A geometric characterization of VES and Kadiyala-type production functions
AA geometric characterization of VES andKadiyala-type production functions
Nicol`o Cangiotti , Mattia Sensi University of Pavia, Department of Mathematics, via Ferrata 5,27100 Pavia (PV), Italy. Email: [email protected] University of Trento, Department of Mathematics, via Sommarive 14,38123 Trento (TN), Italy. Email: [email protected]
Abstract
The basic concepts of the differential geometry are shortly reviewedand applied to the study of VES production function in the spirit ofthe works of Vˆılcu and collaborators. A similar characterization isgiven for a more general production function, namely the Kadiyalaproduction function, in the case of developable surfaces.
Keywords : Production functions; Variable Elasticity of Substitution; Gausscurvature; Production hypersurface; Mean curvature; Flat space.
The study of production functions in the context of neoclassical economyhas a long tradition of research from many fields of knowledge. As pointedout by T. M. Humphrey [9], the first to make a significant contribution tothe development of a mathematically consistent approach to the marginalproductivity theory was the German mathematical economist, location the-orist, and agronomist Johann Heinrich von Th¨unen, in the 19th century (for1 a r X i v : . [ ec on . T H ] A p r urther details on the history of production functions, see the working paperof S. K. Misha [16]).The fortune of this mathematical model came in the 1927, when theeconomist P. Douglas and the mathematics professor C. W. Cobb proposedtheir famous equation, largely used in the textbooks as well as cited in articlesand in surveys [6]. Over the following years, the Cobb-Douglas productionfunction became a key concept of neoclassical economics (for interesting up-dates and testing due to Douglas himself, see [7]). At the end of the 1950s,R. M. Solow introduced a generalization of the Cobb-Douglas productionfunction: the CES (Constant Elasticity of Substitution) production function[21]; his idea was to aggregate the inputs in a single quantity. The functionthat realizes this combination of inputs is the so-called aggregator function .The aggregator function of CES functions has a constant elasticity of sub-stitution . However, a different generalization was developed between the1960s and 1970s by C. A. K. Lovell [13, 14], Y. G. Lu and L. Fletcher [15]and N. S. Revanark [19, 20]: the VES (Variable Elasticity of Substitution)production function. Our analysis stems from the study of this last class offunctions (in particular, from the formalization due to Revanark). The ap-proach which we shall use could be called the differential geometric approach .This particular technique, in connection with the study of production func-tions, was introduced and developed much more recently by A. D. Vˆılcu, andG. E. Vˆılcu [26, 22, 23, 24, 25]. Many contributions are due to B.-Y. Chen[1, 2, 4, 5, 3], and X. Wang [28, 29], as well. The classical theory of produc-tion functions is based on the projections of such functions on a plane, butsuch an approach does not seem exhaustive, at least from the mathematicalpoint of view. Vˆılcu solved this problem by identifying a production function Q : R n → R with Q , the graph of Q ; this turns out to be the nonparametrichypersurface of the ( n + 1)-dimensional Euclidean space R n +1 defined by: G ( x , . . . , x n ) = ( x , . . . , x n , Q ( x , . . . , x n )) ( x , . . . , x n ) ∈ R n + . (1)Thanks to this reinterpretation, one can study the production functions interms of the geometry of their graphs G ⊂ R n +1 . It is appropriate to notice that the interpretation of the Cobb-Douglas productionfunction is still a debated topic, as one can read, e.g., in [12]. We recall that in neoclassical production theory the elasticity of substitution, intro-duced by J. Hicks in the 1930s [8], provides a measure degree of substitutability betweentwo factors of productions. emark 1.1. We are denoting with G : R n → R n +1 a parametrization of thehypersurface G , which is a n -dimensional subset of the ( n + 1) -dimensionalEuclidean space. This distinction is rather important in the formalism ofdifferential geometry. If G is defined in A ⊂ R n , then G = G ( A ) . The aim of the paper is twofold. Starting from basic concepts of differ-ential geometry of surfaces, we shall study the 2-inputs (i.e. 2- dimensional)VES production function, obtaining a result, which essentially agree to thoseachieve by Vˆılcu and collaborators for the generalized Cobb-Douglas produc-tion function, and for the generalized CES production function in relationwith the Gaussian curvature of the corresponding surface. Consequently, weexplore a more general 2-inputs production function introduced by Kadiyalain the 1970s [11]. A renewed interest for the function introduced by Kadiyalaseems to have arisen in recent years, particularly due to the works of C. A.Ioan and G. Ioan [10] and Vˆılcu [27]. For this particular function, which is acombination of Cobb-Douglas, CES and VES production functions, we provea result on the corresponding Gaussian curvature in the case of developablesurfaces.The paper is organized as follows. In Section 2 we present an overviewof the differential geometry of surfaces, with basic definitions and properties.In Section 3 we study the VES production function as a surface, proving aresult which links the returns to scale with the Gaussian curvature (in thesame way as done by Vˆılcu, for instance, in [26]). In Section 4 we showthat the results concerning returns to scale and Gaussian curvature are notvalid for a more general 2-inputs production function, namely the Kadiyalaproduction function. Finally, in Section 5 we draw the conclusion, givingsome suggestions for further developments.
