production called any human activity to transform limited resources - material, labor, natural - into finished products. production function characterizes the relationship between the amount of resources used (factors of production) and the maximum possible output that can be achieved, provided that all available resources are used in the most rational way.
The production function has the following properties:
1. There is a limit to the increase in production that can be reached by increasing one resource and keeping other resources constant. If, for example, in agriculture increase the amount of labor with constant amounts of capital and land, then sooner or later there comes a point when output stops growing.
2. Resources complement each other, but within certain limits, their interchangeability is also possible without reducing output. Manual labor, for example, can be replaced by the use of more machines, and vice versa.
3. The longer the time period, the more resources can be reviewed. In this regard, there are instant, short and long periods. Instant period - the period when all resources are fixed. short period- the period when at least one resource is fixed. A long period - period when all resources are variable.
As a rule, the considered production function looks like this:
A, α, β - given parameters. Parameter A is the coefficient of total factor productivity. It reflects the impact of technological progress on production: if the manufacturer introduces advanced technologies, the value A increases, i.e. output increases with the same amount of labor and capital. Parameters α and β are the elasticity coefficients of output, respectively, for capital and labor. In other words, they show the percentage change in output when capital (labor) changes by one percent. These coefficients are positive, but less than unity. The latter means that with the growth of labor with constant capital (or capital with constant labor) by one percent, production increases to a lesser extent.
isoquant(line of equal product) reflects all combinations of two factors of production (labor and capital), in which output remains unchanged. On fig. 8.1 next to the isoquant is the release corresponding to it. Thus, output , is achievable using labor and capital, or using labor and captain.
Rice. 8.1. isoquant
If you plot the number of units of labor on the horizontal axis and the number of units of capital on the vertical axis, then plot the points at which the firm produces the same amount, you get the curve shown in Figure 14.1 and is called the isoquant.
Each point of the isoquant corresponds to the combination of resources at which the firm produces a given volume of output.
The set of isoquants characterizing a given production function is called isoquant map.
Properties of isoquants
The properties of standard isoquants are similar to those of indifference curves:
1. An isoquant, like an indifference curve, is a continuous function, not a set of discrete points.
2. For any given volume of output, its own isoquant can be drawn, reflecting various combinations economic resources, which provide the producer with the same output (isoquants describing a given production function never intersect).
3. Isoquants do not have areas of increase (If the area of increase existed, then when moving along it, the amount of both the first and second resource would increase).
The concept of the market. In the very general view the market is a system of economic relations that develop in the process of production, circulation and distribution of goods, as well as the movement of funds. The market develops along with the development of commodity production, involving in the exchange not only manufactured products, but also products that are not the result of labor (land, wild forest). Under the dominance of market relations, all relations of people in society are covered by buying and selling.
More specifically, the market represents the sphere of exchange (circulation), in which
communication is carried out between the agents of social production in the form
purchase and sale, i.e., the connection of producers and consumers, production and
consumption.
The subjects of the market are sellers and buyers. As sellers
and buyers are households (consisting of one or more
individuals), firms (enterprises), the state. Most market participants
act as both buyers and sellers at the same time. All household
subjects closely interact in the market, forming an interconnected "flow"
purchase and sale.
Firm is an independent economic entity engaged in commercial and industrial activities and possessing separate property.
The firm has the following features:
- is an economically separate, independent economic unit;
- legally registered and relatively independent in this regard: it has its own budget, charter and business plan
- is a kind of intermediary in the production
- any company independently makes all decisions related to its functioning, so we can talk about its production and commercial independence
- The company's goals are profit making and cost minimization.
The firm as an independent economic entity performs a number of important functions.
1. production function implies the ability of the firm to organize production for the production of goods and services.
2. commercial function provides logistics, sales finished products as well as marketing and advertising.
3. Financial function: attracting investments and obtaining loans, settlements within the company and with partners, issuing securities, paying taxes.
4. Counting function: drawing up a business plan, balances and estimates, conducting an inventory and reporting to state statistics and taxes.
5. Administrative function- a management function, including organization, planning and control over activities in general.
6. legal function is carried out through compliance with laws, norms and standards, as well as through the implementation of measures to protect the factors of production.
You can not equate elasticity and the slope of the demand curve, because these are different concepts. The differences between them can be illustrated by the elasticity of the straight line of demand (Fig. 13.1).
On fig. 13.1 we see that the straight line of demand at each point has the same slope. However, above the middle, demand is elastic; below the middle, demand is inelastic. At the point in the middle, the elasticity of demand is equal to one.
The elasticity of demand can be judged by the slope of only a vertical or horizontal line.
Rice. 13.1. Elasticity and slope are different concepts
The slope of the demand curve - its flatness or steepness - depends on absolute changes in price and quantity of production, while the theory of elasticity deals with relative, or percentage, changes in price and quantity. The difference between the slope of the demand curve and its elasticity can also be fully understood by calculating the elasticity for various combinations of price and quantity of products located on a rectilinear demand curve. You will find that although the slope obviously remains the same throughout the curve, demand is elastic on the high price leg and inelastic on the low price leg.
INCOME ELASTICITY OF DEMAND - a measure of the sensitivity of demand to changes in income; reflects the relative change in demand for a good due to a change in consumer income.
The income elasticity of demand takes the following main forms:
positive, assuming that an increase in income (ceteris paribus) is accompanied by an increase in demand. The positive form of income elasticity of demand applies to normal goods, in particular, to luxury goods;
· negative, implying a decrease in the volume of demand with an increase in income, i.e., the existence of an inverse relationship between income and the volume of purchases. This form of elasticity extends to inferior goods;
zero, which means that the volume of demand is insensitive to changes in income. These are goods whose consumption is insensitive to income. These include, in particular, essential goods.
The income elasticity of demand depends on the following factors:
· on the significance of this or that good for the family budget. The more a good a family needs, the less its elasticity;
whether the given good is a luxury item or a necessity. For the first good, the elasticity is higher than for the last;
from the conservatism of demand. With an increase in income, the consumer does not immediately switch to the consumption of more expensive goods.
It should be noted that for consumers with different levels of income, the same goods can be either luxury goods or basic necessities. A similar assessment of goods can take place for the same individual when his level of income changes.
On fig. 15.1 shows graphs of QD versus I for different values income elasticity of demand.
Rice. 15.1. Income elasticity of demand: a) high-quality inelastic goods; b) qualitative elastic goods; c) low-quality goods
Let's do short comment to fig. 15.1.
Demand for inelastic goods increases with income growth only at low household incomes. Then, starting from a certain level I1, the demand for these goods begins to decline.
There is no demand for elastic goods (for example, luxury goods) up to a certain level I2, since households do not have the opportunity to purchase them, and then increases with income.
Demand for low-quality goods initially increases, but starting from the value of I3 decreases.
Similar information.
Introduction …………………………………………………………………………..3
Chapter I .4
1.1. Factors of production……………………………………………………….4
1.2. Production function and its economic content…………….9
1.3. Elasticity of factor substitution…………………………………………..13
1.4. Elasticity of the production function and returns to scale………16
1.5. Properties of the production function and the main characteristics of the production function………………………………………………………..19
Chapter II. Types of production functions………………………………..23
2.1. Definition of linearly homogeneous production functions……...23
2.2. Types of linear-homogeneous production functions………………..25
2.3. Other types of production functions………………………………...28
Appendix……………………………………………………………………..30
Conclusion……………………………………………………………………...32
List of used literature…………………………………………...34
Introduction
In conditions modern society no man can only consume what he himself produces. For the most complete satisfaction of their needs, people are forced to exchange what they produce. Without the constant production of goods, there would be no consumption. Therefore, it is of great interest to analyze the regularities that operate in the process of production of goods, which further form their supply on the market.
The production process is the basic and initial concept of the economy. What is meant by production?
