The production function of the firm - abstract. Production functions (1)

Production refers to 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 volume of 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 achieved by increasing one resource and keeping other resources constant. If, for example, in agriculture we increase the amount of labor 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 is 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 revised. In this regard, instantaneous, short and long periods are distinguished. Instantaneous period - a period when all resources are fixed. Short period- a period when at least one resource is fixed. A long period - a period when all resources are variable.

Typically, the production function in question looks like this:

A, α, β - specified parameters. Parameter A is the coefficient of total productivity of production factors. It reflects the impact of technological progress on production: if a manufacturer introduces advanced technologies, the value A increases, i.e. output increases with the same quantities of labor and capital. Options α And β are the elasticity coefficients of output for capital and labor, respectively. In other words, they show by how many percent output changes when capital (labor) changes by one percent. These coefficients are positive, but less than one. The latter means that when labor with constant capital (or capital with constant labor) increases by one percent, production increases to a lesser extent.

Isoquant(equal product line) reflects all combinations of two factors of production (labor and capital) for which output remains unchanged. In Fig. 8.1 next to the isoquant the corresponding release is indicated. Thus, output is achievable using labor and capital or using labor and capital.

Rice. 8.1. Isoquant

If we plot the number of units of labor along the horizontal axis, and the number of units of capital along the vertical axis, then designate the points at which the firm produces the same volume, we get the curve shown in Figure 14.1 and called an isoquant.

Each isoquant point corresponds to a 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, and not a set of discrete points.

2. For any given volume of output, its own isoquant can be drawn, reflecting various combinations of economic resources that provide the manufacturer with the same volume of production (isoquants describing a given production function never intersect).

3. Isoquants do not have increasing areas (If an increasing area existed, then when moving along it, the amount of both the first and second resource would increase).

Market concept. In its most general form, a 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 exchange not only manufactured products, but also products that are not the result of labor (land, wild forest). Under the conditions of the dominance of market relations, all relations of people in society are covered by purchase and sale.

More specifically, the market represents the sphere of exchange (circulation), in which

communication is carried out between agents of social production in the form

purchase and sale, i.e. the connection between producers and consumers, production and

consumption.

Market subjects are sellers and buyers. As sellers

and buyers are households (consisting of one or more

persons), firms (enterprises), state. Most market participants

act simultaneously as both buyers and sellers. All household

subjects interact closely in the market, forming an interconnected “flow”

purchase and sale.

Firm is an independent economic entity engaged in commercial and production activities and possessing separate property.

The company has the following characteristics:

is an economically separate, independent economic unit;
legally registered and in this regard relatively independent: it has its own budget, charter and business plan
is a kind of intermediary in 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 to make a profit and minimize costs.

The company, as an independent economic entity, performs a number of important functions.

1. Production function implies the ability of a firm to organize production of goods and services.

2. Commercial function provides logistics, sales of 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, balance sheets and estimates, conducting inventories and reports to state statistics and tax authorities.

5. Administrative function– a management function, including organization, planning and control over activities as a whole.

6. Legal function carried out through compliance with laws, norms and standards, as well as through the implementation of measures to protect production factors.

Elasticity and the slope of the demand curve cannot be equated, because these are different concepts. The differences between them can be illustrated by the elasticity of the straight line of demand (Figure 13.1).

In Fig. 13.1 we see that the straight demand line 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 the 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, while elasticity theory deals with relative, or percentage, changes in price and quantity. The difference between the slope of a demand curve and its elasticity can also be clearly understood by calculating the elasticity for various combinations of price and quantity located on a straight-line demand curve. You will find that although the slope apparently remains constant throughout the curve, demand is elastic in the high-price segment and inelastic in the low-price segment.

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.

Income elasticity of demand appears in the following main forms:

· positive, suggesting that an increase in income (other things being equal) is accompanied by an increase in demand. The positive form of income elasticity of demand applies to normal goods, in particular luxury goods;

· negative, suggesting a reduction 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, meaning 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.

Income elasticity of demand depends on the following factors:

· on the importance of a particular benefit for the family budget. The more a family needs a good, the less elastic it is;

· whether this good is a luxury item or a necessity. For the former good the elasticity is higher than for the latter;

· from the conservatism of demand. As income increases, the consumer does not immediately switch to consuming more expensive goods.

It should be noted that for consumers with different income levels, the same goods can be classified as either luxury goods or basic necessities. A similar assessment of benefits can also take place for the same individual when his level of income changes.

