The concept of the production system and the production process. Technological process and technological set
Ministry of Education and Science of the Russian Federation
Yaroslav the Wise Novgorod State University
Abstract by discipline:
Management
Completed by a student gr.6061 zo
Makarova S.V.
Received by Suchkov A.V.
Velikiy Novgorod
1. PRODUCTION PROCESS AND ITS ELEMENTS.
The basis of the production and economic activity of the enterprise is the production process, which is a set of interrelated labor processes and natural processes aimed at manufacturing certain types of products.
The organization of the production process consists in combining people, tools and objects of labor into a single process for the production of material goods, as well as in ensuring a rational combination in space and time of the main, auxiliary and service processes.
Production processes at enterprises are detailed by content (process, stage, operation, element) and place of implementation (enterprise, redistribution, workshop, department, section, unit).
The set of production processes occurring in the enterprise is a total production process. The process of production of each individual type of product of the enterprise is called private production process. In turn, in a private production process, partial production processes can be distinguished as complete and technologically separate elements of a private production process. production process, which are not the primary elements of the production process (it is usually carried out by workers of different specialties using equipment for various purposes).
As a primary element of the production process should be considered technological operation- a technologically homogeneous part of the production process, performed at one workplace. Technologically separate partial processes are stages of the production process.
Partial production processes can be classified according to several criteria:
For the intended purpose;
The nature of the flow in time;
The method of influencing the object of labor;
The nature of the work involved.
Processes are classified according to purpose. main, auxiliary and service.
Main production processes - processes of transformation of raw materials and materials into finished products, which is the main, profile
products for this company. These processes are determined by the manufacturing technology of this type of product (preparation of raw materials, chemical synthesis, mixing of raw materials, packaging and packaging of products).
Auxiliary production processes are aimed at the manufacture of products or the performance of services to ensure the normal flow of the main production processes. Such production processes have their own objects of labor, different from the objects of labor of the main production processes. As a rule, they are carried out in parallel with the main production processes (repair, packaging, tool facilities).
Serving production processes ensure the creation of normal conditions for the flow of the main and auxiliary production processes. They do not have their own object of labor and proceed, as a rule, sequentially with the main and auxiliary processes, interspersed with them (transportation of raw materials and finished products, their storage, quality control).
The main production processes in the main workshops (sections) of the enterprise form its main production. Auxiliary and service production processes, respectively, in auxiliary and service shops - form an auxiliary economy.
The different role of production processes in the overall production process determines the differences in the management mechanisms of various types of production units. At the same time, the classification of partial production processes according to their intended purpose can only be carried out in relation to a specific private process.
Combining the main, auxiliary, service and other processes in a certain sequence forms the structure of the production process.
The main production process represents the process and production of the main products, which includes natural processes, technological and work processes, as well as inter-operational waiting.
Natural process - a process that leads to a change in the properties and composition of the object of labor, but proceeds without human participation (for example, in the manufacture of certain types of chemical products).
Natural production processes can be considered as necessary technological breaks between operations (cooling, drying, aging, etc.)
Technological process is a set of processes that result in all necessary changes in the object of labor, i.e., it turns into finished products.
Auxiliary operations contribute to the implementation of the main operations (transportation, control, sorting of products, etc.).
Work process - a set of all labor processes (main and auxiliary operations).
The structure of the production process changes under the influence of the technology of the equipment used, the division of labor, the organization of production, etc.
Interoperational laying - breaks provided for by the technological process.
According to the nature of the flow in time, they distinguish continuous and periodical production processes. In continuous processes, there are no interruptions in the production process. Production maintenance operations are carried out simultaneously or in parallel with the main operations. In periodic processes, the execution of basic and maintenance operations occurs sequentially, due to which the main production process is interrupted in time.
According to the method of impact on the object of labor, they distinguish mechanical, physical, chemical, biological and other types of production processes.
According to the nature of the labor used, production processes are classified into automated, mechanized and manual.
The principles of the organization of the production process are the starting points on the basis of which the construction, operation and development of the production process are carried out.
There are the following principles of organization of the production process:
differentiation - the division of the production process into separate parts (processes, operations, stages) and their assignment to the relevant divisions of the enterprise;
combination - the combination of all or part of diverse processes for the manufacture of certain types of products within the same site, workshop or production;
concentration - the concentration of certain production operations for the manufacture of technologically homogeneous products or the performance of functionally homogeneous work at individual workplaces, sites, workshops or production facilities of the enterprise;
specialization - assigning to each workplace and each division a strictly limited range of works, operations, parts and products;
universalization - the manufacture of parts and products of a wide range or the performance of heterogeneous production operations at each workplace or production unit;
proportionality - a combination of individual elements of the production process, which is expressed in their certain quantitative relationship with each other;
parallelism - simultaneous processing of different parts of one batch for a given operation at several workplaces, etc.;
straightness - the implementation of all stages and operations of the production process in the conditions of the shortest path of passage of the object of labor from beginning to end;
Rhythm - repetition through established periods of time of all individual production processes and a single process for the production of a certain type of product.
