Printer Friendly

Memorializing Paul A. Samuelson: a review of his major works, 1915-2009.


We have lost the dean of American economists, the unrivalled leader of neoclassical economics on December 13, 2009. Paul Anthony Samuelson was born on May 15, 1915 in Gary, Indiana to his parents Frank Samuelson and Ella Lipton. The family moved to Chicago, where Paul attended the Hyde Park High School. He entered the University of Chicago at age 16 and took up economics after having heard a lecture on the Reverend T. R. Malthus. After graduating with a BA from there in 1935, he attended Harvard, where he earned an MA in 1936, and a PhD in 1941. He married a fellow student, Marion Crawford, in 1938, and after her death in 1978, he was married to Risha Clay Samuelson.

Samuelson's Ph.D. thesis became the celebrated Foundations of Economic Analysis published in 1947. A year later, he published his famous text, Economics. Those two works bracketed his contribution from the simple to the complex aspects of economics that were imitated by many and had educated over a generation of economists.

Samuelson started teaching as an instructor at Harvard in 1940, but after a month he moved to MIT as an assistant professor. While at MIT, he became an Associate Professor (1944), Professor (1947), and finally Institute Professor (1966). He also received honorary doctoral degrees from the University of Chicago (1961), Oberlin College (1961), Indiana University (1966), and East Anglia University (1966), and was a Ford foundation Research Fellow during 1958-1959. His numerous awards include the David A. Wells Prize in 1941 by Harvard University, the John Bates Clark Medal by the American Economic Association in 1947, and the Nobel Laureate Prize in economics by the Bank of Sweden in 1970 for his scientific contributions to economics.

Because he did not wish to compromise his thinking in economics, Samuelson turned down President Kennedy's requests to serve as the chair of the economic council. Samuelson, however, has been credited with educating the president on Keynesian economics, and he also was the one to encourage the tax cut that was implemented during Johnson's administration.

Goals of Economics

Samuelson's goal was to understand the "... behavior of mixed-economies of the American and Western European type" (Samuelson, CW, 1986, V. 3,728). His means to this goal was to be scientifically honest. He held that "... science consists of descriptions of empirical regularities" (Ibid., 772). Therefore, "... a good economist has good judgment about economic reality" (Ibid., 775). One should not wonder why he often refers to Thomas Kuhn, for Kuhn holds that "economic analysis advances discontinuously. After a great forward step, time must be taken to consolidate the gains achieved" (Samuelson, 1966, V. 2, 1140). Within this research mentality, Samuelson goes after reality with economic models, being well aware that "the science of economics does not provide simple answers to complex social problems" (Ibid., V. 2, 1325). Economics for him was different in degree but not in kind from the physical sciences: "All sciences have the common task of describing and summarizing empirical reality. Economics is no exception" (Ibid., V. 2, 1756). But unlike the falsificationist, he does not look at facts to terminate a theory. Rather, "in economics it takes a theory to kill a theory; facts can only dent the theorist's hide" (Ibid., V. 2, 1568).

Samuelson's representative definition of economics is: "the study of how people and society end up choosing, with or without the use of money, to employ scarce productive resources that could have alternative uses, to produce various commodities and distribute them for consumption, now or in the future, among various persons and groups in society. It analyzes the costs and benefits of improving patterns of resource allocation" (Samuelson, 1980, 2). We also see elements of production, distribution, consumption, and cost benefit analysis in his definition.


Samuelson has evolved the "operational" method of economics. He said: "my work in the theory of revealed preference, in Foundations of Economic: Analysis, and in the several volumes of Collected Scientific Papers, consistently bears out this general methodological procedure" (Samuelson, 1986, V. 5, 793 [Italics original]). Basically, the procedure is " learn what descriptions [new literature and mathematical paradigms] imply for observable data" (Ibid., V. 5,793). With data on the one hand, and logic and theory on the other, operationalism seeks a correspondence of the two sides. "Samuelson's 'correspondence principle between comparative statics and dynamics' ... shows how the problem of deriving operationally meaningful theorems in comparative statics is closely tied up with the problem of stability of equilibrium" (Morishima, 1964, 24).

In the realm of dynamics, Samuelson postulated that dp/dt = H([q.sub.D] - [q.sub.s]), where the term on the left is the rate of change of prices, dp with respect to changes of time, dt. H is a proportional constant, q is quantity, S is supply, and D is demand (Samuelson, 1966, V. 1, 544). Stability is therefore assured if as time goes to infinity, the solution of the differential equation breaks down, which in economic terms means, "... the supply curve cuts the demand curve from below" (Samuelson, 1947, 18).

Areas of interest

Samuelson described himself as a generalist. He once claimed to be the last generalist in economics, writing and teaching such diverse subjects as international trade and econometrics, economic theory and business cycle, demography and labor economics, finance and monopolistic competition, history of doctrines and locational economics (Samuelson, 1986, V. 5, 800). His works were published in five volumes titled Collected Scientific Papers of Paul Samuelson, with two more volumes expected. We sample some of his major works by topics below.


In macroeconomics, Samuelson forged the 'neoclassical synthesis' view, which added the neoclassical economic foundation to Keynesian economic thought. From the fourth (1958) to the eleventh (1980) edition of his Economics, he held that economists have been synthesizing traditional theory with newer Keynesian thoughts on income distribution. He accepted the Keynesian synthesis as a revolutionary way to look "... at how the entire gross national product is determined and how wages and prices and the rate of unemployment are determined with it" (Samuelson, CW, V. 5, 1986, 280). Looking through the Keynesian lenses, Samuelson saw that economic policies should operate from the point of view of a 'mixed-economy', mixed because government works side-by-side with the private sector in economic affairs during business cycles. Such cycles are generated when the traditional concept of the accelerator (constant capital to output in the naive version) interacts with the Keynesian multiplier, a concept first seized by Samuelson.

In 1958, Samuelson introduced the overlapping generation model (OLG), which is seen as a rival to the famous Arrow-Debreu general equilibrium model for the economy, and is a popular model in modern macroeconomic analysis (Samuelson, CW, V. 1, 1966, 219-234). The model allows intergenerational trading such as where a middle-aged person lends his savings to a younger person, expecting from him savings in return in a later period. The model "... breaks each life up into thirds: men produce one unit of product in period 1 and one unit in period 2; in period 3 they retire and produce nothing" (Ibid., 220). U - U([C.sub.1], [C.sub.2], [C.sub.3]) represents the utility function on consumption in each period. The rate of interest is the price of exchange between current and future foods. [R.sub.t] = 1/(1+[i.sub.t]) is the discount rate. Equilibrium is achieved in period three when the discounted value of consumption equals the discounted value of production, or [C.sub.1] + [C.sub.2][R.sub.t] + [C.sub.3][R.sub.t][R.sub.t+1] = 1 + 1[R.sub.t] + 0[R.sub.t][R.sub.t+1] = 1 + 1[R.sub.t] (Ibid., 221).

