Perform a decorative function of their own production. Production function and its characteristics

In conditions modern society no man can only consume what he himself produces. Each individual acts in the market in two roles: as a consumer and as a producer. Without permanent production of goods there would be no consumption. To the well-known question “What to produce?” consumers in the market respond by “voting” with the contents of their wallet for those goods that they really need. To the question "How to produce?" must answer those firms that produce goods on the market.

There are two types of goods in the economy: consumer goods and factors of production (resources) - these are the goods necessary for organizing the production process.

Neoclassical theory traditionally attributed capital, land and labor to the factors of production.

In the 70s of the 19th century, Alfred Marshall singled out the fourth factor of production - organization. Further, Joseph Schumpeter called this factor entrepreneurship.

In this way, production is the process of combining such factors as capital, labor, land and entrepreneurship in order to obtain new goods and services needed by consumers.

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 the production function:

where Q is the maximum volume of a product that can be produced with a given technology and certain production factors; K - capital costs; L - labor costs; M - the cost of raw materials, materials.

For aggregated analysis and forecasting, a production function is used, called the Cobb-Douglas function:

Q = k K L M ,

where Q is the maximum volume of the product for given factors of production; K, L, M - respectively, the costs of capital, labor, materials; k - coefficient of proportionality, or scale; , , , - indicators of the elasticity of the volume of production, respectively, for capital, labor and materials, or growth coefficients Q, per 1% of the growth of the corresponding factor:

+ + = 1

Despite the fact that a combination of different factors is required to produce a particular product, the production function has a number of common properties:

the factors of production are complementary. This means that this production process is possible only with a set of certain factors. The absence of one of these factors will make it impossible to produce the planned product.

there is a certain interchangeability of factors. In the process of production, one factor can be replaced in a certain proportion by another. Interchangeability does not mean the possibility of completely eliminating any factor from the production process.

It is customary to consider 2 varieties of the production function: with one variable factor and with two variable factors.

a) production with one variable factor;

Let us suppose that in general view the production function with one variable factor is:

where y is const, x is the value of the variable factor.

In order to reflect the influence of a variable factor on production, the concepts of total (general), average and marginal product are introduced.

total product (TP) - is the amount of an economic good produced using some amount of a variable factor. This total amount of product produced changes as the use of the variable factor increases.

Average product (AP) (average resource productivity)is the ratio of total product to the amount of variable factor used in production:

marginal product (MP) (marginal resource productivity) usually defined as the increase in total product resulting from an infinitesimal increase in the amount of variable factor used:

The graph shows the ratio of MP, AP and TP.

The total product (Q) will increase with the increase in the use of the variable factor (x) in production, but this growth has certain limits within the framework of a given technology. At the first stage of production (OA), an increase in labor costs contributes to an ever more complete use of capital: marginal and total labor productivity grow. This is expressed in the growth of the marginal and average product, while MP > АР. At point A "the marginal product reaches its maximum. At the second stage (AB), the value of the marginal product decreases and at point B" becomes equal to the average product (MP = AP). If in the first stage (0A) the total product increases more slowly than the amount of the variable factor used, then in the second stage (AB) the total product grows faster than the amount of the variable factor used (Fig. 5-1a). At the third stage of production (BV) MP< АР, в результате чего совокупный продукт растет медленнее затрат переменного фактора и, наконец, наступает четвертая стадия (после точки В), когда MP < 0. В результате прирост переменного фактора х приводит к уменьшению выпуска совокупной продукции. В этом и заключается закон убывающей предельной производительности. He argues that with an increase in the use of any factor of production (while the others remain unchanged), sooner or later a point is reached at which the additional use of a variable factor leads to a decrease in the relative and then absolute volumes of output.

b) production with two variable factors.

Let's assume that in the most general form the production function with two variable factors has the form:

where x and y are the values of the variable factor.

As a rule, 2 simultaneously complementary and interchangeable factors are considered: labor and capital.

This function can be represented graphically using isoquants :

An isoquant, or equal product curve, represents all possible combinations of two factors that can be used to produce a given amount of product.

With an increase in the volume of variable factors used, it becomes possible to produce a larger volume of products. The isoquant, which reflects the production of a larger volume of product, will be located to the right and above the previous isoquant.

The number of used factors x and y can constantly change, respectively, the maximum output of the product will decrease or increase. Therefore, there may be a set of isoquants corresponding to different volumes of output, which form isoquant map.

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. That is, isoquants have properties similar to indifference curves.

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.

Just as indifference curves located at different distances from the origin characterize different level utility for the consumer, and isoquants provide information about different levels of output.

