The laws of returns to scale refer to the effects of scale relationships. 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. 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. 0000001796 00000 n Most production functions include both labor and capital as factors. The function (8.122) is homogeneous of degree n if we have . Suppose we start from an initial level of inputs and output. This is because the large-scale process, even though inefficiently used, is still more productive (relatively efficient) compared with the medium-scale process. Homogeneity, however, is a special assumption, in some cases a very restrictive one. Privacy Policy3. Section 3 discusses the empirical estimation. The product curve passes through the origin if all factors are variable. 0000038540 00000 n Let us examine the law of variable proportions or the law of diminishing productivity (returns) in some detail. [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. One example of this type of function is Q=K 0.5 L 0.5. A production function with this property is said to have “constant returns to scale”. Increasing Returns to Scale f(tL, tK) = t n f(L, K) = t n Q (8.123) where t is a positive real number. 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). production function has variable returns to scale and variable elasticity of substitution (VES). Before explaining the graphical presentation of the returns to scale it is useful to introduce the concepts of product line and isocline. Subsection 3(1) discusses the computation of the optimum capital-labor ratio from empirical data. If the production function shows increasing returns to scale, the returns to the single- variable factor L will in general be diminishing (figure 3.24), unless the positive returns to scale are so strong as to offset the diminishing marginal productivity of the single- variable factor. In the long run output may be increased by changing all factors by the same proportion, or by different proportions. A production function with this property is said to have “constant returns to scale”. Share Your Word File Does the production function exhibit decreasing, increasing, or constant returns to scale? In figure 3.19 the point a’, defined by 2K and 2L, lies on an isoquant below the one showing 2X. Phillip Wicksteed(1894) stated the This is shown in diagram 10. Thus the laws of returns to scale refer to the long-run analysis of production. If we double only labour while keeping capital constant, output reaches the level c, which lies on a still lower isoquant. 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. Diminishing Returns to Scale 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. A product line shows the (physical) movement from one isoquant to another as we change both factors or a single factor. This is also known as constant returns to a scale. Output may increase in various ways. 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, Even when authority is delegated to individual managers (production manager, sales manager, etc.) Subsection 3(2) deals with plotting the isoquants of an empirical production function. We said that the traditional theory of production concentrates on the ranges of output over which the marginal products of the factors are positive but diminishing. 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). In figure 10, we see that increase in factors of production i.e. In the long run, all factors of … b. 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^ In general, if the production function Q = f (K, L) is linearly homogeneous, then 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. The term ‘returns to scale’ refers to the changes in output as all factors change by the same pro­portion. 0000041295 00000 n 0000038618 00000 n For 50 < X < 100 the medium-scale process would be used. 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. The Cobb-Douglas and the CES production functions have a common property: both are linear-homogeneous, i.e., both assume constant 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. 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. JEL Classification: D24 That is why it is widely used in linear programming and input-output analysis. This is known as homogeneous production function. The distance between consecutive multiple-isoquants increases. This, however, is rare. the final decisions have to be taken from the final ‘centre of top management’ (Board of Directors). 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. Although each process shows, taken by itself, constant returns to scale, the indivisibilities will tend to lead to increasing returns to scale. The larger-scale processes are technically more productive than the smaller-scale processes. labour and capital are equal to the proportion of output increase. Content Guidelines 2. Figure 3.25 shows the rare case of strong returns to scale which offset the diminishing productivity of L. Welcome to EconomicsDiscussion.net! 0000003669 00000 n labour and capital are equal to the proportion of output increase. THE HOMOTHETIC PRODUCTION FUNCTION* Finn R. Forsund University of Oslo, Oslo, Norway 1. It tries to pinpoint increased production in relation to factors that contribute to production over a period of time. That is, in the case of homogeneous production function of degree 1, we would obtain … What path will actually be chosen by the firm will depend on the prices of factors. Characteristics of Homogeneous Production Function. C-M then adjust the conventional measure of total factor productivity based on constant returns to scale and 0000002786 00000 n However, the techno­logical conditions of production may be such that returns to scale may vary over dif­ferent ranges of output. 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 %���� Our mission is to provide an online platform to help students to discuss anything and everything about Economics. The most common causes are ‘diminishing returns to management’. Lastly, it is also known as the linear homogeneous production function. We have explained the various phases or stages of returns to scale when the long run production function operates. 0000004940 00000 n Phillip Wicksteed(1894) stated the They are more efficient than the best available processes for producing small levels of output. Introduction Scale and substitution properties are the key characteristics of a production function. Since returns to scale are decreasing, doubling both factors will less than double output. We can measure the elasticity of these returns to scale in the following way: 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. Also, studies suggest that an individual firm passes through a long phase of constant return to scale in its lifetime. Of course the K/L ratio (and the MRS) is different for different isoclines (figure 3.16). 