The above principles of organization of production in practice do not operate in isolation from each other, they are closely intertwined in each production process. The principles of the organization of production develop unevenly - in one period or another, one or another principle comes to the fore or acquires secondary importance.
If the spatial combination of elements of the production process and all its varieties is implemented on the basis of the formation of the production structure of the enterprise and its subdivisions, the organization of production processes in time finds expression in establishing the order of execution of individual logistics operations, the rational combination of execution time various kinds works, determination of calendar and planning standards for the movement of objects of labor.
The basis for building an effective production logistics system is the production schedule, formed on the basis of the task of meeting consumer demand and answering the questions: who, what, where, when and in what quantity will be produced (produced). The production schedule allows you to establish volumetric and temporal characteristics of material flows differentiated for each structural production unit.
The methods used to compile the production schedule depend on the type of production, as well as the characteristics of demand and parameters of orders can be single, small-batch, serial, large-batch, mass.
The characteristic of the type of production is supplemented by the characteristic of the production cycle - this is the period of time between the start and end of the production process in relation to specific products within the logistics system (enterprise).
The production cycle consists of working time and break time in the manufacture of products.
In turn, the working period consists of the main technological time, the time for carrying out transport in control operations and the picking time.
The time of breaks is subdivided into the time of interoperational, inter-sectional and other breaks.
The duration of the production cycle largely depends on the characteristics of the movement of the material flow, which can be sequential, parallel, parallel-serial.
In addition, the duration of the production cycle is also influenced by the forms of technological specialization of production units, the organization system of the production processes themselves, the progressiveness of the technology used and the level of unification of products.
The production cycle also includes waiting time - this is the interval from the moment an order is received to the moment it begins to be executed, to minimize which it is important to initially determine the optimal batch of products - a batch at which the cost per product is the minimum value.
To solve the problem of choosing the optimal batch, it is generally accepted that the cost of production consists of direct manufacturing costs, inventory storage costs, and equipment readjustment and downtime costs when changing batches.
In practice, the optimal lot is often determined by direct calculation, but when forming logistics systems, it is more effective to use mathematical programming methods.
In all areas of activity, but especially in production logistics, the system of norms and standards is of paramount importance. It includes both enlarged and detailed norms for the consumption of materials, energy, use of equipment, etc.
2. Methods for solving the transport problem.
Transport problem (classic)- the problem of the optimal plan for the transportation of a homogeneous product from homogeneous points of availability to homogeneous points of consumption on homogeneous vehicles (predetermined quantity) with static data and a linear approach (these are the main conditions of the problem).
For the classical transport task, two types of tasks are distinguished: the cost criterion (achieving a minimum of transportation costs) or distances and the time criterion (minimum time is spent on transportation).
History of the search for solution methods
The problem was first formalized by the French mathematician Gaspard Monge in 1781 year . The main advance was made in the fields during Great Patriotic War Soviet mathematician and economist Leonid Kantorovich . Therefore, sometimes this problem is called transport task Monge - Kantorovich.
With the help of technological sets, production processes are modeled, which are carried out by the production system. Every system has inputs and outputs:
The production process is presented as a process of unambiguous transformation of factors of production into products of production during a given time interval. During this time interval, the factors completely disappear and products appear.
In such modeling - the transformation of factors into products - the role of internal structure production system, its organization and production management methods.
Observers have access to information about the state of the inputs and outputs of the system. These states are determined, on the one hand, by a point in the space of goods and factors, and on the other hand, 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 the technical way of converting factors of production into products.
A technological process is an ordered set of two vectors , where is the vector of factors of production, is the vector of products. Technological process is an the simplest model space, which is set 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 , i.e. 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.
The linearity of technologies implies an increase in products V with increasing sets of factors U.
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:
Such processes lie on the same beam of production technology.
2. Difference.
Different technological processes lie on different beams and cannot be converted into each other by multiplying by a positive number.
3. Composite technological processes.
A process is called compound if and exist, such that .
A process that is not a composite process is called a base process.
The beam passing through the origin in the direction of the base process is called the base beam. Each base beam corresponds to a base technology, and all points of the base beam reflect similar technological processes.
By definition, a basic workflow cannot be expressed in terms of a linear combination of other workflows.
In the positive octant, one can place a hyperplane that cuts off unit segments from each coordinate.
This allows you to visualize the production technology.
Let us show possible intersections of the hyperplane by technological rays.
1) The only technology available is basic.
2) The emergence of a new additional basic technology.
3) Linear combination of two basic technologies.
4) The third additional basic technology.
5) Possibility of formation of 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 composite technological processes, and the set of all technological processes is called the technological set Z.
Technological sets have the following properties:
1. Not exercising a cornucopia.
(Ø, V) Z, hence, V=Ø.
(Ø, Ø) Z means inaction.
2. The technological set is convex, and the 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 efficient technologies lie on the boundary of this set.
A specific property of technological sets is the existence of inefficient processes.
