Transformation curve

Fig. 1: Production possibilities curve

The transformation curve , also called production possibilities curve or capacity line , is in economics the graphic representation of all efficient combinations of quantities of goods for a given use of resources . It is an economic instrument that serves to show the fundamental problem of scarcity and the alternatives that arise from it.

The reality is represented in the model of the production possibilities in a greatly simplified manner using two goods or two production factors. However, the concept, knowledge and results can be transferred to many goods and entire economies without difficulty.

The closed set under the transformation curve is called the production space or production possibility set. It contains all possible combinations of goods that can be produced with the existing production factors - albeit less efficiently than exactly on the transformation curve.

Classification and description

Both in business administration and in economics , the transformation curve is an important tool for mapping various alternatives of production options. The transformation curve is particularly important in economic foreign trade theory. The curve serves as the basis for further theories and models, such as the Ricardo model , Heckscher-Ohlin model or the Rybczynski theorem . The fact that the production factors available in an economy can only be used alternatively for one or the other use due to their scarcity, that accordingly - with the given production technology - the production potential is limited, justifies the need for the instrument of the transformation curve. The curve could therefore also be called the geometric location of all the maximum possible combinations of quantities of goods , groups of goods or factors of production. In the model of the transformation curve, it is assumed that all resources have been fully used and deployed according to the economic principle .

On the basis of the assumption of scarcity or full capacity utilization, both in a company and in an economy, choices have to be made in order to determine an alternative combination of quantities of goods or production factors. Making a decision always means accepting a waiver. This waiver, more precisely the loss of benefit, is called opportunity costs or costs of the second best alternative. When moving along the transformation curve, there is an increase in the amount of one good, but at the same time a corresponding amount of the other good is dispensed with.

The axes of the production possibilities diagram can depict quantities of goods (e.g. bread or machines), goods groups (e.g. consumer or capital goods), production factors (e.g. labor or capital) and other economic units. For the sake of simplicity, the following considerations are based on a two-goods case.

Fig. 2: Efficiency areas and implementation options in the model

In the literature, the curve is sometimes referred to as the limit of production possibilities. What is meant by this is the limitation of the possible production quantity that is set by the transformation curve. In relation to the graphic (Fig. 2) this means that only those combinations of goods below or to the left of and on the curve are possible. Any combination of quantities outside this range cannot be implemented if the state of technology, knowledge and productivity remain the same . An economy in international trade can show an exception . A comparative advantage of this economy may make it possible to achieve a combination outside or to the right of the production possibilities curve. In general, on the other hand, the transformation curve shifts outwards and thus previously unattainable combinations of quantities become realizable if technical knowledge and / or factors increase in the long term . The related questions are the subject of growth theory . The curve can also be used to assess the efficiency. Efficiency is when all resources are fully used. Only the combinations of quantities of goods on the curve can therefore be called efficient. In contrast, all quantity compositions below the function are considered inefficient because more of one of the two goods could be produced with the same factor input. Another possible explanation is that the possible productive performance is not used to the extent that it would have been possible with the given state of knowledge. This problem is addressed by price theory and allocation theory .

Position and shape of the curve

As can be seen in Fig. 3, a basic distinction is made between two types of production possibility curves, the linear and the concave to the origin. The linear course of the curve shown in Fig. 3 a) is determined by the underlying production functions. Regardless of the level of production, a constant amount of factors per product unit is claimed for both goods. If this condition is not met, the transformation curve generally assumes a concave, i.e. outwardly curved course, as can be seen in Fig. 3 b). Production function , factor intensity and production elasticity thus determine the shape of the capacity line. Unequal elasticities and / or different factor intensities result in a concavity of the function.

Fig. 3: linear and concave course of the transformation curve

An alternative, understandable and practice-oriented approach to the type of shape of the transformation curve is to consider the economies of scale in production. If one assumes, based on the two-goods case, that the combined production of the two goods is associated with certain economies of scope compared to separate production, the curve will assume a concave shape. Economies of scale mean savings in the production factors necessary for manufacturing. In practice, the combined production of two goods could mean, for example, to reduce administrative costs or the purchase price of raw materials used jointly due to volume discounts. If, on the other hand, the combined production of the goods does not result in economies of scope, the production possibilities curve will assume a linear course.