The problem of substitutability of one factor for another can be solved by calculating the marginal rate of technological substitution (MRTS xy or MRTS LK).

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 factors are replaced in the opposite way, the mathematical expression for the MRTS indicator x,y is taken with a minus sign:

MRTS x,y = orMRTS LK=

If we take any point on the 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 x,y:

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 value of MRTS x,y will be large.

As you move down the isoquant, the value of the marginal rate of technological substitution will gradually decrease. This means that in order to increase the factor x by one, a slight decrease in the factor y is required.

In real production processes, there are two exceptional cases in the isoquant configuration:

This is a situation where two variable factors are perfectly interchangeable, With full substitutability of production factors MRTS x,y = const. A similar situation can be imagined with the possibility of full automation of production. Then at point A the entire production process will consist of capital inputs. At point B, all machines will be replaced by working hands, and at points C and D, capital and labor will complement each other.

In a situation with strict complementarity of factors, the marginal rate of technological substitution will be equal to 0 (MRTS x,y = 0). If we take a modern taxi fleet with a constant number of cars (y 1) that require a certain number of drivers (x 1), then we can say that the number of passengers served during the day will not increase if we increase the number of drivers to x 2 , x 3 , ... x n . The volume of the produced product will increase from Q 1 to Q 2 only if the number of used cars in the taxi fleet and the number of drivers increase.

Each producer, acquiring factors for the organization of production, has certain limitations in the means.

Let us assume that labor (factor x) and capital (factor y) act as variable factors. They have certain prices, which remain constant for the analysis period (P x , P y - const).

The manufacturer can purchase the necessary factors in a certain combination, which does not go beyond his budgetary capabilities. Then his cost of acquiring the factor x will be P x · x, the cost of the factor y, respectively, will be P y · y. The total costs (C) will be:

C = P x X + P y Y or
.

For labor and capital:

The graphical representation of the cost function (C) is called isocost (direct equal costs, i.e. these are all combinations of resources, the use of which leads to the same costs spent on production). This straight line is constructed along two points similarly to the budget line (in the equilibrium of the consumer).

The slope of this straight line is determined by:

With an increase in funds for the purchase of variable factors, that is, with a decrease in budget constraints, the isocost line will shift to the right and up:

C 1 \u003d P x X 1 + P y Y 1.

Graphically, the isocosts look the same as the consumer's budget line. At constant prices, the isocosts are straight parallel lines with a negative slope. The greater the budgetary possibilities of the manufacturer, the farther from the origin of coordinates is the isocost.

The isocost graph in the case of a decrease in the price of the factor x will move along the abscissa from the point x 1 to x 2 in accordance with the increase in the use of this factor in the production process (Fig. a).

And if the price of the factor y increases, the producer will be able to attract a smaller amount of this factor into production. The isocost plot along the y-axis will move from point y 1 to y 2 .

Given the production capabilities (isoquants) and the producer's budget constraints (isocosts), an equilibrium can be determined. To do this, we combine the isoquant map with the isocost. That isoquant, in relation to which the isocost takes the position of a tangent, will determine the largest volume of production, given the budget possibilities. The touch point of the isoquant of the isocost will be the point of the most rational behavior of the producer.

When analyzing the isoquant, we found that its slope at any point is determined by the slope of the tangent, or the rate of technological substitution:

MRTS x,y =

The isocost at point E coincides with the tangent. The slope of the isocost, as we determined earlier, is equal to the slope . Based on this, it is possible to determine consumer's equilibrium point as the equality of the ratios between the prices of production factors and the change in these factors.

Bringing this equality to the indicators of the marginal product of the variable factor of production, in this case it is MP x and MP y , we get:

This is the producer's equilibrium or the rule of least cost..

For labor and capital, the producer equilibrium will look like this:

Assume that resource prices remain constant while the producer's budget is constantly increasing. By connecting the intersection points of isoquants with isocosts, we get the line OS - the "path of development" (similar to the line of the standard of living in the theory of consumer behavior). This line shows the growth rate of the ratio between factors in the process of expanding production. In the figure, for example, labor in the course of the development of production is used to a greater extent than capital. The shape of the "development path" curve depends, firstly, on the shape of the isoquants and, secondly, on the prices of resources (the ratio between which determines the slope of the isocosts). The "path of development" line can be straight or curved from the origin.

If the distances between isoquants decrease, this indicates that there is increasing economies of scale, i.e., an increase in output is achieved with a relative economy of resources. And the company needs to increase the volume of production, as this leads to a relative saving of available resources.

If the distances between isoquants increase, this indicates diminishing economies of scale. Decreasing economies of scale indicate that the minimum efficient size of the enterprise has already been reached and further increase in production is not advisable.

