production optimum. The firm's optimum is the minimization of its costs. Task illustrating the concept

I. ECONOMIC THEORY

11. The theory of behavior of the manufacturer. Manufacturer's Optimum

The production function reflects different ways of combining factors to produce a certain amount of output. The information that a production function carries can be represented graphically using isoquants.

isoquant is a curve on which all combinations of production factors are located, the use of which provides the same output (Fig. 11.1).

Rice. 11.1. Isoquant plot

In the long run, when a firm can change any factor of production, production function characterized by such an indicator as the marginal rate of technological substitution factors of production(MRTS)

where DK and DL are changes in capital and labor for a single isoquant, i.e. for constant Q.

The firm is faced with the problem of how to achieve a certain level of production at minimum cost. Assume that the price of labor equals the wage rate (w) and the price of capital equals the rent for equipment (r). Production costs can be represented as isocosts. Isocost includes all possible combinations of labor and capital with equal gross costs

Rice. 11.2. isocost chart

Rewrite the gross cost equation as the equation for straight line, we get

It follows from this that the isocost has a slope equal to

It shows that if a firm forgoes a unit of labor and saves w (c.u.) to acquire a unit of capital at a price of r (c.u.) per unit, then gross production costs remain unchanged.

The equilibrium of the firm occurs when it maximizes profit at a certain volume of production with an optimal combination of production factors that minimize costs (Fig. 11.3).

On the graph, the equilibrium of the firm reflects the point of contact T of the isoquant with the isocost at Q 2 . All other combinations of factors of production (A, B) can produce less output.

Rice. 11.3. consumer equilibrium

Given that the isoquant and the isocost have the same slope at T, and that the slope of the isoquant is measured by the MRTS, the equilibrium condition can be written as

The right side of the formula reflects the utility for the producer of each unit of the factor of production. This utility is measured by the marginal product of labor (MP L) and capital (MP K)

The last equality is the producer's equilibrium. This expression shows that the producer is in equilibrium if 1 ruble invested in a unit of labor is equal to 1 ruble invested in capital.

Production- any human activity aimed at converting resources into necessary goods that are designed to meet needs.

production function- this is the ratio between the resources expended by the firm (labor, capital, land, entrepreneurial ability) and the products or services received. Determines the maximum amount of product produced for each given amount of resources.

Mathematically, the production function is represented as follows: Q=f(K,L,N), where Q is the maximum volume of product that can be produced with a given technology and a certain number of production factors; K, L, N - the spent amount of various types of resources (capital, labor, land).

The production function is always concrete, i.e. reflects the relationship between the maximum possible volume of the product and the amount of resources required for this technology. If will be used new technology- it will be characterized by a new production function.

Graphic image production function is an isoquant - a curve on which all combinations of production factors are located that provide the same output.

Isoquant - is a curve on which all combinations of production factors are located, the use of which provides the same output.

Optimum- Producer equilibrium - a combination of resources that gives the maximum output when they are fully used.

Equilibrium (optimum) the producer is characterized by the point of contact of the isocost and isoquant - point e - the total amount of costs for the production of this output is minimized.

Isocost - a line showing combinations of factors of production that can be bought for the same total amount of money.

The transition from a low isoquant to a higher one indicates an expansion of production (an increase in output)

When prices change, first, the firm's profitability changes; second, the firm can purchase more of the cheaper resource. One can consider decomposing the overall effect of price changes into a substitution effect and an income effect.

Expanding production, the company is faced with the concept of " returns to scale". It shows how much the volume of production increases with an increase in the use of factors of production.

There are: increasing, constant, as well as decreasing returns to scale of production:

Increasing returns to scale- a situation in which a proportional increase in all factors of production leads to an increasing increase in the volume of output. Assume that all factors of production are doubled and output is tripled.

Constant returns to scale- this is a change in the number of all factors of production, which causes a proportional change in the volume of output of the product. Thus, twice the number of factors exactly doubles the output of the product.

