The general technological set of a production element can be. Description of production using a technological set

2. Production sets and production functions

2.1. Production sets and their properties

Let's consider the most important participant in economic processes - an individual manufacturer. The manufacturer realizes his goals only through the consumer and therefore must guess, understand what he wants, and satisfy his needs. We will assume that there are n different goods, the quantity of the nth product is denoted by x n, then a certain set of goods is denoted by X = (x 1, ..., x n). We will consider only non-negative quantities of goods, so that x i  0 for any i = 1, ..., n or X > 0. The set of all sets of goods is called the space of goods C. A set of goods can be treated as a basket in which these goods lie in appropriate quantities.

Let the economy operate in the space of goods C = (X = (x 1, x 2, …, x n): x 1, …, x n  0). The product space consists of non-negative n-dimensional vectors. Let us now consider a vector T of dimension n, the first m components of which are non-positive: x 1, …, x m  0, and the last (n-m) components are non-negative: x m +1, …, x n  0. Vector X = (x 1,…, x m ) let's call cost vector, and vector Y = (x m+1 , …, x n) – release vector. Let's call the vector T = (X,Y) input-output vector, or technology.

In its meaning, technology (X,Y) is a way of processing resources into finished products: by “mixing” resources in the amount of X, we get products in the amount of Y. Each specific manufacturer is characterized by a certain set τ of technologies, which is called production set. A typical shaded set is shown in Fig. 2.1. This manufacturer uses one product to produce another.

Rice. 2.1. Production set

The production set reflects the breadth of the manufacturer's capabilities: the larger it is, the wider its capabilities. The production set must satisfy the following conditions:

it is closed - this means that if the input-output vector T is approximated as accurately as desired by vectors from τ, then T also belongs to τ (if all points of the vector T lie in τ, then Tτ see Fig. 2.1 points C and B) ;

in τ(-τ) = (0), i.e. if Tτ, T ≠ 0, then -Tτ – costs and output cannot be swapped, i.e. production is an irreversible process (set – τ is in the fourth quadrant, where y is 0);

the set is convex, this assumption leads to a decrease in the return on processed resources with an increase in production volumes (to an increase in the rate of expenditure on finished products). So, from Fig. 2.1 it is clear that y/x  decreases as x  -. In particular, the convexity assumption leads to a decrease in labor productivity as output increases.

Often convexity is simply not enough, and then strict convexity of the production set (or some part of it) is required.

2.2. Production Possibilities Curve

and opportunity costs

The concept of production set under consideration is distinguished by a high degree of abstraction and, due to its extreme generality, is of little use for economic theory.

Consider, for example, Fig. 2.1. Let's start with points B and C. The costs for these technologies are the same, but the output is different. The manufacturer, if he is not devoid of common sense, will never choose technology B, since there is a better technology C. In this case (see Fig. 2.1), we find for each x  0 the highest point (x, y) in the production set . Obviously, at cost x, technology (x, y) is the best. No technology (x, b) with b production function. The exact definition of the production function:

Y = f(x)(x, y) τ, and if (x, b)  τ and b  y, then b = x .

From Fig. 2.1 it is clear that for any x  0 such a point y = f(x) is unique, which, in fact, allows us to talk about a production function. But the situation is so simple if only one product is produced. In the general case, for the cost vector X we denote the set M x = (Y:(X,Y)τ). Set M x – is the set of all possible outputs at costs X. In this set, consider the production possibilities “curve” K x = (YM x: if ZM x and Z  Y, then Z = X), i.e. K x – these are many of the best releases, there are none better. If two goods are produced, then this is a curve, but if more than two goods are produced, then this is a surface, a body, or a set of even greater dimension.

So, for any cost vector X, all the best outputs lie on the production possibilities curve (surface). Therefore, for economic reasons, the manufacturer must choose the technology from there. For the case of the release of two goods y 1, y 2, the picture is shown in Fig. 2.2.

If we operate only with physical indicators (tons, meters, etc.), then for a given cost vector X we only have to choose the output vector Y on the production possibilities curve, but which specific output should be chosen cannot yet be decided. If the production set τ itself is convex, then M x is also convex for any cost vector X. In what follows, we will need strict convexity of the set M x. In the case of the output of two goods, this means that the tangent to the production possibilities curve K x has only one common point with this curve.

Rice. 2.2. Production possibility curve

Let us now consider the question of the so-called opportunity costs. Let us assume that output is fixed at point A(y 1 , y 2), see Fig. 2.2. Now there is a need to increase the output of the 2nd product by y 2, using, of course, the same set of costs. This can be done, as can be seen from Fig. 2.2, transferring the technology to point B, for which, with an increase in the output of the second product by y 2, it will be necessary to reduce the output of the first product by y 1.

Imputedcoststhe first product in relation to the second at the point A called
. If the production possibilities curve is given by the implicit equation F(y 1 ,y 2) = 0, then δ 1 2 (A) = (F/y 2)/(F/y 1), where the partial derivatives are taken at the point A. If you look closely at the figure in question, you will find an interesting pattern: when moving down the production possibilities curve from the left, opportunity costs decrease from very large values ​​to very small ones.

