Foundations of Economic Analysis
Encyclopedia
Foundations of Economic Analysis is a book by Paul A. Samuelson published in 1947 (Enlarged ed., 1983) by Harvard University Press
. It sought to demonstrate a common mathematical structure underlying multiple branches of economics from two basic principles: maximizing behavior of agents
(such as of utility by consumers and profits
by firms) and stability of equilibrium
as to economic systems (such as markets or economies). Among other contributions, it advanced the theory of index numbers and generalized welfare economics
. It is especially known for definitively stating and formalizing qualitative
and quantitative versions of the "comparative statics
" method for calculating how a change in any parameter
(say, a change in tax rates) affects an economic system. One of its key insights about comparative statics, called the correspondence principle, states that stability of equilibrium implies testable predictions about how the equilibrium changes when parameters are changed.
Its other stated purpose (p. 3) is to show how operationally meaningful
theorems can be described with a small number of analogous methods. Thus, "a general theory of economic theories" (1983, p. xxvi).
One unifying theme, on the striking formal similarities of analysis in seemingly diverse fields, occurred only in the course of writing on them—from consumer's behavior and production economics of the firm to international trade, business cycles, and income analysis. It dawned on the author that he was prodigal "in proving essentially the same theorems" over and over. His failure of initial intuition, so he suggests, might be less surprising in light of the few economic writings then extant concerned with formulating meaningful theorems – hypotheses about empirical data—that could conceivably be refuted
by empirical data (pp. 3–5).
Samuelson (pp. 5, 21–24) finds three sources of meaningful theorems sufficient to illuminate his purposes:
Part I conjectures that meaningful theorems for economic units and for their respective aggregates are almost all derivable from general conditions of equilibrium. The equilibrium conditions can in turn be stated as maximization conditions. So, meaningful theorems reduce to maximization conditions. The calculus of the relations is at a high level of abstraction but with the advantage of numerous applications. Finally, Part I illustrates that there are meaningful theorems in economics, which apply to diverse fields.
Part II concentrates on aggregation of economic units into equilibrium of the system. But the symmetry conditions required for direct maximization of the system, whether a market or even the simplest model of the business cycle, are lacking, in contrast to an economic unit or its corresponding aggregate. What can be hypothetically derived (or rejected in some cases) is a stable equilibrium of the system. (This is an equilibrium of the system such that, if a variable disturbs equilibrium, the system converges to equilibrium.) Stability of equilibrium is proposed as the principal source of operationally meaningful theorems for economic systems (p. 5).
Analogies from physics (and biology) are conspicuous, such as the Le Chatelier principle
and correspondence principle
, but they are given a nontrivially generalized formulation and application. They and mathematical constructions, such as Lagrangian multipliers
, are given an operational economic interpretation. The generalized Le Chatelier principle is for a maximum condition of equilibrium: where all unknowns of the function are independently variable, auxiliary constraints ("just-binding" in leaving initial equilibrium unchanged) reduce the response to a parameter change. Thus, factor-demand and commodity-supply elasticities
are hypothesized to be lower in the short run than in the long run because of the fixed-cost constraint in the short run. In the course of analysis, comparative statics
, changes in equilibrium of the system that result from a parameter
change of the system, is formalized and most clearly stated (Kehoe, 1987, p. 517). The correspondence principle is that the stability of equilibrium for a system (such as a market or economy) implies meaningful theorems in comparative statics. Alternatively, the hypothesis of stability imposes directional restrictions on the movement of the system (Samuelson, pp. 258, 5). The correspondence is between comparative statics
and the dynamics implied by stability of equilibrium.
Chapter VIII on welfare economics
is described as an attempt "to give a brief but fairly complete survey of the whole field of welfare economics" (p. 252). This Samuelson does in 51 pages, including his exposition of what became known as the Bergson-Samuelson social welfare function. Theorems derived in welfare economics, he notes, are deductive implications of assumptions that are not refutable, thus not meaningful in a certain sense. Still, the social welfare function can represent any index (cardinal
or not) of the economic measures of any logically possible ethical belief system that is required to order any (hypothetically) feasible social configurations as "better than", "worse than", or "indifferent to" each other (p. 221). It also definitively elucidates the notion of Pareto optimality and the "germ of truth in Adam Smith's doctrine of the Invisible Hand
" (Samuelson, 1983, p. xxiv; Fischer, 1987, p. 236).
