Tjalling Charles Koopmans, August 28,
1910February 26, 1985 | By Herbert E. Scarf | Biographical Memoirs
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Tjalling Charles Koopmans
August 28,
1910 February 26, 1985
By
Herbert E. Scarf
|
TJALLING CHARLES KOOPMANS, one of the central figures in
modern economic science, played seminal roles in the
modern theory of the allocation of scarce resources and in the
development of statistical methods for the analysis of economic data. In
both of these areas Koopmans creatively mobilized and developed the
methods of other quantitative disciplines for the purposes of economics:
mathematical statistics became econometrics, and linear programming
became the activity analysis model of production. Koopmans was also one
of the major scholars concerned with the study of economic growth and
the economic consequences of the depletion of nonrenewable resources. He
was a remarkably inspired and inspiring leader of research who combined
his considerable mathematical power with a deep concern for the ultimate
practical applications of his work.
Koopmans was
born in the village of 's Graveland, near the town of Hilversum, in the
Netherlands, on August 28, 1910; he was the third son of Sjoerd Koopmans
and Wijtske van der Zee. Both his mother and father were born in Frisia,
a province in northeastern Holland. Sjoerd's father was the owner of a
small shop in the rural area of Toppenhuizen; Wijtske's father was a
painter of fancy carriages and also an artist who painted many
landscapes and portraits, now owned by his great-grandchildren. The
family in which Sjoerd grew up was severe and Calvinistic, in contrast
to Wijtske's family, which was more relaxed and liberal about religious
matters. At the age of sixteen Sjoerd became the schoolteacher of a
small school in Toppenhuizen and was entrusted with the education
(including bible instruction) of the neighborhood children. He was said
to have been very stern in the classroom, perhaps as a consequence of
the many responsibilities he assumed at so early an age. Wijtske was
also trained as a schoolteacher and, after their marriage, the couple
left Frisia and eventually settled in 's Graveland, where Koopmans's
father became the principal of a much larger "school with the bible."
The family house, as Koopmans described it in an
autobiographical sketch written when he received the Nobel Prize in
economic sciences in 1975,
. . . was squeezed between
two sections of that school. The row of these three buildings was, as
[were] almost all houses in the village, sandwiched between one long
street and a parallel straight and narrow canal marking one of the
village's boundaries. Across the street were large wooded estates each
with meadows and a large mansion. The occupants of the mansions kept
aloof from the life of the village except for the employment of
coachmen, gardeners, servants and contractors.
Every weekday morning at nine, our living quarters
and the narrow strip of garden at the back were engulfed by the sound of
three different hymns sung dutifully, simultaneously, but independently
in true Charles Ives fashion, by the schoolchildren on both sides.
Despite frequent illnesses Koopmans had a happy
childhood in this rural environment, with its many meadows and canals.
His formal education began at his father's school, with its heavy
emphasis on biblical studies, and was followed by five years at the
Christian High School at Hilversum, some ten miles away. At the high
school Tjalling studied Latin, Greek, mathematics, physics, chemistry,
and three modern languages. He was instructed in the theory of evolution
by a teacher who remarked at the end of the course, "the Bible says
otherwise."
The Koopmans family was very musical,
and sang together regularly. Sjoerd played the harmonium, and Tjalling
was taught the violin as a child. He was not entirely satisfied with his
skill on this instrument, and in his later life he replaced the violin
with the piano. Both secular and sacred learning were highly valued in
the Koopmans household. There were prayers before every meal and Bible
reading in the evening, with the servants called from the kitchen to
collect around the dinner table and participate in religious
instruction. Tjalling's father was the dominant influence, and the
atmosphere in the home and the school was a stern and disciplined one.
Tjalling left home for the University of Utrecht
at the age of seventeen. At Utrecht, boarding was arranged with the
minister to the city prisons, whose surname was Couvée. This was
an experience very different for Koopmans from living at home; there
were many young children, some close to Tjalling in age, and much lively
social activity. Due to his post the father of the family had seen a
good deal of the raw life of the city, and, while religious, he was not
strict nor dogmatic. The mother was French, and Tjalling became quite
comfortable with the language. He stayed with the family for two years.
It was customary for a young man to take formal
religious vows at the age of seventeen or eighteen. Koopmans wrestled
with the issue for a considerable period of time, and, in what was a
difficult experience both for himself and his parents, he formally
renounced his ties to the Protestant faith while at the university. But
the moral and educational values of his early home remained with him and
were probably the central source of the great personal integrity and
strong sense of purpose that he displayed throughout his lifetime.
Koopmans's academic abilities must have been apparent
quite early, for he was awarded a generous stipend by a private
foundation--the St. Geertruidsleen--at the age of fourteen. This
scholarship supported his studies until his twenty-sixth birthday and
relieved his family of the financial burden of his education. At the
university Koopmans commenced with the study of mathematics--in
particular, analysis and geometry. He had a vivid geometrical intuition,
and, in many of his subsequent publications, elaborate analytical
arguments are frequently simplified by the use of insightful geometrical
figures. He read widely in other subjects, ranging from physics to
history, psychology, and psychiatry. For a while he contemplated
entering the profession of psychiatry, but, in a somewhat less dramatic
change of field, he moved (in 1930) from pure mathematics to theoretical
physics. This shift in subjects, a first step toward his eventual
decision to take up economics, was "a compromise between my desire for a
subject matter closer to real life and the obvious argument in favor of
a field in which my mathematical training could be put to use."
