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.

Prisoner’s Dilemma. Thought of the Day 64.0

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A system suffering from Prisoner’s Dilemma cannot find the optimal solution because the individual driving forces go against the overall driving force. This is called Prisoner’s Dilemma based on the imaginary situation of two prisoners:

Imagine two criminals, named alphabetically A and B, being caught and put in separate prison cells. The police is trying to get confessions out of them. They know that if none will talk, they will both walk out of there for lack of evidence. So the police makes a proposal to each one: “We’ll make it worth your while. If you confess, and your colleague not, we give you 10 thousand euro and your colleague will get 50 years in prison. If you both confess you will each get 20 years in prison”. The decision table for these prisoners is like this:

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As you can see for yourself, the individual option for A, independent of what B decides to do, is confessing; moving from right column to left column, it is either reducing his sentence from 50 to 20 years, or instead of walking out of there even getting a fat bonus on top. The same applies to B, moving from bottom row to top row of the table. So, they wind up both confessing and getting 20 years in prison. That while it is obvious that the optimal situation is both not talking and walking out of prison scot-free (with the loot!). Because A and B cannot come to an agreement, but both optimize their own personal yield instead, they both get severely punished!

The Prisoner’s Dilemma applies to economy. If people in society cannot come to an agreement, but instead let everybody take decisions to optimize the situation for themselves (as in liberalism), they wind up with a non-optimal situation in which all the wealth is condensed on a single entity. This does not even have to be a person, but the capital itself. Nobody will get anything, beyond the alms granted by the system. In fact, the system will tend to reduce these alms – the minimum wages, or unemployment benefit – and will have all kinds of dogmatic justifications for them, but basically is a strategy of divide-and-conquer, inhibiting people to come to agreements, for instance by breaking the trade unions.

An example of a dogmatic reason is “lowering wages will make that more people get hired for work”. Lowering wages will make the distortion more severe. Nothing more. Moreover, as we have seen, work can be done without human labor. So if it is about competition, men will be cut out of the deal sooner or later. It is not about production. It is about who gets the rights to the consumption of the goods produced. That is also why it is important that people should unite, to come to an agreement where everybody benefits. Up to and including the richest of them all! It is better to have 1% of 1 million than 100% of 1 thousand. Imagine this final situation: All property in the world belongs to the final pan-global bank, with their headquarters in an offshore or fiscal paradise. They do not pay tax. The salaries (even of the bank managers) are minimal. So small that it is indeed not even worth it to call them salary.

Production Function as a Growth Model

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Any science is tempted by the naive attitude of describing its object of enquiry by means of input-output representations, regardless of state. Typically, microeconomics describes the behavior of firms by means of a production function:

y = f(x) —– (1)

where x ∈ R is a p×1 vector of production factors (the input) and y ∈ R is a q × 1 vector of products (the output).

Both y and x are flows expressed in terms of physical magnitudes per unit time. Thus, they may refer to both goods and services.

Clearly, (1) is independent of state. Economics knows state variables as capital, which may take the form of financial capital (the financial assets owned by a firm), physical capital (the machinery owned by a firm) and human capital (the skills of its employees). These variables should appear as arguments in (1).

This is done in the Georgescu-Roegen production function, which may be expressed as follows:

y= f(k,x) —– (2)

where k ∈ R is a m × 1 vector of capital endowments, measured in physical magnitudes. Without loss of generality, we may assume that the first mp elements represent physical capital, the subsequent mh elements represent human capital and the last mf elements represent financial capital, with mp + mh + mf = m.

Contrary to input and output flows, capital is a stock. Physical capital is measured by physical magnitudes such as the number of machines of a given type. Human capital is generally proxied by educational degrees. Financial capital is measured in monetary terms.

Georgescu-Roegen called the stocks of capital funds, to be contrasted to the flows of products and production factors. Thus, Georgescu-Roegen’s production function is also known as the flows-funds model.

Georgescu-Roegen’s production function is little known and seldom used, but macroeconomics often employs aggregate production functions of the following form:

Y = f(K,L) —– (3)

where Y ∈ R is aggregate income, K ∈ R is aggregate capital and L ∈ R is aggregate labor. Though this connection is never made, (3) is a special case of (2).

The examination of (3) highlighted a fundamental difficulty. In fact, general equilibrium theory requires that the remunerations of production factors are proportional to the corresponding partial derivatives of the production function. In particular, the wage must be proportional to ∂f/∂L and the interest rate must be proportional to ∂f/∂K. These partial derivatives are uniquely determined if df is an exact differential.