The final pages of the book (pp. 354–55) outline possible directions analytical methods might take, including for example models that show how:
Samuelson closes by expressing hope in the future use of comparative dynamics to:
Harvard University Press
Harvard University Press is a publishing house established on January 13, 1913, as a division of Harvard University, and focused on academic publishing. In 2005, it published 220 new titles. It is a member of the Association of American University Presses. Its current director is William P...
. It sought to demonstrate a common mathematical structure underlying multiple branches of economics from two basic principles: maximizing behavior of agents
Agent (economics)
In economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....
(such as of utility by consumers and profits
Profit maximization
In economics, profit maximization is the process by which a firm determines the price and output level that returns the greatest profit. There are several approaches to this problem...
by firms) and stability of equilibrium
Economic equilibrium
In economics, economic equilibrium is a state of the world where economic forces are balanced and in the absence of external influences the values of economic variables will not change. It is the point at which quantity demanded and quantity supplied are equal...
as to economic systems (such as markets or economies). Among other contributions, it advanced the theory of index numbers and generalized welfare economics
Welfare economics
Welfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...
. It is especially known for definitively stating and formalizing qualitative
Qualitative economics
Qualitative economics refers to representation and analysis of information about the direction of change in some economic variable as related to change of some other economic variable...
and quantitative versions of the "comparative statics
Comparative statics
In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter....
" method for calculating how a change in any parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
(say, a change in tax rates) affects an economic system. One of its key insights about comparative statics, called the correspondence principle, states that stability of equilibrium implies testable predictions about how the equilibrium changes when parameters are changed.
Introduction
The front page quotes the motto of J. Willard Gibbs: "Mathematics is a language." The book begins with this exacting statement:- The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features. This fundamental principle of generalization by abstraction was enunciated by the eminent American mathematician E. H. MooreE. H. MooreEliakim Hastings Moore was an American mathematician.-Life:Moore, the son of a Methodist minister and grandson of US Congressman Eliakim H. Moore, discovered mathematics through a summer job at the Cincinnati Observatory while in high school. He learned mathematics at Yale University, where he was...
more than thirty years ago. It is the purpose of the pages that follow to work out its implications for theoretical and applied economics.
Its other stated purpose (p. 3) is to show how operationally meaningful
Falsifiability
Falsifiability or refutability of an assertion, hypothesis or theory is the logical possibility that it can be contradicted by an observation or the outcome of a physical experiment...
theorems can be described with a small number of analogous methods. Thus, "a general theory of economic theories" (1983, p. xxvi).
Topical outline
The body of the book is 353 pages. Topics and applications covered (all in terms of theory) include the following.- Part I
- introduction
- equilibrium systemsGeneral equilibriumGeneral equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...
(such as for a marketSupply and demandSupply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...
or economy) - maximizing behavior (such as to profitsProfit maximizationIn economics, profit maximization is the process by which a firm determines the price and output level that returns the greatest profit. There are several approaches to this problem...
by a firm and utilityConsumer theoryConsumer choice is a theory of microeconomics that relates preferences for consumption goods and services to consumption expenditures and ultimately to consumer demand curves. The link between personal preferences, consumption, and the demand curve is one of the most closely studied relations in...
by a consumer) in the calculus
- sales-tax increase on equilibrium for a firm
- comparative staticsComparative staticsIn economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter....
(changes in prices and quantitiesSupply and demandSupply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...
and other equilibrium variables when underlying conditions change) - cost and productionIsoquantIn economics, an isoquant is a contour line drawn through the set of points at which the same quantity of output is produced while changing the quantities of two or more inputs...
- consumer's behaviorConsumer theoryConsumer choice is a theory of microeconomics that relates preferences for consumption goods and services to consumption expenditures and ultimately to consumer demand curves. The link between personal preferences, consumption, and the demand curve is one of the most closely studied relations in...
