Production–possibility frontier

The transformation curve and production possibility curve or line capacity, is in economics, the graphical representation of all efficient freight quantity combinations for a given resource use. It is an economics tool that serves to demonstrate the principle existing problem of scarcity and the consequences of alternatives.

The reality is in the model of production possibilities greatly simplified on the basis of two goods or shown two factors of production. However, the concept, the findings and results without difficulty on many goods and entire economies can be transferred.

The closed amount under the transformation curve is called the production room or Production Possibility Set. It comprises all kinds of goods combinations that can be produced with existing manufacturing factors.

Classification and description

Both in business administration as well as in economics, the transformation curve is an important tool to map different alternatives of production possibilities. The transformation curve is particularly important role in the economic theory of foreign trade. Here, the curve serves as a basis for further theories and models, such as the Ricardian model, the Heckscher -Ohlin model or the Rybczynski theorem. The fact that each available in an economy of production factors can be used because of their scarcity only for either one or the other use that accordingly - for a given production technology - the production potential is limited, justifying the need for the instrument of the transformation curve. The curve could therefore also be described as the locus of all the maximum possible quantity combinations of goods, groups of goods or factors of production. In the model, the transformation curve is assumed here that all the resources have been utilized in full and used in accordance with the economic principle.

Based on the assumption of scarcity or full capacity choice decisions must be made both in a company as well as in an economy, to define an alternative quantity of goods or factors of production combination combination. Come to a decision always means to take a waiver in purchasing. This waiver, or more precisely the rent-stealing, it's called opportunity cost or cost of the second best alternative. With a movement along the transformation curve thus lead to an increase in the amount of one good, but at the same time to give up a corresponding amount of the other good.

The axes of the production possibility diagram can depict amounts of goods (eg bread or machines), groups of goods (eg consumer or capital goods), factors of production (eg labor or capital ) as well as other economic entities. In the following consideration should be assumed for reasons of simplicity of a two-goods case.

In the literature, the curve is sometimes also referred to as a production possibilities frontier. This refers to the restriction of the possible production quantity, which is set by the transformation curve. Based on the graph (Fig. 2 ), this means that only those combinations of goods below or to the left of and on the curve are possible. All quantity combinations outside this range can not be achieved with a constant state of technology, knowledge and productivity. An exception may indicate an economy in international trade. So it may be possible through a comparative advantage in this economy, to achieve a combination of outside and to the right of the production possibilities curve. In general, the transformation curve shifts outward, however, and there are thus far unattainable quantity combinations realized if long-term growing technical knowledge and / or factor endowments. The related questions are the subject of growth theory. Furthermore, the efficiency can be judged from the curve. Efficiency occurs when all the resources are fully utilized. Only the combinations of amounts of goods on the curve can therefore be considered as efficient. In contrast, all amount compositions are below the function as inefficient because at the same factor input more could be produced by one of the two goods. Another possible explanation is that the potential productive capacity is not utilized to the extent as it would have been possible given the state of knowledge. This problem is concerned the theory of price and allocation theory.

Location and shape of the curve

As can be seen in Figure 3, a basic distinction between two types of the course of production possibility curves, the running of linear and convex to the origin. The linear curve shown in Figure 3 a ) of the curve is determined by the underlying production functions. Regardless of the level of production while a constant factor quantity per unit of output in the two goods is claimed. If this condition is not met, the transformation curve usually takes - as shown in Figure 3 b ) to recognize - a convex curve at. Production function, factor intensity and the production elasticity thus determine the shape of the capacity curve. This unequal elasticities and / or different factor intensities have a convexity of the function result.

An alternative understandable and practical approach to the nature of the shape of the transformation curve is the consideration of the scope in production. Referring to the two-goods case relative to that with the combined production of the two goods to a separate production accompanied by certain scope, the curve will assume a concave shape. Economies of my savings for the production of necessary factors of production. In practice, the pooled production of two goods could mean due to Volume Discounts for example, to reduce administrative costs or the purchase price of shared resources. If on the other hand from the combined production of the goods found no scope, the production possibilities curve will assume a linear progression.

