity supplied. The profit for each firm would be determined by multiplying their output and the per unit Market price. Differentiating the profit function with respect to quantity supplied for each firm left a system of linear equations, the simultaneous solution of which gave the equilibrium quantity, price and profits. Cournot's contributions to the mathematization of economics would be neglected for decades, but eventually influenced many of the marginalists. Cournot's models of duopoly and Oligopoly also represent one of the first formulations of non-cooperative games. Today the solution can be given as a Nash equilibrium but Cournot's work preceded modern game theory by over 100 years.
While Cournot provided a solution for what would later be called partial equilibrium, Léon Walras attempted to formalize discussion of the economy as a whole through a theory of general competitive equilibrium. The behavior of every economic actor would be considered on both the production and consumption side. Walras originally presented four separate models of exchange, each recursively included in the next. The solution of the resulting system of equations (both linear and non-linear) is the general equilibrium. At the time, no general solution could be expressed for a system of arbitrarily many equations, but Walras's attempts produced two famous results in economics. The first is Walras' law and the second is the principle of tâtonnement. Walras' method was considered highly mathematical for the time and Edgeworth commented at length about this fact in his review of Éléments d'économie politique pure (Elements of Pure Economics).
Walras' law was introduced as a theoretical answer to the problem of determining the solutions in general equilibrium. His notation is different from modern notation but can be constructed using more modern summation notation. Walras assumed that in equilibrium, all money would be spent on all goods: every good would be sold at the market price for that good and every buyer would expend their last dollar on a basket of goods. Starting from this assumption, Walras could then show that if there were n markets and n-1 markets cleared (reached equilibrium conditions) that the nth market would clear as well. This is easiest to visualize with two markets (considered in most texts as a market for goods and a market for money). If one of two markets has reached an equilibrium state, no additional goods (or conversely, money) can enter or exit the second market, so it must be in a state of equilibrium as well. Walras used this statement to move toward a proof of existence of solutions to general equilibrium but it is commonly used today to illustrate market clearing in money markets at the undergraduate level.
Tâtonnement (roughly, French for groping toward) was meant to serve as the practical expression of Walrasian general equilibrium. Walras abstracted the marketplace as an auction of goods where the auctioneer would call out prices and market participants would wait until they could each satisfy their personal reservation prices for the quantity desired (remembering here that this is an auction on all goods, so everyone has a reservation price for their desired basket of goods).
Only when all buyers are satisfied with the given market price would transactions occur. The market would "clear" at that price—no surplus or shortage would exist. The word tâtonnement is used to describe the directions the market takes in groping toward equilibrium, settling high or low prices on different goods until a price is agreed upon for all goods. While the process appears dynamic, Walras only presented a static model, as no transactions would occur until all markets were in equilibrium. In practice very few markets operate in this manner.
Francis Ysidro Edgeworth
Edgeworth introduced mathematical elements to Economics explicitly in Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences, published in 1881. He adopted Jeremy Bentham's felicific calculus to economic behavior, allowing the outcome of each decision to be converted into a change in utility. Using this assumption, Edgeworth built a model of exchange on three assumptions: individuals are self-interested, individuals act to maximize utility, and individuals are "free to recontract with another independently of...any third party."
Given two individuals, the set of solutions where the both individuals can maximize utility is described by the contract curve on what is now known as an Edgeworth Box. Technically, the construction of the two-person solution to Edgeworth's problem was not developed graphically until 1924 by Arthur Lyon Bowley. The contract curve of the Edgeworth box (or more generally on any set of solutions to Edgeworth's problem for more actors) is referred to as the core of an economy.
Edgeworth devoted considerable effort to insisting that mathematical proofs were appropriate for all schools of thought in economics. While at the helm of The Economic Journal, he published several articles criticizing the mathematical rigor of rival researchers, including Edwin Robert Anderson Seligman, a noted skeptic of mathematical economics. The articles focused on a back and forth over tax incidence and responses by producers. Edgeworth noticed that a monopoly producing a good that had jointness of supply but not jointness of demand (such as first class and economy on an airplane, if the plane flies, both sets of seats fly with it) might actually lower the price seen by the consumer for one of the two commodities if a tax were applied. Common sense and more traditional, numerical analysis seemed to indicate that this was preposterous. Seligman insisted that the results Edgeworth achieved were a quirk of his mathematical formulation. He suggested that the assumption of a continuous demand function and an infinitesimal change in the tax resulted in the paradoxical predictions. Harold Hotelling later showed that Edgeworth was correct and that the same result (a "diminution of price as a result of the tax") could occur with a discontinuous demand function and large changes in the tax rate.