If the production function is (1), this translates into requiring that:

2f/∂xi∂xj = ∂2f/∂xj∂xi ∀i, j —– (4)

which are surely satisfied because all xi are flows so they can be easily reverted. If the production function is expressed by (2), but m = 1 the following conditions must be added to (4):

2f/∂k∂xi2f/∂xi∂k ∀i —– (5)

Conditions 5 are still surely satisfied because there is only one capital good. However, if m > 1 the following conditions must be added to conditions 4:

2f/∂ki∂xj = ∂2f/∂xj∂ki ∀i, j —– (6)

2f/∂ki∂kj = ∂2f/∂kj∂ki ∀i, j —– (7)

Conditions 6 and 7 are not necessarily satisfied because each derivative depends on all stocks of capital ki. In particular, conditions 6 and 7 do not hold if, after capital ki has been accumulated in order to use the technique i, capital kj is accumulated in order to use the technique j but, subsequently, production reverts to technique i. This possibility, known as reswitching of techniques, undermines the validity of general equilibrium theory.

For many years, the reswitching of techniques has been regarded as a theoretical curiosum. However, the recent resurgence of coal as a source of energy may be regarded as instances of reswitching.

Finally, it should be noted that as any input-state-output representation, (2) must be complemented by the dynamics of the state variables:

k ̇ = g ( k , x , y ) —– ( 8 )

which updates the vector k in (2) making it dependent on time. In the case of aggregate production function (3), (8) combines with (3) to constitute a growth model.

Austrian Economics. Ruminations. End Part.

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Mainstream economics originates from Jevons’ and Menger’s marginal utility and Walras’ and Marshall’s equilibrium approach. While their foundations are similar, their presentation looks quite different, according to the two schools which typically represent these two approaches: the Austrian school initiated by Menger and the general equilibrium theory initiated by Walras. An important, albeit only formal, difference is that the former presents economic theory mainly in a literary form using ordinary logic, while the latter prefers mathematical expressions and logic.

Lachmann, who excludes determinism from economics since acts of mind are concerned, connects determinism with the equilibrium approach. However, equilibrium theory is not necessarily deterministic, also because it does not establish relationships of succession, but only relationships of coexistence. In this respect, equilibrium theory is not more deterministic than the theory of the Austrian school. Even though the Austrian school does not comprehensively analyze equilibrium, all its main results strictly depend on the assumption that the economy is in equilibrium (intended as a state everybody prefers not to unilaterally deviate from, not necessarily a competitive equilibrium). Considering both competition and monopoly, Menger examines the market for only two commodities in a barter economy. His analysis is the best to be obtained without using mathematics, but it is too limited for determining all the implications of the theory. For instance, it is unclear how the market for a specific commodity is affected by the conditions of the markets for other commodities. However, interdependence is not excluded by the Austrian school. For instance, Böhm-Bawerk examines at length the interdependence between the markets for labor and capital. Despite the incomplete analysis of equilibrium carried out by the Austrian school, many of its results imply that the economy is in equilibrium, as shown by the following examples.

a) The Gossen-Menger loss principle. This principle states that the price of a good can be determined by analyzing the effect of the loss (or the acquisition) of a small quantity of the same good.

b) Wieser’s theory of imputation. Wieser’s theory of imputation attempts to determine the value of the goods used for production in terms of the value (marginal utility) of the consumption goods produced.

c) Böhm-Bawerk’s theory of capital. Böhm-Bawerk proposed a longitudinal theory of capital, where production consists of a time process. A sequence of inputs of labor is employed in order to obtain, at the final stage, a given consumption good. Capital goods, which are the products obtained in the intermediate stages, are seen as a kind of consumption goods in the process of maturing.

A historically specific theory of capital inspired by the Austrian school focuses on the way profit-oriented enterprises organize the allocation of goods and resources in capitalism. One major issue is the relationship between acquisition and production. How does the homogeneity of money figures that entrepreneurs employ in their acquisitive plans connect to the unquestionable heterogeneity of the capital goods in production that these monetary figures depict? The differentiation between acquisition and production distinguishes this theory from the neoclassical approach to capital. The homogeneity of the money figures on the level of acquisition that is important to such a historically specific theory is not due to the assumption of equilibrium, but simply to the existence of money prices. It is real-life homogeneity, so to speak. It does not imply any homogeneity on the level of production, but rather explains the principle according to which the production process is conducted.

In neoclassical economics, in contrast, production and acquisition, the two different levels of analysis, are not separated but are amalgamated by means of the vague term “value”. In equilibrium, assets are valued according to their marginal productivity, and therefore their “value” signifies both their price and their importance to the production process. Capital understood in this way, i.e., as the value of capital goods, can take on the “double meaning of money or goods”. By concentrating on the value of capital goods, the neoclassical approach assumes homogeneity not only on the level of acquisition with its input and output prices, but also on the level of production. The neoclassical approach to capital assumes that the valuation process has already been accomplished. It does not explain how assets come to be valued originally according to their marginal product. In this, an elaborated historically specific theory of capital would provide the necessary tools. In capitalism, inputs and outputs are interrelated by entrepreneurs who are guided by price signals. In their efforts to maximize their monetary profits, they aim to benefit from the spread between input and output prices. Therefore, money tends to be invested where this spread appears to be wide enough to be worth the risk. In other words, business capital flows to those industries and businesses where it yields the largest profit. Competition among entrepreneurs brings about a tendency for price spreads to diminish. The prices of the factors of production are bid up and the prices of the output are bid down until, in the hypothetical state of equilibrium, the factor prices sum up to the price of the product. A historically specific theory of capital is able to describe and analyze the market process that results – or tends to result – in marginal productivity prices, and can therefore also formulate positions concerning endogenous and exogenous misdirections of this process which lead to disequilibrium prices. Consider Mises,

