In the long run, the firm can change the quantity of all factors used, so the manufacturer needs to determine the optimal combination of inputs used to ensure maximum output. To solve this problem, consider two new economic categories: isoquant (equal output or equal product curve) and isocost (equal cost line).
Isoquant is a curve whose points reflect various combinations of input factors that ensure the same output.
Rice. 2.24. Isoquant map
Let us assume that the firm uses only two factors – labor and capital. Then the isoquant ( Q 1 ) will have the following form (Fig. 2.24):
If we place several isoquants on one graph, we get isoquant map . Equal output curves (by analogy with indifference curves, see Subsection 2.2) have the following properties:
1) isoquants have a negative slope: when moving from a point A to the point B the decrease in the amount of capital must be compensated by an increase in labor input to maintain the same volume of production;
2) isoquants do not intersect;
Q 2 > Q 1 .
The replacement of one factor of production by another while maintaining a constant volume of output reflects the angular coefficient of the slope of the tangent to the isoquant. The absolute value of this coefficient is called the marginal rate of technological substitution ( MRTS) It is determined by the formula:
The marginal rate of technological substitution of capital by labor represents the amount by which capital must be reduced by using one additional unit of labor at a fixed level of output (always treated as a positive quantity and similar to the marginal rate of substitution used in consumer choice theory). The more capital is replaced by labor, the less productive labor becomes, and the use of capital decreases.
more effective. Conversely, the more labor is replaced by capital, the less productive capital becomes and labor more productive.
The entrepreneur buys the factors used on the market and when choosing a combination of them, he must take into account their market prices, as well as the size of his budget.
Isocosta is a straight line, each point of which shows different combinations of two variable factors involved in production at the same costs for their acquisition (Fig. 2.25, line C 1 ).
The isocost equation has the form:
(2.21)
Where C– manufacturer’s budget or costs of purchasing factors of production; r– price of capital; w– price of labor,
where is the angle of inclination of the isocost to the abscissa axis.
The properties of isocosts are similar to the properties of the budget line (see subsection 2.2): negative slope, points of intersection with the axes, line slope angles, changes in the producer budget and prices of production factors.
If there are many combinations of using production factors to achieve a certain volume of output, then the question inevitably arises: which combination of their many will be the most optimal, i.e. allowing to achieve a given output volume with minimal costs?
Rice. 2.26. Optimal combination of production factors used
To determine the optimal combination of production factors used, it is necessary to combine the isoquant map with the isocost (Fig. 2.26). This shows that the isocost at the point E concerns the isoquant. This means that the entrepreneur’s costs for purchasing production factors will be minimal. Other combinations of factors (for example, points A And B) are not optimal, since with the same costs for their acquisition (points A, B, E belong to the same isocost) provide a smaller volume of output (points A And B lie on an isoquant Q 1 , and point E– on an isoquant Q 2 ). Combination of factors corresponding to a point F(which belongs to the same isoquant as the point E, and, therefore, provides the same volume of output Q 2 ) is not available to the manufacturer, since it does not lie on the isocost.
Therefore, the point E – This is the producer's equilibrium point, which corresponds to a combination of production factors that ensures maximum output at minimum costs for the acquisition of production resources.
It should also be noted that at the point E the condition called cost minimization rules when using production factors. This condition has the following form:
Thus, to minimize costs (for a given volume of production), it is advisable for a company to replace one factor with another until the ratio of the marginal product of each factor to the price of a given factor is equal to the value for all factors involved. In other words, equation (2.23) shows that at minimum total costs, each additional monetary unit of input costs adds the same amount of output.
Suppose that only 2 resources are used in production, for example, labor (L) and capital (K) (Fig. 6.7). If we combine all combinations of resources, the use of which will provide the same volume of output, we get isoquants.
Isoquant, or constant (equal) product curve, - a curve representing an infinite number of combinations of production factors that ensure the same output.
The slope of an isoquant expresses the dependence of one factor on another in the production process. At the same time, an increase in one factor and a decrease in another does not cause changes in the volume of output.
A positive slope of an isoquant means that an increase in the use of one factor will require an increase in the use of another factor so as not to reduce output. A negative slope of an isoquant shows that a reduction in one factor (at a given level of production) will always cause an increase in another factor.
