The social utility function

First insertion on Heterodoxe Gazet Sam de Wolff: 20 march 2013

E.A. Bakkum is a blogger for the Sociaal Consultatiekantoor. He loves to reflect on the labour movement.

In 1854 Hermann Gossen was the first person to apply the concept of utility to economics. Half a century later his idea had become the influential neoclassical paradigm. Vilfredo Pareto imposed a radical restriction on the concept of utility by rejecting the cardinal measurement of utility. Since then the economics in the west has abandoned the idea of the social utility function or the welfare function. The East-European plan economists disagreed (the second-hand book shop Helle Panke in Berlin supplies interesting books on the subject).

So the neoclassical paradigm (at least its mainstream) is restricted to the ordinal measurement of utility, which allows only for a qualitative preference. Moreover the ordering of the preferences is an individual act. In this perspective it is inconceivable, that the state as the coordinating institution intervenes in order to increase the efficiency of the economy. The discussion about the property relations is relegated to politics. Incidentally most adherents of the paradigm reject any state interventions, because they would disturb the natural order.

The consequences are so large that it is worth while to devote a critical column to the foundations of the neoclassical paradigm. Here it is obvious to analyse the experiences of the (former) planned economies, notably in Eastern Europe. How do their economists judge the viability of the centrally planned economy? And in what way do they interpret concepts such as utility and profit, and their optimization?

The economic structure in th neoclassical paradigma

In the eighteenth century the economist Adam Smith developed the idea that the economy has an inner tendency, as it were guided by an "invisible hand", to move towards the optimal equilibrium1. In the nineteenth century the economist Léon Walras elaborated on this diea, and developed a mathematical equilibrium model. In the beginning the model was simply an exchange economy, but later it was extended with a production system. This is now called the competitive equilibrium, because it is assumed that the producers maximize their profit.

Vilfredo Pareto, the successor of Walras, showed that in the general equilibrium of Walras the available means are used in an optimal way. Within the ruling property relations all resources are allocated efficiently. The consumers optimize their utility, and the producers their profit. The resulting situation is called Pareto optimal, or Pareto efficient. In short, as a reminder to the informed reader: the general market equilibrium of Walras with Pareto efficiency rests on three pillars:

It is often ignored that the neoclassical paradigm restricts the optimization to the good use of all opportunities within the ruling order. The starting point of the equilibrium model is an already existing distribution of wealth, and she can be very unjust. Even if the very skewed property relations would compulsively cause a revolution, then the ancien régime can still be Pareto efficient. Reversely, the state can modify the property relations (for instance, by means of democratic decisions), and subsequently create a Pareto optimal situation.

So the neoclassical paradigm can not determine when the optimum of the social welfare has been reached. The reason for this deficit is that the paradigm considers merely the experienced utility of the individual. It is impossible (and this opinion has been voiced first by Pareto) to measure the utility in a quantitative manner (cardinal measurement). Only the order of preferences can be determined (ordinal measurement). Moreover, the experienced utility of two individuals can not be compared. A sandwich during a plentiful dinner can be as useful as (or more than) a sandwich for a hungry homeless person2. A consequence of the statement of Pareto is that a social "cardinal" utility function could not exist3.

Heine and Herr in their excellent introductory textbook Volkswirtschaftslehre4 express their disappointment about the scientific foundation of the Walrassian general market equilibrium with Pareto efficiency. For the time being, this foundation has been completed in the works of Arrow and Debreu. Their model shows that an economically optimal situation (defined by the full employment of all resources including the factor labour) is merely possible in exceptional conditions. Even under these assumptions the optimum of the social welfare remains controversial and the subject of a political debate. It is impossible to choose between all those Pareto optimal situations, because within the neoclassical paradigm the social utility function is missing. Heine and Herr give an interesting summary of the deficits, which accompany the idea of the universal equilibrium.

Decentralized socialism and planning

In the interesting but fullfledged neoclassical book Welfare economics and social choice theory6 A.M. Feldman argues that a centrally planned economy can never be successful. Here central planning is defined as the direct hierarchical management of the material production and distribution. The whole economic system of manufacture and trade must follow the commands of the central agency. Feldman identifies two obstacles for such an approach. First, the central agency is obliged to process all relevant economic information, before a decision can be made. This is an impossible task for humans. Second, the executive is not responsible, so that it misses any incitement to improve the processes and procedures7.