Koopmans's professor at Utrecht was Hans Kramers, the
leading theoretical physicist in Holland at the time. He admired Kramers
enormously and described him as "a humane and inspiring person with a
gentle wit." In 1933 Koopmans wrote an important paper on quantum
mechanics, which is still frequently cited by physicists many years
after its publication. But, of course, these were the years of the Great
Depression, and theoretical physics must have seemed remote from the
distress of daily economic life. As Koopmans later said, "It dawned on
me that the economic world order was unreliable, unstable, and most of
all, iniquitous." He began, at the suggestion of fellow students, to
read the works of Karl Marx; this was his first exposure to abstract
economic reasoning. While he was not persuaded by Marxian economic
analysis, he felt deeply moved by Marx's description of the plight of
workers during the Industrial Revolution.
It was
at this point that Koopmans was introduced to Jan Tinbergen, who was
seven years older and already one of the leaders in the new field of
mathematical economics. Tinbergen, who was to share the first Nobel
Prize in economic science with Ragnar Frisch in 1969, had been trained
in mathematical physics as a student of Ehrenfest. He had been a
conscientious objector to military service at the age of eighteen and,
as an alternative obligation, was required to spend some time at the
Statistical Office in the Hague, where he became acquainted with and
concerned about social and economic issues. Despite his change in
interest Tinbergen continued to work with Ehrenfest; his Ph.D. thesis,
written in 1929 at Leiden, was on the topic of minimization problems in
both physics and economic theory. After receiving his degree Tinbergen
began to develop the elements of a mathematical theory of business
cycles and to construct a formal mathematical model of the Dutch
economy.
Koopmans decided to affiliate himself
with Tinbergen. He moved from Utrecht to Amsterdam in January of 1934
and joined a group of Tinbergen's young disciples, among them Truus
Wanningen, whom Koopmans was to court and, finally, marry in October
1936.
Tinbergen offered a weekly lecture in
economics, which Koopmans attended. As he later said in his Nobel
biographical sketch,
In the first half of that year
[1934], I had the privilege of almost weekly private tutoring from him
over lunch after his lecture. I have been deeply impressed by his
selflessness, his abiding concern for economic well-being and greater
equality among all of mankind, his unerring priority at any time for
problems then most crucial to these concerns, his ingenuity in economic
modeling and his sense of realism and wide empirical knowledge of
economic behavior relations.
Tinbergen instructed
Koopmans in many aspects of mathematical economics and econometrics. He
suggested that Koopmans read the works of the theorists Cassel and
Wicksell and that he become familiar with the field of statistics and
its applications to economic problems.
Tinbergen
had a profound influence on Koopmans's professional career, and it may
be useful to make a brief digression about Tinbergen's work on business
cycles and macro-economic models. In order to place this work in
perspective, let me describe a fundamental distinction between two
attitudes toward dynamic models in economic theory. We are all familiar
with the basic idea that prices are determined so as to equate the
supply and demand for goods and services. In its most elementary form,
the demand for a particular commodity may be thought of as a function of
its price (and perhaps the prices of other competing commodities) and
demand declines as the price rises. Similarly, the supply brought forth
by producers of a particular commodity may be viewed as a function of
the price at which the commodity may be sold (and the prices of the
factors of production required to manufacture the commodity); typically,
the supply of a commodity rises as its price increases. The static
equilibrium price is at the intersection of these two curves.
Suppose that we wish to examine a dynamic variant in
which the commodity is produced and consumed at a sequence of
consecutive points of time. On the one hand, we can imagine that the
production and consumption decisions are made in the presence of perfect
futures markets and with the full knowledge of the prices that are
expected to prevail over time. Making use of this information, producers
purchase factors of production and consumers purchase outputs at times
when they are inexpensive and store them for future use, seeking to
smooth their production and consumption plans over time. On the other
hand, we can imagine that the imperfections of financial institutions
require that such choices be made in a myopic fashion, attending only to
those prices and values of other significant economic variables that
prevail today.
In the first version, prices would
clear both spot and futures markets instantaneously; the model would
describe an economic situation of full dynamic equilibrium with no
underemployment of resources. In the latter variant, markets would
respond sluggishly to previous signals and the evolution of the economy
might best be described by a mathematical system in which the future
values of major economic variables are an extrapolation of their past
values.
Clearly, the depression years of the early
1930s could not be accurately described by a classical model in which
all economic resources are fully employed. Tinbergen was drawn to the
alternative formulation, which had played an important role in the
analysis of business cycles and which was ultimately to lead to the
Keynesian model. For example, Tinbergen published a paper in 1931 in
which cycles in shipbuilding are analyzed by means of a simple
difference-differential equation stating that the increase in available
shipping tonnage at a particular time is related linearly to the stock
of tonnage with a fixed time delay. There is no explicit consideration
of freight rates or the costs of constructing new shipping. Freight
rates are examined in subsequent papers but not in the neoclassical
manner as those prices that equilibrate the demand for shipping services
with its supply. Instead, Tinbergen engaged in skillful curve fitting;
he fitted a regression of freight rates to a pair of indices purporting
to measure the demand and supply of shipping services and the cost of
coal.
