## Enrico Barone

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## Barone, Enrico

# Barone, Enrico

Enrico Barone (1859–1924) was an Italian mathematical economist who made lasting contributions to modern international trade theory, the theory of the firm, welfare economics, and the theory of general economic equilibrium. Barone’s mathematical background was superior to that of Walras and Pareto, who were laying the foundations of modern mathematical economics in the last quarter of the nineteenth century; Barone was therefore able to refine some of their important analyses. Moreover, his deliberate subordination of personal antagonisms to the desire to develop a scientific economics (see Walras [1857–1909] 1965, pp. 661–662) led him to draw upon British economics for inspiration to a greater extent than Walras or Pareto, although he did his most significant work within their context of general economic equilibrium. However, he cannot be viewed merely as an embellisher of the constructions of other economists or as a mediating agent between different architectural modes in economic theory.

Barone was a career army officer until the age of 48, attaining early eminence in this profession. When he was only 35, he was appointed a professor at the War College in Turin, which prepared officers destined for general staff duty. He wrote extensively in the field of military history and strategy. However, like Pareto, he had been introduced to economic theory by Maffeo Pantaleoni, and his involvement with it during his army career was far greater than that even of the passionately devoted amateur. He began as early as 1895 to seek a chair in economics in order to devote his full attention to the field, but not until 1907, when he received a professorship at the Istituto di Scienze Economiche in Rome, was he to achieve his desire. (At this time he resigned his commission as general staff colonel.) Paradoxically, all his lasting contributions to economics were made in the period of his army career.

**The theory of international trade.** Barone joined Marshall and Pareto in moving the analysis of international trade away from the classical labor quantity theory, or barter basis, toward the determination of trade flows by price differentials. Employing consumer surplus reasoning, Barone demonstrated rigorously that protective tariffs always reduce consumer welfare, yet he added that the imposition of such tariffs with their attendant reduction of welfare may prevent the imposition of policies even more damaging to welfare (Barone 1894–1924, vol. 2).

**The theory of the firm.** Barone played a significant role in developing marginal productivity analysis as the basis of the firm’s supply functions for output and demand functions for inputs (see the discussion of the theory of general equilibrium, below). However, Barone never truly resolved the dilemma which plagued the analysis of the entire Lausanne school: in pure competition, how can the firm be maximizing profit when by the very nature of the market structure it must earn zero profits in equilibrium? As a consequence of the inability to solve this seeming paradox, Walras, Pareto, Barone, and their followers confused the maximum profit conditions on marginal value productivities with minimum cost conditions and the maximum profit requirement that price be equated to marginal cost with the condition that it be equal to average cost (Kuenne 1963, pp. 138–140, 180).

Barone’s full role in the marginal productivity controversy during the last quarter of the nineteenth century is difficult to judge because much of his analysis seems not to have been published and his papers were not preserved. However, at least a translation of a most revealing document did escape destruction and has in recent years become available. It is a review of Philip Wicksteed’s *Essay on the Coordination of the Laws of Distribution* which Barone wrote for the *Economic Journal,* but which was rejected in 1896 (for unclear reasons) by its editor, F. Y. Edgeworth. Fortunately, although the original manuscript was lost, a careful French translation by Walras survived (see Barone 1896; Walras [1857–1909] 1965, pp. 644–648).

In this review Barone attempted to generalize the exhaustion-of-product proposition for a linear homogeneous production function to the case of nonhomogeneous production functions in purely competitive environments. It is well known, following Euler’s theorem, that for functions which are homogeneous of degree one, if every factor of production is paid its marginal product, the entire product will be exhausted in such payments. Homogeneity of this degree implies that for a fixed set of allowable input prices, average costs are independent of the scale of output if all factors are variable in quantity and, consequently, that average costs equal marginal costs.

Barone, in his attempt to show that this condition holds in the nonhomogeneous case, began with the assumption that pure competition implies the equation of price and average cost of a product, and therefore the absence of profits for the firm. Let *x* be the amount of output for the firm, P the price of the output per unit, X_{i} the firm’s demand for input *i*, and P_{i} the price per unit of input *i*. Then, if price equals average cost,

Output is related to inputs by the following continuous smooth function:

To maximize profits for an interior maximum and under proper constraints concerning the shape of the production function in (2), it is necessary to equate the “marginal value product” of every input to its price. That is, it is necessary that

and Barone derived these conditions. By substituting them into (1) and eliminating P, he derived the relation of Euler’s theorem:

He believed he had thus freed Wicksteed’s analysis from the restrictive assumption of linear homogeneity for (2).

