PRODUCTION FUNCTIONS

1 PRODUCTION FUNCTIONS1. ALTERNATIVEREPRESENTATIONS OFTECHNOLOGYThe technology that is available to a firm can be represented in a variety ofways. The most general are those based on correspondences and technology set for a given PRODUCTION process is de-fined asT={(x,y) : x Rn+,y Rm:+x can produce y}where x is a vector of inputs and y is a vector of outputs. The set consists ofthose combinations of x and y such that y can be produced from the given Output Correspondence and the Output is often convenient to define a PRODUCTION correspondence andthe associated output :The output correspondence P, maps inputs x Rn+into subsets of outputs, , P:Rn+ 2Rm+.

2 A correspondence is different from a function in that agiven domain is mapped into a set as compared to a single real variable(or number) as in a :The output set for a given technology, P(x), is the set of all output vectorsy Rm+that are obtainable from the input vector x Rn+. P(x) is then theset of all output vectors y Rm+that are obtainable from the input vectorx Rn+. We often write P(x) for both the set based on a particular value ofx, and the rule (correspondence) that assigns a set to each vector between P(x) and T(x,y).P(x)=(y:(x, y) T) of P(x). :Inaction and No Free Lunch. 0 P(x) x Rn+.

3 :y6 P(0), y> :Input Disposability. x Rn+, P(x) P( x), :Strong Input Disposability. x, x Rn+,x x P(x) P(x ). :Output Disposability. x Rn+,y P(x) and 0 1 y P(x). :Strong Output Disposability. x Rn+,y P(x) y P(x), 0 y :Boundedness. P(x) is bounded for all x Rn+. :T is a closed set P:Rn+ 2Rm+is a closed correspondence, , if [x` x0,y` y0and y` P(x`), `] then y0 P(x0). :Attainability. If y P(x), y 0 and x 0, then 0, 0 suchthat y P( x).Date: August 29, :P(x) is convexP(x) is a convex set for all x Rn+. This is equivalent to the correspon-dence V:<n+ 2<m+being :P is correspondence P is quasi-concave onRn+which means x, x Rn+,0 1, P(x) P(x ) P( x + (1- )x ).

4 This is equivalent to V(y)being a convex :Convexity of T. P is concave onRn+which means x, x Rn+,0 1, P(x)+(1- )P(x ) P( x+(1- )x ) Input Correspondence and Input (Requirement) than representing a firm s technology with the technol-ogy set T or the PRODUCTION set P(x), it is often convenient to define an input corre-spondence and the associated input requirement :The input correspondence maps outputs y Rm+into subsets of inputs,V:Rm+ 2Rn+. A correspondence is different from a function in that agiven domain is mapped into a set as compared to a single real variable(or number) as in a :The input requirement set V(y) of a given technology is the set of all com-binations of the various inputs x Rn+that will produce at least the levelof output y Rm+.

5 V(y) is then the set of all input vectors x Rn+thatwill produce the output vector y Rm+. We often write V(y) for both theset based on a particular value of y, and the rule (correspondence) thatassigns a set to each vector between V(y) and T(x,y).V(y)=(x:(x, y) T) between Representations: V(y), P(x) and T(x,y).The technol-ogy set can be written in terms of either the input or output {(x, y):x Rn+,y Rm+,such that x will produce y}(1a)T={(x, y) Rn+m+:y P(x),x Rn+}(1b)T={(x, y) Rn+m+:x V(y),y Rm+}(1c)We can summarize the relationships between the input correspondence, theoutputcorrespondence, and the PRODUCTION possibilitiesset in the following P(x) x V(y) (x,y) T2.

6 Of a PRODUCTION this point we have described thefirm s technology in terms of a technology set T(x,y), the input requirement setV(y) or the output set P(x). For many purposes it is useful to represent the re-lationship between inputs and outputs using a mathematical function that mapsvectors of inputs into a single measure of output. In the case where there is a singlePRODUCTION FUNCTIONS3output it is sometimes useful to represent the technology of the firm with a math-ematical function that gives the maximum output attainable from a given vectorof inputs. This function is called a PRODUCTION function and is defined asf(x) = maxy[y:(x, y) T]= maxy[y:x V(y)]= maxy P(x)[y](2)Once the optimization is carried out we have a numerically valued function ofthe formy=f(x1,x2.)

7 ,xn)(3)Graphically we can represent the PRODUCTION function in two dimensions as infigure PRODUCTION FunctionxyfHxLIn the case where there is one output, one can also think of the PRODUCTION func-tion as the boundary of P(x), , f(x) = Eff P(x). and the Induced PRODUCTION the produc-tion function exist. If it exists, is the output correspondence induced by it the sameas the original output correspondence from which f was derived? What propertiesdoes f(x) inherit from P(x)?a:To show that PRODUCTION function exists and is well defined, let x Rn+.By axiom , P(x)6= . By axioms and , P(x) is compact.

8 Thus P(x)contains a maximal element and f(x) is well defined. NOte that only thesethree of the axioms on P are needed to define the PRODUCTION :The output correspondence induced by f(x) is defined as followsPf(x)=[y R+:f(x) y],x Rn+(4)4 PRODUCTION FUNCTIONSThis gives all output levels y that can be produced by the input vectorx. We can show that this induced correspondence is equivalent to theoutput correspondence that produced f(x). We state this in a (x)=P(x), x Rn+. y Pf(x), x Rn+. By definition, y f(x). This means that y max{z: z P(x)}. Then by , y P(x). Now show the other way. Let y P(x).

9 By the definition of f, y max{z: z P(x)}= f(x). Thus y Pf(x). Properties , , and are sufficient to yield the induced pro-duction between P(x) and f(x).We can summarize the relation-ship between P and f with the following proposition:Proposition P(x) f(x) y, x Rn+ of PRODUCTION function for the PRODUCTION technology for corn ona per acre basis. The inputs might include one acre of land and various amountsof other inputs such as tillage operations made up of tractor and implement use,labor, seed, herbicides, pesticides, fertilizer, harvesting operations made up of dif-ferent combinations of equipment use, etc.

10 If all but the fertilizer are held fixed,we can consider a graph of the PRODUCTION relationship between fertilizer and cornyield. In this case the PRODUCTION function might be written asy = f (land,tillage,labor,seed,fertilizer,..)( 5) PRODUCTION a PRODUCTION function with twoinputs given by y = f(x1,x2). A Cobb-Douglas [4] [5] represention of technologyhas the following 11x 22=5x131x142(6)Figure 2 is a graph of this PRODUCTION 3 shows the contours of this a single output and input, a Cobb-Douglas PRODUCTION function has theshape shown in figure PRODUCTION often approximate a PRODUCTION functionusing polynomials.