A number of themes that appear in these
early works of Tinbergen became major influences in Koopmans's later
research agenda. Tinbergen's concerns with the shipping industry were to
stimulate Koopmans's subsequent interest in formal mathematical models
of transportation. Tinbergen's use of statistical analysis opened up a
series of questions that were to preoccupy Koopmans and other scholars
for many years, and Koopmans's fundamental research in economic growth
theory very probably had its roots in the early dynamic models of
Tinbergen.
Koopmans's Ph.D. dissertation, titled
"Linear Regression Analysis of Economic Time Series," was supervised
jointly by Tinbergen and Kramers; the degree was granted in November
1936. In retrospect, this thesis can be seen as an important step in the
development of modern econometric methodology. By the 1930s economists
had already been exposed to the use of regression analysis and other
statistical techniques in analyzing the relationship between the demand
for a particular good and its price and in the study of business cycles.
The parameters in Tinbergen's model of the Dutch economy had been
estimated using multiple correlation analysis with a degree of care and
detail not seen in previous economic reports, and Frisch had developed
his own ingenious statistical methods. But the new paradigm for
statistics offered by R. A. Fisher had not yet found its way into
econometric analysis prior to Koopmans's thesis.
The major innovation suggested by Fisher was an
assessment of the merits of various statistical methods based on a
formal probabilistic model. To take an important example, consider a set
of observations
(
y
i
,
x
i
)
i=1,...,T
of a
dependent variable
y
and an independent variable
x
. A
linear relationship,
y =
x +
, between
these two variables can be obtained by a least squares regression of
y
on
x
. But such a regression is essentially an exercise
in curve fitting, and the parameters could equally well be found by
other contending methods, such as one that minimizes the sum of the
absolute values of the deviations, rather than the sum of their squares.
In order to justify the use of one particular method, Fisher introduced
an underlying probabilistic model that is assumed to generate the
observed data. For example, assume that the observations
y
i
are independently drawn from normal distributions
with means
ax
i
+
b
, and with a common standard
deviation
. Given the parameters
a
,
b
, and
and the sequence of values of the independent variable
x
= (
x
1
,...,
x
T
), the probability
of observing the sequence
y
=
(
y
1
,...,
y
T
) can be expressed as a
function
F
(y|
a
,
b
,
;
x
). For the
observed sequence (
y
,
x
), Fisher suggests that the
parameters
a
,
b
, and
be selected so as to maximize
this likelihood function, that is, to select those parameters that give
the highest probability to the sequence of observed data.
Economic data are distinctly different in at least two
very significant ways from those arising in the agricultural experiments
that motivated Fisher's analysis. Economic data are similar to
astronomical observations in the sense that they are natural
observations that do not arise in experimental laboratories. The
independent variables
x
, which might represent temperature and
other experimental parameters in Fisher's controlled experiments could,
in an econometric study, become the prices at which a sequence of
commodity demands were observed. But even if prices were thought of as
being independent variables in the sense that the price of food would
cause a certain level of demand for food to arise, these prices could
not be set by the experimenter and would, themselves, be measured with
error.
After an exposition of Fisher's program,
Koopmans's thesis contains a lucid set of proposals for accommodating
the particular econometric problem that all of the relevant variables
might be measured with error. He does not, at this point, address a
second major problem, that is, the fact that causal connections are far
from obvious in economics and the values of many economic variables
might very well be considered to be simultaneously determined. This is a
point that will arise again.
In the period 1936-38
Tinbergen was called to the League of Nations at Geneva to find out,
with the aid of statistics, which theory of the business cycle was
closest to reality. At Geneva Tinbergen also prepared a business-cycle
model of the United States. Koopmans took over the teaching of his class
in mathematical economics at the Netherlands School of Economics in
Rotterdam. During this time Koopmans embarked on a lengthy study of the
relationship between freight rates and the construction of oil tankers.
The study was not based on a formal mathematical model, but it did
display a sure grasp of economic theory and a detailed knowledge of the
tanker industry that was remarkable for a young scholar recently
preoccupied with mathematical physics. The work was published as a
monograph titled
Tanker Freight Rates and Tankship Building
by
the Netherlands Economic Institute in 1939. There is a clear
foreshadowing in the monograph of Koopmans's subsequent interest in the
construction of optimal transportation routes.
In
1938 Tinbergen and Koopmans exchanged places. Tinbergen returned to
Rotterdam and Dr. and Mrs. Koopmans moved to Geneva, where Koopmans was
assigned the task of constructing a mathematical model of the United
Kingdom's economy. In early 1939 he attended a conference on Tinbergen's
work at Oxford University. At the conference Koopmans met a number of
economists, including Jacob Marschak, with whom he was to have a long
and significant relationship. Later in the year the Koopmans went on a
leisurely vacation, traveling through the French Alps by bus. As Mrs.
Koopmans later related to me, "We had a good time and I became
pregnant." Their first child, Anne, was born prematurely in April of
1940.
It was, of course, a time when the signs of
war were everywhere; the invasion of Poland took place during the
Koopmans's vacation. In April 1940 the Germans invaded Norway, and the
Koopmans family decided to leave Europe for the United States. As Mrs.
