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The concept of the production system and the production process. Technological process and technological set. Manufacturer Behavior

concept is familiar to every person, since he is born and lives among a set of things that is characteristic of the material culture of his society. Even the entire economic theory begins with a description of the subject set, which he gave in his work, by comparing the number and quantity of objects and the number of professions (technologies), which determined the wealth of a particular state. Another thing is that all previous theories accepted this position axiomatically, but along with the loss of interest in the concept, they understood meaning of subject- technological set only in connection with a separate .

Therefore, it is still a discovery that PTM associated with, which can only sometimes coincide with the economy of the state. The phenomenon of subject-technological set turned out to be not as simple as it seemed to economists. In this article about the subject-technological set the reader will find not only description of the subject-technological set like but also a history of recognition PTM as a measure for comparing the development of countries.

subject-technological set

The people themselves are the product of a fairly high standard of living that the steppe hominids have achieved due to the appearance of some stable ones in their flocks. If for primates - gathering, as a way of obtaining resources from the territory of the natural complex, did not require the combined efforts of several individuals, then hunting for large ungulates, which became the main way to ensure the existence of hominids during the development of the steppes, was a complexly organized activity with a division of roles among several participants.

At the same time, the small size of the steppe hominids did not allow them to kill a large animal without hunting tools, even as part of a group. However, in the steppes, stones of a suitable shape are not everywhere and it is difficult to find a pointed stick, so the hominids had to carry hunting tools with them. Along with the clothes that appeared along with upright walking, the result of which was deprivation of hair, and simply - due to the cool climate of the steppes, STAI-TRIBES acquire a certain set, in other words - many- items, the presence of which provides members with a starvation level of existence.

People, on the other hand, appear along with luxury, that is, objects that hominids had no time for before - neither simply appropriate the objects that interested them from Nature, nor make them by labor, since there was neither the need nor the opportunity to constantly carry with them. Luxury goods include all improved tools, after all, for people, as one of the mammalian species, a set of life goods is enough for life, the production of which fully provided the subject set that hominids had in flocks. As a biological being, man already millions of years ago could and lived above the level of hominids with the same set of objects, but people are so strong that people did not stop at the level of hominids, as it should have been for an animal species that reached a level of prosperity. People did not have the opportunity to improve their living conditions in the natural environment, so they begin to create their own artificial environment from the objects of labor.

In the tribes of people, he continued to act, inherited from the hominids, in the flocks of which the first consumer of any luxury (beautiful feathers as an example of "charm") could only be the leader. When the leader had a lot of feathers, he gave them to his close associates - members with a high status. Such gift practice the rest of the tribe gave rise to the belief that the possession of a thing from the everyday life of the leader raises the status of the owner in the hierarchy. Consumption according to status forced high-ranking members of society to demand the most luxurious things.

At the same time, many low-ranking members are ready to sacrifice a lot in order to get things from the everyday life of the hierarchs, since the possession of these things allows them to feel an increase in their status in front of the rest. So things that first appear in the everyday life of hierarchs, in copies, became the subject of consumption of high-status members, and desire from other members with a strong hierarchical instinct led to mass production, which lowered the price, making the thing accessible to any member of the community. This race for prestige has gone on for thousands of years, multiplying the number of items, so that we now live surrounded by millions of items that make people's lives ONLY MUCH MORE COMFORTABLE than the lifestyle of the hominid ancestor.

But biologically man is still the same hominid with a hierarchical instinct, which he realizes in a field called -. Subject-technological set is another difference between man and animals - this is a new artificial habitat that man creates thanks to scientific and technological progress, which is driven by. As you can see, there is nothing sacred in ECONOMIC DEVELOPMENT, only satisfaction is one of the instincts.

We can say that it is familiar to every person, since he is born and lives surrounded by many objects, but the idea of a subject-technological set appeared when they decided compare wealth of different states. And here subject-technological set turned out to be a clear indicator of wealth or the degree of development. In one case, it is possible to compare by assortment - i.e. by the number of different subjects, which makes it possible to characterize the development of the same society over a certain period of time (which is described in the topic of scientific and technological progress). Otherwise, we can say that one society is richer than another, but then it is necessary to add a characteristic of the quality and technological perfection of the compared items to the assortment parameter (this is studied in the topic -). But, as a rule, fundamentally new objects appear in the subject set of a richer society, in the manufacture of which new technologies were used. The connection between more advanced and fundamentally new products and new technologies is quite obvious, therefore, which a certain society has, implies not just a list of items, but also technology set, which allows in the sphere of production of this society to produce these products.

For the old economic theories- the unit of the economy is the economy of a sovereign state. It is the population of the state that is considered the community, the subject-technological set of which is determined by the ability of the economy of this state to produce all these items. And the connection with technologies is assumed to be mechanical - literally, if the state has technologies, then nothing prevents the production of products corresponding to them.

However, with the advent of the global system of division of labor, the inaccuracy of identifying the economy of one country with the community of people that has such an attribute as subject-technological set. The fact is that in countries participating in the international division of labor, most of the components, parts and spare parts from which finished products are assembled here can even not produced in the territory of this state and vice versa - only parts are produced, but final products are not produced.