In Fig. Figure 15.1 shows graphs of QD versus I for various values of income elasticity of demand.

Rice. 15.1. Income elasticity of demand: a) high-quality inelastic goods; b) high-quality elastic goods; c) low-quality goods

Let us make a brief comment on Fig. 15.1.

Demand for inelastic goods increases with income only when household incomes are low. Then, starting from a certain level I1, the demand for these goods begins to decline.

The demand for elastic goods (for example, luxury goods) is absent up to a certain level I2, since households do not have the opportunity to purchase them, and then increases with increasing income.

The demand for low-quality goods initially increases, but starting from the value of I3 it decreases.

Related information.

Manufacturing is actually the process of transforming one product into another. In the process of which, from a combination of simple things, something more complex in essence is obtained. The Cobb-Douglas production function, like any other, reflects the existing relationship between the result obtained and the combination of factors that were used to achieve it. The differences between different models lie in the depth of their coverage of the real state of affairs. The simplest is linear, which reflects the relationship between the number of workers and real output. The Cobb-Douglas production model no longer considers only labor as a resource for obtaining results, but also capital. The most complex are modern multifactor models. They include land, entrepreneurial abilities, and even information.

Production as a process

Production is essentially the transformation of various material and intangible investments (plans, know-how) to create items intended for consumption. It is the process of creating a product or service that is useful to individuals. Increased production means improved economic well-being. This is because all products are directly or indirectly used to satisfy human needs. And the latter, as you know, are limitless. Therefore, the economic well-being of a state is often assessed by the degree to which the needs of its citizens are satisfied. Its increase is associated with two factors: an improvement in the quality-price ratio of available products and an increase in the purchasing power of people due to more efficient market production.

Source of economic wealth

There are mainly only two processes in the economy: production and consumption. And there are just as many types of actors. Manufacturers produce products to satisfy consumer needs. Economic well-being thus consists of two components. The first is efficient production, the second is the interaction between factors. The welfare of consumers depends on the products they can afford, and producers - on the income they receive as compensation for their labor and the tangible and intangible assets invested in the production process.

Product creation process

Every enterprise deals with many separate activities in the course of its work. However, to make it easier to understand production, it is customary to distinguish five main processes, each of which has its own logic, goals, theory and key figures. And it is important to study them not only as a whole, but also separately. Thus, during production the following processes are distinguished:

Economic definition

The production function is the relationship between output and the combination of factors used to produce it. The main one is labor. A simple linear model considers only this. The Cobb-Douglas production function, an example of which will be discussed below, takes into account not only labor, but also capital as a factor in the production process. Other models additionally take into account land (P) and entrepreneurial ability (H). Thus, production is a function of the combination of these indicators or Q = f (K, L, P, H). Each sector of the economy or even a separate enterprise has its own characteristics. Therefore, an infinite number of production functions can be invented.

Simple linear model

The Cobb-Douglas production function takes into account two factors, as is common in neoclassical theories. However, it is much easier to consider only one. Adam Smith's theory of absolute advantage, with which virtually all modern economics began, was based only on labor as a factor of production. David Ricardo did not escape this assumption either. And only in the 60s of the last century, Swedish economists Eli Heckscher and Bertil Ohlin took it upon themselves to begin to consider another factor - capital. The simplest production model is linear. It describes the relationship between the quantity of labor and output. Her equation includes only one independent variable. Thus, the linear production function has the following form: Q = a * L, where Q is the volume of output, a is a parameter, L is the number of workers employed in production. Let's look at a separate example. One worker can make 10 chairs per day. In this case, the equation will look like this: Q = 10 * L.

Law of Diminishing Returns

Let's continue with the example given above. A linear function implies that an increase in the number of workers always leads to an increase in output. One master can make 10 chairs a day, five - 50, one hundred - 1000. However, in reality, everything is a little more complicated. In such models, fixed capital funds and diminishing returns must be taken into account. Therefore, an additional parameter appears in the equation - b. It lies between zero and one, which follows from its economic essence. Now the relationship between the volume of output and the number of workers can be described as follows: Q = a * L b. The equation from the previous example in reality will look like this: Q = 10 * L 0.5. And this means that one worker produces 10 chairs, and five do not produce 50, but only 22. A hundred craftsmen can actually make not a thousand products, but only a hundred. And this is the law of diminishing returns in action.