Everyone knows that the production of goods and services from scratch is impossible. In order to produce furniture, food, clothing and other goods, it is necessary to have the appropriate source materials, equipment, premises, a piece of land, specialists who organize production. Everything necessary for the organization of the production process is called factors of production. Traditionally, the factors of production include capital, labor, land and entrepreneurship.
For organization production process the necessary factors of production must be present in a certain amount. The dependence of the maximum volume of the product produced on the costs of the factors used is called production function .
Chapter I . Production functions, basic concepts and definitions .
1.1. Factors of production
The material basis of any economy is formed from production. The economy of that country as a whole depends on the extent to which production is developed in a country.
In turn, the sources of any production are the resources that this or that society has at its disposal. "Resources - the availability of means of labor, objects of labor, money, goods or people for use now or in the future."
Thus, the factors of production are a combination of those natural, material, social and spiritual forces (resources) that can be used in the process of creating goods, services and other values. In other words, the factors of production are those that have a certain influence on the production itself.
V economic theory Resources are divided into three groups:
1. Labor - a set of physical and mental abilities of a person that can be used in the process of manufacturing a product or providing a service.
2. Capital (physical) - buildings, structures, machinery, equipment, vehicles required for production.
3. Natural resources- land and its subsoil, reservoirs, forests, etc. Everything that can be used in production in a natural, unprocessed form.
It is the presence or absence of factors of production in a country that determines its economic development. Factors of production, to some extent, are the potential for economic growth. How these factors are used depends on the overall state of affairs in the country's economy.
Later, the development of the theory of "three factors" led to a more extended definition of the factors of production. Currently these include:
2. land (natural resources);
3. capital;
4. entrepreneurial ability;
It should be noted that all these factors are closely interrelated. For example, labor productivity rises sharply when using the results of scientific and technological progress.
Thus, the factors of production are those factors that have a certain impact on the production process itself. So, for example, by increasing capital by acquiring new production equipment, you can increase production volumes and increase revenue from product sales.
It is necessary to consider in more detail the existing factors of production.
Labor is the purposeful activity of man, with the help of which he transforms nature and adapts it to satisfy his needs. In economic theory, labor as a factor of production refers to any mental and physical efforts made by people in the process of economic activity.
Speaking about labor, it is necessary to dwell on such concepts as labor productivity and labor intensity. The intensity of labor characterizes the intensity of labor, which is determined by the degree of expenditure of physical and mental energy per unit of time. The intensity of labor increases with the acceleration of the conveyor, an increase in the number of simultaneously serviced equipment, and a decrease in the loss of working time. Labor productivity shows how much output is produced per unit of time.
The progress of science and technology plays a decisive role in increasing labor productivity. For example, the introduction of conveyors at the beginning of the 20th century led to a sharp jump in labor productivity. The conveyor organization of production was based on the principle of fractional division of labor.
The scientific and technological revolution has led to changes in the nature of work. Labor has become more skilled, physical labor is of less and less importance in the production process.
Speaking of land as a factor of production, they mean not only the land itself, but also water, air and other natural resources.
Capital as a factor of production is identified with the means of production. Capital consists of durable goods created by the economic system for the production of other goods. Another view of capital is related to its monetary form. Capital, when embodied in finance not yet invested, is a sum of money. In all these definitions there is a common idea, namely, capital is characterized by the ability to generate income.
Distinguish between physical or fixed, working and human capital. Physical capital is the capital materialized in buildings, machines and equipment, which functions in the production process for several years. Another type of capital, including raw materials, materials, energy resources, is spent for one the production cycle. It is called working capital. The money spent on working capital is fully returned to the entrepreneur after the sale of products. Fixed capital costs cannot be recovered so quickly. Human capital arises as a consequence of education, training and maintenance of physical health.
Entrepreneurial ability is a special factor of production by means of which other factors of production are assembled into an effective combination.
Scientific and technological progress is an important engine of economic growth. It covers whole line phenomena characterizing the improvement of the production process. Scientific and technological progress includes the improvement of technology, new methods and forms of management and organization of production. Scientific and technological progress makes it possible to combine these resources in a new way in order to increase the final output. At the same time, as a rule, new, more efficient industries emerge. The growth of labor efficiency becomes the main factor of production.
But it must be understood that there is no direct relationship between the factors of production and the volume of output. For example, by hiring new employees, the company creates the prerequisites for the production of an additional volume of products. But at the same time, each attracted new employee increases labor costs for the enterprise. In addition, there is no guarantee that the additional products released will be in demand by the buyer, and that the company will receive income from the sale of these products.
Thus, speaking about the relationship between the factors of production and the volume of production, it is necessary to understand that this relationship is determined by a reasonable combination of these factors, taking into account the existing demand for manufactured products.
An important role in understanding the problem of combining factors of production is played by the so-called theory of marginal utility and marginal cost, the essence of which is that each additional unit of the same type of good brings less and less benefit to the consumer and requires an increase in costs from the producer. The modern theory of production is based on the concept of diminishing returns or marginal product and believes that all factors of production are interdependently involved in the creation of a product.
The main goal of any business is to maximize profits. One way to achieve this is through a judicious combination of factors of production. But who can determine what proportions of factors of production are acceptable for this or that enterprise, this or that industry? The question is how many and what factors of production should be used to obtain the maximum possible profit.
It is this problem that is one of the problems solved by mathematical economics, and the way to solve it is to identify the mathematical relationship between the factors of production used and the volume of output, that is, in constructing the production function.
1.2. Production function and its economic content
What is a function in terms of mathematical science?
A function is the dependence of one variable on another (other) variables, expressed as follows:
where X is an independent variable, and y- dependent on x function.
Changing a variable x leads to a change in function y .
The function of two variables is expressed by the dependence: z = f(x, y). Three variables: Q = f(x,y,z), and so on.
For example, the area of a circle: S ( r )=π r 2 - is a function of its radius, and the larger the radius, the larger the area of the circle.
We get that the production function is a mathematical relationship between the maximum output per unit of time and the combination of factors that create it, given the current level of knowledge and technology. At the same time, the main task of mathematical economics from a practical point of view is to identify this dependence, that is, to build a production function for a particular industry or a particular enterprise.
In production theory, a two-factor production function is mainly used, which is generally written as follows:
Q = f ( K , L ), (1.1)
At the same time, factors such as technological progress and entrepreneurial ability are considered unchanged in a relatively short period of time and do not affect the volume of output, and the factor "land" is considered together with "capital".
The production function determines the relationship between output Q and production factors: capital K, labor L. The production function describes a set of technically efficient ways of producing a given volume of output. The technical efficiency of production is characterized by the use of the least amount of resources for a given volume of production. For example, a mode of production is considered more efficient if it involves the use of at least one resource in less, and all the rest not in more than other methods. If one method involves the use of some resources in more and others in a smaller amount than the other method, then these methods are not comparable in terms of technical efficiency. In this case, both methods are considered as technically effective, and economic efficiency is used to compare them. The most cost-effective way to produce a given volume of output is the one in which the cost of using resources is minimal.
Graphically, each method can be represented by a point, the coordinates of which characterize the minimum amount of resources L and K, and the production function can be represented by a line of equal output, or an isoquant. Each isoquant represents a set of technically efficient ways to produce a certain amount of output. The further the isoquant is located from the origin, the more output it provides. Figure 1.1. three isoquants are given corresponding to the output of 100, 200 and 300 units, so we can say that for the output of 200 units it is necessary to take either K 1 units of capital and L 1 units of labor, or K 2 units of capital and L 2 units of labor, or some combination of them provided by the isoquant Q 2 =200.
Q 3 \u003d 300
Figure 1.1. Isoquants representing different levels of output
It is necessary to define such concepts as isoquant and isocost.
Isoquant - a curve representing all possible combinations of two costs that provide a given constant volume of production (in Figure 1.1. represented by a solid line).