It must be said here that discrepancy THE AVAILABILITY of technology and the POSSIBILITY to produce some products on its basis - there was also BEFORE the international division of labor, but the old economic science discrepancy I didn’t even notice, even more - in the understanding of previous theories - the economies of all states were equal (the difference was accepted only in size - one can be more or less than the other) and as soon as technology is given, the POSSIBILITY to produce anything appears immediately.
The fact that practice refuted these theoretical assumptions did not interfere with the old economics give recipes for developing countries to build production facilities of any technological complexity. A very common example is Romania, which, according to economists, has no barriers to reaching the level of the United States of America, at least in the field of production, although it is clear that in order for Romania's subject-technological set to become as large as in USA, it is necessary to have at least as many people in production. However, if the assortment of the subject-technological set of the United States exceeds the number of a resident of Romania, then it is not clear who on the territory of Romania will be able to produce so many items.
There are objective restrictions for development - and they come down rather not only to the size of the division of labor system that can be created in the country (for example, India, where the population theoretically allows you to create the largest in the world, but from a theoretical possibility - India has not become richer) , and in . For example, Finland is short term managed to take the place of the most advanced country in the production mobile phones. But after all, the manufactured Nokia phones did not all remain within the subject-technological set of Finland, they replenished the subject sets of many countries. Therefore, we must conclude - power of subject technological set specific is determined not so much by the number of people employed in production, but to a greater extent by the size of the market (the number of products depends on it), and most importantly, by the presence of a mass solvent DEMAND for the product.

As you can see now - the concept of subject-technological set not as easy as it seems. First, we now understand that subject-technological set rather associated with a certain system of division of labor, and not with the state (in the sense, although historically subject-technological set we deduce from the subject set , which was the first ). This system can be inside or external supersystem in relation to the population. Second, present subject-technological set we can, if it has a countable assortment - otherwise, the number of different items in it is finite, which implies a countable limited number of people in the community. If we mean by a community having PMT, a system of division of labor, then we must talk about its CLOSENESS, since objects from a multitude are both produced and consumed in this system.

Own scientific value subject-technological set receives with the opening new object in the economy, which is called , which represents closed, in which those items that are produced are also consumed in it. An example of a reproductive complex is in, but the following - such as, and especially - could have a combination of several.

The term subject-technological set used already in the first works on , when he was interested in the interaction of developed and developing countries. That's when I started using term subject-technological set, as a certain characteristic of the systems of division of labor that have developed in different countries. Then it was not very clear what entity it was connected with. PMT, that's why term subject-technological set was used to characterize states when comparing them. Tut followed the founder of political economy, who in his work compared the welfare of countries as a comparison of the number and volume of products that are produced by the labor of citizens.

Eligibility of use PMT concepts to the state - remained, but the reader must remember - subject-technological set characterizes closed system of division of labor, which in some models may mean economy of one independent state.

Another question directly related to the forecast of the present is Can the subject-technological set decrease? The answer is, of course, it can, although it seems to many that scientific and technological progress can only increase power of the subject-technological set, if you look at it as an attribute of the state. It is clear that some objects naturally leave the life of people, others are so improved that they no longer resemble their historical prototype. This natural process is associated with the emergence of new technologies, but, as the history of the Roman Empire has shown - subject-technological set can shrink along with the oblivion of all technological achievements, if the system of division of labor that replaces it is not able to ensure the reproduction PTM in all volume.

At the beginning of our era, a demographic crisis begins in Europe, so that the tribes cannot bud, and the desire to remove the excess population leads to land. On the periphery of the Roman Empire, states begin to turn, and it turns out that Ancient Rome (like Ancient Greece) was a branch of the eastern empire on the European continent. Indigenous Europe comes to a natural state of the period of formation of states, which in Europe, due to the initial small population of its master, has shifted centuries later than it was in the EAST. The Roman Empire did not have a chance to resist the desire of the tribes to expand, and the loss of territories destroyed the existing system of division of labor, the collapse of which led to the disappearance of demand for the former everyday products of the Romans. The collapse of the subject set was so great that many Roman technologists were completely forgotten and they were rediscovered only after a millennium, and the standard of living that existed in the cities of Ancient Rome was re-achieved in Europe only in the 19th century, for example - plumbing in the upper floors of multi-storey buildings.

I outlined the main nuances of the concept subject-technological set, but must lead definition of subject-technological set from the official Glossary of Neoconomics:

THE CONCEPT OF SUBJECT-TECHNOLOGICAL SET (PTM)

it SUBJECT-TECHNOLOGICAL SET consists of items (products, parts, types of raw materials) that actually exist in a certain system of division of labor, that is, they are produced by someone and, accordingly, consumed - sold on the market or distributed. As for the details, they may not be goods, but be part of the goods.

Another part of this set is a set of technologies, that is, methods for the production of goods sold on the market - from and / or with - using the items included in this set. That is, knowledge of the correct sequences of actions with the material elements of the set.

In every period of time we have subject-technological set(PTM) different in power. As the division of labor deepens PTM expands.

The importance of this concept lies in the fact that PTM determines the possibility scientific and technological progress. When poor PTM new inventions, even if they can be implemented in the form of prototypes, as a rule, do not have a chance to be mass-produced if they require certain products or technologies that are not available in PTM. They just turn out to be too expensive.

Related materials

Only in front of you excerpt from Chapter 8 of The Age of Growth, in which gives description of the subject-technological set:

Let's introduce concept of subject-technological set. This set consists of items (products, parts, types of raw materials) that actually exist, that is, they are produced by someone and, accordingly, are sold on the market. As for the details, they may not be goods, but be part of the goods. The second part of this set is technologies, that is, methods for the production of goods sold on the market from and with the help of items included in this set. That is knowledge of the correct sequences of actions with the material elements of the set.