Isoquants are convex in the direction of the origin, because Although factors can be replaced by one another, they are not absolute substitutes.
Fig.6.7. Isoquants
The curvature of the isoquant illustrates the elasticity of substitution of factors in producing a given volume of product and reflects how easily one factor can be replaced by another. In the case where the isoquant is similar to a right angle, the probability of replacing one factor with another is extremely small. If the isoquant looks like a straight line with a downward slope, then the probability of replacing one factor with another is significant.
An isoquant lying above and to the right of another represents a larger volume of output. A set of isoquants, each of which shows the maximum output achieved using certain combinations of resources, is called isoquant map.
An increase in the cost of factor F1 (labor) compensates for a decrease in the cost of factor F2 (capital). Limit rate of technical substitution or technological replacement ( MRTS ) - the amount of one resource that can be reduced in exchange for a unit of another resource while maintaining the total volume of output unchanged.
The slope of the isoquant measures the marginal rate of technological substitution.
Isoquants can have different configurations: linear, rigid complementarity, continuous substitutability, broken isoquant.
Linear isoquant is an isoquant expressing the perfect substitutability of production factors (MRTS LK = const). The graph of such an isoquant is similar to the isocost graph.
Strict complementarity of production factors represents a situation in which labor and capital are combined in the only possible ratio, when the marginal rate of technical substitution is equal to zero (MRTS LK = 0), the so-called Leontief-type isoquant.
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Fig.6.8. Isocost (a) and producer equilibrium (b)
The isocost (direct line of equal costs) allows you to maximize output at given costs. Isocosta is a straight line showing all combinations of resources whose use requires the same costs.
An increase in the manufacturer's budget or a decrease in resource prices shifts the isocost to the right, and a budget reduction or increase in prices shifts it to the left (Fig. 6.8a). The tangency of the isoquant with the isocost determines the equilibrium position of the producer (point T), since allows you to achieve maximum production volume with the limited funds available.
Let's assume that resource prices remain constant and the producer's budget is constantly growing. By connecting the intersection points of isoquants with isocosts, we get the “development path” line (Fig. 6.9). This line shows the growth rate of the ratio between factors in the process of expanding production.
Fig.6.9. Development Path Curve
The shape of the “development path” curve depends on the shape of the isoquants and on resource prices (the ratio between which determines the slope of the isocosts). The development path line can be a straight line or a curve starting 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 relative savings in resources. If the distance between isoquants increases, this indicates diminishing economies of scale . In the case where an increase in production requires a proportional increase in resources, they say about constant economies of scale.
Thus, the isoquant allows not only to economically use available resources to achieve a given volume of production, but also to determine the minimum effective size of an enterprise in the industry. In the case of increasing economies of scale, the company needs to increase production volume, because this leads to relative savings in resources.
Output analysis using isoquants allows determine the technical efficiency of production. The intersection of isoquants with isocost allows us to determine economic efficiency , i.e. choose a technology (labor- or capital-saving, energy- or material-saving) that allows for maximum production output with the funds available to the manufacturer.
The task of any manufacturer is minimize financial losses and achieve maximum volume of output.
To do this, you need to correctly combine all resources, especially for a long-term period of work, when external factors are constantly changing.
In order to solve this problem, new economic categories were introduced: isoquant, isocost, isoprofit. Let's look at each of them in detail.
Isoquant is the equal output/equal product curve. It represents a line connecting points that depict various options for combining factors to maintain the production of a product at the same level.
Let's assume that the company uses two main factors: labor and capital resources. Then the isoquant will look like this (in Fig. 1. Designated Q1):
Fig. 1 - Isoquant graph
A diagram showing several such lines is called an isoquant map.
Let's consider properties of equal product curves (isoquants):
The angular coefficient of the slope of the tangent line to the isoquant is an indicator indicating the replacement of a production factor with another when producing the same amount of goods. Its numerical value is calculated using the formula: MRTS= -K/L. This indicator is called the maximum rate of technical substitution.
In our example rate of substitution limit is the amount by which capital must be reduced when additional labor units are added. With such substitution, labor is less productive and capital investments are used more efficiently.
The manufacturer acquires these factors on the labor market, taking into account possible financial costs and market prices for resources.