If exists, then any technological processes satisfying the condition (for factors), (for products) are possible.
Exist ( ,Ø) Z, which means the complete destruction of the factors of production. No products appear 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.
Methods for describing technologies.
Production is the main area of activity of the company. Firms use production factors, which are also called input (input) factors of production. For example, a bakery owner uses inputs such as labor, raw materials such as flour and sugar, and capital invested in ovens, mixers, and other equipment to produce products such as bread, pies, and confectionery.
We can subdivide factors of production into major categories- labor, materials and capital, each of which includes narrower groupings. For example, labor production factor through the indicator of labor intensity, it combines both skilled (carpenters, engineers) and unskilled labor (agricultural workers), as well as the entrepreneurial efforts of firm managers. Materials include steel, plastic materials, electricity, water, and any other product that a firm purchases and turns into a finished product. Capital includes buildings, equipment and inventories.
The set of all technologically available net output vectors for a given firm is called the production set and denoted by Y.
PRODUCTION SET- the set of admissible technological ways given economic system (X,Y ) , where X - aggregate cost vectors, a Y - aggregate release vectors.
P. m. is characterized by the following features: it closed and convex(cm. A bunch of), the cost vectors are necessarily non-zero (one cannot produce something without spending anything), the components of the P. m. - costs and outputs - cannot be interchanged, because production is an irreversible process. The convexity of P. m. shows, in particular, the fact that the return on processed resources decreases with an increase in the volume of processing.
Properties of production sets
Consider an economy with l goods. It is natural for a particular firm to consider some of these goods as factors of production and some as output. It should be noted that such a division is rather arbitrary, since the company has sufficient freedom in choosing the range of products and cost structure. When describing the technology, we will distinguish between output and costs, representing the latter as output with a minus sign. For the convenience of presenting the technology, products that are neither consumed nor produced by the firm will be referred to as its output, and the volume of production of this product is assumed to be 0. In principle, the situation in which the product produced by the firm is also consumed by it in the production process is not ruled out. In this case, we will consider only a clean release this product, i.e., its output minus costs.
Let the number of factors of production be n and the number of outputs be m, so that l = m + n. Let us denote the cost vector (according to absolute value) through r 2 Rn+, and output volumes through y 2 Rm+
The vector (−r, yo) will be called the vector of net outputs. The set of all technologically admissible net output vectors y = (−r, yo) constitutes the technological set Y . Thus, in the case under consideration, any technological set is a subset of Rn − × Rm+
This description of production is of a general nature. At the same time, it is possible not to adhere to a rigid division of goods into products and factors of production: the same good can be spent with one technology, and produced with another.
Let us describe the properties of technological sets, in terms of which the description of concrete classes of technologies is usually given.
1. Non-emptiness. The technological set Y is non-empty. This property means the fundamental possibility of carrying out production activities.
2. Closure. The technological set Y is closed. This property is rather technical; it means that the technology set contains its boundary, and the limit of any sequence of technologically feasible net output vectors is also a technologically feasible net output vector.
3. Freedom of spending. This property can be interpreted as having the ability to produce the same amount of output at a higher cost, or less output at the same cost.
4. The absence of a “horn of plenty” (“no free lunch”). if y 2 Y and y > 0, then y = 0. This property means that the production of products in a positive quantity requires costs in a non-zero volume.
< _ < 1, тогда y0 2 Y. Иногда это свойство называют (не совсем точно) убывающей отдачей от масштаба. В случае двух благ, когда одно затрачивается, а другое производится, убывающая отдача означает, что (максимально возможная) average performance the factor spent 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.
fifty . Non-diminishing returns to scale: if y 2 Y and y0 = _y, where _ > 1, then y0 2 Y.
In the case of two goods, where one is spent and the other is produced, increasing returns means that the (maximum possible) average productivity of the input factor does not decrease.
500 . Constant returns to scale - the situation when the technological set satisfies conditions 5 and 50 simultaneously, i.e. if y 2 Y and y0 = _y0, then y0 2 Y 8_ > 0.
Geometrically constant returns to scale means that Y is a cone (possibly not containing 0). In the case of two goods, where one is consumed and the other produced, constant returns mean that the average productivity of the factor input does not change with the change in output.
5. Nonincreasing returns to scale: if y 2 Y and y0 = _y, where 0< _ < 1, тогда y0 2 Y. Иногда это свойство называют (не совсем точно) убывающей отдачей от масштаба. В случае двух благ, когда одно затрачивается, а другое производится, убывающая отдача означает, что (максимально возможная) средняя производительность затрачиваемого фактора не возрастает. Если за час вы можете решить в лучшем случае 5 однотипных задач по микроэкономике, то за два часа в условиях убывающей отдачи вы не смогли бы решить более 10 таких задач.
fifty . Non-diminishing returns to scale: if y 2 Y and y0 = _y, where _ > 1, then y0 2 Y. In the case of two goods, where one is spent and the other is produced, increasing returns mean that the (maximum possible) average productivity of the input factor does not decrease.
