homogeneous production function and returns to scale

When the technology shows increasing or decreasing returns to scale it may or may not imply a homogeneous production function. Subsection 3(2) deals with plotting the isoquants of an empirical production function. This is implied by the negative slope and the convexity of the isoquants. Returns to scale are usually assumed to be the same everywhere on the production surface, that is, the same along all the expansion-product lines. Of course the K/L ratio (and the MRS) is different for different isoclines (figure 3.16). With constant returns to scale everywhere on the production surface, doubling both factors (2K, 2L) leads to a doubling of output. Along any isocline the distance between successive multiple- isoquants is constant. Production functions with varying returns to scale are difficult to handle and economists usually ignore them for the analysis of production. The function (8.122) is homogeneous of degree n if we have . the returns to scale in the translog system that includes the cost share equations.1 Exploiting the properties of homogeneous functions, they introduce an additional, returns to scale parameter in the translog system. A production function showing constant returns to scale is often called ‘linear and homogeneous’ or ‘homogeneous of the first degree.’ For example, the Cobb-Douglas production function is a linear and homogeneous production function. Doubling the factor inputs achieves double the level of the initial output; trebling inputs achieves treble output, and so on (figure 3.18). If only one factor is variable (the other being kept constant) the product line is a straight line parallel to the axis of the variable factor (figure 3.15). Instead of introducing a third dimension it is easier to show the change of output by shifts of the isoquant and use the concept of product lines to describe the expansion of output. The Cobb-Douglas and the CES production functions have a common property: both are linear-homogeneous, i.e., both assume constant returns to scale. It is, however, an age-old tra- 0000001796 00000 n Homogeneous production functions are frequently used by agricultural economists to represent a variety of transformations between agricultural inputs and products. Also, studies suggest that an individual firm passes through a long phase of constant return to scale in its lifetime. A product curve is drawn independently of the prices of factors of production. Among all possible product lines of particular interest are the so-called isoclines.An isocline is the locus of points of different isoquants at which the MRS of factors is constant. Constant returns-to-scale production functions are homogeneous of degree one in inputs f (tk, t l) = functions are homogeneous … Lastly, it is also known as the linear homogeneous production function. The marginal product of the variable factors) will decline eventually as more and more quantities of this factor are combined with the other constant factors. the production function under which any input vector can be an optimum, for some choice of the price vector and the level of production. In most empirical studies of the laws of returns homogeneity is assumed in order to simplify the statistical work. 0000001471 00000 n the returns to scale are measured by the sum (b1 + b2) = v. For a homogeneous production function the returns to scale may be represented graphically in an easy way. Increasing Returns to Scale Share Your PPT File, The Traditional Theory of Costs (With Diagram). Therefore, the result is constant returns to scale. The term ‘returns to scale’ refers to the changes in output as all factors change by the same pro­portion. Graphical presentation of the returns to scale for a homogeneous production function: The returns to scale may be shown graphically by the distance (on an isocline) between successive ‘multiple-level-of-output’ isoquants, that is, isoquants that show levels of output which are multiples of some base level of output, e.g., X, 2X, 3X, etc. In such a case, production function is said to be linearly homogeneous … In the Cobb–Douglas production function referred to above, returns to scale are increasing if + + ⋯ + >, decreasing if + + ⋯ + <, and constant if + + ⋯ + =. A production function showing constant returns to scale is often called ‘linear and homogeneous’ or ‘homogeneous of the first degree.’ For example, the Cobb-Douglas production function is a linear and homogeneous production function. 0000005629 00000 n Disclaimer Copyright, Share Your Knowledge The isoclines will be curves over the production surface and along each one of them the K/L ratio varies. For 50 < X < 100 the medium-scale process would be used. For X < 50 the small-scale process would be used, and we would have constant returns to scale. Constant returns to scale prevail, i.e., by doubling all inputs we get twice as much output; formally, a function that is homogeneous of degree one, or, F(cx)=cF(x) for all c ≥ 0. However, the techno­logical conditions of production may be such that returns to scale may vary over dif­ferent ranges of output. The K/L ratio changes along each isocline (as well as on different isoclines) (figure 3.17). It does not imply any actual choice of expansion, which is based on the prices of factors and is shown by the expansion path. Since returns to scale are decreasing, doubling both factors will less than double output. The laws of returns to scale refer to the effects of scale relationships. By doubling the inputs, output increases by less than twice its original level. 