Diminishing returns to scale- this is a situation in which a balanced increase in the volume of all factors of production leads to an ever smaller increase in the volume of output. In other words, the volume of output increases to a lesser extent than the inputs of production factors. For example, all factors of production have tripled, but the volume of production has only doubled.

Positive returns to scale can be achieved through the following factors:

1) division of labor

2) improved management

3) an increase in the scale of production most often does not require a proportional increase in the cost of all resources.

Reasons for negative returns to scale:

1) significant inertia and loss of flexibility in large enterprise;

2) the exit of the enterprise beyond the threshold of manageability - its significant size creates a cumbersome management system prone to bureaucracy, which negatively affects production efficiency.

For a profit maximizing firm, the best combination of factors will be the one that provides the lowest cost. Therefore, the task of the firm is to ensure the minimization of costs for each given volume of production. An isoquant is used to identify all possible combinations.

Isoquant (constant (equal) product curve)- a curve representing an infinite number of combinations of factors of production (resources) that provide the same output.

Properties of an isoquant: have a negative slope, are convex about the origin, and never intersect each other.

A set of isoquants, each of which shows the maximum output achieved using certain combinations of resources is called isoquant map.

With the help of the slope of isoquants, one can determine the degree of substitution of one factor of production by another. The slope of the isoquant shows us how a given substitution occurs. Therefore, the absolute value of this coefficient characterizes technical limit (or technological) substitution - MRTS.

The marginal rate of technological substitution is directly related to the marginal products of production factors. By reducing the amount of one of the factors, such as capital (ΔK), the firm thereby reduces the volume of output by a certain amount. This value is equal to the product of the marginal product of capital (MR K) and changes in its quantity (ΔK):

Δ Q=MP K  (-ΔK) (7.1),

where: Δ Q- change in the volume of output; MR To is the marginal product of capital; Δ K- change in the amount of capital used.

In order to stay on the same isoquant, the decrease in output must be compensated by an increase in the amount of labor applied (ΔL), i.e.

Δ Q=MP L  ∆L (7.2),

where: MP L is the marginal product of labor Δ L- change in the amount of labor used.

This means that the absolute value of ΔQ in equations (7.1) and (7.2) must be the same. Therefore, we can write:

MRTS KL = –  K / L.

And
Zoquants can have different forms depending on the degree of interchangeability of resources:

1) resources may have absolute interchangeability. This means that a given volume of output can be provided both by using any one of the two variable resources, and by their combinations. In this case, the isoquant will look like a straight line, and the MRTS will be a constant value;

2
) resources have the property absolute complementarity. This means that the two variable resources used to produce a given product share the same specific proportion. In other words, a given production function assumes that there is only one possible combination of resources. In this case, MRTS will be equal to 0, and the isoquant will look like a right angle;

3
) isoquants reflecting partial interchangeability resources. In this case, the production of products can be carried out with the obligatory use of two variable resources, for example, labor and capital. However, their combinations can be very different in accordance with a given production function. This form isoquant is the most common, and it is considered to be the standard.

D
To obtain the optimum, you need to make sure that the costs are minimal and the income is maximum.

To maximize output at given costs allows isocost (straight line of equal costs). If a R K is the price K, and R L is the price L, then, having a certain budget B, our manufacturer can buy K units of capital and L units of labor:

B=P K  K+P L  L.

The slope of the isocost is equal to the ratio of the prices of the factors used multiplied by (-1), since the isocost has a negative slope. In other words, if a firm increases the amount of one factor, then it must correspondingly reduce the use of another in order to keep the total cost of acquiring factors unchanged, i.e. P L  ΔL = - (P K  ΔK). From this it follows that: – ∆K/ ΔL = P L / P K .

Any change in the price of one of the two resources used leads to a change in the slope of the isocost.

To Touching an isoquant with an isocost determines the position equilibrium manufacturer, because it allows you to achieve the maximum volume of production with the limited funds available that can be spent on the purchase of resources.