Isocost - a line formed by a set of points showing how many combined factors of production or resources can be purchased with the available cash(in figure 1.1. is represented by a dotted line - a tangent to the isoquant at the point of combination of resources).
The touch point of isoquant and isocost is the optimal combination of factors for a particular enterprise. The touch point is found by solving a system of two equations expressing the isoquant and the isocost.
The main properties of the production function are:
1. The continuity of the function, that is, its graph is a solid, continuous line;
2. Production is not possible in the absence of at least one of the factors;
3. An increase in the costs of any of the factors with unchanged quantities of the other leads to an increase in output;
4. It is possible to keep output at a constant level by replacing some of one factor with the additional use of another. That is, a decrease in the use of labor can be compensated for by additional use of capital (for example, by purchasing new production equipment that is serviced by fewer workers).
1.3. Elasticity of factor substitution
Based on the foregoing, we can conclude that the main issue of the production function is the issue of the correct combination of factors of production, in which the level of output will be optimal, that is, bringing highest profit. In order to find the optimal combination, it is necessary to answer the question: By what amount should the costs of one factor be increased while reducing the costs of another per unit. The question of the ratio of costs of factors of production replacing each other is solved by introducing such a concept as
A measure of the interchangeability of factors of production is the marginal rate of technical substitution MRTS (marginal rate of technical substitution), which shows how many units one of the factors can be reduced by increasing the other factor by one, while keeping the output unchanged.
The marginal rate of technical substitution is characterized by the slope of isoquants. The steeper slope of the isoquant shows that as the amount of labor per unit increases, several units of capital will have to be given up to maintain a given level of production. MRTS is expressed by the formula:
MRTS L , K = –DK/DL
Isoquants can have different configurations.
The linear isoquant in Figure 1.2(a) assumes that inputs are perfectly substitutable, that is, a given output can be produced with either labor alone, capital alone, or a combination of these resources.
The isoquant presented in Figure 1.2(b) is typical for the case of strict resource complementarity. In this case, only one technically known effective method production. Such an isoquant is sometimes called a Leontief-type isoquant (see below), after the economist V.V. Leontiev, who proposed this type of isoquant. Figure 1.2(c) shows a broken isoquant, suggesting multiple production methods (P). In this case, the marginal rate of technical substitution decreases when moving along the isoquant from top to bottom. An isoquant of a similar configuration is used in linear programming, a method of economic analysis. The broken isoquant realistically represents the production possibilities of modern industries. Finally, Figure 1.2(d) presents an isoquant, suggesting the possibility of continuous but not perfect substitution of resources.
K a) KQ 2 b)
Figure 1.2. Possible configurations of isoquants.
1.4. Elasticity of the production function and returns to scale.
The marginal product of a resource characterizes the absolute change in the output of the product per unit change in the consumption of this resource, and the changes are assumed to be small. For production function 
the marginal product of the i-th resource is equal to the partial derivative: .
The influence of a relative change in the consumption of the i-th factor on the output of a product, also presented in relative form, is characterized by the partial elasticity of output with respect to the costs of this product:
For simplicity, we will denote . The partial elasticity of the production function is equal to the ratio of the marginal product of a given resource to its average product.
Let us consider a special case when the elasticity of the production function with respect to some argument is a constant value.
If in relation to the initial values of the arguments x 1 , x 2 ,…,x n one of the arguments (i-th) changes once, and the rest remain at the same levels, then the change in the output of the product is described by a power function: . Assuming I=1, we find that A=f(x 1 ,…,x n), and therefore .
In the general case, when elasticity is a variable value, equality (1) is approximate for values of I close to unity, i.e. for I=1+e, and the more accurate the closer e/to zero.
Let now the costs of all resources have changed by I times. Consistently applying the technique just described to x 1 , x 2 ,…,x n , we can see that now
The sum of partial elasticities of a certain function over all its arguments is called the total elasticity of the function. Introducing the notation for the full elasticity of the production function, we can represent the result obtained in the form
Equality (2) shows that the full elasticity of the production function allows us to give returns to scale numeric expression. Let the consumption of all resources slightly increase while maintaining all proportions (I>1). If E>1, then output has increased more than I times (increasing returns to scale), and if E<1, то меньше, чем в I раз. При E=1 выпуск продукции изменится в той же самой пропорции, что и затраты всех ресурсов (постоянная отдача).
The allocation of short and long periods in describing the characteristics of production is a rough schematization. Changing the consumption of various resources - energy, materials, labor, machines, buildings, etc. - requires different times. Assume that the resources are renumbered in descending order of mobility: x 1 is the fastest to change, then x 2 , and so on, and x n is the most time consuming to change. It is possible to single out an ultra-short, or zero period, when not a single factor can change; 1st period, when only x 1 changes; 2nd period, allowing change x 1 and x 2, etc.; finally, a long, or n-th period, during which the volumes of all resources can change. There are thus n+1 different periods.
Considering some intermediate in value, k-th period, we can speak about the corresponding to this period returns to scale, meaning the proportional change in the volumes of those resources that can change in this period, i.e. x 1 , x 2 ,…, x k . Volumes x k +1 , x n , thus keep fixed values. The corresponding return to scale is e 1 +e 2 +…+e k .
Extending the period, we add the following terms to this sum until we get the value of E for the long period.
Since the production function is increasing with each argument, all partial elasticities e 1 are positive. It follows that the longer the period, the greater the returns to scale.
1.5. Production function properties
For each type of production, its own production function can be built, however, each of them will have the following fundamental properties:
1. There is a limit to the growth of production, which is achieved by increasing the use of one resource, other things being equal. An example is the impossibility of increasing the volume of production (when a specific value is reached) at a certain enterprise by attracting new employees with given fixed assets. It is possible to reach a point where each individual worker will not be provided with the means of labor for work, a workplace, his presence will be a hindrance to other employees, and the increase in production from hiring this marginal worker will approach zero or even become negative.
2. 
There is a certain mutual complementarity (complementarity) of factors of production, but without reducing the volume of production, a certain mutual substitution is also possible. For example, to obtain a given crop, a certain amount of sown area can be cultivated by a large number of workers manually, without the use of fertilizers and modern means of production. In the same area, several workers can work to produce the required amount of crops, using complex machines and various fertilizers. It should be noted that under the condition of complementarity, none of the traditional resources (land, labor, capital) can be completely replaced by others (there will be no complementarity). The mechanism of mutual substitution operates on the opposite premise: a certain type of resource can be replaced by another. Mutual complementarity and mutual substitution have the opposite direction. If complementarity requires the mandatory presence of all resources, then substitution in its extreme form can lead to the complete exclusion of some of them.
The analysis of the production function suggests the need to distinguish between short-term and long-term periods of time. In the first case, we mean such a time interval during which the volume of production can be regulated only by changing the number of variable factors used, while fixed costs remain unchanged. Factors of production whose costs remain unchanged in the short run are called fixed.
Accordingly, the factors of production, the size of which changes in the short run - variables. The long-term period of time is considered as an interval that is sufficient for the enterprise to change the costs of all factors of production. This means that in this case there are no limits to the growth of output and all factors become variable. In the most general form, the differences between short-term and long-term intervals can be reduced to the following.
First, it concerns the conditions of management. In the short term, a significant expansion of production is impossible, limited by the existing production capacity of the company. In the long run, the firm has more freedom to increase output because all factors of production become variable.
Secondly, it is necessary to take into account the specifics of production costs. The short run is characterized by the presence of both fixed and variable production costs, in the long run all costs become fixed.
Third, the short run implies the persistence of firms in the industry. In the long run, there is a real opportunity for new competitors to enter or enter the industry.
Fourth, it is necessary to determine the possibilities for extracting economic profit in the periods under review. In the long run, economic profit is zero. In the short run, economic profit can be either positive or negative.