2.3. Production functions and their properties

A production function is an analytical relationship that connects variable values ​​of costs (factors, resources) with the amount of output. Historically, one of the first works on the construction and use of production functions was work on the analysis of agricultural production in the United States. In 1909, Mitscherlich proposed a nonlinear production function: fertilizers - yield. Independently, Spillman proposed an exponential yield equation. On their basis, a number of other agrotechnical production functions were built.

Production functions are designed to model the production process of a certain economic unit: a separate company, industry or the entire economy of the state as a whole. With the help of production functions the following problems are solved:

assessing the return of resources in the production process;

forecasting economic growth;

developing options for a production development plan;

optimizing the functioning of a business unit subject to a given criterion and resource limitations.

General form of the production function: Y = Y(X 1, X 2, ..., X i, ..., X n), where Y is an indicator characterizing production results; X – factor indicator of the i-th production resource; n – number of factor indicators.

Production functions are determined by two groups of assumptions: mathematical and economic. Mathematically, the production function is expected to be continuous and doubly differentiable. Economic assumptions are as follows: in the absence of at least one production resource, production is impossible, i.e. Y(0, X 2, ..., X i, ..., X n) =

Y(X 1 , 0, …, X i , …, X n) = …

Y(X 1, X 2, …, 0, …, X n) = …

Y(X 1, X 2, …, X i, …, 0) = 0.

However, it is not possible to satisfactorily determine the only output Y for given costs X using natural indicators: our choice has narrowed only to the production possibilities “curve” K x . For these reasons, only the theory of production functions of producers has been developed, the output of which can be characterized by one value - either the volume of output, if one product is produced, or the total value of the entire output.

The cost space is m-dimensional. Each point in the cost space X = (x 1, ..., x m) corresponds to a single maximum output (see Fig. 2.1) produced using these costs. This relationship is called the production function. However, the production function is usually understood less restrictively and any functional relationship between inputs and output is considered a production function. In what follows, we will assume that the production function has the necessary derivatives. The production function f(X) is assumed to satisfy two axioms. The first of these states that there is a subset of cost space called economic area E, in which an increase in any type of input does not lead to a decrease in output. Thus, if X 1, X 2 are two points of this region, then X 1  X 2 implies f(X 1)  f(X 2). In differential form, this is expressed in the fact that in this region all the first partial derivatives of the function are non-negative: f/x 1 ≥ 0 (for any increasing function the derivative is greater than zero). These derivatives are called marginal products, and the vector f/X = (f/x 1 , …, f/x m) – vector of marginal products (shows how many times production output will change when costs change).

The second axiom states that there is a convex subset S of the economic domain for which the subsets (XS:f(X)  a) are convex for all a  0. In this subset S, the Hessian matrix composed of the second derivatives of the function f(X) , is negative definite, therefore,  2 f/x 2 i

Let us dwell on the economic content of these axioms. The first axiom states that the production function is not some completely abstract function invented by a mathematical theorist. It, albeit not throughout its entire domain of definition, but only in part of it, reflects an economically important, indisputable and at the same time trivial statement: VIn a reasonable economy, an increase in costs cannot lead to a decrease in output. From the second axiom we will explain only the economic meaning of the requirement that the derivative  2 f/x 2 i be less than zero for each type of cost. This property is called in economics behindThe Law of Diminishing Returns or Diminishing Returns: as costs increase, starting from a certain moment (when entering the region S!), bymarginal product begins to decrease. The classic example of this law is the addition of more and more labor to the production of grain on a fixed piece of land. In what follows, it is assumed that the production function is considered on a region S in which both axioms are valid.

You can create a production function for a given enterprise without even knowing anything about it. You just need to place a counter (either a person or some kind of automatic device) at the gate of the enterprise, which will record X - imported resources and Y - the amount of products that the enterprise has produced. If you accumulate a sufficient amount of such static information and take into account the operation of the enterprise in various modes, then you can predict output, knowing only the volume of imported resources, and this is knowledge of the production function.

2.4. Cobb-Douglas production function

Let's consider one of the most common production functions - the Cobb-Douglas function: Y = AK  L , where A, ,  > 0 are constants,  + 

Y/K = AαK α -1 L β > 0, Y/L = AβK α L β -1 > 0.

The negativity of second partial derivatives, i.e., decreasing marginal products: Y 2 /K 2 = Aα(α–1)K α -2 L β 0.

Let's move on to the main economic and mathematical characteristics of the Cobb-Douglas production function. Average labor productivity is defined as y = Y/L – the ratio of the volume of product produced to the amount of labor expended; average capital productivity k = Y/K – ratio of the volume of produced product to the value of funds.

For the Cobb-Douglas function, average labor productivity y = AK  L  , and due to condition , with increasing labor costs, average labor productivity decreases. This conclusion allows for a natural explanation - since the value of the second factor K remains unchanged, it means that the newly attracted labor force is not provided with additional means of production, which leads to a decrease in labor productivity (this is also true in the most general case - at the level of production sets).

Marginal labor productivity Y/L = AβK α L β -1 > 0, which shows that for the Cobb-Douglas function, marginal labor productivity is proportional to average productivity and is less than it. The average and marginal capital productivity are determined similarly. For them, the indicated ratio is also valid - the marginal capital productivity is proportional to the average capital productivity and is less than it.