In this section we recall some basic concepts of differential geometry of 2-dimensional surfaces in R (we refer to [17] for further readings); these con-cepts can easily be generalized to n -dimensional hypersurfaces in R n +1 , forwhich we refer to [26]; however, it is unnecessary for the purpose of this ar-ticle, in which we focus on functions of two variables, namely the VES andthe Kadiyala production functions.Let U be an open set in R , and let f : U → R be a (smooth) function.3et F : R → R , defined as F ( x , x ) = ( x , x , f ( x , x )) , be the parametrization of the surface F = { ( x , x , f ( x , x )) ∈ R | ( x , x ) ∈ U } . (2)Denote with (cid:104)· , ·(cid:105) the natural inner product on R , and with (cid:107)·(cid:107) the norm itinduces. With this notation, we can give the following: Definition 2.1.
The first fundamental form g of the surface F is given by g := (cid:88) i =1 g ii d x i + 2 (cid:88) ≤ i The second fundamental form h of the surface F is givenby h := (cid:88) i =1 h ii d x i + 2 (cid:88) ≤ i Definition 2.4. The Gaussian curvature of a point x of the surface is givenby K ( x ) = det[ II ]( x )det[ I ]( x ) . (8) Definition 2.5. We call developable a surface having zero Gaussian curva-ture in all its points. We are particularly interested in developable surfaces because they canbe flattened on a plane by projection, without losing essential informationabout their geometry, hence easing their study.The main results of the paper are two theorems, which give conditions onthe VES and Kadiyala production functions, which ensure the correspondingsurfaces are developable. This section is devoted to the study of the VES production function, intro-duced by N. S. Revankar in [18, 19, 20]: Q ( u, v ) = ku δ (1 − βρ ) (( ρ − u + v ) βδρ . (9)We shall assume the following set of hypotheses:( (cid:63) ) k > , < β < , < βρ < , ( ρ − u + v > ,δ > . In this settings, δ is the parameter of return to scale.5igure 1: Plot of Q ( u, v ) for a random choice of the parameters β and ρ satisfying ( (cid:63) ). Remark 3.1. We recall that a VES production function has constant, in-creasing or decreasing returns to scale if δ = 1 , δ > , or δ < , respectively. Remark 3.2. The assumptions ( (cid:63) ) allows us to exclude degenerate casesin which, for instance, one of the two inputs is removed. Moreover, theassumption ( ρ − u + v > it is necessary when ρ < to ensure the well-posedness of Q ( u, v ) (it is clear that if ρ > , since u, v > , this condition isredundant). We notice also that for ρ < we can rewrite that condition as uv < − ρ , or, equivalently, vu > − ρ. (10)By using the same notation of Sect. 1, we introduce the following VESsurface parametrized by G ( u, v ) = ( u, v, Q ( u, v )) . (11)6 emark 3.3. In [18], Revankar proved that the elasticity of substitution afor the VES production function is σ ( u, v ) = 1 + ρ − − βρ uv . Hence, the VES production function varies linearly with the capital-labor ratio u/v . In [20], Revankar assumes σ > obtaining, as an additional constraintfor the economically relevant region of the variables domain, vu > − ρ − βρ , which, since − βρ < , is stricter than (10). We can now present and prove the main theorem of this section. Theorem 3.4. Let us consider the parametrization of a VES surface definedin Eq. (11) , with Q ( u, v ) satisfying conditions ( (cid:63) ) . • The VES production function has constant return to scale if and onlyif the VES surface is developable. • The VES production function has decreasing return to scale if and onlyif the VES hypersurface has positive Gaussian curvature. • The VES production function has increasing return to scale if and onlyif the VES hypersurface has negative Gaussian curvature.Proof. We can write the Gaussian curvature (defined as in Sec. 2) of theVES surface explicitly, using Eq. (11), obtaining: K = β ( δ − δ k ρ ( βρ − u βδρ + δ +1) (( ρ − u + v ) βδρ +2 (Den F ( u, v )) , (12)whereDen F ( u, v ) = δ k u δ (cid:0) u (cid:0) ρ (cid:0) β ρ + ρ − (cid:1) + 1 (cid:1) − ρ − uv ( βρ − v ( βρ − (cid:1) (( ρ − u + v ) βδρ + (( ρ − u + v ) u βδρ +2 = δ k u δ (cid:0) β ρ u + (( ρ − u − v ( ρβ − (cid:1) (( ρ − u + v ) βδρ +(( ρ − u + v ) u βδρ +2 . It is easy to see that Den F ( u, v ) (cid:54) = 0 for u, v > 0. The claim follows immedi-ately, keeping in mind assumptions ( (cid:63) ).7 In the 1970s, Kadiyala introduced an interesting generalization of produc-tion functions [11] (see also the works already mentioned by Ioan [10] andVˆılcu [27].), which in this section we shall study with a differential geometryapproach. The production function is given by: P ( u, v ) = (cid:0) k u β + β + 2 k u β v β + k v β + β (cid:1) δβ β . (13)We shall assume the following set of hypotheses:( (cid:63)(cid:63) ) k + 2 k + k = 1 ,k i ≥ , i = 1 , , , ( k , k ) (cid:54) = (0 , , ( k , k ) (cid:54) = (0 , ,β ( β + β ) > ,β ( β + β ) > ,δ > . As in the previous section, δ > P ( u, v ) for a random choice of the parameters β , β and δ < 1. 8igure 3: Visualization of P ( u, v ) for a random choice of the parameters β , β and δ > Remark 4.1. We are assuming k + 2 k + k = 1 , without loss of generality.The function P ( u, v ) is homogeneous of degree one (in the inputs u and v )when δ = 1 (i.e., for constant returns to scale). We also assume that β and β have the same sign as β + β . In this way, we ensure that themarginal products are non-negative. The two conditions ( k , k ) (cid:54) = (0 , and ( k , k ) (cid:54) = (0 , exclude the possibility of the elimination of one input, whichwould lead to a degenerate production function. Remark 4.2. We notice that for k = 0 we recover a CES-type productionfunction (setting also β + β < ); for k = 0 we obtain the Lu-Fletcher-typeproduction function; for k = 0 , k = 0 , and δ = 1 we obtain a Cobb-Douglas-type production function ; finally, for β = ρµ − , β = 1 , k = 0 we get aVES-type production function back. By using the same notation as Sect. 1, we introduce the following We are referring here to the classical Cobb-Douglas function with constant return toscale ( δ = 1): Q ( u, v ) = Au − α v α , α = β β + β . To obtain increasing/decreasing returns to scale the reader could keep δ as free parameter. adiyala surface parametrized by G ( u, v ) = ( u, v, P ( u, v )) . (14)Analogously to the previous section, we can now state the main result. Theorem 4.3. Let us consider the Kadiyala surface with the parametrizationgiven by Eq. (14) , with P ( u, v ) satisfying conditions ( (cid:63)(cid:63) ) . Then the Kadiyalasurface is developable if and only if one of the following conditions holds: • δ = 1 (i.e. the Kadiyala production function has constant returns toscale). • k = 0 and β + β = 1 • β = β = 1 and k − k k = 0 .In particular the last two cases implies that the Kadiyala production functionis a perfect substitutes production function .Proof. Firstly, we explicitly calculate the Gaussian curvature of the Kadiyalasurface and we obtain: K = T ( u, v ) · T ( u, v )(Den G ( u, v )) , where T ( u, v ) = ( β + β ) ( δ − δ u β +2 v β +2 ·· (cid:0) k u β + β + v β (cid:0) k u β + k v β (cid:1)(cid:1) δβ β +2 ,T ( u, v ) = ( β + β ) k u β (cid:16) β − β k u β + ( β ++ (2 β − β + ( β − β ) k v β (cid:17) − β k v β ·· (cid:16) β k u β − ( β − 1) ( β + β ) k v β (cid:17) , and Den G ( u, v ) = A + A + A + A + A , A = ( β + β ) k v u β + β ) ·· (cid:16) δ (cid:0) k u β + β + v β (cid:0) k u β + k v β (cid:1)(cid:1) δβ β + u (cid:17) A = ( β + β ) k u v β + β ) ·· (cid:16) δ (cid:0) k u β + β + v β (cid:0) k u β + k v β (cid:1)(cid:1) δβ β + v (cid:17) A =4 ( β + β ) k k u β +2 v β +2 β ·· (cid:16) β (cid:16) δ (cid:0) k u β + β + v β (cid:0) k u β + k v β (cid:1)(cid:1) δβ β + v (cid:17) + β v (cid:17) A =4 k u β v β ·· (cid:18) β v (cid:16) δ (cid:0) k u β + β + v β (cid:0) k u β + k v β (cid:1)(cid:1) δβ β + u (cid:17) + β u ·· (cid:16) δ (cid:0) k u β + β + v β (cid:0) k u β + k v β (cid:1)(cid:1) δβ β + v (cid:17) + 