Equilibrium Market Prices are Unique – Convexity and Concavity Utility Functions on a Linear Case. Note Quote + Didactics.

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Consider a market consisting of a set B of buyers and a set A of divisible goods. Assume |A| = n and |B| = n′. We are given for each buyer i the amount ei of money she possesses and for each good j the amount bj of this good. In addition, we are given the utility functions of the buyers. Our critical assumption is that these functions are linear. Let uij denote the utility derived by i on obtaining a unit amount of good j. Thus if the buyer i is given xij units of good j, for 1 ≤ j ≤ n, then the happiness she derives is

j=1nuijxij —— (1)

Prices p1, . . . , pn of the goods are said to be market clearing prices if, after each buyer is assigned an optimal basket of goods relative to these prices, there is no surplus or deficiency of any of the goods. So, is it possible to compute such prices in polynomial time?

First observe that without loss of generality, we may assume that each bj is unit – by scaling the uij’s appropriately. The uij’s and ei’s are in general rational; by scaling appropriately, they may be assumed to be integral. By making the mild assumption that each good has a potential buyer, i.e., a buyer who derives nonzero utility from this good. Under this assumption, market clearing prices do exist.

It turns out that equilibrium allocations for Fisher’s linear case are captured as optimal solutions to a remarkable convex program, the Eisenberg–Gale convex program.

A convex program whose optimal solution is an equilibrium allocation must have as constraints the packing constraints on the xij’s. Furthermore, its objective function, which attempts to maximize utilities derived, should satisfy the following:

  1. If the utilities of any buyer are scaled by a constant, the optimal allocation remains unchanged.
  2. If the money of a buyer b is split among two new buyers whose utility functions are the same as that of b then sum of the optimal allocations of the new buyers should be an optimal allocation for b.

The money weighted geometric mean of buyers’ utilities satisfies both these conditions:

max (∏i∈Auiei)1/∑iei —– (2)

then, the following objective function is equivalent:

max (∏i∈Auiei) —– (3)

Its log is used in the Eisenberg–Gale convex program:

maximize, ∑i=1n’eilogui

subject to

ui = ∑j=1nuijxij ∀ i ∈ B

i=1n’ xij ≤ 1 ∀ j ∈ A

xij ≥ 0 ∀ i ∈ B, j ∈ A —– (4)

where xij is the amount of good j allocated to buyer i. Interpret Lagrangian variables, say pj’s, corresponding to the second set of conditions as prices of goods. Optimal solutions to xij’s and pj’s must satisfy the following:

    1. ∀ j ∈ A : p≥ 0
    2. ∀ j ∈ A : p> 0 ⇒ ∑i∈A xij = 1
    3. ∀ i ∈ B, j ∈ A : uij/pj ≤ ∑j∈Auijxij/ei
    4. ∀ i ∈ B, j ∈ A : xij > 0 ⇒ uij/pj = ∑j∈Auijxij/ei

From these conditions, one can derive that an optimal solution to convex program (4) must satisfy the market clearing conditions.

For the linear case of Fisher’s model:

  1. If each good has a potential buyer, equilibrium exists.
  2. The set of equilibrium allocations is convex.
  3. Equilibrium utilities and prices are unique.
  4. If all uij’s and ei’s are rational, then equilibrium allocations and prices are also rational. Moreover, they can be written using polynomially many bits in the length of the instance.

Corresponding to good j there is a buyer i such that uij > 0. By the third condition as stated above,

pj ≥ eiuij/∑juijxij > 0

By the second condition, ∑i∈A xij = 1, implying that prices of all goods are positive and all goods are fully sold. The third and fourth conditions imply that if buyer i gets good j then j must be among the goods that give buyer i maximum utility per unit money spent at current prices. Hence each buyer gets only a bundle consisting of her most desired goods, i.e., an optimal bundle.

The fourth condition is equivalent to

∀ i ∈ B, j ∈ A : eiuijxij/∑j∈Auijxij = pjxij

Summing over all j

∀ i ∈ B : eij uijxij/∑j∈Auijxij = pjxij

⇒ ∀ i ∈ B : ei = ∑jpjxij

Hence the money of each buyer is fully spent completing the proof that market equilibrium exists. Since each equilibrium allocation is an optimal solution to the Eisenberg-Gale convex program, the set of equilibrium allocations must form a convex set. Since log is a strictly concave function, if there is more than one equilibrium, the utility derived by each buyer must be the same in all equilibria. This fact, together with the fourth condition, gives that the equilibrium prices are unique.

