CES Production . . . How This Ubiquity Is . . . Second Requirement Main Idea Behind a . . . How to Explain Ubiquity of Derivation of the CES . . . Constant Elasticity of Groups and Abelian . . . Discussion Substitution (CES) Let Us Use Homogeneity Possible Application to . . . Production and Utility Home Page Functions Without Explicitly Title Page Postulating CES ◭◭ ◮◮ ◭ ◮ Olga Kosheleva 1 , Vladik Kreinovich 1 , and Page 1 of 21 Thongchai Dumrongpokaphan 2 Go Back 1 University of Texas at El Paso, El Paso, TX 79968, USA olgak@utep.edu, vladik@utep.edu Full Screen 2 Department of Mathematics, Faculty of Science Chiang Mai University, Thailand, tcd43@hotmail.com Close Quit
CES Production . . . How This Ubiquity Is . . . 1. Outline Second Requirement • The dependence of production on various factors is of- Main Idea Behind a . . . ten described by CES functions. Derivation of the CES . . . Groups and Abelian . . . • These functions are usually explained by postulating Discussion two requirements: Let Us Use Homogeneity – that the formulas should not change if we change a Possible Application to . . . measuring unit, and Home Page – a less convincing CES requirement. Title Page • In this paper, we show that the CES requirement can ◭◭ ◮◮ be replaced by a more convincing requirement: ◭ ◮ – that the combined effect of all the factors Page 2 of 21 – should not depend on the order in which we com- Go Back bine these factors. Full Screen Close Quit
CES Production . . . How This Ubiquity Is . . . 2. CES Production Functions and CES Utility Second Requirement Functions Are Ubiquitous Main Idea Behind a . . . • Most observed data about production y is well de- Derivation of the CES . . . scribed by the CES production function Groups and Abelian . . . � n � 1 /r Discussion � a i · x r y = . Let Us Use Homogeneity i i =1 Possible Application to . . . Home Page • Here x i are the numerical measures of the factors that influence production, such as: Title Page ◭◭ ◮◮ – amount of capital, – amount of labor, etc. ◭ ◮ Page 3 of 21 • A similar formula describes how the person’s utility y depends on different factors x i such as: Go Back – amounts of different types of consumer goods, Full Screen – utilities of other people, etc. Close Quit
CES Production . . . How This Ubiquity Is . . . 3. How This Ubiquity Is Explained Now Second Requirement • The current explanation for the empirical success of Main Idea Behind a . . . CES function is based on two requirements. Derivation of the CES . . . Groups and Abelian . . . • The first requirement is that the corresponding func- Discussion tion y = f ( x 1 , . . . , x n ) is homogeneous : Let Us Use Homogeneity f ( λ · x 1 , . . . , λ · x n ) = λ · f ( x 1 , . . . , x n ) . Possible Application to . . . Home Page • Meaning: we can describe different factors by using Title Page different monetary units. ◭◭ ◮◮ • The results should not change if we replace the original unit by a one which is λ times smaller. ◭ ◮ Page 4 of 21 • After this replacement, the numerical value of each fac- tor changes from x i to λ · x i and y is replace by λ · y . Go Back • So, we get exactly the above requirement. Full Screen Close Quit
CES Production . . . How This Ubiquity Is . . . 4. Second Requirement Second Requirement • The second requirement is that f ( x 1 , . . . , x n ) should Main Idea Behind a . . . provide constant elasticity of substitution (CES). Derivation of the CES . . . Groups and Abelian . . . • The requirement is easier to explain for the case of two Discussion factors n = 2. Let Us Use Homogeneity • In this case, this requirement deals with “substitution” Possible Application to . . . situations in which: Home Page – we change x 1 and then Title Page – change the original value x 2 to the new value x 2 ( x 1 ) ◭◭ ◮◮ – so that the overall production or utility remain the ◭ ◮ same. Page 5 of 21 • The corresponding substitution rate can then be cal- Go Back = dx 2 def culated as s . dx 1 Full Screen Close Quit
CES Production . . . How This Ubiquity Is . . . 5. Second Requirement (cont-d) Second Requirement • The substitution function x 2 ( x 1 ) is explicitly defined Main Idea Behind a . . . by the equation f ( x 1 , x 2 ( x 1 )) = const, then Derivation of the CES . . . Groups and Abelian . . . s = − f , 1 ( x 1 , x 2 ) = ∂f def f , 2 ( x 1 , x 2 ) , where f ,i ( x 1 , x 2 ) ( x 1 , x 2 ) . Discussion ∂x i Let Us Use Homogeneity • The requirement is that: Possible Application to . . . – for each percent of the change in ratio x 2 Home Page , x 1 Title Page – we get the same constant number of percents ◭◭ ◮◮ ds � = const . change in s : � x 2 ◭ ◮ d x 1 Page 6 of 21 • Problem: the CES condition is too mathematical to be Go Back convincing for economists. Full Screen • We provide: more convincing arguments. Close Quit