Modern mathematical economics
From the later-1930s, an array of new mathematical tools from the differential calculus and differential equations, convex sets, and graph theory were deployed to advance economic theory in a way similar to new mathematical methods earlier applied to physics. The process was later described as moving from mechanics to axiomatics.
Vilfredo Pareto analyzed microeconomics by treating decisions by economic actors as attempts to change a given allotment of goods to another, more preferred allotment. Sets of allocations could then be treated as Pareto efficient (Pareto optimal is an equivalent term) when no exchanges could occur between actors that could make at least one individual better off without making any other individual worse off. Pareto's proof is commonly conflated with Walrassian equilibrium or informally ascribed to Adam Smith's Invisible hand hypothesis. Rather, Pareto's statement was the first formal assertion of what would be known as the first fundamental theorem of welfare economics. These models lacked the inequalities of the next generation of mathematical economics.
In the landmark treatise Foundations of Economic Analysis (1947), Paul Samuelson identified a common paradigm and mathematical structure across multiple fields in the subject, building on previous work by Alfred Marshall. Foundations took mathematical concepts from physics and applied them to economic problems. This broad view (for example, comparing Le Chatelier's principle to tâtonnement) drives the fundamental premise of mathematical economics: systems of economic actors may be modeled and their behavior described much like any other system. This extension followed on the work of the marginalists in the previous century and extended it significantly. Samuelson approached the problems of applying individual utility maximization over aggregate groups with comparative statics, which compares two different equilibrium states after an exogenous change in a variable. This and other methods in the book provided the foundation for mathematical economics in the 20th century.
Restricted models of general equilibrium were formulated by John von Neumann in 1937. Unlike earlier versions, the models of von Neumann had inequality constraints. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of Brouwer's fixed point theorem. Von Neumann's model of an expanding economy considered the matrix pencil A - λ B with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation
along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the rate of growth of the economy, which equals the interest rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann. Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices. The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics.
In 1936, the Russian–born economist Wassily Leontief built his model of input-output analysis from the 'material balance' tables constructed by Soviet economists, which themselves followed earlier work by the physiocrats. With his model, which described a system of production and demand processes, Leontief described how changes in demand in one economic sector would influence production in another. In practice, Leontief estimated the coefficients of his simple models, to address economically interesting questions. In production economics, "Leontief technologies" produce outputs using constant proportions of inputs, regardless of the price of inputs, reducing the value of Leontief models for understanding economies but allowing their parameters to be estimated relatively easily. In contrast, the von Neumann model of an expanding economy allows for choice of techniques, but the coefficients must be estimated for each technology.
In mathematics, mathematical optimization (or optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives. In the simplest case, an optimization problem involves maximizing or minimizing a real function by selecting input values of the function and computing the corresponding values of the function. The solution process includes satisfying general necessary and sufficient conditions for optimality. For optimization problems, specialized notation may be used as to the function and its input(s). More generally, optimization includes finding the best available element of some function given a defined domain and may use a variety of different computational optimization techniques.
Economics is closely enough linked to optimization by agents in an economy that an influential definition relatedly describes economics qua science as the "study of human behavior as a relationship between ends and scarce means" with alternative uses. Optimization problems run through modern economics, many with explicit economic or technical constraints. In microeconomics, the utility maximization problem and its dual problem, the expenditure minimization problem for a given level of utility, are economic optimization problems. Theory posits that consumers maximize their utility, subject to their budget constraints and that firms maximize their profits, subject to their production functions, input costs, and market demand.
Economic equilibrium is studied in optimization theory as a key ingredient of economic theorems that in principle could be tested against empirical data. Newer developments have occurred in dynamic programming and modeling optimization with risk and uncertainty, including applications to portfolio theory, the economics of information, and search theory.
Optimality properties for an entire market system may be stated in mathematical terms, as in formulation of the two fundamental theorems of welfare economics and in the Arrow–Debreu model of general equilibrium (also discussed below). More concretely, many problems are amenable to analytical (formulaic) solution. Many others may be sufficiently complex to require numerical methods of solution, aided by software. Still others are complex but tractable enough to allow computable methods of solution, in particular computable general equilibrium models for the entire economy.