Surplus. What All Could Social Activists Do, But Debate?

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The social surplus is a basic concept of classical political economy which has been revived in the post-war period by Paul Baran and Paul Sweezy. They defined it as

.. the difference between what a society produces and the costs of producing it. The size of a surplus is an index of productivity and wealth, and of how much freedom a society has to accomplish whatever goals it may set for itself. The composition of the surplus shows how it uses that freedom: how much it invests in expanding its productive capacity, how much it consumes in various forms, how much it wastes and in what ways.

The surplus can be calculated in alternative ways. One is to estimate the necessary costs of producing the national product, and to deduct the costs from the national product. This raises the conceptual problem of calculating the necessary costs of production. Some of the outlays recorded as costs by firms (such as outlays for superficial product differentiation and advertising) may be unnecessary from the social viewpoint. Hence the determination of the necessary costs is crucial for this first method. A second method is to estimate the various expenditures absorbing the surplus (non-essential consumption, investment etc.) and to add them up.

The re-elaboration of the surplus concept in the post-war period is connected to the evolution of certain features of capitalism. In Monopoly Capital Baran and Sweezy argued that capitalism had made a transition from a competitive phase to a monopolistic phase in the twentieth century. In their view, the concentration of capital in giant corporations enables them to fix prices, in contrast to nineteenth century capitalists who worked under more intense competition. These giant corporations set their sales prices by adding mark-ups to production costs. Such price setting gives the corporations control over the partition of the value added with their workers. Corporations also strive to increase their profits by reducing their production costs. On the macroeconomic plane, the general endeavour to reduce production costs (inclusive of labor costs) tends to raise the share of the surplus in GDP. This rising surplus can be sustained only if it is absorbed. The consumption of capitalists, the consumption of employees in non-productive activities (e.g. superficial product differentiation, advertising, litigation etc.), investment and some part of government expenditure (e.g. public investment, military outlays) are the main outlets for absorbing the surplus.

As almost sixty years have elapsed since the above framework was formulated, it is legitimate to ask: has the increasing ratio of trade to global output impaired the diagnosis of Baran and Sweezy with regard to the monopolization of capital, and with respect to the inclination for the surplus in GDP to increase? Has increasing trade and integration of markets raised competitive pressures so as to restrict the pricing latitude of industrial conglomerates?

The immediate effect of global trade expansion obviously must be to increase overall competition, as greater numbers of firms would come to compete in formerly segregated markets. But a countervailing effect would emerge when large firms with greater financial resources and organizational advantages eliminate smaller firms (as happens when large transnationals take on firms of peripheral countries in opened markets). Another countervailing trend to the competition-enhancing effect of trade expansion is mergers and acquisitions, on which there is evidence in the core countries. A powerful trend increase in the extent of firm level concentration of global markets share could be observed in industries as diverse as aerospace and defence, pharmaceuticals, automobiles, trucks, power equipment, farm equipment, oil and petrochemicals, mining, pulp and paper, brewing, banking, insurance, advertising, and mass media. Indications are that the competition-enhancing effect of trade is balanced (perhaps even overwhelmed) by the monopolizing effect of the centralization of capital, which may sustain the ability of large corporations to control the market prices of their products.

On the other hand, if mergers and acquisitions imply an increase in the average size of the workforce of corporations, this could stimulate a counterbalance to corporate power by higher unionization and worker militancy. However, the increasing mobility of capital, goods and services on the one hand, and unemployment on the other is weakening unionization in the core countries, and making workers accept temporary employment, part-time employment, flexibility in hiring and dismissing, flexible working days and weeks, and flexibility in assigning tasks in the workplace. Increasing flexibility in labor relations shifts various risks related to the product markets and the associated costs from firms onto workers. Enhanced flexibility cannot but boost gross profits. Hence the trend towards increased flexibility in labor practices clearly implies increased surplus generation for given output in individual countries.

The neoliberal global reform agenda also includes measures to increase surplus generation through fiscal and institutional reforms, both in developed and underdeveloped countries. Lowering taxes on corporate profits, capital gains and high incomes; increasing taxes on consumption; raising fees on public services and privatization of these services, of utilities and of social security – all these policies aim at disburdening the high income earners and property owners of contributing to financing essential services for the maintenance of the labor force. These reforms also contribute to increasing the share of surplus in total output.

In brief, in the era of neoliberal policies evidence does not seem to suggest that the tendency for the share of surplus in GDP to rise in individual countries may have waned. If so, what is happening to the surplus generated in international production?

