# Thünen, Johann Heinrich von

# Thünen, Johann Heinrich von

Johann Heinrich von Thünen (1783-1850) was a German economist of great originality who contributed highly significant concepts and techniques to economic theory. He was the first to develop an exact definition of marginal productivity in the modern sense (although he did not use the term) and to apply the principle generally in the theory of production and distribution. He was a founder of mathematical economics and of econometrics, combining systematic empirical research with a genius for abstract reasoning and generalization. Most of his economic theory is based upon models of general static equilibrium, often expressed in terms of systems of equations, but he also contributed to the theory of capital formation and economic growth. Other areas in which he pioneered include theories of economic location, rent, and enterprise profit and some of the more practical aspects of agricultural economics. He emphasized far more than did his English contemporaries the central economic problem of allocating scarce resources so as to maximize the achievement of defined goals.

Von thünen was born on his father’s estate in the Grand Duchy of Oldenburg. His education included training in practical agriculture as well as in the academic disciplines, particularly mathematics. He attended the Agricultural College at Gross-Flottbeck near Hamburg and spent two semesters at the University of Gottingen. In 1810 he purchased Tellow, an estate in Mecklenburg, where he carried on the extensive experiments and systematic data-gathering that became the empirical basis for his great work, *Der isolierte Staat.*

Von thünen’s book won him considerable recognition during his lifetime. According to Schumacher (1868), Rodbertus credited von thünen with bringing to economics the rare combination of a most exact method and a humane heart, and the British Parliament used von thünen’s calculations of the grain production of the European continent in its deliberations on the corn laws. But the voluminous proportions of the work, its seemingly formidable mathematics, and its unusual originality appear to have prevented it from being either widely read or understood until the rediscovery of marginal analysis and the introduction of mathematical formulation into the mainstream of economic theory more than twenty years after von thünen’s death. Alfred Marshall acknowledged a major debt to von thünen.

**Rent, profit, and the “isolated state.”** For theoretical purposes, von thünen set up a model economy which he called the “isolated state,” consisting of a single city in the center of a large plain of uniformly fertile land. The inhabitants of the city exchange manufactured goods for the agricultural products of the plain, with the wagon as the sole means of transportation. Various agricultural prod ucts are grown in concentric circles around the city. The location at which each crop is grown is determined by the cost of transportation to the city, in accordance with what we would call the principle of “opportunity cost.” Land use ranges from the growing of garden vegetables nearest the city to grazing in the most distant circle, with the land beyond that occupied by forest which can be cleared if the economy expands. Von thünen changed the conditions of the model from time to time, to facilitate comparison and to illustrate various principles, always in full awareness of its abstract character.

Von thünen’s theory of land rent is based upon the same fundamental principles as that of Ricardo. In the “isolated state” the rent of each unit of land is determined exclusively by the difference between the cost of transporting the crop of that unit to the market city and the cost of transporting there an equal quantity of produce from the most distant land in use that yields no rent. Estate revenue consists of interest on the transferable capital invested in improvements, which must be paid at the current rate if the improvements are to be maintained, and payment for the land itself, which is the only true rent. If total estate revenue falls below the interest on transferable capital, payment for the land may for a time be negative, but under these circumstances the estate will sooner or later be abandoned.

Von thünen’s distinction between enterprise profit and interest on capital is modern, and it contrasts with the then prevailing concept of “profit” as both enterprise profit and interest. The entrepreneur’s profit consists only of that part of revenue which remains after he has deducted interest on his capital, an insurance premium against all insurable risks, and his own salary for management. This profit remains to the entrepreneur despite competition, because there is no insurance against such uncertainties as adverse price movements. Although the chances of gain are often as great as the chances of loss, the former do not compensate wholly for the latter, since the entrepreneur’s pleasure from doubling his fortune is less intense than his suffering from its complete loss.

**Wages, interest, and marginal analysis.** Von thünen’s application of marginal analysis to the theories of wages, interest, and resource allocation —the most important of his contributions to economic theory—appears in Volume 2 of *Der isolierte Staat* and was worked out during the period 1826 to 1848. He used a number of mutually corroborating approaches to the problem, with verbal, numerical, and algebraic modes of expression. A few brief examples follow.