However, an alternative approach of planning exists, which explicitely uses the neoclassical paradigm. It has been developed notably by the Polish economist Oskar Lange8. Feldman calls this alternative approach the decentralized socialism. Just like in centralized socialism the central agency plans a certain size and structure of the national nett product QN. But instead of steering the executive by means of direct commands, she dictates a certain wage- and price-structure (w, p). Now the executive is free to manage the production, as long as it satisfies the boundary conditions.

The loyal reader will recognize here the vector notation. The quantity QN is a vector with n components, which each denote a certain quantity, as much as the number of different types of goods, that are produced. The total product, including the required means of production, is represented by the vector Q, and the vector p represents the corresponding piece prices. The wage level is w. According to the second fundamental theorem of welfare economics, at least in the version of Feldman9, there exists a price vector p in such a way, that in the competitive general equilibrium the nett product will equal QN. In other words, when for a given price system p the executive tries to maximize its profit, then their production will of its own accord equal the desired nett product.

The decentralized socialism has the evident advantage, that the executive (producers and sellers) become responsible for the results. The central planning agency is relieved from this burden. Besides, there is now an incentive to make a profit, which stimulates cost-effective actions. Nevertheless, the decentralized socialism of Lange looks a bit suspicious, because the previous paragraph raised doubts about the idea of the general equilibrium10. Indeed the second fundamental theorem is only valid under rather restricting conditions11.

Actually the neoricardian paradigm gives a better understanding of the theorem than the neoclassical paradigm. The interested reader can consult the column about the choice of technique in the model of Sraffa. There it is shown that the profit is maximal on the technology frontier. That is to say, when the wage level w is given, and thus also the capital efficiency r, then under those conditions a single technique excels over the others (apart from the situation in the switching points). Suppose that the central planning agency imposes the price system, that corresponds with this technique. Then the producers are forced to use that technique, because they would make a loss with the other production techniques.

In addition the imposed price system will force the consumers to create a certain effective demand, namely precisely the nett product QN, which is preferred by the planning agency. Then the only remaining question is, how, starting from a certain initial situation, that general equilibrium can be reached. Feldman states in imitation of Lange, that the central planning agency must apply the method of trial and error. The planning agency makes consecutive changes in the price system p, until finally the desired QN is realized.

The reader may agree with the distrust, that has been voiced just now. The approach of Lange, and in his footsteps the one of Feldman, can not succeed. For as long as the planning agency is forced to make arbitrary price changes, it has no idea about the direction of these changes. The price system would change indefinitely, without ever reaching the general equilibrium (with the exception of a miraculous hit)! No, the neoclassical paradigm does not provide a solution for the planned economy. A material planning may not be very attractive, it still looks better than decentralized socialism.

Practical experiences with centrally planned economies

The viability of planned economies is beyond debate. For many decades the planned economies in Eastern Europe have participated in a systems race with the western capitalist production. Besides, since the twentieth century the system in the west is a mixed economy, which at times, when it is required by the conditions, bears a strong resemblance to a planned economy. So it is interesting and relevant to study the design of the system in Eastern Europe.

In fact the planned economies in Eastern Europe have chosen a combination of planning with both physical quantities and aggregated value (money sums, based on prices). The material balances and intertwined tables form the core of the plan. In previous columns such tables have already been discussed. Moreover, the value balances and value-based intertwined tables are used in a systematic and empathic manner for inciting the executive to obey the commands of the central agency.

The importance, that is attributed to the exchange value, is expressed clearly in several citations from the popular East-German textbook Einführung in die politische Ökonomie des Sozialismus12. It argues among others (p.207): "The costs, prices, profit and their derived indices express essential aspects of the realized or expected utility, they allow to compare expenses and yields, and they make the efficiency visible". And on p.208: "The measurement- and distribution-function of the value categories ... appeal to the material interests of the enterprises and of each worker, and they are applied on purpose in order to stimulate a high efficiency". A previous column has already described the price formation in these states.

In the planned economies of Eastern Europe the philosophy of the competitive general equilibrium is translated into an interaction between the production and the satisfaction of the needs. This concept is discussed in among others the book Bedürfnisse, Bedarf, Planung13, again an acquisition from the stocks of Helle Panke. The central planning agency dictates the optimal welfare in the form of the plan for the total social consumption. The plan is the compromise between on the one hand the productive capacity (supply) and on the other hand the needs of the people (demand).

