Production Function as a Growth Model

Cobb-Douglas_Production_Function

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.

Advertisement

Austrian Economics. Some Ruminations. Part 1.

austrian-school-of-economics

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.

…….