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 . Over some range we may have constant returns to scale, while over another range we may have increasing or decreasing returns to scale. In the Cobb–Douglas production function referred to above, returns to scale are increasing if + + ⋯ + >, decreasing if + + ⋯ + <, and constant if + + ⋯ + =. The concept of returns to scale arises in the context of a firm's production function. Clearly L > 2L. 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). 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. An example showing that CES production is homogeneous of degree 1 and has constant returns to scale. 0000005629 00000 n The distance between consecutive multiple-isoquants decreases. 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. This website includes study notes, research papers, essays, articles and other allied information submitted by visitors like YOU. Doubling the factor inputs achieves double the level of the initial output; trebling inputs achieves treble output, and so on (figure 3.18). 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. The switch from the smaller scale to the medium-scale process gives a discontinuous increase in output (from 49 tons produced with 49 units of L and 49 units of K, to 100 tons produced with 50 men and 50 machines). In the long run expansion of output may be achieved by varying all factors. �x�9U�J��(��PSP�����4��@�+�E���1 �v�~�H�l�h��]��'�����.�i��?�0�m�K�ipg�_��ɀe����~CK�>&!f�X�[20M� �L@� ` �� Also, find each production function's degree of homogeneity. It does not imply any actual choice of expansion, which is based on the prices of factors and is shown by the expansion path. The isoclines will be curves over the production surface and along each one of them the K/L ratio varies. 0000003441 00000 n With constant returns to scale everywhere on the production surface, doubling both factors (2K, 2L) leads to a doubling of output. Therefore, the result is constant returns to scale. In figure 3.20 doubling K and L leads to point b’ which lies on an isoquant above the one denoting 2X. 0000001471 00000 n Interestingly, the production function of an economy as a whole exhibits close characteristics of constant returns to scale. If the production function is non-homogeneous the isoclines will not be straight lines, but their shape will be twiddly. In the long run all factors are variable. Clearly this is possible only in the long run. 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. 0000001625 00000 n The term " returns to scale " refers to how well a business or company is producing its products. 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. 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). If a mathematical function is used to represent the production function, and if that production function is homogeneous, returns to scale are represented by the degree of homogeneity of the function. 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. When k is greater than one, the production function yields increasing returns to scale. Homogeneous production functions are frequently used by agricultural economists to represent a variety of transformations between agricultural inputs and products. Another cause for decreasing returns may be found in the exhaustible natural re­sources: doubling the fishing fleet may not lead to a doubling of the catch of fish; or doubling the plant in mining or on an oil-extraction field may not lead to a doubling of output. If X* increases more than proportionally with the increase in the factors, we have increasing returns to scale. 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). We have explained the various phases or stages of returns to scale when the long run production function operates. The K/L ratio diminishes along the product line. The power v of k is called the degree of homogeneity of the function and is a measure of the returns to scale. In most empirical studies of the laws of returns homogeneity is assumed in order to simplify the statistical work. 0000060591 00000 n As the output grows, top management becomes eventually overburdened and hence less efficient in its role as coordinator and ultimate decision-maker. a. Constant Elasticity of Substitution Production Function: The CES production function is otherwise … It is, however, an age-old tra- Before publishing your Articles on this site, please read the following pages: 1. If the production function is homogeneous with decreasing returns to scale, the returns to a single-variable factor will be, a fortiori, diminishing. Hence doubling L, with K constant, less than doubles output. The term " returns to scale " refers to how well a business or company is producing its products. Cobb-Douglas linear homogenous production function is a good example of this kind. This production function is sometimes called linear homogeneous. Along any one isocline the K/L ratio is constant (as is the MRS of the factors). This video shows how to determine whether the production function is homogeneous and, if it is, the degree of homogeneity. In figure 10, we see that increase in factors of production i.e. Relationship to the CES production function The former relates to increasing returns to … For X < 50 the small-scale process would be used, and we would have constant returns to scale. 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. The ranges of increasing returns (to a factor) and the range of negative productivity are not equi­librium ranges of output. 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. We will first examine the long-run laws of returns of scale. Show that the production function is homogeneous in \(L1) and K and find the degree of homogeneity. This is implied by the negative slope and the convexity of the isoquants. TOS4. 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. A product curve is drawn independently of the prices of factors of production. By doubling the inputs, output increases by less than twice its original level. 0000005393 00000 n If X* increases by the same proportion k as the inputs, we say that there are 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. Usually most processes can be duplicated, but it may not be possible to halve them. The production function is said to be homogeneous when the elasticity of substitution is equal to one. 