Basic Notions of Production FunctionsOverview of the Production FunctionSlide 3Slide 4One Product, One-Variable Factor RelationshipsSlide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Stage of ProductionSlide 15Elasticity of ProductionSlide 17Slide 18Slide 19Slide 20Slide 21One Product, Two-Variable Factor RelationshipsSlide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Isoquants, Isoclines and RidgelinesSlide 31Slide 32Slide 33Slide 34Slide 35Slide 36Basic Notions of Production Functions Lecture IOverview 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.One way to write the production function is as a function map: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. :n mf R R+ +�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.One Product, One-Variable Factor Relationships 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. where y is the scalar (single) output and x is a vector (multiple) inputs. ( )y f x=Focusing for a moment on the single output case, we could simplify the above representation to: or we are interested in examining the relationship between x1 and y given that all the other factors of production are held constant. ( )1 2y f x x=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, ( )f xyAPPx x= =Marginal physical product–defined as the rate of change in total physical product at a specific input level. Mathematically, ( ) ( )( )d TPP d f xdyMPP f xdx dx dx�= = = =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”. 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: The average function sets v2 = 1.0, a=.0005433, and b=.01794.  1,213121vvbExpvavv25 50 75 100 125 150 20 40 60 80 100 12025 50 75 100 125 150 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Marginal Physical Product Average Physical ProductStage 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.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%%dyy dy x MPPyEdxx dx y APPxD= = = =D25 50 75 100 125 150 0.5 1 1.5 2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Following through on this relationship, we have 0d APPMPP APPdx= � =0 10 10 1d APPMPP APP Edxd APPMPP APP Edxd APPMPP APP Edx> � > � >= � = � =< � < � <In addition, we know thatThus, 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 function is in stage III. 0 0, is maximum0 0E MPP TPPE MPP= � =< � <One Product, Two-Variable Factor Relationships Expanding the production, we start by considering the case of two inputs producing one output. In the general functional mapping notation: 2 1:f R R+ +�( )1 2,y f x x=501001500.80.911.11.20100200501001500.80.911.11.2These functions still have average physical products and marginal physical products, but they are conditioned on the level of other inputs: ( )( )1 211 11 222 2,,f x xyAPPx xf x xyAPPx x= == =Similarly, the marginal physical products are defined by the partial derivatives: ( )( )1 211 11 222 2,,f x xyMPPx xf x xyMPPx x��= =� ���= =� �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 ( )( )( ) ( )[ ]( ) ( )( ) ( )11 2 1 20 01 2 1 222 22 21 2 1 221 1 211 22 221 2 1 222 1 2, ,, ,, ,12, ,dxf x x f x xf x x f x xdxx xf x x f x xx x xdxdx dxdxf x x f x xx x x� �� �� �� + +� �� �� �� �� �� �� �� �� ��� �� �� �� �� �� �� �� � �� �� �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 ( ) ( )1 2 1 21 1 2 21 2, ,f x x f x xdy f dx f dxx x� � � �� �� � � �� = +� � � �� �� � � ���The Linear Production Function The Quadratic Production Function 1 1 2 2y b x b x= +( )[ ] [ ]2 21 1 2 2 11 1 12 1 2 22 21 11 12 11 2 1 22 21 22 212212y a x a x A x A x x A xx A A xy