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Faculty Working Papers 
 
 ON THE USE OF COBB-DOUGLAS SPLINES 
 
 Dale J. Poirier 
 
 #223 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana-Champaign 
 
FACULTY WORKING PAPERS 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 December 5, 1974 
 
 ON THE USE OF COBB-DOUGLAS SPLINES 
 
 Dale J. Poirier 
 
 0223 
 
On the Use of Cobb-Douglas Splines 
 Dale J. Poirier 
 
 1. Introduction 
 
 Since the legendary work of Cobb and Douglas [1], Cobb-Douglas pro- 
 duction functions and (to a somewhat lesser degree) Cobb-Douglas utility 
 functions have been popular tools of economists. This popularity can be 
 attributed both to the simplicity and to the wide-applicability of these 
 functions. However, 'these functions are of course subject to rather severe 
 restrictions. For example, in the production function context returns to 
 scale are non-varying, hence, U-shaped average cost curves are ruled out. 
 Also, homotheticity and unitary elasticities of substitution are required. 
 
 This study develops the idea of continuous piecewise Cobb-Douglas 
 functions along the spline function lines discussed in Poirier [10] - [12 ] • 
 This development will permit U-shaped average cost curves and "piecewise- 
 homotheticity" at the expense of differentiability of the functions along 
 lines parallel to the input axes. However, the unitary elasticities of 
 substitution requirements will remain. 
 
 Implicit in this discussion is the belief expressed in Poirier [10 ] - 
 [12] that for a wide range of problems in economics, the added generality 
 of more complicated functional forms is often best achieved by using 
 continuous piecewise functions. Briefly, the rationale is two-fold. First, 
 
 The author is an Assistant Professor of Economics at the University 
 of Illinois at Urbana-Champaign. The contents of this study rely heavily on 
 Poirier [10, Chapter 4]. Thanks are due Joeteph Hotz, Steven Garber, William 
 Greene, and Diane Christenqen of the University of Wisconsin at Madison for 
 their data handling and programming assistance which contributed greatly to 
 section 5. Of course any errors are the sole responsibility of the author. 
 

 
 
 
 
 
 
 
 
-2- 
 since within each "piece" such functions have simple and familiar forms, 
 the analysis proceeds quite straightforwardly. Indeed all economists are 
 familiar with C >bb-Douglas functions and their properties, and so this 
 previous kr. in be easily apolied in analyzing piecewise Cobb-Dougles 
 functions. Second, changes jn ^he behevic r of such functions as one passes 
 from one piece to another are often of primary concern In economic analysis 
 (e.g., the changes in expansion paths diecussed in Section j) . Indeed the 
 testability for the existence of such "structural changes" will be an im- 
 portant consideration throughout this discussion. 
 
 Organizationally, we will proceed as follows. Section 2 defines a 
 Cobb-Douglas spline. For the sake of simplicity, but not at the expense of 
 generality, a production function context with two inputs, labor and capital, 
 •ill be used. Section 3 discusses the properties of Cobb-Doublas spline pro- 
 duction functionr, lenvlng to section 4 a discussion of the properties of 
 Cobb-Douglas spline utility functions. Finally, section 5 contains an emr J " J 
 application of a CDS production fraction, 
 
 2. Definition of a Cobb-Douglas Spline 
 
 Let the --ts A^ - [L- < L, <...< L_ ,} and A_ - (*.< K* 2 <...< *• j) 
 
 be meshes defining intervals in the labor (L) and capital (K) dimensions. 
 The elements in A and A. are crlled knots and they define a rectangular 
 grid in the positive quadrant that conflicts of IJ rectangles (see Figure 1). 
 A Cobb-Douglas spline (CDS) is a function Q(L,K) which can be defined as 
 
 (1) Q(L,K / - e^ 
 

 
 
 
 
 
 
 
 
 

 K, - 
 
 O 
 
 c. 
 o 
 O 
 
 K,-ih- 
 
 K 
 
 1 
 
 (i,J) ' • 
 
 * • 
 
 i 
 
 (i f J) 
 
 • • 
 
 r 
 
 (I,J) 
 
 • 
 • 
 • 
 
 ti.j) 
 
 • • 
 
 • 
 
 • 
 
 i 
 
 
 • 
 • 
 • 
 
 (••J) 
 
 i 
 
 • * 
 
 • 
 
 (I. J) 
 
 • 
 
 • 
 * 
 
 • 
 
 • 
 > 
 
 
 • 
 • 
 • 
 
 (i,i) '< ■ 
 
 . • 
 
 i 
 
 (i,l) ! . . 
 