Barone’s analysis fails first of all because it implicitly assumes that linear homogeneity holds exactly or at least approximately at the equilibrium. Equation (1) is not the fundamental relation of price and cost in purely competitive equilibrium. His belief that pure competition implies the equality of price and average cost led him to substitute into (1) from ( 3 ) to eliminate the P_{i} and to obtain (4). But (1) need not hold in short-run pure competition, profits may be earned, and (1) is not a universal implication of that market structure.

There is, however, a second source of Barone’s failure. Even if one consents to deal only with long-run pure competition, in which event (1) is a correct assumption, Barone did not establish the exhaustion-of-product proposition in the general case. The crucial price—cost relationship in pure competition is that of the equality of price and *marginal* cost. This relationship is necessary in order to derive (3)—a fact which escaped Barone. Therefore, given (1) and (3), Barone’s proof of exhaustion-of-product requires that price equal both average and marginal cost. This set of conditions rules in Wicksteed’s case of linear homogeneity (exactly) or in the case where all firms are operating at the minimum point of a U-shaped average cost curve (approximately). In the general case, however, it is impossible to have the firm equate price to both average and marginal costs.

It is symptomatic of the analytical uncertainty in this field at the time that Barone was writing that he affirmed the necessity of (3) in the firm’s equilibrium, but that instead of explaining it as an *internal* adjustment of the firm necessary to maximize profits, he asserted that it must hold if factors are to be allocated properly among uses. This is true and its interpretation in this manner was quite insightful, but this external aspect of the firm’s adjustment has little relevance to its own discretionary decision making.

The confusion was aggravated by Walras, who, in a letter to Barone, dated October 30, 1895 (Walras [1857–1909] 1965, pp. 650–651), insisted that the conditions of (3) were imposed to guarantee minimum average cost for the firm. Barone dropped his former interpretation of them and agreed, too readily, with Walras that this was indeed their genesis (Walras [1857–1909] 1965, pp. 652–653). From this time forward Barone seems invariably to have identified pure competition with the firm’s equilibrium at zero profits and minimum average cost.

We now know that it is possible to state the firm’s maximum profit conditions in this form only if the conditions of linear homogeneity hold exactly or approximately. If a firm is constrained to earn zero profits under these conditions and it is minimizing its costs, it is in fact maximizing its profits at zero. In this method of stating the conditions, therefore, Barone was once more implicitly assuming special cases, not developing the general theory, which permits profits to be earned.

**The theory of general equilibrium.** Yet, in a letter to Walras, written September 20, 1894, Barone had indeed already stated the problem correctly and solved it for the general case (Walras [1857–1909] 1965, pp. 619–621). He recognized the possibility of profit in pure competition, the role of the conditions of (3) in explaining it, and, most importantly, the method of deriving supply curves of output and demand curves for inputs from these conditions. In three short paragraphs, Barone there indicated the manner in which profit maximization under marginal productivity considerations determines the values of variable coefficients of production for the firm, supply-of-output and demand-for-input functions for the firm, and their integration into the functions and equations of general equilibrium.

These three rich paragraphs indicate that Barone had by 1894 discovered the proper method of eliminating constant coefficients of production from the Walrasian structure and of making the desired integration described above. Indeed, the firm was given a real existence within the industry for the first time in general equilibrium theory, since it was allowed determinate outputs and input demands. It is an interesting question why he did not persist in this interpretation of the firm’s decision making; perhaps it was because he was misled by his attempt to bring the entrepreneurial factor under the regime of marginal productivity.

**Collectivist economic planning.** There can be no doubt that Barone’s most widely known contribution is his demonstration that a ministry of production in a collectivist economy would be able, through the use of instrumental “shadow prices” for the establishment of purely competitive equilibrium conditions, to plan production rationally. Although Barone was not himself sympathetic to socialism, his famous article, published in 1908 but translated into English only in 1935, furnished a theoretical basis to counter the arguments of those who urged that such planning is impossible in the absence of market prices.