Koopmans described it to me:
Not a stitch of work was
being done because everybody foreign to Switzerland was struggling
desperately to get away. We ourselves were scrambling for a visa--to the
U.S., Canada, Cuba, even to Martinique. We were lucky; we had an
invitation to come to Princeton, arranged for us by Professor Samuel
Wilks, with whom we had become very friendly the year before, and we had
gotten a visitors' visa. Furthermore, because Tjalling's term at the
League of Nations was coming to an end, we had already arranged for
passage on a Dutch ship for Genoa to the U.S. Somehow that passage on
the Dutch ship was converted into passage on an American ship almost on
the spot. I believe that happened in Bordeaux.
The chance to get away came up suddenly, so I had
hurriedly packed a small trunk with necessities and clothes, and a
suitcase with diapers and milk powder for our 6-weeks-old baby. Then we
got word that the U.S. ship (the
Washington
) was ordered to
Bordeaux instead of to Genoa after Italy entered the war. We heard that
at 9 a.m. on June 4; at 12:00 noon, we were on the train to Bordeaux.
The Polak family had given us a travel basket for the baby; others
supplied us with sleeping bags; Tjalling carried his briefcase, the
luggage and gas masks; I carried the baby. We never saw our trunk again.
Because we had a baby, we were given a small cabin to ourselves while
the rest of the ship slept dormitory style. The vessel was only half
full in Bordeaux--the day after we left Switzerland France closed all
its borders--and many Americans who had been booked to sail were
stranded in Italy and Spain. But while we were en route, the ship was
ordered to Lisbon to pick up many people there, so that then the ship
was filled to its capacity of 1,000 passengers. After that, we went to
pick up more Americans in Galway, Ireland. Our adventure was not over
for on the way to Ireland we were halted by German submarines and
ordered into the lifeboats. Fortunately, it got across to the Germans
that the ship was an American one, and America had not entered the war
yet, so after some 4 hours of terror in the water, we were on our way
again. In Galway, we took aboard another 1,000 persons. The rest of the
trip was uneventful. We learned of the fall of Paris while at sea and we
arrived in New York with only the clothes on our back, the child in her
basket and some borrowed money. We had nothing else whatsoever.
The next several years were to be peripatetic. The
departure from Europe was sudden, and long-term employment could not be
arranged before arriving in this country. In 1940-41 Koopmans was
engaged as a research assistant at Princeton and, simultaneously, taught
a course in statistics at NYU. During this time, Koopmans worked on a
celebrated problem of mathematical statistics in the tradition of
earlier work by R. A. Fisher: the exact distribution of the serial
correlation coefficient in normal samples. Koopmans derived a
representation for this distribution by means of a contour integral and
illustrated the use of an ingenious smoothing approximation that
facilitated numerical computations. His paper, titled "Serial
Correlation and Quadratic Forms in Normal Variables," was published in
the
Annals of Mathematical Statistics
. It remains a permanent
contribution to a problem that was never fully solved analytically yet
absorbed the interest of many of the world's leading mathematical
statisticians throughout the 1940s.
After a year
the jobs at Princeton and NYU were terminated, and Koopmans took a
position as an economist at the Penn Mutual Life Insurance Company in
Philadelphia. A paper, "The Risk of Interest Fluctuations in Life
Insurance Operations," which does not seem to have been published, was
written at this time.
In 1942 the family left
Philadelphia for Washington, where Koopmans was to be employed for two
years as a statistician for the British Merchant Shipping Mission. The
work was interesting though routine, and Koopmans found the time to
initiate a line of inquiry about the economics of cargo routing. This
was eventually to be of great significance in the development of linear
programming and in the study of the activity analysis model of
production.
Koopmans's problem can be described in
the following way. Given a list of ports, the flows of a homogeneous
ship-borne cargo can be described by a graph, whose vertices are the
ports and whose edges are marked by the tonnage shipped between that
pair of ports. Given also a fixed set of supplies at some ports and
demands at others, an increase in the amount shipped from one particular
port to another will cause compensating changes in the matrix of flows
between other pairs of ports. In the paper, "Exchange Ratios Between
Cargos in Various Routes," written in 1942, Koopmans showed how to
calculate these compensating changes and their consequences for the
total cost expressed in ton-miles.
The problem of
determining the shipping plan that minimizes total cost, given a
preassigned pattern of availabilities of supplies and demands, is known
as the transportation problem. It is one of the most elementary examples
of a linear programming problem, that is, the maximization of a linear
function of several variables, subject to a series of linear inequality
constraints. But in 1942 the concept of linear programming had not yet
been proposed in the West, and Koopmans was unable to see his work as an
instance of this more general problem.
In 1939
Jacob Marschak, whom Koopmans had previously met in Oxford, left Europe
to become a professor at the New School for Social Research. There he
organized a seminar in mathematical economics and econometrics, and the
relationship between the two scholars was renewed when Koopmans attended
the seminar on a regular basis in 1940 and 1941. In 1943 Marschak was
appointed director of research at the Cowles Commission for Research in
Economics at Chicago, and in 1944 Koopmans wrote to Marschak about his
desire to leave Washington. Soon after, Koopmans accepted Marschak's
invitation to join the staff of the Cowles Commission, and thus began a
long association--both with Marschak and the commission--that was to
prove extraordinarily productive.