If savings is income less consumption, we can specify the model from the savings and the savers' points of view. [S.sub.p]([R.sub.t], [R.sub.t+1]) is the saving function for p periods: 1, 2, 3. The population, B, at time, t, can be written as [B.sub.t]. The population at different periods can be represented by a first period, [B.sub.t], a second period [B.sub.t-1], and a third period [B.sub.t-2]. The equilibrium condition can then be represented as [B.sub.t][S.sub.1]([R.sub.t], [R.sub.t+1]) + [B.sub.t-1][S.sub.2]([R.sub.t-1], [R.sub.t]) + [B.sub.t- 2][S.sub.3]([R.sub.t-2], [R.sub.t-1]) = 0 (Ibid., 222). If all the B's are the same (i.e. a stationary population), then [R.sub.t] = 1. If B grows exponentially, then the equality of the discount rate with the population growth rate is also a solution. The solution being that there would be harmony between current and future savings and consumption plans across generations, with the model addressing how they would grow alongside zero and the actual growth rate in population. Considering further developments, Solow elevated this model in the history of economic thought: "... this innocent little device of Samuelson's has been developed into a serious and quite general modeling strategy that uncovers equilibrium possibilities not to be found in standard Walrasian formulations" (Solow, 2006, 40).

Cambridge Controversy

In 1962, Samuelson derived a Walrasian-like production function by creating an index for capital from the many equations that capture the technique of production of firms that populate the economy (Samuelson, 1966, V. 1,325-337). Using this surrogate capital index, he derived a surrogate production function to capture certain stylized facts about neoclassical economics. According to C. E. Ferguson, (1975, 245)"The essential issue may be put this way. If production functions are smoothly continuous and everywhere continuously differentiable, the neoclassical results hold (possible in a somewhat attenuated form if one allows for heterogeneous capital goods)."

The problem of measuring capital came up with David Ricardo, to which the roots of marginal analysis are traced. "The whole marginal analysis was born of Ricardo's attempt to explain the share of rent in the national income, and to show why some rents on some land were higher than on other...'land' could be acres...adding together different acres weighted by their relative prices... the same could be done for labor...using relative wages as the basis of weights. With capital the problem was an entirely different one, since there was no unit in which we could reduce capital to homogeneous units" (Lutz and Hague, 1961, 305). As K. Wicksell (1911, V. 1,149) puts it:
 Whereas labor and land are measured each in
 terms of its own technical unit (e.g. working
 days or months, acre per an- num) capital ... is
 reckoned ... as a sum of exchange value-whether
 in money or as an average of
 products, in other words, each particular
 capital-good is measured by a unit extraneous
 to itself. [This] is a theoretical anomaly which
 disturbs the correspondence which would
 otherwise exist between all the factors of
 production. The productive contribution of a
 piece of technical capital, such as a steam
 engine, is determined not by its cost but by
 the horse-power which it develops, and by the
 excess or scarcity of similar machines. If capital
 were to be measured in technical units,
 the defect would be remedied and the correspondence
 would be complete. But, in that
 case, productive capital would have to be
 distributed into as many categories as there
 are kinds of tools, machinery, and materials,
 etc., and a unified treatment of the role of
 capital in production would be impossible.
 Even then we should only know the yield of
 the various objects at a particular moment,
 but nothing at all about the value of the goods
 themselves, which it is necessary to know in
 order to calculate the rate of interest, which in
 equilibrium is the same on all capital.

The measurement of capital is therefore, among the top two problems in capital theory, the other being the capital-output ratio (Lutz, 1961, 9). Leon Walras' view on capital is also foundational, although his model is different from Ricardo's (see Nell, 1967, 15-26). Walras postulated "... a capital goods market, where capital goods are bought and sold...They are demanded because of the land-services, labor and capital-services they render, or better, because the rent, wages and interest which these services yield" (Walras, 1954, 267). The price of the capital goods depends on the price of its services or its income. Net income is the gross income, p, less the price of the capital goods, P, adjusted for depreciation, It, and insurance premiums, v, i.e, [pi] = [p - ([Mu] + v)P]. (Ibid., 268). From this we get the rate of net income i = [pi]/P, suggesting that p - ([MU] + v)P = iP, from which we can get the price of all capital goods (Ibid., 269). Walras suggested that we should not deduct depreciation or insurance charges for land, nor for personal faculties (human capital) because they are natural and known. Land and personal faculties are hired in kind in the capital market, but capital is usually hired in the form of money in the money market. The proper capital goods are artificial (not natural), and are subject to cost of production, depreciation and insurance premiums (Ibid., 271). "Capital formation consists ... in the transformation of services into new capital goods, just as production consists in the transformation of services into consumer goods" (Ibid., 282). Putting it all together, "Once the equilibrium has been established in principle (through groping), exchange can take place immediately. Production, however, requires a certain lapse of time ... equilibrium in production ... will be established effectively through the reciprocal exchange between services employed and products manufactured within a given period of time during which no change in the data is allowed" (Ibid., 242). A similar situation holds for capital formation (Ibid., 282). In the end, "Capital formation in a market ruled by free competition is an operation by which the excess of income over consumption can be transformed into such types and quantities of new capital goods proper as are best suited to yield the greatest possible satisfaction of wants" (Ibid., 305).

With this Walrasian background, we can appreciate Joan Robinson's position that "a piece of equipment or a stock of raw materials, regarded as a product, has a price, like any other product, made up of prime cost plus a gross margin. These costs (direct and indirect) are composed of wages, rents, depreciation and net profit. The amount of net profit entering into the price of the product is, obviously, influenced by the general rate of profit prevailing in the industries concerned. Thus the value of capital depends upon the rate of profit. There is no way of presenting a quantity of capital in any realistic manner apart from the rate of profit, so that to say that profits measure, or represent or correspond to the marginal product of capital is meaningless" (Robinson, 1971, 601). Yet we find attempts to construct an aggregate production function in a J. B. Clark and Frank Ramsey style, where the definition of capital represents a challenge. From the macroeconomic point of view, the aggregate production function is written as Y = f(K, L) which is read that output is a function of capital and labor, respectively. In equilibrium, the marginal product of labor is the wage rate. The marginal product of capital is set equal to the rate of interest, which is at the center of the controversy.

Joan Robinson sets up the controversy this way: "In 1961 I encountered Professor Samuelson on his home ground; in the course of an argument I happened to ask him 'When you define the marginal product of labor, what do you keep constant?' He seemed disconcerted, as though none of his pupils had ever asked that question, but the next day he gave a clear answer. Either the physical inputs other than labor are kept constant, or the rate of profit on capital is kept constant. I found this satisfactory, for it destroys the doctrine that wages are regulated by marginal productivity" (Robinson, 1970, 310).

The problem has to do with time, or the time period of production. "Wicksell never used the term K ... but always inserted the term T on the grounds that it is by allowing labor to use roundabout, time-consuming processes of production that capital raises the productivity of labor and thus is itself production" (Lutz, 1961, 10). But in a letter to Alfred Marshall, Wicksell (1905, V. 3, 102 [Italics original]) wrote: "... the theory of capital and interest cannot be regarded as complete yet. As I have tried to show several times ... so long as capital is defined as a sum of commodities (or of value) the doctrine of the marginal productivity of capital as determining the rate of interest is never quite true and often not true at all--it is true individually, but not in respect of the whole capital of society."

Samuelson wrote several articles on the topic of capital theory leading to his 1962 milestone article on the surrogate production function. His 1937 article, "Some Aspects of The Pure Theory of Capital" probed the aspect of time and timeless analysis of the production function (Samuelson, CW, V. 1, 1966, 161-188). He posited relationships for "constant rate of interest" and "the rate of interest itself in an unrestricted function of time" (Ibid., 163). Treating the rate of interest, r, as a constant, one can capitalize an income stream at the beginning and at the end of a period, returning values V(0, r), and V(t, r) for time t = 0, and t = t, respectively. The internal rate of return, f, makes the initial value of the investment zero, V(0, F) = 0 (Ibid., 165-167). We can show " any instant of time the value of every investment account is unequivocally determined" (Ibid., 169). The income stream so determined will vary in the real world due to uncertainty, imputation of income and market imperfections (Ibid., 170).