500 . Constant returns to scale - the situation when the technological set satisfies conditions 5 and 50 simultaneously, i.e. if y 2 Y and y0 = _y0, then y0 2 Y 8_ > 0.
Geometrically constant returns to scale means that Y is a cone (possibly not containing 0).
In the case of two goods, where one is consumed and the other produced, constant returns mean that the average productivity of the factor input does not change with the change in output.
6. Convexity: The convexity property means the ability to "mix" technologies in any proportion.
7. Irreversibility
Let 5 bearings be produced from a kilogram of steel. Irreversible means that it is impossible to produce a kilogram of steel from 5 bearings.
8. Additivity. if y 2 Y and y0 2 Y , then y + y0 2 Y. The additivity property means the ability to combine technologies.
9. Permissibility of inactivity:
Theorem 44:
1) From the non-increasing returns to scale and the additivity of the technological set, its convexity follows.
2) Non-increasing returns to scale follow from the convexity of the technological set and the permissibility of inactivity. (The converse is not always true: with non-increasing returns, the technology may be non-convex)
3) A technological set has the properties of additivity and non-increasing returns to scale if and only if it is a convex cone.
Not all eligible technologies are equally important from an economic point of view.
Efficient technologies stand out among the admissible ones. An admissible technology y is called efficient if there is no other (different from it) admissible technology y0 such that y0 > y. Obviously, this definition of efficiency implies implicitly that all goods are desirable in some sense. Efficient technologies constitute the effective frontier of the technological set. Under certain conditions, it is possible to use the effective frontier in the analysis instead of the entire technological set. Here it is important that for any admissible technology y there is an efficient technology y0 such that y0 > y. In order for this condition to be satisfied, it is required that the technological set be closed, and that within the technological set it is impossible to increase the output of one good to infinity without reducing the output of other goods.
TECHNOLOGICAL METHOD - general concept combining two: T. s. production (production method, technology) and T. s. consumption; set of basic characteristics ( ingredients) production process (respectively - consumption) one or another product. AT economic and mathematical model T. s., or technology (activity), is described by a system of numbers inherent in it ( vector): e.g. cost rates and release various resources per unit of time or per unit of production, etc., including coefficients material consumption, laboriousness, capital intensity, capital intensity.
For example, if x = (x 1 , ..., x m) - the vector of resource costs (listed under numbers i = 1, 2, ..., m), a y = (y 1 , ..., y n) - vector of production volumes of products j= 1, 2, ..., n, then technologies, technological processes, methods of production can be called pairs of vectors ( x,y ). Technological acceptability means here the ability to obtain from the consumed (used) vector ingredients x product vector y .
The set of all possible admissible technologies ( XY) forms technological or production set given economic system.
VECTOR- an ordered set of a certain number of real numbers (this is one of the many definitions - the one that is accepted in economic and mathematical methods). For example, the daily plan of the workshop can be written as a 4-dimensional vector (5, 3, -8, 4), where 5 means 5 thousand parts of one type, 3 - 3 thousand parts of the second type, (-8) - metal consumption in t, and the last component, for example, saving 4 thousand kW. h electricity. As you can see, the number of components ( coordinates) B. arbitrarily (in this case, the plan of the workshop may consist not of four, but of any other number of indicators); they are unacceptable to be interchanged; they can be both positive and negative.
Vectors can be multiplied by a real number (for example, if you increase the plan by 1.2 times for all indicators, you get a new vector with the same number of components). Vectors containing an equal number of additive components of the same name, respectively, can be added and subtracted.
It is customary to highlight the letter designation V. in bold type (although this is not always observed).
The sum of vectors x = (x 1 ,..., x n) and y = (y 1 , ..., y n) is also B. ( x + y ) = (x 1 + y 1 , ..., xn+yn).
Dot product of vectors x and y a number equal to the sum of the products of the corresponding components of these V. is called:
Vectors x and y called orthogonal if their dot product is zero.
Equality V. - component, i.e., two V. are equal if their corresponding components are equal.
Vector 0 - (0, ..., 0) null;
n-dimensional V. - positive ( x > 0) if all its components x i Above zero, non-negative (x ≥ 0) if all its components x i greater than 0 or equal to zero, i.e. x i≤ 0; and semi-positive, if at least one component x i≥ 0 (notation x ≥ 0); if V. have an equal number of components, their ordering (full or partial) is possible, i.e., introduction on a set of vectors binary relation “> ”: x > y , x ≥y , x ≥ y depending on whether the difference is positive, semi-positive or non-negative x-y.
THE LAW OF DIMENSIONING RECOMMENDATIONS- a statement that if the use of any one factor of production and at the same time the costs of all other factors are preserved (they are called fixed), then the physical volume marginal product, produced with the help of the specified factor, will (at least from a certain stage) decrease.
PRODUCTION BEAM- locus of points representing a proportional increase in the number resources when using a specific technological method with increasing intensity.