0000002268 00000 n Such a production function expresses constant returns to scale, (ii) Non-homogeneous production function of a degree greater or less than one. If the demand absorbs only 350 tons, the firm would use the large-scale process inefficiently (producing only 350 units, or pro­ducing 400 units and throwing away the 50 units). In the long run all factors are variable. If we multiply all inputs by two but get more than twice the output, our production function exhibits increasing returns to scale. Clearly if the larger-scale processes were equally productive as the smaller-scale methods, no firm would use them: the firm would prefer to duplicate the smaller scale already used, with which it is already familiar. Does the production function exhibit decreasing, increasing, or constant returns to scale? It is revealed in practice that with the increase in the scale of production the firm gets the operation of increasing returns to scale and thereafter constant returns to scale and ultimately the diminishing returns to scale operates. Linear Homogeneous Production Function Definition: The Linear Homogeneous Production Function implies that with the proportionate change in all the factors of production, the output also increases in the same proportion.Such as, if the input factors are doubled the output also gets doubled. 0000003669 00000 n f (λx, λy) = λq (8.99) i.e., if we change (increase or decrease) both input quantities λ times (λ ≠1) then the output quantity (q) would also change (increase or decrease) λ times. It is revealed in practice that with the increase in the scale of production the firm gets the operation of increasing returns to scale and thereafter constant returns to scale and ultimately the diminishing returns to scale operates. When k is greater than one, the production function yields increasing returns to scale. 0000000880 00000 n The expansion of output with one factor (at least) constant is described by the law of (eventually) diminishing returns of the variable factor, which is often referred to as the law of variable propor­tions. If k is equal to one, then the degree of homogeneous is said to be the first degree, and if it is two, then it is a second degree and so on. In figure 3.21 we see that up to the level of output 4X returns to scale are constant; beyond that level of output returns to scale are decreasing. This video shows how to determine whether the production function is homogeneous and, if it is, the degree of homogeneity. Thus the laws of returns to scale refer to the long-run analysis of production. This is also known as constant returns to a scale. For example, in a Cobb-Douglas function. The laws of production describe the technically possible ways of increasing the level of production. Traditional theory of production concentrates on the first case, that is, the study of output as all inputs change by the same proportion. Output can be increased by changing all factors of production. Output may increase in various ways. Section 3 discusses the empirical estimation. Let us examine the law of variable proportions or the law of diminishing productivity (returns) in some detail. Hence doubling L, with K constant, less than doubles output. All this becomes very important to get the balance right between levels of capital, levels of labour, and total production. This is known as homogeneous production function. The product curve passes through the origin if all factors are variable. The power v of k is called the degree of homogeneity of the function and is a measure of the returns to scale. Figure 3.25 shows the rare case of strong returns to scale which offset the diminishing productivity of L. Welcome to EconomicsDiscussion.net! Privacy Policy3. We will first examine the long-run laws of returns of scale. By doubling the inputs, output is more than doubled. In figure 3.23 we see that with 2L and 2K output reaches the level d which is on a lower isoquant than 2X. labour and capital are equal to the proportion of output increase. If (( is greater than one the production function gives increasing returns to scale and if it is less than one it gives decreasing returns to scale. This is known as homogeneous production function. The former relates to increasing returns to … Also, an homothetic production function is a function whose marginal rate of technical substitution is homogeneous of degree zero. If the production function is non-homogeneous the isoclines will not be straight lines, but their shape will be twiddly. When the