Lecture01-2011 - Basic Notions of Production Functions...

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Unformatted text preview: Basic Notions of Production Functions Charles B. Moss 1 University 1 of Florida August 23, 2011 Charles B. Moss (University of Florida) Production Functions August 23, 2011 1 / 35 Outline 1 Overview of Production Economics 2 Overview of the Production Function Chambers on Production Functions Production Function Mapping 3 One Product, One Input Derived Relationships Estimated Production Functions Stages of Production 4 Elasticity of Production APP and MPP 5 One Product, Two Variable Input Relationships Empirical Production Functions 6 Isoquants, Isoclines, and Ridgelines Factor-Factor Relationships Charles B. Moss (University of Florida) Production Functions August 23, 2011 2 / 35 Overview of Production Economics Overview of Production Economics The single chapter which follows summarizes the scope and method of agricultural production economics. The treatment does not do justice to this important field of applied economics, which is surpassed by no other in its accomplishment, its breadth, and its interesting history (Heady p.1). Charles B. Moss (University of Florida) Production Functions August 23, 2011 3 / 35 Overview of Production Economics Agricultural production economics is the oldest and most widespread specialization in the applied field of science know as agricultural economics; the science of agricultural economics originated as a study in farm production economics. The field has continued to grow until it now embraces more workers than any other specialization. In the field of agricultural economics the majority of outstanding names has been associated with some phase of production economics, both in farm and in national use of resources. Charles B. Moss (University of Florida) Production Functions August 23, 2011 4 / 35 Overview of Production Economics The list in the United States has included such prominent persons as G.F. Warren, H.C. Taylor, T.N. Carver, W.J. Spillman, L.C. Grey, C.L. Holmes, J.D. Black, E.C. Young, H. Tolley, M. Ezekiel, A. Boss, O.V. Wells, H.C.M. Case, S.E. Johnson, T.W. Schultz, G.E Forster, W.W. Wilcox, and W. G. Murray. The European list includes such well known names as J. Von Thunen, M. Kitchen, F Aereboe, T. Brinkman, A.W. Ashby, G. Lauer, R. Cohen and others. All of these scholars have been known for their findings or particular methods of analysis. In this list are not only persons who pioneered the field of agricultural economics but also those who contributed both to theory and to empirical knowledge. The central objective of their work has been to increase the efficiency with which farm resources are used (Heady p.1). Charles B. Moss (University of Florida) Production Functions August 23, 2011 5 / 35 Overview of Production Economics Production economists who focus their attention on agriculture are concerned with choice and decision-making in the use of the capital, labor, land, and management resources in the farming industry. The goals of agricultural production economics are twofold: (1) to provide guidance to individual farmers in using their resources most efficiently, and (2) to facilitate the most efficient use of resources from the standpoint of the consuming economy (Heady p.3). Charles B. Moss (University of Florida) Production Functions August 23, 2011 6 / 35 Overview of the Production Function Overview of the Production Function The production function is a technical relationship depicting the technical transformation of inputs into outputs. The production function in and of itself is devoid of economic content. In the development of production functions, we are interested in certain characteristics that make it possible to construct economic models based on optimizing behavior. Charles B. Moss (University of Florida) Production Functions August 23, 2011 7 / 35 Overview of the Production Function Chambers on Production Functions Chambers on Production Functions The production function (and indeed all representations of technology) is a purely technical relationship that is void of economic content. Since economists are usually interested in studying economic phenomena, the technical aspects of production are interesting to economists only insofar as they impinge upon the behavior of economic agents.... Because the economist has no inherent interest in the production function, if it is possible to portray and to predict economic behavior accurately without direct examination of the production function, so much the better. This principle, which sets the tone for much of the following discussion, underlies the intense interest that recent developments in duality have aroused (Chambers p.7). Charles B. Moss (University of Florida) Production Functions August 23, 2011 8 / 35 Overview of the Production Function Chambers on Production Functions The point of these two statements is that economists are not engineers and have no insights into why technologies take on any particular shape. We are only interested in those properties that make the production function that make the production function consistent with optimizing behavior. Charles B. Moss (University of Florida) Production Functions August 23, 2011 9 / 35 Overview of the Production Function Production Function Mapping Production Function Mapping One way to write the production function is as a function map + + f : Rn → Rm (1) 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. 