Technological set and its properties. Manufacturer behavior
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) that do not upset the balance, that is, economically and socially safe for society. Let's imagine all this in the form of a diagram (see diagram 4.1.).
The main standard models for optimizing the production activities of a company with a linear technological set, statistical and dynamic models for planning production investments, issues of economic and mathematical analysis of business decisions based on the use of the apparatus of dual assessments are considered. The main approaches to the problem of assessing the quality of production investments are outlined, as well as methods and indicators for assessing their effectiveness.
Let us consider the case, which is very important for model applications, when the technological set of a production system is a linear convex set, i.e., the production model turns out to be linear.
Comment. Together, assumptions 2.1 and 2.2 mean that the technological set is a convex cone. Assumption 2.3, highlighting linear technologies, means that this cone is a convex polyhedron in a half-space
Is it possible to say that in the economic area 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?
Relationship (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 general technological set of a production element can be obtained as a result of combining all input-output vectors acceptable 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 technology of an element leads to the need to supplement it with a number of features. We will look at 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. Taking into account 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 nonlinear 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 specifying a set of acceptable costs and a set of acceptable outputs for each cost level. Descriptions of this kind are convenient in problems such as optimal resource allocation, in which, for given levels of resource consumption, it is necessary to determine the acceptable and most effective (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 specified by the plan and it is necessary to determine the acceptable and minimum levels of costs of the elements. Problems of this kind can be conventionally called problems of optimal implementation of the planned production program. In such problems, it is convenient to apply the inverse sequence of describing the technological set of a production element, first specifying the set U of permissible outputs and g = U, and then for each acceptable level of output - the set V (and) of allowable costs v E = V (and).
The general technological set Y of the production element has the form
In Fig. 3.4 this constraint is satisfied by all points of the technological set located above the segment EC or lying on it.
For the most part, material 4.21 is also original. An assessment of the effectiveness of market mechanisms ensuring the existence of a unified equilibrium control 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. A well-known 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 market equilibrium exists. As a consideration of the auction scheme in a market system shows, this situation 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 competitive economic models, the existence of equilibrium has been established by a number of authors under various assumptions. Typically the proof assumes the convexity of the utility functions (or preferences) of consumers and the technological sets of producers. A generalization of the Arrow-Debreu model for the case of a continuum of players is given. At the same time, it was possible to abandon assumptions about the convexity of consumer preference functions.
Each manufacturer (firm) j is characterized by a technological set Y. - a set of technologically feasible l-dimensional vectors of costs - output; their positive components correspond to the quantities produced, and the negative ones correspond to the quantities expended. It is assumed that the manufacturer chooses the input-output vector so as to obtain maximum profit. At the same time, he, like the consumer, does not try to influence prices, accepting them as given. Thus, its choice is a 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 consumer is strictly monotonic 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 excess demand functions, the uniqueness of equilibrium is also ensured by the condition of a 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's model. When choosing production volumes yj = y к, 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 = УЗ e Rl /,(%) > 0. Another convenient representation (when only one good h is produced) is in the form of an explicit production function y 0.
Technological set and its properties
TECHNOLOGICAL SET - see Production set, Technological method.
We will consider a description of one specific type of technological set for a production element that consumes several types of inputs 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 way of 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 Eth of dimension m O E Y d Em containing the zero element.
The methods for representing technological sets of production elements discussed in the previous paragraph characterize their properties, but do not explicitly specify the description. For single-product production elements, an explicit description of the technological set can be specified using the concept of production function. In 1.2 we already touched on this concept and its use, in this section we will continue to consider these issues.
Using single-product production functions to describe the technological set of a multi-product element. If a multi-product element produces certain types of products, while consuming /gevx types of inputs, then its input and output vectors have the form v = (i>i, vz,..., Vy x) and u = (m1g w2,.. . , itvykh) respectively.
It corresponds to a part of the technological set, limited by the curved triangle AB (marked with shading in Fig. 3.4).
The Arrow-Deb-re-McKsnzie model of a decentralized economy. The general model of a decentralized economy describes production, consumption and decentralized
With the help of technological sets, production processes that are carried out by the production system are modeled. Each system has inputs and outputs:
The production process is presented as a process of unambiguous transformation of production factors into production products within a given time interval. During this time interval, factors completely disappear and products appear.
With such modeling - the transformation of factors into products - the role of the internal structure of the production system, its organization and production management methods is completely hidden.
Observers have access to information about the state of the system inputs and outputs. These states are determined, on the one hand, by a point in the space of goods and factors, and on the other, the state of outputs is determined by a point in the space of outputs.
Space models include many space factors, many space parameters, and many available technologies.
Technology is a technical way of converting factors of production into products.
A technological process is an ordered set of two vectors, where is the vector of production factors and is the vector of products. The technological process is the simplest model of space, which is specified from a number of elements:
Thus, the technological process is described by a set of (n+ m) numbers: .
For example, let’s take a computer of type A and , that is, one computer is produced, then this technological process is described 7+1=8 numbers.
