Definition and Properties of the Production FunctionOverview of the Production FunctionSlide 3A Brief Brush with DualitySlide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Production Function DefinedSlide 13Slide 14Properties of the Production FunctionSlide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Law of Variable ProportionsSlide 32Slide 33Slide 34Elasticity of ScaleSlide 36Slide 37Slide 38Slide 39Slide 40Definition and Properties of the Production Function Lecture IIOverview of the Production Function “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.” (Chambers p. 7).“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).A Brief Brush with DualityThe 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 useful in economic analysis, or those properties that make the system solvable.One approach would be to estimate a production function, say a Cobb-Douglas production function in two relevant inputs: 1 2y x xa b=Given this production function, we could derive a cost function by minimizing the cost of the two inputs subject to some level of production: 1 21 1 2 2,1 2min. .x xw x w xs t y x xa b+=( )1 1 2 2 1 21 211 11 222 21 2000L w x w x y x xx xLwx xx xLwx xLy x xa ba ba ba bllll= + + -�= - =��= - =��= - =�1 1 2 21 22 1 12Lx w x wx xLw x wx��� = � =��( )1*2 12 2 2 1 21 20 , ,w wLy x x x w w y yw waaa bba bl++� � � ��� - = � =� � � ��� � � �( )1*21 1 21, ,wx w w y ywba ba b++� �=� �� �( )1 12 11 2 1 21 212 112, ,w wC w w y w y w yw ww wywwbaa b a ba b a bbaa b a ba ba ba ba b+ ++ ++ ++++� �� �� � � �� �� �= +� � � �� �� �� � � �� �� �� �� �� �= +� �� �� �Thus, in the end, we are left with a cost function that relates input prices and output levels to the cost of production based on the economic assumption of optimizing behavior. Following Chamber’s critique, recent trends in economics skip the first stage of this analysis by assuming that producers know the general shape of the production function and select inputs optimally. Thus, economists only need to estimate the economic behavior in the cost function.Following this approach, economists only need to know things about the production function that affect the feasibility and nature of this optimizing behavior. In addition, production economics is typically linked to Sheppard’s Lemma that guarantees that we can recover the optimal input demand curves from this optimizing behavior.Production Function Defined Following our previous discussion, we then define a production function as a mathematical mapping function: :n mf R R+ +�However, we will now write it in implicit functional form This notation is sometimes referred to as a netput notation where we do not differentiate inputs or outputs. ( )0Y z =Following the mapping notation, we typically exclude the possibility of negative outputs or inputs, but this is simply a convention. In addition, we typically exclude inputs that are not economically scarce such as sunlight.Finally, I like to refer to the production function as an envelope implying that the production function characterizes the maximum amount of output that can be obtained from any combination of inputs. ( ), 0Y y x =Properties of the Production Function Monotonicity and Strict Monotonicity: ( ) ( )If , then (monotonicity)x x f x f x� �� �( ) ( )If then (strict monotonicity)x x f x f x� �> >Quasi-Concavity and Concavity ( ) ( ){ }: is a convex set (qausi-concave)V y x f x y= �( )( ) ( )( )( )0 * 0 *1 1 for any 0 1 (concave)f x x f x f xq q q q q+ - � + - � �Weakly essential and strictly essential inputs ( )0 0, where 0 is the null vector (weakly essential)n nf =( )1 1 1, ,0, , 0 for all (strict esstential)i i n if x x x x x- +=L LThe set V(y) is closed and nonempty for all y > 0.f(x) is finite, nonnegative, real valued, and single valued for all nonnegative and finite x. Continuity f(x) is everywhere continuous; and f(x) is everywhere twice-continuously differentiable.Properties (1a) and (1b) require the production function to be non-decreasing in inputs, or that the marginal products be nonnegative. In essence, these assumptions rule out stage III of the production process, or imply some kind of assumption of free-disposal. One traditional assumption in this regard is that since it is irrational to operate in stage III, no producer will choose to operate there. Thus, if we take a dual approach (as developed above) stage III is irrelevant.Properties (2a) and (2b) revolve around the notion of isoquants or as redeveloped here input requirement sets. The input requirement set is defined as that set of inputs required to produce at least a given level of outputs, V(y). Other notation used to note the same concept are the level set.Strictly speaking, assumption (2a) implies that we observe a diminishing rate of technical substitution, or that the isoquants are negatively sloping and convex with respect to the origin.1x2x( )V yAssumption (2b) is both a stronger version of assumption (2a) and an extension. For example, if we choose both points to be on the same input requirement set, then the graphical depiction is simply1x2x( )V y( )( ) ( )( )( )0 0 0 01 1f x x f x f xq q q q+ - � + -If we assume that the inputs are on two different input requirement sets, then Clearly, letting  approach zero yields f(x) approaches f(x*), however, because of the inequality, the left-hand side is less than the right hand side.