Using his model of the “isolated state,” he assumed free, homogeneous, mobile laborers, who receive wages in excess of minimum subsistence and who are capable of producing products either for current consumption or for capital formation. In his analysis of capital formation essentially independent of the capitalist-laborer relationship, the workers either accumulate the excess of their wages over subsistence until they can devote a year’s labor to capital formation while living on their accumulated stocks, or they form groups of workers, some of which produce capital while the rest simultaneously produce subsistence for the entire group, all members then sharing equally in the ownership of the new capital. Von thünen demonstrated that in competitive equilibrium the interest per unit of capital equals the increment of product value resulting from an increment of capital, all other inputs remaining constant. The same principle is then applied to labor. Both labor and capital are shown to be subject to diminishing incremental returns, but the marginal product of labor increases as the quantity of capital per worker increases. Von thünen emphasized the importance of using infinitesimal increments in the formal analysis. He also pointed out that the natural rates of wages and interest prevail only under conditions of ideal competition and resource mobility that unfortunately do not prevail in the real world. He demonstrated by verbal and algebraic statement and by numerous examples the principles of cost minimization (Marshall’s “principle of substitution") and net revenue maximization, with emphatic assertions of their general applicability and their fundamental economic importance. Productive agents can be substituted for one another, and an optimum is reached when the ratio of their respective marginal products equals the ratio of their respective unit costs. Net revenue is maximized when the value-product of the last added unit of each agent just equals its cost.

An appreciation of von thünen’s method of analysis and an understanding of his interesting blunders can be achieved only by sampling his mathematical reasoning. Let *a + y* be the annual wage, in commodity units, of a working family, where *a* is one family’s annual subsistence and *y* is surplus over subsistence available for capital accumulation. Let *q* be the average quantity of capital per working family, measured in units equal to the annual wage of one working family, *a + y;* p be the average annual product of one working family when assisted by *q* units of capital; and z be the rate of interest in percentage per annum. Then on no-rent land at the boundary of the isolated state, *a + y*+ *q(a + y)z =* p, and with perfect competition and free mobility of labor and capital the rate of interest throughout the state is z = [p *— (a + y)]/q(a +* y). Von thünen assumed that each working family converts its annual surplus, *y,* into capital and that each wishes to maximize the annual revenue on that capital, expressed as *zy — y[p*— (a *+ y)]/q(a + y),* which is maximized when

Solving this equation for the “natural wage,” he obtained

Von thünen applied the marginal productivity principle as follows. One worker with *q* units of capital produces p units of product, and with *q + 1/n* units of capital he produces p + *β* units of product, where *n* is some large number. Then *1/n* units of capital yield an annual revenue of *β* units of product, and one unit of capital yields *nβ* or a units. The annual wage is then p — *αq,* and the value of the capital used by one worker is *q(p — αq).* The interest rate is then given by z = *αq/q(p — αq) — α/(p*— *αq),* a worker’s annual surplus by *y = p — aq — a,* and the revenue by *zy = a(p — aq — a)/(p — aq).* Since *a = f(q),* von thünen differentiated this expression for *zy* with respect to *a* to find the value for *q* which maximizes *zy,* thus:

and solved for the wage p — *aq = .* Here the wage is determined as the remainder of the product p after interest (determined as the marginal product of capital) has been paid. To determine the wage on the basis of the marginal product of labor, he assumed an enterprise hiring *n* workers at a wage A and using *nq* units of capital. Revenue is *n(p*— *A).* With one less worker, each remaining worker uses *nq/(n*- 1) units of capital and produces *p + v* units of product. Total product is now (n — 1) (p *+ v),* the wage bill is (n — 1)A, and the revenue of the enterprise is *(n — 1) (p + v) — (n — 1)A.* If in equilibrium the wage equals the marginal product of labor, the discharge of a worker should leave the revenue unchanged, or *np - nA*= (n - 1) (p + *v)*- (n - 1 )A. Solving for the wage, A = *p*- (n - 1 *)v.* If n is very large, this expression approximates p — *nv,* and the capital per worker approximates *q + q/n.* Thus, each remaining worker has additional capital of *q/n* and produces additional product of *v.* By the previous example, when a worker has additional capital of *1/n* units, his product increases by *fl.* Therefore, *v — Bq,* and since *nB = a, nv*= *aq.* Substituting in the expression for the wage, A = p — *nv = p*— *aq,* which agrees with previous results and in equilibrium equals

Although von thünen was so impressed with his “natural wage” as the geometric mean of a worker’s subsistence and the average product of labor that he had engraved upon his tombstone, it is today no more than an intellectual curiosity. Neither *a* nor p can be analytically defined. There is no reason to suppose that a rational worker would want to maximize the annual interest on one year’s savings. Measurement of a unit of capital by its wage cost alone is inconsistent with the assumption that each worker must use *q* units of capital, interest on which is part of the cost of the unit of capital. The importance of the work lies in its method of analysis and in the kinds of problems to which that method is applied.

Arthur H. Leigh

*[See also* Rent*and* Spatial economics. *Other relevant material may be found in the biographies of* Bohm-bawerk; Launhardt; Marshall; Moore, Henry L.]