- transformations, elasticitiesElasticity (economics)In economics, elasticity is the measurement of how changing one economic variable affects others. For example:* "If I lower the price of my product, how much more will I sell?"* "If I raise the price, how much less will I sell?"...
, composite commoditiesComposite goodIn economics, demand for a good is often the focus as to a change in its price. A composite good is an abstraction used in economics that represents all goods in the relevant budget besides the one in question.-Purpose:...
, index numbersIndex (economics)In economics and finance, an index is a statistical measure of changes in a representative group of individual data points. These data may be derived from any number of sources, including company performance, prices, productivity, and employment. Economic indices track economic health from...
, and rationing - cardinal utilityCardinal utilityIn economics, cardinal utility refers to a property of mathematical indices that preserve preference orderings uniquely up to positive linear transformations...
, constancy of the marginal utility of income, and consumer's surplus - welfare economicsWelfare economicsWelfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...
- comparative statics
- Part II
- stability of equilibrium systemsGeneral equilibriumGeneral equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...
, dynamics (disturbances in equilibrium), and comparative statics
- stability of equilibrium systems
- the Keynesian systemIS/LM modelThe IS/LM model is a macroeconomic tool that demonstrates the relationship between interest rates and real output in the goods and services market and the money market...
- linear and nonlinear systems
- Malthusian and optimum population
- dynamics
- the business cycle
- endogenousEndogenousEndogenous substances are those that originate from within an organism, tissue, or cell. Endogenous retroviruses are caused by ancient infections of germ cells in humans, mammals and other vertebrates...
models - mixed exogenousExogenousExogenous refers to an action or object coming from outside a system. It is the opposite of endogenous, something generated from within the system....
-endogenousEndogenousEndogenous substances are those that originate from within an organism, tissue, or cell. Endogenous retroviruses are caused by ancient infections of germ cells in humans, mammals and other vertebrates...
theories - mixed systems of a linear-stochasticStochasticStochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
type- conclusions (on neoclassical theoryGeneral equilibriumGeneral equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...
from WalrasWalrasWalras is surname of:* Auguste Walras , French school administrator and economist* Léon Walras * Walras' law...
to hints of the futureExogenous growth modelThe neoclassical growth model, also known as the Solow–Swan growth model or exogenous growth model, is a class of economic models of long-run economic growth set within the framework of neoclassical economics...
in comparative dynamics, the comparative-statics counterpart of dynamic systems)
- conclusions (on neoclassical theory
Methods and analysis
Samuelson's Foundations demonstrates that economic analysis benefits from the parsimonious and fruitful language of mathematics. In its original version as a dissertation submitted to the David A. Wells Prize Committee of Harvard University in 1941, it was subtitled "The Observational Significance of Economic Theory" (p. ix).One unifying theme, on the striking formal similarities of analysis in seemingly diverse fields, occurred only in the course of writing on them—from consumer's behavior and production economics of the firm to international trade, business cycles, and income analysis. It dawned on the author that he was prodigal "in proving essentially the same theorems" over and over. His failure of initial intuition, so he suggests, might be less surprising in light of the few economic writings then extant concerned with formulating meaningful theorems – hypotheses about empirical data—that could conceivably be refuted
The Logic of Scientific Discovery
The Logic of Scientific Discovery is a 1934 book by Karl Popper. It was originally written in German and titled Logik der Forschung. Then Popper reformulated his book in English and republished it in 1959. This forms the rare case of a major work to appear in two languages, both written and one...
by empirical data (pp. 3–5).
Samuelson (pp. 5, 21–24) finds three sources of meaningful theorems sufficient to illuminate his purposes:
- maximizing behavior of economic units (as to utility for a consumer and profit for a firm)
- economic systems (including markets and economies) in stable equilibrium
- qualitative propertiesQualitative economicsQualitative economics refers to representation and analysis of information about the direction of change in some economic variable as related to change of some other economic variable...
between two or more variables, such as an alleged technological relation or psychological law (indexed by the sign of the relevant functional relationship).