The treatment of a constant rate of interest represents a condition of stationary society without capital accumulation. Making the rate of interest a function of time, r = r(t), requires the consideration of an average rate, p. The relation of p to r is the same as the relationship of the margin to the average. The constancy of the rate of interest is seen as a special case of the variation of the rate of interest with time. The same relationship between perpetual income and value will hold for the constant and variable view of the interest rate (Ibid., 177).

In 1939, Samuelson furthered the analysis to show "...some of the forces which help to determine the market rate of interest at which all can borrow or lend under ideal conditions" (Ibid., 189200). He used discrete periods, when the interest rate in any period equilibrates total asset holdings with the total assets of all enterprises. The approach he took "... does not require any definition of capital as a physical quantity" (Ibid., 199).

In 1943, Samuelson considered dynamic, static, and stationary state conditions for the rate of interest to be zero (Samuelson, CW, 1966, V. 1,201-211). After considering cases where capital should have a zero net productivity (according to Frank Ramsey), and where the maximum output is never attained for a finite value of capital (according to Frank Knight), Samuelson took the position "... not to reify the limit by asking what really happens at a zero rate of interest, but rather to concentrate upon the dynamic path toward this limiting condition" (Samuelson, Ibid., 211).

In 1956, Samuelson and Robert Solow extended consideration of the Ramsey zero interest and one capital-good model to heterogeneous capital goods (Samuelson, CW, 1966, V. 1,261-286). This article was setting the stage to "... reconstruct the composition of its diverse capital goods so that there may remain great heuristic value in the simpler J. B. Clark-Ramsey models of abstract capital substance" (Ibid., 261-262). Broadly speaking, the treatment of capital, fixed or circulating, can be looked at either as inputs or efficient outputs or consumption it generates. Simplified, the Ramsey model maximizes all future utility of consumption, U(C), constrained by a saving function, f(s). The model, however, does not consider "... differences between different kinds of goods and different kinds of labor, and suppose them to be expressed in terms of fixed standards, so that we can speak simply of quantities of capital, consumption and labor without discussing their particular forms" (Ramsey, 1928, 544). Samuelson's and Solow's effort was to drop that assumption.

In a Clark-Ramsey framework, capital enters as a constraint, f(s) in the initial period. In his 1961 paper on "The Evaluation of Social Income: Capital Formation and Wealth", Samuelson (CW, V. 1, 1966, 299-324) elaborated on this capital constraint. There was one output, produced from inputs, F(K, L) which can be invested or consumed. Capital K can be changed to say K2 with twice the value of the original capital, which will no longer be produced. The production function for gross output is now F(K + [K.sub.2], L), which is netted for depreciation at a rate equal to m. He used this model to answer an old "... 1935 debate between Pigou and Hayek as to the meaning of maintaining capital intact" (Ibid., 302). From this model also, the factors are rewarded their marginal products (Ibid., 304). The article "... dealt with the problem of efficiency by valuing all capital in terms of new capital" (Lutz and Hague, 1960, 314). His treatment of depreciation seems to be dealing with mortality or accident as in the Walrasian insurance premium case (Ibid., 314). The income article though, did not address the problem of measuring capital. The model avoided the "... heterogeneity of capital goods. He had been able, with his methods, to avoid the problem of whether one was dealing with four-or five-year-old cars, or with different qualities of land as in Ricardo's theory" (Ibid., 315).

In a milestone article of 1962, Samuelson wanted to show that the surrogate production function, represented by a w - r frontier, can be derived from heterogeneous capital goods as well as from an aggregate homogenous capital good (Ferguson, 1972, 169). If we start with a heterogeneous set of capital-goods, each associated with labor, then "one need never speak of the production function, but rather should speak of a great number of separate production functions, which correspond to each activity and which have no smooth substitutability properties" (Samuelson, CA, V. 1, 166, 326). These have a linear programming-like structure that Samuelson and others explored in Linear Programming and Economic Analysis (Dorfman, Samuelson, and Solow, 1958), and in an independent piece on the subject (Samuelson, CW, V. 1, 1966, 287-298).

In general, production has two sectors, one producing consumption goods, and one producing capital goods--A). [P.sub.k] = [a.sub.k]W + [b.sub.k] (r + [delta])[P.sub.k], where p is the price, W is the wage, r is the interest rate, [delta] is the depreciation, k represents the capital sector, a and b are labor requirements per unit of output, and B). [P.sub.c] = [a.sub.c]W + [b.sub.c] (r + [delta])[P.sub.c]. Given the rate of interest, we can find the wage rate, and vice versa. One method in solving them is to eliminate prices in each equation and then set the results equal to each other. The linearity comes out if we assume with Samuelson that [a.sub.k] = [a.sub.c]; [b.sub.k] = [b.sub.c], for then we get W/[P.sub.c] = [1 - [b.sub.k](r + [delta])]/[a.sub.k], which is linear (Samuelson, CW, V. 1, 1966, 337).

In stationary or steady state conditions, a tradeoff frontier between wage and profit emerges. For a certain rate and wage level, we get a point on the w - r frontier. The slope will be constant for the fixed proportion of labor and capital, yielding a straight line. Many such relationships exist for various capital goods, yielding many negatively sloped straight lines in the w-r plane.

If we parametize the coordinates of these points to time, then we will find that the more roundabout a production process is, the steeper will be its w - r frontier. This is because one process will be used at very high interest or profit rate in preference to another. As the interest rate is lowered, society will consider using one process rather than the other. This way an envelope of all the straight lines will be formed, representing a piece-wise linear factor-price frontier. Samuelson wants to demonstrate that even in the "... discrete-activity fixed-coefficient model of heterogeneous physical capital goods, the factor price (wage and interest rate) can still be given various long-run marginalism (i.e., partial derivative) interpretations" (Ibid., 322). Garegnani (1970, 412) has shown that if we make the parameters of the A) and B) defined for an interval in which the value of the functions are positive, then the curves will have a 'smooth' envelope enclosing them. This continuous feature makes good comparison for a smooth frontier derived from the Clark-Ramsey model.

Samuelson then used the Clark-Ramsey homogenous capital model to approximate the w - r of the discrete heterogeneous capital model as close as we like. Let output depend on labor and capital: Q = F(L, J). If the function is homogenous, we can factor out an input, yielding: Q = LF(I, J/L). In equilibrium, w = [partial derivative]Q/[partial derivative]L, and r = [partial derivative]Q/[partial derivative]J. Taking their derivatives, and forming the ratio yields: dw/dr = -(J/L), the slope of the frontier. The heterogeneous discrete capital case for deriving w - r can be made arbitrary close to approximate the homogenous smooth capital case of deriving w - r. In his 1966 article, "A Summing Up," Samuelson (CW, V. 3, 1972, 236) wrote "the fact of possible reswitching teaches us to suspect the simplest neoclassical parables." Reswitching is a situation where one technique is feasible at two different levels of the rate of interest. It can occur if two frontiers intersect. Both Samuelson (CW, V. 4, 136) and Pasinetti (2006, 151-152) point to Pierro Sraffa's work as fundamental to the origin of the reswitching debate. According to Pasinetti, the basic conclusion of the debate ends with Samuelson's admission that reswitching is possible.

Trade Theory

During his lifetime, Samuelson had made major contributions to trade theory. He put the production possibility and indifference curves to work in a general equilibrium framework. He kept that framework in focus through his works on trade, but had integrated his discovery of preference theory into the framework. He not only brought trade theory under the general equilibrium (GE) framework, but had reversed the traditional approach to trade theory. The reversal worked from autarky to trade (Krugman, 1995, 1245).