Linear and nonlinear programming have profoundly affected microeconomics, which had earlier considered only equality constraints. Many of the mathematical economists who received Nobel Prizes in Economics had conducted notable research using linear programming: Leonid Kantorovich, Leonid Hurwicz, Tjalling Koopmans, Kenneth J. Arrow, Robert Dorfman, Paul Samuelson and Robert Solow. Both Kantorovich and Koopmans acknowledged that George B. Dantzig deserved to share their Nobel Prize for linear programming. Economists who conducted research in nonlinear programming also have won the Nobel prize, notably Ragnar Frisch in addition to Kantorovich, Hurwicz, Koopmans, Arrow, and Samuelson.
Linear programming was developed to aid the allocation of resources in firms and in industries during the 1930s in Russia and during the 1940s in the United States. During the Berlin airlift (1948), linear programming was used to plan the shipment of supplies to prevent Berlin from starving after the Soviet blockade.
Extensions to nonlinear optimization with inequality constraints were achieved in 1951 by Albert W. Tucker and Harold Kuhn, who considered the nonlinear optimization problem:
In allowing inequality constraints, the Kuhn–Tucker approach generalized the classic method of Lagrange multipliers, which (until then) had allowed only equality constraints. The Kuhn–Tucker approach inspired further research on Lagrangian duality, including the treatment of inequality constraints. The duality theory of nonlinear programming is particularly satisfactory when applied to convex minimization problems, which enjoy the convex-analytic duality theory of Fenchel and Rockafellar; this convex duality is particularly strong for polyhedral convex functions, such as those arising in linear programming. Lagrangian duality and convex analysis are used daily in operations research, in the scheduling of power plants, the planning of production schedules for factories, and the routing of airlines (routes, flights, planes, crews).
Variational calculus and optimal control
Economic dynamics allows for changes in economic variables over time, including in dynamic systems. The problem of finding optimal functions for such changes is studied in variational calculus and in optimal control theory. Before the Second World War, Frank Ramsey and Harold Hotelling used the calculus of variations to that end.
Following Richard Bellman's work on dynamic programming and the 1962 English translation of L. Pontryagin et al.'s earlier work, optimal control theory was used more extensively in economics in addressing dynamic problems, especially as to economic growth equilibrium and stability of economic systems, of which a textbook example is optimal consumption and saving. A crucial distinction is between deterministic and stochastic control models. Other applications of optimal control theory include those in finance, inventories, and production for example.
It was in the course of proving of the existence of an optimal equilibrium in his 1937 model of economic growth that John von Neumann introduced functional analytic methods to include topology in economic theory, in particular, fixed-point theory through his generalization of Brouwer's fixed-point theorem. Following von Neumann's program, Kenneth Arrow and Gérard Debreu formulated abstract models of economic equilibria using convex sets and fixed–point theory. In introducing the Arrow–Debreu model in 1954, they proved the existence (but not the uniqueness) of an equilibrium and also proved that every Walras equilibrium is Pareto efficient; in general, equilibria need not be unique. In their models, the ("primal") vector space represented quantities while the "dual" vector space represented prices.
In Russia, the mathematician Leonid Kantorovich developed economic models in partially ordered vector spaces, that emphasized the duality between quantities and prices. Kantorovich renamed prices as "objectively determined valuations" which were abbreviated in Russian as "o. o. o.", alluding to the difficulty of discussing prices in the Soviet Union.
Even in finite dimensions, the concepts of functional analysis have illuminated economic theory, particularly in clarifying the role of prices as normal vectors to a hyperplane supporting a convex set, representing production or consumption possibilities. However, problems of describing optimization over time or under uncertainty require the use of infinite–dimensional function spaces, because agents are choosing among functions or stochastic processes.
Differential decline and rise
John von Neumann's work on functional analysis and topology broke new ground in mathematics and economic theory. It also left advanced mathematical economics with fewer applications of differential calculus. In particular, general equilibrium theorists used general topology, convex geometry, and optimization theory more than differential calculus, because the approach of differential calculus had failed to establish the existence of an equilibrium.