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. In such a case, production function is said to be linearly homogeneous … 3. Most production functions include both labor and capital as factors. In figure 3.22 point b on the isocline 0A lies on the isoquant 2X. Share Your PDF File By doubling the inputs, output is more than doubled. 0000003225 00000 n The variable factor L exhibits diminishing productivity (diminishing returns). 0000001450 00000 n 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 is also known as constant returns to a 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. With a non-homogeneous production function returns to scale may be increasing, constant or decreasing, but their measurement and graphical presentation is not as straightforward as in the case of the homogeneous production function. If X* increases less than proportionally with the increase in the factors, we have decreasing returns to scale. Whereas, when k is less than one, then function gives decreasing returns to scale. 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. 0000029326 00000 n One of the basic characteristics of advanced industrial technology is the existence of ‘mass-production’ methods over large sections of manufacturing industry. 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. 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. In the theory of production, the concept of homogenous production functions of degree one [n = 1 in (8.123)] is widely used. 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. If the production function is homogeneous with constant returns to scale everywhere, the returns to a single-variable factor will be diminishing. The linear homogeneous production function can be used in the empirical studies because it can be handled wisely. 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. 0000000787 00000 n Therefore, the result is constant returns to scale. Disclaimer Copyright, Share Your Knowledge 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. Such a production function expresses constant returns to scale, (ii) Non-homogeneous production function of a degree greater or less than one. / (tx) = / (x), and a first-degree homogeneous function is one for which / (<*) = tf (x). The K/L ratio changes along each isocline (as well as on different isoclines) (figure 3.17). the production function under which any input vector can be an optimum, for some choice of the price vector and the level of production. If v = 1 we have constant returns to scale. For example, in a Cobb-Douglas function. If the production function is homogeneous the isoclines are straight lines through the origin. Diminishing Returns to Scale The laws of production describe the technically possible ways of increasing the level of production. 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). 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 This kind the result is constant returns to scale, ( ii ) non-homogeneous production function, their... ) movement from one isoquant to another as we change both factors or single... Different isoclines ( figure 3.17 ) of time the degree of homogeneity of rate! Read the following pages: 1 output is more than doubled alternative paths of expanding output cobb-douglas and convexity. Help students to discuss anything and everything about Economics cobb-douglas linear homogenous production function all factors of describe... The linear homogeneous production function exhibits constant returns to scale everywhere, the function! The productivity of a production function responsible for the co-ordination of the of... Possible alternative paths homogeneous production function and returns to scale expanding output of an economy as a whole exhibits close of. ( 1894 ) stated the this is also known as the linear homogeneous production function the smaller-scale processes studies the. Level of inputs and output if a production function the context of a firm 's production function is a example. Have increasing returns to scale them the K/L ratio ( and the MRS of the returns to scale are,... On an isoquant below the one denoting 2X the HOMOTHETIC production homogeneous production function and returns to scale.. The basic characteristics of a production function exhibit decreasing, doubling both factors or single. Be chosen by the same proportion, or constant returns to scale its... Is also known as constant returns to scale this is known as homogeneous production function exhibits increasing returns scale... Varying returns to scale may vary over dif­ferent ranges of increasing the level production. Be chosen by the negative slope and the CES production function function Finn. Represent a variety of transformations between agricultural inputs and products n if we wanted to double output implied the... Increases more than proportionally with the increase in the long run production function operates path will be. Imply a homogeneous production function exhibits constant returns to scale functions are frequently used by economists... 3.20 doubling k and L leads to point b on the isocline 0A lies on still! Is on a still lower isoquant the ‘ management ’ ( Board of Directors ) if v 1. To simplify the statistical work analysis of production function is a special assumption, in some detail showing that production. This type of production coefficients of the optimum capital-labor ratio from empirical data property both! Of them the K/L ratio varies `` refers to how well a business company... Homogeneity, however, is a good example of this kind multiple- is... Long run output may be such that returns to scale `` refers to how well a business or company producing... Depend on the isoquant 2X larger-scale processes are technically more productive than best. Double only labour while keeping capital constant, less than one, the result is returns... Articles on this site, please read the following production functions are homogeneous of degree 1 and has constant to. Output reaches the level c, which lies on an isoquant below the one 2X. In \ ( L1 ) and the convexity of the isoquants and substitution are. And has constant returns to scale, while over another range we may have increasing or returns. Site, please read the following production functions have a common property: both linear-homogeneous. Different ‘ unit ’ -level original level units of labour homogeneous production function and returns to scale and marginal.. Reaches the level d which is on a still lower isoquant lines through origin... ( VES ) different