The combination of factors at point A will provide the lowest cost with the volume of output equal to Q 1 ; at point B - a volume equal to Q 2; at point C - a volume equal to Q 3. All other possible combinations of factors belonging to isoquants with production volumes Q 1 , Q 2 , Q 3 , respectively, lie on higher lines of the budget constraint. By connecting points A, B, C, we get a curve showing the optimal combinations of resources at existing prices for them for each given volume of output. When deciding on the volume of production, the firm will move along a given curve, which is commonly called growth trajectory.

The fact that cost minimization is achieved at the point where the isocost and isoquant touch, allows us to conclude that, as is known, the slope of the isocost is equal to the ratio of factor prices (P L / P K), and the slope of the isoquant is equal to MRTS KL . At the point of contact, the slope of the isocost is equal to the slope of the isoquant. Therefore, equilibrium is reached when the ratio of the prices of the factors is equal to the ratio of their marginal products, i.e. P L / P K = MP L / MP K .

The intersection of isoquants with an isocost 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:

- if MP L / P L > MR To / R K, the firm minimizes its costs by replacing capital with labor. During this replacement, the marginal product of labor will decrease and the marginal product of capital will increase. The substitution will be carried out until the equality of the factors weighted at the corresponding prices of the marginal products is reached;

- if MP L / P L < MP K / R To, then the firm should replace labor with capital to achieve equality

The optimum will be reached if MP L / P L = MP K / P K is the rule of cost minimization.

MRP L / P L = MRP K / P K = 1 is the profit maximization rule.

Compliance with this condition means that the company is functioning efficiently, i.e. the optimal combination of factors is ensured, minimizing production costs, with the only possible output that maximizes profit.

Optimization principle: the firm seeks to choose the best set of factors of production (K ,L) from among those that it can afford.

Balance principle: the firm purchases labor and capital at prices and combines these factors in such a way as to achieve an equilibrium between the quantity supplied and the quantity demanded for its products.

3.1. Equilibrium of the producer in the short run.

To solve this problem, tools are used: isoquant and isocost.

isoquant is a curve reflecting all the different combinations of resources that can be used to produce a given volume of output. The isoquant shows the multivariance of the production of a given volume of output. Highly mechanized technology can be used, or, on the contrary, technology that uses a minimum of technology (in the economic sense of capital) and a maximum of labor. Isoquants are similar to indifference curves. Just as indifference curves represent alternative consumer product choices that provide a given level of utility, isoquants represent alternative cost combinations to produce a given quantity of output.

For simplicity of analysis, as before, we will assume that:

The investigated function of production depends on two factors: labor and capital;

Factors of production will be interchangeable within certain limits;

The technology of production during the entire period under review does not change.

Let's present this function in the form of a table for values u from 1 to 4.

As can be seen from the table, there are several combinations of labor and capital that provide, within certain limits, a given volume of output. An example can be obtained using a combination of (1.4), (4.1) and (2.2).

If you plot the number of units of labor on the horizontal axis and the number of units of capital on the vertical axis, then plot the points at which the firm produces the same amount, you get the curve shown in Figure 14.1 and is called the isoquant.

Each point of the isoquant corresponds to the combination of resources at which the firm produces a given volume of output.

The set of isoquants characterizing a given production function is called isoquant map.

Properties of isoquants

The properties of standard isoquants are similar to those of indifference curves:

An isoquant, like an indifference curve, is a continuous function, not a set of discrete points.

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

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

Marginal rate of technological substitution one resource to another (for example, labor to capital) shows the degree of substitution of labor by capital, in which the volume of output remains unchanged.

An algebraic expression showing the degree to which a producer is willing to reduce the amount of capital in exchange for an increase in labor sufficient to maintain the same output is: .

As you can see in the figure above, when moving from point to point, the volume of production remains unchanged. This means that the decrease in output as a result of a decrease in the cost of capital is compensated by an increase in output due to the use of an additional amount of labor. .

The decrease in output as a result of a decrease in capital input is equal to the product of the marginal product of capital, or. The increase in output due to the use of additional labor is in turn equal to the product times the marginal product of labor, or.

Thus, it can be written that . Let's write this expression in a different way: or.

The production function, which links the amount of capital, labor and output, also allows us to calculate the marginal rate of technological substitution through the derivative of this function: .