Baran and Sweezy argued that the surplus of underdeveloped countries had been and was being drained away to the centers of the world-system. Their description of core firms‘ overseas activities in Monopoly Capital can be read as a description of offshore outsourcing activities today if one replaces subsidiary with suppliers:

What they [giant multinational corporations] want is monopolistic control over foreign sources of supply and foreign markets, enabling them to buy and sell on specially privileged terms, to shift orders from one subsidiary to another, to favour this country or that depending on which has the most advantageous tax, labour and other policies…

The authors’ view was that imperialism had a two-fold function with respect to the surplus: finding cheap foreign sources of supply (which increases the surplus in the home country), and using other countries‘ markets as outlets (which helps absorb the surplus of the home country). A major motive of transnational companies in their current practice of outsourcing parts of production to underdeveloped countries is to cut production costs, hence to increase gross profits. When the corporation of a core country decides to outsource its production to a peripheral country, or when it shifts its sources of supply of intermediate inputs to a peripheral country, this increases global surplus creation. Global output remains the same, the costs of producing it decline. For the firm, the effect of offshore outsourcing is the same as if it were to reduce its own (in-house) costs of production, or were to outsource to a cheap supplier in the home economy. If the workers in the core country dismissed due to the offshore outsourcing find newly created jobs and continue to produce surplus, then global output increases and surplus creation increases a fortiori. If the workers dismissed due to the outsourcing remain unemployed, then their consumption (provided by family, unemployment benefits etc.) absorbs part of the surplus produced by other workers in employment. Should the supplier in the peripheral country expand her production to meet the order under subcontract, there will also be some increase in surplus creation in the peripheral country. In this case the total increase in surplus may accrue to both countries  economies – in indeterminate proportions.

It is worth noting that the effect of offshore outsourcing on productivity in the core economies is ambiguous. The formula

Productivity = (Sales Revenue – Material Input Cost) / Number of Workers

shows that an increase in material input cost (due to the increase in outsourced inputs) and a reduction of the in-house workforce (due to outsourcing) may ultimately affect the outsourcing firm‘s productivity either way. The gains that motivate firms to outsourcing are not gains in labor productivity (which arguably could legitimize outsourcing from a social viewpoint), but gains in gross profits – i.e. in surplus appropriation.

It emerges that the basic tendencies in the production and growth of the social surplus described by Baran and Sweezy have not changed under globalizing capitalism. New economic policies, corporate strategies and international rules of conduct appear to promote increasing surplus transfers from the periphery to the core of the world-system. In order to lift itself out of destitution the periphery is exhorted to remove restrictions on trade and capital flows, and to compete for advantageous positions in global value chains controlled by transnationals by improving quality, reducing costs, innovating etc. The export-led growth economic strategy compels peripheral producers to individually compete for exportation by repressing wages, and conceding much of the surplus produced to their trade partners in the core countries. Part of the surplus accruing to the periphery is consumed by transnational élites imitating the consumption of the well-to-do in the core societies. On the other hand dollarization, capital flight and official reserve accumulation exert downward pressure (a pressure unrelated to trade balances) on the exchange rate of peripheral currencies. The undervaluation of peripheral currencies, reflected in deteriorating terms of trade, translates into a loss of surplus to the core countries, and reduces the capacity of poor countries to import capital goods from the core. The resulting meager per capita fixed capital formation in the underdeveloped countries bodes grim prospects for the welfare of future generations of working people in the periphery. These trends are maintained by the insertion of millions of workers in Asian hinterlands into global production networks, and by the willingness of peripheral states governed by transnational élites to continue free trade and capital transactions policies, and to accumulate foreign exchange reserves. Africa’s poor populations await their turn to be drawn into the world labor market, to eke out a subsistence and produce a surplus, of which a large part will likely flow to the core.

In order to prevent the drift of the victims of globalizing capitalism to irrational reaction (religious or nationalist fanaticism, clash of civilizations etc.) and to focus their attention on the real issues, social scientists and activists should open to debate the social and economic consequences of the export-led growth idea, all the theories and policies that give precedence to global efficiency over national saving and investment, and the social psychology of consumerism. There is pressing need to promote socio-economic programs based on the principle of self-sufficient and self-reliant national development, wherein the people can decide through democratic procedures how they will dispose the social surplus they produce (how they will distribute it, how much they will save, invest, export) under less pressure from world markets dominated by transnational companies, and with less interefence from international institutions and core states. Within the framework of the capitalist world-system, there is little hope for solving the deep social contradictions the system reproduces. The solution, reason shows, lies outside the logic of the system.