When an increase in production requires a proportional increase in resources, one speaks of permanent economies of scale.

Thus, the analysis of output using isoquants makes it possible to determine the technical efficiency of production. The intersection of isoquants with isocosts makes it possible to determine not only technological, but also economic efficiency, i.e., to choose a technology (labor- or capital-saving, energy- or material-saving, etc.) Money ah, which the manufacturer has to organize production.

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, the amount of labor in agriculture is increased 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 more cars 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 BUT is the coefficient of total factor productivity. It reflects the influence technical progress for production: if the manufacturer introduces advanced technologies, the value BUT increases, i.e. output increases with the same amount of labor and capital. Options 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 we 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 volume, we 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, providing the producer with the same volume of production (isoquants describing a given production function never intersect).

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

The concept of the market. 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 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 characteristics:

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 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 valuable papers, 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 straight-line 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 income levels, the same goods can be either luxury items or essential items. 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 us make a brief comment on 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 are unable 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.

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 fully and efficiently.

Production function properties:

1. there is a limit to the increase in production, which can be achieved with an increase in one resource and the constancy of other resources. If, for example, in agriculture the amount of labor is increased with constant amounts of capital and land, then sooner or later the moment comes when output ceases to grow;

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;

3. the longer the time period, the more resources can be reviewed. In this regard, there are instant, short-term and long-term periods.

Instant period- the period when all resources are fixed.

short term- the period when at least one resource is fixed.

Long term- the period when all resources are variable.

General view of the production function:

Q= f (KL)

· Q- a given output volume;

· L- the amount of labor used;

· K- the amount of capital used;

· f is the functional dependence of the given volume of output on the amount of the resource.

The production function graph is an isoquant.

isoquant(Greek "iso" - the same, Latin "quanto" - quantity) is a line (of constant output), which reflects all combinations of two factors of production (labor and capital), in which output remains unchanged. (Fig. 3.1).

Rice. 1.13. Isoquant.

Properties of an isoquant:

1. Isoquant shows the minimum amount of resources involved in the production process.

2. All combinations of resources on segment AB reflect technologically effective ways production of a given volume of output.

3. The isoquant is always concave (has a negative slope); the degree of concavity depends on the marginal rate of technological replacement, i.e. on the ratio of the marginal productivity of labor and capital. When moving from top to bottom along the isoquant, the marginal rate of technological replacement decreases all the time, as evidenced by the decreasing slope of the isoquant.

Marginal rate of technological replacement of one resource by another- is the amount of another resource that can be replaced by this resource to obtain the same output:

o MRTS LK - marginal rate of technological replacement of labor by capital;

o MP L - marginal labor productivity;

o MP K - marginal productivity of capital;

o ∆L – increment of labor;

o ∆K – capital increment.

If we reduce the capital gain by the value of ∆K, then this reduction will reduce the volume of production by the corresponding amount (- ∆K × MP K).

If we attract a unit of labor, then this increment of labor will increase the volume of production by the value (∆L × MRL).

Therefore, for a given volume of production, the equality is true:

MRTS LK = MP L × ∆L = MP K × ∆K

This equality can be justified as follows. Let the marginal product of labor be 10 and the marginal product of capital be 5. This means that by hiring another worker, the firm increases output by 10 units, and by giving up one unit of capital, it loses 5 units of output. Therefore, to keep output the same, the firm can replace two units of capital with one worker.

With infinitely small changes in L and K, it is the limiting rate of technological replacement is the derivative of the isoquant function at a given point:

Geometrically, it is the slope of the isoquant (Fig. 1.14):

Rice. 1.14. Marginal rate of technological replacement

There are two ways to produce a given volume of products: technologically efficient and cost-effective.

Technologically efficient way of production- production of a given volume of output with the least amount of labor and capital.

Cost effective way of production-production of a given volume of products at the lowest cost.

Figure 1.15. Technologically efficient and inefficient production

o production method A - technologically efficient compared to the way AT, because it requires the use of at least one resource in a smaller amount.

o production method B is technologically inefficient in comparison with A (the dashed line segment reflects all technologically inefficient production methods).

Technologically inefficient modes of production are not used by rational entrepreneurs and do not belong to the production function. Consequently, An isoquant cannot have a positive slope.(Fig. 1.16):

Isoquant map- a set of isoquants (Fig. 1.16).

Rice. 1.16. Isoquant map.

o q 1 ; q 2 - isoquants on the isoquant map;

o the isoquant located to the right and above the previous one (q 2) corresponds to a larger volume of output.

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 - raw materials, energy, labor, 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 the production function.