An important characteristic is such as capital-labor ratio f = K/L, showing the volume of funds per employee (per unit of labor).

Let us now find the labor elasticity of production:

(Y/L):(Y/L) = (Y/L)L/Y = AβK α L β -1 L/(AK α L β) = β.

So the meaning is clear parameter - This elasticity (ratio of marginal labor productivity to average labor productivity) of output by labor. The labor elasticity of production means that to increase output by 1%, it is necessary to increase the volume of labor resources by %. Has a similar meaning parameter – is the elasticity of production across funds.

And one more meaning seems interesting. Let  +  = 1. It is easy to check that Y = (Y/K)/K + (Y/L)L (substituting the previously calculated Y/K, Y/L into this formula ). Let us assume that society consists only of workers and entrepreneurs. Then income Y is divided into two parts - the income of workers and the income of entrepreneurs. Since at the optimal size of the firm the value Y/L - the marginal product of labor - coincides with wages (this can be proven), then (Y/L)L represents the income of workers. Similarly, the value Y/K is the marginal return on capital, the economic meaning of which is the rate of profit, therefore, (Y/K)K represents the income of entrepreneurs.

The Cobb-Douglas function is the most famous among all production functions. In practice, when constructing it, sometimes some requirements are waived (for example, the sum  +  can be greater than 1, etc.).

Example 1. Let the production function be the Cobb-Douglas function. To increase output by a = 3%, it is necessary to increase fixed assets by b = 6% or the number of employees by c = 9%. Currently, one worker produces products worth M = 10 4 rubles per month . , and the total number of employees is L = 1000. Fixed assets are valued at K = 10 8 rubles. Find the production function.

Solution. Let's find the coefficients , :  = a/b = 3/6 = 1/2,  = a/c = = 3/9 = 1/3, therefore, Y = AK 1/2 L 1/3. To find A, we substitute the values ​​K, L, M into this formula, keeping in mind that Y = ML = 1000 . 10 4 = 10 7 – – 10 7 = A(10 8) 1/2 1000 1/3. Hence A = 100. Thus, the production function has the form: Y = 100K 1/2 L 1/3.

2.5. Theory of the firm

In the previous section, when analyzing and modeling the behavior of the manufacturer, we used only natural indicators and did without prices, but we could not finally solve the problem of the manufacturer, i.e., indicate the only course of action for him in the current conditions. Now let's consider prices. Let P be a price vector. If T = (X,Y) is a technology, i.e., an input-output vector, X is costs, Y is output, then the scalar product PT = PX + PY is the profit from using technology T (costs are negative quantities) . Now let us formulate a mathematical formalization of the axiom that describes the behavior of the manufacturer.

Manufacturer's problem: The manufacturer selects a technology from its production set, aiming to maximize profits . So, the manufacturer solves the following problem: PT→max, Tτ. This axiom greatly simplifies the situation of choice. So, if prices are positive, which is natural, then the “output” component of the solution to this problem will automatically lie on the production possibilities curve. Indeed, let T = (X,Y) be some solution to the manufacturer’s problem. Then there exists ZK x , Z  Y, therefore, P(X, Z)  P(X, Y), which means that point (X, Z) is also a solution to the manufacturer’s problem.

For the case of two types of products, the problem can be solved graphically (Fig. 2.3). To do this, you need to “move” a straight line perpendicular to the vector P in the direction where it points; then the last point, when this straight line still intersects the production set, will be the solution (in Fig. 2.3 this is point T). As is easy to see, the strict convexity of the required part of the production set in the second quadrant guarantees the uniqueness of the solution. The same reasoning applies in the general case, for a larger number of types of inputs and outputs. However, we will not follow this path, but use the apparatus of production functions and call the manufacturer a firm. So, the firm's output can be characterized by one value - either the volume of output, if one product is produced, or the total value of the entire output. The cost space is m-dimensional, the cost vector X = (x 1, ..., x m). Costs uniquely determine output Y, and this relationship is the production function Y = f(X).

Rice. 2.3. Solving the manufacturer's problem

In this situation, let us denote by P the vector of prices for goods-costs and let v be the price of a unit of manufactured goods. Therefore, profit W, which is ultimately a function of X (and prices, but they are considered constant), is W(X) = vf(X) – PX→max, X  0. Equating the partial derivatives of the function W to zero, we obtain:

v(f/x j) = p j for j = 1, …, m or v(f/X) = P (2.1)

We will assume that all costs are strictly positive (zero ones can simply be excluded from consideration). Then the point given by relation (2.1) turns out to be internal, i.e., an extremum point. And since the Hessian matrix of the production function f(X) is also assumed to be negatively defined (based on the requirements for production functions), this is the maximum point.

