Lecture notes on constant elasticity functions thomas f. Likewise, what is now well known as the leontief production function was formulated. Second, technological change is purely laboraugmenting in the long run. A brief history of production functions munich personal repec. American journal of business education april 2011 volume 4. It encompasses the cobbdouglas, the leontief and the linear production functions as its special cases. Request pdf a note on the constantelasticityofsubstitution. The estimation of the elasticity of substitution of a ces.
The shape of production functions and the direction of. Cobbdouglas production function it is the most frequently used production function to estimate production relationship it is written as. A key parameter in economics is the elasticity of substitution between capital and labor. Two important limiting cases of the homothetic constant elasticity of substitution ces production function developed by arrow, chenery, minhas, and solow acms are the cobb douglas and the leontief production functions. Two input leontief production function with isoquants in economics, the leontief production function or fixed proportions production function is a production function that implies the factors of production will be used in fixed technologically predetermined proportions, as there is no substitutability between factors. Substituting back into equation 1 shows that, for any commodity i, x ip,y pr. One input if the function has only one input, the form can be represented using the following formula. This lecture note aims at being a reference for algebraic computations of the leontief and cobbdouglas functions by taking limits of ces functions both in discrete and. Productivity growth and efficiency under leontief technology. A note on the constantelasticityofsubstitution production function.
Second, i show how the ces function nests some particular functional forms as special cases. It implies that if the inputoutput ratio is independent of the scale of production, there is existence of leontief production function. Domar leontief production function, which is based on the assumption that there is no substitutability between factors, and the cobbdouglas production function, which assumes unitary factor substitution elasticity. Note on the estimation of substitution elasticities with three. How do we get cobbdouglas and leontief functions from ces. When measuring the elasticity of substitution between two factors when there are other factors in the production function, one must take care of controlling for possible cross effects. Formally, the elasticity of substitution measures the percentage change in factor proportions due to a change in marginal rate of technical substitution. What is the concept of elasticity of factor substitution. Pdf an application of the shephard duality theorem. Constant elasticity of substitution ces production function. A generalized leontief model to calculate substitution elasticities defining p p and p o as prices of peak and offpeak electricity and es p and es o as shares of electricity expenditure in peak and offpeak hours, then the twocommodity homothetic generalized leontief gl indirect utility function and budget shares for electricity can be. The shape of production functions and the direction of technical. Substitution elasticities in a ces production framework. One other restriction on the cobbdouglas production function involved the elasticity of substitution.
Production function is the mathematical representation of relationship between physical inputs and physical outputs of an organization. So revenue is an increasing function of pif p 1 and a decreasing function of pif elastic. Leontief and the linear production functions are only the special cases of the ces. A note on numerical estimation of satos twolevel ces. Figure16 shows different types of production function. Technological choice generating normalized ces production functions. Two mutually interrelated difficulties with the ces production function came to light very soon. Generalized leontief an overview sciencedirect topics. However, the aggregate production function, which does much the same thing on the production side of the economy was left largely unexamined. Constant elasticity of substitution production function. Featured on meta meta escalationresponse process update marchapril 2020 test results, next.
Wassily leontif, uses fixed proportion of inputs having no substitutability between them. Pdf estimate of the elasticities of substitution of the. The first is in the constancy of the elasticity of substitution between. Relative demand function y x fp x p y the elasticity of this function is the elasticity of substitution in consumption. First, i derive a number of conditions such as the optimal demand schedule when aggregation technology is ces. The leontief production function as a limiting case of the ces. Production functions david rezza baqaee ucla emmanuel farhi harvard july 22, 2019 abstract aggregate production functions are reducedform relationships that emerge en. This production function exhibits constant returns to scale. Extending the concept of the elasticity of substitution from a twoinput production function into one with three or more inputs invites complications. Review of microeconomics algebraic formulation of mrs in. The elasticity of substitution parameter a tends to 1 in the former case and to 0 in the latter. Functions featuring constant elasticity of substitution ces are widely used in applied economics and. An assessment of ces and cobbsdouglas production functions.
It is a tedious but straightforward application of lagrangian calculus to demonstrate. There are different types of production functions that can be classified according to the degree of substitution of one input by the other. The shape of the isoquant is related to its elasticity of substitution. Growth, leontief, pareto, capital share, equipment prices, steady. In other words, an assumption kortum 1997 suggests we make if we want a model to exhibit steadystate growth leads to important predictions about the shape of production functions and the direction of technical change. A constantelasticityofsubstitution production function in which the. A generalized leontief production function article pdf available in journal of political economy 793. Leontief production function wolfram demonstrations project. Sector specific multifactor ces elasticities of substitution and the corresponding productivity growths are jointly measured by. Elasticity of factor substitution is zero for leontief function, one for cobb douglas function and constant for linear and ces function. B activity analysis technology c puttyclay technology 4 the elasticity of substitution a measuring substitutability b elasticity of substitution under constant returns to scale c cobbdouglas production functions. The microeconomic foundations of aggregate production. The procedure requires that, as we know, before the. Q total production the monetary value of all goods produced in a year l labor input k capital input a total factor productivity and are the output elasticities of labor and capital.
The linear production function is the simplest form of a production function. An early alternative to the cobbdouglas production function is the constant elasticity of substitution ces production function 1. We derive the efficient frontier as the convex hull of leontief production functions. In other words, for our canonical production function, y k, l, the elasticity of substitution between capital and labor is given by. Doyne farmer 5 1 institute for new economic thinking at the oxford martin school, university of oxford, uk 2 mathematical institute, university of oxford, uk 3 complexity science hub vienna, austria. Production functions linear vs leontief vs cobbdouglas. The magnitude of the elasticity of substitution can be assessed by looking at the curvature of isoquants. Constant elasticity of substitution ces, in economics, is a property of some production functions and utility functions specifically, it arises in a particular type of aggregator function which combines two or more types of consumption goods, or two or more types of production inputs into an aggregate quantity. The microeconomic foundations of aggregate production functions. Introduction a macroeconomic production function is a mathematical expression that describes a systematic relationship between inputs and output in an economy, and the cobbdouglas and constant elasticity of substitution ces are two functions that have been used extensively. Browse other questions tagged productionfunction leontief or ask your own question. Contents a measuring substitutability b elasticity of substitution under constant returns to scale c cobbdouglas production functions d constant elasticity of substitution ces production functions e elasticities of substitution in multiinput cases a measuring substitutability let us now turn to the issue of measuring the degree of substitutability between any pair of factors.
In the singleoutput case we can describe a production set z in two other waysthat are sometimes more convenient. To shift from one optimal solution to another, a producer has to change both factors in the established proportion. In economics, the leontief production function or fixed proportions production function is a production function that implies the factors of production will be used in fixed technologically predetermined proportions, as there is no substitutability between factors. Based on annual data, we estimate the elasticity of substitution. A constantreturns multifactor ces production function of. J ones this paper views the standard production function in macroeconomics as a reduced form and derives its properties from microfoundations. It assumed the value for this elasticity was equal to unity. But for our rst crack at the story it is helpful to consider the case where there are just two inputs and the production function is homogeneous of some degree k0. The regression results of the ces production function table 11. Leontief function marginal product of laborcapital. D the intensive production function 3 technology and substitution a leontief fixed proportions technology. In response to this result, we provide a comprehensive set of consistently estimated substitution elasticities covering 35 sectors.
The shape of this production function is governed by the distribution of ideas. It is regarded as the limiting case for constant elasticity of substitution. Mar 14, 2019 a production function is an equation that establishes relationship between the factors of production i. Leontief production function uses fixed proportion of inputs having no substitutability between them. There are three main types of production functions.