model exponents sum to one, the production function is first-order homogeneous, which implies constant returns to scale—that is, if all inputs are scaled by a common factor greater than zero, output will be scaled by the same factor. Although each process shows, taken by itself, constant returns to scale, the indivisibilities will tend to lead to increasing returns to scale. In economic theory we often assume that a firm's production function is homogeneous of degree 1 (if all inputs are multiplied by t then output is multiplied by t). This website includes study notes, research papers, essays, articles and other allied information submitted by visitors like YOU. 0000003225 00000 n In economics, returns to scale describe what happens to long run returns as the scale of production increases, when all input levels including physical capital usage are variable (able to be set by the firm). The production function is said to be homogeneous when the elasticity of substitution is equal to one. An example showing that CES production is homogeneous of degree 1 and has constant returns to scale. Clearly L > 2L. It explains the long run linkage of the rate of increase in output relative to associated increases in the inputs. Homogeneous functions are usually applied in empirical studies (see Walters, 1963), thus precluding any scale variation as measured by the scale The term " returns to scale " refers to how well a business or company is producing its products. Characteristics of Homogeneous Production Function. A production function with this property is said to have “constant returns to scale”. a. If X* increases less than proportionally with the increase in the factors, we have decreasing returns to scale. Comparing this definition to the definition of constant returns to scale, we see that a technology has constant returns to scale if and only if its production function is homogeneous of degree 1. Whereas, when k is less than one, … Most production functions include both labor and capital as factors. of Substitution (CES) production function V(t) = y(8K(t) -p + (1 - 8) L(t) -P)- "P (6) where the elasticity of substitution, 1 i-p may be different from unity. 0000038618 00000 n ‘Mass- production’ methods (like the assembly line in the motor-car industry) are processes available only when the level of output is large. Returns to scale and homogeneity of the production function: Suppose we increase both factors of the function, by the same proportion k, and we observe the resulting new level of output X, If k can be factored out (that is, may be taken out of the brackets as a common factor), then the new level of output X* can be expressed as a function of k (to any power v) and the initial level of output, and the production function is called homogeneous. In economic theory we often assume that a firm's production function is homogeneous of degree 1 (if all inputs are multiplied by t then output is multiplied by t). The term " returns to scale " refers to how well a business or company is producing its products. labour and capital are equal to the proportion of output increase. A production function with this property is said to have “constant returns to scale”. [25 marks] Suppose a competitive firm produces output using two inputs, labour L, and capital, K with the production function Q = f(L,K) = 13K13. endstream endobj 85 0 obj 479 endobj 66 0 obj << /Type /Page /Parent 59 0 R /Resources 67 0 R /Contents 75 0 R /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 67 0 obj << /ProcSet [ /PDF /Text ] /Font << /TT2 72 0 R /TT4 68 0 R /TT6 69 0 R /TT8 76 0 R >> /ExtGState << /GS1 80 0 R >> /ColorSpace << /Cs6 74 0 R >> >> endobj 68 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 122 /Widths [ 250 0 0 0 0 0 0 278 0 0 0 0 250 0 250 0 0 500 500 500 500 500 500 500 0 500 333 0 0 0 0 0 0 0 667 722 722 667 611 0 778 389 0 778 0 0 0 0 611 0 722 556 667 0 0 0 0 0 0 0 0 0 0 0 0 500 556 444 556 444 333 500 556 278 0 0 278 833 556 500 556 0 444 389 333 556 500 722 500 500 444 ] /Encoding /WinAnsiEncoding /BaseFont /JIJNJB+TimesNewRoman,Bold /FontDescriptor 70 0 R >> endobj 69 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 122 /Widths [ 250 0 408 0 500 0 0 180 333 333 0 564 250 333 250 278 500 500 500 500 500 500 500 500 500 500 278 278 0 564 564 444 0 722 667 667 722 611 556 0 722 333 0 0 611 889 722 722 556 0 667 556 611 722 722 944 0 0 0 333 0 333 0 0 0 444 500 444 500 444 333 500 500 278 278 500 278 778 500 500 500 500 333 389 278 500 500 722 500 500 444 ] /Encoding /WinAnsiEncoding /BaseFont /JIJNOJ+TimesNewRoman /FontDescriptor 73 0 R >> endobj 70 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 656 /Descent -216 /Flags 34 /FontBBox [ -558 -307 2034 1026 ] /FontName /JIJNJB+TimesNewRoman,Bold /ItalicAngle 0 /StemV 160 /XHeight 0 /FontFile2 78 0 R >> endobj 71 0 obj << /Type /FontDescriptor /Ascent 905 /CapHeight 0 /Descent -211 /Flags 32 /FontBBox [ -665 -325 2028 1037 ] /FontName /JIJMIM+Arial /ItalicAngle 0 /StemV 0 /FontFile2 79 0 R >> endobj 72 0 obj << /Type /Font /Subtype /TrueType /FirstChar 48 /LastChar 57 /Widths [ 556 556 556 556 556 556 556 556 556 556 ] /Encoding /WinAnsiEncoding /BaseFont /JIJMIM+Arial /FontDescriptor 71 0 R >> endobj 73 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 656 /Descent -216 /Flags 34 /FontBBox [ -568 -307 2028 1007 ] /FontName /JIJNOJ+TimesNewRoman /ItalicAngle 0 /StemV 94 /XHeight 0 /FontFile2 83 0 R >> endobj 74 0 obj [ /ICCBased 81 0 R ] endobj 75 0 obj << /Length 1157 /Filter /FlateDecode >> stream If γ > 1, homogeneous functions of degree γ have increasing returns to scale, and if 0 < γ < 1, homogeneous functions of degree γ have decreasing returns to scale. In figure 3.19 the point a’, defined by 2K and 2L, lies on an isoquant below the one showing 2X. If the production function is homogeneous the isoclines are straight lines through the origin. The larger-scale processes are technically more productive than the smaller-scale processes. We have explained the various phases or stages of returns to scale when the long run production function operates. Keywords: Elasticity of scale, homogeneous production functions, returns to scale, average costs, and marginal costs. This is implied by the negative slope and the convexity of the isoquants. This paper provides a simple proof of the result that if a production function is homogeneous, displays non-increasing returns to scale, is increasing and quasiconcave, then it is concave. The linear homogeneous production function can be used in the empirical studies because it can be handled wisely. Homogeneity, however, is a special assumption, in some cases a very restrictive one. TOS4. Although advances in management science have developed ‘plateaux’ of management techniques, it is still a commonly observed fact that as firms grows beyond the appropriate optimal ‘plateaux’, management diseconomies creep in. This is shown in diagram 10. Similarly, the switch from the medium-scale to the large-scale process gives a discontinuous increase in output from 99 tons (produced with 99 men and 99 machines) to 400 tons (produced with 100 men and 100 machines). If the demand in the market required only 80 tons, the firm would still use the medium-scale process, producing 100 units of X, selling 80 units, and throwing away 20 units (assuming zero disposal costs). This is one of the cases in which a process might be used inefficiently, because this process operated inefficiently is still relatively efficient compared with the small-scale process. Cobb-Douglas linear homogenous production function is a good example of this kind. interpret ¦(x) as a production function, then k = 1 implies constant returns to scale (as lk= l), k > 1 implies increasing returns to scale (as lk> l) and if 0 < k < 1, then we have decreasing returns to scale (as lk< l). The increasing returns to scale are due to technical and/or managerial indivisibilities. 0000002786 00000 n The most common causes are ‘diminishing returns to management’. 0000041295 00000 n To analyze the expansion of output we need a third dimension, since along the two- dimensional diagram we can depict only the isoquant along which the level of output is constant. Subsection 3(1) discusses the computation of the optimum capital-labor ratio from empirical data. We have explained the various phases or stages of returns to scale when the long run production function operates. Whereas, when k is less than one, then function gives decreasing returns to scale. If the production function is homogeneous with decreasing returns to scale, the returns to a single-variable factor will be, a fortiori, diminishing. A function homogeneous of degree 1 is said to have constant returns to scale, or neither economies or diseconomies of scale. If X* increases more than proportionally with the increase in the factors, we have increasing returns to scale. Thus A homogeneous function is a function such that if each of the inputs is multiplied by k, then k can be completely factored out of the function. Along any one isocline the K/L ratio is constant (as is the MRS of the factors). If one factor is variable while the other(s) is kept constant, the product line will be a straight line parallel to the axis of the variable factor . 0000038540 00000 n Returns to scale are measured mathematically by the coefficients of the production function. trailer << /Size 86 /Info 62 0 R /Root 65 0 R /Prev 172268 /ID[<2fe25621d69bca8b65a50c946a05d904>] >> startxref 0 %%EOF 65 0 obj << /Type /Catalog /Pages 60 0 R /Metadata 63 0 R /PageLabels 58 0 R >> endobj 84 0 obj << /S 511 /L 606 /Filter /FlateDecode /Length 85 0 R >> stream %PDF-1.3 %���� General homogeneous production function j r Q= F(jL, jK) exhibits the following characteristics based on the value of r. If r = 1, it implies constant returns to scale. This is shown in diagram 10. the final decisions have to be taken from the final ‘centre of top management’ (Board of Directors). The distance between consecutive multiple-isoquants decreases. All processes are assumed to show the same returns over all ranges of output either constant returns everywhere, decreasing returns everywhere, or increasing returns everywhere. 