Let's consider situations in which The equal production curve looks unusual:
A line consisting of points that show different combinations of two non-constant factors used in production, with the same price for their purchase, is called isocost.
Let's consider the so-called isocost map(Fig.2)
Rice. 2 – Isocost map
Isocost formula: С=rK+wL.
C is the cost of production factors, r is the cost of capital, w is the cost of labor.
Isocosts have the same properties as budget lines:
It is beneficial for the manufacturer to select the right combination of production factors that will allow the production of the specified volume of product with the least financial losses.
To correctly combine resources, isoquant and isocost maps are combined (Fig. 3.)
Rice. 3 - Combined isocost and isoquant map
E on this graph - the point of tangency of two lines. It is called the equilibrium point of production. It is at this value that the manufacturer will receive the minimum costs when purchasing resources. Other points of the image (For example, A and B) are not optimal, because they show a smaller volume of product output at the same costs. At point F, the purchase of resources is generally impossible, because it does not belong to the isocost.
The condition reached at point E on the graph is called minimization production costs .
A combination of optimal points for production, created for variable production volumes and costs, while maintaining a stable cost of resources, determines the development trajectory of the enterprise. The trajectory can take many forms and is usually considered over the long term. It allows you to conclude whether production is labor-intensive or capital-intensive and select technologies for the uniform use of all resources.
Conclusion: to minimize costs, it is profitable for the company to replace one production factor others until the ratio of the volumes of all resources to the prices of these resources becomes equal.
To maintain profit maximization, every company must adhere to two important rules that can be used for any market conditions :
The main condition for obtaining the maximum possible income is the opportunity to make a profit from all units of production produced. To study the factors on which a company's income depends, concepts such as marginal, average and total income are used.
In general, profit can be calculated as the difference between total income and total costs. Formula: TP=TR-TC.
The equation for the profit function in production with two main resources and one type of product: TP=TR-TC=PQ-(rK+wL).
K here is the volume of capital, L is the number of labor units, r is the cost of one capital unit, w is the cost of a labor unit.
Using the equation of the profit function, you can construct its graph. For this purpose, we express the quantity of products produced through the amounts of income and costs:
Q=TP/P+rK/P+wL/P.
Let us assume that the amount of capital used is constant in the short term. Then we depict on the graph the dependence of product output volumes on the variable values of labor units. We get parallel inclined lines - isoprofits. (Fig.4) The angle between these lines and the horizontal coordinate axis is calculated using the formula w/P, the equation for the point of intersection of them with the vertical: TP/P+rK/P.
Rice. 4 - Isoprofits
Another name for isoprofits– equal profit curve. This is a set of points showing the combination of the volume of output of a product and the amount of a variable resource at which one level of income is achieved.
Using a company's production function and production curve, it is easy to figure out what level of production and level of resource use is needed to generate maximum revenue.
Rice. 5 - Making the most profit
Let's look at Fig.5. It shows that the company receives the greatest profit at the point of intersection of the highest isoprofit with the production schedule.
In long-run production, all factors are variable, as is the income function. Mathematically, this can be expressed as follows: the function is maximum if the first two derivatives have zero value.
Using isoprofit you can construct Cournot oligopoly model. The latter is a variant of market competition and is named after a French scientist. Let us briefly explain the essence of this model:
All participants should know the number of companies present on the market. Each of them considers the output volumes of other firms to be constant. Costs may vary.
A special case is a duopoly (two organizations participate in the process). Under equilibrium conditions, each duopolist, producing its product, fulfills 1/3 of the market's needs. Having together covered 2/3 of demand, production participants provide the greatest profit for themselves, but not for the entire industry. They could maximize total income if they took into account their errors in calculating each other's output and entered into a formal or informal agreement to form a monopoly. This situation would divide the market in half, and each company would cover 1/4 of the demand.
The Cournot duopoly model has been criticized more than once, because its participants make incorrect assumptions about the competitor’s behavior, technical costs cannot be zero, and the number of enterprises is constant, which does not lead to equilibrium.
Some of these disadvantages may disappear with adding response curves to the Cournot model. But before that, you need to pay attention to the equal profit curves - isoprofits. In this model, they represent a set of points showing the combination of outputs of both duopolists, at which one of the participants achieves a constant level of profit. For the second duopoly, isoprofit has a similar meaning.