For example, if the combination of 3 units. capital (funds) and 2 units. labor (i.e. a combination of 3 K + 2L) gives 10 units. some product, then combinations 6 K + 4L, 9K + 6L, giving respectively 20 and 30 units. etc., will lie on the graph on a straight line called P. l. or technology beam. With a different combination of factors P. l. will have a different slope. Due to the indivisibility of many factors of production the number of technological methods and, accordingly, P. l. accepted as final.
For example, if a team of three miners is working in a coal lava and one more is added to them, the output will increase by a quarter, and if a fifth, sixth, seventh one is added, the increase in output will decrease, and then stop altogether: miners in cramped conditions will simply interfere each other.
Key Concept here - the marginal productivity of labor (more widely - marginal productivity of a factor of production δ Y/δ x). For example, if two factors are considered, then with an increase in the costs of one of them (the first or the second), its marginal productivity falls.
The law is applicable in the short term and for this technology (its revision changes the situation).
Consider an economy with l goods. It is natural for a particular firm to consider some of these goods as factors of production and some as output. It should be noted that such a division is rather arbitrary, since the company has sufficient freedom in choosing the range of products and cost structure. When describing the technology, we will distinguish between output and costs, representing the latter as output with a minus sign. For the convenience of presenting the technology, products that are neither consumed nor produced by the firm will be referred to as its output, and the volume of production of this product is assumed to be 0. In principle, the situation in which the product produced by the firm is also consumed by it in the production process is not ruled out. In this case, we will consider only the net output of a given product, i.e., its output minus costs.
Let the number of factors of production be n and the number of outputs be m, so that l = m + n. Let's denote the cost vector (in absolute value) as r Rn + , and the output volumes as y Rm + . The vector (−r, yo ) will be called net issues vector. The set of all technologically feasible net output vectors y = (−r, yo ) is technological set Y . Thus, in the case under consideration, any technological set is a subset of Rn − × Rm + .
This description of production is of a general nature. At the same time, it is possible not to adhere to a rigid division of goods into products and factors of production: the same good can be spent with one technology, and produced with another. In this case Y Rl .
Let us describe the properties of technological sets, in terms of which the description of concrete classes of technologies is usually given.
1. Non-emptiness
The technological set Y is non-empty.
This property means the fundamental possibility of carrying out production activities.
2. Closure
The technological set Y is closed.
This property is rather technical; it means that the technology set contains its boundary, and the limit of any sequence of technologically feasible net output vectors is also a technologically feasible net output vector.
3. Freedom of spending:
if y Y and y0 6 y, then y0 Y.
This property can be interpreted as having the ability to produce the same amount of output at a higher cost, or less output at the same cost.
4. Lack of “cornucopia” (“no free lunch”)
if y Y and y > 0, then y = 0.
This property means that for the production of products in a positive quantity, costs in a non-zero volume are necessary.
Rice. 4.1. Technological set with increasing returns to scale.
5. Nonincreasing returns to scale:
if y Y and y0 = λy, where 0< λ < 1, тогда y0 Y.
This property is sometimes called (not exactly) diminishing returns to scale. In the case of two goods, where one is spent and the other is produced, diminishing returns means that the (maximum possible) average productivity of the input factor does not increase. If in an hour you can solve at best 5 similar problems in microeconomics, then in two hours under conditions of diminishing returns you could not solve more than 10 such problems.
fifty . Non-diminishing returns to scale:
if y Y and y0 = λy, where λ > 1, then y0 Y.
In the case of two goods, where one is spent and the other is produced, increasing returns means that the (maximum possible) average productivity of the input factor does not decrease.
500 . Constant returns to scale - the situation when the technological set satisfies conditions 5 and 50 simultaneously, i.e.
if y Y and y0 = λy0 , then y0 Y λ > 0.
Geometrically constant returns to scale means that Y is a cone (possibly not containing 0).
In the case of two goods, where one is consumed and the other produced, constant returns mean that the average productivity of the factor input does not change with the change in output.
Rice. 4.2. Convex technology set with diminishing returns to scale
The property of convexity means the ability to "mix" technologies in any proportion.
7. Irreversibility
if y Y and y 6= 0, then (−y) / Y.
Let 5 bearings be produced from a kilogram of steel. Irreversible means that it is impossible to produce a kilogram of steel from 5 bearings.
8. Additivity.
if y Y and y0 Y , then y + y0 Y.
The property of additivity means the ability to combine technologies.
9. Permissibility of inactivity:
Theorem 44:
1) From the non-increasing returns to scale and the additivity of the technological set, its convexity follows.
2) From the convexity of the technological set and the permissibility of inactivity, non-increasing returns to scale follow. (The converse is not always true: with non-increasing returns, technology can be non-convex, see Fig. 4.3 .)
3) The technological set has the properties of additivity and non-increasing
returns to scale if and only if it is a convex cone.
Rice. 4.3. Non-convex technological set with non-increasing returns to scale.