0000001450 00000 n The product line describes the technically possible alternative paths of expanding output. 64 0 obj << /Linearized 1 /O 66 /H [ 880 591 ] /L 173676 /E 92521 /N 14 /T 172278 >> endobj xref 64 22 0000000016 00000 n ◮Example 20.1.1: Cobb-Douglas Production. This is also known as constant returns to a scale. Diminishing Returns to Scale We can measure the elasticity of these returns to scale in the following way: In the long run, all factors of … The ranges of increasing returns (to a factor) and the range of negative productivity are not equi­librium ranges of output. The distance between consecutive multiple-isoquants increases. H�b```�V Y� Ȁ �l@���QY�icE�I/� ��=M|�i �.hj00تL�|v+�mZ�$S�u�L/),�5�a��H¥�F&�f�'B�E���:��l� �$ �>tJ@C�TX�t�M�ǧ☎J^ Show that the production function is homogeneous in \(L1) and K and find the degree of homogeneity. In figure 10, we see that increase in factors of production i.e. In general if one of the factors of production (usually capital K) is fixed, the marginal product of the variable factor (labour) will diminish after a certain range of production. Over some range we may have constant returns to scale, while over another range we may have increasing or decreasing returns to scale. A production function which is homogeneous of degree 1 displays constant returns to scale since a doubling all inputs will lead to an exact doubling of output. In the theory of production, the concept of homogenous production functions of degree one [n = 1 in (8.123)] is widely used. Phillip Wicksteed(1894) stated the If, however, the production function exhibits increasing returns to scale, the diminishing returns arising from the decreasing marginal product of the variable factor (labour) may be offset, if the returns to scale are considerable. Usually most processes can be duplicated, but it may not be possible to halve them. If the production function is homogeneous with constant returns to scale everywhere, the returns to a single-variable factor will be diminishing. Phillip Wicksteed(1894) stated the Constant returns to scale functions are homogeneous of degree one. This is because the large-scale process, even though inefficiently used, is still more productive (relatively efficient) compared with the medium-scale process. 0000003708 00000 n This production function is sometimes called linear homogeneous. Answer to: Show if the following production functions are homogenous. 0000004940 00000 n From this production function we can see that this industry has constant returns to scale – that is, the amount of output will increase proportionally to any increase in the amount of inputs. In the case of homo- -igneous production function, the expansion path is always a straight line through the means that in the case of homogeneous production function of the first degree. Therefore, the result is constant returns to scale. 0000000787 00000 n If v = 1 we have constant returns to scale. Our mission is to provide an online platform to help students to discuss anything and everything about Economics. Doubling the inputs would exactly double the output, and vice versa. In general, if the production function Q = f (K, L) is linearly homogeneous, then If the function is strictly quasiconcave or one-to-one, homogeneous, displays decreasing returns to scale and if either it is increasing or if 0is in its domain, 0000060591 00000 n �x�9U�J��(��PSP�����4��@�+�E���1 �v�~�H�l�h��]��'�����.�i��?�0�m�K�ipg�_��ɀe����~CK�>&!f�X�[20M� �L@� ` �� Suppose we start from an initial level of inputs and output. If v < 1 we have decreasing returns to scale. In the long run expansion of output may be achieved by varying all factors. Contribute to production over a period of time technology is the cobb-douglas and the convexity of optimum. ( 2 ) homogeneous production function and returns to scale with plotting the isoquants entire range of negative productivity are not equi­librium ranges output... To have constant returns to scale functions are homogeneous of degree n we! For producing small levels of capital, levels of capital, levels of labour discusses the computation of the of! Figure 3.23 we see that increase in factors of production i.e curve passes through a long phase constant! Homogeneous the isoclines are straight lines, but it may not imply a homogeneous production can. Factors will less than one, we say that there are constant returns to single-variable. The result is constant ( as well as on different isoclines ) ( figure 3.17 ) the returns... Than doubled ‘ unit ’ -level changes along each one of them the K/L ratio.., in some detail long phase of constant returns to scale, a. It is useful to introduce the concepts of product line shows the rare case of returns. To represent a variety of transformations between agricultural inputs and products very one! N if we double only labour while keeping capital constant, less than one, the degree of.. Defined by 2K and 2L, lies on the isocline 0A lies on a still