In the practice of modeling real production systems, the hypothesis of linear technologies is used as a first approximation.
Linearity of technology implies an increase in products V with increasing sets of factors U.
Let's consider the main properties of technological processes:
1. Similarity.
The technological process is similar, i.e. ~ if the condition is met: 
, which means that this is the same technological process, but proceeding with intensity:
For such processes, the system of equalities is fulfilled:
Similar processes lie on the same line of production technology.
2. Difference.
Different technological processes lie on different rays and cannot be converted into each other by multiplying by a positive number.
3. Composite technological processes.
A process is called composite if there exist and , that .
A process that is not compound is called basic.
The ray passing through the origin in the direction of the base process is called the base ray. Each base beam corresponds to a base technology, and all points on the base beam reflect similar technological processes.
By definition, a basic technological process cannot be expressed through a linear combination of other technological processes.
In the positive octant, you can place a hyperplane that cuts off unit segments from each coordinate.
This allows you to visualize production technologies.
Let us show possible intersections of the hyperplane with technological rays.
1) The only technology available is basic.
2) The emergence of new additional basic technology.
3) Linear combination of two basic technologies.
4) Third additional basic technology.
5) The possibility of forming technologies lying inside the triangular area.
6) Two triangular areas with six basic technologies.
7) Combining technologies - a convex hexagon.
8) The case with an infinite number of basic technologies is possible.
In these graphic images, all internal and boundary points, with the exception of the vertices, reflect the constituent technological processes, and the set of all technological processes is called the technological set Z.
Technological sets have the following properties:
1. Not realizing the cornucopia.
(Ø, V) Z, hence, V= Ø.
(Ø, Ø) Z means inaction.
2. The technological set is convex, and processes whose rays lie on the boundary of this set can mix with each other.
3. The technological set is limited from above due to limited economic resources.
4. The technological set is closed, and effective technologies lie on the border of this set.
A specific property of technological sets is the existence of ineffective processes.
If , then any technological processes are possible that satisfy the condition (for factors) (for products).
Exists ( ,Ø) Z, which means the complete destruction of factors of production. No products arise in it at all.
The technological process is more efficient than if and/or.
PRODUCTION FUNCTION.
The mathematical description of an efficient process can be converted into a production function by aggregating the factors of production as well as aggregating the products of production into a single product.
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 Tτ see Fig. 2.1 points C and B) ;
in τ(-τ) = (0), i.e. if Tτ, T ≠ 0, then -Tτ – 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 = (YM x: if ZM 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 (XS: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, Tτ. 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 ZK 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, vf/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:
Features of inflation processes in modern Russia.
1. The concept of production and PF. Production set.
2. Profit maximization problem
3. Producer equilibrium. Technical progress
4. Cost minimization problem.
5. Aggregation in production theory. Equilibrium of the firm and industry in the d/s period
(independently) proposal of competitive firms having alternative goals
Production– activities aimed at producing the maximum amount of material goods depend on the number of production factors used, specified by the technological aspect of production.
Any technological process can be represented using a vector of net outputs, which we will denote by y. If, according to this technology, a company produces the i-th product, then the i-th coordinate of the vector y will be positive. If, on the contrary, the i-th product is spent, then this coordinate will be negative. If a certain product is not consumed and produced according to this technology, then the corresponding coordinate will be equal to 0.
We will call the set of all technologically accessible vectors of net outputs for a given firm the production set of the firm and denote it Y.
Properties of production sets:
1. The production set is not empty, i.e. At least one technological process is available to the company.
2. The production set is closed.
3. Absence of a “cornucopia”: if y 0 and y ∊Y, then y=0. You cannot produce something without spending anything (no y<0, т.е. ресурсов).
4. Possibility of inaction (liquidation): 0∊Y. in reality, there may be sunk costs.
5. Freedom of spending: y∊Y and y` y, then y`∊Y. The production set includes not only optimal technologies, but also technologies with lower output/resource consumption.
6. irreversibility. If y∊Y and y 0, then –y Y. If from 2 units of the first good it is possible to produce 1 of the second, then the reverse process is not possible.
7. Convexity: if y`∊Y, then αy + (1-α)y` ∊ Y for all α∊. Strict convexity: for all α∊(0,1). Property 7 allows you to combine technologies to obtain other available technologies.
8. Returns to scale:
If, in percentage terms, the volume of factors used has changed by ∆ N, and the corresponding change in output was ∆Q, then the following situations occur:
- ∆N = ∆Q there is a proportional return (an increase in the number of factors led to a corresponding increase in output)
- ∆ N< ∆Q there are increasing returns (positive economies of scale) – i.e. output increased in greater proportion than the number of factors consumed increased
- ∆N > ∆Q there are diminishing returns (diseconomies of scale) – i.e. an increase in costs leads to a smaller percentage increase in output
Economies of scale are relevant in the long term. If an increase in the scale of production does not lead to a change in labor productivity, we are dealing with constant returns to scale. Diminishing returns to scale are accompanied by a decrease in labor productivity, while increasing returns are accompanied by an increase.