In each period of time we have a different power subject-technological set (PTM). By the way, it can not only expand. Some items cease to be produced, some technologies are lost. Maybe the drawings and descriptions remain, but in reality, if suddenly necessary, the restoration of elements PTM may be a complex project, in fact - a new invention. They say that when, in our time, they tried to reproduce the Newcomen steam engine, they had to expend great efforts in order to make it at least somehow work. But in the 18th century, hundreds of these machines worked quite successfully.

But, in general, PTM while expanding. Let's highlight two extreme cases of how this expansion can occur. The first is pure innovation, that is, a completely new item created using a previously unknown technology from completely new raw materials. I don’t know, I suspect that in reality this case has never occurred, but let’s assume that it can be so.

The second extreme case is when new set elements are formed as combinations of already existing elements. PTM. Such cases are just not uncommon. Already Schumpeter viewed innovations as new combinations of what already exists. Take the same personal computers. In a sense, it cannot be said that they were "invented". All of their components already existed, and were simply combined in a certain way.

If we can talk about some kind of discovery here, then it lies in the fact that the initial hypothesis: "they will buy this thing" - was completely justified. Although, if you think about it, then it was not at all obvious, and the greatness of the discovery lies precisely in this.

We understand that most of the new elements PTM are a mixed case: closer to the first or second. So, the historical trend, it seems to me, is that the share of inventions close to the first type is declining, while the share of the second type is increasing.

In general, in the light of my story about the devices of the series BUT and device B it is clear why this happens. For more details, see Chapter 8 of the book at the click of a button:

It is characterized by variables that take an active part in changing the production function (capital, land, labor, time). Neutral technical progress is determined by such technical changes (autonomous or material form) that do not disturb the balance, that is, they are economically and socially safe for society. Let's present all this in the form of a diagram (see diagram 4.1.).

The main typical models for optimizing the production activity of a company with a linear technological set, statistical and dynamic models for planning production investments, issues of economic and mathematical analysis of economic decisions based on the use of the apparatus of dual estimates are considered. The main approaches to the problem of assessing the quality of industrial investments, as well as methods and indicators for assessing their effectiveness are outlined.

Let us consider the case, which is very important for modeling applications, when the technological set of the production system is a linear convex set , i.e., the production model turns out to be linear.

Comment. Assumptions 2.1 and 2.2 together mean that the technological set is a convex cone. Assumption 2.3, distinguishing linear technologies, means that this cone is a convex polyhedron in the half-space

Can it be argued that in the economic field of a company with a linear technological set, the production function is monotonic How is the definition of the production function related to the optimality criterion in the Kantorovich problem

Relation (3.26) makes it possible to indicate a specific type of production function for a model of a production system with a linear technological set (model (1.1) - (1.6) considered above)

The state of each production element will continue to be specified by the input-output vector yt = (vt, u), and the constraint model by the technological set Yt yt = (Vi, ut) e YI.

The general technological set of a production element can be obtained as a result of the union of all cost-output vectors admissible from the point of view of conditions (2.1.2) and (2.1.3)

The description of the technological set of a single-product element given in the previous paragraph is the simplest. Taking into account the additional properties of the element technology leads to the need to supplement it with a number of features. We will consider some of them in this paragraph. Of course, the above considerations do not exhaust all the possibilities available in this direction.

Separable convex production model. Accounting for the nonlinearity factor in the model of production constraints described in the previous example leads to a nonlinear separable model of a multi-product element. Nonlinearity is taken into account by introducing non - linear separable production functions . The technological set of a multi-product element with such production functions has the form

In the considered technological models of production elements, the description of the technological set is given by setting the set of allowable costs and the set of allowable outputs for each level of costs. Descriptions of this kind are convenient in problems such as the optimal distribution of resources, in which, for given levels of resource consumption, it is necessary to determine the permissible and most efficient (in the sense of one or another criterion) output levels. At the same time, in practice (especially in a planned economy), there is also a kind of inverse problem, when the level of output of the elements is given by the plan and it is necessary to determine the allowable and minimum levels of costs of the elements. Problems of this kind can be conditionally called problems of optimal execution of the planned output program. In such problems, it is convenient to apply the reverse sequence of describing the technological set of a production element, first set the set U of allowable outputs and g = U, and then for each allowable level of outputs - the set V (u) of allowable costs v E = V (u).

The general technological set Y of the production element in this case has the form

On fig. 3.4 this restriction is satisfied by all points of the technological set located above the EC segment or lying on it.

For the most part, material 4.21 is also original. An assessment of the effectiveness of market mechanisms that ensure the existence of a single equilibrium management was carried out in the works. Material 4.21 is an extension of these works. Consideration of the auction scheme in the market system is carried out according to. famous model, considered as an example in this paragraph, is the market economy model. A detailed discussion of it can be found, for example, in the works. In 4.21 we assumed that a market equilibrium exists. As an examination of the auction scheme in a market system shows, this may not always be the case. Consideration of issues related to the existence of equilibrium in market models is one of the central issues of mathematical economics. In relation to models of a competitive economy, the existence of equilibrium has been established by a number of authors under various assumptions. Usually the proof assumes the convexity of utility functions (or preferences) of consumers and technology sets of producers. In the generalization of the Arrow-Debré model for the case of a continuum of players is given. At the same time, it was possible to abandon the assumptions about the convexity of consumer preference functions.

