Given a cone e in the euclidean space equation and an ordering. However, the aggregate production function, which does much the same thing on the production side of the economy was left largely unexamined. Homogeneous production functions have the property that f. 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. Give an example of a homogeneous function of degree 1. Ppt homogeneous and homothetic functions powerpoint. The aim of this work is to investigate a production model recently studied in under the name of quasihomogeneous twofactor production function, as a natural generalization of the family of classical homogeneous production functions with two inputs labor and capital. A specific homogeneous functional form which economists frequently use as a production or utility function is the cobbdouglas function. Wilson mathematics for economists may 7, 2008 homogeneous functions for any r, a function f. It is clear that homothetiticy is ordinal property. Variable elasticity substitution production function. Constant elasticity of substitution production function and 4. In their studies the main emphasis has been put on examining the curvature of the ppl, under general conditions 5, 7 and under specific assumptions about the underlying production functions, namely when functions are homogeneous of a certain degree 2, 7, 9, homothetic 3, or even nonhomothetic 4.
An interesting and important class of production functions is the homothetic pro. Classification of hhomogeneous production functions with constant elasticity of substitution article pdf available in tamkang journal of mathematics 432. A homogeneous production function is also homotheticrather, it is a special case of homothetic production functions. Note that for the purpose of production functions we need to make sure that gxisapositive monotonic function for x. Shephard developed the notion of a homothetic production function and employed the idea in function and formulating the concept of homothetic separability. Anidentifying characteristic of a homothetic function is that its mrts is the same at zand tz. Letfx 0 be a continuously differentiable, homothetic function. Homogeneous implies homothetic, but not conversely. Show that, if f is homogeneous of degree 1, then the hessian h fx is degenerate at every x60. A linearly homogeneous production function with inputs capital and labour has the properties that the marginal and average physical products of both capital and labour can be expressed as functions of the capitallabour ratio alone. Since m1 e al and f is the inf of a set of linear homogeneous concave functions, f e y.
Homogeneous productions functions and returns to scale. Can a function be homothetic if its not homogeneous. Homogeneous and homothetic functions 1 homogeneous and homothetic functions 2 homogeneous functions. One important purpose of the production function is to address. Also, an homothetic production function is a function whose marginal rate of technical substitution is homogeneous of degree zero. Notice that the cost function, being linearly homogeneous, is necessarily homothetic with respect to prices. Mathematical economics econ 471 lecture 5 homogeneous. Assumption of homotheticity simplifies computation, derived functions have homogeneous properties, doubling prices and income doesnt change demand, demand functions are homogenous of degree 0.
Homogeneous functions ucsbs department of economics. Homogeneous functions homogeneous of degree k applications in economics. Homothetic and homothetically separable functions are commonly used in models of consumer preferences and rm production. The nonhomothetic aspect of the production function may be best characterized by the existence of the nonhomotheticity coefficient or parameter for the marginal rate of substitution. This is a monotone transformation of a homogenous function, so it is homothetic. It follows from above that any homogeneous function is a. A function is said to be homogeneous of degree r, if multiplication of each of its independent variables by a constant j will alter the value of the function by the proportion jr, that is, if. A homothetic function is a production function of the form fx 1x n fhx 1x n. C cv, w, q minimum total cost is a function of input prices and output quantity.
The properties and generation of homothetic production. The relationship between homogeneous production functions and eulers t heorem is presented. Homogeneous functions are a subset of the set of homothetic functions. Constant returns production functions possess the property that the rts depends only on the ratio k to l, not on the level of output scale of production. Homothetic production functions and the shape of the. The function f of two variables x and y defined in a domain d is said to be homogeneous of degree k if, for all x,y in d ftx, ty tk fx,y multiplication of both variables by a positive factor t will thus multiply the value of the function. The microeconomic foundations of aggregate production. What is the distinction between homogeneous and homothetic. Consider the following idea related to production functions, the returns to scale. Advanced microeconomicshomogeneous and homothetic functions.