However, the decline of differential calculus should not be exaggerated, because differential calculus has always been used in graduate training and in applications. Moreover, differential calculus has returned to the highest levels of mathematical economics, general equilibrium theory (GET), as practiced by the "GET-set" (the humorous designation due to Jacques H. Drèze). In the 1960s and 1970s, however, Gérard Debreu and Stephen Smale led a revival of the use of differential calculus in mathematical economics. In particular, they were able to prove the existence of a general equilibrium, where earlier writers had failed, because of their novel mathematics: Baire category from general topology and Sard's lemma from differential topology. Other economists associated with the use of differential analysis include Egbert Dierker, Andreu Mas-Colell, and Yves Balasko. These advances have changed the traditional narrative of the history of mathematical economics, following von Neumann, which celebrated the abandonment of differential calculus.
John von Neumann, working with Oskar Morgenstern on the theory of games, broke new mathematical ground in 1944 by extending functional analytic methods related to convex sets and topological fixed-point theory to economic analysis. Their work thereby avoided the traditional differential calculus, for which the maximum–operator did not apply to non-differentiable functions. Continuing von Neumann's work in cooperative game theory, game theorists Lloyd S. Shapley, Martin Shubik, Hervé Moulin, Nimrod Megiddo, Bezalel Peleg influenced economic research in politics and economics. For example, research on the fair prices in cooperative games and fair values for voting games led to changed rules for voting in legislatures and for accounting for the costs in public–works projects. For example, cooperative game theory was used in designing the water distribution system of Southern Sweden and for setting rates for dedicated telephone lines in the USA.
Earlier neoclassical theory had bounded only the range of bargaining outcomes and in special cases, for example bilateral monopoly or along the contract curve of the Edgeworth box. Von Neumann and Morgenstern's results were similarly weak. Following von Neumann's program, however, John Nash used fixed–point theory to prove conditions under which the bargaining problem and noncooperative games can generate a unique equilibrium solution. Noncooperative game theory has been adopted as a fundamental aspect of experimental economics, behavioral economics, information economics, industrial organization, and political economy. It has also given rise to the subject of mechanism design (sometimes called reverse game theory), which has private and public-policy applications as to ways of improving economic efficiency through incentives for information sharing.
In 1994, Nash, John Harsanyi, and Reinhard Selten received the Nobel Memorial Prize in Economic Sciences their work on non–cooperative games. Harsanyi and Selten were awarded for their work on repeated games. Later work extended their results to computational methods of modeling.
Agent-based computational economics
Agent-based computational economics (ACE) as a named field is relatively recent, dating from about the 1990s as to published work. It studies economic processes, including whole economies, as dynamic systems of interacting agents over time. As such, it falls in the paradigm of complex adaptive systems. In corresponding agent-based models, agents are not real people but "computational objects modeled as interacting according to rules" ... "whose micro-level interactions create emergent patterns" in space and time. The rules are formulated to predict behavior and social interactions based on incentives and information. The theoretical assumption of mathematical optimization by agents markets is replaced by the less restrictive postulate of agents with bounded rationality adapting to market forces.
ACE models apply numerical methods of analysis to computer-based simulations of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use. Starting from specified initial conditions, the computational economic system is modeled as evolving over time as its constituent agents repeatedly interact with each other. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of the economy. In contrast to other standard modeling methods, ACE events are driven solely by initial conditions, whether or not equilibria exist or are computationally tractable. ACE modeling, however, includes agent adaptation, autonomy, and learning. It has a similarity to, and overlap with, game theory as an agent-based method for modeling social interactions. Other dimensions of the approach include such standard economic subjects as competition and collaboration, market structure and industrial organization, transaction costs, welfare economics and mechanism design, information and uncertainty, and macroeconomics.
The method is said to benefit from continuing improvements in modeling techniques of computer science and increased computer capabilities. Issues include those common to experimental economics in general and by comparison and to development of a common framework for empirical validation and resolving open questions in agent-based modeling. The ultimate scientific objective of the method has been described as "test[ing] theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher's work building appropriately on the work that has gone before."
Mathematicization of economics
Over the course of the 20th century, articles in "core journals" in economics have been almost exclusively written by economists in academia. As a result, much of the material transmitted in those journals relates to economic theory, and "economic theory itself has been continuously more abstract and mathematical." A subjective assessment of mathematical techniques employed in these core journals showed a decrease in articles that use neither geometric representations nor mathematical notation from 95% in 1892 to 5.3% in 1990. A 2007 survey of ten of the top economic journals finds that only 5.8% of the articles published in 2003 and 2004 both lacked statistical analysis of data and lacked displayed mathematical expressions that were indexed with numbers at the margin of the page.