Each producer (firm) j is characterized by a technological set Y. - a set of technologically admissible l-dimensional vectors of costs - output, their positive components correspond to produced quantities, and negative - spent. It is assumed that the manufacturer chooses the cost-output vector in such a way as to maximize profit. At the same time, he, like the consumer, does not try to influence prices, taking them as given. Thus, his choice is the solution to the following problem

From (16) the weak axiom of revealed preference also follows. Inequality (16) is certainly satisfied if the demand of each of the consumers is strictly monotonous, and no special requirements are imposed on technological sets. An interpretation of the monotonicity condition and a number of related results are given in . For smooth functions of excess demand, the uniqueness of the equilibrium is also ensured by the condition of the dominant diagonal. This condition means that the module of the derivative of demand for each product at the price of this product is greater than the sum of the modules of all derivatives of demand for the same

manufacturer model. When choosing production volumes yj = y k, each firm j e J is limited by its technological set YJ with 1R1. These sets of admissible technologies can be specified, in particular, in the form of (implicit) production functions fj(yj) YJ = UZ e Rl /,(%) > 0 . Another convenient representation (when only one good h is produced) is as an explicit production function y 0.

Technological set and its properties

TECHNOLOGICAL SET - see Production set, Technological way.

Description of one specific type Let us consider a technological set for a production element that consumes several types of costs and produces products of only one type (single-product production element). The state vector of such an element has the form yt-(vtl, viz, . . . , v. x, ut). A well-known method for describing the technological set of a single-product element is based on the concept of a production function and is as follows.

It is usually assumed that the technological set of an element is a convex, closed subset of the Euclidean space Ет containing a zero element of dimension m О Е Y d Em.

The methods of representation of technological sets of production elements considered in the previous paragraph characterize their properties, but do not specify a description in an explicit form. For one-product production elements, an explicit description of the technological set can be given using the concept of production function . In 1.2 we have already touched on this concept and its use, in this section the consideration of these issues will be continued.

Using single-product production functions to describe the technological set of a multi-product element. If a multi-commodity element produces goods types of products, while consuming / gewx types of inputs, then its input and output vectors have the form , itvy), respectively.

It corresponds to a part of the technological set, limited by a curved triangle AB (marked with hatching in Fig. 3.4).

The Arrow-Deb-re-McKnzie decentralized economy model. The general model of a decentralized economy describes production, consumption and decentralized

Consider an economy with l goods. It is natural for a particular firm to consider some of these goods as factors of production and some as output. It should be noted that such a division is rather arbitrary, since the company has sufficient freedom in choosing the range of products and cost structure. When describing the technology, we will distinguish between output and costs, representing the latter as output with a minus sign. For the convenience of presenting the technology, products that are neither consumed nor produced by the firm will be referred to as its output, and the volume of production of this product is assumed to be 0. In principle, the situation in which the product produced by the firm is also consumed by it in the production process is not ruled out. In this case, we will consider only the net output of a given product, i.e., its output minus costs.

Let the number of factors of production be n and the number of outputs be m, so that l = m + n. Let's denote the cost vector (in absolute value) as r Rn + , and the output volumes as y Rm + . The vector (−r, yo ) will be called net issues vector. The set of all technologically feasible net output vectors y = (−r, yo ) is technological set Y . Thus, in the case under consideration, any technological set is a subset of Rn − × Rm + .

This description of production is general character. At the same time, it is possible not to adhere to a rigid division of goods into products and factors of production: the same good can be spent with one technology, and produced with another. In this case Y Rl .

Let us describe the properties of technological sets, in terms of which the description of concrete classes of technologies is usually given.

1. Non-emptiness

The technological set Y is non-empty.

This property means the fundamental possibility of implementing production activities.

2. Closure

The technological set Y is closed.

This property is rather technical; it means that the technology set contains its boundary, and the limit of any sequence of technologically feasible net output vectors is also a technologically feasible net output vector.

3. Freedom of spending:

if y Y and y0 6 y, then y0 Y.

This property can be interpreted as having the ability to produce the same amount of output, but through high costs, or less output at the same cost.

4. Lack of “cornucopia” (“no free lunch”)

if y Y and y > 0, then y = 0.

This property means that the production of a product in a positive quantity requires costs in a non-zero volume.

Rice. 4.1. Technological set with increasing returns to scale.

5. Nonincreasing returns to scale:

if y Y and y0 = λy, where 0< λ < 1, тогда y0 Y.

This property is sometimes called (not exactly) diminishing returns to scale. In the case of two goods, where one is spent and the other is produced, diminishing returns means that the (maximum possible) average productivity of the input factor does not increase. If in an hour you can solve at best 5 of the same type of problems in microeconomics, then in two hours under conditions of diminishing returns you could not solve more than 10 such problems.

fifty . Non-diminishing returns to scale:

if y Y and y0 = λy, where λ > 1, then y0 Y.

In the case of two goods, where one is spent and the other is produced, increasing returns means that the (maximum possible) average productivity of the input factor does not decrease.

500 . Constant returns to scale - the situation when the technological set satisfies conditions 5 and 50 simultaneously, i.e.

if y Y and y0 = λy0 , then y0 Y λ > 0.

Geometrically constant returns to scale means that Y is a cone (possibly not containing 0).

In the case of two goods, where one is consumed and the other produced, constant returns mean that the average productivity of the factor input does not change with the change in output.

