Another common production function is the Cobb-Douglas production function. Our mission is to provide an online platform to help students to discuss anything and everything about Economics. 0000005629 00000 n This is because the large-scale process, even though inefficiently used, is still more productive (relatively efficient) compared with the medium-scale process. Usually most processes can be duplicated, but it may not be possible to halve them. In the long run expansion of output may be achieved by varying all factors. Relationship to the CES production function 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. It tries to pinpoint increased production in relation to factors that contribute to production over a period of time. H��VKs�6��W�-d�� ��cl�N��xj�<=P\$d2�A A�Q~}w�!ٞd:� �����>����C��p����gVq�(��,|y�\]�*��|P��\�~��Qm< �Ƈ�e��8u�/�>2��@�G�I��"���)''��ș��Y��,NIT�!,hƮ��?b{�`��*�WR僇�7F��t�=u�B�nT��(�������/�E��R]���A���z�d�J,k���aM�q�M,�xR�g!�}p��UP5�q=�o�����h��PjpM{�/�;��%,s׋X�0����?6. 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 In figure 3.22 point b on the isocline 0A lies on the isoquant 2X. This is known as homogeneous production function. Introduction Scale and substitution properties are the key characteristics of a production function. If v = 1 we have constant returns to scale. As the output grows, top management becomes eventually overburdened and hence less efficient in its role as coordinator and ultimate decision-maker. For X < 50 the small-scale process would be used, and we would have constant returns to scale. This is implied by the negative slope and the convexity of the isoquants. However, the techno­logical conditions of production may be such that returns to scale may vary over dif­ferent ranges of output. 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. The concept of returns to scale arises in the context of a firm's production function. 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. 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. A production function with this property is said to have “constant returns to scale”. Therefore, the result is constant returns to scale. The distance between consecutive multiple-isoquants decreases. In such a case, production function is said to be linearly homogeneous … That is why it is widely used in linear programming and input-output analysis. 0000001625 00000 n In figure 10, we see that increase in factors of production i.e. If the production function is homogeneous with constant or decreasing returns to scale everywhere on the production surface, the productivity of the variable factor will necessarily be diminishing. The laws of returns to scale refer to the effects of scale relationships. 3. Suppose we start from an initial level of inputs and output. Before explaining the graphical presentation of the returns to scale it is useful to introduce the concepts of product line and isocline. If a production function is homogeneous of degree one, it is sometimes called "linearly homogeneous". Does the production function exhibit decreasing, increasing, or constant 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. If v > 1 we have increasing returns to scale. and we increase all the factors by the same proportion k. We will clearly obtain a new level of output X*, higher than the original level X0. f(tL, tK) = t n f(L, K) = t n Q (8.123) where t is a positive real number. Production functions with varying returns to scale are difficult to handle and economists usually ignore them for the analysis of production. When k is greater than one, the production function yields increasing returns to scale. THE HOMOTHETIC PRODUCTION FUNCTION* Finn R. Forsund University of Oslo, Oslo, Norway 1. The K/L ratio diminishes along the product line. %PDF-1.3 %���� The ranges of increasing returns (to a factor) and the range of negative productivity are not equi­librium ranges of output. Answer to: Show if the following production functions are homogenous. 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. C-M then adjust the conventional measure of total factor productivity based on constant returns to scale and hM�4dr;c�6����S���dB��'��Ķ��[|��ziz�F7���N|.�/�^����@V�Yc��G���� ���g*̋1����-��A�G%�N��3�|1q��cI;O��ө�d^��R/)�Y�o*"�\$�DGGػP�����Qr��q�C�:��`�@ b2 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. This is also known as constant returns to a scale. Along any isocline the distance between successive multiple- isoquants is constant. Traditional theory of production concentrates on the first case, that is, the study of output as all inputs change by the same proportion. 0000004940 00000 n The variable factor L exhibits diminishing productivity (diminishing returns). We have explained the various phases or stages of returns to scale when the long run production function operates. The product line describes the technically possible alternative paths of expanding output. 0000029326 00000 n In general, if the production function Q = f (K, L) is linearly homogeneous, then If we wanted to double output with the initial capital K, we would require L units of labour. The larger-scale processes are technically more productive than the smaller-scale processes. Whereas, when k is less than one, … Subsection 3(1) discusses the computation of the optimum capital-labor ratio from empirical data. The former relates to increasing returns to … Lastly, it is also known as the linear homogeneous production function. 0000000880 00000 n 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. 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. 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. Although each process shows, taken by itself, constant returns to scale, the indivisibilities will tend to lead to increasing 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. 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, the production function under which any input vector can be an optimum, for some choice of the price vector and the level of production. Since returns to scale are decreasing, doubling both factors will less than double output. The power v of k is called the degree of homogeneity of the function and is a measure of the returns to scale. Cobb-Douglas linear homogenous production function is a good example of this kind. [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 of the basic characteristics of advanced industrial technology is the existence of ‘mass-production’ methods over large sections of manufacturing industry. If we multiply all inputs by two but get more than twice the output, our production function exhibits increasing returns to scale. One example of this type of function is Q=K 0.5 L 0.5. When the technology shows increasing or decreasing returns to scale it may or may not imply a homogeneous production function. 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. Show that the production function is homogeneous in \(L1) and K and find the degree of homogeneity. The linear homogeneous production function can be used in the empirical studies because it can be handled wisely. 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. 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). Section 3 discusses the empirical estimation. 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. labour and capital are equal to the proportion of output increase. The term " returns to scale " refers to how well a business or company is producing its products. If X* increases by the same proportion k as the inputs, we say that there are constant returns to scale. Clearly L > 2L. Diminishing Returns to Scale 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^ By doubling the inputs, output is more than doubled. Over some range we may have constant returns to scale, while over another range we may have increasing or decreasing returns to scale. Privacy Policy3. 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. 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. the final decisions have to be taken from the final ‘centre of top management’ (Board of Directors). 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. JEL Classification: D24 Share Your PDF File 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. 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. 0000001796 00000 n Constant Elasticity of Substitution Production Function: The CES production function is otherwise … Even when authority is delegated to individual managers (production manager, sales manager, etc.) In the short run output may be increased by using more of the variable factor(s), while capital (and possibly other factors as well) are kept constant. This is known as homogeneous production function. Also, studies suggest that an individual firm passes through a long phase of constant return to scale in its lifetime. Along any one isocline the K/L ratio is constant (as is the MRS of the factors). With constant returns to scale everywhere on the production surface, doubling both factors (2K, 2L) leads to a doubling of output. If the production function is homogeneous with constant returns to scale everywhere, the returns to a single-variable factor will be diminishing. 0000003708 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. This video shows how to determine whether the production function is homogeneous and, if it is, the degree of homogeneity. Output may increase in various ways. Subsection 3(2) deals with plotting the isoquants of an empirical production function. 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. 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