Multifactor models

The Cobb-Douglas production function is: Q = a * L b * K c . As can be seen from the formula, we are already dealing with three parameters (a, b, c) and two factors (L, K). It takes into account not only labor resources (number of workers), but also capital resources (number of saws at disposal). The parameters of the Cobb-Douglas production function depend not only on the industry sector, but also on the technology used in an individual enterprise. We must not forget about the effect of the law of diminishing returns from any factor used. Our equation from the example above can be expanded as follows: Q = 10 * L 0.5 * K. The Cobb-Douglas production function is used most often in modern neoclassical theories because of its relative simplicity and closeness to reality. More complex models are just beginning to become widespread.

Fixed proportions

Suppose the only way to produce a chair is to give each worker a saw. In this case, extra tools are simply useless. This means that the release of a product requires a certain ratio of capital and labor resources. In this case, the production volume is determined by the “weak link”. For this case, economists came up with a special function. It has the following form: min (L, K). If to create a chair you need two workers and one saw, then min (2L, K).

Ideal substitutes

If one factor can be replaced by another, then this will have an effect on the shape of the production function. For example, suppose robots could be used instead of carpenters. The formula from the example will then look like this: Q = 10 * L + 10 * R. Or more generally: Q = a * L + d * R, where a, d are parameters, and L and R are the number of carpenters and robots. If machines are 10 times faster than workers, then the formula will look like this: Q = 10 * L + 100 * R.

Cobb-Douglas production function: properties

Let's start looking at the most popular neoclassical model with its main features:

1. Cobb-Douglas production functions take into account two factors: labor and capital.

2. Positively decreasing marginal product.

3. Constant elasticity of output equal to b for L and c for K.

4. The Cobb-Douglas production function has the form: Q = a * L b * K c.

5. Constant economies of scale equal to the sum of b and c.

Historical information

The basis of any economic theory is the factors of production. The Cobb-Douglas production function considers two of the four basic ones: labor and capital. Today, for each enterprise, you can come up with separate examples of it. The solution to the Cobb-Douglas production functions did not occur without the work of Knut Wicksell (1851-1926). It was he who first designed this model. Charles Cobb and Paul Douglas, after whom it was later named, only tested it in practice. In 1928, their book was published, which described the economic growth of the United States in 1899-1922. Scientists explained it using two factors: labor resources used and capital invested. Of course, economic growth is influenced by many other parameters, but statistics have proven that the decisive ones are the two that Knath Wicksell identified.

According to Paul Douglas, the first formulation of the function appeared in 1927. At this time, he tried to derive a mathematical expression for the relationship between workers and capital. He turned to his colleague Charles Cobb. The latter managed to derive a modern equation, which, as it turned out, was previously used in his works by Knath Wicksell. Using the least squares method, scientists were able to derive the exponent of labor (0.75). Its importance has been confirmed by data from the National Bureau of Economic Research. In the 1940s, scientists moved away from constants and declared that exponents could change over time.

Model assumptions

If output is a derivative of two factors (labor and capital), then the elasticity of the entire function will depend on the marginal productivity of each of them. Thus, Cobb and Douglas based their model on the following assumptions:

Production cannot continue in the absence of one of the factors. Labor and capital are not substitutes that can replace each other in the output process. Additional saws cannot create chairs without the participation of carpenters.
The marginal productivity of each factor is proportional to the volume of output per unit.

Release elasticity

Obviously, reducing the volume of materials used leads to a reduction in the volume of products. The Cobb-Douglas production function deals with marginal output. Elasticity in economics is the percentage change in the value of one indicator in response to a decrease or increase in another associated with it. The Cobb-Douglas production function assumes that b and c are constants. If b is equal to 0.2 and the number of workers increases by 10%, then output will increase by 2%.

Economies of scale

To actually increase output, the volume of factors of production used must increase proportionally. If this happens, then we say that we are using economies of scale. The Cobb-Douglas production function, the properties of which we have already examined, takes it into account. If b + c = 1, then this means that we are dealing with a constant effect of scale, >1 - increasing,<1 - уменьшающимся.

Time factor

The Cobb-Douglas production function model is often used to describe the medium- and long-term outlook. Obviously, it is often much easier to hire new people than to increase capital resources. Therefore, some economists argue that a simple linear model is best suited to describe short periods of business operation. The company owns a certain size of premises, a limited number of machines, which can only be changed with the help of long-term planning. The time period it takes may vary from one plant to another, as can the elasticity of the Cobb-Douglas production function.