Rice. 4.2. Convex technology set with diminishing returns to scale

The property of convexity means the ability to "mix" technologies in any proportion.

7. Irreversibility

if y Y and y 6= 0, then (−y) / Y.

Let 5 bearings be produced from a kilogram of steel. Irreversible means that it is impossible to produce a kilogram of steel from 5 bearings.

8. Additivity.

if y Y and y0 Y , then y + y0 Y.

The property of additivity means the ability to combine technologies.

9. Permissibility of inactivity:

Theorem 44:

1) From the non-increasing returns to scale and the additivity of the technological set, its convexity follows.

2) From the convexity of the technological set and the permissibility of inactivity, non-increasing returns to scale follow. (The converse is not always true: with non-increasing returns, technology can be non-convex, see Fig. 4.3 .)

3) The technological set has the properties of additivity and non-increasing

returns to scale if and only if it is a convex cone.

Rice. 4.3. Non-convex technological set with non-increasing returns to scale.

Not all eligible technologies are equally important from an economic point of view. Among the admissible ones stand out efficient technologies. An admissible technology y is called efficient if there is no other (different from it) admissible technology y0 such that y0 > y. Obviously, this definition of efficiency implies implicitly that all goods are desirable in some sense. Efficient technologies make up effective frontier technological set. At certain conditions it turns out to be possible to use the effective frontier in the analysis instead of the entire technological set. Here it is important that for any admissible technology y there is an efficient technology y0 such that y0 > y. In order for this condition to be satisfied, it is required that the technological set be closed, and that within the technological set it is impossible to increase the output of one good to infinity without reducing the output of other goods. It can be shown that if technological

Rice. 4.4. Effective frontier of the technological set

set has the freedom of spending property, then the effective boundary uniquely defines the corresponding technological set.

Initial courses and courses of intermediate complexity, when describing the behavior of a producer, are based on the representation of its production set by means of a production function. The question is, under what conditions production set such a representation is possible. Although it is possible to give a broader definition of the production function, however, hereinafter we will only talk about “single-product” technologies, i.e. m = 1.

Let R be the projection of the technological set Y onto the space of cost vectors, i.e.

R = ( r Rn | yo R: (−r, yo ) Y ) .

Definition 37:

The function f( ) : R 7→R is called production function, representing technology Y , if for each r R the value f(r) is the value of the following problem:

yo → max

(−r, yo ) Y.

Note that any point of the effective boundary of the technological set has the form (−r, f(r)). The reverse is true if f(r) is an increasing function. In this case, yo = f(r) is the effective boundary equation.

The following theorem gives the conditions under which a technological set can be represented??? production function.

Theorem 45:

Let for technological set Y R × (−R) for any r R the set

F (r) = ( yo | (−r, yo ) Y )

closed and bounded from above. Then Y can be represented by a production function.

Note: The fulfillment of the conditions of this statement can be guaranteed, for example, if the set Y is closed and has the properties of non-increasing returns to scale and the absence of a cornucopia.

Theorem 46:

Let the set Y be closed and have the properties of non-increasing returns to scale and the absence of a cornucopia. Then for any r R the set

F (r) = ( yo | (−r, yo ) Y )

closed and bounded from above.

Proof: The closedness of the sets F (r) follows directly from the closedness of Y . Let us show that F (r) are bounded from above. Let this not be the case, and for some r R

there is an infinitely increasing sequence (yn ) such that yn F (r). Then, due to non-increasing returns to scale (−r/yn , 1) Y . Therefore (due to closedness), (0, 1) Y , which contradicts the absence of a cornucopia.

We also note that if the technological set Y satisfies the free spending hypothesis, and there exists a production function f( ) representing it, then the set Y is described by the following relationship:

Y = ( (−r, yo ) | yo 6 f(r), r R ) .

Let us now establish some relationships between the properties of the technological set and the production function representing it.

Theorem 47:

Let the technological set Y be such that for all r R the production function f(·) is defined. Then the following is true.

1) If the set Y is convex, then the function f( ) is concave.

2) If the set Y satisfies the free spending hypothesis, then the converse is also true, i.e., if the function f(·) is concave, then the set Y is convex.

3) If Y is convex, then f( ) is continuous on the interior of R.

4) If the set Y has the free spending property, then the function f( ) does not decrease.

5) If Y has the cornucopia-free property, then f(0) 6 0.

6) If the set Y has the inactivity admissibility property, then f(0) > 0.

Proof: (1) Let r0 , r00 R. Then (−r0 , f(r0 )) Y and (−r00 , f(r00 )) Y , and

(−αr0 − (1 − α)r00 , αf(r0 ) + (1 − α)f(r00 )) Y α ,

since the set Y is convex. Then by definition of the production function

αf(r0 ) + (1 − α)f(r00 ) 6 f(αr0 + (1 − α)r00 ),

which means that f( ) is concave.

(2) Since the set Y has the property of free spending, then the set Y (up to the sign of the cost vector) coincides with its subgraph. And the subgraph of a concave function is a convex set.

(3) The fact to be proved follows from the fact that the concave function is continuous in the internal

sti of its domain of definition.

(4) Let r 00 > r0 (r0 , r00 R). Since (−r0 , f(r0 )) Y , then by the freedom of spending property (−r00 , f(r0 )) Y . Hence, by the definition of the production function, f(r00 ) > f(r0 ), that is, f( ) does not decrease.

(5) The inequality f(0) > 0 contradicts the assumption that there is no cornucopia. Hence, f(0) 6 0.