The demand side of the consumption depends on the historic and social factors. Therefore she can be determined with statistical methods, by means of public opinion polls. Information is gathered about the composition of the population: the professional groups, the types of households, the age distribution, the income categories, etcetera. In short, this is simply the collection of facts, which is done in the Netherlands by the Centraal Bureau voor de Statistiek and by the Centraal Plan Bureau. Subsequently, for each professional group the normal consumptive pattern is identified, in the form of consumer normatives.

Incidentally, according to the economist Jan Pen such consumer normatives have also been developed in the Netherlands14. He objects to their subjective nature, which makes them an instrument of ideology. A scientific formulation is partly possible, for instance with respect to healthy food and housing. Indeed many needs are determined by tradition, and change only slowly. Also aspects like the discarding and replacing of durable consumer goods can be analyzed by means of statistics. But in the end there are always some controversial needs. For instance, must each household dispose of a micro computer? In the states of Eastern Europe the debate was hypothetical, because the decisions were always made by the leaders of the Leninist party.

The party leaders were eager to discourage some types of consumption, which were in conflict with the ideological preferences. For the needs are considered to be a social phenomenon, because they take on their concrete form only as the result of human interaction. The utility of a product (in the marxist terminology: the social use value) is determined by and for the society as a whole, and not by individuals in isolation. Indeed this seems to be a logical point of view. It provides the foundation for the pervasive criticism of the unbridled advertising.

The supply side of the consumer goods is limited by the availability of the productive resources, and by their efficient allocation and use. The enterprises have even in the planned economies a significant responsibility for the continuous improvement of the production processes, The plan and the structure encourage them to innovate. For the number of conceivable production techniques is limited, because not all combinations of production factors yield a workable production process. The substitution of production factors is by no means perfect. Within each branch just a few production methods are available, each with its own efficiency. So it is quite conceivable, that not all resources are actually put to use. This phenomenon is illustrated with examples in the column about the substitution of production factors. At the same time, that column shows that thanks to the optimization the planned economy makes a better use of the capacity than the private market. As a concessions she accepts a lower capital efficiency.

The product value plays an important role in the distribution of the total social product. That is to say, the incomes of the household must be attuned to the available material supply. It is said that the fund for purchases and the commodity fund must be at unity15. One recognizes here the budget line of the neoclassical paradigm. The match is made by a careful planning and elaboration of the incomes of all categories of households, for now and the near future (the planning horizon).

Finally a well-known problem of central planning must be mentioned, namely that the consumer is expected to behave according to the social norm. Deviating needs and tastes can not be satisfied. The central planning agency does meet the development of the needs, and often even encourages this process. But since the development is always rather unpredictable, the demand management leads to regular shortages in the supply (and presumably also to surpluses). These deviations are accepted for the sake of the efficiency, that is realized in the planned production.

It is obvious that the poor reliability of the deliveries causes some frustration for the disappointed consumers. In the GDR the dissatisfaction of the eager to buy was acknowledged by the party. She could be voiced in the cabaret (among others Distel, still performing in Berlin) and in satirical magazines, such as Eulenspiegel. The interested reader will find in the footnote16 a small anthology from the jokes, jests and back-talk, that were published in Eulenspiegel. It was all possible. One back-talk tells more than a handfull of formulas.

The welfare function

In Eastern Europe the science had to support and advance the planned economic growth. These states did not engage in the western controversy about the social utility function or welfare function. It was assumed that she exists, as a logical consequence of the solid philosophical idea of the social determination of needs. An interesting model is presented by H.-D. Anders en K. Schwarz in the two volumes Gesetzmäßigkeiten der intensiv erweiterten Reproduktion17.

Anders and Schwarz give the following formula for the welfare function:

(1)     U(Q) = U(Q1, Q2, ...., Qn)

In the formula 1 the value of U respresents the social utility. There are n different end products, and Qj is the quantity of product j, which is available for the society as a whole. So the formula 1 is not individual, like in the neoclassical paradigm. The authors adhere to the cardinal interpretation of utility, but they do not elaborate on this point. The essence is the practical applicability! By the way, it is worth noting that they define the utility as the social use value18.

The formula 1 shows how the central planning agency makes a choice in favour of a certain nett product QN or of a certain social consumption K. All possible combinations of end products are simply compared, and the combination with the largest U value is preferred. That combination determines the optimal QN or K. U as such represents mainly the demand side. The supply side is expressed by the limitations, which the technical and material availability of resources impose on the combinations Q. Here the loyal reader recognizes the interplay between the objective function and the corresponding inequalities (the boundary conditions, which are imposed to the economic quantities) in a problem of optimization.

