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# UF AEB 6184 - Basic Notions of Production Functions

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1 Basic Notions of Production Functions Lecture I I. Overview of the Production Function A. The production function is a technical relationship depicting the technical transformation of inputs into outputs. 1. The production function in and of itself is devoid of economic content. 2. In the development of production functions, we are interested in certain characteristics that make it possible to construct economic models based on optimizing behavior. B. One way to write the production function is as a function map: :nmfRR++→ which states that the production function (f) is a function that maps n inputs into m outputs. By convention, we are only interested in positive input bundles that yield positive output bundles. C. The first lecture will focus on the production function as a continuous function as students have probably seen it in previous courses. The next lecture will develop the concept of the production function more rigorously. II. One Product, One-Variable Factor Relationships A. A commonly used form of the production function is the “closed form” representation where the total physical product is depicted as a function of a vector of inputs. ()yfx= where y is the scalar (single) output and x is a vector (multiple) inputs. B. Focusing for a moment on the single output case, we could simplify the above representation to: ()12yfxx= or we are interested in examining the relationship between 1x and y given that all the other factors of production are held constant. Using this relationship, we want to identify three primary relationships: 1. Total physical product–which is the original production function. 2. Average physical product–defined as the average output per unit of input. Mathematically, ()fxyAPPxx== 3. Marginal physical product–defined as the rate of change in total physical product at a specific input level. Mathematically, ()()()dTPP df xdyMPP f xdx dx dx′====AEB 6184 – Production Economics Lecture I Professor Charles B. Moss Fall 2005 2C. Given these notions of a production function, we can introduce the classical shape of the production function: 0204060801001201401601800 20 40 60 80 100 120 140 160 180Nitrogen (lbs./acre)Corn (bu./acre)High Yield FunctionAverage Yield FunctionLow Yield Function This set of production functions are taken from Moss and Schmitz “Investing in Precision Agriculture”. 1. This shape is referred to as a “sigmoid” shape. 2. The exact functional form in this figure can be attributed to Zellner, Arnold, (1951) “An Interesting General form for a Production Function”, Econometrica, 19, 188-89. The exact mathematical form of the function is: ()1,213121−=vvbExpvavvφ The average function sets 21.0v=, 0.0005433a=, and 0.01794b =. The total physical product graph given by Mathematica 4.0.AEB 6184 – Production Economics Lecture I Professor Charles B. Moss Fall 2005 3 25 50 75 100 125 15020 40 60 80 100 120 The marginal physical product and average physical product graphs for this production function become 25 50 75 100 125 150 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Marginal Physical Product Average Physical Product D. Given these relationships, we can define the stages of production. 1. While the production function itself is devoid of economic content, we use the physical relationships to define the economically valid production region. 2. Stage of Production a. Stage I: This stage of the production function is defined as that region where the average physical product is increasing. In this region, the marginal physical product is greater than the average physical product. In this region, each additional unit of input yields relatively more output on average. b. Stage II: This stage of the production process corresponds with the economically feasible region of production. MarginalAEB 6184 – Production Economics Lecture I Professor Charles B. Moss Fall 2005 4physical product is positive and each additional unit of input produces less output on average. c. Stage III: This stage of production implies negative marginal return on inputs. 3. Mathematically, these stages of production imply certain restrictions on the shape of the production function. a. The production function is a positive valued, initially increasing function. Further, around the point of optimality, the production function is concave in variable inputs. 4. Elasticity of Production a. Elasticities are often used in economics to produce a unit-free indicator of the shape of a function. Most are familiar with the elasticity of consumer demand. b. In defining the production function, we are interested in the factor elasticity. The factor elasticity is defined as %%dyydyxMPPyEdxxdx y APPx∆====∆ For the Zellner production function, the factor elasticity can be depicted as 25 50 75 100 125 150 0.5 1 1.5 2 c. There is a specific relationship between the average physical product and marginal physical product when the average physical product is maximized. Mathematically, d TPP d x APP d APPMPP APP xdx dx d x== =+ Thus, when the APP is maximized 0dAPPMPP APPdx=⇒ =AEB 6184 – Production Economics Lecture I Professor Charles B. Moss Fall 2005 5Following through on this relationship, we have 010101dAPPMPP APP EdxdAPPMPP APP EdxdAPPMPP APP Edx>⇒ > ⇒ >=⇒=⇒=<⇒<⇒< In addition, we know that 00,is maximum00EMPPTPPEMPP=⇔ =<⇔ < Thus, if 1E> then the production function is in stage I. If 10E>>, then the production function is in stage II. If 0E<, then the production function is in stage III. III. One Product, Two-Variable Factor Relationships A. Expanding the production, we start by considering the case of two inputs producing one output. In the general functional mapping notation: 21:fRR++→ B. This class of production functions can also be depicted as ()12,yfxx= 1. In general, the univariate production functions are simply “slices” out of the multivariate production functions. 501001500.80.911.11.20100200501001500.80.911.11.2 a. This graph depicts the more general version of the Zellner production function where both 1x and 2x are allowed to change: ()311212,exp 1axxxxbxφ=−AEB 6184 – Production Economics Lecture I Professor Charles B. Moss Fall 2005 62. These functions still have average physical products and marginal physical products, but they are conditioned on the level of other inputs:

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