In theory, convex or linear courses consisting of several sections are also conceivable. The compound linear variant is caused when the factor intensities of two goods with linear limitational production functions are unequal. Convexity can result from a superlinearity of a production function or from a negative effect in combined production. In this case, negative impact means that combined production incurs disadvantages such as higher costs in contrast to the separate production of the goods.

Slope - limit rate of transformation

The slope of the production possibility curve, also referred to as the limit rate of transformation, typically has a negative slope, i.e. a slope falling from top left to bottom right (see Fig. 3a). The opportunity costs - the inevitable foregoing a certain amount of one good while producing an additional unit of quantity of the other good - explain the falling slope of the curve. This means that there is an inverse relationship between the goods X and Y shown in Figure 3; this could also be described as a “ trade-off ” between the goods. From a mathematical point of view, the marginal rate of the transformation corresponds to the transformation ratio, consequently the ratio of the two product quantity changes of good X and good Y using their differentials and . Considered as a formula: ${\displaystyle \mathrm {d} x}$ ${\displaystyle \mathrm {d} y}$

${\displaystyle GRT=-{\frac {\mathrm {d} y}{\mathrm {d} x}}}$.

• ${\displaystyle GRT}$: Marginal rate of transformation
• ${\displaystyle \mathrm {d} }$: Differential

In order to obtain a positive amount for the limit rate, a negative sign is placed in front of the ratio. At the same time, this ratio indicates the marginal waiver or alternative costs.

In the case of a linear function, the transformation ratio remains unchanged over the entire course, so there are constant alternative costs. In relation to the two-goods case, both goods would always have relative costs in the same ratio, that is, they could be substituted at every point on the function in the same ratio of the marginal rate of transformation.

Fig. 4: Slope - limit rate of transformation

In many cases, however, the case of the outwardly curved (= concave) production possibility curve described in Fig. 3b occurs. Here, the amount of the gradient changes degressively along the falling course of the function. The reason for this is the "law of decreasing alternative costs" in connection with the income law (= "law of decreasing income growth"). It is assumed that the yields from additional input decrease the more factor output is already available in a certain production. In Fig. 4 the law of income can be seen graphically, if you do without one unit of good Y, you can produce ∆X1 units more of good X - this is equivalent to a return of ∆X1, if you do without a unit of Good Y. If further units of good Y are dispensed with, the yields (∆X2… ∆X4 in the graphic) decrease continuously. One possible reason for this is that the more production factors for the production of good X are deducted from the production of good Y, the less suitable they are for the additional production of good X. It would also be conceivable that one of the two products would be constant and the other product has declining economies of scale or both products show declining economies of scale. In the two-goods case, both goods would always have relative costs in a different ratio along the curve, that is, they would be substitutable at every point on the function in the respective ratio of the limit rate of transformation.

Derivation

Graphic derivation

Fig. 5: Derivation of the transformation curve from the Edgeworth box

To make the derivation easier to understand, use the box diagram in the upper part of Fig. 5, also called the Edgeworth box . It assumes the production of two goods and only limited production factors. Under this condition, the isoquants for the two goods to be represented X and Y are plotted in the diagram. If you now connect the tangential points, i.e. the points of contact of the isoquants (in the graphic A, B, C and D), you get the line of efficient production, also known as the contract curve. The position of the touching isoquants to the respective origin represents the different production quantities at these points. The slope of the isoquants is determined by the ratio of the marginal productivities . Since the slope of the respective isoquants (here goods X and Y) is identical at the points of contact, the ratio of the marginal productivities of the two goods must also assume the same value. Thus, good X can no longer be produced without foregoing a certain amount of good Y - the tangential points consequently represent, through the isoquants, certain Pareto-optimal quantity combinations of goods X and Y. These goods quantity combinations A, shown in the Edgeworth box B, C and D can be translated into a known XY product quantity diagram. In Fig. 5 this transfer takes place vertically downwards, please note at this point that the transfer lines do not have to run completely vertically, as the axes of both diagrams are denoted by different variables. If you connect the transferred points A, B, C and D in the diagram below, the production possibilities curve is created. This runs concave in the figure, for example, other courses are also conceivable (see position and shape of the curve).