Barone limited himself to consideration of production conditions by assuming that income distribution in the collectivist state is determined by ethical considerations. Income redistribution is effected by differential payments to classes of consumers from the earnings of state-owned means of production (and, he should have added, profits) to supplement private earnings. In Barone’s treatment, profits occur only as differential rents to managerial ability, which he incorrectly argued should be eliminated by reductions in average cost of the product over the industry. As already pointed out, he should have included the usual concept of short-run profits in his collective income.

Barone’s first step is to show that there exists a function, Φ, whose dimension is “units of money per period,” such that when the conditions of pure competition rule throughout the economy it is true that *d*Φ = 0 and *d*^{2}Φ ≤ 0, when prices are treated as fixed. Continuing to assume that there exist *m* factor service inputs *i,* that *n* consumer goods *j* are produced, that *m* capital goods Z_{i} exist to produce the respective factor services *i,* and that *E* is the value of new capital goods produced per period, Barone then defined

where the D-terms define total consumer demand. Further, on the assumption that new capital goods are eternal and will sell in equilibrium at the value of their services rendered per period capitalized at the rate of interest, r, he obtained

Under conditions of pure competition, (*a*) all consumers are adjusting marginal rates of substitution between goods to the same price ratios, (*b*) the prices of all goods equal minimum marginal costs, and (*c*) the prices to which firms adjust are the same as those to which consumers adjust, *d*Φ = 0, and, with well-behaved functions, *d*^{2}Φ ≤ 0. This in turn implies that for the given distribution of income, small changes in the goods produced taken in conjunction with the implied changes in the factor services consumed or small changes in some factor services and compensating changes in the opposite direction for others with outputs constant will result in no change in consumer welfare, and larger changes will take the economy away from Pareto optimality.

Despite the fact that Barone, in his analysis of the collective economy, refused to employ marginal utility or even indifference curve concepts, preferring to remain with demand-and-supply functions, he implicitly relied upon the equality of marginal rates of substitution and prices for all consumers in his interpretation of this function (Barone [1908] 1935, pp. 254–255). Barone asserted, properly, that when the conditions of pure competition rule, and the differentials of Φ meet the above conditions, consumers cannot all benefit from such changes as were discussed above, nor can the gainers benefit to the extent of having net gains after compensating the losers.

He then showed how the ministry of production in a collectivist economy can set shadow prices on goods and services in such fashion as to employ all resources and, by iterative corrections of the prices within these resource constraints, lead the economy to the position where the equivalent of Φ has a first-order differential equal to zero. With the setting of the interest rate in the public interest by the ministry and the distribution of collective income, the equilibrium will resemble that of pure competition; that is, the ministry will have maximized social welfare when, subject to the availabilities of resources, all goods-prices equal minimum marginal costs and all capital goods earn the same net return as a percentage of costs. During the approach to an equilibrium, as well as at equilibrium, consumers are given the freedom within their earnings and social dividends to purchase what they wish.

Reflecting the unsatisfactory state of analysis of the Paretian school in the area of savings and investment, Barone’s analysis of the state’s role in crucial savings–investment decisions is vague. Indeed, his analysis of these processes even in the purely competitive economy is almost impossible to decipher, particularly as it concerns the nature of “new working capital” and the determination of the interest rate. It does seem clear that he believed that in the collective economy individuals are required to deliver their savings to the state, that they are entitled to the earnings of their savings, and that the state sets an interest rate in the public interest.

Barone expressed some pessimism concerning the practical operational feasibility of a ministry of production performing the necessary iterative process, reserving his most severe misgivings for the case of variable coefficients of production. What he did succeed in doing, however, at a higher level of abstraction, was to point to the logical consistency of the ministry’s tasks, the possibility and indeed rational necessity of using prices to guide allocation, the ability to use numerical analytical iterative techniques to approach an operationally definable equilibrium rather than having to solve many equations simultaneously, and the formal similarity of the solutions for the market and collective economies.