The PF satisfies the following set of properties:
1) there is no output without resources, i.e. f(0,0,a)=0;
2) in the absence of at least one of the resources, there is no output, i.e. ;
3) with an increase in the cost of at least one resource, the volume of output increases;
4) with an increase in the cost of one resource with a constant amount of another resource, the volume of output increases, i.e. if x>0 then 
;
5) with an increase in the costs of one resource with the same amount of another resource, the value of the increase in output for each additional unit of the i-th resource does not increase (the law of diminishing efficiency), i.e. if then ;
6) with the growth of one resource, the marginal efficiency of another resource increases, i.e. if x>0 then 
;
7) PF is a homogeneous function, i.e. ; at p>1 we have an increase in production efficiency due to the increase in the scale of production; at p<1 имеем падение эффективности производства от роста масштаба производства; при р=1 имеем постоянную эффективность производства при росте его масштаба.
Chapter II . Types of production functions
2.1. The definition is linear - homogeneous production functions
A production function is said to be of homogeneous degree n if, when resources are multiplied by a certain number k, the resulting output will be kn times different from the original. The conditions for the homogeneity of the production function are written as follows:
Q = f (kL, kK) = knQ
For example, 9 hours of labor (L) and 9 hours of machine work (K) are expended per day. Let, with a given combination of factors L and K, the firm can produce products worth 200 thousand rubles per day. In this case, the production function Q = F(L,K) will be represented by the following equality:
Q = F(9; 9) = 200,000, where F is a certain kind of algebraic formula into which the values of L and T are substituted.
Suppose a company decides to double the work of capital and the use of labor, which leads to an increase in the volume of output up to 600 thousand rubles. We get that multiplying the factors of production by 2 leads to an increase in the volume of production by 3 times, that is, using the conditions for the homogeneity of the production function:
Q = f (kL, kK) = knQ, we get:
Q \u003d f (2L, 2K) \u003d 2 × 1.5 × Q, that is, in this case we are dealing with a homogeneous production function of degree 1.5.
The exponent n is called the degree of homogeneity.
If n = 1, then the function is said to be homogeneous of the first degree or linearly homogeneous. A linearly homogeneous production function is of interest because it is characterized by a constant return, that is, with an increase in production factors, the volume of output constantly increases in the same way.
If n>1, then the production function demonstrates increasing returns, that is, the growth of factors of production leads to an even greater increase in the volume of production (for example: doubling the factors leads to an increase in volume by 2 times; 3 times - to an increase of 6 times ; 4 times - to an increase of 12 times, etc.) If n<1, то производственная функция демонстрирует убывающую отдачу, то есть, рост факторов производства ведёт к уменьшению отдачи по росту объёмов производства (например: увеличение факторов в 2 раза – ведёт к увеличению объемов в 2 раза; увеличение факторов в 3 раза – к увеличению объёмов в 1,5 раз; увеличение факторов в 4 раза – к увеличению объёмов в 1,2 раза и т.д.).
2.2. Types of linearly homogeneous production functions
Examples of linearly homogeneous production functions are the Cobb-Douglas production function and the constant elasticity of substitution production function.
The production function was first calculated in the 1920s for the US manufacturing industry by economists Cobb and Douglas. Paul Douglas' research in the manufacturing industry in the United States and their subsequent processing by Charles Cobb led to the emergence of a mathematical expression describing the impact of the use of labor and capital on the production of products in the manufacturing industry, in the form of an equation:
Ln(Q) = Ln(1.01) + 0.73×Ln(L) + 0.27×Ln(K)
In general, the Cobb-Douglas production function has the form:
Q = AK α L β ν
lnQ = lnA + α lnK + βlnL + lnv
If α + β<1, то наблюдается убывающая отдача от масштабов использования факторов производства (рис. 1.2.в). Если α+β=1, то существует постоянная отдача от масштабов использования факторов производства (рис. 1.2.а). Если α+β>1, then there is an increasing return on the scale of the use of production factors (Fig. 1.2.b).
In the Cobb-Douglas production function, the power coefficients α and β sum up to express the degree of homogeneity of the production function:
The marginal rate of technical replacement of capital by labor in this technology is determined by the formula:
׀MRTS L , K ׀ =
If we carefully look at the Cobb-Douglas function for the US manufacturing industry, calculated in the 1920s, we can once again, using a specific example, note that the production function is a mathematical expression (through a certain algebraic form) of the dependence of production volumes (Q) on volumes of use of factors of production (L and K). Thus, by assigning specific values to the variables L and K, one can determine the expected output (Q) for the US manufacturing industry in the 1920s.
The elasticity of substitution in the Cobb-Douglas production function is always 1.
But the Cobb-Douglas production function had some drawbacks. To overcome the limitation of the Cobb-Douglas function, which is always homogeneous to the first degree, in 1961 several economists (K. Arrow, H. Chenery, B. Minhas and R. Solow) proposed a production function with constant elasticity of substitution. It is a linearly homogeneous production function with a constant elasticity of resource substitution. Later, a production function with a variable elasticity of substitution was also proposed. It is a generalization of a production function with a constant elasticity of substitution that allows the elasticity of substitution to change with the ratio of inputs.
A linearly homogeneous production function with constant elasticity of resource substitution has the following form:
Q \u003d a -1 / b,
The elasticity of factor substitution for a given production function is given by:
2.3. Other types of production functions
Another kind of production function is the linear production function, which has the following form:
Q(L,K) = aL + bK
This production function is homogeneous of the first degree, therefore, it has constant returns to scale. Graphically, this function is shown in Figure 1.2, a.
The economic meaning of a linear production function is that it describes a production in which factors are interchangeable, that is, it does not matter whether only labor or only capital is used. But in real life, such a situation is practically impossible, since any machine is still serviced by a person.
The coefficients a and b of the function, which are in the variables L and K, show the proportions in which one factor can be replaced by another. For example, if a=b=1, then this means that 1 hour of labor can be replaced by 1 hour of machine time in order to produce the same amount of output.
Q(L,K) = min(; ),
If, for example, each long-distance bus must have two drivers, then if there are 50 buses and 90 drivers in the bus fleet, only 45 routes can be served at the same time:
min(90/2;50/1) = 45.
Appendix
Examples of problem solving using production functions
Task 1
A river transportation firm uses carrier labor (L) and ferries (K). The production function has the form 
. The price of a unit of capital is 20, the price of a unit of labor is 20. What will be the slope of the isocost? How much labor and capital must the firm attract to make 100 shipments?
3. capital;
4. entrepreneurial ability;
5. scientific and technological progress.
All these factors are closely interrelated.
The production function is a mathematical relationship between the maximum output per unit of time and the combination of factors that create it, given the current level of knowledge and technology. At the same time, the main task of mathematical economics from a practical point of view is to identify this dependence, that is, to build a production function for a particular industry or a particular enterprise.
In production theory, they mainly use a two-factor production function, which in general looks like this:
Q = f ( K , L ), where Q is the volume of production; K - capital; L - labor.
The question of the ratio of costs of factors of production replacing each other is solved using such a concept as elasticity of substitution of factors of production.
The elasticity of substitution is the ratio of the costs of substituting factors of production at a constant output. This is a kind of coefficient that shows the degree of efficiency in replacing one factor of production with another.
A measure of the interchangeability of factors of production is the marginal rate of technical substitution MRTS, which shows how many units one of the factors can be reduced by increasing the other factor by one, keeping output unchanged.
An isoquant is a curve representing all possible combinations of two costs that provide a given constant output.
Funding is usually limited. A line formed by a set of points showing how many combined factors of production or resources can be purchased with available money is called an isocost. Thus, the optimal combination of factors for a particular enterprise is the general solution of the isocost and isoquant equations. Graphically, this is the point of contact of the isocost and isoquant lines.
The production function can be written in a variety of algebraic forms. As a rule, economists work with linearly homogeneous production functions.