The Cowles
Commission for Research in Economics was founded in 1932 by Alfred
Cowles, the president of Cowles and Company, an investment counseling
firm with offices in Colorado Springs, Colorado. Mr. Cowles's initial
motivation in establishing the commission was to assemble a group of
mathematicians, statisticians, and economists whose combined efforts
might provide a rational basis for investment choices. The formal
charter of the organization, however, allowed for a broader mandate and
contained the phrase, "The particular purpose and business for which
said corporation is formed is to educate and benefit its members and
mankind, and to advance the scientific study and development . . . of
economic theory in its relation to mathematics and statistics." It was
this broader mandate that was ultimately adopted by the commission,
which, during its long history, was to become a primary vehicle for the
elaboration and dissemination of quantitative methods in economics.
During the last half-century, the subject of economics has been
transformed by the introduction of quantitative techniques, and the
Cowles Commission has played a major role in this process. I know of no
other example in the history of science in which a research institution,
founded and nourished by a private patron, has had so profound an impact
on an intellectual discipline.
Initially the
organization was located in Colorado Springs, with a small research
staff headed by Charles A. Roos, who became the commission's first
director of research in 1934. Starting in 1935, summer conferences were
held regularly, with an ever-widening research agenda and group of
participants from the United States and abroad. As pleasant as the
location was for summer conferences, however, Mr. Cowles found it
difficult to attract permanent staff to Colorado Springs, and he
arranged for the commission to move to Chicago, where it became
affiliated with the University of Chicago in 1939. Theodore Yntema, the
first director of research at Chicago, was succeeded by Jacob Marschak
in 1943.
Marschak was a scholar of great
intellectual force, curiosity, and initiative. As director he continued
the program of summer conferences, but now there was a dramatic increase
in the number of visitors and the size of the resident staff. Marschak
organized a series of weekly seminars, as well, and initiated the
practice of disseminating research results as discussion papers and
reprints. Leonid Hurwicz had been recruited by Yntema, and in the next
several years Trygve Haavelmo, Koopmans, Herman Rubin, Lawrence Klein,
Theodore Anderson, Kenneth J. Arrow, Herman Chernoff, Herbert Simon, and
other distinguished statisticians and economists were to be associated
with the commission in one way or another. The early research agenda,
set by Marschak, was primarily concerned with the particular statistical
problems arising in the estimation of parameters in a set of
simultaneous equations.
The idea that the
relationships among economic variables are best described by a set of
simultaneous equations is a time-honored concept of economic theory. The
price of a given commodity and the quantity purchased may be depicted by
the intersection of a demand curve and a supply curve--the first
relating the demand for the commodity to its price (given the incomes of
consumers), and the second relating the supply of the commodity to its
price (given the prices of the factors used in its production). Each of
these equations will involve various parameters whose estimation is
required if the system is to be used for the prediction of future values
of price and quantity. The naive approach is to estimate the parameters
in each equation separately using ordinary least square regressions. The
question was: How good are the naive methods?
In
several extremely important publications, Trygve Haavelmo, previously a
student of Frisch, laid the groundwork for answering this question.
Using the probabilistic methods of R. A. Fisher, Haavelmo assumed that
the observed series of economic variables satisfied a system of, say,
linear equations with stochastic errors governed by specific probability
distributions with unknown parameters. Given the parameters of the error
terms and of the equations themselves, any particular set of possible
values will have a well-defined probability. The maximum likelihood
estimates of the unknown parameters are those that give the highest
probability to the values of the economic variables actually observed.
As Haavelmo had shown, these maximum likelihood estimates could differ
substantially from ordinary least squares estimates.
At an even more basic level, the structure of the
system of equations may make estimation of the unknown parameters
impossible. If, for example, prices and quantities are derived from the
intersection of demand and supply curves, there may not be enough
information to ascertain the separate slopes of each of these curves. It
was the study of these statistical problems that Koopmans took up as his
major area of concern soon after arriving at the Cowles Commission. A
first paper concerned the bias arising from an ordinary least squares
regression of the parameters of a single equation, if the equation is,
in reality, part of a larger system. A second paper, written with the
assistance of Herman Rubin and Roy Leipnik, provided a complete solution
to the problem of "identification," that is, a description of the
necessary and sufficient conditions that permit the structural
parameters of a linear system to be determined uniquely from the
probability distributions of the data and hence amenable to statistical
estimation. This latter paper also developed systems of maximum
likelihood estimators and derived their large sample statistical
properties. The theoretical advances in this paper proved to be of
lasting significance. Its results are still the core of the theory of
simultaneous equations and endure in every textbook treatment of the
subject.
In addition to his research on these and
other aspects of econometrics, Koopmans organized a Cowles Commission
Conference (in early 1945) devoted to the statistical problems arising
from a system of simultaneous equations. He also edited the report of
the conference, published as Cowles Commission Monograph No. 10, in
1950. This volume eventually became a classic in the field, and its
themes have been fundamental in both the teaching of econometrics and
subsequent research.
Koopmans became the
acknowledged leader of that school of econometrics, focusing on the
problem of simultaneity and insisting on a complete probabilistic model
of the data to be analyzed. In 1947 he took the battle to the profession
as a whole in his review of the volume,
Measuring Business
Cycles
, authored by Arthur F. Burns and Wesley C. Mitchell. Koopmans
found this work, written by two senior economists associated with the
National Bureau of Economic Research, deficient in several respects.