Application problems

Although the two-factor production function has gained widespread acceptance and has been statistically tested by Cobb and Douglas, some economists still doubt its accuracy across industries and time periods. The main assumption of this model is the constancy of the elasticity of labor and capital in developed countries. However, is this really so? Neither Cobb nor Douglas provided a theoretical basis for its existence. The constancy of the coefficients b and c greatly simplifies the calculations, and that's all. At the same time, scientists knew nothing about engineering, technology and production process management. In addition, the possibility of its application at the micro level does not indicate its correctness in macroeconomic conditions, and vice versa.

Criticism has dogged the Cobb-Douglas production function since its introduction in 1928. At first, it upset scientists so much that they wanted to quit working on it. But then they decided to continue. In 1947, Douglas came forward with further confirmation of its correctness as president of the American Economic Association. The scientist was unable to continue working on it due to health problems. The production function was later refined by Paul Samuelson and Robert Solow, forever changing the way we study macroeconomics.

The Cobb-Douglas production function is one of the most important concepts today. It describes the relationship between the input factors and the resulting result. Unlike simple linear models, which are only suitable for describing a short period of the life of an enterprise, it can be used for long-term planning. However, we must not forget about a number of assumptions and problems associated with its application.

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)

Text of the lecture on the topic “Production functions”

in the discipline "Econometrics"

Plan:

Introduction

Concept of a single variable production function

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

Tests for control of learned material

Literature

Introduction

In modern society, no person can consume only what he himself produces. To most fully satisfy their needs, people are forced to exchange what they produce. Without constant production of goods there would be no consumption. Therefore, it is of great interest to analyze the patterns operating in the process of production of goods, which subsequently shape their supply on the market.

The production process is the basic and original concept of economics. 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 appropriate raw materials, equipment, premises, a piece of land, and specialists who organize production. Everything necessary to organize the production process is called factors of production. Traditionally, factors of production include capital, labor, land and entrepreneurship.

To organize the production process, the necessary factors of production must be present in a certain quantity. The dependence of the maximum volume of a product produced on the costs of the factors used is called production function.

Concept of a single variable production function

We will begin our consideration of the concept of “production function” with the simplest case, when production is determined by only one factor. In this case Pproduction function – This is a function whose independent variable takes the values of the resource used (factor of production), and the dependent variable takes 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 single-resource or single-factor. Its domain of definition is the set of non-negative real numbers. The symbol f is a characteristic of a production system that converts a resource into an output. In microeconomic theory, it is generally accepted that y is the maximum possible volume of output if the resource is expended or used in the amount of x units. In macroeconomics, this understanding is not entirely correct: perhaps, with a different distribution of resources between the structural units of the economy, output could have been greater. In this case, PF is a statistically stable relationship between resource costs and output. The symbolism is more correct

where a is the vector of PF parameters.

Example 1. Let us take the PF f in the form f(x)=ax b, where x is the amount of resource expended (for example, working time), f(x) is the volume of products produced (for example, the number of refrigerators ready for shipment). Values a and b are parameters of PF f. Here a and b are positive numbers and the number b1, the parameter vector is a two-dimensional vector (a,b). PF у=ax b is a typical representative of a wide class of single-factor PFs.

The PF chart is shown in Figure 1

The graph shows that as the amount of resource spent increases, y increases. however, each additional unit of resource gives an increasingly smaller increase in the volume y of output. The noted circumstance (an increase in volume y and a decrease in the increase in volume y with an increase in 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 a farmer's production of an agricultural product. Let all factors of production, such as the size of land, the farmer’s availability of agricultural machinery, seed, and the amount of labor invested in the production of the product, remain constant from year to year. Only one factor changes - the amount of fertilizer used. Depending on this, the size 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 fertilizers used, 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 at both the micro and macroeconomic levels. Let's first look at the microeconomic level. PF y=ax b , discussed above, can be used to describe the relationship between the amount of resource x spent or used 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 or an intersectoral production complex can also act as a production system. MIPFs are built and used mainly to solve 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 plays the role of the production system - we have a macroeconomic level and a macroeconomic PF (MAPF). MAPFs are built and actively used to solve all three types of problems (analysis, planning and forecasting).