So, under natural assumptions on production functions (these assumptions are met for a producer with common sense and in a reasonable economy), relation (2.1) gives a solution to the firm’s problem, i.e., it determines the volume X * of processed resources, resulting in output Y * = f(X *) Point X *, or (X *,f(X *)) will be called the optimal solution of the company. Let us dwell on the economic meaning of relation (2.1). As stated, (f/X) = (f/x 1 ,…,f/x m) is called marginal product vector, or vector of marginal products, and f/x i is called the i-th marginal product, or release response to change i -th item costs. Therefore, vf/x i dx i is price i -th marginal product additionally obtained from dx i units i th resource. However, the cost of dx i units of the i-th resource is equal to р i dx i , i.e., an equilibrium has been obtained: it is possible to involve additional dx i units of the i-th resource into production, spending р i dx i on its purchase, but there will be no gain, t Because after processing the products, we will receive exactly the same amount as we spent. Accordingly, the optimal point given by relation (2.1) is an equilibrium point - it is no longer possible to squeeze more out of goods-resources than was spent on their purchase.

Obviously, the increase in the firm's output occurred gradually: at first, the cost of marginal products was less than the purchase price of the goods and resources required for their production. Production volumes increase until relation (2.1) begins to be fulfilled: equality of the value of marginal products and the purchase price of goods and resources required for their production.

Let us assume that in the firm’s problem W(X) = vf(X) – PX → max, X  0, the solution X * is unique for v > 0 and P > 0. Thus, we obtain the vector function X * = X * ( v, P), or functions x * I = x * i (v, p 1 , p m) for i = 1, …, m. These m functions are called resource demand functions at given prices for products and resources. In essence, these functions mean that if the prices P for resources and the price v for the produced goods have been established, a given manufacturer (characterized by a given production function) determines the volume of processed resources using the functions x * I = x * i (v, p 1, p m) and asks for these volumes on the market. Knowing the volumes of processed resources and substituting them into the production function, we obtain output as a function of prices; let's denote this function by q * = q * (v,P) = f(X(v,P)) = Y * . It is called product supply function depending on the price v for products and prices P for resources.

A-priory, i-th type resource called of little value, if and only if,x * i /v i.e., when the price of a product increases, the demand for a low-value resource decreases. It is possible to prove an important relation: q * /P = -X * /v or q * /p i = -x * i /v, for i = 1, …, m. Consequently, an increase in the price of a product leads to an increase (decrease) in demand for a certain type of resource if and only if an increase in payment for this resource leads to a reduction (increase) in optimal output. This shows the main property of low-value resources: an increase in payment for them leads to an increase in output! However, it is possible to strictly prove the existence of such resources, an increase in payment for which leads to a decrease in output (i.e., all resources cannot be of low value).

It is also possible to prove that x * i /p i are complementary if x * i /p j are interchangeable if x * i /p j > 0. That is, for complementary resources, an increase in the price of one of them leads to a fall demand for another, and for interchangeable resources, an increase in the price of one of them leads to an increase in the demand for the other. Examples of complementary resources: a computer and its components, furniture and wood, shampoo and conditioner for it. Examples of fungible resources: sugar and sugar substitutes (for example, sorbitol), watermelons and melons, mayonnaise and sour cream, butter and margarine, etc.

Example 2. For a company with a production function Y = 100K 1/2 L 1/3 (from example 1), find the optimal size if the depreciation period of fixed assets is N = 12 months, the employee’s salary per month is a = 1000 rubles.

Solution. The optimal size of output or production volume is found from relation (2.1). In this case, output is measured in monetary terms, so v = 1. The cost of monthly maintenance of one ruble of funds is 1/N, i.e. we obtain a system of equations

, solving which we find the answer:
, L = 8 . 10 3, K = 144. 10 6.

2.6. Tasks

1. Let the production function be the Cobb-Douglas function. To increase output by 1%, it is necessary to increase fixed assets by b = 4% or the number of employees by c = 3%. Currently, one worker produces products worth M = 10 5 rubles per month . , and the total number of workers is L = 10 4 . Fixed assets are valued at K = 10 6 rubles. Find the production function, average capital productivity, average labor productivity, capital-labor ratio.

2. A group of “shuttles” in the amount of E decided to unite with N sellers. Profit from a day of work (revenue minus expenses, but not wages) is expressed by the formula Y = 600(EN) 1/3. The shuttle worker’s salary is 120 rubles. per day, seller - 80 rubles. in a day. Find the optimal composition of the group of “shuttles” and sellers, i.e. how many “shuttles” there should be and how many sellers.

3. A businessman decided to found a small trucking company. Having familiarized himself with the statistics, he saw that the approximate dependence of daily revenue on the number of cars A and the number N is expressed by the formula Y = 900A 1/2 N 1/4. Depreciation and other daily expenses for one machine are 400 rubles, the daily salary of a worker is 100 rubles. Find the optimal number of workers and vehicles.

4. The businessman decided to open a beer bar. Let us assume that the dependence of revenue Y (minus the cost of beer and snacks) on the number of tables M and the number of waiters F is expressed by the formula Y = 200M 2/3 F 1/4. The cost for one table is 50 rubles, the waiter’s salary is 100 rubles. Find the optimal size of the bar, i.e. the number of waiters and tables.

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 was given in the 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 the meaning of the subject-technological set only in connection with the separate .

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

subject-technological set

People themselves are a product of a fairly high standard of living, which the steppe hominids achieved thanks to the appearance of some stable ones in their flocks. If for primates gathering, as a way of obtaining resources from the territory of a natural complex, did not require the combined efforts of several individuals, then hunting for large ungulates, which became the main way of ensuring 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 suitable shapes are not scattered everywhere and it is difficult to find a sharpened stick, so hominids had to carry hunting tools with them. Together with clothing, which appeared along with upright walking, the consequence of which was the loss of hair, and simply because of the cool climate of the steppes, Flocks-TRIBES acquire a certain set, in other words - many- items, the presence of which provides members with a hunger-free level of existence.