But an enterprise can implement in different ways manufacturing process, 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 production options are those that cannot be improved either by increasing the production of a product without increasing the consumption of resources, or by reducing the costs of any resource without reducing output and without increasing the costs of other resources.

The production function takes into account only technically efficient options. Its value is the maximum amount of product that the enterprise can produce with given volumes 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. So, the enterprise, spending a resource in the amount of x, can produce a product in the amount of q.

Production function:

establishes a relationship between these quantities. Note that here, as in other lectures, all volumetric quantities are quantities of the flow type: the volume of resource costs is measured by the number of units of the resource per unit of time, and the volume of output is measured by the number of units of the product per unit of time.

On Fig. 1 shows the graph of the production function for the case under consideration. All points on the graph correspond to technically efficient options, in particular points A and B. Point C corresponds to an inefficient option, and point D to an unattainable option.

Rice. one.

The production function of the form (1), which establishes the dependence of the volume of production on the volume of costs of a single resource, can be used not only for illustrative purposes. It is also useful when the consumption of only one resource can change, and the costs of all other resources, for one reason or another, must be considered fixed. In these cases, the dependence of the volume of production on the costs of a single variable factor is of interest.

A much greater variety appears when considering a production function that depends on the volumes of two consumed resources:

q \u003d f (x 1, x 2) (2)

An analysis of such functions makes it easy to pass to the general case, when the number of resources can be arbitrary.

In addition, the production functions of two arguments are widely used in practice, when the researcher is interested in the dependence of the volume of product output on the most important factors - labor costs (L) and capital (K):

q = f(L, K). (3)

A graph of a function of two variables cannot be drawn in a plane.

The production function of the form (2) can be represented in a three-dimensional Cartesian space, two coordinates of which (x 1 and x 2) are plotted on the horizontal axes and correspond to resource costs, and the third (q) is plotted on the vertical axis and corresponds to the product output (Fig. 2) . The graph of the production function is the surface of the "hill", rising with the growth of each of the coordinates x 1 and x 2 . The construction in Fig. 1 in this case can be considered as a vertical section of the "hill" by a plane parallel to the x 1 axis and corresponding to a fixed value of the second coordinate x 2 = x * 2 .

Rice. 2.

The horizontal section of the "hill" combines production options characterized by a fixed output of the product q = q * with various combinations of costs of the first and second resources. If the horizontal section of the surface of the "hill" is depicted separately on a plane with coordinates x 1 and x 2, a curve will be obtained that combines such combinations of resource costs that make it possible to obtain a given fixed volume of product output (Fig. 3). Such a curve is called the isoquant of the production function (from the Greek isoz - the same and the Latin quantum - how much).

Rice. 3.

Let us assume that the production function describes the output depending on the inputs of labor and capital. The same amount of output can be obtained with different combinations of inputs of these resources.

It is possible to use a small number of machines (i.e., to make do with a small outlay of capital), but at the same time a large amount of labor must be expended; it is possible, on the contrary, to mechanize certain operations, to increase the number of machines, and thereby to reduce labor costs. If for all such combinations the largest possible output remains constant, then these combinations are represented by points lying on the same isoquant.

By fixing the output of a product at a different level, we get a different isoquant of the same production function.

After performing a series of horizontal cuts at various heights, we obtain the so-called isoquant map (Fig. 4) - the most common graphical representation of the production function of two arguments. It is similar to a geographical map, on which the terrain is depicted by contour lines (otherwise - isohypses) - lines connecting points that lie at the same height.

Rice. four.

It is easy to see that the production function is in many ways similar to the utility function in consumption theory, the isoquant is similar to the indifference curve, the isoquant map is similar to the indifference map. Later we will see that the properties and characteristics of the production function have many analogies in consumption theory. And it's not just a matter of similarity. In relation to resources, the firm behaves like a consumer, and the production function characterizes precisely this side of production - production as consumption. This or that set of resources is useful for production insofar as it allows you to get the appropriate amount of output of the product. We can say that the values of the production function express the utility for the production of the corresponding set of resources. Unlike consumer utility, this "utility" has a well-defined quantitative measure - it is determined by the volume of products produced.

The fact that the values of the production function refer to technically efficient options and characterize the largest output when consuming a given set of resources also has an analogy in consumption theory.

The consumer can use the acquired goods in different ways. The usefulness of a purchased set of goods is determined by the way they are used in which the consumer receives the greatest satisfaction.

However, with all the noted similarities between consumer utility and "utility" expressed by the values of the production function, these are completely different concepts. The consumer himself, based only on his own preferences, determines how useful this or that product is for him - by buying or rejecting it.