Between the world wars, advances in mathematical statistics and a cadre of mathematically trained economists led to econometrics, which was the name proposed for the discipline of advancing economics by using mathematics and statistics. Within economics, "econometrics" has often been used for statistical methods in economics, rather than mathematical economics. Statistical econometrics features the application of linear regression and time series analysis to economic data.
Ragnar Frisch coined the word "econometrics" and helped to found both the Econometric Society in 1930 and the journal Econometrica in 1933. A student of Frisch's, Trygve Haavelmo published The Probability Approach in Econometrics in 1944, where he asserted that precise statistical analysis could be used as a tool to validate mathematical theories about economic actors with data from complex sources. This linking of statistical analysis of systems to economic theory was also promulgated by the Cowles Commission (now the Cowles Foundation) throughout the 1930s and 1940s.
The roots of modern econometrics can be traced to the American economist Henry L. Moore. Moore studied agricultural productivity and attempted to fit changing values of productivity for plots of corn and other crops to a curve using different values of elasticity. Moore made several errors in his work, some from his choice of models and some from limitations in his use of mathematics. The accuracy of Moore's models also was limited by the poor data for national accounts in the United States at the time. While his first models of production were static, in 1925 he published a dynamic "moving equilibrium" model designed to explain business cycles—this periodic variation from overcorrection in supply and demand curves is now known as the cobweb model. A more formal derivation of this model was made later by Nicholas Kaldor, who is largely credited for its exposition.
Much of classical economics can be presented in simple geometric terms or elementary mathematical notation. Mathematical economics, however, conventionally makes use of calculus and matrix algebra in economic analysis in order to make powerful claims that would be more difficult without such mathematical tools. These tools are prerequisites for formal study, not only in mathematical economics but in contemporary economic theory in general. Economic problems often involve so many variables that mathematics is the only practical way of attacking and solving them. Alfred Marshall argued that every economic problem which can be quantified, analytically expressed and solved, should be treated by means of mathematical work.
Economics has become increasingly dependent upon mathematical methods and the mathematical tools it employs have become more sophisticated. As a result, mathematics has become considerably more important to professionals in economics and finance. Graduate programs in both economics and finance require strong undergraduate preparation in mathematics for admission and, for this reason, attract an increasingly high number of mathematicians. Applied mathematicians apply mathematical principles to practical problems, such as economic analysis and other economics-related issues, and many economic problems are often defined as integrated into the scope of applied mathematics.
This integration results from the formulation of economic problems as stylized models with clear assumptions and falsifiable predictions. This modeling may be informal or prosaic, as it was in Adam Smith's The Wealth of Nations, or it may be formal, rigorous and mathematical.
Broadly speaking, formal economic models may be classified as stochastic or deterministic and as discrete or continuous. At a practical level, quantitative modeling is applied to many areas of economics and several methodologies have evolved more or less independently of each other.
Example: The effect of a corporate tax cut on wages
The great appeal of mathematical economics is that it brings a degree of rigor to economic thinking, particularly around charged political topics. For example, during the discussion of the efficacy of a corporate tax cut for increasing the wages of workers, a simple mathematical model proved beneficial to understanding the issues at hand.
Where the variables in this equation are:
The standard choice for the production function is the Cobb-Douglas production function:
where is the factor of productivity - assumed to be a constant. A corporate tax cut in this model is equivalent to a tax on capital. With taxes, firms look to maximize:
where is the capital tax rate, is wages per worker, and is the exogenous interest rate. Then the first-order optimality conditions become:
Therefore, the optimality conditions imply that:
Define total taxes . This implies that taxes per worker are:
Then the change in taxes per worker, given the tax rate, is:
To find the change in wages, we differentiate the second optimality condition for the per worker wages to obtain:
Assuming that the interest rate is fixed at , so that , we may differentiate the first optimality condition for the interest rate to find:
For the moment, let's focus only on the static effect of a capital tax cut, so that . If we substitute this equation into equation for wage changes with respect to the tax rate, then we find that:
Therefore, the static effect of a capital tax cut on wages is:
Based on the model, it seems possible that we may achieve a rise in the wage of a worker greater than the amount of the tax cut. But that only considers the static effect, and we know that the dynamic effect must be accounted for. In the dynamic model, we may rewrite the equation for changes in taxes per worker with respect to the tax rate as:
Recalling that , we have that:
Using the Cobb-Douglas production function, we have that:
Therefore, the dynamic effect of a capital tax cut on wages is:
If we take , then the dynamic effect of lowering capital taxes on wages will be even larger than the static effect. Moreover, if there are positive externalities to capital accumulation, the effect of the tax cut on wages would be larger than in the model we just derived. It is important to note that the result is a combination of:
This result showing that, under certain assumptions, a corporate tax cut can boost the wages of workers by more than the lost revenue does not imply that the magnitude is correct. Rather, it suggests a basis for policy analysis that is not grounded in handwaving. If the assumptions are reasonable, then the model is an acceptable approximation of reality; if they are not, then better models should be developed.
