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` �� 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. By doubling the inputs, output increases by less than twice its original level. 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). 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. Whereas, when k is less than one, … Whereas, when k is less than one, then function gives decreasing returns to scale. 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). 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). Homogeneous functions are usually applied in empirical studies (see Walters, 1963), thus precluding any scale variation as measured by the scale 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). In general, if the production function Q = f (K, L) is linearly homogeneous, then 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 are usually assumed to be the same everywhere on the production surface, that is, the same along all the expansion-product lines. Homogeneous production functions are frequently used by agricultural economists to represent a variety of transformations between agricultural inputs and products. When k is greater than one, the production function yields increasing returns to scale. Traditional theory of production concentrates on the first case, that is, the study of output as all inputs change by the same proportion. When the technology shows increasing or decreasing returns to scale it may or may not imply a homogeneous production function. f(tL, tK) = t n f(L, K) = t n Q (8.123) where t is a positive real number. The concept of returns to scale arises in the context of a firm's production function. Let us examine the law of variable proportions or the law of diminishing productivity (returns) in some detail. TOS4. This is shown in diagram 10. 0000002268 00000 n
Privacy Policy3. This, however, is rare. 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). 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 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. This is also known as constant returns to a scale. 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. 0000003020 00000 n
If X* increases less than proportionally with the increase in the factors, we have decreasing returns to scale. If we wanted to double output with the initial capital K, we would require L units of labour. The ‘management’ is responsible for the co-ordination of the activities of the various sections of the firm. 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. 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. endstream
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