Samuelson's scientific approach to trade theory started with a hypothesis elaborating gains from trade in 1938. This was followed by an attempt to use numbers to validate and find possible counter examples in his 1939 paper, a problem with the prediction of the model in his Stolper-Samuelson approach in 1941, and a proof of the propositions in the 1940s.

In his 1938 paper (Samuelson, 1966, V. 2, Item 60) he assumed given taste and technology, one person or one country model, two goods, (x, y), and two productive services, (a, b). Given a, b, y, one can find the maximum amount of x that is produced. Given a, b, and x, the maximum amount of y that is produced can be estimated. The general representation is: [phi])(x, y, a, b) = 0. The Production Possibility Curve can be derived by setting values for a and b, then solving for y = f(x). To get the indifference curve, assign values for x and y and solve for b = f(a). The equilibrium condition for each country under autarky occurs at the tangency of these curves for their respective countries. For gains from trade to take place in this model, a person or a country can perform with less productive services (a or b) in trade, foregoing one commodity for another to attain a higher indifference curve, or one can gain by moving to a higher position on its preference scale at the expense of the other. The argument for gains from trade is inconclusive because we need a welfare utility function to measure gains or losses. Samuelson (1966, V. 2, 775) concluded that "it is demonstrable that free trade (pure competition) leads to an equilibrium in which each country is better off than in the absence of trade.... Nevertheless, this does not prove that each country is better off than under any other kind of trade; indeed, if all others are free trading, it always pays a single country not to trade freely."

The objective in his 1939 paper was to refine the conclusion of the 1938 paper to show that "... free trade or some trade is to be preferred to no trade at all" (Ibid., 781). The theorem investigated was stated as follows:

Samuelson Theorem I. (1939): "the introduction of outside (relative) prices differing from those which would be established in our economy in isolation will result in some trade, and as a result every individual will be better off than he would be at the prices which prevailed in the isolated state" (Samuelson, 1966, V. 2, 786 [Italics original]).

Samuelson tries to find counterexamples for this theorem. He pondered whether there are any numbers for which the theorem is true. Using numbers for prices, p, and quantities, q, Samuelson examines three commodities and two factors prices, w, and factor quantities, a, by creating four scenarios or cases. The cases were used to validate the hypothesis that

[summation]p'x' - [summation]w'a' [greater than or equal to] [summation]p'[bar.x] - [summation]w'a (1)

s.t. [summation]px = [summation]p[bar.x] (2)

where the prime indicates preassigned prices and quantities, and the bar indicates optimal values. The summation runs over n commodities and s factors. The "subject to" condition of equation 2 requires that exports must equal import (Ibid., 784).

With his 1938 and 1939 papers, Samuelson was committed to the free trade doctrine. He even announced that proofs were forthcoming. In those regards, his Review of Economic Studies, 1941 piece with Wolfgang F. Stolper was a moment of pause. The "... argument seems to have relevance to the American discussion of protection versus free trade ... labor is the relatively scarce factor in the American economy, it would appear that trade would necessarily lower the relative position of the laboring class as compared to owners of other factors of production" (Ibid., 832). The Stolper-Samuelson Model assumes there exists perfect competition between two countries, I and II, where two homogenous goods, wheat (A) and watches (B), have relative prices, [P.sub.a]/[P.sub.b]. There also exist two fixed factors, Labor (L) and Capital (C), assuming full employment with perfect factor mobility and the same production functions (Ibid., 835). Four additional assumptions (two relating to the HO model and two for case studies) are incorporated.

a) Capital is abundant and labor is scarce,

b) Capital is more important in the production of wheat (A) than in the production of watches (B), and

1) Capital is relatively more important in wheat (A) as the wage good.

2) Capital is relatively more important in watches (B) as the wage good (Ibid., 838).

The equations provided are:

[L.sub.a] + [L.sub.b] = L (1)

[C.sub.a] + [C.sub.b] = C (2)

A = A([L.sub.a], [C.sub.a]) (3)

B = B([L.sub.b], [C.sub.b]) (4)

where the subscripts of the factors indicate the factor amount required to produce wheat (A) or watches (B). The model predicts that "... the introduction of trade lowers the proportion of capital to labor in each line and the prohibition of trade, as by a tariff, necessarily raises the proportion of capital to labor in each industry" (Ibid., 841). These have the consequence that: (1) trade will lower the real wage of the scarce factor of production, and (2) protection will increase the real wage of the scarce factor of production.

To determine the predictions of the model, a consistent set of equations needs to be solved for their unknowns. Given the production functions above, equilibrium conditions require that the factor prices must equal their marginal productivities. Since we assumed that the production functions for the two goods are the same, instead of writing two equations for labor (i.e., a wage rate equation for wheat, and a wage rate equation for watches), we can write one wage rate equation for labor, namely w = [P.sub.a][partial derivative]A/[partial derivative][L.sub.a] = [P.sub.b][partial derivative]A/[partial derivative][L.sub.b]. A similar reason for capital can also be applied. Instead of writing two separate equations for the return of capital in each industry, we can write one equation for the return on capital, namely r = [P.sub.a][partial derivative]A/[partial derivative][L.sub.a] = [P.sub.b][partial derivative]A/[partial derivative][L.sub.b]. Along with the four equations above, we now have eight additional equations for the production side of the economy. We need equations on the demand side to close the system to create a solution. We can add at least two more equations, one representing the demand for wheat and the other representing the demand for watches. With ten equations at hand, one is redundant according to Walras's law, leaving nine independent equations to solve for nine variables--two for the quantity of labor in each industry, two for the quantity of capital used in each industry, one for the total amount of watches and wheat, the real wage rate, the real return on capital, and the relative price of the two goods (Burmeister and Dorbell, 1970. Ch. 4; Takayama, 1972, 47).

Samuelson continued to develop his contributions to trade theory with papers in 1948, 1949, 1953, and 1967. The purpose of the 1948 paper was to probe the proof of Ohlin's partial equalization theorem that "(1) free mobility of commodities in international trade can serve as a partial substitute for factor mobility and (2) will lead to a partial equalization of relative (and absolute)factor prices" (Ibid., 847 [Italics original]). Samuelson enunciated four propositions, of which the first two are proven, and the latter two are derived. The two main propositions are: 1) Given partial specialization and each country producing some of the two goods, factor prices will be equalized, absolutely and relatively by free trade, and 2) If factor endowments are not too unequal, commodity mobility will always substitute perfectly for factor mobility (Ibid., 853).

In his 1949 paper, Samuelson admitted that his 1948 paper argued that "... free commodity trade will, under specified conditions, inevitably lead to complete factor-price equalization and appears to be in need of further amplification" (Ibid., 869). In the 1948 paper he advanced a relationship between wage/rent to the labor/land ratio that was tied to two areas, Europe and the U.S. (Ibid., 857). In 1949 he added a wage/rent to the commodity price ratio, (Ibid., 876) and used a more integrated model, treating the world itself as a country.

Figure 1 below is an abridged diagram of Samuelson's 1949 paper, representing a generalization of the Stolper-Samuelson theorem where L is labor, T is Land, C is clothing, F represents food, w is the wage rate, r is rent, and P indicates commodity prices. Clothing is labor intensive, and food is land intensive.