In theory, convex or linear patterns composed of several sections also are conceivable. The composite linear variation caused when the intensity factor of two materials are non- linear functions limitational production. Convexity can follow with combined production due to linearity over a production function or negative effect. Negative effect means in the event that in the folded manufacture disadvantages such as higher costs as opposed to the separate production of goods.

Slope - marginal rate of transformation

The slope of the curve production possibility, also referred to as a limit rate of transformation, typically has a negative slope, that is, a top-left to right falling below curve (see Figure 3a). The opportunity costs - the unavoidable absence of a specific quantity of one good while producing an additional unit of good to others - explain the falling slope of the curve. This means between the goods shown in Figure 3 X and Y there is an inverse relationship, you could this as a " trade-off " call between the goods. Mathematically equivalent to the marginal rate of transformation of the transformation ratio, hence the ratio of the two product changes in quantity of Good X and Good Y using their differentials and. Viewed as a formula results in:

• : Marginal rate of transformation
• : Differential

In order to obtain a positive amount for the marginal rate, a negative sign is placed before the relationship. At the same time, this ratio is the marginal waiver or alternative costs.

For a linear function, the transformation ratio remains the complete course over unchanged, there are so constant alternative costs. Based on the two-goods case goods have both always relative costs in the same proportion, that is, they would be at any point on the function in the constant ratio of the marginal rate of transformation substitutable.

In many cases, however, there is the case of outwardly curved ( = concave ) described in Figure 3b on the production possibility curve. Here, the amount of the slope along the falling curve of the function changes gradually. The reason is the " law of diminishing opportunity costs " in connection with the law of diminishing returns ( = " law of diminishing returns "). It is assumed that the returns are diminished by additional input, the more power factor in one production already exists. In Figure 4, the law of diminishing returns is graphically to realize is dispensed with one unit of Good Y, can you? X 1 units produce more of good X - which is equated with an income of? X 1, with a waiver of a unit of Good Y. If now dispense with further units of good Y, so take the income (in the graphic ΔX2 ... ΔX4 ) continuously. One possible reason for this is that the more factors of production for the production of Good X from the production of Good Y are deducted, this is it, the less suitable for the additional production of good X. would also conceivable that one of the two products and constant has the other product declining returns to scale or both products have declining returns to scale. In the two-goods case had both goods along the curve always relative costs in a differents ratio, that is if they were at any point on the function in the respective ratio of the marginal rate of transformation substitutable.

Derivation

Graphical derivation

To make the derivation more comprehensible, one uses the upper part of Figure 5 to be recognized box diagram, also called Edgeworth box. This is based on the production of two goods and limited available production factors. Under this condition, the isoquants for the two displayed goods X and Y are plotted in the chart. If you connect up the tangent points, ie the points of contact of the isoquants ( in the graph A, B, C and D), we obtain the line of efficient production, also called contract curve. The location of the contacting isoquants for each origin represents the various production volumes in these points. The slope of the isoquant is determined by the ratio of marginal productivities. Since at the contact points, the slope of each isoquant ( the goods here X and Y) is identical, therefore must assume the same value, the ratio of marginal productivities of the two goods. Thus, of good X are no longer manufactured, without giving up a certain amount of good Y - the points of tangency represent therefore the isoquants certain Pareto- optimal amount of combinations of goods X and Y. These depicted in the Edgeworth box goods quantity combinations A, B, C and D can be transferred into a known amount of product XY chart. In Figure 5, this transmission occurs vertically downwards, it should be noted at this point that the transmission lines must not extend completely vertical, since the axes of both graphs are denoted by different variables. If you connect up again the transferred points A, B, C and D in the lower diagram, created the production possibility curve. This example extends concavely in the figure, possible paths are other (see position and shape of the curve).