2 β β u v (cid:19) A =2 ( β + β ) k u β + β v β +2 ·· (cid:18) ( β + β ) k u v β + 2 k u β ·· (cid:16) β (cid:16) δ (cid:0) k u β + β + v β (cid:0) k u β + k v β (cid:1)(cid:1) δβ β + u (cid:17) + β u (cid:17) (cid:19) Den G ( u, v ) is clearly positive for ( u, v ) (cid:54) = (0 , .To prove the thesis we need to describe the parameter set in which K ≡ K ≡ ⇐⇒ T ( u, v ) ≡ ∨ T ( u, v ) ≡ . If δ = 1, we immediately get T ( u, v ) ≡ 0, and hence K ≡ 0. Let us assume δ (cid:54) = 1. In this case T ( u, v ) (cid:54) = 0, so we can conclude that T ( u, v ) ≡ ⇐⇒ δ = 1 . We shall now study T ( u, v ). Firstly, we rewrite T ( u, v ), collecting powers We recall that ( (cid:63)(cid:63) ) implies that β , β and β + β have the same sign. u and v as follows: T ( u, v ) = (cid:0) β + 2 β β − β − β β (cid:1) k k u β + β − β β k u β v β + (cid:0) β + 3 β β − β + 3 β β − β β + β − β (cid:1) k k u β v β + (cid:0) β + 2 β β − β − β β (cid:1) k k v β + β . If k = 0 (or, by symmetry, if k = 0), we obtain T ( u, v ) (cid:54) = 0 (in the firstquadrant). In the case k = 0, we have T ( u, v ) = (cid:0) β + 3 β β − β + 3 β β − β β + β − β (cid:1) k k u β v β = ( β + β − 1) ( β + β ) k k u β v β We get T ( u, v ) ≡ β + β = 1.Finally, let we assume k i (cid:54) = 0 for i = 1 , , 3. If β (cid:54) = β it is impossible to obtain T ( u, v ) = 0. Thus, let us fix β = β = β . So T ( u, v ) becomes: T ( u, v ) =4 β ( β − k k u β +4 β (cid:0) (2 β − k k − k (cid:1) u β v β +4 β ( β − k k v β , which is equal to 0 if and only if β = 1 and k k = k . The proof iscompleted by noting that T ( u, v ) ≡ ⇐⇒ ( k = 0 ∧ β + β = 1) ∨ (cid:0) β = β = 1 ∧ k k = k (cid:1) . In conclusion, we notice that for k = 0 and β + β = 1 we have the followingfunction: P ( u, v ) = ( k u + k v ) δ . Moreover, for β = β = 1 and k − k k = 0 we obtain P ( u, v ) = (cid:16)(cid:112) k u + (cid:112) k v (cid:17) δ . Thus, both cases lead to a perfect substitutes production function. The solution β = − β is forbidden by ( (cid:63)(cid:63) ). Because of, e.g., the term − β β k u β v β . The solution β = 0 is forbidden by ( (cid:63)(cid:63) ). Summary and conclusions In this paper we analyze two production functions from the point of view ofdifferential geometry.In particular, in accordance with the approach of Vˆılcu, we give a charac-terization of the (2-input) VES function in terms of curvature of the relatedsurface. This result is analogous (as we expected) to the results obtained byVˆılcu for the Cobb-Douglas and the CES production function.The second part of the paper is devoted to another kind of productionfunction, which could be seen as a combination of the most famous (2-inputs)production functions. We call it Kadyiala production function. For the lat-ter, computations become more cumbersome, but it is still possible to give acharacterization connected with the Gaussian curvature of the correspondingsurface, at least in the case of developable surfaces. The constant returnsto scale is a necessary condition if we suppose that the Kadyiala produc-tion function is not a perfect substitutes production function; this result isconsistent with the previous works of Vˆılcu, as well.We conclude with a short outlook on possible research perspectives: anatural successive step in our analysis would be to study in detail the sign ofthe curvature of the Kadyiala production function, its dependence on specificchoices of the parameters and the interpretation of such picks. Anotherlogical path to follow would be to generalize the results presented in thispaper for functions of a generic number of inputs n ; however, one wouldneed to propose a clever way of analyzing such a function, since computationsproved to be cumbersome even for the 2-dimensional case; in this regard, itwould be particularly interesting to study the connections between our workand recent papers by Ioan [10] and Vˆılcu [27]. Declarations of interest: none. 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