This 1949 based Figure 1 incorporates Quadrant I of his 1948 paper. The new diagram in Quadrant II shows the relationship of factor prices to commodity prices. In the diagram the distance, OM, has a similar weighted average interpretation as equation (5) below, where the commodities are clothing and food, and the factors are labor and land (Samuelson, 1966, 858). Samuelson equates the overall C/L = [bar.k] to a weighted average of the [C.sub.a]/[L.sub.a] = [k.sub.a] for the wheat industry, and [C.sub.b]/[L.sub.b] = [k.sub.b] for the watch industry. The weights are the labor share for the wheat and watch industry, i.e., [L.sub.a]/([L.sub.a] + [L.sub.a]) == [[lambda].sub.a], [L.sub.b]/([L.sub.b] + [L.sub.b]) == [[lambda].sub.b], respectively. We therefore have

[[lambda].sub.a][k.sub.a] + [[lambda].sub.b][k.sub.b] = [bar.k] (5)


The distance OM is a new expression for equation (5) involving the following terms:

OM = total labor/total land = (food land/total land x food labor/food land) + (clothing land/total land x clothing labor/clothing land) (5')

Figure 1 indicates that OM's range is between M' and M". In between those points, both commodities, food and clothing, will be produced, marking the case of incomplete specialization. Outside of that range, only one commodity will be produced in each country implying complete specialization.

The QQ line in Figure 1 represents a situation where the wage/rent ratio is the same for each industry. This underscores that the factor proportion will have to be the same in both industries. If the factor price ratios were not equal, it would be necessary to draw two such lines, say QQ and Q'Q' not shown, which would represent different factor price ratios and different factor proportions. The different factor mix will result in different marginal productivities, which, in equilibrium, will yield different factor prices. With equal factor prices, however, there will be equal factor proportions in the first quadrant, corresponding to a unique commodity price ratio in the second quadrant. Samuelson summarized his 1949 paper with the finding that "within any country." (a) an increase in the ratio of wages to rents will cause a defined decrease in the proportion of labor to land in both industries; (b) to each determinate state of factor proportion in the two industries there will correspond one, and only one, commodity price ratio and a unique configuration of wages and rent; and (c) the change in wage/rents factor proportions incident to an increase in wage must be followed by a one-directional increase in clothing prices relative to food prices (Samuelson, V. 2, 1966, 875 [Italics original]).

The rest of the 1949 paper lays out the mathematics behind Figure 1. Food and clothing are made homogenous functions of the inputs labor and land, namely F = [T.sub.f]f([L.sub.f]/[T.sub.f]) and C = [T.sub.c]c([L.sub.c]/[T.sub.c]) respectively. Partial derivatives for the marginal physical products for (1) labor in food, (2) land in food, (3) labor in clothing, and (4) land in clothing are taken. These marginal productivities are converted into values by multiplying them by their respective prices. The values of labor in food and clothing industries are equated to form one equation. The values of land in both industries are then equated to form a second equation. We now have two equations with three variables. From Figure 1, the variables are [L.sub.c]/[T.sub.c], [L.sub.f]/[T.sub.f], and [P.sub.f]/[P.sub.c]. Given prices, it is now possible to solve for the other two variables.

To solve the 2 x 2 matrix described above, Samuelson looked for a condition on the determinants to guarantee a solution. The condition is that the determinant of the Jacobian matrix must not vanish. Technically, the Jacobian matrix is derived from a set of differentiable functions. Given a set of equations: [y.sub.1] = 5[x.sub.2] and [y.sub.2] = [x.sub.1][x.sub.2], the determinant of the Jacobian matrix can be written as:


If the determinant is zero, then the two equations are dependent, and no solution exists for the system. If the determinant is not equal to zero, then the equations are independent, meaning that it is possible to solve for the unknown variables.

For Samuelson's 2 x 2 case, the determinant is derived from the commodity price ratio, the second derivatives of the two homogenous production functions, and the differences in factor intensities. The following equation shows the Jacobian matrix and its determinant for the Samuelson's 2 x 2 case:


In equation (6), the f" and c" are the second derivatives of the food and clothing production functions, respectively, they are also negative because the production functions exhibit diminishing returns. Prices and quantities are positive, and clothing is more labor intensive, making the square bracket items positive. Because the Jacobian determinant did not vanish, Samuelson therefore concluded that "the equilibrium is unique" (Ibid., 880).

In his 1953 article, however, Samuelson recognized A. Turing for informing him that the equilibrium condition is not true globally or "in the large" (Ibid., 909). Global conditions are more complicated than local conditions because local conditions deal with the value of a function in the neighborhood of a point, whereas global conditions deal with the behavior of a function over the domain (i.e. an interval). For instance, between two points, b > a on the x-axis, a function can rise or fall many times with varying amplitudes. Locally, many maximums and minimums may exist in the interval. Globally, over the whole interval, b - a, only one highest peak or one lowest trough is likely to exist (Frisch, 1965, 4).

In the 1953 article, Samuelson began to address more general cases beyond the 2 x 2 trade model. This meant that he had to look for global conditions for a unique solution of relationships between commodity and factor prices. He began by relating these prices with equations of the following form:

[P.sub.i] = [A.sub.i]([w.sub.1] .... [w.sub.r]) and [[partial derivative][p.sub.i]/[partial derivative][w.sub.j]] = [[a.sub.ij]] (7)

Where p is the commodity price with i = 1 ... n, w is the factor price with j = 1 ... r, and the coefficient, [a.sub.ij], represents the required amount of input, j, to produce a unit of the good, i (Ibid., 889-890).

Samuelson considered three cases in interpreting equation (7). Case (i): Equal goods and factors (n = r). This case deals with the situation where the number of factors and goods are equal, (the n x n case). The conclusion is that "... if two countries have the same production functions, and if they do produce in common as many different goods as there are factors, and if the goods differ in their "factor intensities," and if there are no barriers to trade to produce commodity price differentials, then the absolute returns of every factor must be fully equalized" (Ibid., 893).

Case (ii): More Goods than Factors (n > r): In this case, the number of factors is less than the number of goods, resulting in more commodity equations than factors to be determined. This may be called an over determined system. Samuelson argued that if prices are arbitrarily fixed, then certain industries will shut down, reducing Case (ii) to Case (i). If the market determines prices, however, r factor prices will adjust to the market price, and the (n-r) prices will require factor endowments to be determined.

Case (ii) can be looked at from the point of view of the least-square problems in regression analysis where the number of observations is greater than the coefficients to be determined. In such a case, the best solution can be determined. As in this statistical over-determined system, we project the observation onto a line, and similarly we can imagine projecting the factor prices space into the commodity price space. We can think of the price space as the four walls of a room, and the factor space as just one wall. The projection is therefore a projection of the w-space to a subspace of the p-space. This projection restricts the commodity space from its (n-r) dimension, to be compatible with the r-dimension of the factor space.

Case (iii): More Factors than Goods (n < r): This is an under-determined system characterized by less equations than unknowns to be solved. Samuelson proposed the adding of an equation for endowments to enable a solution.

Solution for ease (i) where the number of commodity and goods prices are the same Essentially, equation (7) is a mapping between commodity prices and factor prices, namely:

f : p [right arrow] w or [p.sub.i] = [f.sub.i](w) (8)

If [J.sub.f](w) = [partial derivative][f.sub.i](w)/[partial derivative][w.sub.j], then we can find w = [f.sup.-1.sub.j] p for j = 1 ... n (McKenzie, 1967, 272).