(6) By the assumption of the admissibility of inactivity (0, 0) Y . So, by definition

Assuming the existence of a production function, the properties of technology can be described directly in terms of this function. We will show this with the example of the so-called elasticity of scale.

Let the production function be differentiable. At a point r, where f(r) > 0, we define

local scale elasticity e(r) as:

If at some point e(r) is equal to 1, then it is considered that at this point constant returns to scale if more than 1 then increasing returns, less - diminishing returns to scale. The above definition can be rewritten as follows:

P ∂f(r) e(r) = i ∂r i r i .

Theorem 48:

Let the technological set Y be described by the production function f( ) and

in point r, e(r) > 0. Then the following is true:

1) If the technological set Y has the property of diminishing returns to scale, then e(r) 6 1.

2) If the technological set Y has the property of increasing returns to scale, then e(r) > 1.

3) If Y has the property of constant returns to scale, then e(r) = 1.

Proof: (1) Consider the sequence (λn ) (0< λn < 1), такую что λn → 1. Тогда (−λn r, λn f(r)) Y , откуда следует, что f(λn r) >λnf(r). Let's rewrite this inequality as:

	f(λn r) − f(r)
Passing to the limit, we have		λn − 1
Passing to the limit, we have
			∂ri	ri 6 f(r).



Thus e(r) 6 1.
Properties (2) and (3) are proved similarly.

Technological sets Y can be specified as implicit production functions g(·). By definition, a function g( ) is called an implicit production function if technology y belongs to technology set Y if and only if g(y) >

Note that such a function can always be found. For example, a function is suitable such that g(y) = 1 for y Y and g(y) = −1 for y / Y . Note, however, that this function is not differentiable. Generally speaking, not every technological set can be described by a single differentiable implicit production function, and such technological sets are not something exceptional. In particular, technological sets, considered in elementary microeconomics courses, are often such that two (or more) inequalities with differentiable functions are needed to describe them, since additional constraints on the non-negativity of factors of production must be taken into account. To take into account such restrictions, one can use vector implicit

Let us continue the study of models of balanced economic growth at a more general level and move on to models of economic well-being close to them. The latter, like growth models, are normative models.

Speaking about the welfare economy, they mean such a development when all consumers evenly reach the maximum of their utility. However, in practice, such an ideal situation rarely occurs, since the well-being of some is often achieved at the expense of the deterioration of the condition of others. Therefore, it is more realistic to speak of such a level of distribution of goods, when no consumer can increase his welfare without infringing on the interests of other consumers.

If along the trajectory of equilibrium growth, no consumer, like no producer, can acquire more without additional costs (no profit in equilibrium), then with the development of the economy along the trajectory of such “welfare”, no consumer can become richer without impoverishing while another.

It follows from the previous section that taking into account time factors in mathematical models of the economy helps to reveal a completely logical connection between economic processes and the natural growth of production and consumer opportunities. Under the conditions of linear models, under certain assumptions, the rate of such growth is equal to the percentage of capital, and the corresponding process of economic expansion is characterized by a balanced growth in the output intensities of all products and a balanced decrease in their prices. In this section, we formulate a general dynamic model of production, covering the previously considered linear models as special cases, and study the issues of balanced growth in it.

The generality of the model considered here lies in the fact that the production process is described not by the production function in general, and the linear production function (as in the Leontiev and Neumann models) in particular, but by the so-called technological set.

technological set(we denote it by the symbol ) is the set of such transformations of the economy, when the production of goods at costs is technologically possible if and only if . The couple is called production process, so the set is the set of all production processes possible with this technology. For example, in the Leontief model, the technological set j-th industry has the form where - gross output j-th product, and - j-th column of the technological matrix A. Therefore, the technological set in the Leontief model as a whole is and in the Neumann model -

The production process, generally speaking, may contain such products that are both consumed and produced at the same time (for example, fuels and lubricants, flour, meat, etc.). In economic-mathematical models, for greater generality, it is often assumed that each product from can be both spent and produced (for example, in the Leontiev and Neumann models). In this case the vectors x and y have the same dimension and their respective components denote the same products.

Let - the amount spent i-th product, and - its output volume. Then the difference is called net release in the process . Therefore, instead of the production process, the vector of net output is often considered, characterizing this difference as flow(or intensity), i.e. net output per unit of time. At the same time, the technological set is understood as the set of all possible pure outputs. and the vector is called process with thread.

We list some properties of the technological set, which are a reflection of the fundamental laws of production.

Different production processes can be compared both in terms of efficiency and profitability.

A process is said to be more efficient than a process if , . The process is called effective, unless it contains more efficient processes than .

Let be a price vector. They say that the process more profitable than the process if the value is not less than the value .

These two options for in-kind and cost evaluation of processes are actually equivalent.

Theorem 6.1. Let be a technological set. Then a) if for the price vector the process maximizes profit on the set , then it is an efficient process; b) if is convex and is efficient in the process, then there is such a price vector , that the profit reaches a maximum at

Let us determine the structure of the technological set for those models that take into account the time factor. Consider a planning period with discrete points Let the economy be characterized by a stock of goods in a year (i.e. at the beginning of the planning period ) In this case, the economy is said to be in state . By the end of the period, the economy reaches a different state, which is predetermined by the previous state. In this case, we say that the production process has been implemented where is a given technological set. Here, the vector is considered as the costs incurred at the beginning of the period , and - as the output corresponding to these costs, produced with a time lag of one year. On the next steps we have production etc. In this way it is carried out dynamics of economic development. Such a movement of the economy is self-sustaining, since the products in the system are reproduced without any influx from outside.