CES Production . . . How This Ubiquity Is . . . 6. Main Idea Behind a New Explanation Second Requirement • In our explanation, we will use the fact that in most Main Idea Behind a . . . practical situations, we combine several factors. Derivation of the CES . . . Groups and Abelian . . . • We can combine these factors in different order. For Discussion example: Let Us Use Homogeneity – we can first combine the effects of capital and labor Possible Application to . . . into a single characteristic, Home Page – and then combine it with other factors. Title Page • Alternatively: ◭◭ ◮◮ – we can first combine capital with other factors, ◭ ◮ – and only then combine the resulting combined fac- Page 7 of 21 tor with labor, etc. Go Back • The result should not depend on the order in which we perform these combinations. Full Screen • We show that this idea implies the CES functions. Close Quit
CES Production . . . How This Ubiquity Is . . . 7. Derivation of the CES Functions from the Second Requirement Above Idea Main Idea Behind a . . . • Let us denote a function that combines factors i and j Derivation of the CES . . . into a single quantity x ij by f i,j ( x i , x j ). Groups and Abelian . . . Discussion • Similarly, let’s denote a function that combines x ij and Let Us Use Homogeneity x kℓ into a single quantity x ijkℓ by f ij,kℓ ( x ij , x kℓ ). Possible Application to . . . • In these terms, the requirement that the resulting val- Home Page ues do not depend on the order means that Title Page f 12 , 34 ( f 1 , 2 ( x 1 , x 2 ) , f 3 , 4 ( x 3 , x 4 )) = f 13 , 24 ( f 1 , 3 ( x 1 , x 3 ) , f 2 , 4 ( x 2 , x 4 )) . ◭◭ ◮◮ • In both production and utility situations, for each i ◭ ◮ and j , f i,j ( x i , x j ) is increasing in x i and x j . Page 8 of 21 • It is also reasonable to require that: Go Back – the function f i,j ( x i , x j ) is continuous, and Full Screen – when one of the factors tends to infinity, the result also tends to infinity. Close Quit
CES Production . . . How This Ubiquity Is . . . 8. Derivation (cont-d) Second Requirement • Under these assumptions, f ( a, b ) is invertible : Main Idea Behind a . . . Derivation of the CES . . . – for every a ∈ A and for every c ∈ C , there exists a Groups and Abelian . . . unique value b ∈ B for which c = f ( a, b ); Discussion – for every b ∈ B and for every c ∈ C , there exists a Let Us Use Homogeneity unique value a ∈ A for which f ( a, b ) = c . Possible Application to . . . • It is known that: Home Page – for every set of invertible operations that satisfy the Title Page generalized associativity requirement, ◭◭ ◮◮ – there exists an Abelian group G and 1-1 mappings ◭ ◮ r i : X i → G , r ij : X ij → G and r X : X → G Page 9 of 21 – for which, for all x i ∈ X i and x ij ∈ X ij , we have Go Back f ij ( x i , x j ) = r − 1 ij ( g ( r i ( x i ) , r j ( x j ))) and Full Screen f ij,kl ( x ij , x kℓ ) = r − 1 X ( g ( r ij ( x ij ) , r kℓ ( x kℓ ))) . Close Quit
CES Production . . . How This Ubiquity Is . . . 9. Groups and Abelian Groups: Reminder Second Requirement • A set G with an associative operation g ( a, b ) and a unit Main Idea Behind a . . . element e ( g ( a, e ) = g ( e, a ) = a ) is called a group Derivation of the CES . . . Groups and Abelian . . . – if every element is invertible, i.e., Discussion – if for every a , there exists an a ′ for which Let Us Use Homogeneity g ( a, a ′ ) = e. Possible Application to . . . Home Page • A group in which the operation g ( a, b ) is commutative Title Page is known as Abelian . ◭◭ ◮◮ ◭ ◮ Page 10 of 21 Go Back Full Screen Close Quit
CES Production . . . How This Ubiquity Is . . . 10. Discussion Second Requirement • All continuous 1-D Abelian groups with order- Main Idea Behind a . . . preserving operations are isomorphic to (I R , +). Derivation of the CES . . . Groups and Abelian . . . • Here, (I R , +) is the additive group of real numbers, Discussion with Let Us Use Homogeneity g ( a, b ) = a + b. Possible Application to . . . • Thus, we can conclude that all combining operations Home Page have the form Title Page f ij ( x i , x j ) = r − 1 ij ( r i ( x i ) + r j ( x j )) . ◭◭ ◮◮ ◭ ◮ • Equivalently, f ij ( x i , x j ) = y means that Page 11 of 21 r ij ( y ) = r i ( x i ) + r j ( x j ) . Go Back Full Screen Close Quit
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