Mathematical derivation

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

(1) , ${\displaystyle X=K_{x}^{1/2}L_{x}^{1/2}}$

(2) . ${\displaystyle Y=2K_{y}^{1/2}L_{y}^{1/2}}$

• ${\displaystyle X}$: Good X
• ${\displaystyle Y}$: Good Y
• ${\displaystyle K}$: Production factor capital
• ${\displaystyle L}$: Labor as a factor of production

As already explained, the quantity combinations of goods are Pareto-optimal, the slope of the isoquants - thus the marginal rates of technical substitution - of goods X and Y must be identical in the tangent points. Hence:

(3) , ${\displaystyle {GRTS}_{X}={\frac {{GP}_{L,X}}{{GP}_{K,X}}}={\frac {{dX/dL}_{X}}{{dX/dK}_{X}}}={\frac {{(1/2)X/L}_{X}}{{(1/2)X/K}_{X}}}={\frac {K_{X}}{L_{X}}}}$

(4) . ${\displaystyle {GRTS}_{Y}={\frac {{GP}_{L,Y}}{{GP}_{K,Y}}}={\frac {{dY/dL}_{Y}}{{dY/dK}_{Y}}}={\frac {{(1/2)Y/L}_{Y}}{{(1/2)Y/K}_{Y}}}={\frac {K_{Y}}{L_{Y}}}}$

• ${\displaystyle GRTS}$: Marginal rate of technical substitution
• ${\displaystyle GP}$: Marginal productivity
• ${\displaystyle d}$: Differential

If the relationships (3) and (4) are equated, the desired optimum for the present example is obtained:

${\displaystyle {\frac {K_{X}}{L_{X}}}={\frac {K_{Y}}{L_{Y}}}}$.

Then for:

${\displaystyle \operatorname {\mathit {K_{Y}}} ={K_{g}}-{K_{X}}}$and as well as and ${\displaystyle \operatorname {\mathit {L_{Y}}} ={L_{g}}-{L_{X}}}$ ${\displaystyle \operatorname {\mathit {K_{X}}} ={K_{g}}-{K_{Y}}}$ ${\displaystyle \operatorname {\mathit {L_{X}}} ={L_{g}}-{L_{Y}}}$

• ${\displaystyle K_{g}}$: Total capital production factor
• ${\displaystyle L_{g}}$: Total labor production factor

inserted and reshaped, you get:

(5) , ${\displaystyle K_{X}={\frac {K_{g}}{L_{g}}}L_{X}}$

(6) . ${\displaystyle K_{Y}={\frac {K_{g}}{L_{g}}}L_{Y}}$

Equations (5) and (6) clearly show a linear relationship between the factors to be used. For reasons of simplification, linear production functions were used in the underlying example. This also results in the contract curve to be seen in Fig. 5 in the Edgeworth box - for the line connecting the tangential points A to D - a linear course with the ratio as the slope. In the following step to derive the transformation function, the expression of the efficient use of capital (5) is placed in the production function of good X (1): ${\displaystyle \operatorname {\mathit {K_{g}}} /{L_{g}}}$

(7) . ${\displaystyle X=\left({\frac {K_{g}}{L_{g}}}L_{X}\right)^{1/2}L_{x}^{1/2}=\left({\frac {K_{g}}{L_{g}}}\right)^{1/2}L_{x}}$

Function (7) converted to leads to the labor demand of: ${\displaystyle \operatorname {\mathit {L_{X}}} }$

(8) . ${\displaystyle L_{X}=\left({\frac {L_{g}}{K_{g}}}\right)^{1/2}X}$

If you insert the into the production function of good Y (2) for the expression of function (6) and replace it again with , you get the following function expression : ${\displaystyle \operatorname {\mathit {K_{Y}}} }$ ${\displaystyle \operatorname {\mathit {L_{Y}}} }$ ${\displaystyle \operatorname {\mathit {L_{Y}}} ={Lg}-{L_{X}}}$

${\displaystyle Y=2\left({\frac {K_{g}}{L_{g}}}\right)^{1/2}\left({L_{g}}-{L_{X}}\right)}$

and after the onset of labor demand (8) for and some transformations, ultimately: ${\displaystyle \operatorname {\mathit {L_{X}}} }$

(9) . ${\displaystyle Y=2\left({K_{g}}{L_{g}}\right)^{1/2}-2X}$

In relation to the preceding production functions (1) and (2), equation (9) represents the associated transformation function. The above mathematical derivation is only to be regarded as an example for the two fictitiously assumed production functions and thus for a special linear production possibility curve. Analogous to the procedure outlined above - but mathematically more demanding - both a concave transformation curve as shown in Fig. 5 and all individually running capacity lines can be derived analytically.