 • 
 
 1 
 
 (1,1) 
 
 Labor 
 Figure 1: obor- Capitol 
 

where L and , positi- 
 
 .'0 con r 
 
 I ty 
 reqi at imp 
 
 (2) In 9 (1+1)j a In C t i" a i+i ) ln ^i 3 ,2 ,. . . ,1-1 
 for all j, and 
 
 (3) ln6 1(J+1) -li + ( V 6 j+i> ln *J J" 1 . 2 ----' 
 
 for all i. In this context the output elasticities of labor (the a. 's) and 
 capital (the <5 s) are step functions over the meshes A and A^,, respective 
 Often it is more convenient to work with a CDS in its logarithmic 
 rm 
 
 (4) In Q - ln 6 . . + a, in L + 5 In R 
 
 i j 
 
 for L and K in rectangle In formulation (4) ln Q ia "he sum of lwo 
 linear splines (see Poirier [10, Chapter 2]) one in the In L dimension, 
 and one in the ln K dimension. erms of Poirier [12], (4) is also a 
 bilinear spline with no interaction era ^fines a 
 
 piecewise-planar or "roof-like* 
 
 tentative] uity condltl _) and (3) can be. ed 
 
 to representation (A) wing manner. Define ables 
 
 L ± - max [(:. , 0] 
 
 - max )] j - 
 
 Then for all L and 
 
 J-] ~ « 
 M + a ] ]n ' $ 5 1 **i + 5 6 1 K 1' 
 
 J-2 
 
 2 
 
 milar comparisons in terms of grafted polynomials , see Fuller ( 
 
 TOd Gallant and Fuller [J]. 
 

wh* 
 
 chat i 
 
 Rep 
 
 are of part 
 
 In I ed in a production 
 
 theory context, it will be i CDS pr . m, and when it 
 
 is usee! in c utility theory context, it will be called a CDS utility 
 function . 
 
 CDS Production Functions 
 
 Of foremest importance * ussing the theoretical properties o 
 
 the CDS product! laoquants. For a fi.ced outi vel 
 
 the isoquait over rectangle (i,j) is 
 (6) 
 
 % ^ 
 
 L 9 U J 
 
 l/6 d 
 
 ughout the labor-capita: xsions the isoquants are continuous, 
 -ever, the;, have "< Long t ^st result f.) 
 
 from the continuif- he productio .ion, and the latter ifl ° result 
 
 f the nrodu' i the grid 
 
 lines. The iso y convex iff 
 
 and 
 
 > a 1+1 
 
 j+1 J 
 
 •ther word s are . ai 
 
 is a decreasing f its respect! - 
 

 
(ci . + " 
 CO! nditii 
 
 and 
 
 + 6> u + 6 m-t t i+l,. . . ,1; n- ,J 
 I ) — m n 
 
 As shown in Figure 2, this 
 
 scale over ill rectangles beloi (i,j), and 
 
 smaller returns tn sc rectangles above and to the right of 
 
 rectangle Compai to rectangles nbove and c- 
 
 *angle . and beli e right ot rectangle (i,j) are inconclusive 
 
 It is well knovn that Cobb-Douglas isoquants give I • a strai, 
 line expansion path since the production function is homothetic. The 
 Cobb-Douglas ipline production function has straight line expansion path 
 pegments over individual rectangle, however, these paths exhibit uniqee 
 behavior along C 
 
 To see this suppose the convexity conditior no\d and consider the 
 slope of isoquant (6) l r ingle ( ->t alon£ its border: 
 
 The margi lot well- 
 
 defined icts arc not well-r 
 
 there as a result of the ontlnuities in the output elasticities. 
 