People appear along with luxury, that is, objects for which hominids previously did not have time - either to simply appropriate objects from Nature that interested them, or to produce them with labor, since there was neither the need nor the opportunity to constantly carry with them. Luxury items include all improved tools, after all, for people, as one of the species of mammals, a set of vital goods is sufficient for life, the production of which was fully ensured by the variety of objects that hominids had in packs. As a biological being, man, already millions of years ago, could and did live above the hominid level with the same variety of objects, but in humans it is so strong that people did not stop at the hominid level, as it should have been for an animal species that had reached a level of prosperity. People did not have the opportunity to improve living conditions in the natural environment, so they begin to create their own artificial environment from objects of labor.

In human tribes, the influence continued to operate, inherited from hominids, in whose flocks 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 associates - members with high status. Such gift giving practice among the remaining members of the tribe, it gave rise to the belief that possessing an item from the leader’s use increases the status of the owner in the hierarchy. Consumption in accordance with 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 hierarchs' use, since the possession of these things allows them to feel an increase in their status in front of others. Thus, things that first appeared in the everyday life of hierarchs, in copies, became objects of consumption for high-status members, and lust on the part of 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 prestigious things has continued for thousands of years, increasing the variety of objects, so that now we live surrounded by millions of objects that make people's lives ONLY MUCH MORE COMFORTABLE than the lifestyle of the hominid ancestor.

But biologically, a person is still the same hominid with a hierarchical instinct, which he realizes in a field called -. Subject-technological set is another difference between humans and animals - this is a new artificial habitat that humans create thanks to scientific and technological progress, the driving force of which is. As we 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 a multitude of objects, but the idea of ​​​​an object-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 degree of development. In one case, a comparison by assortment is possible - i.e. by the number of different objects, 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). In another case, we can say that one society is richer than another, but then you have to add to the assortment parameter a characteristic of the quality and technological excellence of the items being compared (this is studied in the topic -). But, as a rule, in the object set of a richer society, fundamentally new objects appear, 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, presupposes not just a list of items, but also set of technologies, allowing the production of these products in the sphere of production of this society.

For old economic theories, the unit of economy is the economy of a sovereign state. It is the population of the state that is considered the community whose subject-technological set is determined by the ability of the economy of a given state to produce all these items. And the connection with technology 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 division of labor system, the inaccuracy of identifying the economy of one country with that 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 may even not be produced in the territory of this state and, conversely, only parts are produced, but final products are not produced.

Here it must be said that inconsistency THE AVAILABILITY of technology and the POSSIBILITY to produce some products based on it - existed BEFORE the international division of labor, but the old economic science inconsistency I didn’t notice, even more - in the understanding of previous theories - the economies of all states were equivalent (the difference was accepted only in size - one could be larger or smaller than the other) and as soon as technology was given, the POSSIBILITY to produce anything immediately appeared.

The fact that practice refuted these theoretical assumptions did not prevent the old economic science from giving recipes for developing countries to build production facilities of any technological complexity. A very common example is that of Romania, which, according to economists, has no obstacles to reaching the level of the United States of America, at least in the sphere of production, although it is clear that in order for the subject-technological variety of Romania 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 variety of the United States exceeds the number of residents of Romania, then it is not clear who on the territory of Romania will be able to produce so many items.

There ARE objective limitations to development - and they most likely come down 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 the theoretical possibility - India has not become richer) , and in . For example, Finland for a short time managed to take the place of the most advanced country in the production of mobile phones. But 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 A specific product 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 mass effective DEMAND for the product.

As you can now see - concept of subject-technological set is not as simple as it seems. Firstly, we now understand that subject-technological set rather connected with some system of division of labor, and not with the state (in the sense, although historically subject-technological set we derive from the objective set , which was the first). This system can be internal part or external supersystem in relation to the population. Secondly, imagine subject-technological set we can, if it has a countable assortment - otherwise, the number of different objects in it is finite, which implies at a particular moment in time countable limited number of people in the community. If we mean by community having PMT, a system of division of labor, then we must talk about its CLOSEDNESS, since objects from the set are both produced and consumed in this system.

Yours scientific meaning subject-technological set receives with opening new object in the economy, which 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 his first works on, when he became interested in the interaction between developed and developing countries. That's when I started using term subject-technological set, as a certain characteristic of the division of labor systems 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. Here I 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 - remains, but the reader must remember - subject-technological set characterizes closed a 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 - Can the subject-technological variety decrease? The answer is, of course, it can, although many people think that scientific and technological progress can only increase power of subject-technological set, if you look at it as an attribute of the state. It is clear that some objects naturally disappear from people’s everyday life, 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 capable of ensuring reproduction PTM in its entirety.