If the set of goods that are produced is different from the set of resources that are used, and only one product is produced, then the production set can be described using a production function.
Production function(PF) - reflects the relationship between maximum output and a certain combination of factors (labor and capital) and at a given level of technological development of society.
Q=f(f1,f2,f3,…fn)
where Q is the firm's output for a certain period of time;
fi is the amount of the i-th resource used in the production of products;
Typically, there are three factors of production: labor, capital and materials. We will limit ourselves to the analysis of two factors: labor (L) and capital (K), then the production function takes the form: Q =f(K, L).
Types of PF may vary depending on the nature of the technology, and can be presented in three types:
A linear PF of the form y = ax1 + bx2 is characterized by constant returns to scale.
Leontief PF - in which resources complement each other, their combination is determined by technology and production factors are not interchangeable.
PF Cobb-Douglas– a function in which the factors of production used have the property of being interchangeable. General view of the function:
Where A is the technological coefficient, α is the labor elasticity coefficient, and β is the capital elasticity coefficient.
If the sum of the exponents (α + β) is equal to one, then the Cobb-Douglas function is linearly homogeneous, that is, it demonstrates constant returns when the scale of production changes.
The production function was first calculated in the 1920s for the US manufacturing industry, in the form of the equality
For the Cobb-Douglas PF:
1. Since a< 1 и b < 1, предельный продукт каждого фактора меньше среднего продукта (МРК < АРК и MPL < APL).
2. Since the second derivatives of the production function for labor and capital are negative, it can be argued that this function is characterized by a decreasing marginal product of both labor and capital.
3. As the value of MRTSL decreases, K gradually decreases. This means that the isoquants of the production function have a standard form: they are smooth isoquants with a negative slope, convex to the origin.
4. This function is characterized by a constant (equal to 1) elasticity of substitution.
5. The Cobb-Douglas function can characterize any type of returns to scale, depending on the values of parameters a and b
6. The function under consideration can serve to describe various types of technical progress.
7 The power-law parameters of the function are the coefficients of output elasticity with respect to capital (a) and labor (b), so that the equation for the growth rate of output (8.20) for the Cobb-Douglas function takes the form GQ = Gz + aGK + bGL. Parameter a, thus, characterizes the “contribution” of capital to the increase in output, and parameter b characterizes the “contribution” of labor.
PF is based on a number of “production features”. They concern the effect of output in three cases: (1) a proportional increase in all costs, (2) a change in the cost structure with constant output, (3) an increase in one factor of production with the rest unchanged. case (3) refers to the short-term period.
The production function with one variable factor has the form:
We see that the most effective change in the variable factor X is observed on the segment from point A to point B. Here the marginal product (MP), having reached its maximum value, begins to decrease, the average product (AP) still increases, the total product (TP) receives the greatest growth.
Law of Diminishing Returns(law of diminishing marginal product) - defines a situation in which the achievement of certain production volumes leads to a decrease in the output of finished products per additionally introduced unit of resource.
Typically, a given volume can be produced through various production methods. This is due to the fact that factors of production are interchangeable to a certain extent. It is possible to draw isoquants corresponding to all production methods necessary to produce a given volume. As a result, we obtain an isoquant map, which characterizes the relationship between all possible combinations of inputs and output levels and, therefore, is a graphical illustration of the production function.
Isoquant ( line of equal output - isoquant) – a curve reflecting all combinations of production factors that ensure the same output.
A set of isoquants, each of which shows the maximum output achieved by using certain combinations of resources, is called an isoquant map. The further the isoquant is located from the origin, the more resources are involved in the production methods located on it and the larger the output sizes that are characterized by this isoquant (Q3> Q2> Q1).
The isoquant and its form reflect the dependence specified by the PF. In the long term, there is a certain mutual complementarity (completeness) of production factors, however, without a decrease in output, a certain interchangeability of these production factors is also likely. Thus, various combinations of resources can be used to produce a good; it is possible to produce this good using less capital and more labor, and vice versa. In the first case, production is considered technically efficient in comparison with the second case. However, there is a limit to how much labor can be replaced by more capital without reducing production. On the other hand, there is a limit to the use of manual labor without the use of machines. We will consider the isoquant in the technical substitution zone.
The level of interchangeability of factors is reflected by the indicator maximum rate of technical substitution. – the proportion in which one factor can be replaced by another while maintaining the same output volume; reflects the slope of the isoquant.
MRTS=- ∆K / ∆ L = MP L / MP K
In order for output to remain unchanged when the quantity of factors of production used changes, the quantities of labor and capital must change in different directions. If the amount of capital decreases (AK< 0), то количество труда должно увеличиваться (AL >0). Meanwhile, the marginal rate of technical substitution is simply the proportion in which one factor of production can be replaced by another, and, as such, is always a positive quantity.