For n = r = 2, the global association between wages and prices were kept in alignment by their factor intensity. For instance, we could argue that a rise in the price of a good that was produced by a labor-intensive technique will lead to an increase in the price of labor to produce that good. The non-vanishing of the Jacobian determinant discussed in equation (7) satisfied that factor intensity condition. In general, Alan Turing pointed out to Samuelson that the determinant of the Jacobian may not vanish. The reason for non-vanishing Jacobian determinants in the large includes the occurrence of factor-intensity reversal. But as Ivor Pearce put it, "A 3 x 3 determinant can easily be zero for a great many reasons totally unconnected with factor intensities" (Pearce, 1979, 496).

In 1953, Samuelson refocused his attention on the Jacobian determinant of the [a.sub.ij] of equation (7) to satisfy the inverse requirements of equation (8). Samuelson wrote: "Fortunately, the economics of the situation was clearer than my mathematical analysis; because all the elements of the Jacobian represented inputs or a' s, they were essentially one-signed; and this condition combined with the non-vanishing determinant, turns out to be sufficient to guarantee uniqueness in the large" (Samuelson, 1966, V. 2, 903). He proceeded to give sufficient conditions for a unique solution for the global case. He first renumbered the p' s and w' s in the differentiable equation (7), such that the successive principal minor of the partial derivatives are non-vanishing for all w' s (Ibid., 903). To refresh the terminology, given an element of a matrix, a minor is a matrix form by deleting the row and column that is associated with the given element (Strand, 1988, 226). Let A be an m x n matrix, where i = 1 ,..., m and j = 1,..., n. If i = j, [A.sub.ij] is called a principal minor. These minors always involve the successive members on the principal diagonal, which is the diagonal running from the northeast to the southwest. Equation (9) below lists all the principal minors for the 2 x 2 case.


According to Kenneth Arrow and Frank Hahn (1971, 242), Samuelson's 1953 proposition "... paid insufficient attention to the domain of the mappings" between the p' s and w' s of equation (7). Gale and Nikaido (1965, 82), provided a counter example to illustrate this point. Given a mapping by the equations f(x, y) = [e.sup.2x] - [y.sup.2] + 3, and g(x, y) = 4[e.sup.2x,] y - [y.sup.3], then [D.sub.1] = 2[e.sup.2x], and [D.sub.2] = 2[e.sup.2x] (4[e.sup.2x] + 5[y.sup.2]) (which are both positive). But for the domain points (0,2) and (0,-2) the functions are mapped to the origin which is zero.

Gale and Nikaido (Ibid., 681) provided the domain element by arguing that "... if all principal submatrices of the Jacobian matrix have positive determinants, the mapping is univalent in any rectangular region." Formally speaking, the Gale-Nikaido theorem can be stated as follows:

Given a map F : [R.sup.n] [right arrow] [R.sup.n]. Let the domain of the map be rectangular, i.e., [omega] = {x [member of] [R.sup.n] : [p.sub.i] [less than or equal to] x [less than or equal to] [q.sub.i], (i = 1,2,...,n)}. Let the components of the map in that domain, F(x) = [f.sub.i](x) be differentiable ([C.sub.1]) (i.e. its total differential exist at each point x of [omega]). Then the mapping F: [omega] [right arrow] [R.sup.n] is univalent (one-to-one) if the Jacobian matrix, J(x) has strictly positive principal minors, strictly negative principal minors, or is positive quasi-definite everywhere in a convex set [omega].

In the Gale-Nakaido Theorem, the Jacobian matrix is called a P - matrix. Positive quasi-definite means that for each vector, x [not equal to] 0 and x' Ax > 0. Univalence of the mapping means that for each element in the domain there can be only one element (image) in the range. As a practical matter, one can test this concept in the plane by using a vertical line on the graph to see if the line cuts the graph at only one point. But a function defined globally may not refer to a point on a graph such as when the graph is sketched, or find its extrema over the real numbers, or when we look for undefined points for each element in the domain.

The 1965 Samuelson postscript proceeds to accommodate Gale and Nikaido's findings through "... a naturally ordered set of principal minors ... everywhere in the Euclidean n-space, bordered by two positive numbers" (Samuelson, V. 2, 1966, 908). What is new here is that the sequence of principal minors has determinants that are bounded away from zero. This condition was earlier foreshadowed by McKenzie's dominant diagonal (DD) matrix (1960, 49). He argued that "... an n x n matrix A is said to have a dominant diagonal if [absolute value of [a.sub.ij]] > [[summation].sub.i [not equal to] j] [absolute value of [a.sub.ij]] for each j." In simple terms, the DD condition states that in each row of the [a.sub.ij] matrix, the main diagonal element must be greater than the sum of the other row elements when the comparison is in absolute value terms. A dominant diagonal "... means that each good can be identified with a factor that is uniquely important in the production of that good" (Ibid., 54). Gale and Nikaido provided a P-matrix that includes the DD matrix as a special case. "If a matrix with a dominant diagonal has positive diagonal entries, then it is a P-matrix (Gale and Nikaido, 1965, 84).

One problem with the Gale Nikaido theorem is that it is "over sufficient". According to Pearce, "... for each condition of a rectilinear region satisfying the Gale-Nikaido condition, and hence possessing an inverse, it is possible to construct an infinity of mapping not satisfying the conditions which nevertheless posses an inverse also" (Pearce, 1970, 525). Andreu Mas-Collel has also given two propositions in the direction of weakening the strong assumptions on the univalence condition of the Gale-Nikaido theorem. His two propositions are considered as a generalization of the Gale-Nikaido theorem discussed above. The two propositions are listed as follows:

Proposition I (Samuelson-Nikaido-Mas-Collel): The restriction on the principal minor of the w 0f'(w) is irrelevant. All input share function, w/[f.sub.i](w) x [partial derivative][f.sup.i](w)/[partial derivative][w.sup.j], that matters is that the determinant of the function be uniformly bounded away from zero in order to attain global univalence within the strictly positive orthant, [[R.sup.l].sub.++].

Proposition II (Gale-Nikaido-Mas-Collel): The condition of the cost function within the [R.sup.l.sub.+] domain can be weakened, based on the general [C.sup.1] function on compact polyhedral (MasCollell, 1979b, 324).

Following his 1965 postscript, Samuelson answered a question raised by Jagdish Bhagwati regarding the difference between the rental rate on capital and the interest rate, which is a capitalization of the rental rate on capital. Working with the equation that GNP = NNP + Depreciation, he rewrote the price-cost equation for each industry (capital goods, food, and clothing) as [p.sub.i] = [a.sub.i]w + [b.sub.i]r [p.sub.o] + [m.sub.i] [b.sub.i] [p.sub.0], where the a's, b's and m's are labor, capital, and depreciation coefficients respectively, and [p.sub.0] is the price of capital goods. Samuelson then proceeded to apply the method of his modified Gale-Nikaido ideas in his postscript to discuss the solution (Samuelson, 1966, V. 2, 912-915). The result indicated that the rate of interest is inversely related, and the real wage directly related to the wage-rental ratio (Ibid., 916).

In 1967, Samuelson summarized the factor-price equalization literature. He then pursued the development of the use of factor endowments to bring about a solution. The model was now convened to a maximization problem using Lagrange multipliers (Samuelson, V. 3, 1972, 351). The Hessian matrix formed for the partials of the Lagrangian equations does not satisfy the Gale-Nikaido conditions. "The fact that its principal minors formed from crossing out any r < [n.sup.2] of the first [n.sup.2] rows and columns are of the sign needed for the maximum suffices, I believe, to assure univalence of the equation set" (Ibid.,).

Regarding the factor price equalization theorem, John Hicks considered two sets of equations for two countries, namely, ar + bw = ar'+ bw', and cr + dw = cr' +dw'.