At the beginning of our era, a demographic crisis begins in Europe, so that tribes cannot bud together, and the desire to remove the excess population leads to land grabs. States begin to develop on the periphery of the Roman Empire, and it turns out that Ancient Rome (like Ancient Greece) was a branch of the eastern empire on the European continent. Indigenous Europe is entering the natural state of the period of state formation, which in Europe, due to the initial small number of the population developing it, has shifted centuries later than it was in the EAST. The Roman Empire had no chance to resist the desire of the tribes to expand, and the loss of territories destroyed the established 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 were rediscovered only after a millennium, and the standard of living that existed in the cities of Ancient Rome was again achieved in Europe only in the 19th century, for example, running water in the upper floors of multi-story 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 MULTIPLE (PTM)

This SUBJECT-TECHNOLOGICAL MULTIPLE consists of objects (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 parts, they may not be goods, but be part of the goods.

Another part of this set is a set of technologies, that is, methods of producing goods sold on the market - from and/or with - using 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 is expanding.

The importance of this concept is determined by the fact that PTM determines the possibility of 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 of going into series if they require certain products or technologies that are not available in PTM. They simply turn out to be too expensive.

Related materials

In front of you is only excerpt from Chapter No. 8 of the book The Age of Growth, in which gives description of subject-technological set:

Let's introduce concept of subject-technological set. This set consists of objects (products, parts, types of raw materials) that actually exist, that is, produced by someone and, accordingly, sold on the market. As for the parts, they may not be goods, but be part of the goods. The second part of this set consists of technologies, that is, methods of producing 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 material elements of the set.

In each period of time we have different power subject-technological set (PTM). By the way, it can not only expand. Some items are no longer 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, essentially a new invention. They say that when in our time they tried to reproduce Newcomen's steam engine, they had to expend enormous efforts in order to make it somehow work. But in the 18th century, hundreds of these machines worked quite successfully.

But, in general, PTM For now it is 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 previously unknown technology from completely new raw materials. I don’t know, I suspect that this case has never happened in reality, but let’s assume that this could be the case.

The second extreme case is when new elements of the set are formed as combinations of already existing elements PTM. Such cases are not uncommon. Schumpeter already saw innovation as new combinations of what already exists. Let's take the same personal computers. In a sense, they cannot be said to have been “invented.” All of their components already existed, and were simply combined in a certain way.

If we can talk about any discovery here, it is 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.

As we understand it, most of the new items PTM represent 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 decreasing, and those close to the second are increasing.

In general, in light of my story about the devices of the series A and device B It's clear why this happens. For more details, see Chapter 8 of the book by clicking on the button:

A formalizing set of all technologically feasible vectors of net outputs.

Definition

Let the economy have $N$ good In the process of production of them $n$ benefits are spent. Let us denote the vector of these benefits (costs) $x$(vector dimension $n$). Other $m=N-n$ goods are released in the production process (the dimension of the vector is $m$). Let us denote the vector of these benefits $y$. Then the vector $z=(-x,y)$(dimension - $N$) is called a vector net issues. The totality of all technologically feasible vectors of net outputs is technological set. In fact, this is some subset of space $R^N$.

For readers who have difficulty with vector concepts, there are many:

vector - a list of goods, each good is described by its quantity, a set of numbers;

all goods consumed in production are recorded at the beginning of the net output vector z with a minus sign (-x), those produced with a plus sign (y);

all combinations possible for production form a technological set (production combinations).

Properties

• Non-emptiness: the technological set is not empty. Non-emptiness means the fundamental possibility of production.
• Acceptability of inactivity: the zero vector belongs to the technological set. This formal property means that zero output at zero input is acceptable.
• Closedness: the technological set contains its own boundary and the limit of any sequence of technologically feasible vectors of net outputs also belongs to the technological set.
• Freedom to spend: if the given vector $z$ belongs to the technological set, then any vector belongs to it $z"\backslash leqslant\; z$. This means that formally the same volume of output can be produced at higher costs.
• Absence of a "cornucopia": of the non-negative vectors of net output, only the zero vector belongs to the technological set. This means that non-zero costs are required to produce a positive quantity of output.
• Irreversibility: for any valid vector $z$, opposite vector $-z$ does not belong to the technological set. That is, it is impossible to produce resources from manufactured products in the same quantities as they are used to produce these products.
• Additivity: The sum of two valid vectors is also a valid vector. That is, a combination of technologies is allowed.
• Properties related to returns to scale of production:
  • Non-increasing returns to scale: for anyone $\backslash lambda\; \backslash in\; (0;1)$ $\backslash lambda\; z$
  • Non-diminishing returns to scale: for anyone $\backslash lambda\; >1$ if z belongs to the technological set, then $\backslash lambda\; z$ also belongs to the technological set.
  • Constant returns to scale: simultaneous fulfillment of the two previous properties, that is, for any positive $\backslash lambda$ If $z$ belongs to the technological set, then $\backslash lambda\; z$ also belongs to the technological set. The property of constant return means that the technological set is a cone.

8. Convex: for any two valid vectors $z\_1,\; z\_2$ Any vectors are also valid $\backslash alpha\; z\_1\; +(1-\backslash alpha)z\_2$, Where . The convexity property means the ability to “mix” technologies. In particular, it is fulfilled if the technological set has the property of additivity and non-increasing returns to scale. Moreover, in this case the technological set is a convex cone.