A finite sequence of vectors is called acceptable trajectory of the economy(described by the technological set Z) on the time interval , if each pair of its two successive terms belongs to the set Z, i.e.

Denote by the set of all admissible trajectories on the interval corresponding to the initial state

Let The trajectory is called more efficient than if the trajectory is called effective trajectory, if there is no more efficient trajectory than . The trajectory is called more profitable than if

Methods for describing technologies.

Production is the main area of activity of the company. Firms use production factors, which are also called input (input) factors of production. For example, a bakery owner uses inputs such as labor, raw materials such as flour and sugar, and capital invested in ovens, mixers, and other equipment to produce products such as bread, pies, and confectionery.

We can subdivide the factors of production into large categories - labor, materials and capital, each of which includes more narrow groupings. For example, labor production factor through the indicator of labor intensity, it combines both skilled (carpenters, engineers) and unskilled labor (agricultural workers), as well as the entrepreneurial efforts of firm managers. Materials include steel, plastic materials, electricity, water and any other product that the company acquires and turns into finished goods. Capital includes buildings, equipment and inventories.

The set of all technologically available net output vectors for a given firm is called the production set and denoted by Y.

PRODUCTION SET- the set of admissible technological ways given economic system (X,Y ) , where X - aggregate cost vectors, a Y - aggregate release vectors.

P. m. is characterized the following features: it closed and convex(cm. Lots of), the cost vectors are necessarily non-zero (one cannot produce something without spending anything), the components of the P. m. - costs and outputs - cannot be interchanged, because production is an irreversible process. The convexity of P. m. shows, in particular, the fact that the return on processed resources decreases with an increase in the volume of processing.

Properties of production sets

Let the number of factors of production be n and the number of outputs be m, so that l = m + n. Let us denote the cost vector (in absolute value) as r 2 Rn+, and the output volumes as y 2 Rm+

The vector (−r, yo) will be called the vector of net outputs. The set of all technologically admissible net output vectors y = (−r, yo) constitutes the technological set Y . Thus, in the case under consideration, any technological set is a subset of Rn − × Rm+

This description of production is of a general nature. At the same time, it is possible not to adhere to a rigid division of goods into products and factors of production: the same good can be spent with one technology, and produced with another.

Let us describe the properties of technological sets, in terms of which the description of concrete classes of technologies is usually given.

1. Non-emptiness. The technological set Y is non-empty. This property means the fundamental possibility of carrying out production activities.

2. Closure. The technological set Y is closed. This property is rather technical; it means that the technology set contains its boundary, and the limit of any sequence of technologically feasible net output vectors is also a technologically feasible net output vector.

3. Freedom of spending. This property can be interpreted as having the ability to produce the same amount of output at a higher cost, or less output at the same cost.

4. The absence of a “horn of plenty” (“no free lunch”). if y 2 Y and y > 0, then y = 0. This property means that the production of products in a positive quantity requires costs in a non-zero volume.

< _ < 1, тогда y0 2 Y. Иногда это свойство называют (не совсем точно) убывающей отдачей от масштаба. В случае двух благ, когда одно затрачивается, а другое производится, убывающая отдача означает, что (максимально возможная) средняя производительность затрачиваемого фактора не возрастает. Если за час вы можете решить в лучшем случае 5 однотипных задач по микроэкономике, то за два часа в условиях убывающей отдачи вы не смогли бы решить более 10 таких задач.

fifty . Non-diminishing returns to scale: if y 2 Y and y0 = _y, where _ > 1, then y0 2 Y.

In the case of two goods, where one is spent and the other is produced, increasing returns means that the (maximum possible) average productivity of the input factor does not decrease.

500 . Constant returns to scale - the situation when the technological set satisfies conditions 5 and 50 simultaneously, i.e. if y 2 Y and y0 = _y0, then y0 2 Y 8_ > 0.

Geometrically constant returns to scale means that Y is a cone (possibly not containing 0). In the case of two goods, where one is consumed and the other produced, constant returns mean that the average productivity of the factor input does not change with the change in output.

5. Nonincreasing returns to scale: if y 2 Y and y0 = _y, where 0< _ < 1, тогда y0 2 Y. Иногда это свойство называют (не совсем точно) убывающей отдачей от масштаба. В случае двух благ, когда одно затрачивается, а другое производится, убывающая отдача означает, что (максимально возможная) средняя производительность затрачиваемого фактора не возрастает. Если за час вы можете решить в лучшем случае 5 однотипных задач по микроэкономике, то за два часа в условиях убывающей отдачи вы не смогли бы решить более 10 таких задач.

fifty . Non-diminishing returns to scale: if y 2 Y and y0 = _y, where _ > 1, then y0 2 Y. In the case of two goods, where one is spent and the other is produced, increasing returns mean that the (maximum possible) average productivity of the input factor does not decrease.

500 . Constant returns to scale - the situation when the technological set satisfies conditions 5 and 50 simultaneously, i.e. if y 2 Y and y0 = _y0, then y0 2 Y 8_ > 0.

Geometrically constant returns to scale means that Y is a cone (possibly not containing 0).

In the case of two goods, where one is consumed and the other produced, constant returns mean that the average productivity of the factor input does not change with the change in output.