Part I conjectures that meaningful theorems for economic units and for their respective aggregates are almost all derivable from general conditions of equilibrium. The equilibrium conditions can in turn be stated as maximization conditions. So, meaningful theorems reduce to maximization conditions. The calculus of the relations is at a high level of abstraction but with the advantage of numerous applications. Finally, Part I illustrates that there are meaningful theorems in economics, which apply to diverse fields.
Part II concentrates on aggregation of economic units into equilibrium of the system. But the symmetry conditions required for direct maximization of the system, whether a market or even the simplest model of the business cycle, are lacking, in contrast to an economic unit or its corresponding aggregate. What can be hypothetically derived (or rejected in some cases) is a stable equilibrium of the system. (This is an equilibrium of the system such that, if a variable disturbs equilibrium, the system converges to equilibrium.) Stability of equilibrium is proposed as the principal source of operationally meaningful theorems for economic systems (p. 5).
Analogies from physics (and biology) are conspicuous, such as the Le Chatelier principle
Le Châtelier's principle
In chemistry, Le Chatelier's principle, also called the Chatelier's principle, can be used to predict the effect of a change in conditions on a chemical equilibrium. The principle is named after Henry Louis Le Chatelier and sometimes Karl Ferdinand Braun who discovered it independently...
and correspondence principle
Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....
, but they are given a nontrivially generalized formulation and application. They and mathematical constructions, such as Lagrangian multipliers
Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
, are given an operational economic interpretation. The generalized Le Chatelier principle is for a maximum condition of equilibrium: where all unknowns of the function are independently variable, auxiliary constraints ("just-binding" in leaving initial equilibrium unchanged) reduce the response to a parameter change. Thus, factor-demand and commodity-supply elasticities
Elasticity (economics)
In economics, elasticity is the measurement of how changing one economic variable affects others. For example:* "If I lower the price of my product, how much more will I sell?"* "If I raise the price, how much less will I sell?"...
are hypothesized to be lower in the short run than in the long run because of the fixed-cost constraint in the short run. In the course of analysis, comparative statics
Comparative statics
In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter....
, changes in equilibrium of the system that result from a parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
change of the system, is formalized and most clearly stated (Kehoe, 1987, p. 517). The correspondence principle is that the stability of equilibrium for a system (such as a market or economy) implies meaningful theorems in comparative statics. Alternatively, the hypothesis of stability imposes directional restrictions on the movement of the system (Samuelson, pp. 258, 5). The correspondence is between comparative statics
Statics
Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...
and the dynamics implied by stability of equilibrium.
| The starting point of the analysis is the postulate of maximizing behavior. The point is not (or not only) that everyone is out to maximize (Fischer, 1987, p. 235), even if true. Rather, first- and in particular higher-order (derivative) conditions of equilibrium at the maximum imply local behavioral relations (Samuelson, p. 16). The stability of equilibrium with sufficient other hypothetical qualitative restrictions then generates testable hypotheses (pp. 16, 28–29). Even where there is no context for purposive maximizing behavior, reduction to a maximization problem may be a convenient device for developing properties of the equilibrium, from which, however, no "teleological Teleology A teleology is any philosophical account which holds that final causes exist in nature, meaning that design and purpose analogous to that found in human actions are inherent also in the rest of nature. The word comes from the Greek τέλος, telos; root: τελε-, "end, purpose... or normative Normative economics Normative economics is that part of economics that expresses value judgments about economic fairness or what the economy ought to be like or what goals of public policy ought to be.... welfare significance" is warranted (pp. 52–53). |
Chapter VIII on welfare economics
Welfare economics
Welfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...
is described as an attempt "to give a brief but fairly complete survey of the whole field of welfare economics" (p. 252). This Samuelson does in 51 pages, including his exposition of what became known as the Bergson-Samuelson social welfare function. Theorems derived in welfare economics, he notes, are deductive implications of assumptions that are not refutable, thus not meaningful in a certain sense. Still, the social welfare function can represent any index (cardinal
Cardinal utility
In economics, cardinal utility refers to a property of mathematical indices that preserve preference orderings uniquely up to positive linear transformations...
or not) of the economic measures of any logically possible ethical belief system that is required to order any (hypothetically) feasible social configurations as "better than", "worse than", or "indifferent to" each other (p. 221). It also definitively elucidates the notion of Pareto optimality and the "germ of truth in Adam Smith's doctrine of the Invisible Hand
Invisible hand
In economics, invisible hand or invisible hand of the market is the term economists use to describe the self-regulating nature of the marketplace. This is a metaphor first coined by the economist Adam Smith...