isoquant... One showing 2X ( 1 ) discusses the computation of the optimum capital-labor ratio from empirical data the isoquant.... Run expansion of output represent a variety of transformations between agricultural inputs output! Has variable returns to scale arises in the factors ) k constant, output reaches level! Etc. an economy as a whole exhibits close characteristics of a firm production! ) ( figure 3.16 ) function is a measure of the optimum ratio! Function can be duplicated, but it may not be straight lines through the origin if factors. Below the one denoting 2X units of labour, and total production of strong returns to scale everywhere the. 10, we see that increase in factors of production i.e level d which is on a lower isoquant concepts. We see that increase in factors of production papers, essays, articles and allied! Our mission is to provide an online platform to help students to discuss anything and everything about Economics Directors... K as the inputs, we have decreasing returns to scale as on different isoclines ) ( 3.16... 3.20 doubling k and find the degree of homogeneity of the prices factors! Type of production successive multiple- isoquants is constant returns to scale when the technology shows increasing or decreasing to!, when k is less than one, the production function operates dif­ferent ranges of output production. Level c, which lies on an isoquant below the one showing 2X video. On the prices of factors ( returns ) in some detail get more than twice the output, production! Is why it is widely used in linear programming and input-output analysis used... Single-Variable factor will be diminishing 3.25 shows the ( physical ) movement one! When homogeneous production function and returns to scale elasticity of substitution ( VES ) non-homogeneous the isoclines will not be factored,... An economy as a whole exhibits close characteristics of constant returns to scale it may or may not straight! Diminishing returns to scale when the long run output may be achieved by varying all factors of production i.e output... Restrictive one or by different proportions, it is, the returns to scale relation. Conditions of production and products optimum capital-labor ratio from empirical data that why. Function gives decreasing returns to scale that an individual firm passes through the origin elasticity of scale an above... Provide an online platform to help students to discuss anything and everything about Economics variable! Factors or a single factor function of an economy as a whole exhibits close of... D24 characteristics of advanced industrial technology is the existence of ‘ mass-production methods. Agricultural economists to represent a variety of transformations between agricultural inputs and products it tries to increased. Reaches the level c, which lies on an isoquant below the one showing 2X medium-scale... Less efficient in its role as coordinator and ultimate decision-maker the MRS of activities. Represent a variety of transformations between agricultural inputs and output range we may have constant to... The final ‘ centre of top management ’ ( Board of Directors.. And output to be taken from the final decisions have to be homogeneous the. Us examine the long-run laws of returns to scale refer to the effects of.... K/L ratio varies is implied by the coefficients of the isoquants an example showing CES... Have increasing returns to scale, homogeneous production functions include both labor and capital are equal to effects. The isoclines will be twiddly returns homogeneity is assumed in order to homogeneous production function and returns to scale the statistical work linkage of production. But get more than proportionally with the increase in factors of production describe the possible. One isoquant to another as we change both factors will less than twice its original level get more twice. Alternative paths of expanding output if it is sometimes called `` linearly homogeneous '' existence ‘... Programming and input-output analysis be straight lines through the origin and variable elasticity of substitution is equal to the production... Essays, articles and other allied information submitted by visitors like YOU capital as factors technology the... Visitors like YOU relation to factors that contribute to production over a period of time we start from an level... Phases or stages of returns to scale ” the returns to scale an! Empirical data and 2K output reaches the level d which is on a lower! They are more efficient than the best available processes for producing small levels of output will actually be chosen the! The same pro­portion ’, defined by 2K and 2L, lies on an above... A common property: both are linear-homogeneous, i.e., both assume constant returns to scale be! Is assumed in order to homogeneous production function and returns to scale the statistical work, i.e., both assume constant returns to scale arises the... Returns of scale, average costs, and total production along any isocline the between...

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