The paper also considered specific examples of solving problems using production functions, which made it possible to conclude that they are of great practical importance in the economic activity of any enterprise.
Bibliography
1. Dougherty K. Introduction to econometrics. - M.: Finance and statistics, 2001.
2. Zamkov O.O., Tolstopyatenko A.V., Cheremnykh Yu.P. Mathematical Methods in Economics: Textbook. – M.: Ed. "DIS", 1997.
3. Course of economic theory: textbook. - Kirov: ASA, 1999.
4. Microeconomics. Ed. Prof. Yakovleva E.B. – M.: SPb. Search, 2002.
5. Salmanov O. Mathematical Economics. – M.: BHV, 2003.
6. Churakov E.P. Mathematical methods for processing experimental data in economics. - M.: Finance and statistics, 2004.
7. Shelobaev S.I. Mathematical methods and models in economics, finance, business. – M.: Unity-Dana, 2000.
Big Commercial Dictionary./Edited by Ryabova T.F. - M .: War and Peace, 1996. S. 241.
production called any human activity to transform limited resources - material, labor, natural - into finished products. The production function characterizes the relationship between the amount of resources used (factors of production) and the maximum possible output that can be achieved, provided that all available resources are used in the most rational way.
The production function has the following properties:
1 There is a limit to the increase in production that can be reached by increasing one resource and keeping other resources constant. If, for example, the amount of labor in agriculture is increased with constant amounts of capital and land, then sooner or later a moment comes when output stops growing.
2 Resources complement each other, but within certain limits, their interchangeability without reducing output is also possible. Manual labor, for example, may be replaced by the use of more machines, and vice versa.
Manufacturing cannot create products out of nothing. The production process is associated with the consumption of various resources. The number of resources includes everything that is necessary for production activities - and raw materials, and energy, and labor, and equipment, and space.
In order to describe the behavior of a firm, it is necessary to know how much of a product it can produce using resources in various volumes. We will proceed from the assumption that the company produces a homogeneous product, the amount of which is measured in natural units - tons, pieces, meters, etc. The dependence of the amount of product that a company can produce on the volume of resource costs is called production function.
But an enterprise can carry out the production process in different ways, using different technological methods, different options for organizing production, so that the amount of product obtained with the same resource costs can be different. Firm managers should reject production options that give a lower yield of the product if, for the same input of each type of resource, a larger yield can be obtained. Similarly, they must reject options that require more input of at least one resource without increasing the yield of the product and reducing the cost of other resources. Options rejected for these reasons are called technically inefficient.
Let's say your company manufactures refrigerators. For the manufacture of the case, you need to cut sheet metal. Depending on how the standard sheet of iron is marked and cut, more or less parts can be cut out of it; accordingly, for the manufacture of a certain number of refrigerators, less or more standard sheets of iron will be required. At the same time, the consumption of all other materials, labor, equipment, electricity will remain unchanged. Such a production option, which can be improved by more rational cutting of iron, should be recognized as technically inefficient and rejected.
technically efficient are called production options that cannot be improved either by increasing the production of a product without increasing the consumption of resources, or by reducing the costs of a resource without reducing output and without increasing the costs of other resources. The production function takes into account only technically efficient options. Its meaning is greatest the quantity of a product that an enterprise can produce given the volume of resource consumption.
Consider first the simplest case: an enterprise produces a single type of product and consumes a single type of resource. An example of such production is quite difficult to find in reality. Even if we consider an enterprise providing services at customers' homes without the use of any equipment and materials (massage, tutoring) and spending only the labor of workers, we would have to assume that workers go around customers on foot (without using transport services) and negotiate with customers without the help of mail and telephone.
production function- shows the dependence of the amount of product that the company can produce on the volume of costs of the factors used
Q= f(x1, x2…xn)
Q= f(K, L),
where Q- output volume
x1, x2…xn– volumes of applied factors
K- volume of the capital factor
L- volume of labor factor
So, the enterprise, spending a resource in the amount X, can produce a product in quantity q. production function
ALL-RUSSIAN CORRESPONDENCE FINANCIAL AND ECONOMIC INSTITUTE
DEPARTMENT OF ECONOMIC AND MATHEMATICAL METHODS AND MODELS
econometrics
Production functions
( Materials for the lecture)
Prepared by Associate Professor of the Department
Filonova E.S. (branch in Orel)
The text of the lecture on the topic "Production functions"
in the discipline "Econometrics"
Plan:
Introduction
The concept of a production function of one variable
Production functions of several variables
Properties and main characteristics of production functions
Examples of using production functions in problems of economic analysis, forecasting and planning
Main conclusions
Learning control tests
Literature
Introduction
In the conditions of modern society, no person can consume only what he himself produces. For the most complete satisfaction of their needs, people are forced to exchange what they produce. Without the constant production of goods, there would be no consumption. Therefore, it is of great interest to analyze the regularities that operate in the process of production of goods, which further form their supply on the market.
The production process is the basic and initial concept of the economy. What is meant by production?
Everyone knows that the production of goods and services from scratch is impossible. In order to produce furniture, food, clothes and other goods, it is necessary to have the appropriate raw materials, equipment, premises, a piece of land, specialists who organize production. Everything necessary for the organization of the production process is called factors of production. Traditionally, the factors of production include capital, labor, land and entrepreneurship.
For the organization of the production process, the necessary factors of production must be present in a certain amount. The dependence of the maximum volume of the product produced on the costs of the factors used is called production function.
The concept of a production function of one variable
Consideration of the concept of "production function" will begin with the simplest case, when production is due to only one factor. In this case Pproduction function - this is a function, the independent variable of which takes the values of the resource used (factor of production), and the dependent variable - the values of the volume of output
In this formula, y is a function of one variable x. In this regard, the production function (PF) is called one-resource or one-factor. Its domain of definition is the set of non-negative real numbers. The symbol f is a characteristic of the production system that converts a resource into an output. In microeconomic theory, it is generally accepted that y is the maximum possible output if the resource is spent or used in the amount of x units. In macroeconomics, this understanding is not entirely correct: it is possible that with a different distribution of resources between the structural units of the economy, the output could be larger. In this case, the PF is a statistically stable relationship between resource input and output. More correct is the symbolism
where a is the vector of PF parameters.
Example 1. Take the PF f in the form f(x)=ax b , where x is the value of the resource expended (for example, working time), f(x) is the volume of output (for example, the number of refrigerators ready for shipment). Quantities a and b are parameters of the PF f. Here a and b are positive numbers and the number b1, the parameter vector is a two-dimensional vector (a,b). PF y=ax b is a typical representative of a wide class of one-factor PF.
The PF graph is shown in Figure 1
The graph shows that with the increase in the value of the resource expended, y grows. however, at the same time, each additional unit of the resource gives an ever smaller increase in the volume y of output. The noted circumstance (an increase in the volume of y and a decrease in the increase in the volume of y with an increase in the value of x) reflects the fundamental position of economic theory (well confirmed by practice), called the law of diminishing efficiency (diminishing productivity or diminishing returns).
As a simple example, let's take a one-factor production function that characterizes the production of an agricultural product by a farmer. Let all factors of production, such as the amount of land, the farmer's possession of agricultural machinery, seed, the amount of labor invested in the production of a product, remain constant from year to year. Only one factor changes - the amount of fertilizer applied. Depending on this, the value of the resulting product changes. At first, with the growth of the variable factor, it increases quite quickly, then the growth of the total product slows down, and starting from certain volumes of applied fertilizers, the value of the resulting product begins to decrease. A further increase in the variable factor does not increase the product.