Efficient technology set boundary

Acceptable technology $z$ called effective, if there is no other acceptable technology different from it $z"\backslash geqslant\; z$. Many effective technologies form efficient frontier technological set.

If the condition of freedom of spending and closedness of the technological set is met, then it is impossible to endlessly increase the production of one good without reducing the output of others. In this case, for any acceptable technology $z$ there is effective technology $z"\; \backslash geqslant\; z$. In this case, instead of the entire technological set, only its effective boundary can be used. Typically, the efficient frontier can be given by some production function.

Production function

Let's consider single-product technologies $(-x,y)$, Where $y$- dimension vector $m=1$, A $x$- dimension cost vector $n$. Consider the set $X$, which includes all possible cost vectors $x$, such that for everyone $x$ exists $y$, such that the net output vectors $(-x,y)$ belong to the technological set.

Numeric function $f(x)$ on $X$ called production function, if for each given cost vector $x$ meaning $f(x)$ defines the maximum value of the allowed output $y$(such that the net output vector (-x,y) belongs to the technological set).

Any point of the effective boundary of the technological set can be represented in the form $(-x,f(x))$, and the opposite is true if $f(x)$ is an increasing function (in this case $y=f(x)$- equation of the effective boundary). If a technological set has the property of freedom of expenditure and can be described by a production function, then the technological set is determined on the basis of the inequality $y\backslash leqslant\; f(x)$.

In order for a technological set to be specified using a production function, it is sufficient that for any $x$ a bunch of $F(x)$ permissible outputs at given costs $x$, was limited and closed. In particular, this condition is satisfied if the technological set has the properties of closure, non-increasing returns to scale and the absence of a cornucopia.

If the technological set is convex, then the production function is concave and continuous on the interior of the set $X$. If the condition of freedom of expenditure is satisfied, then $f(x)$ is a non-decreasing function (in this case, the concavity of the function also implies the convexity of the technological set). Finally, if both the condition of the absence of a cornucopia and the admissibility of inactivity are simultaneously satisfied, then $f(0)=0$.

If the production function is differentiable, then it is possible to define a local elasticity of scale in the following equivalent ways:

$e(x)=\backslash frac\; (d\; f(\backslash lambda\; x))(d\; \backslash lambda)\; \backslash cdot\; \backslash frac\; (\backslash lambda)(f(x))|\_(\backslash lambda=1)=\backslash frac\; (f"(x\; )x)(f(x))$

Where $f"(x)$ is the gradient vector of the production function.

Having thus determined the elasticity of scale, it can be shown that if a technological set has the property of constant returns to scale, then $e(x)=1$, if there are diminishing returns to scale, then $e(x)\backslash leqslant\; 1$, if increasing returns, then $e(x)\backslash geqslant\; 1$.

Manufacturer's challenge

If the price vector is given $p$, then the product $pz$ represents the producer's profit. The manufacturer’s task comes down to finding such a vector $z$, so that for a given price vector the profit is maximum. We denote the set of prices of goods at which this problem has a solution $P$. It can be shown that for a non-empty, closed technological set with non-increasing returns to scale, the manufacturer’s problem has a solution on the set of prices $P$, giving negative profit on the so-called recessive directions (these are vectors $z$ technological set, for which, for any non-negative $\backslash lambda$ vectors $\backslash lambda\; z$ also belong to the technological set). In particular, if the set of recessive directions coincides with $R^N\_-$, then a solution exists for any positive prices.

Profit function $\backslash pi(p)$ defined as $pz(p)$, Where $z(p)$- solving the manufacturer’s problem at given prices (this is the so-called supply function, possibly multi-valued). The profit function is positively homogeneous (of the first degree), that is $\backslash pi(\backslash lambda\; p)=\backslash lambda\; \backslash pi(p)$ and continuous on the inside $P$. If the technological set is strictly convex, then the profit function is also continuously differentiable. If the technological set is closed, then the profit function is convex on any convex subset of acceptable prices $P$.

Sentence function (display) $z(p)$ is positively homogeneous of degree zero. If the technological set is strictly convex, then the supply function is single-valued on P and continuous on the interior $P$. If a supply function is twice differentiable, then the Jacobian matrix of this function is symmetric and non-negative definite.

If the technological set is represented by a production function, then profit is defined as $pf(x)-wx$, Where $w$- vector of prices for production factors, $p$ in this case, the price of manufactured products. Then for any internal solution (that is, belonging to the interior $X$) the producer's problem is fair: the equality of the marginal product of each factor to its relative price, that is, in vector form $f"(x)=w/p$.