This means that graphically, at any point of the isoquant, the limiting degree of technological substitution is equal to the tangent of the slope of the tangent to the isoquant at this point.

Example 14.2 Finding the MRTS for a Given Function

Condition: Let the production function look like .

Define: Approx.

Solution:

Obviously, the degree of substitution of labor by capital does not remain constant when moving along the isoquant. When moving down the curve, the absolute value of the MRTS of labor by capital decreases, since an increasing amount of labor has to be used to compensate for the decrease in capital costs (So, in the above example, at L=1 MRTS=-10, and at L=10 MRTS=- 0.1.)

In the future, MRTS reaches its limit (MRTS=0), and the isoquant becomes horizontal. It is obvious that a further reduction in capital costs will only lead to a reduction in output. The amount of capital at point E is the minimum allowable for a given volume of production (similarly, the minimum allowable amount of labor for the production of a given volume takes place at point A).

Decreasing marginal rate of technological substitution

Decreasing MRTS of one resource by another is typical for most production processes and is typical for all isoquants of the standard form.

Special cases of the production function (non-standard isoquants)

Perfect interchangeability of resources

If the resources used in the production process are absolutely replaceable, then it is constant at all points of the isoquant, and the isoquant map looks like in Figure 14.2. (An example of such production is a production that allows both full automation and manual production of a product).

Fixed structure of resource usage

If the technological process excludes the substitution of one factor for another and requires the use of both resources in strictly fixed proportions, the production function has the form of a Latin letter, as in Figure 14.3.

An example of this kind is the work of a digger (one shovel and one person). An increase in one of the factors without a corresponding change in the amount of the other factor is irrational, therefore only angular combinations of resources will be technically effective (the corner point is the point where the corresponding horizontal and vertical lines intersect).

Isocost - a line, all points of which reflect a combination of labor and capital, having the same total value, i.e. all combinations of factors of production with equal total costs.

As we have already found out earlier, a set of isoquants of an individual firm (isoquant map) show the technically possible combinations of resources that provide the firm with the appropriate output volumes. However, when choosing the optimal combination of resources, the manufacturer must take into account not only the technology available to him, but also his financial resources, as well as prices for relevantfactors of production.

The combination of the last two factors determines area of economic resources available to the producer.

The producer's budget constraint can be written as an inequality:

If the manufacturer fully spends its money on the acquisition of these resources, then we get the equality:

The resulting equation is called isocost equation.

isocost line shown in Figure 14.4 shows the set of combinations of economic resources (in this case, labor and capital) that a firm can acquire given market prices for resources and fully utilizes its budget. The slope of the isocost line is determined by the ratio of market prices for labor and capital (- Р L /Р K), which follows from the isocost equation.

Optimal combination of resources

The firm's commitment to efficient production encourages it to achieve the maximum possible output at a given cost of resources, or, what is the same, to minimize costs in the production of a given volume of output.

The combination of resources that provides the minimum level of the firm's total costs is called optimal and lies at the point of contact of the isocost and isoquant lines.

By combining isoquats and isocosts, one can determine the optimal position of the firm. The point at which the isoquant touches the isocost indicates the cheapest combination of factors required to produce a given volume of output.

American economists Douglas and Solow found that a 1% increase in labor costs provides 3/4 of the increase in output, and a 1% increase in capital costs makes it possible to increase the amount of output by 1/4.

These indices (3/4 and 1/4) were called aggregate, and the relationship between output and factors of production came to life under the name of the aggregate function of production. which suggests that investments in human capital have a greater effect in increasing production than the growth of means of production.

Cost minimization rule

The touch point of the isocost and isoquant determines the optimum (equilibrium) of the producer in a short period. In this case, the following conditions must be met:

The firm must fully spend the budget intended for the acquisition of resources;

The firm should allocate allocated for the acquisition resource means, so that the marginal rate of technological replacement of capital by labor is equal to the ratio of the price of labor to the price of capital MRTS LK =P L /R to .