The exact interpretation of the concepts of expended or used resource and output, as well as the choice of units of measurement, depend on the nature and scale of the production system, the characteristics of the problems being solved, and the availability of initial data. At the microeconomic level, inputs and output can be measured both in natural and in monetary units (indicators). Annual labor costs can be measured in man-hours or in rubles of wages paid; Product output can be presented in pieces or other natural units or in the form of its value.

At the macroeconomic level, costs and output are measured, as a rule, in cost terms and represent cost aggregates, that is, the total values of the products of the volumes of resources expended and output products and their prices.

Production functions of several variables

Let us now move on to consider the production functions of several variables.

Production function of several variables is a function whose independent variables take on the values of the volumes of resources expended 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 ,…,x n). (2)

In formula (2) y (y 0) is a scalar, and x is a vector quantity, x 1,...,x n are the coordinates of the vector x, that is, f(x 1,...,x n) is a numerical function of several variables x 1,...,x n. In this regard, the PF f(x 1,...,x n) is called multi-resource or multi-factor. The following symbolism is more correct: f(x 1,...,x n,a), where a is the vector of PF parameters.

In economic terms, all variables of this function are non-negative, therefore, the domain of definition of a multifactorial PF is a set of n-dimensional vectors x, all coordinates x 1,..., x n of which are non-negative numbers.

For an individual enterprise (firm) producing a homogeneous product, the PF f(x 1 ,...,x 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, and fixed capital. PFs of this type characterize the current technology of an enterprise (firm).

When constructing the PF for a region or country as a whole, the total product (income) of the region or country, usually calculated in constant rather than current prices, is often taken as the value of annual output Y; fixed capital (x 1 (= K) is considered as resources - the volume of fixed capital used during the year) and living labor (x 2 (=L) - the number of units of living labor spent during the year), usually calculated in value terms. Thus, a two-factor PF Y=f(K,L) is constructed. From two-factor PFs they move to three-factor ones. In addition, if the PF is constructed using time series data, then technical 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 year number, t=0,1,…,T; t= 0 – base year of the time period covering years 1,2,…,T.

Example 2. To model a separate region or a 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= is often used
, where a 0, a 1, and 2 are the PF parameters. These are positive constants (often a 1 and a 2 are such that a 1 + a 2 = 1). The PF of the type just given is called the Cobb-Douglas PF (Cobb-Douglas PF) after the two American economists who proposed its use in 1929.

PFKD is actively used to solve a variety of 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 = K is equal to the volume of fixed capital used (the volume of fixed assets used - in domestic terminology),
- costs of living labor, then the PFKD takes on the form often used in the literature:

Y=
.

Historical reference

In 1927, Paul Douglas, an economist by training, discovered that if one plots the logarithms of real output against time (Y), capital investments (K) and labor costs (L), then the distances from the points on the graph of output indicators to the points on the graphs of indicators of labor and capital inputs will be a constant proportion. He then turned to mathematician Charles Cobb with a request to find a mathematical relationship that had this feature, and Cobb proposed the following function:

This function had been proposed about 30 years earlier by Philip Wicksteed, as noted by C. Cobb and P. Douglas in their classic work (1929), but they were the first to use empirical data to construct it. The authors do not describe how they actually fitted the function, but presumably they used a form of regression analysis since they referred to “least squares theory.”

Example 3. Linear PF (LPF) has the form:
(two-factor) and (multifactor). LPF belongs to the class of so-called additive PF (APF). The transition from a multiplicative PF to an additive one is carried out using the logarithm operation. For a two-factor multiplicative PF

this transition has the form: . By introducing the appropriate substitution, we obtain an 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 labor productivity and capital-labor ratio, respectively. Using new symbols, we get

those. from the two-factor PFCD we obtain a formally single-factor PFCD. Due to the fact that 0 1

Note that the fraction called capital productivity or capital productivity, reciprocal fractions
are called capital intensity and labor intensity of output, respectively.

PF is called dynamic, If:

time t appears as an independent variable (as if an independent factor of production) influencing the volume of output;

PF parameters and its characteristic f depend on time t.

Note that if the PF parameters were estimated using 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 in advance.

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 capital productivity: 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 scientific and technological progress.

Example 4. Let us present a version of the PFKD taking into account NTP

The calculation of the numerical values of the parameters of such a function is carried out using correlation and regression analysis.

Selecting the analytical form of the PF
is dictated primarily by theoretical considerations, which must take into account the peculiarities of the relationships between specific resources or economic patterns. Estimation of PF parameters is usually carried out using the least squares method.