Some geometric applications of allens matrices of the homothetic production functions are also given. Differentiability conditions consider the scaling function 1. The derivative of c y in the production function 3 and also in the marginal. American journal of business education april 2011 volume 4. The constant elasticity of substitution ces production. Homothetic and nonhomothetic ces production functions. Chapter 8 cost functions done university of tennessee. The historical role of the production function in economics and business david gordon, university of saint francis, usa. Thus, the c function represents the minimum cost necessary to produce output q with fixed input prices. A homothetic function is a production function of the form fx fhx1. A constant returns production function is homogeneous of degree 1 linear homogeneous. Pdf classification of hhomogeneous production functions. Homothetic functions are production functions whose marginal technical rate of substitution is homogeneous of degree zero 9, 12, 16.
This video explains how to determine if a function is homogeneous and if it is homogeneous, what is the degree of the homogeneous function. V college, university of delhi page 2 homogeneous and homothetic function 1 dc1 semesterii paperiv. Moreover, in this case if each input is paid at a rate equal. Show that the elasticity of substitution for any homothetic production function is equal to the elasticity of substitution for its linear homogeneous part alone. The exponent, n, denotes the degree of homogeneity. Jorgenson, from the forward to the 1981 reprint of shephards cost and production functions, first published in 1953. Pdf classification of homothetic functions with constant elasticity. More interesting is whether a utility function is homothetic. It is important to highlight that the quasihomogeneity property of production functions was originally considered in.
On homogeneous production functions with proportional. Page 1 homogeneous and homothetic function 1 dc1 semesterii paperiv. A production function is homogeneous of degree n if when inputs are multiplied by some constant, say. Returns to scale, homogeneous functions, and eulers theorem. E homogeneous and homothetic functions springerlink.
Pdf on quasihomogeneous production functions researchgate. For example, a function is homogeneous of degree 1 if, when all its arguments are multiplied by any number t. These classifications generalize some recent results of c. The results obtained in this article motivate further studies to. Econ501 advanced microeconomic theory 1 fall semester. We completely classify homogeneous production functions with proportional marginal rate of substitution and with constant elasticity of labor and capital, respectively. Douglas and acms production functions are always regular. The cost function can be derived from the production function for the bundle of inputs defined by the expansion path conditions. We can measure the elasticity of these returns to scale in the following way. Put more formally, if there is a monotonic transformation such that y7. Let q fx be a production function on rn, that is homogeneous of degree k.
I know that a homogeneous function of positive degree is homothetic, but can a function that is not homogeneous be homothetic. Rna function is homogeneous if it is homogeneous of degree. The production function is the central part of production theory and as such there is a theoretical interest in its estimates. In consumer theory, a consumers preferences are called homothetic if they can be represented by a utility function which is homogeneous of degree 1 146 for example, in an economy with two goods, homothetic preferences can be represented by a utility function that has the following property. Homogeneous and homothetic function lesson developer. So it is interesting to ask if a production function is homogeneous.
Production function wikimili, the best wikipedia reader. A homogeneous production function is also homothetic rather, it is a special case of homothetic production functions. R is called homothetic if it is a monotonic transformation of a homogenous function, that is there exist a strictly increasing function g. The production function is one of the key concepts of mainstream neoclassical theories, used to define marginal product and to distinguish allocative efficiency, a key focus of economics. But this makes a homothetic function a monotonic transformation of a homogeneous function. M, x r, m, defined for all y, x c 9, x 9p, by iy, x min. Such as, if the input factors are doubled the output also gets doubled. Linear homogeneous production function business jargons. A function is homogeneous of degree k if, when each of its arguments is multiplied by any number t 0, the value of the function is multiplied by t k. Therefore, not all monotonic transformations preserve the homogeneity property of a utility function. All this becomes very important to get the balance right between levels of capital, levels of labour, and total production. In fact, it turns out that the reason why we are interested in homothetic functions is that they preserve the property seen in homogeneous functions that along a ray through the origin, the isoquants have the same slope. Now, homogeneous functions are a strict subset of homothetic functions. Homogeneity of degree one is constant returns to scale.
900 513 1338 239 718 637 595 93 1526 878 367 78 1274 1227 1521 1396 380 78 1274 1015 421 416 277 1466 378 69 1052 947 50 872 420