A set of production resources will ultimately prove useful to the extent that the product produced using these resources is approved by the consumer.

Since the production function is inherent in the most general properties utility function, we can further consider its main properties, without repeating the detailed arguments given in Part II.

We will assume that an increase in the costs of one of the resources, while the costs of the other remain unchanged, allows us to increase the output. This means that the production function is an increasing function of each of its arguments. A single isoquant passes through each point of the resource plane with coordinates x 1 , x 2 . All isoquants have a negative slope. The isoquant corresponding to a higher yield of the product is located to the right and above the isoquant for a lower yield. Finally, all isoquants will be considered convex in the direction of the origin.

On Fig. 5 shows some isoquant maps characterizing various situations arising from the production consumption of two resources. Fig. 1. 5a corresponds to the absolute mutual substitution of resources. In the case shown in Fig. 5b, the first resource can be completely replaced by the second one: the isoquant points located on the x2 axis show the amount of the second resource, which makes it possible to obtain one or another product output without using the first resource. The use of the first resource reduces the cost of the second, but it is impossible to completely replace the second resource with the first.

Rice. 5 ,c depicts a situation in which both resources are needed and neither can be completely replaced by the other. Finally, the case shown in Fig. 5d is characterized by absolute complementarity of resources.

Rice. 5.

The production function, which depends on two arguments, has a fairly visual representation and is relatively easy to calculate. It should be noted that the economy uses the production functions of various objects - enterprises, industries, national and world economies. Most often, these are functions of the form (3); sometimes a third argument is added - the cost of natural resources (N):

q = f(L, K, N). (3)

This makes sense if the amount of natural resources involved in production activities, is variable.

In applied economic research and in economic theory, different types of production functions are used. Their features and differences will be discussed in Section 3. In applied calculations, the requirements of practical computability make it necessary to limit ourselves to a small number of factors, and these factors are considered on an enlarged basis - "labor" without subdivision according to professions and qualifications, "capital" without taking into account its specific composition, etc. e. In the theoretical analysis of production, one can abstract from the difficulties of practical computability. The theoretical approach requires that each type of resource be considered absolutely homogeneous. Raw materials of different grades should be considered as different kinds resources, just like cars of various brands or labor, differing in professional and qualification characteristics.

Thus, the production function used in the theory is a function of a large number of arguments:

q \u003d f (x 1, x 2, ..., x n). (four)

The same approach was used in the theory of consumption, where the number of types of consumed goods was not limited in any way.

Everything that was said earlier about the production function of two arguments can be transferred to a function of the form (4), of course, with reservations regarding dimension.

The isoquants of function (4) are not flat curves, but n-dimensional surfaces. Nevertheless, we will continue to use "flat isoquants" - both for illustrative purposes and as a convenient means of analysis in cases where the costs of two resources are variable, and the rest are considered fixed.

Each company, undertaking the production of a particular product, seeks to achieve maximum profit. The problems associated with the production of products can be divided into three levels:

An entrepreneur may be faced with the question of how to produce a given quantity of products in a particular enterprise. These problems relate to the issues of short-term minimization of production costs;
the entrepreneur can decide on the production of the optimal, i.e. bringing more profit, the number of products at a particular enterprise. These questions are about long-term profit maximization;
the entrepreneur may be faced with finding out the most optimal size of the enterprise. Similar questions pertain to long-term profit maximization.

You can find the optimal solution based on an analysis of the relationship between costs and production volume (output). After all, profit is determined by the difference between the proceeds from the sale of products and all costs. Both revenue and costs depend on the volume of production. Economic theory uses the production function as a tool for analyzing this dependence.

The production function determines the maximum amount of output for each given amount of resources. This function describes the relationship between resource input and output, allowing you to determine the maximum possible output for each given amount of resources, or the minimum possible amount of resources to provide a given output. The production function summarizes only technologically efficient methods of combining resources to ensure maximum output. Any improvement in production technology that contributes to an increase in labor productivity leads to a new production function.

PRODUCTION FUNCTION - a function that displays the relationship between the maximum volume of the product produced and the physical volume of production factors at a given level of technical knowledge.

Since the volume of production depends on the volume of resources used, the relationship between them can be expressed as the following functional notation:

Q = f(L,K,M),

where Q is the maximum volume of products produced with a given technology and certain production factors;
L - labor; K - capital; M - materials; f is a function.

The production function with this technology has properties that determine the relationship between the volume of production and the number of factors used. For different types production production functions are different, however? they all have common properties. Two main properties can be distinguished.