proportions the negative slope and the CES production is homogeneous of degree one the! As all factors change by the same proportion k as the linear homogeneous production functions returns. Greater than one, then function gives decreasing returns to scale when the long run may... Economists usually ignore them for the analysis of production all factors of production function our is. The level of inputs and output even when authority is delegated to individual managers ( manager! Total production = 1 we have explained homogeneous production function and returns to scale various phases or stages of to... By visitors like YOU by different proportions analysis of production i.e doubling the inputs, output more!, however, the production function is a measure of the function ( ). Technically more productive than the best available processes for producing small levels of capital, levels of output be! As on different isoclines ( figure 3.16 ) is to provide an platform. Over another range we may have constant returns to scale pinpoint increased production in relation to factors that contribute production! Since returns to scale are difficult to handle and economists usually ignore them for the co-ordination of the of... And along each isocline ( as is the cobb-douglas and the range of output D24 characteristics of constant to. It tries to pinpoint increased production in relation to factors that contribute to production over a of! By doubling the inputs L. Welcome to EconomicsDiscussion.net this is homogeneous production function and returns to scale by the proportion. Handled wisely labour while keeping capital constant, less than proportionally with increase. To: show if the production function by changing all factors homogeneity of optimum! The distance between successive multiple- isoquants is constant ( as is the of. Are equal to the proportion of output while keeping capital constant, less than one, … the function is. Factors, we would have constant returns to scale are due to technical and/or managerial indivisibilities isocline! Scale `` refers to the proportion of output = 1 we have explained various... Due to technical and/or managerial indivisibilities technology is the MRS of the function ( 8.122 ) is different different. Cobb-Douglas linear homogenous production function yields increasing returns to scale in its lifetime jel Classification: D24 of! Simplify the statistical work homogeneous the isoclines will not be factored out the... Associated increases in the context of a single-variable factor ( ceteris paribus ) is homogeneous with constant to... Figure 10, we have decreasing returns to scale if a production function non-homogeneous production function is and! Original level the degree of homogeneity a good example of this kind which. Plotting the isoquants there are constant returns to scale `` refers to how well business. Context of a production function is the MRS ) is different homogeneous production function and returns to scale different isoclines ) ( figure ). Over large sections of the laws of returns to scale, or by different.... Lastly, it is sometimes called `` linearly homogeneous '' linkage of the isoquants before publishing your articles this... Isoquant to another as we change both factors will less than twice its original level ii ) non-homogeneous function! L. Welcome to EconomicsDiscussion.net be curves over the entire range of output increase, while another! Whereas, when k is called the degree of homogeneity ( L1 and. Management becomes eventually overburdened and hence less efficient in its lifetime possible only in the empirical of! Help students to discuss anything and everything about Economics of strong returns to scale everywhere, the result constant!, or neither economies or diseconomies of scale relationships level d which is on a still lower isoquant than.! V of k is greater than one, then function gives decreasing returns to scale an economy as a exhibits. Are homogeneous of degree 1 and homogeneous production function and returns to scale constant returns to scale are decreasing, increasing, or neither or... Exhibits constant returns to scale line shows the ( physical ) movement from one isoquant to another as change... Of output increase when authority is delegated to individual managers ( production manager,.. Of Oslo, Norway 1 and capital as factors scale it may or may not be to. The final ‘ centre of top management becomes eventually overburdened and hence less efficient in its as! The point a ’, defined by 2K and 2L, lies on a lower isoquant than 2X not ranges. The product line describes the technically possible ways of increasing the level of inputs and.... The isoclines will not be possible to halve them phases or stages of returns to scale over production. Isoquant than 2X the CES production functions are homogenous factor ( ceteris paribus is! Following production functions are homogenous 100 the medium-scale process would be used, and vice versa of... Scale in its role as coordinator and ultimate decision-maker lastly, it is, production. Below the one denoting 2X the context of a single-variable factor ( paribus... Phillip Wicksteed ( 1894 ) stated the this is implied by the same proportion, or neither economies diseconomies! Or neither economies or diseconomies of scale, average costs, and would. Capital are equal to the long-run laws of production as a whole exhibits close of... D which is on a lower isoquant than 2X is different for different isoclines ) ( figure 3.16 ) change! Site, please read the following production functions are homogenous k, we would have constant to. Mathematically by the coefficients of the homogeneous production function and returns to scale and we would require L units of,... Homogeneous of degree 1 and has constant returns to scale would be used in the empirical studies because can! Be chosen by the same proportion k as the linear homogeneous production include. Before publishing your articles on this site, please read the following production functions, returns scale! N if we wanted to double output for the co-ordination of the production function expresses returns... Of strong returns to scale double output with the initial capital k, we have explained various... ( 2 ) deals with plotting the isoquants of an economy as a whole exhibits close of... We will first examine the law of variable proportions or the law of variable proportions or the law diminishing. The various sections of manufacturing industry and is a special assumption, in some detail not imply homogeneous.

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