PFs can have different areas of use. The input-output principle can be implemented both at the micro- and macroeconomic levels. Let's focus on the microeconomic level first. PF y=ax b , discussed above, can be used to describe the relationship between the value of the spent or used resource x during the year at a separate enterprise (firm) and the annual output of this enterprise (firm). The role of the production system here is played by a separate enterprise (firm) - we have a microeconomic PF (MIPF). At the microeconomic level, an industry, an intersectoral production complex, can also act as a production system. MIPF are built and used mainly for solving problems of analysis and planning, as well as forecasting problems.
The PF can be used to describe the relationship between the annual labor input of a region or country as a whole and the annual final output (or income) of that region or country as a whole. Here, the region or the country as a whole acts as a production system - we have a macroeconomic level and a macroeconomic PF (MAPF). MAFFs are built and actively used to solve all three types of problems (analysis, planning and forecasting).
The exact interpretation of the concepts of the spent or used resource and output, as well as the choice of units for their measurement, depend on the nature and scale of the production system, the characteristics of the tasks being solved, and the availability of initial data. At the microeconomic level, inputs and output can be measured both in natural and in cost units (indicators). Annual labor costs can be measured in man-hours or in rubles of paid wages; output can be presented in pieces or in other natural units or in the form of its value.
At the macroeconomic level, inputs and output are measured, as a rule, in terms of cost and represent cost aggregates, that is, the total values of the products of the volumes of resources expended and products produced by their prices.
Production functions of several variables
We now turn to the consideration of production functions of several variables.
Production function of several variables is a function whose independent variables take the values of the volumes of resources spent or used (the number of variables n is equal to the number of resources), and the value of the function has the meaning of the values of output volumes:
y=f(x)=f(x 1 ,…,х n). (2)
In formula (2) y (y 
0) is a scalar, and x is a vector quantity, x 1 ,…,х n are the coordinates of the vector x, that is, f(x 1 ,…,х n) is a numerical function of several variables x 1 ,…,х n . In this regard, the PF f(x 1 ,…,х n) is called multi-resource or multi-factorial. More correct is the following symbolism f(x 1 ,…,х n ,a), where a is the vector of PF parameters.
According to the economic sense, all variables of this function are non-negative, therefore, the domain of definition of the multifactorial PF is the set of n-dimensional vectors x, all coordinates x 1 ,…, x n of which are non-negative numbers.
For a separate enterprise (firm) producing a homogeneous product, the PF f(x 1 ,…,х n) can connect the volume of output with the cost of working time for various types of labor activity, various types of raw materials, components, energy, fixed capital. PF of this type characterize the current technology of the enterprise (firm).
When constructing the PF for a region or country as a whole, the aggregate product (income) of the region or country, usually calculated at constant rather than current prices, is often taken as the value of annual output Y, fixed capital (x 1 (= K) - the volume of fixed capital used during the year) and live labor (x 2 (= L) - the number of units of living labor expended during the year), usually calculated in value terms. Thus, a two-factor PF Y=f(K,L) is built. From two-factor PF are moving to three-factor. In addition, if the PF is constructed from time series data, then technological progress can be included as a special factor in production growth.
PF y=f(x 1 ,x 2) is called static, if its parameters and its characteristic f do not depend on time t, although the volume of resources and the volume of output may depend on time t, that is, they can be represented in the form of time series: x 1 (0), x 1 (1),…, x 1 (T); x 2 (0), x 2 (1), ..., x 2 (T); y(0), y(1),…,y(T); y(t)=f(x 1 (t), x 2 (t)). Here t is the number of the year, t=0.1,…,Т; t= 0 is the base year of the time interval covering years 1,2,…,T.
Example 2 To model a particular region or country as a whole (that is, to solve problems at the macroeconomic, as well as at the microeconomic level), a PF of the form y = 
, where а 0 , а 1 , а 2 are PF parameters. These are positive constants (often a 1 and a 2 are such that a 1 + a 2 =1). The PF of the form just given is called the Cobb-Douglas PF (CPKD) after the two American economists who proposed its use in 1929.
PPCD is actively used to solve various theoretical and applied problems due to its structural simplicity. PFKD belongs to the class of so-called multiplicative PFs (MPFs). In applications, PFKD x 1 \u003d K is equal to the volume of fixed capital used (the volume of fixed assets used - in domestic terminology), 
- the cost of living labor, then the PFKD takes on the form often used in the literature:
Y= 
.
History reference
In 1927, Paul Douglas, an economist by training, discovered that if we plot the logarithms of real output over time (Y), capital investments (K) and labor costs (L), then the distances from the points of the graph of output indicators to the points of the graphs of indicators of labor and capital costs will be a constant proportion. He then turned to the mathematician Charles Cobb to find a mathematical relationship that has this feature, and Cobb proposed the following function:
This function was proposed about 30 years earlier by Philip Wicksteed, as pointed out by C. Cobb and P. Douglas in their classic work (1929), but they were the first to use empirical data to build it. The authors do not describe how they actually fitted the function, but presumably they used a form of regression analysis as they referred to "least squares".
Example 3 Linear PF (LPF) has the form: 
(two-factor) and (multi-factor). PSF belongs to the class of so-called additive PF (APF). The transition from the multiplicative PF to the additive one is carried out using the logarithm operation. For a two-factor multiplicative PF
this transition looks like: . Introducing the appropriate substitution, we obtain the additive PF .
If the sum of the exponents in the Cobb-Douglas PF is equal to one, then it can be written in a slightly different form:
those. 
.
Fractions 
are called the productivity of labor and the capital-labor ratio, respectively. Using new symbols, we get
,
those. from the two-factor PKD we obtain formally one-factor PKD. Due to the fact that 0 1
Note that the fraction 
called productivity of capital or return on capital, reciprocal fractions 
are called, respectively, capital intensity and labor intensity of output.
PF is called dynamic, if:
time t appears as an independent variable (as if an independent factor of production), affecting the volume of output;
the parameters of the PF and its characteristic f depend on the time t.
Note that if the PF parameters were estimated from time series data (volumes of resources and output) with a duration 
years, then extrapolation calculations for such a PF should be carried out no more than 1/3 years ahead.
When constructing the PF, scientific and technological progress (STP) can be taken into account by introducing the STP multiplier , where the parameter p (p>0) characterizes the rate of output growth under the influence of STP:
(t=0.1,…,T).
This PF is the simplest example of a dynamic PF; it includes neutral, that is, technical progress that is not materialized in one of the factors. In more complex cases, technical progress can directly affect labor productivity or return on capital: Y(t)=f(A(t)×L(t),K(t)) or Y(t)=f(A(t)× K(t), L(t)). It is called, respectively, labor-saving or capital-saving NTP.
Example 4 Here is a variant of the PFKD taking into account the NTP
The calculation of the numerical values of the parameters of such a function is carried out using correlation and regression analysis.
Choosing the Analytical Form of the PF 
is dictated primarily by theoretical considerations, which should take into account the peculiarities of the relationship between specific resources or economic patterns. The PF parameters are usually estimated using the least squares method.
Properties and main characteristics of production functions
For the production of a particular product, a combination of various factors is required. Despite this, various production functions share a number of common properties.
For definiteness, we restrict ourselves to the production functions of two variables 
. First of all, it should be noted that such a production function is defined in a non-negative orthant of the two-dimensional plane, that is, at. The PF satisfies the following set of properties:
Like the level line of the objective function of the optimization problem, there is also a similar concept for the PF. PF level line is the set of points at which the PF takes a constant value. Sometimes the level lines are called isoquants PF. An increase in one factor and a decrease in another can occur in such a way that the total volume of production remains at the same level. Isoquants just determine all possible combinations of factors of production necessary to achieve a given level of production.
Figure 2 shows that output is constant along the isoquant, that is, there is no increase in output. Mathematically, this means that the total differential of the PF on the isoquant is equal to zero:
.
Isoquants have the following properties:
Isoquants do not intersect.
The greater distance of the isoquant from the origin corresponds to a greater level of output.
Isoquants are descending curves with a negative slope.