Robert E. Kuenne

[For *the historical context of Barone’s work, see the biographies of*Marshall; Pantaleoni; Pareto; Walras; Wicksteed; *for discussion of the subsequent development of Barone’s ideas, see*Firm, theory of the; Production; Welfare economics.]

## WORKS BY BARONE

(1894–1924) 1936–1937 *Le opere economiche.* 3 vols. Bologna (Italy): Zanichelli. → Volume 1: *Scritti vari,* 1894–1924. Volume 2: *Principi di economia politica,* 1908. Volume 3: *Principi di economia finanziaria,* 1911–1912.

(1896) 1964 Sur un livre récent de Wicksteed. *Cahiers Vilfredo Pareto* 3:68-73. → Léon Walras’ French translation of Barone’s unpublished review of 1896.

(1908) 1935 The Ministry of Production in the Collectivist State. Pages 245–290 in Frederick A. von Hayek (editor), *Collectivist Economic Planning: Critical Studies on the Possibilities of Socialism by N. G. Pierson, Ludwig von Mises, Georg Hahn, and Enrico Barone.* London: Routledge. → First published as “II ministro della produzione nello stato collettivista.”

## SUPPLEMENTARY BIBLIOGRAPHY

Bergson, Abram 1938 A Reformulation of Certain Aspects of Welfare Economics. *Quarterly Journal of Economics* 52:310–334.

JaffÉ, William 1964 New Light on an Old Quarrel. *Cahiers Vilfredo Pareto* 3:61–102.

Kuenne, Robert E. 1963 *The Theory of General Economic Equilibrium.* Princeton Univ. Press.

Samuelson, Paul A. (1947) 1958 *Foundations of Economic Analysis.* Harvard Economic Studies, Vol. 80. Cambridge, Mass.: Harvard Univ. Press.

Schumpeter, Joseph A. (1954) 1960 *History of Economic Analysis.* Edited by E. B. Schumpeter. New York: Oxford Univ. Press.

Walras, LÉon (1857–1909) 1965 *The Correspondence of Léon Walras and Related Papers.* 3 vols. Selected and edited by William Jaffé. Amsterdam: North-Holland Pub. Co. → See especially Volume 2: *1884–1897.*

Walras, LÉon (1874–1877) 1954 *Elements of Pure Economics: Or, the Theory of Social Wealth.* Translated by William Jaffé. Homewood, III.: Irwin; London: Allen & Unwin. → First published in French as *Éléments d’économie politique pure.*

## Barone, Enrico

# BARONE, ENRICO

(1859–1924), Italian soldier, politician, and economist with strong mathematical training.

Enrico Barone was a contemporary and interlocutor of both Leon Walras and Vilfredo Pareto, best known for his careful formulation of the equilibrium system that would have to be solved by central planners in a socialist economy. Published in 1908 as "Il Ministro della Produzione nello Stato Collettivista" in the journal *Giornale delgi Economisti,* and reprinted in English in F. A. Hayek's edited volume, *Collectivist Economic Planning,* his formulation provided an analytic foundation for arguments supporting the feasibility of socialist calculation, socialist central planning, and ultimately "market socialism." In it he provided a Walrasian (general equilibrium) system of equations whose solutions would resolve the valuation and coordination quandary for socialist central planners—a system of economic rationality without markets for production inputs and capital. Socialist economists such as Oscar Lange, Fred Taylor, and Maurice Dobb, have taken this as a refutation of Ludwig von Mises's critique of the possibility of economic rationality under socialist planning. In particular, it is argued that his formulation shows how modern high-speed multiprocessor computing can be used to find optimal scarcity valuations and prices for all products and assets in an economy, thereby allowing rational formulation of an economic plan by social planners.

Barone also contributed to general equilibrium theory by showing Walras how to incorporate variable production techniques into his equilibrium system of equations (the Walrasian system). This contributed to the development of marginal productivity theory, a central part of neoclassical economic analysis. Finally, he made notable contribution to the economics of taxation in his three studies of public finance in 1912.

*See also:* command administrative economy; market socialism; socialism

## bibliography

Samuelson, P. A. (1947). *Foundations of Economic Analysis.* Cambridge, MA: Harvard University Press.

Schumpeter, J. A. (1954). *History of Economic Analysis.* New York: Oxford University Press.

Richard E. Ericson