CES production function
Now let's assume that instead of the Cobb-Douglas production function we have a more general constant elasticity of substitution (CES) production function:
where ; is the elasticity of substitution between capital and labor. The relevant quantity we want to calculate is , which may be derived as:
Therefore, we may use this to find that:
Therefore, under a general CES model, the dynamic effect of a capital tax cut on wages is:
We recover the Cobb-Douglas solution when . When , which is the case when perfect substitutes exist, we find that - there is no effect of changes in capital taxes on wages. And when , which is the case when perfect complements exist, we find that - a cut in capital taxes increases wages by exactly one dollar.
The Handbook of Mathematical Economics series (Elsevier), currently 4 volumes, distinguishes between mathematical methods in economics, v. 1, Part I, and areas of economics in other volumes where mathematics is employed.
Another source with a similar distinction is The New Palgrave: A Dictionary of Economics (1987, 4 vols., 1,300 subject entries). In it, a "Subject Index" includes mathematical entries under 2 headings (vol. IV, pp. 982–3):
A widely used system in economics that includes mathematical methods on the subject is the JEL classification codes. It originated in the Journal of Economic Literature for classifying new books and articles. The relevant categories are listed below (simplified below to omit "Miscellaneous" and "Other" JEL codes), as reproduced from JEL classification codes#Mathematical and quantitative methods JEL: C Subcategories. The New Palgrave Dictionary of Economics (2008, 2nd ed.) also uses the JEL codes to classify its entries. The corresponding footnotes below have links to abstracts of The New Palgrave Online for each JEL category (10 or fewer per page, similar to Google searches).
Criticisms and defences
Adequacy of mathematics for qualitative and complicated economics
Friedrich Hayek contended that the use of formal techniques projects a scientific exactness that does not appropriately account for informational limitations faced by real economic agents. 
Heilbroner stated that "some/much of economics is not naturally quantitative and therefore does not lend itself to mathematical exposition."
Testing predictions of mathematical economics
Philosopher Karl Popper discussed the scientific standing of economics in the 1940s and 1950s. He argued that mathematical economics suffered from being tautological. In other words, insofar as economics became a mathematical theory, mathematical economics ceased to rely on empirical refutation but rather relied on mathematical proofs and disproof. According to Popper, falsifiable assumptions can be tested by experiment and observation while unfalsifiable assumptions can be explored mathematically for their consequences and for their consistency with other assumptions.
Sharing Popper's concerns about assumptions in economics generally, and not just mathematical economics, Milton Friedman declared that "all assumptions are unrealistic". Friedman proposed judging economic models by their predictive performance rather than by the match between their assumptions and reality.
Mathematical economics as a form of pure mathematics
Defense of mathematical economics
In response to these criticisms, Paul Samuelson argued that mathematics is a language, repeating a thesis of Josiah Willard Gibbs. In economics, the language of mathematics is sometimes necessary for representing substantive problems. Moreover, mathematical economics has led to conceptual advances in economics. In particular, Samuelson gave the example of microeconomics, writing that "few people are ingenious enough to grasp [its] more complex parts... without resorting to the language of mathematics, while most ordinary individuals can do so fairly easily with the aid of mathematics."
Some economists state that mathematical economics deserves support just like other forms of mathematics, particularly its neighbors in mathematical optimization and mathematical statistics and increasingly in theoretical computer science. Mathematical economics and other mathematical sciences have a history in which theoretical advances have regularly contributed to the reform of the more applied branches of economics. In particular, following the program of John von Neumann, game theory now provides the foundations for describing much of applied economics, from statistical decision theory (as "games against nature") and econometrics to general equilibrium theory and industrial organization. In the last decade, with the rise of the internet, mathematical economists and optimization experts and computer scientists have worked on problems of pricing for on-line services --- their contributions using mathematics from cooperative game theory, nondifferentiable optimization, and combinatorial games.
Prominent mathematical economists include, but are not limited to, the following (by century of birth).