Cost minimization rule for a given volume of output, due to the optimal combination of resources: the company must allocate funds for the acquisition of resources in such a way that every last ruble spent on each resource brings an equal increase in output:

We must understand that cost minimization involves the purchase of such a set of labor (L) and capital (K), which allows the firm to maximize profits.

The firm's desire for efficient production encourages it to achieve the maximum possible output for a given cost of resources, or, equivalently, to minimization of costs in the production of a given volume of output.

The combination of resources that provides the minimum level of total costs of the firm is called balanced (optimal) and lies at the point of contact of the isocost and isoquant lines, as shown in Figure 9.

Fig.9 Optimum point

Optimal combination of resources involves the implementation following conditions:

1) the equilibrium combination of resources (K*,L*) always lies on the isocost line, and not below it. This means that in order to minimize costs, the firm must make full use of the means for the purchase of resources .

2) at the equilibrium point, the slope of the isoquant curve is equal to the slope of the isocost line.

Since tg of the slope of the isoquant curve = ,

tg isocost line slope = -PL/PK,

then, consequently, the second optimum condition assumes such a distribution of the costs of the firm, under which the marginal rate of technological replacement of one resource by another is equal to the ratio of their prices.

The economic meaning of this condition:

MRTS defines opportunity technological substitution capital by labor. Price ratio reflects economic the ability of the producer to replace capital with labor. Until these opportunities are equalized, changes in the ratio of resources used will lead to an increase in output or a decrease in the firm's total costs.

Second maximization condition can be written as

When n amount of resources, the expression takes the form

This means that the firm must allocate its budget in such a way as to obtain the same surplus product per ruble spent to acquire each resource.

8.3.4. Path (trajectory) of development and returns to scale.

Let us assume that the prices of resources remain unchanged, while the financial resources of the producer, which he has, are constantly growing - this is expressed in a parallel shift of the isocost to the right-up. By connecting the touch points of isoquant and isocost, we get a line - " path (trajectory) of development". The set of optimum points of the producer, built for a changing volume of production, and therefore, changing costs (TC) of the company with the prices of resources unchanged, reflects development trajectory firms (Figure 10). This line shows the growth rate of the ratio between factors in the process of expanding production.

Fig.10 Development trajectory

The shape of the development trajectory is considered, as a rule, in the long term and allows us to identify capital-intensive (Fig. 11a), labor-intensive (Fig. 11b) production methods, as well as technologies that involve a uniform increase in the use of both labor and capital (Fig. 11c).

Fig.11abc Various forms development trajectories

If the distances between isoquants decrease, this indicates increasing returns to scale- increase in output, due to the relative saving of resources. (Fig.12)

Rice. 13 Diminishing returns to scale.

In the case when an increase in production requires a proportional increase in resources, one speaks of constant returns to scale. (Fig.14)

Rice. 14 Constant returns to scale.

In this way, isoquant as an analysis tool allows not only to economically use available resources to achieve a given production volume, but also to determine minimum efficient enterprise size in branch.

In the case of increasing returns to scale firm it is necessary to increase the volume of production, as this leads to a relative saving of available resources.

Decreasing returns to scale indicate that the minimum efficient enterprise size has already been reached. and further increase in production is impractical.

Lecture 9. Firm as a subject market economy:

production costs, income, profit; behavior in the short and long term.

The nature of costs. Total income. External and internal costs. Economic and accounting profit. Search for profit and search for rent. The firm's costs short term: fixed and variable costs. Average and marginal costs. Gross, average and marginal revenue of the firm. Goals and tasks solved by the company when entering the market in a short time interval. Scale effect and costs of the firm in the long run.

In the previous topic, the firm was analyzed as a production unit that transforms inputs into a new product from the standpoint of technological and economic efficiency in the short and long term. Now consider the firm as a business unit acquiring the necessary inputs used to produce a new product and bearing themes most production costs, hoping to sell a new product at high prices and get revenue(total income) exceeding production costs. The main questions of our research in this topic will be: different kinds costs that make up the cash outflow of the firm; various types of income that make up the firm's cash flow; the ratio of the corresponding types of costs and income: profit (positive excess of income over costs) and losses (excess of costs over income).

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