6. Convexity: The convexity property means the ability to "mix" technologies in any proportion.

7. Irreversibility

Let 5 bearings be produced from a kilogram of steel. Irreversible means that it is impossible to produce a kilogram of steel from 5 bearings.

8. Additivity. if y 2 Y and y0 2 Y , then y + y0 2 Y. The additivity property means the ability to combine technologies.

9. Permissibility of inactivity:

Theorem 44:

1) From the non-increasing returns to scale and the additivity of the technological set, its convexity follows.

2) Non-increasing returns to scale follow from the convexity of the technological set and the permissibility of inactivity. (The converse is not always true: with non-increasing returns, the technology may be non-convex)

3) A technological set has the properties of additivity and non-increasing returns to scale if and only if it is a convex cone.

Not all eligible technologies are equally important from an economic point of view.

Efficient technologies stand out among the admissible ones. An admissible technology y is called efficient if there is no other (different from it) admissible technology y0 such that y0 > y. Obviously, this definition of efficiency implies implicitly that all goods are desirable in some sense. Efficient technologies constitute the effective frontier of the technological set. Under certain conditions, it turns out to be possible to use the effective frontier in the analysis instead of the entire technological set. Here it is important that for any admissible technology y there is an efficient technology y0 such that y0 > y. In order for this condition to be satisfied, it is required that the technological set be closed, and that within the technological set it is impossible to increase the output of one good to infinity without reducing the output of other goods.

TECHNOLOGICAL METHOD - general concept combining two: T. s. production (production method, technology) and T. s. consumption; set of basic characteristics ( ingredients) production process (respectively - consumption) one or another product. AT economic and mathematical model T. s., or technology (activity), is described by a system of numbers inherent in it ( vector): e.g. cost rates and release various resources per unit of time or per unit of production, etc., including coefficients material consumption, laboriousness, capital intensity, capital intensity.

For example, if x = (x 1 , ..., x m) - the vector of resource costs (listed under numbers i = 1, 2, ..., m), a y = (y 1 , ..., y n) - vector of production volumes of products j= 1, 2, ..., n, then technologies, technological processes, methods of production can be called pairs of vectors ( x,y ). Technological acceptability means here the ability to obtain from the consumed (used) vector ingredients x product vector y .

The set of all possible admissible technologies ( XY) forms technological or production set given economic system.

VECTOR- an ordered set of a certain number of real numbers (this is one of the many definitions - the one that is accepted in economic and mathematical methods). For example, the daily plan of the workshop can be written as a 4-dimensional vector (5, 3, -8, 4), where 5 means 5 thousand parts of one type, 3 - 3 thousand parts of the second type, (-8) - metal consumption in t, and the last component, for example, saving 4 thousand kW. h electricity. As you can see, the number of components ( coordinates) B. arbitrarily (in this case, the plan of the workshop may consist not of four, but of any other number of indicators); they are unacceptable to be interchanged; they can be both positive and negative.

Vectors can be multiplied by a real number (for example, if you increase the plan by 1.2 times for all indicators, you get a new vector with the same number of components). Vectors containing an equal number of additive components of the same name, respectively, can be added and subtracted.

It is customary to highlight the letter designation V. in bold type (although this is not always observed).

The sum of vectors x = (x 1 ,..., x n) and y = (y 1 , ..., y n) is also B. ( x + y ) = (x 1 + y 1 , ..., xn+yn).

Dot product of vectors x and y a number equal to the sum of the products of the corresponding components of these V. is called:

Vectors x and y called orthogonal if their dot product is zero.

Equality V. - component, i.e., two V. are equal if their corresponding components are equal.

Vector 0 - (0, ..., 0) null;

n-dimensional V. - positive ( x > 0) if all its components x i Above zero, non-negative (x ≥ 0) if all its components x i greater than 0 or equal to zero, i.e. x i≤ 0; and semi-positive, if at least one component x i≥ 0 (notation x ≥ 0); if V. have an equal number of components, their ordering (full or partial) is possible, i.e., introduction on a set of vectors binary relation “> ”: x > y , x ≥y , x ≥ y depending on whether the difference is positive, semi-positive or non-negative x-y.

THE LAW OF DIMENSIONING RECOMMENDATIONS- a statement that if the use of any one factor of production and at the same time the costs of all other factors are preserved (they are called fixed), then the physical volume marginal product, produced with the help of the specified factor, will (at least from a certain stage) decrease.

PRODUCTION BEAM- locus of points representing a proportional increase in the number resources when using a specific technological method with increasing intensity.

For example, if the combination of 3 units. capital (funds) and 2 units. labor (i.e. a combination of 3 K + 2L) gives 10 units. some product, then combinations 6 K + 4L, 9K + 6L, giving respectively 20 and 30 units. etc., will lie on the graph on a straight line called P. l. or technology beam. With a different combination of factors P. l. will have a different slope. Due to the indivisibility of many factors of production the number of technological methods and, accordingly, P. l. accepted as final.

For example, if a team of three miners is working in a coal lava and one more is added to them, the output will increase by a quarter, and if a fifth, sixth, seventh one is added, the increase in output will decrease, and then stop altogether: miners in cramped conditions will simply interfere each other.

Key Concept here - ultimate performance labor (more broadly - marginal productivity of a factor of production δ Y/δ x). For example, if two factors are considered, then with an increase in the costs of one of them (the first or the second), its marginal productivity falls.

The law is applicable in the short term and for this technology (its revision changes the situation).

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