There is a limit to the growth in output that can be achieved by increasing the cost of one resource, other things being equal. So, in a firm with a fixed number of machines and production facilities, there is a limit to the growth of output by increasing additional workers, since the worker will not be provided with machines for work.
There is a certain complementarity (completeness) of factors of production, however, without a decrease in the volume of output, a certain interchangeability of these factors of production is also likely. Thus, various combinations of resources can be used to produce a good; it is possible to produce this good by using less capital and more labor, and vice versa. In the first case, production is considered technically efficient in comparison with the second case. However, there is a limit to how much labor can be replaced by more capital without reducing production. On the other hand, there is a limit to the use of manual labor without the use of machines.

In graphical form, each type of production can be represented by a point, the coordinates of which characterize the minimum resources necessary for the production of a given volume of output, and the production function can be represented by an isoquant line.

Having considered the production function of the firm, let's move on to characterizing the following three important concepts: total (cumulative), average and marginal product.

Rice. a) Curve of the total product (TR); b) curve of average product (AP) and marginal product (MP)

On fig. the curve of the total product (TP) is shown, which varies depending on the value of the variable factor X. Three points are marked on the TP curve: B is the inflection point, C is the point that belongs to the tangent coinciding with the line connecting this point with the origin, D – point of maximum TP value. Point A moves along the TP curve. Connecting point A to the origin, we get the line OA. Dropping the perpendicular from point A to the abscissa axis, we get the triangle OAM, where tg a is the ratio of the side AM to OM, i.e., the expression for the average product (AP).

Drawing a tangent through point A, we get the angle P, the tangent of which will express the marginal product MP. Comparing the triangles LAM and OAM, we find that up to a certain point the tangent P is greater than tg a. Thus, marginal product (MP) is greater than average product (AP). In the case when point A coincides with point B, the tangent P takes on a maximum value and, therefore, the marginal product (MP) reaches the largest volume. If point A coincides with point C, then the value of the average and marginal product are equal. The marginal product (MP), having reached its maximum value at point B (Fig. 22, b), begins to decline and at point C it intersects with the graph of the average product (AP), which at this point reaches its maximum value. Then both the marginal product and the average product decrease, but the marginal product decreases at a faster rate. At the point of maximum total product (TP), marginal product MP = 0.

We see that the most effective change in the variable factor X is observed in the segment from point B to point C. Here, the marginal product (MP), having reached its maximum value, begins to decrease, the average product (AR) still increases, the total product (TR) receives the greatest growth.

Thus, the production function is a function that allows you to determine the maximum possible output for various combinations and quantities of resources.

In production theory, a two-factor production function is traditionally used, in which the volume of production is a function of the use of labor and capital resources:

Q = f(L, K).

It can be presented as a graph or curve. In the theory of the behavior of producers, under certain assumptions, there is a unique combination of resources that minimizes the cost of resources for a given volume of production.

The calculation of the firm's production function is a search for an optimum, a choice among many options that provide for various combinations of factors of production, one that gives the maximum possible output. In the face of rising prices and cash costs, the firm, i.e. the cost of acquiring factors of production, the calculation of the production function is focused on finding such an option that would maximize profits at the lowest cost.

The calculation of the firm's production function, seeking to achieve an equilibrium between marginal cost and marginal revenue, will focus on finding such a variant that will provide the required output at minimum production costs. The minimum costs are determined at the stage of calculating the production function by the method of substitution, the displacement of expensive or increased in price factors of production by alternative, cheaper ones. Substitution is carried out with the help of a comparative economic analysis of interchangeable and complementary factors of production at their market prices. A satisfactory option would be one in which the combination of factors of production and a given volume of output meets the criterion of the lowest production costs.

There are several types of production function. The main ones are:

Nonlinear PF;
Linear PF;
Multiplicative PF;
PF "input-output".

Production function and selection of the optimal production size

A production function is the relationship between a set of factors of production and the maximum possible amount of product produced by this set of factors.

The production function is always concrete, i.e. intended for this technology. New technology is a new productivity feature.

The production function determines the minimum amount of input needed to produce a given amount of product.

Production functions, no matter what kind of production they express, have the following general properties:

An increase in production due to an increase in costs for only one resource has a limit (you cannot hire many workers in one room - not everyone will have places).
Factors of production can be complementary (workers and tools) and interchangeable (production automation).

In its most general form, the production function looks like this:

Q = f(K,L,M,T,N),

where L is the volume of output;
K - capital (equipment);
M - raw materials, materials;
T - technology;
N - entrepreneurial abilities.

The simplest is the two-factor model of the Cobb-Douglas production function, which reveals the relationship between labor (L) and capital (K). These factors are interchangeable and complementary.