Application, example

Fig. 6: Model economy in the two-goods case

Finally, the various aspects of the production possibilities curve should be explained again in a more understandable way using a standard example often used in economic literature. In the example mentioned, a model economy is assumed that can only produce two goods - cannons and butter. The products represent the categories of consumer goods and defense goods. Fig. 6 shows different production possibilities of the economy under consideration, so the production is restricted in such a way that either only 10 million pieces of cannons or 10 million pounds of butter can be produced. This illustrates the scarcity of input factors that exist in reality. So a decision has to be made as to whether one of the extreme cases - only consumer goods (butter) or only defense goods (cannons) - should be chosen, or one of the numerous efficient combinations of quantities of goods on the transformation curve (points B-D). It is also possible to produce all inefficient combinations below the curve (for example point F). In the previous consideration it was already stated that making a decision also means accepting a waiver in the sense of opportunity costs. It is assumed that in the initial situation one is in the application example in point B, i. H. 9 million cannons and 4 million pounds of butter are produced. Due to the growing population, there is now a greater need for food - it is therefore decided to produce 3 million pounds more butter. If the graphic in Fig. 6 is used as an aid, it can be seen that point B is already on the curve, so that all available resources have been used. In order to be able to produce 3 million pounds more of the butter, all that remains is the combination of quantities of goods in point C, which means a waiver of 2 million cannons - this waiver of cannons is viewed as an opportunity cost.

If there is a movement on the curve, quantities of one product (cannons) are transformed into quantities of the other product (butter). Since the production possibilities curve in Fig. 6 is curved outwards, the transformation ratio of cannons in butter changes along the function. The example clearly shows the law of decreasing yield growth. While changing from the quantity combination A to B for a waiver of 1 million pieces of cannons equal 4 million pounds of butter can be substituted, if you change from point D to E for a waiver of 4 million pieces of cannons, only one remains Yield of 1 million pounds of butter. The alternative costs change in line with the decreasing income.

If the macroeconomic goal is to increase macroeconomic production, for example to achieve the combination of quantities of goods in point G, this cannot be achieved under constant conditions. In this case it would be necessary to increase the input or the factor endowment . An expansion of production capacities through an increase in 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.

literature

• Ulrich Bröse: Introduction to Economics - Microeconomics . Oldenbourg, Munich / Vienna 1997, ISBN 3-486-23699-7 .
• Horst Demmler: Introduction to Economics. 7th edition. Oldenbourg, Munich / Vienna 2001, ISBN 3-486-25623-8 .
• Gustav Dieckheuer: International economic relations. 5th edition. Oldenbourg, Munich / Vienna 2001, ISBN 3-486-25806-0 .
• Ulrich Fehl, Peter Oberender: Fundamentals of Microeconomics. 9th edition. Munich 2004, ISBN 3-8006-3107-5 .
• Rainer Fischbach: Economics - Introduction and Basics. 8th edition. Oldenbourg, Munich 1994, ISBN 3-486-22792-0 .
• Gabriele Hildmann: Microeconomics - Intensive Training. In: Volker Drosse, Ulrich Vossebein (Hrsg.): Repetitorium Wirtschaftswwissenschaften. 2nd Edition. Munich 2005, ISBN 3-409-22620-6 .
• Gerhard Kolb: Fundamentals of Economics . Munich 1991, ISBN 3-8006-1568-1 .
• Paul R. Krugman, Maurice Obstfeld: - Theory and politics of foreign trade. 7th edition. Munich 2006, ISBN 3-8273-7199-6 .
• Robert S. Pindyck, Daniel L. Rubinfeld: Microeconomics. 5th edition. Munich 2003, ISBN 3-8273-7025-6 .
• Paul A. Samuelson, William D. Nordhaus: Economics - Fundamentals of Macro and Microeconomics. 8th edition. Cologne 1987, ISBN 3-7663-0986-2 .
• Klaus Schöler: Fundamentals of Microeconomics. 2nd Edition. Munich 2004, ISBN 3-8006-3065-6 .
• Joseph E. Stiglitz: Economics. 2nd Edition. Munich / Vienna 1999 3-486-23379-3
• Alfred Stobbe: Microeconomics. 2nd Edition. Berlin / Heidelberg / New York 1991, ISBN 3-540-54136-5 .
• Artur Woll: General Economics. 14th edition. Munich 2003, ISBN 3-8006-2973-9 .
• Hal R. Varian: Fundamentals of Microeconomics. 6th edition. Munich / Vienna 2004, ISBN 3-486-27453-8 .
• Andreas Zenthöfer: Basics of microeconomics. In: HP Richter (Ed.): Basic economic courses . Kiel 2006, ISBN 3-935150-51-2 .