The prime is used to distinguish the equation for the other country. A simultaneous solution of both equations yields r' - r = a/b (w - w') = d/c(w - w'), where the ratios indicate capital-intensities. Since capital-intensities can differ, they can only differ if (w - w') is the same, yielding, r = r' (Hicks, 1983, 226). Hicks concluded that "... the analysis which emerges does not sound to be so unrealistic. It sounds to me like ringing tree" (Ibid., 233).

According to Findlay (1995, 7), although the Factor-price model is also credited to Abba Lerner, it was Samuelson who first introduced it to the economic profession. Findlay quoted a rare citation of Samuelson, pointing out another of Samuelson's novel contribution in regards to the H-O model as well. "Already in 1924 Ohlin has melded Heckscher and Walras. But neither then, nor in 1933 and 1967, did Ohlin descend from full generality to strong and manageable cases--such as two factors of production and two or more goods. What a pity. Not only did Ohlin leave to my generation these easy pickings, but in addition he would for the first time have really understood his own system had he played with graphable versions" (Ibid., 7).


Samuelson's major contribution to microeconomics is in the area of consumer choice. Traditional theory predicts a consumer choice from assumptions about their tastes and preferences. Samuelson began with the assumption of choice (i.e., let the consumer select one item over another). This has become the theory of "revealed preference" (Varian, 2006, 99). Preferences are deduced for the choices consumers make in the market place, based on commodity prices and consumer's income, thereby moving the analysis from the realm of the unobservable taste and preference to the world of observable choices.

In his original paper, Samuelson gave three postulates: (1) A single-value function on prices and income, subject to a budget constraint; (2) Homogeneity of order zero so as to make consumer behavior independent of the units of measurement of prices. (Given vectors of two goods, [PSI] and [PSI]', with their respective price vectors, p and p', forming their inner product yields: [[PSI]p], and [[PSI]'p'] [[PSI]'p']), and (3) "if this cost [[PSI]'p] is less than or equal to the actual expenditures in the first period when the first batch of goods [[psi]p] was actually bought, then it means that the individual could have purchased the second batch of goods with the price and income of the first situation, but did not choose to do so. That is, the first batch ([psi]) was selected instead of ([psi]')" (Samuelson, 1966, V. 2, 7).

In making choices, Samuelson assumed consumers need to be consistent. "If an individual selects batch one over batch two, he does not at the same time select two over one" (Ibid.,). In a latter note, Samuelson integrates the first two propositions with the third concluding that "postulates 1 and 2 are already implied in postulate 3, and hence may be omitted" (Ibid., 13).

With those assumptions, Samuelson pronounced that "... even within the framework of the ordinary utility- and indifference- curve assumptions, it is believed to be possible to derive already known theorems quickly, and also to suggest new sets of conditions. Furthermore ... the transitions from individual to market demand functions are considerably expedited" (Ibid., 23). But the revealed preference theory matured into an even more powerful rival research paradigm.

In 1950, Samuelson wrote "I suddenly realized that we could dispense with almost all notions of utility; starting from a few logical axioms of demand consistency; I could derive the whole of the valid utility analysis as corollaries" (Samuelson, 1966, V. 1, 90). He proceeded to make the following axioms:

* "Weak Axiom: If, at the price and income of situation A you could have bought the goods actually bought at a different point B and if you actually chose not to, then A is defined to be 'revealed to be better than' B. The basic postulate is that B is never to reveal itself to be also 'better than' A" (Samuelson, 1966, V. 1, 90).

* "Strong Axiom: If A reveals itself to be 'better than' B, and if B reveals itself to be 'better than' C, and if C reveals itself to be 'better than' D, etc...., then I extend the definition of 'revealed preference' and say that A can be defined to be 'revealed to be better than' Z, the last in the chain. In such cases it is postulated that Z must never also be revealed to be better than A" (Ibid.,).

In 1953, Samuelson then elevated the revealed preference theory to the empirical domain: "... consumption theory does definitely have some refutable empirical implications", or we can "score the theory of revealed preference" (Ibid., 106). Samuelson required a benchmark to allow refutation/scoring, for which he postulated this fundamental theorem: "Any good (simple or composite) that is known always to increase in demand when money income alone rises must definitely shrink in demand when its price alone rises" (Ibid., 107). He then proceeded "to show that within the framework of the narrowest version of revealed preference the important fundamental theorem, stated above, can be directly demonstrated (a) in commonsense words, (b) in geometrical argument and (c) by general analytic proof" (Ibid., 108).

Hildenbrand, a modern mathematical economist, appraised the revealed preference theory as follows: "Instead of deriving demand in a given wealth-price situation from the preferences, considered as the primitive concept, one can take the demand function (correspondence) directly as the primitive concept. If the demand function f reveals a certain 'consistency' of choices ... one can show that there exists a preference relation ... which will give rise to the demand function f" (Hildenbrand, 1974, 95).


Many attempts to look at Samuelson's contribution to economics have only been able to examine his findings. Writers pick the fruits of his erudition but ignore the tree that generated them. Elsewhere, we have looked at various ways to know Samuelson by looking at his character. We have also looked at him as a Wunderkind, and what of his views will survive in the 21st century. Here, we went behind his major topical contributions to feel the depth of his thoughts, particularly in the areas of capital and trade, and to use a Newtonian expression, leaving the sea of his discovery for others to investigate.

This memoriam took a peek at some trunks of the tree, not going off on its many branches. We looked at trade because Samuelson thought that it is the one theory that is true but cannot be proved. We looked at capital theory because it has engaged some of the best minds in economics for the last half a century on both sides of the Atlantic. Research into trade and capital theory is still ongoing, leaving room for more research and debate. No one will doubt that from the touch of Samuelson's hand, economics has become highly transparent and knowledge elevated, more so in some areas than in others.


Arrow, Kenneth J., and F. H. Hahn, General Competitive Analysis, (San Francisco, CA: Holden-Day, 1971).

Bhagwati, Jagdish, Essays in International Economic Theory: The Theory of Commercial Policy, edited by Robert C. Freenstra, Vol. I, (Cambridge, MA: The MIT Press, 1983).

Burmeister, E., and A. R. Dorbell, Mathematical Theories of Economic Growth, (New York: The Macmillan Company, 1970), Chapter 4.

Dorfman, Robert, Paul A. Samuelson and Robert M. Solow, Linear Programming and Economic Analysis, (New York: Dover Publication, Inc., 1958).

Ferguson, C. E., "Transformation Curve in Production Theory: A Pedagogical Note," Southern Economic Journal, Vol. 29, No. 2 (Oct., 1962), 96-102.

Ferguson, C. E., The Neoclassical Theory of Production and Distribution, (New York: Cambridge University Press, 1975).

Findlay, Ronald, Factor Proportions, Trade, and Growth, (Cambridge, MA: The MIT Press, 1995).

Frisch, Ragnar, Maxima and Minima: Theory and Economic Applications, in collaboration with A. Nataf, (Chicago, Ill.: Rand McNally and Company, 1966).

Gale, David and Hukukane Nikaido, "The Jacobian Matrix and Global Univalence of Mappings," Mathematische Annalen, 1965, 68-80, reprinted in Peter Newman, edited, Readings in Mathematical Economics, (Baltimore, MD: Johns Hopkins University Press, 1968), 81-93.

Garegnani, P., "Heterogeneous Capital, the Production function and the Theory of Distribution," Review of Economic' Studies, Vol. 37, 1970, pp. 407-436.