Not all eligible technologies are equally important from an economic point of view. Among the admissible ones stand out efficient technologies. An admissible technology y is called efficient if there is no other (different from it) admissible technology y0 such that y0 > y. Obviously, this definition of efficiency implies implicitly that all goods are desirable in some sense. Efficient technologies make up effective frontier technological set. Under certain conditions, it turns out to be possible to use the effective frontier in the analysis instead of the entire technological set. Here it is important that for any admissible technology y there is an efficient technology y0 such that y0 > y. In order for this condition to be satisfied, it is required that the technological set be closed, and that within the technological set it is impossible to increase the output of one good to infinity without reducing the output of other goods. It can be shown that if technological
Rice. 4.4. Effective frontier of the technology set
set has the freedom of spending property, then the effective boundary uniquely defines the corresponding technological set.
Initial courses and courses of intermediate complexity, when describing the behavior of a producer, are based on the representation of its production set by means of a production function. It is appropriate to ask 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, however, hereinafter we will only talk about “single-product” technologies, i.e. m = 1.
Let R be the projection of the technological set Y onto the space of cost vectors, i.e.
R = ( r Rn | yo R: (−r, yo ) Y ) .
Definition 37:
The function f( ) : R 7→R is called production function, representing technology Y , if for each r R the value f(r) is the value of the following problem:
yo → max
(−r, yo ) Y.
Note that any point of the effective boundary of the technological set has the form (−r, f(r)). The reverse is true if f(r) is an increasing function. In this case, yo = f(r) is the effective boundary equation.
The following theorem gives the conditions under which a technological set can be represented??? production function.
Theorem 45:
Let for technological set Y R × (−R) for any r R the set
F (r) = ( yo | (−r, yo ) Y )
closed and bounded from above. Then Y can be represented by a production function.
Note: The fulfillment of the conditions of this statement can be guaranteed, for example, if the set Y is closed and has the properties of non-increasing returns to scale and the absence of a cornucopia.
Theorem 46:
Let the set Y be closed and have the properties of non-increasing returns to scale and the absence of a cornucopia. Then for any r R the set
F (r) = ( yo | (−r, yo ) Y )
closed and bounded from above.
Proof: The closedness of the sets F (r) follows directly from the closedness of Y . Let us show that F (r) are bounded from above. Let this not be the case, and for some r R
there is an infinitely increasing sequence (yn ) such that yn F (r). Then, due to non-increasing returns to scale (−r/yn , 1) Y . Therefore (due to closedness), (0, 1) Y , which contradicts the absence of a cornucopia.
We also note that if the technological set Y satisfies the free spending hypothesis, and there exists a production function f( ) representing it, then the set Y is described by the following relationship:
Y = ( (−r, yo ) | yo 6 f(r), r R ) .
Let us now establish some relationships between the properties of the technological set and the production function representing it.
Theorem 47:
Let the technological set Y be such that for all r R the production function f(·) is defined. Then the following is true.
1) If the set Y is convex, then the function f( ) is concave.
2) If the set Y satisfies the free spending hypothesis, then the converse is also true, i.e., if the function f( ) is concave, then the set Y is convex.
3) If Y is convex, then f( ) is continuous on the interior of R.
4) If the set Y has the free spending property, then the function f( ) does not decrease.
5) If Y has the cornucopia-free property, then f(0) 6 0.
6) If the set Y has the inactivity admissibility property, then f(0) > 0.
Proof: (1) Let r0 , r00 R. Then (−r0 , f(r0 )) Y and (−r00 , f(r00 )) Y , and
(−αr0 − (1 − α)r00 , αf(r0 ) + (1 − α)f(r00 )) Y α ,
since the set Y is convex. Then by definition of the production function
αf(r0 ) + (1 − α)f(r00 ) 6 f(αr0 + (1 − α)r00 ),
which means that f( ) is concave.
(2) Since the set Y has the property of free spending, then the set Y (up to the sign of the cost vector) coincides with its subplot. And the subgraph of a concave function is a convex set.
(3) The fact to be proved follows from the fact that the concave function is continuous in the internal
sti of its domain of definition.
(4) Let r 00 > r0 (r0 , r00 R). Since (−r0 , f(r0 )) Y , then by the freedom of spending property (−r00 , f(r0 )) Y . Hence, by the definition of the production function, f(r00 ) > f(r0 ), that is, f( ) does not decrease.
(5) The inequality f(0) > 0 contradicts the assumption that there is no cornucopia. Hence, f(0) 6 0.
(6) By the assumption of the admissibility of inactivity (0, 0) Y . So, by definition
Assuming the existence of a production function, the properties of technology can be described directly in terms of this function. We will show this with the example of the so-called elasticity of scale.
Let the production function be differentiable. At a point r, where f(r) > 0, we define
local scale elasticity e(r) as:
If at some point e(r) is equal to 1, then it is considered that at this point constant returns to scale if more than 1 then increasing returns, smaller - diminishing returns to scale. The above definition can be rewritten as follows:
P ∂f(r) e(r) = i ∂r i r i .
Theorem 48:
Let the technological set Y be described by the production function f( ) and
in point r, e(r) > 0. Then the following is true:
1) If the technological set Y has the property of diminishing returns to scale, then e(r) 6 1.