Properties and main characteristics of production functions

To produce a particular product, a combination of various factors is required. Despite this, various production functions have a number of common properties.

For definiteness, we restrict ourselves to 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 a two-dimensional plane, that is, at. The PF satisfies the following series of properties:

Similar to the level line of the objective function of the optimization problem, a similar concept also applies to the PF. PF level line is the set of points at which the PF takes a constant value. Sometimes 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 precisely determine all possible combinations of production factors necessary to achieve a given level of production.

From Figure 2 it is clear that along the isoquant, output is constant, 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 the distance of the isoquant from the origin of coordinates corresponds to a greater level of output.

Isoquants are decreasing curves that have 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 isoquants is explained by the fact that an increase in the use of one factor for a certain volume of product output will always be accompanied by a decrease in the amount of another factor. The slope of the isoquant is characterized by the marginal rate of technological substitution of production factors (MRTS) . Let's 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 ratio, the mathematical expression of 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 CM to it, then the tangent of the angle will give us the MRTS value:

It can be noted that in the upper part of the isoquant the angle will be quite large, which indicates that significant changes in factor y are required to change factor x by one. Therefore, in this part of the curve the MRTS value will be high. As you move down the isoquant, the value of the marginal rate of technological substitution will gradually decrease. This means that an increase in the x factor by one would require a slight decrease in the y factor. With complete substitutability of factors, isoquants from curves are converted into straight lines.

One of the most interesting examples of the use of PF isoquants is the study economies of scale of production (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 unequivocally, giving priority to industrial giants. With the transition to a market economy, widespread disaggregation of previously created associations began. Where is the golden mean? A demonstrative answer to this question can be obtained by examining the effect of scale in production.

Let's imagine that at a shoe factory the management decided to allocate a significant part of the profit received to the development of production in order to increase the volume of products produced. Let's assume that 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 to the volume of output?

From the analysis of Figure 5

There are three answer options:

The quantity of production will double (constant returns to scale);

Will more than double (increasing returns to scale);

It will increase, but less than twice (diminishing returns to scale).

Constant returns to scale of production 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 operates or two small ones are created instead.

With diminishing returns to scale, it is unprofitable to create large-scale production. The reason for low efficiency in this case, as a rule, is the additional costs associated with managing such production and the difficulty of coordinating large-scale production.

Increasing returns to scale, as a rule, are characteristic of those industries where widespread automation of production processes and the use of production and conveyor lines are possible. But we need to be very careful with the trend of increasing returns to scale. Sooner or later it turns into 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 PFs 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 for the i-th resource. First partial derivative of the PF
(i=1,2) is called the marginal productivity of the i-th resource or the marginal output of the i-th resource. This limiting quantity is sometimes interpreted using a close approximation of the ratio of small finite quantities
. Approximately, it shows by how many units the volume of output y will increase if the volume of expenditure of the i-th resource increases by one (sufficiently small) unit while the volume of another resource spent remains unchanged.

For example, in the PFKD, for the average productivity of fixed capital u/K and labor u/L, the terms capital productivity and labor productivity are used, respectively:

Let us determine the marginal productivity of factors for this function:

And
.

Thus, if
, That
(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 productivity is called the elasticity of output with respect to the i-th factor of production

or approximately

Thus, the elasticity of output (production volume) for a certain factor (elasticity coefficient) is approximately defined 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 constant volumes of another resource.

Sum +=E called elasticity of production. For example, for PFKD = , 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 evaluate the average and marginal efficiency of various production resources, the elasticity of output for various resources, marginal rates of resource substitution, economies of scale in production, and much more.

Example 1. Let us assume that the production process is described using the output function

Let us evaluate the main characteristics of this function for the production method in which K = 400 and L = 200.

Solution.

Marginal productivity of factors.

To calculate these quantities, we determine the partial derivatives of the function for each of the factors:

Thus, the marginal productivity of the labor factor is four times higher than that of the capital factor.

Elasticity of production.

The elasticity of production is determined by the sum of the elasticities of output for each factor, that is

Marginal rate of resource substitution.

Above in the text this value was denoted
and equaled
. Thus, in our example

that is, four units of capital resources are needed to replace a unit of labor at this point.

Isoquant equation.