In the preceding paragraphs it has become clear, that the welfare function U is based on statistical-scientific data, such as the consumer normatives. At the same time those paragraphs explain that the prices pj of the products j (j = 1, ..., n) are a convenient instrument for the central management of the planned economy. The total social product can be written as

(2)     Q = Q(B, p)

In the formula 2 p is the price vector with components pj, and B is the social budget. In other words, B is the available room for expenditure.

Now Anders and Schwarz employ the laws of Gossen19 (see also the first paragraph about the neoclassical paradigm):

(3)     ∂U/∂Qj = λ(B, p) × pj

In the formula 3 the quantity λ is a constant, which expresses a proportional relation. Suppose now that the central planning agency takes the quantity Q0 as its starting point. Let the possible variation of the quantities around this starting point be ΔQ. Now the formula 1 can be developed in a Taylor series around the starting point, by means of the formula 3 20:

(4)     U(Q) = U(Q0) + λ(B, p) × p · ΔQ + ...

In the formula 4 one has Q = Q0 + ΔQ. The interesting aspect of the formula 4 is, that the welfare function is transformed into an objective function. For the piece prices of the end products appear as the expression of the value, which society attaches to each product. Incidentally, in 1976, when Anders and Schwarz published their text, this method for utility functions with prices had not yet been introduced in the planning practice. But in any case it gives an impression of the philosophy, which underpins the planning by means of optimization methods and their value-based objective functions.