2) If the technological set Y has the property of increasing returns to scale, then e(r) > 1.
3) If Y has the property of constant returns to scale, then e(r) = 1.
Proof: (1) Consider the sequence (λn ) (0< λn < 1), такую что λn → 1. Тогда (−λn r, λn f(r)) Y , откуда следует, что f(λn r) >λnf(r). Let's rewrite this inequality as:
f(λn r) − f(r) |
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Passing to the limit, we have |
λn − 1 |
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∂ri |
ri 6 f(r). |
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Thus e(r) 6 1. |
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Properties (2) and (3) are proved similarly. |
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Technological sets Y can be specified as implicit production functions g(·). By definition, a function g( ) is called an implicit production function if technology y belongs to technology set Y if and only if g(y) >
Note that such a function can always be found. For example, a function is suitable such that g(y) = 1 for y Y and g(y) = −1 for y / Y . Note, however, that this function is not differentiable. Generally speaking, not every technological set can be described by a single differentiable implicit production function, and such technological sets are not something exceptional. In particular, technological sets considered in elementary microeconomics courses are often such that two (or more) inequalities with differentiable functions are needed to describe them, since additional constraints on the non-negativity of factors of production must be taken into account. To take into account such restrictions, one can use vector implicit
Technology description: production function, set of production factors used, isoquant map.
production function - technological dependence between the cost of resources and output.
Expressed formally, the production function looks like this:
Let us assume that the production function describes the output depending on the costs of labor and capital, that is, consider a two-factor model. The same amount of output can be obtained with different combinations of inputs of these resources. Can be used a small amount of machines (that is, to do with a small investment of capital), but at the same time a large amount of labor will have to be spent; it is possible, on the contrary, to mechanize certain operations, to increase the number of machines, and thereby to reduce labor costs. If for all such combinations the largest possible volume of output remains constant, then these combinations are represented by points lying on the same isoquante. That is, an isoquant is a line of equal output or quantity. In the graph, x1 and x2 are the resources used.
Having fixed a different quantity of manufactured products, we get a different isoquant, that is, the same production function has isoquant map.
Properties of isoquants:
isoquants have a negative slope. There is an inverse relationship between resources, that is, by reducing the amount of labor, it is necessary to increase the amount of capital in order to remain at the same level of production.
isoquants are convex with respect to the origin. As already mentioned, with a decrease in the use of one resource, it is necessary to increase the use of another resource. The convexity of the indifference curve with respect to the origin is a consequence of the falling marginal rate of technological substitution (MRTS). About MRTS in the third ticket is described in detail. A gentle descent of the isoquant indicates a decrease in the rate of substitution of one resource for another as the share of this good in production decreases.
the absolute value of the slope of the isoquant is equal to the marginal rate of technological substitution. The slope of the isoquant at a given point shows the rate at which one resource can be replaced by another without gaining or losing the amount of good produced.
isoquants do not intersect. The same level of output cannot be characterized by several isoquants, which contradicts their definition.
Mathematical justification and economic meaning of the decrease in the marginal rate of technological substitution.
Consider (substitution of CAPITAL BY WORK). That is, how much capital is the producer willing to give up in order to obtain 1 unit of labor. It is necessary to prove that this indicator decreases. 
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But since Q=const, therefore dQ=0
As you know, the marginal product of labor decreases (since a rational producer works in the second stage of production), therefore, with an increase in labor, MPL will decrease, and MPK will increase, since the amount of capital decreases, therefore, it will decrease.
The economic reason for the decrease in MRTS is that in most industries the factors of production are not completely interchangeable: they complement each other in the production process. Each factor can do what another factor of production cannot or can make worse.
Elasticity of substitution of factors of production (usual and logarithmic representation). Isoquant Curvature and Technology Flexibility
Elasticity of substitution of factors of production - used in economic theory an indicator showing by what percentage it is necessary to change the ratio of factors of production when their marginal rate of substitution changes by 1% in order for the output to remain unchanged.
Let us determine the marginal rate of substitution of capital by labor under technology 
Then from the previous ticket follows:
When plotting graphically MRTS corresponds to the tangent of the slope of the tangent to the isoquant at the point indicating the necessary volumes of labor and capital to produce a given volume of output.
For a given technology, each value of the capital-labor ratio (a point on the isoquant) corresponds to its own ratio between the marginal productivity of production factors. In other words, one of the specific characteristics of technology is how much the ratio of the marginal productivity of capital and labor changes with a small change in the capital-labor ratio, that is, the amount of capital used. Graphically, this is shown by the degree of curvature of the isoquant. A quantitative measure of this property of technology is the elasticity of substitution of factors of production, which shows by how many percent the capital-labor ratio must change so that when the ratio of factor productivity changes by 1%, output remains unchanged. Let's denote ; then the elasticity of substitution of factors of production
atQ=
const
Here is the logarithmic representation. Pzdts)
Let us designate - the marginal rate of substitution of the -th factor -th factor, and - the ratio of the number of these factors used in production. Then the elasticity of substitution will be:
At the same time, it can be shown that
The only thing I could not find is the output of this “…”.