To determine the form of the isoquant, it is necessary to fix the value of the output volume (Y). Let, for example, Y=500. For convenience, we take L to be a function of K, then the isoquant equation takes the form

The marginal rate of resource substitution determines the tangent of the angle of inclination of the tangent to the isoquant at the corresponding point. Using the results of step 3, we can say that the point of tangency is located in the upper part of the isoquan, since the angle is quite large.

Example 2. Let us consider the Cobb-Douglas function in general form

Let's assume that K and L are doubled. Thus, the new output level (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 =0.8, and =1.74, then Y increases less than twice.

Thus, in example 1 there could be a constant effect of scale in production.

Historical reference

In their first article, C. Cobb and P. Douglas initially assumed constant returns to scale. They subsequently relaxed this assumption, preferring to estimate returns to scale.

The main task of production functions is still to provide source material for the most effective management decisions. Let us illustrate the issue of making optimal decisions based on the use of production functions.

Example 3. Let a production function be given that relates the volume of output of an enterprise to the number of workers , production assets and the volume of machine hours used

where do we get the solution from?
, at which y=2. Since, for example, the point (0,2,0) belongs to the admissible region and in it y = 0, we conclude that the point (1,1,1) is a global maximum point. The economic conclusions from 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, economic forecasting begins with an assessment of the forecast values of production factors. In this case, you can use the method of economic forecast that is most suitable in each individual case.

Main conclusions

Tests to check the material learned

Choose the correct answer.

What does the production function characterize?

A) the total volume of production resources used;

B) the most effective way of technological organization of production;

C) the relationship between costs and maximum output;

D) a method of minimizing profits while minimizing costs.

Which of the following equations is the Cobb-Douglas production function equation?

D) y=
.

3. What does a production function with one variable factor characterize?

A) the dependence of production volume on factor prices,

B) a dependence in which factor x changes, and all others remain constant,

C) a relationship in which all factors change, but factor x remains constant,

D) the relationship between factors x and y.

4. Isoquant map is:

A) a set of isoquants showing output under 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 productivity of these factors.

The Cobb-Douglas function is a production function that shows the maximum output using labor and capital.

There is no limit to the growth of the product produced with one variable factor of production.

An isoquant is an equal product curve.

An isoquant shows all possible combinations of using two variable factors to obtain 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.: Publishing house. "DIS", 1997.

Economic theory course: textbook. – Kirov: “ASA”, 1999.

Microeconomics / Ed. Prof. Yakovleva E.B. – M.: St. Petersburg. Search, 2002.

World economy. Classroom options for teachers. – M.: VZFEI, 2001.

Ovchinnikov G.P. Microeconomics. – St. Petersburg: Publishing house named after. Volodarsky, 1997.

Political Economy; economic encyclopedia. – M.: Publishing house. “Owl. Encyclopedia", 1979.

Answer

Entrepreneurs purchase factors of production on markets, organize production and produce products. Production function is a technological relationship between the number of factors of production used and the maximum possible output produced during a certain period of time. Such a technological connection exists for each specific level of technological development. The production function expresses the maximum output for each combination of factors of production. A function can be presented as a table, a graph, or analytically as an equation.

If the entire set of resources necessary for production is represented as the costs of labor, capital and materials, then the production function will take the following form:

Q = F (T, K, M),

where Q is the maximum volume of products produced using a given technology in a given ratio: labor - T, capital - K, materials - M.

The production function shows the relationship between factors and makes it possible to determine the share of each in the creation of goods and services.

Graphically, the relationship between factors of production can be depicted as an isoquant. An isoquant is a curve reflecting various combinations of resources that can be used to produce a certain volume of output. The set of isoquants forms an isoquant map that shows the alternatives to the production function. Isoquants have the following properties:

Isoquants cannot intersect, because are the geometric locus of equal outputs;

Isoquants are strictly convex to the origin and have a negative slope;

The higher and to the right the isoquant, the greater the volume of output it characterizes.

The production function can only be determined empirically (experimentally), i.e. through measurements based on actual performance.

Question 7. Production capabilities of the economy

Answer

A common property of economic resources is their limited quantity, so the economy is constantly faced with the question of alternative choice: increasing the production of one product (commodity set) means refusing to produce part of another. Society strives to ensure full employment and full production in order to satisfy its needs as much as possible. Concept full employment characterizes the economically feasible use of all resources. Under full volume production implies the efficient allocation of resources, ensuring the highest output.