  1. The contents of this paragraph is only understandable for readers, who are insiders. It follows from the discussion, which M. Heine and H. Herr give in their excellent introductory textbook Volkswirtschaftslehre (2003, Oldenbourg Wissenschafts Verlag GmbH). See p.175 and further. Incidentally, recently a new edition has appeared. The explanation of the neoclassical equilibrium model can naturally also be found in any other textbook about general economics. On must be aware that many of them do not mention the many deficits and misunderstandings in their evaluation of the model. One must really be suspicious. (back)
  2. Each right-minded person will immediately understand, that Pareto defends a senseless idea. The utility for the homeless person is naturally larger. And in general the textbooks present indeed the utility as a quantitative mathematical function! (back)
  3. The economist B.M.S.van Praag defends in his valedictory address the measurement of the cardinal utilities. Several decades before, Sam de Wolff applied the concept of cardinal utility, among other in his model of the national product. (back)
  4. See p.189 and further in Volkswirtschaftslehre. (back)
  5. See p.141 and further in Volkswirtschaftslehre. (back)
  6. See p.77 and further in Welfare economics and social choice theory (1980, Martinus Nijhoff Publishing) by A.M. Feldman. (back)
  7. The same rather worn-out objections are mentioned by P. Bofinger on p.57 of Grundzüge der Volkswirtschafslehre (2011, Pearson Studium). Bofinger refers to the principal-agent problem. The principal is higher in the hierarchy than the agent, and therefore can command the latter. However, the agent has his own interests, and these do not by necessity coincide with those of the principal. Therefore the agent tries to take advantage of the asymmetric distribution of information: he possesses information, which he can select or hide from the principal. This will undermine the central plan, and according to Bofinger the agent will even try to minimize his own efforts. (back)
  8. See among others p.259 and further in Ökonomisch-theoretische Studien (1977, Europäische Verlags Anstalt GmbH). Oskar Lange was a Polish economist, who during the Second Worldwar had fled to the USA. After the war he returned to Poland, and contributed to the establishment of a centrally planned economy. (back)
  9. In a rather loose interpretation of Feldman on p.78 of Welfare economics and social choice theory it is argued: Suppose that for all producers the optimal production Qj rises in a uniform manner for increasing quantities qij of production factors. Suppose that this rise weakens according as the quantities qij increase (Feldman calls this a convex behaviour). Define QjN as the nett product belonging to the optimal production plan Qj. Then there is a price vector pj such that QjN will be produced automatically in the competitive general equilibrium. So that equilibrium will be determined by the combination (Q, p). (back)
  10. Indeed on p.13 of Ökonomisch-theoretische Studien Wlodzimiers Brus wonders why Lange has not commented on the criticism of the neoricardian paradigm. Brus is unable to give a satisfactory explanation. Perhaps a reason is that some findings of Sraffa had already been published by Soviet economists. A typical example is V.S. Nemchinov (in the German language W.S. Nemtschninow), who among other published Ökonomisch-mathematische Methoden und Modelle (1962; German translation 1965, Verlag Die Wirtschaft). Due to the ideological struggle between the economic systems the Leninist block was not eager to praise western economists like P. Sraffa. (back)
  11. The economist J.E. Woods shows on p.154 and further in The production of commodities (1990, MacMillan Education Ltd.), that the optimal production does not always follow a rising trend, when the means of production increase in quantity. The deviations are caused by the "reversing" of capital. That occurs always, when a reswitching of the production technique takes place. (back)
  12. See chapter 8 in Einführung in die politische Ökonomie des Sozialismus (1974, Dietz Verlag), written by a collective of authors and edited by W. Becker, G. Schulz en K.-H. Stiemerling. (back)
  13. See Bedürfnisse, Bedarf, Planung (1976, Verlag Die Wirtschaft), by H.-D. Haustein and G. Manz. (back)
  14. See p.96 and further in Harmonie en conflict (1968, De Bezige Bij) by J. Pen. The publications of the NIBUD also remind of consumer normatives. (back)
  15. See p.48 in Bedürfnisse, Bedarf, Planung. (back)
  16. "Der Kunde, der bekanntlich nicht arbeitet, sondern nur immerzu einkauft, ist eigentlich ein asoziales Element und darüber entsprechend zu belehren"
    "Ach, du bist's, Kollege - ich dachte schon, ich hätt' 'nen Dieb erwischt!"
    "Pech, endlich haben wir ein Kraftfutter entwickelt, und nun sind die Dinosaurier ausgestorben".
    "Herr Ober, wäre es nicht besser, Ihre Kollegen würden mit bedienen, statt zu musizieren?"
    "Als technisch hochstehendes Volk interessiert uns natürlich, wie der Trabant diesen wunderschönen blauen Qualm produziert".
    "Freunde, ihr seid zu gut für mich. Hier bin ich doch nicht der Leiter der Abteilung Handel und Versorgung, sonder einfach euer Wassja!"
    "Eingeschaltet und gleich kaputt? Das behaupten alle, die hier stehen, Angeklagter".
    "Wozu technischer Entwicklung und modernem Design hinterher rennen? Warten wir, bis die Nostalgiewelle auch uns erreicht hat".
    "Der Tag wird kommen, da sind auch unsere Erzeugnisse von solcher Qualität, daß wir Torkontrollen durchführen müssen".
    "Als alter Fuchs habe ich dem Tod beweisen können, daß der Termin nicht realisierbar ist". (back)
  17. See chapter 9 in the second volume of Gesetzmäßigkeiten der intensiv erweiterten Reproduktion bei der weiteren gestaltung der entwickelten sozialistischen Gesellschaft (1976, Akademie-Verlag), edited by W. Heinrichs and H. Maier. Also this work has been supplied by the second-hand bookshop Helle Panke in Berlin. (back)
  18. Your columnist is still puzzled by the relation between the concepts use value and utility. The concept of use value originates from K.H. Marx. The concept utility has been developed mainly in England, thanks to the work of W.S. Jevons. It pertains to individual preferences. According to Anders and Schwarz there is a difference between material and social use value. The material use value is the natural ability of the product of itself to perform a certain task. For instance, nails can be used for the repair of a chair. The social use value is the need, which due to the historical and technological developments exists in society for that ability. For instance, nails will not be very useful to the society, when in general glue yields a better result. The social use value resembles the concept of utility. (back)
  19. Until now your columnist has not found a similar concrete application of the laws of Gossen in the Leninist literature. A possible explanation is the resentment, which most economists in Eastern Europe apparently felt with regard to the use of mathematical formulas. They feared that model building might blur the insight in the real social processes (class struggle etcetera). Another explanation is that the laws of Gossen are difficult to apply to practical situations. (back)
  20. According to Anders en Schwarz the method has been invented by E.E. Sluckij. The authors calculate also the second-order derivatives in the Taylor series. For this purpose the differentiate the formula 3 with respect to B and to pk. The result is Σi=1n ∂²U/∂Qi∂Qj × ∂Qi/∂B = ∂λ/∂B × pj en Σi=1n ∂²U/∂Qi∂Qj × ∂Qi/∂pk = ∂λ/∂pk × pj + λ × δjk. Here δjk is the mathematical Kronecker delta. From these two sets of equations∂²U/∂Qi∂Qj and ∂λ/∂pk can be calculated. Next their values can be substituted into the second-order terms of the Taylor series. (back)