Q = AK α * L β ,

where A is a production coefficient showing the proportionality of all functions and changes when the basic technology changes (in 30-40 years);
K, L - capital and labor;
α, β are the elasticity coefficients of the volume of production in terms of capital and labor costs.

If = 0.25, then a 1% increase in capital costs increases output by 0.25%.

Based on the analysis of elasticity coefficients in the Cobb-Douglas production function, we can distinguish:

a proportionally increasing production function when α + β = 1 (Q = K 0.5 * L 0.2).
disproportionately - increasing α + β > 1 (Q = K 0.9 * L 0.8);
decreasing α + β< 1 (Q = K 0,4 * L 0,2).

The optimal sizes of enterprises are not absolute in nature, and therefore cannot be established outside of time and outside the location area, since they are different for different periods and economic regions.

The optimal size of the projected enterprise should provide a minimum of costs or a maximum of profit, calculated by the formulas:

Ts + S + Tp + K * En_ - minimum, P - maximum,

where Tc - the cost of delivery of raw materials and materials;
C - production costs, i.e. production cost;
Tp - the cost of delivering finished products to consumers;
K - capital costs;
En is the normative coefficient of efficiency;
P is the profit of the enterprise.

In other words, the optimal size of enterprises is understood as those that provide the targets for the plan for output and increase in production capacity minus the reduced costs (taking into account capital investments in related industries) and the maximum possible economic efficiency.

The problem of optimizing production and, accordingly, answering the question of what should be the optimal size of the enterprise, with all its acuteness, also confronted Western entrepreneurs, presidents of companies and firms.

Those who failed to achieve the necessary scale found themselves in the unenviable position of high-cost producers, doomed to exist on the brink of ruin and ultimately bankruptcy.

Today, however, those US companies that are still striving to compete by saving on concentration are gaining rather than losing. In modern conditions, this approach initially leads to a decrease not only in flexibility, but also in production efficiency.

In addition, entrepreneurs remember that small businesses mean less investment and therefore less financial risk. As for the purely managerial side of the problem, American researchers note that enterprises with more than 500 employees become poorly managed, clumsy and poorly responsive to emerging problems.

Therefore, a number of American companies in the 60s went to the downsizing of their branches and enterprises in order to significantly reduce the size of the primary production links.

In addition to the simple mechanical disaggregation of enterprises, the organizers of production carry out a radical reorganization within enterprises, forming command and brigade org. structures instead of linear-functional ones.

When determining the optimal size of the enterprise, firms use the concept of the minimum effective size. It is simply the lowest level of output at which a firm can minimize its long-run average cost.

Production function and the choice of the optimal production size.

Production is called any human transformation of 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:

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 there comes a point when output stops growing.
Resources complement each other, but within certain limits, their interchangeability is also possible without reducing output. Manual labor, for example, may be replaced by the use of more machines, and vice versa.
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. A short period is a period when at least one resource is fixed. The long period is the period when all resources are variable.

Usually in microeconomics, a two-factor production function is analyzed, reflecting the dependence of output (q) on the amount of labor used ( L) and capital ( K). Recall that capital refers to the means of production, i.e. the number of machines and equipment used in production, measured in machine hours. In turn, the amount of labor is measured in man-hours.

As a rule, the considered production function looks like this:

q = AK α L β

A, α, β - given parameters. Parameter A is the coefficient of total productivity of production factors. It reflects the impact of technological progress on production: if the manufacturer introduces advanced technologies, the value of A increases, i.e., output increases with the same amount of labor and capital. The parameters α and β are the elasticity coefficients of output with respect to capital and labor, respectively. 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.

Building an isoquant

The above production function says that the producer can replace labor with capital and capital with labor, leaving the output unchanged. For example, in agriculture in developed countries, labor is highly mechanized, i.e. there are many machines (capital) for one worker. On the contrary, in developing countries the same output is achieved through a large amount of labor with little capital. This allows you to build an isoquant (Fig. 8.1).

The 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. Yes, release q 1, achievable using L1 labor and K1 capital or using L 2 labor and K 2 capital.

Rice. 8.1. isoquant

Other combinations of the amounts of labor and capital required to achieve a given output are also possible.

All combinations of resources corresponding to this isoquant reflect technically efficient methods of production. Production method A is technically efficient compared to method B if it requires the use of at least one resource in a smaller amount, and all the others not in large quantities compared to method B. Accordingly, method B is technically inefficient compared to A. Technically inefficient modes of production are not used by rational entrepreneurs and do not belong to the production function.

It follows from the above that an isoquant cannot have a positive slope, as shown in Fig. 8.2.