First of all, it was a detailed analysis of a great volume of data
relating to business cycles, but its categories were not based on an
underlying theoretical model incorporating maximizing behavior of the
individual agents in the economy. Second, the statistical approach was
eclectic, with no formal probabilistic model to account for the data and
to justify the use of the author's statistical techniques. The
methodology used by Burns and Mitchell was descriptive, Koopmans
maintained, rather than flowing from the logical and analytical stance
toward economic data that was at the heart of the Cowles program.
A passionate rebuttal to Koopmans's review was offered
by Rutledge Vining, who stressed the merits of a synthetic approach
capable of suggesting tentative hypotheses in an important area of
economic discourse lacking a formal model. There was much jockeying
about on the issue of whether economics was currently in the Tycho
Brahé phase--simply codifying and mastering unstructured masses
of data--or in the Keplerian and Newtonian phase in which a parsimonious
and robust paradigm was available for explanation and illumination. Both
the review and the rebuttal were written with such lucidity,
scholarship, and care for these eternal economic concerns as to commend
them to the general reader some four decades later.
At the Cowles Commission, Koopmans continued his study
of the transportation problem that he had initiated in 1942. By the end
of 1946 he realized that his earlier problem of transporting a
homogeneous commodity from a set of origins to a set of destinations so
as to minimize the total cost of transportation could be formulated as a
problem of minimizing a linear function of a number of variables,
subject to a set of linear inequalities constraining the values assumed
by these variables. He also proposed a method of solution based on an
economic idea that was to become of central importance in his subsequent
research.
A particular instance of the
transportation problem is specified by the supply at each origin, the
demand at each destination, and a matrix of unit costs for shipping from
each origin to each destination. Koopmans observed that a vector of
prices, one for each location, could be associated with the optimal
shipping plan. The prices would meet the condition that each route in
use would make a profit of zero, in the sense that the price at the
destination would equal the price at the origin plus the unit cost of
shipping along that route. The routes not in use would, moreover, have a
profit less than or equal to zero. He also demonstrated that if such a
system of prices could be associated with an arbitrary feasible solution
to the constraints of the transportation problem, the feasible solution
would indeed be the optimal solution. The arguments made use of the
theory of convex sets, which were to become of great importance in the
study of the general linear programming problem.
Koopmans presented these ideas at a meeting of the
International Statistical Conference in Washington in September 1947.
Several months earlier he had a consequential meeting with George B.
Dantzig, who was the first Western scholar to study the general linear
programming problem. Dantzig had initiated his work on linear
programming while employed by the U.S. Department of the Air Force, and
in the summer of 1947 he developed the details of the simplex method, an
algorithm for their solution. The simplex method is a remarkably
effective computational technique that converges to the optimal solution
in a relatively small number of iterations, even for problems of
substantial size. The method makes use of a system of dual
variables--one for each inequality--that are used at each step of the
algorithm to test whether some of those activities not currently in use
should be introduced. In the special case of the transportation problem,
these dual variables are precisely those prices previously employed by
Koopmans.
Subsequent to his meeting with Dantzig,
Koopmans extended his observations about the relationship between prices
and optimality to the general activity analysis model of production. In
an activity analysis model the possible techniques of production
available to a firm, or to the economy as a whole, are given by a finite
list of elementary activities that can be used simultaneously and at
arbitrary non-negative levels. The resulting production possibility set
is a polyhedral cone, approximating the smooth transformation sets of
neoclassical economics to an arbitrary degree of accuracy. The activity
analysis model, a generalization of the Leontief input/output model, can
be used to generate a large number of distinct linear programs,
depending on the objective function to be chosen and on the specific set
of factor endowments.
Koopmans demonstrated that
an efficient plan--a plan for which no alternative existed using less
inputs and providing no less of any output--would be associated with a
vector of prices with a special property. The prices, intimately related
to Dantzig's dual variables, would yield a zero profit for the
activities used in that plan and a profit less than or equal to zero for
all the remaining activities. Conversely, a feasible production plan
associated with such a vector of prices would in fact be efficient. This
permitted Koopmans to make the fertile suggestion that if the correct
prices were known the optimal selection of activities could be
accomplished in a decentralized fashion by managers who were mindful of
their private considerations of profit maximization. In this way
Koopmans gave precision to the intuitive beliefs of economists, from
Adam Smith onwards, that a decentralized competitive economy achieves
socially optimal results "as if by an invisible hand."
In 1948 Koopmans succeeded Marschak as the director of
the Cowles Commission. A conference on activity anal-ysis was sponsored
by the commission in 1949, and the results of the conference appeared in
Cowles Commission Monograph No. 13 in 1951. The monograph, edited by
Koopmans, contained a paper by Dantzig on linear programming as well as
a lengthy exposition of the activity analysis model by the editor. In
this paper and in a nontechnical essay published in
Econometrica
,
Koopmans demonstrated a sharp awareness of the relationship of these
ideas to the fascinating discussion of socialist economic planning in
the 1930s.
His strong convictions regarding the
importance of the activity analysis model for economic planning in
Eastern Europe led Koopmans to make extended trips to the Soviet Union
in 1965 and 1970. There he met Leonid Kantorovich, a Soviet
mathematician who independently initiated the study of linear
programming in 1939. Kantorovich, who was to share the Nobel Prize with
Koopmans in 1975, had developed a test for optimality and an outline of
an algorithm for linear programming that was similar to but more
cumbersome than the simplex method. In Kantorovich's work the problem of
the optimal allocation of resources was approached not only from the
point of view of a pure mathematician, but also with the economist's
appreciation of the fundamental role played by prices in reaching an
optimal decision.