Response curves- this is a set of points of the highest profit possible for one duopolist, with a fixed value of the output of the other.
Thus, the market is in a state of equilibrium only when each enterprise does not change its strategy alone, but can only respond to changes in the behavior of competitors in the market.
For simplicity of analysis, as before, we will assume that:
Let's present it in the form of a table this function for values and from 1 to 4.
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→ |
1 | 2 | 3 | 4 |
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 6 | 8 |
| 3 | 3 | 6 | 9 | 12 |
| 4 | 4 | 8 | 12 | 16 |
As can be seen from the table, there are several combinations of and that provide a given output volume within certain limits. For example, you can get it using a combination of (1,4), (4,1) and (2,2).
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, called an isoquant.
Each isoquant point corresponds to a combination at which the firm produces a given volume of output.
The set of isoquants characterizing a given is called isoquant map.
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 reduction in output resulting from a decrease in capital expenditure is compensated by an increase in output due to the use of additional labor.
The reduction in output as a result of a decrease in capital expenditure is equal to the product of marginal product capital, or . The increase in output due to the use of additional labor is in turn equal to the product of the marginal product of labor, or .
Thus, we can write that . Let's write this expression differently: or .
The production function, which connects 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 angle of inclination of the tangent to the isoquant at this point.
Example 14.2 Finding the MRTS for a given functionCondition: Let the production function have the form .
Define: with for .
Solution:
It is obvious that the degree of labor substitution with capital does not remain constant when moving along the isoquant. When moving down the curve, the absolute value of MRTS of labor by capital decreases, since more and more labor has to be used to compensate for the decrease in capital costs (So, in the above example, with L=1 MRTS=-10, and with L=10 MRTS=- 0.1.)
Subsequently, MRTS reaches its limit (MRTS = 0), and the isoquant takes on a horizontal form. It is obvious that a further reduction in capital costs will only lead to a reduction in output volumes. The amount of capital at point E is the minimum allowable amount for a given volume of production (in the same way, the minimum amount of labor acceptable for the production of a given volume occurs at point A).
A decrease in the MRTS of one resource by another is typical for most production processes and is characteristic of all isoquants of the standard form.
If the resources used in the production process are absolutely replaceable, then the isoquant is constant at all points, and the isoquant map looks like in Figure 14.2. (An example of such production would be production that allows both full automation and handmade any product).
If 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 another factor is irrational, therefore, only angular combinations of resources will be technically effective (an angular point is the point where the corresponding horizontal and vertical lines intersect).
The combination of the last two factors determines area of economic resources available to the producer.
The manufacturer's budget constraint can be written as an inequality:
If the manufacturer spends all of his funds on purchasing these resources, then we get the equality:
The resulting equation is called isocost equation.
Isocost line presented in Figure 14.4 shows the set of combinations of economic resources (in this case, labor and capital) that a firm can purchase, taking into account market prices for resources and using its full budget.
The slope of the isocost line is determined by the ratio of market prices for labor and capital (- РL/РK), which follows from the isocost equation.
Manufacturer isocost line
The company's desire to efficient production encourages it to achieve the maximum possible output at given resource costs, or, what is the same, to minimize costs in producing a given volume of output.
The combination of resources that ensures the minimum level of total costs for the company is called optimal and lies at the point of tangency between the isocost and isoquant lines.
By combining isoquats and isocosts, the firm's optimal position can be determined. The point at which the isoquant touches the isocost means the cheapest combination of factors necessary to produce a certain volume of output.
American economists Douglas and Solow found that an increase in costs by 1% provides 3/4 of the increase in output, and an increase in costs by 1% 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 production function. which allows us to assert that investments in , have a greater effect in increasing production than growth.
The set of optimal points of the manufacturer, constructed for a changing production volume, and, consequently, changing costs () of the company with constant prices for resources, reflects the development trajectory of the company. Figure 14.6.
The shape of the development trajectory is usually considered in the long term and allows us to identify capital-intensive (Figure 14.7a), labor-intensive (Figure 14.7b) production methods, as well as technologies that involve a uniform increase in the use of both labor and capital (Figure 14.7c).
