We can measure the elasticity of these returns to scale in the following way: Another common production function is the Cobb-Douglas production function. If we multiply all inputs by two but get more than twice the output, our production function exhibits increasing returns to scale. The concept of returns to scale arises in the context of a firm's production function. 0000003669 00000 n
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. 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. 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, 0000029326 00000 n
Interestingly, the production function of an economy as a whole exhibits close characteristics of constant returns to scale. 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). This website includes study notes, research papers, essays, articles and other allied information submitted by visitors like YOU. 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. For 50 < X < 100 the medium-scale process would be used. The ranges of increasing returns (to a factor) and the range of negative productivity are not equilibrium ranges of output. 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). 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 . By doubling the inputs, output increases by less than twice its original level. If k cannot be factored out, the production function is non-homogeneous. f(tL, tK) = t n f(L, K) = t n Q (8.123) where t is a positive real number. 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. 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. 0000003441 00000 n
Our mission is to provide an online platform to help students to discuss anything and everything about Economics. Characteristics of Homogeneous Production Function. We have explained the various phases or stages of returns to scale when the long run production function operates. If X* increases less than proportionally with the increase in the factors, we have decreasing returns to scale. The K/L ratio diminishes along the product line. Constant returns-to-scale production functions are homogeneous of degree one in inputs f (tk, t l) = functions are homogeneous … 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. If the production function is homogeneous with decreasing returns to scale, the returns to a single-variable factor will be, a fortiori, diminishing. 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. If X* increases by the same proportion k as the inputs, we say that there are constant returns to scale. 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, Does the production function exhibit decreasing, increasing, or constant returns to scale? the final decisions have to be taken from the final ‘centre of top management’ (Board of Directors). 0000002268 00000 n
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Also, studies suggest that an individual firm passes through a long phase of constant return to scale in its lifetime. Output may increase in various ways. 0000000787 00000 n
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That is, in the case of homogeneous production function of degree 1, we would obtain … 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. Most production functions include both labor and capital as factors. In the long run, all factors of … Production functions with varying returns to scale are difficult to handle and economists usually ignore them for the analysis of production. Therefore, the result is constant returns to scale. Before explaining the graphical presentation of the returns to scale it is useful to introduce the concepts of product line and isocline. Most production functions include both labor and capital as factors. [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. 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. It tries to pinpoint increased production in relation to factors that contribute to production over a period of time. Answer to: Show if the following production functions are homogenous. Constant Elasticity of Substitution Production Function: The CES production function is otherwise … %PDF-1.3
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All this becomes very important to get the balance right between levels of capital, levels of labour, and total production. 0000004940 00000 n
labour and capital are equal to the proportion of output increase. 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. Cobb-Douglas linear homogenous production function is a good example of this kind. 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. For example, in a Cobb-Douglas function. Whereas, when k is less than one, … Cobb-Douglas linear homogenous production function is a good example of this kind. 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. 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. 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. In figure 3.19 the point a’, defined by 2K and 2L, lies on an isoquant below the one showing 2X. 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). 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. We have explained the various phases or stages of returns to scale when the long run production function operates. 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. Figure 3.25 shows the rare case of strong returns to scale which offset the diminishing productivity of L. Welcome to EconomicsDiscussion.net! A function g : R — R is said to be a positive monotonie transformation if g is a strictly increasing function; that is, a function for which x > y implies that g(x) > g(y). A production function with this property is said to have “constant returns to scale”. An example showing that CES production is homogeneous of degree 1 and has constant returns to scale. The term " returns to scale " refers to how well a business or company is producing its products. 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. This is shown in diagram 10. 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. If we double only labour while keeping capital constant, output reaches the level c, which lies on a still lower isoquant. Homogeneous production functions are frequently used by agricultural economists to represent a variety of transformations between agricultural inputs and products. With constant returns to scale everywhere on the production surface, doubling both factors (2K, 2L) leads to a doubling of output. Clearly L > 2L. Subsection 3(2) deals with plotting the isoquants of an empirical production function. 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). 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. The product curve passes through the origin if all factors are variable. The ‘management’ is responsible for the co-ordination of the activities of the various sections of the firm. If we wanted to double output with the initial capital K, we would require L units of labour. 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. b. 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. The product line describes the technically possible alternative paths of expanding output. In figure 3.20 doubling K and L leads to point b’ which lies on an isoquant above the one denoting 2X. 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. Increasing Returns to Scale TOS4. Lastly, it is also known as the linear homogeneous production function. Since returns to scale are decreasing, doubling both factors will less than double output. 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. 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