Hayek, F. A., Collected Works, Vol. 4, (University of Chicago Press, 1992).

Hicks, John, Collected Essays on Economic Theory: Classics and Moderns, Vol. III, (Cambridge, MA: Harvard University Press, 1983).

Hildenbrand, Werner, Core and Equilibria of A Large Economy, (Princeton University Press, 1974).

Krugman, Paul, "Increasing Returns, Imperfect Competition and The Positive Theory of International Trade," in Handbook of International Economics: Vol. 3. Edited by Gene M. Grossman and Kenneth Rogoff, (North Holland: Elsevier, 1995), 1243-1277.

Lutz, F. A, "The Essentials of Capital Theory," in Friedrich Lutz and D. C. Hague, edited, The Theory of Capital, (New York: Macmillan 1961), 3-17.

Marshall, Alfred, The Correspondence of Alfred Marshall: Vol. Three: Towards the Close, 1903-1924, Edited by John K. Whitaker, (New York: Cambridge University Press, 1996).

Mas-Collel, Andreu, The Theory of General Economic Equilibrium: A Differentiable Approach, (New York: Cambridge University Press, 1985).

Mas-Collel, Andreu, "Homeomorphism of Compact, Convex Sets and the Jacobian Matrix," Siam Journal of Math. Anal., Vol. 10, No. 6, November 1979a, 1105-1109.

Mas-Collel, Andreu, "Two Propositions on the Global Univalence of Systems of Cost Functions", in General Equilibrium Growth and Trade, J. Green and J. Scheinkman, editors, (New York: Academic Press, 1979b), 323-331.

McKenzie, "Matrices with Dominant Diagonals and Economic Theory," in Kenneth J. Arrow, Samuel Karlin, and Patrick Suppes, edited, Mathematical Methods in the Social Sciences, (Stanford, CA: Stanford University Press, 1960), 47-62.

McKenzie, Lionel W., "The Inversion of Cost Functions: A Counter-Example," International Economic Review, Vol. 8, No. 3 (Oct., 1967), 271-278.

Morishima, Michio, Equilibrium, Stability and Growth: A Multi-Sectoral Analysis, (Oxford, U. K.: Oxford University Press, Inc., 1964).

Lerner, Abba, "Factor Prices and International Trade," in Abba P. Lerner, Essays in Economic" Analysis, (London: Macmillan, 1953, 67-84).

Mas-Colell, A. 1979. "Two propositions on the global univalence of systems of cost functions" in General Equilibrium, Growth, and Trade, J. Green and J. Scheinkman, editors, (New York: Academic Press, 1979), 323-311.

Nell, E. J., "Theories of Growth and Theories of Value," Economic Development and Cultural Change, Vol. 16, 1967, 15-26.

Pasinetti, Luigi L., "Paul Samuelson and Piero Sraffa-Two Prodigious Minds at the Opposite Poles," in Michael Szenberg, Lall Ramrattan, and Aron A. Gottesman, editor, Samuelsonian Economics and the Twenty-First Century, (New York: Oxford University Press, 2006) 146-164.

Pearce, I. F., International Trade: A Survey of Principles and Problems of the International Economy, (New York: W. W. Norton and Company, 1970).

Ramsey, F. P., "A Mathematical Theory of Saving," The Economic" Journal, Vol. 38, No. 152 (Dec., 1928), 543-559.

Robinson, Joan, "The Measure of Capital: The End of the Controversy," The Economic Journal, Vol. 81, No. 323 (Sep., 1971), 597-602.

Robinson, Joan, "Capital Theory up to Date," The Canadian Journal of Economics, Vol. 3, No. 2 (May, 1970), 309-317.

Samuelson Paul A. The Collected Scientific Papers of Paul A. Samuelson, edited by Joseph E. Stiglitz. Vols. 1 and 2, (Cambridge, MA: MIT Press, 1966).

Samuelson Paul A. The Collected Scientific Papers of Paul A. Samuelson, edited by Robert C. Merton. Vol. 3, (Cambridge, MA: MIT Press, 1986).

Samuelson Paul A. The Collected Scientific Papers of Paul A. Samuelson, edited by H. Nagatani and K. Cowley. Vol. 4, (Cambridge, MA: MIT Press, 1986).

Samuelson Paul A. The Collected Scientific Papers of Paul A. Samuelson, edited by Kate Crowley. Vol. 5, (Cambridge, MA: MIT Press, 1986).

Samuelson Paul A. Foundations of Economic Analysis. Harvard Economic Studies, Vol. 80 (1947), References are to the Seventh Reprint, (New York: Atheneum, 1974).

Samuelson Paul A. "My Life Philosophy: Policy Credos and Working Ways." In Eminent Economists, edited by Michael Szenberg, (New York: Cambridge University Press, 1992).

Samuelson. Paul A., Economics, (New York: McGraw-Hill Book Company, 1980).

Samuelson, Paul A., and William D. Nordhaus. Economics. 15th ed., (New York: McGraw Hill, 1995).

Solow, Robert, "Overlapping-Generation" in Michael Szenberg, Lall Ramrattan, and Aron A. Gottesman, edited, Samuelsonian Economics and the Twenty-First Century, (New York: Oxford University Press, 2006) 35-41.

Szenberg, Michael, Lall Ramrattan and Aron A. Gottesman, Samuelsonian Economics and the Twenty-First Century, edited, (New York: Oxford University Press, 2006a).

Szenberg, Michael and Lall Ramrattan, Paul A. Samuelson, Dictionary of American Economists, (Devon: Thoemmes Press, 2006b).

Szenberg, Michael, Aron Gottesman, and Lall Ramrattan, On Being an Economist, (New York: Jorge Pinto Books, New York, 2005).

Szenberg, Michael, Lall Ramrattan and Aron Gottesman, "Ten Ways to Know Paul Samuelson, in Economics of Education Review, (Spring 2006).

Szenberg, Michael, Lall Ramrattan and Aron Gottesman, "Paul A. Samuelson: Philosopher and Theorist," International Journal of Social Economics", Vol. 32, No. 4, 2005, 325-338.

Takayama, Akira, International Trade: An Approach to the Theory, (New York: Holt, Rinehart and Winston, 1972).

Varian, Hal R. "Revealed Preference", in Michael Szenberg, Lall Ramrattan, and Aron A. Gottesman, edited, Samuelsonian Economics and the Twenty-First Century, (New York: Oxford University Press, 2006).

Walras, Leon, Elements of Pure Economics, trans. William 1 Jaffe, (New York: Augustus M. Kelley, Reprints, 1926 [1969]).

Wicksell, Knut, Lectures on Political Economy, Vol. I, (London: George Routledge, 1911 [1934]).

Wicksell, Knut, "Letter to Alfred Marshall, June 1, 1905," in The Correspondence of Alfred Marshall, Volume Three: Towards the Close, 1903-1924, edited by John K. Whitaker, (New York: Cambridge University Press, 1996), 102.

Lall Ramrattan, University of California, Berkeley Extension

Michael Szenberg, Corresponding author: Lubin School of Business, Pace University,
COPYRIGHT 2010 Omicron Delta Epsilon
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2010 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Ramrattan, Lall; Szenberg, Michael
Publication:American Economist
Article Type:Report
Geographic Code:1U2NY
Date:Sep 22, 2010
Previous Article:The effects of option introduction on analyst coverage and earnings estimates.
Next Article:Actuarially fair premia for deductible insurance policies.

Terms of use | Privacy policy | Copyright © 2019 Farlex, Inc. | Feedback | For webmasters