The segment marked with a dotted line reflects all technically inefficient methods of production. In particular, in comparison with method A, method B to ensure the same output ( q 1) requires the same amount of capital but more labor. It is obvious, therefore, that way B is not rational and cannot be taken into account.

Based on the isoquant, it is possible to determine the marginal rate of technical replacement.

The marginal rate of technical replacement of factor Y by factor X (MRTS XY) is the amount of the factor Y(for example, capital), which can be abandoned by increasing the factor X(for example, labor) by 1 unit so that the output does not change (we remain on the same isoquant).

Rice. 8.2. Technically efficient and inefficient production

Consequently, the marginal rate of technical replacement of capital by labor is calculated by the formula
For infinitely small changes in L and K, it is
Thus, the marginal rate of technical replacement is the derivative of the isoquant function at a given point. Geometrically, it is the slope of the isoquant (Fig. 8.3).

Rice. 8.3. Marginal rate of technical replacement

When moving from top to bottom along the isoquant, the marginal rate of technical replacement decreases all the time, as evidenced by the decreasing slope of the isoquant.

If the producer increases both labor and capital, then this allows him to achieve a higher output, i.e. move to a higher isoquant (q2). An isoquant located to the right and above the previous one corresponds to a larger output. The set of isoquants forms an isoquant map (Fig. 8.4).

Rice. 8.4. Isoquant map

Special cases of isoquants

Recall that the given isoquants correspond to a production function of the form q = AK α L β. But there are other production functions. Let us consider the case when there is a perfect substitution of factors of production. Let us assume, for example, that skilled and unskilled loaders can be used in warehouse work, and the productivity of a skilled loader is N times higher than that of an unskilled one. This means that we can replace any number of skilled movers with unskilled ones at a ratio of N to one. Conversely, one can replace N unskilled loaders with one qualified one.

The production function then looks like: q = ax + by, where x- the number of skilled workers, y- the number of unskilled workers, a and b- constant parameters reflecting the productivity of one skilled and one unskilled worker, respectively. The ratio of the coefficients a and b is the marginal rate of technical replacement of unskilled movers by qualified ones. It is constant and equal to N: MRTSxy=a/b=N.

Let, for example, a qualified loader be able to process 3 tons of cargo per unit time (this will be the coefficient a in the production function), and an unskilled one - only 1 ton (coefficient b). This means that the employer can refuse three unskilled loaders, additionally hiring one qualified loader, so that the output (total weight of the handled load) remains the same.

The isoquant in this case is linear (Fig. 8.5).

Rice. 8.5. Isoquant under perfect substitution of factors

The tangent of the slope of the isoquant is equal to the marginal rate of technical replacement of unskilled movers by qualified ones.

Another production function is the Leontief function. It assumes a rigid complementarity of factors of production. This means that the factors can only be used in a strictly defined proportion, the violation of which is technologically impossible. For example, an air flight can normally be operated with at least one aircraft and five crew members. At the same time, it is impossible to increase aircraft-hours (capital) while simultaneously reducing man-hours (labor), and vice versa, and to keep output unchanged. Isoquants in this case have the form of right angles, i.e. the marginal rates of technical replacement are zero (Fig. 8.6). At the same time, it is possible to increase output (the number of flights) by increasing both labor and capital in the same proportion. Graphically, this means moving to a higher isoquant.

Rice. 8.6. Isoquants in the case of rigid complementarity of factors of production

Analytically, such a production function has the form: q = min (aK; bL), where a and b are constant coefficients reflecting the productivity of capital and labor, respectively. The ratio of these coefficients determines the proportion of the use of capital and labor.

In our flight example, the production function looks like this: q = min(1K; 0.2L). The fact is that the productivity of capital here is one flight for one aircraft, and the productivity of labor is one flight for five people, or 0.2 flights for one person. If an airline has a fleet of 10 aircraft and 40 flight personnel, then its maximum output will be: q = min( 1 x 8; 0.2 x 40) = 8 flights. At the same time, two aircraft will be idle on the ground due to a lack of personnel.

Let us finally look at the production function, which assumes the existence of a limited number of production technologies for the production of a given amount of output. Each of them corresponds to a certain state of labor and capital. As a result, we have a number of reference points in the “labor-capital” space, connecting which, we get a broken isoquant (Fig. 8.7).

Rice. 8.7. Broken isoquants in the presence of a limited number of production methods

The figure shows that the output in the volume q1 can be obtained with four combinations of labor and capital, corresponding to points A, B, C and D. Intermediate combinations are also possible, achievable in cases where the enterprise jointly uses two technologies to obtain a certain total release. As always, by increasing the amount of labor and capital, we move to a higher isoquant.