Web links

• Transformation curve - definition in the Gabler business dictionary
• Transformation curve - W. Lorenz: mikrooekonomie.de - complete entry-level article
  • Up, above and below the transformation curve - deepening the entry-level article
  • Transformation curve and opportunity costs - deepening the entry-level article

Individual evidence

1. ↑ ^{a b} Hal R. Varian: Fundamentals of microeconomics . 2004, p. 590.
2. ↑ Gerhard Kolb: Fundamentals of Economics . 1991, pp. 42-43.
3. ^ Artur Woll: General Economics . 2003, p. 60.
4. ^ Paul R. Krugman, Maurice Obstfeld: International Economy - Theory and Politics of Foreign Trade . 2006, p. 59.
5. ↑ Robert S. Pindyck, Daniel L. Rubinfeld: Microeconomics . 2003, p. 814.
6. ↑ Robert S. Pindyck, Daniel L. Rubinfeld: Microeconomics . 2003, p. 821.
7. ^ ^{A b} Artur Woll: General Economics . 2003, p. 59.
8. ^ Paul A. Samuelson, William D. Nordhaus: Economics - Fundamentals of Macro and Microeconomics . 1987, pp. 66-67.
9. ↑ Ulrich Fehl, Peter Oberender: Fundamentals of Microeconomics . 2004, pp. 265-269.
10. ↑ Andreas Zenthöfer : Fundamentals of microeconomics . 2006, pp. 55-56.
11. ^ Gabriele Hildmann: Microeconomics - intensive training . 2005, pp. 92-95.
12. ↑ Ulrich Fehl, Peter Oberender: Fundamentals of Microeconomics . 2004, p. 267.
13. ↑ Ulrich Bröse: Introduction to Economics - Microeconomics . 1997, p. 61.
14. ^ Alfred Stobbe: Microeconomics . 1991, p. 195.
15. ^ Paul A. Samuelson, William D. Nordhaus: Economics - Fundamentals of Macro and Microeconomics . 1987, pp. 75-80.
16. ↑ Gerhard Kolb: Fundamentals of Economics . 1991, p. 43.
17. ^ Alfred Stobbe: Microeconomics . 1991, p. 197.
18. ↑ Klaus Schöler: Fundamentals of microeconomics . 2004, p. 168.
19. ^ Gabriele Hildmann: Microeconomics - intensive training . 2005, pp. 93-94.
20. ↑ Klaus Schöler: Fundamentals of microeconomics . 2004, pp. 168-169.
21. ↑ Klaus Schöler: Fundamentals of microeconomics . 2004, pp. 169-170.
22. ↑ Robert S. Pindyck, Daniel L. Rubinfeld: Microeconomics . 2003, pp. 282-283.
23. ↑ Klaus Schöler: Fundamentals of microeconomics . 2004, pp. 169-171.
24. ^ Joseph E. Stiglitz : Economics . 1999, p. 42.
25. ↑ Ulrich Bröse: Introduction to Economics - Microeconomics . 1997, p. 59.
26. ^ Paul A. Samuelson, William D. Nordhaus: Economics - Fundamentals of Macro and Microeconomics . 1987, p. 65.
27. ^ Joseph E. Stiglitz: Economics . 1999, p. 44.
28. ↑ Ulrich Bröse: Introduction to Economics - Microeconomics . 1997, pp. 60-61.

This version was added to the list of articles worth reading on May 7, 2008 .