" (Samuelson, 1983, p. xxiv; Fischer, 1987, p. 236).
The final pages of the book (pp. 354–55) outline possible directions analytical methods might take, including for example models that show how:
- deficit financingDeficit spendingDeficit spending is the amount by which a government, private company, or individual's spending exceeds income over a particular period of time, also called simply "deficit," or "budget deficit," the opposite of budget surplus....
could produce positive short-run effects on the economy that are swamped by adverse long-run effects on capital accumulation (seriously reconsidered later as crowding outCrowding out (economics)In economics, crowding out occurs when Expansionary Fiscal Policy causes interest rates to rise, thereby reducing private spending. That means increase in government spending crowds out investment spending....
) - declines in age-specific mortalityMortality rateMortality rate is a measure of the number of deaths in a population, scaled to the size of that population, per unit time...
affect the net reproductive rateDemographyDemography is the statistical study of human population. It can be a very general science that can be applied to any kind of dynamic human population, that is, one that changes over time or space...
(whose the implications for population growthPopulation growthPopulation growth is the change in a population over time, and can be quantified as the change in the number of individuals of any species in a population using "per unit time" for measurement....
are less abstract than they might first appear).
Samuelson closes by expressing hope in the future use of comparative dynamics to:
- aid in the attack upon diverse problems – from the trivial behavior of a single small commodity, to the fluctuations of important components of the business cycle, and even to the majestic problems of economic development.
Appendices
There are two mathematical appendices totalling 83 pages. The first gathers and develops "very briefly" and "without striving for rigor" results on maximization conditions and quadratic forms used in the book and not conveniently collected elsewhere (p. 389). The other is on difference equations ("for the dynamic economist") and other functional equations.Enlarged edition
The 1983 Enlarged edition includes an additional 12-page "Introduction" and a new 145-page appendix with some post-1947 developments in analytical economics, including how conclusions of the book are affected by them.Assessments
- Kenneth ArrowKenneth ArrowKenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
(1983, p. 19) describes Foundations as "the only example I know of a doctoral dissertation that is a treatise, perhaps I should say of a treatise that has so much originality in every part that it is entitled to be accepted as a thesis." - Richard N. CooperRichard N. CooperRichard Newell Cooper is an American economist, policy adviser, and academic.Cooper graduated from Oberlin College in 1956 and received a master's degree in economics from the London School of Economics and Political Science as a Marshall Scholar in 1958. He received his Ph.D...
(1997) writes that the book "drastically redirected the advanced study of economics toward greater and more productive use of mathematics." - Notwithstanding the important workSocial Choice and Individual ValuesKenneth Arrow's monograph Social Choice and Individual Values and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor...
of Arrow, Kotaro SuzumuraKotaro Suzumurais a Japanese economist and professor emeritus of Hitotsubashi University. He graduated from Hitotsubashi University in 1966. His research interests have included social choice theory and welfare economics.- Selected publications :...
(1987, p. 420) affirms the Bergson-Samuelson social welfare function as "logically impeccable." - The Nobel Prize citation is applicable to Foundations: "for the scientific work through which [Samuelson] has developed static and dynamic economic theory and actively contributed to raising the level of analysis in economic science."
See also
- Economic methodologyEconomic methodologyEconomic methodology is the study of methods, especially the scientific method, in relation to economics, including principles underlying economic reasoning...
- Mathematical economicsMathematical economicsMathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...
- Neoclassical economicsNeoclassical economicsNeoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...
- Paul SamuelsonPaul SamuelsonPaul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...
- Social welfare functionSocial welfare functionIn economics, a social welfare function is a real-valued function that ranks conceivable social states from lowest to highest. Inputs of the function include any variables considered to affect the economic welfare of a society...
External links
- "Maximum Principles in Analytical Economics", link to Nobel Prize lecture
- Nobel Prize presentation speech, in which the first 3 areas discussed apply to Foundations