Research in econometric
methodology continued at the Cowles Commission, but under Koopmans's
leadership and guidance new lines of activity in economic theory were
initiated. The modern study of the general equilibrium model, in which
the theory of production is united with a description of consumer
preferences, was inaugurated by Arrow and Gerard Debreu; Arrow's classic
Social Choice and Individual Values
was in the making. At the
same time Harry Markowitz was working on portfolio analysis; Arrow,
Theodore Harris, and Marschak were writing an optimal inventory policy,
and formal theories of decision-making under uncertainty were proposed.
In 1955 the commission left the University of
Chicago for Yale University, where it was renamed the Cowles Foundation
for Research in Economics. James Tobin, whom the commission had earlier
tried to lure to Chicago, assumed the directorship in New Haven. Moving
along with Koopmans were Debreu, Marschak, Roy Radner, and Martin
Beckmann.
The last several years at Chicago were
charged with intellectual disagreements between the staff of the Cowles
Commission and members of the Department of Economics. Tjalling felt
under considerable pressure and began to compose music. The Koopmans and
their three children, Anne, Henry, and Helen, spent two summers at
Bennington, visiting with friends and attending a composers' conference
in which instruction in composition was given and the members of the
group had their works played and recorded. The children were small and
the family--which was of great importance to Tjalling--enjoyed swimming,
hiking, and other outdoor activities.
Koopmans's
strong desire to make the results of theoretical and mathematical
analysis available to a wide audience of nonspecialists is revealed in
the remarkable volume,
Three Essays on the State of Economic
Science
, published in 1957. The relationship between prices and
economic efficiency in both static and dynamic models of production and
the role played by the assumption of convexity in welfare economics are
discussed by means of simple geometric diagrams and with a lucidity
rarely attained by an active research scientist. A second expository
tour de force was his paper, "Selected Topics in Economics Involving
Mathematical Reasoning," written jointly with Bausch, which appeared in
1959.
In the decade of the 1960s Koopmans's major
research preoccupation was the theory of economic growth, in which he
directly addressed questions of efficiency and optimality in dynamic
models of production. He published a masterful paper, "On the Concept of
Optimal Economic Growth," in which his original presentation of the
calculus of variations was used to study the maximization of an
objective function given by a discounted sum of utilities. In the model
the input of labor is assumed to be exogenously growing. Output, which
can be allocated between consumption and investment, is specified by a
production function based on inputs of capital and labor. In several
other publications he introduced a class of stationary utility functions
that properly included the previous discounted sum of utilities, and he
used this larger class to study the concept of "impatience": roughly
speaking, a preference for current rather than postponed consumption.
The analysis was based on a sophisticated generalization of the concept
of Haar measure independently arrived at by Koopmans and his
collaborator, Richard Williamson.
In the
autobiographical sketch written when he received the Nobel Prize,
Koopmans says, "In most of my Yale period my research, chiefly on
optimal allocation over time, had more of a solitary character." But
this is only in contrast to the Chicago days, when the energies of the
entire Cowles team were focused on specific projects. In Chicago the
commission was engaged in a methodological revolution involving the use
of formal mathematics in economic theory and econometrics. By 1960 the
battle had been won; the troops no longer had to be massed for assaults
on exposed positions. Mathematical reasoning had become an accepted mode
of exposition for economic arguments, and the members of the Cowles
Foundation felt freer to pursue their own individual substantive
interests.
By the early 1970s Koopmans may have
felt that the mathematical revolution led by him had been too
successful--that elaborate mathematical arguments were being advanced
throughout the profession to the neglect of more immediate practical
concerns. He began to apply the techniques of growth theory to the study
of exhaustible resources and, in particular, those resources used in the
provision of energy. A lengthy study of copper supplies was initiated,
in collaboration with William Nordhaus, his colleague in the Department
of Economics, and Robert Gordon and Brian Skinner, both geologists at
Yale. He took on the chairmanship of a committee of the National Academy
of Sciences devoted to the study of alternative energy systems. This was
followed by a one-year visit to the International Institute for Applied
Systems Analysis (IIASA), in Laxenburg, Austria, where he succeeded
George Dantzig (in the second half of 1974) as the leader of the
Methodology Group.
On the morning in October 1975
when his Nobel Prize was announced, I visited Tjalling and Truus
Koopmans at their home. The prize was shared with Kantorovich for their
independent work on the optimal allocation of resources. Much of our
conversation was taken up by Tjalling's distress about the fact that
George Dantzig had not shared the prize. In a characteristic gesture
involving a fine blend of morality and precise computation, Tjalling
told me that he had decided to devote one-third of his prize to the
establishment of a fellowship in honor of Dantzig at IIASA. As we left
the house for a press conference at Cowles, Tjalling said, with a
certain shy amusement about what was awaiting him, "Now I have become a
public man."
In 1978 Koopmans agreed to assume the
presidency of the American Economics Association, after the death of his
longtime friend, Marschak, who had been president-elect. His
presidential address, "Economics Among the Sciences," was devoted to a
discussion of the differences in outlook of economists, engineers, and
natural scientists engaged in interdisciplinary collaboration. The
paper, written with Tjalling's characteristic conceptual clarity and
mastery of the facts, was illustrated by his work on energy modeling and
other topics addressed in recent reports of the National Research
Council.