Alternative choice in economics can be characterized using production possibilities curve, each point of which reflects the maximum possible volume of production of two products with given resources. Society determines which combination of these products it chooses. The functioning of an economy on the production possibilities frontier indicates its efficiency and the correctness of the choice of the method of producing a good. Points outside the production possibilities curve contradict the accepted condition.

The number of other products that must be sacrificed in order to obtain any quantity of a given product is called alternative ( opportunity) production costs of this product. It is necessary to distinguish between the opportunity costs of an additional unit of goods and the total (or total) opportunity costs. The absence of perfect elasticity or interchangeability of resources has been established. It follows from this that when switching resources from the production of one product to another, each additional unit of product will require the involvement of an increasing number of additional products. This phenomenon is called law of increasing opportunity costs. Thus, law of opportunity costs reflects the process of constant increase in opportunity costs.

The theory of opportunity costs and the production possibilities curve are used to justify investment programs and projects, as well as to formulate the optimal structure of products, study consumer behavior and solve other issues requiring the redistribution of resources.

Question 8. Stages of social production

Answer

Factors of production (funds or capital) go through three stages: purchase of factors of production; the production process, where the means of production and labor are combined; selling goods and making a profit.

A continuously repeating production process is called reproduction. Distinguish prime(descending) And expanded reproduction. Simple reproduction ensures the recreation of the previously achieved state of the economy - this is production on an unchanged scale. Decreasing production is typical for crisis states of the economy. With it, the scale of production is reduced. Expanded manufacturing is characterized by a constant increase in the scale of production. There are intensive and extensive types of expanded reproduction. At intensive type, expansion of the scale of production is achieved through qualitative improvement and better use of production factors, the use of more efficient technologies, and increased labor productivity. Extensive type is characterized by a quantitative increase in factors of production.

The sequential passage of production assets (capital) through three stages forms circulation of production assets. The circulation of production assets, considered as a continuously repeating process, is called turnover of funds (capital). The turnover time of funds consists of production time And time of appeal. The turnover of funds (capital) ends when, in the process of selling goods, the owner of the funds fully reimburses the capital advanced into factors of production.

Depending on the specifics of turnover, production assets are divided into basic, serving for a long time, and negotiable, which are consumed during one production cycle.

Distinguish physical And obsolescence fixed production assets. The process of compensating for the depreciation of fixed production assets by gradually including their value in the production costs of the created goods is called depreciation. The ratio of the amount of annually transferred depreciation deductions to the cost of labor instruments as a percentage is called depreciation rate.

Circulation funds enterprises include finished products and enterprise cash. Together with working production assets they form working capital enterprises. Working capital turnover is an important indicator of the efficiency of their use.

Production efficiency in In general, it is determined by the relationship between the effect (result) and the cause that causes it. The most important indicators of production efficiency are: labor productivity, labor intensity, capital-labor ratio, capital productivity, capital intensity, material intensity.

Question 9. Product as a result of production

Answer

Product represents the result of the purposeful activity of people - labor (thing or service) and at the same time acts as a condition for the flow of the labor process. The product ensures the reproduction of personal and material factors of production.

There are material and social aspects of the product. Natural - real the side of a product is the totality of its properties (mechanical, chemical, physical, etc.) that make this product a useful thing that can satisfy human needs. This property of the product is called consumer value. Public side product is that each product, being the result of human labor, accumulates a certain amount of this labor.

A product manufactured by a separate manufacturer acts as single or individual product. The result of all social production is public a product that represents the entire mass of use values created in society and serves as the basis of its material and spiritual life.

According to its natural-material form, the social product is divided into means of production and items of personal consumption. Means of production returned during production. They serve to replace worn-out production assets and to increase (expand) them. Personal items finally leave the sphere of production and enter the sphere of consumption. The division of the social product into means of production and personal consumption items allows us to divide all material production into two large divisions: production of means of production(1 division) and production of personal consumption goods(2nd division).

In a commodity economy, the social product has a value, the external manifestation of which is price. The cost of a product is determined by the total (total) costs of its production, i.e., the costs of past (materialized) labor and the costs of living labor. In Western literature, instead of the term “product,” the term “good” is often used.

Production function– dependence of production volumes on the quantity and quality of available production factors, expressed using a mathematical model. The production function makes it possible to identify the optimal amount of costs required to produce a certain portion of goods. At the same time, the function is always intended for a specific technology - the integration of new developments entails the need to review the dependency.