## WORKS BY THÜNEN

*Der isolierte Staat in Beziehung auf Landwirthschaft und Nationalokonomie.* 3 vols. Jena (Germany): Fischer, 1930. *-** Volume 1: *Untersuchungen iiber den Ein-fluss, den die Getreidepreise, der Reichtum des Bodens und die Abgaben auf den Ackerbau ausüben* (1826). Volume 2: Der *naturgemasse Arbeitslohn und dessen Verhdltnis zum Zinsfuss und zur Landrente* (1850— 1863). Volume 3: *Grundsdtze zur Bestimmung der Bodenrente, der vorteilhaftesten Umtriebszeit und des Wertes der Holzbestdnde von verschiedenem Alter fiir Kieferwaldungen* (1863). A partial translation of Volume 2 as “The Natural Wage and Its Relation to the Rate of Interest and to Economic Rent” is included in B. W. Dempsey, *The Frontier Wage: The Economic Organization of Free Agents,* published by the Loyola University Press, Chicago, 1960.

## SUPPLEMENTARY BIBLIOGRAPHY

Ackermann, Konrad P. 1954 *Der ethisch-okonomische Erklarungsdualismus in der Lehre vom naturgemds-sen Arbeitslohn und Kapitalzins bei Joh. H. v. thünen unter Beriicksichtigung des Grenzproduktivitdtsge-dankens: Mit einem Anhang iiber das thünen-Archiv und die neuere thünen-Forschung.* Winterthur (Switzerland): Keller.

BÖhm-bawerk, Eugen von (1884-1912) 1959 *Capital and Interest.* 3 vols. South Holland, 111.: Libertarian Press. -” First published as *Kapital und Kapitalzins.*

Bulow, Friedrich 1950 thünen als Raumdenker. Welt-*wirtschaftliches Archiv* 65:1-24.

Bulow, Friedrich 1958 Johann Heinrich von thünen als forstwirtschaftlicher Denker. *Weltwirtschaftliches Archiv* 80:183-234.

Clark, John B. (1899) 1902 *The Distribution of Wealth: A Theory of Wages, Interest and Profits.* New York and London: Macmillan. -” Reprinted by Kelley in the “Reprints of Economic Classics.”

Engelhardt, Werner 1953 Die Theorien der Produktion, des Preises und der Verteilung bei J. H. v. thünen: Analyse seines Werkes unter verandertem Blickpunkt. *Schmollers Jahrbuch fur Gesetzgebung, Verwaltung und Volkswirtschaft im Deutschen Reiche* 73:1-33.

Hoffman, Friedrich 1950 J. H. v. thünen im Blick-feld des deutschen Kameralismus. *Weltwirtschaftliches Archiv* 65:25-40.

Johann Heinrich von thünen. 1958 *Zeitschrift fur Agrargeschichte und Agrarsoziologie* 6 (Special Issue). -” The entire issue is devoted to von thünen.

Krzymowsk, Richard 1928 Graphical Presentation of thünen’s Theory of Intensity. *Journal of Farm Economics* 10:461-482.

Leigh, Arthur H. 1946 Von thünen’s Theory of Distribution and the Advent of Marginal Analysis. *Journal of Political Economy* 54:481-502.

Marshall, Alfred (1925) 1956 *Memorials of Alfred Marshall.* New York and London: Macmillan. -> Reprinted by Kelley in the “Reprints of Economic Classics.”

Melamid, Alexander 1955 Some Applications of thünen’s Model in Regional Analysis of Economic Growth. Regional Science Association, *Proceedings* 1.L1-L5.

Moore, Henry L. 1895 Von thünen’s Theory of Natural Wages. *Quarterly Journal of Economics* 9:291-304, 388-408.

Ohlin, Bertil G. 1935 Some Aspects of the Theory of Rent: Von thünen vs. Ricardo. Pages 171-183 in *Economics, Sociology and the Modern World: Essays in Honor of T. N. Carver.* Edited by Norman E. Himes. Cambridge, Mass.: Harvard Univ. Press.

Petersen, Asmus 1944 *thünens isolierter Staat: Die Landwirtschaft als Glied der Volkswirtschaft.* Berlin: Parey.

Petersen, Asmus 1959 Johann Heinrich v. thünen, Volume 10, pages 386-392 in *Handworterbuch der Sozialwissenschaften.* Stuttgart (Germany): Fischer.

Schneider, Erich 1934 Johann Heinrich von thünen. *Econometrica* 2:1-12.

Schumacher, H. 1868 *Johann Heinrich von thünen: Ein Forscherleben.* Rostock (Germany): Leopold.

Wicksell, Knut (1901) 1951 Lectures *on Political Economy.* Volume 1: General Theory. London: Rout-ledge. -” Translated from the third Swedish edition. See especially pages 216-217 on von thünen’s theory of capital accumulation.

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