Looking back, one can see a pattern in
Koopmans's professional career. He would invest himself for an extended
period of time in a particular area of study in which his analytical
capabilities could be used to clarify a large issue of potential
practical value. He would gather together a group of collaborators,
scholars with diverse backgrounds, and energize them with his benignly
patriarchal sense of purpose and direction. He would make personal
friendships with his intellectual associates, play chess with them,
listen to music with them, and take them on canoe trips and long walks.
The customary anxieties of the isolated research scholar would be handed
over to Tjalling, the leader of the group, whose confidence and resolve
would provide comfort and quiet any doubts. But, at the same time, he
himself would be engaged in an internal debate about the merits of the
collaborative activity--and, if the reckoning so indicated, he could
deliberately take leave of the activity and prepare himself for the next
venture.
Tjalling suffered a series of cerebral
strokes in the last months of 1984. In the short time between then and
his death on February 26, 1985, at the age of seventy-four, he was still
capable of intellectual and social interaction with his family and with
the loving friends who surrounded him.
I AM VERY GRATEFUL for many conversations with Truus
Koopmans and for the advice and assistance given to me by Kenneth J.
Arrow, Gerard Debreu, George Dantzig, Leo Hurwicz, Alvin Klevorick,
Peter Phillips, Martin Shubik, Herbert Simon, T. N. Srinivasan, Jan
Tinbergen, and James Tobin.
- 1933
- Uber
die Zuordnung von Wellenfunktionen und Eigenwerten zu den Einzelnen
Elektronen eines Atoms.
Physica
1:104-13.
- 1937
- Linear Regression Analysis of
Economic Time Series
. Publication No. 20, Netherlands Economic
Institute. Haarlem: De Erven Bohn.
- 1939
- Tanker Freight Rates and Tankship Building
.
Publication No. 27. Netherlands Economic Institute. Haarlem: De Erven
Bohn.
- 1942
- Serial
correlation and quadratic forms in normal variables.
Ann. Math.
Stat.
13:14-34.
- Exchange ratios between cargoes on
various routes (non-refrigerating dry cargoes). In
Memorandum for the
Combined Shipping Adjustment Board
, pp. 1-12.
- 1945
- Statistical estimation of simultaneous
economic relations.
J. Am. Stat. Assoc.
40:448-66.
- 1947
- Measurement without
theory. (Review of Burns and Mitchell, "Measuring Business Cycles.")
Rev. Econ. Stat.
29:161-72.
- 1949
- Identification problems in economic model construction.
Econometrica
17:125-44.
- Optimum utilization of
the transportation system.
Proceedings of the International
Statistics Conference, 1947
, vol. 5., pp. 136-46.
- 1950
- (Editor and contributor.)
Statistical Inference in Dynamic Economic Models
. Cowles
Commission Monograph No. 10. New York: John Wiley & Sons.
- 1951
- (Editor and
contributor.)
Activity Analysis of Production and Allocation:
Proceedings of a Conference
. Cowles Commission Monograph No. 13. New
York: John Wiley & Sons.
- Efficient allocation of
resources.
Econometrica
19:455-65.
- 1953
- (Editor with W. C. Hood and
contributor.)
Studies in Econometric Method
. Cowles Commission
Monograph No. 14. New York: John Wiley & Sons.
- 1957
- Three Essays on the State of
Economic Science.
New York: McGraw-Hill.
- Water
storage policy in a simplified hydroelectric system.
Proceedings of
the First International Conference on Operational Research
, pp.
193-227. Bristol, U.K.: The Stonebridge Press.
- 1959
- With A. Bausch. Selected topics in
economics involving mathematical reasoning.
SIAM Review
1:79-148.
- 1960
- Stationary ordinal
utility and impatience.
Econometrica
28:287-309.
- 1964
- With P. Diamond and R.
Williamson. Stationary utility and time perspective.
Econometrica
32:82-100.
- 1969
- With R.
Beals. Maximizing stationary utility in a constant technology.
SIAM
J. Appl. Math.
17:1001-15.
- 1972
- "Representation of Preference Orderings with Independent
Components of Consumption" and "Representation of Preference Orderings
over Time." In
Decision and Organization, A Volume in Honor of Jacob
Marschak
, eds. C. B. McGuire and R. Radner, pp. 57-100. New York:
North-Holland.
- With T. Hansen. On the definition and
computation of a capital stock invariant under optimization.
J. Econ.
Theory
5:487-523.
- 1975
- Concepts of optimality and their uses. (Nobel lecture,
December 11, 1975, Stockholm.)
Am. Econ. Rev.
67:261-74;
Math.
Programming
11:212-28;
The Scandinavian J. Econ.
78:542-60;
Les Prix Nobel
275-98.
- 1978
- Energy Modeling for an Uncertain Future. Supporting Paper 2,
Report of the Modeling Resource Group, Synthesis Panel of the Committee
on Nuclear and Alternative Energy System, National Research Council,
National Academy of Sciences, Washington, D.C.
- 1979
- Economics among the sciences.
Am.
Econ. Rev.
69:1-13.
- 1987
- With R. B. Gordon, W. D. Nordhaus, and B. J. Skinner.
Toward a New Iron Age? Quantitative Modeling of Resource
Exhaustion
. Cambridge: Harvard University Press.