If the profit function is given $\backslash pi(p)$, which is a twice continuously differentiable, convex and positively homogeneous (first degree) function, then it is possible to restore the technological set as a set containing for any non-negative price vector $p$ clean release vectors $z$, satisfying the inequality $pz\backslash leqslant\backslash pi(p)$. It can also be shown that if the supply function is positively homogeneous of degree zero and the matrix of its first derivatives is continuous, symmetric and non-negative definite, then the corresponding profit function satisfies the above requirements (the converse is also true).

see also

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Excerpt characterizing the Technological set

Suddenly Prince Hippolyte stood up and, stopping everyone with hand signs and asking them to sit down, spoke:
- Ah! aujourd"hui on m"a raconte une anecdote moscovite, charmante: il faut que je vous en regale. Vous m"excusez, vicomte, il faut que je raconte en russe. Autrement on ne sentira pas le sel de l"histoire. [Today I was told a charming Moscow joke; you need to teach them. Sorry, Viscount, I will tell it in Russian, otherwise the whole point of the joke will be lost.]
And Prince Hippolyte began to speak Russian with the accent that the French speak when they have been in Russia for a year. Everyone paused: Prince Hippolyte so animatedly and urgently demanded attention to his story.
– There is one lady in Moscow, une dame. And she's very stingy. She needed to have two valets de pied [footmen] for the carriage. And very tall. It was to her liking. And she had une femme de chambre [maid], still very tall. She said…
Here Prince Hippolyte began to think, apparently having difficulty thinking straight.
“She said... yes, she said: “girl (a la femme de chambre), put on the livree [livery] and come with me, behind the carriage, faire des visites.” [make visits.]
Here Prince Hippolyte snorted and laughed much earlier than his listeners, which made an unfavorable impression for the narrator. However, many, including the elderly lady and Anna Pavlovna, smiled.
- She went. Suddenly there was a strong wind. The girl lost her hat and her long hair was combed...
Here he could no longer hold on and began to laugh abruptly and through this laughter he said:
- And the whole world knew...
That's the end of the joke. Although it was not clear why he was telling it and why it had to be told in Russian, Anna Pavlovna and others appreciated the social courtesy of Prince Hippolyte, who so pleasantly ended Monsieur Pierre’s unpleasant and ungracious prank. The conversation after the anecdote disintegrated into small, insignificant talk about the future and the past ball, performance, about when and where they would see each other.

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

Let the number of factors of production be equal to n, and the number of types of output equal to m, so that l = m + n. Let us denote the vector of costs (in absolute value) by r Rn + , and the volume of output by y Rm + . We will call the vector (−r, yo ) vector of net issues. The set of all technologically feasible vectors of net outputs 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 in nature. At the same time, it is possible not to adhere to a strict 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 specific classes of technologies are usually described.

1. Non-emptiness

The technological set Y is non-empty.

This property means the fundamental possibility of carrying out production activities.

2. Closedness

The technological set Y is closed.

This property is rather technical; it means that the technological 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 to spend:

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

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

4. No “cornucopia” (“no free lunch”)

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

This property means that to produce a product in a positive quantity, costs are required in a non-zero volume.

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

5. Non-increasing returns to scale:

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

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

50 . Non-decreasing returns to scale:

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

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

500. Constant returns to scale is a 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 mean that Y is a cone (possibly not containing 0).

In the case of two goods, where one is input and the other is produced, constant output means that the average productivity of the input does not change as output changes.

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

The convexity property means the ability to “mix” technologies in any proportion.

7. Irreversibility

if y Y and y 6= 0, then (−y) / Y.

Let's say you can produce 5 bearings from a kilogram of steel. Irreversibility 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. Acceptability of inactivity:

Theorem 44:

1) From the non-increasing returns to scale and 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, 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. A 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 permissible ones, special ones stand out efficient technologies. An admissible technology y is usually called effective if there is no other (different from it) admissible technology y0 such that y0 > y. Obviously, this definition of efficiency implicitly implies that all goods are in some sense desirable. Effective technologies constitute efficient frontier technological set. Under certain conditions, it becomes possible to use the effective frontier in the analysis instead of the entire technological set. In this case, it is important that for any admissible technology y there is an effective technology y0 such that y0 > y. In order for this condition to be met, 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 indefinitely without reducing the output of other goods. It can be shown that if technological

Rice. 4.4. Efficient technology set boundary

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

Introductory and intermediate courses, when describing the behavior of a producer, are based on the representation of his production set through a production function. A relevant question is under what conditions on the production set such a representation is possible. Although it is possible to give a broader definition of the production function, 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 on the effective boundary of the technological set has the form (−r, f(r)). The converse is true if f(r) is an increasing function. In this case, yo = f(r) is the effective frontier equation.

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

Theorem 45:

Let for a 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 so and for some r R there is

there exists 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 closure), (0, 1) Y , which contradicts the absence of a cornucopia.

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

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 the set R.

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

5) If Y has the property of lacking a cornucopia, then f(0) 6 0.

6) If the set Y has the property of permissible inactivity, 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 f(·) is concave.

(2) Since the set Y has the property of free spending, 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 a concave function is continuous internally.

the size of its domain of definition.

(4) Let r 00 > r0 (r0 , r00 R). Since (−r0 , f(r0 )) Y , then by the property of freedom of spending (−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 of the absence of a cornucopia. So 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 a technology can be described directly in terms of this function. Let us demonstrate this using the example of the so-called elasticity of scale.

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

local elasticity of scale 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

V at point r we have 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) >λn f(r). Let us rewrite this inequality as:

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

Note that such a function can always be found. For example, a suitable function is 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 one differentiable implicit production function, and such technological sets are not something exceptional. In particular, the technological sets considered in initial microeconomics courses are often such that their description requires two (or more) inequalities with differentiable functions, since it is necessary to take into account additional restrictions on the non-negativity of factors of production. To account for such restrictions, one can use vector implicit