The curvature of an isoquant illustrates the elasticity of substitution of factors for a given volume of product and reflects how easily one factor can be replaced by another. In the case when the isoquant is similar to a right angle, the probability of substituting one factor for another is extremely small. If the isoquant has the form of a straight line with a downward slope, then the probability of replacing one factor with another is significant. (for more details see different kind functions in the fifth ticket)
Moreover, when the isoquant is continuous, it characterizes the flexibility of the technology. That is, the company has a huge number of production options.
For an excellent understanding of this shit, check out the 5th, everything is spelled out there.
Special types of production functions (linear, Leontief, Cobb-Douglas, CES): analytical, graphical and economic representation; the economic meaning of the coefficients; returns to scale; the elasticity of output with respect to factors of production; elasticity of substitution of factors of production.
Perfect interchangeability of resources or linear production function
If the resources used in the production process are absolutely replaceable, then it is constant at all points of the isoquant, and the isoquant map looks like in Figure 14.2. (An example of such a production is a production that allows both full automation and manual production any product).
Q=a*K+b*L, where K:L=b/a is the proportion of one resource being replaced by another (b-point of intersection Q1 of the OK axis, a-axis OL)
Constant returns to scale, elasticity of substitution of resources is infinite, MRTSlk=-b/a, elasticity of output for labor - in, for capital - a.
Fixed resource usage structure, also known as the Leonov function
If the technological process excludes the replacement of one factor by another and requires the use of both resources in strictly fixed proportions, the production function has the form of a Latin letter, as in Figure 14.3.
An example of this kind is the work of a digger (one shovel and one person). An increase in one of the factors without a corresponding change in the amount of the other factor is irrational, therefore only angular combinations of resources will be technically effective (the corner point is the point where the corresponding horizontal and vertical lines intersect).
Q=min(aK;bL); Constant returns to scale, K:L=b:a complement proportion, MRTSlk=0, elasticity of substitution 0, elasticity of output 0.
Cobb-Douglas function
A-characterizes the technology.
Elasticity of substitution of factors can be any, returns to scale (1-constant, less than one - decreasing, more than one - increasing), elasticity of output by factors of production for capital - alpha, for labor - beta, elasticity of substitution of factors 
FunctionCES
The CES function (CES - Eng. Constant Elastisity of Substitution) is a function used in economic theory that has the property of constant elasticity of substitution. Sometimes it is also used to model a utility function. This function is primarily used to model the production function. Some other popular production functions are special or limiting cases of this function.
Returns to scale depend on: greater than 1, increasing returns to scale, less than 1, decreasing returns to scale, equal to 1, constant returns to scale.
FOR THIS TICKET I COULD NOT FIND THE ELASTICITY OF THE RELEASE AT ALL NORMAL ANYWHERE
The concept of economic costs. Isocosts, their economic meaning.
Opportunity costs arise in a world of limited resources, and therefore all the desires of people cannot be satisfied. If resources were unlimited, then no action would be carried out at the expense of another, i.e., the opportunity cost of any action would be equal to zero. Obviously, in the real world of limited resources, the opportunity cost is positive.
Based on the concept opportunity cost, we can say that economic costs - these are the payments that the firm is obliged to make, or the income that the firm is obliged to provide to the supplier of resources in order to divert these resources from use in alternative industries.
These payments can be either external or internal.
External costs are payments for resources (raw materials, fuel, transport services- everything that the company does not produce itself to create any product) to suppliers who are not among the owners of this company.
In addition, the firm may use certain resources that belong to itself. The costs of own and self-used resource are unpaid, or internal, costs. From the point of view of the firm, these internal costs are equal to the cash payments that could be received for a self-used resource with the best - of possible ways- its application. Internal costs also include normal profit as the minimum remuneration of an entrepreneur, necessary for him to continue his business and not switch to another. Thus, the economic costs look like this:
Economic cost = External cost + Internal cost (including normal profit)
Isocost- a straight line showing all combinations of factors of production at a fixed amount of total costs.
A set of isoquants of an individual firm (isoquant map) show the technically possible combinations of resources that provide the firm with the appropriate output volumes.
When choosing the optimal combination of resources, the manufacturer must take into account not only the technology available to him, but also their financial resources , as well as prices of the relevant factors of production.
The combination of these two factors determines the area of economic resources available to the producer (its budget constraint).
B The producer's budget constraint can be written as an inequality:
P K *K+P L *L TC, where
P K , P L - the price of capital, the price of labor;
TC is the firm's total cost of acquiring resources.
If the manufacturer (firm) fully spends its funds on the acquisition of these resources, we get the following equality:
P K *K+P L *L=TC
On the graph, the isocost is determined in the axes L, K, therefore, for plotting, it is convenient to bring the equality into the following form:
isocost equation.
The slope of the isocost line is determined by the ratio of market prices for labor and capital: (- P L / P K)
K
L