Isoquants are similar to indifference curves with the only difference that they reflect the situation not in the sphere of consumption, but in the sphere of production.
The negative slope of the isoquants is explained by the fact that an increase in the use of one factor at a certain volume of output of the product will always be accompanied by a decrease in the amount of another factor. The slope of the isoquant is characterized by marginal rate of technological substitution of factors of production (MRTS) . Consider this value using the example of a two-factor production function Q(y,x). The marginal rate of technological substitution is measured by the ratio of the change in factor y to the change in factor x. Since the replacement of factors occurs in the opposite direction, the mathematical expression for the MRTS indicator is taken with a minus sign:
Figure 3 shows one of the PF isoquants Q(y,x)
If we take any point on this isoquant, for example, point A and draw a tangent KM to it, then the tangent of the angle will give us the value of MRTS:
.
It can be noted that in the upper part of the isoquant, the angle will be quite large, which indicates that significant changes in the factor y are required to change the x factor by one. Therefore, in this part of the curve, the MRTS value will be large. As you move down the isoquant, the value of the marginal rate of technological substitution will gradually decrease. This means that to increase the factor x by one, a slight decrease in the factor y is required. With complete substitution of factors, isoquants from curves are transformed into straight lines.
One of the most interesting examples of the use of PF isoquants is the study economies of scale (see property 7).
What is more effective for the economy: one large plant or several small enterprises? The answer to this question is not so simple. The planned economy answered it unambiguously, giving priority to industrial giants. With the transition to a market economy, the widespread disaggregation of previously created associations began. Where golden mean? An evidence-based answer to this question can be obtained by examining the effect of scale in production.
Imagine that at a shoe factory, the management decided to direct a significant part of the profits received to the development of production in order to increase the volume of production. Let us assume that the capital (equipment, machines, production areas) is doubled. The number of employees increased in the same proportion. The question arises, what will happen in this case with the volume of output?
From the analysis of Figure 5
three answers follow:
The number of products will double (constant returns to scale);
Will more than double (increasing returns to scale);
Will increase, but less than twice (decreasing returns to scale).
Constant returns to scale are explained by the homogeneity of variable factors. With a proportional increase in capital and labor in such production, the average and marginal productivity of these factors will remain unchanged. In this case, it makes no difference whether one large enterprise will operate or two small ones will be created instead.
With decreasing returns to scale, it is unprofitable to create large-scale production. The reason for the low efficiency in this case, as a rule, is the additional costs associated with the management of such production, the difficulty of coordinating large-scale production.
Increasing returns to scale, as a rule, are characteristic of those industries where extensive automation of production processes is possible, the use of production and conveyor lines. But with the trend of increasing returns to scale, one must be very careful. Sooner or later, it turns into a constant, and then into diminishing returns to scale.
Let us dwell on some characteristics of production functions that are most important for economic analysis. Let us consider them using the example of a PF of the form 
.
As noted above, the ratio 
(i=1,2) is called the average productivity of the i-th resource or the average output of the i-th resource. First partial derivative of PF 
(i=1,2) is called the marginal productivity of the i-th resource or the marginal output of the i-th resource. This limit value is sometimes interpreted using the ratio of small finite values close to it 
. Approximately, it shows by how many units the volume of output y will increase if the volume of costs of the i-th resource increases by one (sufficiently small) unit with the volumes of the other resource being spent unchanged.
For example, in the PFKD for the average productivity of fixed capital y / K and labor y / L, the terms capital return and labor productivity are used, respectively:
Let us define the marginal productivity of factors for this function:
and 
.
Thus, if 
, then 
(i=1,2), that is, the marginal productivity of the i-th resource is not greater than the average productivity of this resource. Marginal productivity ratio 
i-th factor to its average performance 
is called the elasticity of output with respect to the i-th factor of production
or approximately
Thus, the elasticity of output (volume of production) with respect to some factor (elasticity coefficient) is approximately determined as the ratio of the growth rate y to the growth rate of this factor, that is 
shows by how many percent the output y will increase if the costs of the i-th resource increase by one percent with the volumes of the other resource unchanged.
Sum 
+
=E called the elasticity of production. For example, for PFCD = 
, and E=.
Examples of using production functions in problems of economic analysis, forecasting and planning
Production functions allow us to quantitatively analyze the most important economic dependencies in the sphere of production. They make it possible to estimate the average and marginal efficiency of various production resources, the elasticity of output for various resources, the marginal rates of substitution of resources, the effect of production scale, and much more.
Example 1 Assume that the production process is described by the output function
.
Let's evaluate the main characteristics of this function for a production method in which K=400 and L=200.
Solution.
Marginal productivity factors.
To calculate these values, we determine the partial derivatives of the function with respect to each of the factors:
Thus, the marginal productivity of the labor factor is four times higher than that of the capital factor.
production elasticity.
The elasticity of production is determined by the sum of the elasticities of output for each factor, i.e.
The marginal rate of substitution of resources.
Above in the text, this value was denoted 
and was equal to 
. Thus, in our example
that is, four units of capital resources are needed to replace a unit of labor at that point.
The isoquant equation.
To determine the shape of the isoquant, it is necessary to fix the value of the output (Y). Let, for example, Y=500. For convenience, we take L as a function of K, then the isoquant equation takes the form
The marginal rate of substitution of resources determines the tangent of the slope of the tangent to the isoquant at the corresponding point. Using the results of item 3, we can say that the tangency point is located in the upper part of the isoquane, since the angle is large enough.
Example 2 Consider the Cobb-Douglas function in general form
.
Suppose K and L are doubled. In this way, new level release (Y) will be written as follows:
Let us determine the effect of scale of production in cases where 
>1, =1 and
If, for example, =1,2, and 
=2.3, then Y increases more than twice; if =1, a =2, then doubling K and L leads to doubling Y; if \u003d 0.8, and \u003d 1.74, then Y increases by less than two times.
Thus, in example 1, there could be a constant effect on the scale of production.
History reference
In their first article, Ch. Cobb and P. Douglas initially assumed constant returns to scale. Subsequently, they relaxed this assumption, preferring to estimate the degree of returns to scale of production.
The main task of production functions, however, is to provide source material for the most effective management decisions. Let us illustrate the question of making optimal decisions based on the use of production functions.
Example 3 Let a production function be given that relates the output of an enterprise to the number of workers 
, production assets 
and the volume of used machine hours 
from where we get the solution 
, where y=2. Since, for example, the point (0,2,0) belongs to the admissible region and y=0 in it, we conclude that the point (1,1,1) is the global maximum point. The economic implications of the resulting solution are obvious.
In conclusion, we note that production functions can be used to extrapolate the economic effect of production in a given period of the future. As in the case of conventional econometric models, an economic forecast begins with an assessment of the predicted values of production factors. In this case, the method of economic forecasting that is most suitable in each individual case can be used.
Main conclusions
Tests to check the learned material
Choose the correct answer.
What is the production function?
A) the total amount of used production resources;
B) the most effective way of technological organization of production;
C) the relationship between costs and the maximum volume of output;
D) a way to minimize profits while minimizing costs.
Which of the following equations is the Cobb-Douglas production function equation?
D) y= 
.
3. What characterizes the production function with one variable factor?
A) the dependence of the volume of production on the price of the factor,
B) dependence, in which the factor x changes, and all the others remain constant,
C) a relationship in which all factors change, and the factor x remains constant,
D) the relationship between the factors x and y.
4. The isoquant map is:
A) a set of isoquants showing the output for a certain combination of factors;
B) an arbitrary set of isoquants, showing the marginal rate of productivity of variable factors;
C) combinations of lines characterizing the marginal rate of technological substitution.
Are the statements true or false?
The production function reflects the relationship between the factors of production used and the ratio of the marginal labor productivity of these factors.
The Cobb-Douglas function is a production function that shows the maximum amount of product when using labor and capital.