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. Charles B. Moss (University of Florida) Production Functions August 23, 2011 10 / 35 One Product, One Input One Product, One Variable Factor A commonly used form of the production function is the ”closed form” representation where total physical product is depicted as a function of a vector of inputs y = f (x ) (2) where y is the scalar (single) output and x is a vector of inputs. Focusing for a moment on the single output case, we could simplify the above representation to y = f ( x 1 | x2 ) (3) or we are interested in examining the relationship between x1 and y given that all the other factors of production are held constant. Charles B. Moss (University of Florida) Production Functions August 23, 2011 11 / 35 One Product, One Input Derived Relationships Derived Relationships Using this relationship, we want to identify three primary relationships: Total Physical Product - which is the original production function. Average Physical Product - defined as the average output per unit of input. Mathematically, APP = y f (x ) = x x (4) Marginal Physical Product - defined as the rate of change in the total physical product at a specific input level. Mathematically, MPP = Charles B. Moss (University of Florida) d (TPP ) dy d f (x ) = = = f ￿ (x ) dx dx dx Production Functions August 23, 2011 (5) 12 / 35 One Product, One Input Estimated Production Functions Estimated Production Functions 150 Corn 100 50 50 100 150 Nitrogen Charles B. Moss (University of Florida) Production Functions August 23, 2011 13 / 35 One Product, One Input Estimated Production Functions This set of production functions are taken from Moss and Schmitz Investing in Precision Agriculture. This shape is referred to as a sigmoid shape. 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: φ ( v 1 , v2 ) = 3 av1 ￿ v1 exp b −1 v2 ￿ (6) The average function sets v2 = 1.0, a = 0.0005433, and b = 0.01794. Charles B. Moss (University of Florida) Production Functions August 23, 2011 14 / 35 One Product, One Input Estimated Production Functions Corn(bushel/acre) 120 100 80 60 40 20 50 100 150 Nitrogen(lbs/acre) Charles B. Moss (University of Florida) Production Functions August 23, 2011 15 / 35 Corn One Product, One Input 1.0 Estimated Production Functions A 0.5 B 50 100 150 Nitrogen AveragePhysicalProduct Charles B. Moss (University of Florida) MarginalPhysicalProduct Production Functions August 23, 2011 16 / 35 One Product, One Input Stages of Production Stages of Production 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. Stage II: This stage of the production process corresponds with the economically feasible region of production. Marginal physical product is positive and each additional unit of input produces less output on average. Stage III: This stage of production implies negative marginal return on inputs. Charles B. Moss (University of Florida) Production Functions August 23, 2011 17 / 35 One Product, One Input Stages of Production B Corn(bushel/acre) 120 100 A 80 60 40 20 50 100 150 Nitrogen(lbs/acre) Charles B. Moss (University of Florida) Production Functions August 23, 2011 18 / 35 Elasticity of Production Elasticity of Production 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. In defining the production function, we are interested in the factor elasticity. The factor elasticity is defined as dy %∆y dy x MPP y E= = = = dx %∆x dx y APP x Charles B. Moss (University of Florida) Production Functions (7) August 23, 2011 19 / 35 Elasticity of Production FactorElasticity 2.0 1.5 1.0 0.5 50 100 150 Nitrogen(lbs/acre) Charles B. Moss (University of Florida) Production Functions August 23, 2011 20 / 35 Elasticity of Production APP and MPP APP and MPP 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 APP = = APP + x dx dx dx Thus, when the APP is maximized MPP = d APP = 0 ⇒ MPP = APP dx Charles B. Moss (University of Florida) Production Functions (8) (9) August 23, 2011 21 / 35 Elasticity of Production APP and MPP Following through this relationship, we have d APP > 0 ⇒ MPP > APP ⇒ E > 1 dx d APP = 0 ⇒ MPP = APP ⇒ E = 1 dx (10) d APP < 0 ⇒ MPP < APP ⇒ E < 1 dx In addition, we know that if E = 0 ⇔ MPP = 0, TPP is maximum and E < 0 ⇔ MPP < 0. Thus, if E > 1 then the production function is in Stage I. If 1 > E > 0, then the production function is in Stage II. If E < 0, then the production functio is in Stage III. Charles B. Moss (University of Florida) Production Functions August 23, 2011 22 / 35 One Product, Two Variable Input Relationships One Product, Two Variable Input Relationships Expanding the production, we start by considering the case of two inputs producing one output. In the general functional mapping notation + + f : R2 → R1 y = f ( x1 , x2 ) Charles B. Moss (University of Florida) Production Functions (11) August 23, 2011 23 / 35 One Product, Two Variable Input Relationships Figure: Two Inputs Charles B. Moss (University of Florida) Production Functions August 23, 2011 24 / 35 One Product, Two Variable Input Relationships These functions still have average physical products and marginal physical products, but they are conditioned on the level of other inputs APP1 = y f ( x1 , x2 ) = x1 x1 (12) y f ( x1 , x2 ) APP2 = = x2 x2 Similarly, the marginal physical products are defined by the partial derivatives MPP1 = ∂y ∂ f ( x1 , x2 ) = ∂ x1 ∂ x1 (13) ∂y ∂ f ( x1 , x2 ) MPP2 = = ∂ x2 ∂ x2 Charles B. Moss (University of Florida) Production Functions August 23, 2011 25 / 35 One Product, Two Variable Input Relationships It may be useful at this point to briefly visit the notion of the Taylor expansion. Taking the second-order expansion of the production function yields ￿ 0 0￿ ￿ f ( x 1 , x 2 ) = f x1 , x 2 + ∂ f ( x 1 , x2 ) ∂ x1 1￿ dx1 2 Charles B. Moss ∂ 2 f ( x1 , x2 ) 2 ￿ dx2 2 ∂ x1 ∂ f ( x1 , x2 ) ∂ x1 ∂ x2 (University of Florida) ∂ f ( x1 , x2 ) ∂ x2 ￿ ￿ dx ￿ 1 + dx2 ∂ 2 f ( x1 , x2 ) ￿ ￿ ∂ x1 ∂ x2 dx1 ∂ 2 f (x1 , x2 ) dx2 2 ∂ x2 Production Functions August 23, 2011 (14) 26 / 35 One Product, Two Variable Input Relationships This approximation is exact in the case of either linear or quadratic production functions. However, if we focus on a quadratic production function, it is clear that ￿ ￿ ￿ ￿ ∂ f ( x1 , x2 ) ∂ f ((x1 , x2 ) dy ≈ f1 = dx1 + f2 = dx2 (15) ∂ x1 ∂ x2 Charles B. Moss (University of Florida) Production Functions August 23, 2011 27 / 35 One Product, Two Variable Input Relationships Empirical Production Functions Empirical Production Functions Linear Production Function y = b 1 x1 + b 2 x2 (16) Quadratic Production Function ￿ 1￿ 2 2 A11 x1 + 2A12 x1 x2 + A22 x2 2 ￿ ￿ ￿ ￿￿ ￿ ￿ x1 ￿ A11 A12 1￿ x1 a2 x1 x 2 + x2 A21 A22 x2 2 y = a 1 x1 + a 2 x 2 + y= ￿ a1 (17) Cobb-Douglas Production Function bb y = Ax1 1 x2 2 Charles B. Moss (University of Florida) Production Functions (18) August 23, 2011 28 / 35 One Product, Two Variable Input Relationships Empirical Production Functions Transcendental Production Function a a y = Ax1 1 e b1 x1 x2 2 e b2 x2 (19) Constant Elasticity of Substitution (CES) Production Function ￿ ￿ −v g − − y = A bx1 g (1 − b ) x2 g Charles B. Moss (University of Florida) Production Functions (20) August 23, 2011 29 / 35 Isoquants, Isoclines, and Ridgelines Isoquants, Isoclines, and Ridgelines Given the multivariate nature of the production function, it is possible to define isoquants, or the relationship that depicts the combinations of inputs that yield the same output. Starting from the basic production function y = f ( x 1 , x 2 ) ⇒ x2 = f ∗ ( x1 , y ) (21) That is we are interested in constructing a functional mapping of x2 based on the level of x1 and y . Next, we hold the level of output constant and derive the levels of x2 for any level of x1 . Charles B. Moss (University of Florida) Production Functions August 23, 2011 30 / 35 Isoquants, Isoclines, and Ridgelines x2 y3 y2 y1 x1 Charles B. Moss (University of Florida) Production Functions August 23, 2011 31 / 35 Isoquants, Isoclines, and Ridgelines The isoquants are useful in defining the rate of technical substitution or the rate at which one input must be traded for the other input dy = f1 dx1 + f2 dx2 = 0 ⇒ Charles B. Moss (University of Florida) Production Functions dx1 f1 =− dx2 f1 (22) August 23, 2011 32 / 35 Isoquants, Isoclines, and Ridgelines Ridgeline Isocline Charles B. Moss (University of Florida) Production Functions August 23, 2011 33 / 35 Isoquants, Isoclines, and Ridgelines Factor-Factor Relationships Factor-Factor Relationships Factor independence: Two factors are independent if the MPP of one factor is not a function of the MPP of the other factor. The simplest example of this is a quadratic production function with A12 = A21 = 0. In this case, the isoquants are circles (or elipses). ∂y = a + A x ￿ 1￿ 1 11 1 2 2 x1 y = a 1 x1 + a 2 x2 + A11 x1 + A22 x2 ⇒ ∂y 2 x2 = a2 + A22 x2 (23) Case I: If f12 > 0, then x1 and x2 are technically complementary. ￿ ∂y ∂ 2 ∂y ∂ x1 = ∂ x1 ∂ x2 ∂ x2 Charles B. Moss (University of Florida) ￿ = f12 > 0 Production Functions (24) August 23, 2011 34 / 35 Isoquants, Isoclines, and Ridgelines Factor-Factor Relationships Case II: If f12 = 0, then x1 and x2 are technically independent. Case III: If f12 < 0, then x1 and x2 are technically competitive. Charles B. Moss (University of Florida) Production Functions August 23, 2011 35 / 35 ...
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