Mathematical derivation

Mathematically, the transformation function from the production functions of the two goods X and Y are derived. This will be shown using a sample statement under certain assumptions. Here, it is assumed that there are two linearly homogeneous production functions with the same partial production elasticities of ½. There is also analogous to the graphical derivation of only two factors of production, capital and labor. The production functions for goods X and Y are:

• : Good X
• : Good Y
• : Factor of production capital
• : Factor of production labor

As already explained, the quantity of goods combinations are Pareto- optimal, the slope of the isoquant - thus the marginal rates of technical substitution - of goods X and Y must be identical to the tangent points. Therefore, we have:

• : Marginal rate of technical substitution
• : Marginal productivity
• : Differential

If the relations (3) and (4 ) was made to obtain the desired optimum for the present example:

Subsequently, for:

• : Production factor capital total
• : Production factor labor total

Used and transformed, is thus obtained:

Equations ( 5) and (6 ) clearly show a linear relationship of the factors to be used. Linear production functions were used sake of simplicity, in the underlying sample. So also results for the in Figure 5 in the Edgeworth box to be recognized contract curve - for the line joining the tangency points A to D - a linear relationship with the ratio of the slope. In the following step to derive the transformation function, the expression of the efficient use of capital ( 5) into the production function of good X (1 ) is set:

Changed via the function (7) leads to the demand for labor by:

Substituting in the production function of good Y ( 2) for the expression of the function (6 ) and replaced in turn by, we obtain the following function expression:

And after the onset of labor demand (8) and some rearrangements ultimately:

Based on the production functions prefixed ( 1) and (2) the equation (9 ) the corresponding transformation function dar. above provides mathematical derivation is just as exemplary of the two imaginary, assumed production functions and thus to consider for a special linear production possibility curve. Similar to the above procedure - but mathematically sophisticated - can shown both a concave transformation curve as shown in Figure 5, as well as all individual running capacity lines are derived analytically.

Application / example

Finally, to be explained once again comprehensible by means of a standard example often used in the economic literature, the various aspects of the production possibility curve. In the example it is assumed that a model economy that only two goods - can produce - guns and butter. The products are thereby representative of the categories Consumer Goods and defense equipment. Figure 6 shows different production possibilities of this economy, output is restricted to the extent that either only 10 million pieces of cannon, or 10 million pounds of butter can be produced. This illustrates the existing in reality scarcity of input factors. So there must be a decision whether to one of the extreme cases - only consumer goods (butter ) or only defense equipment (guns ) - selects or one of the many efficient freight quantity combinations on the transformation curve (point B -D). In addition, it is possible to produce all combinations inefficient under the curve (e.g., point F). In the previous consideration has been determined that a decision also means taking a waiver within the meaning of opportunity cost in purchasing. It is assumed in the baseline situation you find yourself in the application example in point B, that is, there are 9 million guns and 4 million pounds of butter produced. Due to increasing population, a higher demand for food now is a - it is therefore decided to produce 3 million pounds more butter. Considering the graph in Figure 6 to help, it can be seen that point B is already on the curve, so all available resources are used. In order to be able to produce 3 million pounds more of the butter, so only the quantity of goods combination is at point C, which means a waiver in the amount of 2 million pieces of cannon - the waiver of guns is seen as opportunity costs.

Find a motion on the curve instead, so quantities of one product (guns ) can be transformed (butter ) in amounts of another product. Since the production possibility curve is curved in Figure 6 to the outside, the transformation ratio of guns changed in butter along the function. The example shows so clearly the law of diminishing returns. While a change of the quantity combination A to B for a waiver of one million pieces of cannon 4 million pounds of butter can be substituted for the same, with a change from point D to E for a waiver of 4 million pieces of cannon remains just a income of 1 million pounds of butter. Analog of diminishing returns change the opportunity cost.

Should be the economic goal of increasing the overall economic production to reach for example, for goods quantity combination at point G, this is not feasible under constant conditions. In this case it would be necessary to increase the input or factor endowments. An expansion of production capacity by increasing the production factors labor, capital or knowledge, especially through immigration of guest workers, capital inflow from abroad or new technical research findings can solve this problem.