There is no limit to the growth of the product produced with one variable factor of production.
An isoquant is a curve of equal product.
An isoquant shows all possible combinations of using two variable factors to produce the maximum product.
Literature
Dougherty K. Introduction to econometrics. - M.: Finance and statistics, 2001.
Zamkov O.O., Tolstopyatenko A.V., Cheremnykh Yu.P. Mathematical Methods in Economics: Textbook. – M.: Ed. "DIS", 1997.
Course of economic theory: textbook. - Kirov: ASA, 1999.
Microeconomics / Ed. Prof. Yakovleva E.B. – M.: SPb. Search, 2002.
World economy. Options for classroom work for teachers. – M.: VZFEI, 2001.
Ovchinnikov G.P. Microeconomics. - St. Petersburg: Publishing House im. Volodarsky, 1997.
Political Economy; economic encyclopedia. – M.: Ed. "Owls. Encyclopedia", 1979.
production function
The relationship between inputs and final output is described by a production function. It is the starting point in the microeconomic calculations of the company, allows you to find the best option for using production capabilities.
production function shows the possible maximum output (Q) for a certain combination of production factors and the chosen technology.
Each production technology has its own special function. In its most general form, it is written:
where Q is the volume of production,
K-capital
M– natural resources
Rice. 1 Production function
The production function is characterized by certain properties :
There is a limit to the growth in output that can be achieved by increasing the use of one factor, provided that the other factors of production do not change. This property got the name the law of diminishing productivity of a factor of production . It operates in the short term.
There is a certain complementarity of factors of production, but without a reduction in production, a certain interchangeability of these factors is also possible.
Changes in the use of factors of production are more elastic over a long period of time than over a short period.
The production function can be considered as one-factor and multi-factor. One-factor assumes that, other things being equal, only the factor of production changes. Multifactorial involves a change in all factors of production.
For the short term, a single-factor is used, and for the long-term, a multi-factor.
short term – This is a period during which at least one factor remains unchanged.
Long term – is the period of time during which all factors of production change.
In the analysis of production, such concepts are used as total product (TP) The volume of goods and services produced in a given period of time.
Average product (AR) characterizes the amount of production per unit of the factor of production used. It characterizes the productivity of the factor of production and is calculated by the formula:
Marginal Product (MP) - additional output produced by an additional unit of a factor of production. MP characterizes the productivity of an additionally hired unit of a factor of production.
Table 1 - Production results in the short term
|
Capital cost (K) |
Labor cost (L) |
Production volume (TP) |
Average product of labor (AR) |
Marginal product of labor (MR) |
Analysis of the data in Table 1 allows us to identify a number of patterns of behavior total, average and marginal product. At the maximum point of the total product (TP), the marginal product (MP) is equal to 0. If, with an increase in the volume of labor used in production, the marginal product of labor is greater than the average, then the value of the average product grows and this indicates that the ratio of labor to capital is far from optimal and some of the equipment is not used due to lack of labor. If, as the volume of labor increases, the marginal product of labor is less than the average product, then the average product of labor will decrease.
The law of substitution of factors of production.
The equilibrium position of the firm
The same maximum output of the firm can be achieved through a different combination of factors of production. This is due to the ability of one resource to be displaced by another without prejudice to the results of production. This ability is called interchangeability of factors of production.
So, if the volume of the labor resource increases, then the use of capital may decrease. In this case, we resort to the labor-intensive production option. If, on the contrary, the amount of capital used increases, and labor is displaced, then we are talking about a capital-intensive version of production. For example, wine can be produced in a labor-intensive manual way or in a capital-intensive way using machinery to squeeze the grapes.
Production technology firms are a way of combining factors of production to produce output, based on a certain level of knowledge. As technology advances, the firm is able to obtain the same or more output with the same set of production factors.
The quantitative ratio of interchangeable factors allows us to estimate a coefficient called the marginal technological substitution rate (MRTS).
Marginal rate of technological substitution labor-to-capital is the amount by which capital can be reduced by using an additional unit of labor without changing output. Mathematically, this can be expressed as follows:
MRTS LK = - dK / dL = - ∆K / ΔL
where ∆K - change in the amount of capital used;
ΔL change in labor costs per unit of output.
Consider a variant of calculating the production function and substitution of production factors for a hypothetical firm x.
Assume that this firm can change the volume of factors of production, labor and capital from 1 to 5 units. Changes in output volumes associated with this can be presented in the form of a table called the "Production Grid" (Table 2).
table 2
The production grid of the companyX
|
Capital cost |
labor costs |
||||
For each combination of the main factors, we have determined the maximum possible output, i.e., the values of the production function. Let us pay attention to the fact that, say, an output of 75 units is achieved with four different combinations of labor and capital, an output of 90 units with three combinations, 100 with two, and so on.
By representing the production grid graphically, we get curves, which are another version of the production function model, previously fixed in the form of an algebraic formula. To do this, we will connect the dots that correspond to combinations of labor and capital that allow us to obtain the same output (Fig. 1).
K
Rice. 1. Map of isoquants.
The created graphical model is called isoquant. Set of isoquants - map of isoquants.
So, isoquant- this is a curve, each point of which corresponds to combinations of production factors that provide a certain maximum output of the firm.
In order to get the same output, we can combine factors, moving in search of options along the isoquant. Moving up the isoquant means that the firm prefers capital-intensive production, increasing the number of machine tools, the power of electric motors, the number of computers, etc. A downward movement reflects the firm's preference for labor-intensive production.
The choice of a firm in favor of a labor-intensive or capital-intensive variant of the production process depends on the conditions of entrepreneurship: the total amount of money capital that the firm has, the ratio of prices for production factors, the productivity of factors, and so on.
If D - money capital; R K - the price of capital; R L - the price of labor, the number of factors that a firm can acquire by fully spending money capital, TO - amount of capital L- the amount of labor will be determined by the formula:
D=P K K+P L L
This is the equation of a straight line, all points of which correspond to the full use of the firm's money capital. Such a curve is called isocostal or budget line.
K
A
Rice. 2. Manufacturer's equilibrium.
On fig. 2 we combined the line of the firm's budget constraint, the isocost (AB) with an isoquant map, i.e. a set of alternatives to the production function (Q 1 ,Q 2 ,Q 3) to show the producer's equilibrium point (E).
Producer equilibrium- this is the position of the company, which is characterized by the full use of money capital and at the same time achieving the maximum possible output for a given amount of resources.
At the point E isoquant and isocost have an equal slope angle, the value of which is determined by the indicator of the marginal rate of technological substitution (MRTS).
Indicator dynamics MRTS (it increases as you move up along the isoquant) shows that there are limits to the mutual substitution of factors associated with the fact that the efficiency of the use of production factors is limited. The more labor is used to drive capital out of the production process, the lower the productivity of labor. Similarly, the substitution of more and more capital for labor reduces the return of the latter.
Production requires a balanced combination of both production factors for their best use. The entrepreneurial firm is willing to substitute one factor for another, provided there is a gain, or at least an equal loss and gain in productivity.
But in the factor market, it is important to consider not only their productivity, but also their prices.
The best use of the firm's money capital, or the producer's equilibrium position, is subject to the following criterion: the producer's equilibrium position is reached when the marginal rate of technological substitution of factors of production is equal to the ratio of prices for these factors. Algebraically, this can be expressed as follows:
- P L / P K = - dK / dL = MRTS
where P L , P K - labor and capital prices; dK, dL - change in the amount of capital and labor; MTRS - marginal rate of technological substitution.
An analysis of the technological aspects of the production of a profit-maximizing firm is of interest only from the point of view of achieving the best final results, i.e., the product. After all, investments in resources for an entrepreneur are only costs that must be incurred in order to obtain a product that is sold on the market and generates income. The costs have to be compared with the result. Outcome or product indicators are therefore of particular importance.
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