In balance sheets and in profit-and-loss statements, […] it is necessary to enter the estimated money equivalent of all assets and liabilities other than cash. These items should be appraised according to the prices at which they could probably be sold in the future or, as is especially the case with equipment for production processes, in reference to the prices to be expected in the sale of merchandise manufactured with their aid.

According to this, not the monetary costs of the assets, which can be verified unambiguously, but their values are supposed to be the basis of entrepreneurial calculation. As the words indicate, this procedure involves a tremendous amount of uncertainty and can therefore only lead to fair values if equilibrium conditions are assumed.

Austrian Economics. Some Ruminations. Part 1.

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Keynes argued that by stimulating spending on outputs, consumption, goods and services, one could increase productive investment to meet that spending, thus adding to the capital stock and increasing employment. Hayek, on the other hand furiously accused Keynes of insufficient attention to the nature of capital in production. For Hayek, capital investment does not simply add to production in a general way, but rather is embodied in concrete capital items. Rather than being an amorphous stock of generalized production power, it is an intricate structure of specific interrelated complementary components. Stimulating spending and investment, then, amounts to stimulating specific sections and components of this intricate structure. Before heading out to Austrian School of Economics, here is another important difference between the two that is cardinal, and had more do with monetary system. Keynes viewed the macro system as vulnerable to periodic declines in demand, and regarded micro adjustments such as wage and price declines as ineffective to restore growth and prosperity. Hayek viewed the market as capable of correcting itself by taking advantages of competitions, and regarded government and Central Banks’ policies to restore growth as sources of more instability.

The best known Austrian capital theorist was Eugen von Böhm-Bawerk, though his teacher Carl Menger is the one who got the ball rolling, providing the central idea that Böhm-Bawerk elaborated. For the Austrians, the general belief lay in the fact that production takes time, and more roundabout the process, the more delay production needs to anticipate. Modern economies comprise complex, specialized processes in which the many steps necessary to produce any product are connected in a sequentially specific network – some things have to be done before others. There is a time structure to the capital structure. This intricate time structure is partially organized, partially spontaneous (organic). Every production process is the result of some multiperiod plan. Entrepreneurs envision the possibility of providing (new, improved, cheaper) products to consumers whose expenditure on them will be more than sufficient to cover the cost of producing them. In pursuit of this vision the entrepreneur plans to assemble the necessary capital items in a synergistic combination. These capital combinations are structurally composed modules that are the ingredients of the industry-wide or economy-wide capital structure. The latter is the result then of the dynamic interaction of multiple entrepreneurial plans in the marketplace; it is what constitutes the market process. Some plans will prove more successful than others, some will have to be modified to some degree, some will fail. What emerges is a structure that is not planned by anyone in its totality but is the result of many individual actions in the pursuit of profit. It is an unplanned structure that has a logic, a coherence, to it. It was not designed, and could not have been designed, by any human mind or committee of minds. Thinking that it is possible to design such a structure or even to micromanage it with macroeconomic policy is a fatal conceit. The division of labor reflected by the capital structure is based on a division of knowledge. Within and across firms specialized tasks are accomplished by those who know best how to accomplish them. Such localized, often unconscious, knowledge could not be communicated to or collected by centralized decision-makers. The market process is responsible not only for discovering who should do what and how, but also how to organize it so that those best able to make decisions are motivated to do so. In other words, incentives and knowledge considerations tend to get balanced spontaneously in a way that could not be planned on a grand scale. The boundaries of firms expand and contract, and new forms of organization evolve. This too is part of the capital structure broadly understood.

Hayek emphasizes that,

the static proposition that an increase in the quantity of capital will bring about a fall in its marginal productivity . . . when taken over into economic dynamics and applied to the quantity of capital goods, may become quite definitely erroneous.

Hayek stresses chains of investments and how earlier investments in the chains can increase the return to the later, complementary investments. However, Hayek is primarily concerned with applying those insights to business cycle phenomena. Also, Hayek never took the additional step that endogenous growth theory has in highlighting the effects of complementarities across intangible investments in the production of ideas and/or knowledge. Indeed, Hayek explicitly excludes their consideration:

It should be quite clear that the technical changes involved, when changes in the time structure of production are contemplated, are not changes due to changes in technical knowledge. . . . It excludes any changes in the technique of production which are made possible by new inventions.

…….