 Inc usive 
 
 K 
 
 <« 
 
 K.- 
 
 ri 
 
 I 
 
 Greater 
 
 Smaller 
 
 Inconclusive 
 
 Figure 
 
 Li- 
 te S 
 
 
 orisons 
 
I 
 
 isoq 
 
 dK 
 rtL^ 
 
 • 
 ' L •* L 
 
 " 6 J 
 
 
 dK 
 dL" 
 
 limit 
 
 ■ T~ 
 
 
 Letting w denote the 
 for all price ratios w/r such t 
 
 ,d r the price of capita] 
 
 (7) 
 
 dK 
 
 .v 
 
 dK 
 
 dL~ 
 
 the first order conditions for the firm's output maximization or cost 
 
 nation problem are not £ati6fi< ?., the marginal rate of tech.. 
 substitution does not equal the price ratio. Hence, compensated price 
 
 nges within the boundt of (7) do not change the optimal input combi- 
 nation for producing a fixed level of output. 
 
 Fixing w/r and expanding output, Ln rectangle 
 
 |) approa f ne gri does no at a croical level K* 
 
 satisf 
 
 dL~ 
 
 _i 
 
 or solving f 
 
 K* 
 
 6 L 
 
 'u »K*) 
 
 Since the outpu 
 tha > J ) , 
 
 □el by in er the righr 
 
 equals ty of capil 
 

 drops, i. 
 
 
 
 
 
 solving f 
 
 K** 
 
 
 I r 
 
 then the I ion pat H.J) *oi i » K * 
 
 is a straight th a si \ T) 
 
 of the expansion path In I). Tlv lllust 
 
 for the simp. I, i ■ j • 1. 
 
 Hot if the grid line V is reach-. -t, then the 
 
 path for outputs Ql ,K ) continues along L - L ± 
 
 (9) 
 
 L - L ± + |dK . !i±l 21 . 
 
 
 limit 
 
 K * 
 
 
 dK 
 dL 
 
 ^. 
 
 It 
 
 (10) 
 
 
 
 5. 
 
 < - < 
 
 Lt 
 
 L i 
 
 
 hold;. 
 
 . 
 
 
 the ab ted. 
 

 K 
 
 K 
 K 
 
 1 
 
 7 
 
 La 
 (a) 
 
 K** 
 
 K* 
 
 / 
 
 Li 
 
 (b) 
 
 K- 
 
 K" - 
 
 '*». 
 
 K—'Kx 
 
 K" - 
 
 ) 
 
 Figure 3: Expansion P 
 
'11- 
 
 >d in 
 
 Jg ous to the 
 
 po • -a at 
 proceed along either 
 
 ■ training p / not d by ( (10) is I 
 
 
 (ID 
 
 w i+1 
 
 J+l K 
 
 slope C6 x Wi 
 
 If (11) holds, then the expansion pauh in rectangle (i,j 
 
 :aancipacing from ( 
 function associated with a f>DS production function 
 
 continuous piec pieces 
 
 -ingle. tor 
 cost fu. 
 
 
 
 
 
 
 
 
ano the < 
 
 
 , 
 
 uij 
 
 
 In 
 as Q ■» Q< 
 
 ■ ■ ■ 
 by dividing eacl I to (12) , the average c 
 
 functi ^irtg, con decreasing, depending on whether Q 
 
 produced j.n a rectangle with deer-- , constant, or increasing retu' 
 scale. Since the cost functioi inuous , so is the average c 
 function. 
 
 marginal cost function associated with (12) - tainec 
 
 .differentiating each e<ju. ^3pect At outp'> 
 
 an path marginal cost is e- 
 fined sine* the left and right erivat. b re not In gene. 
 Of course wi (12), the marginal cost sagme 
 
 CDS UtilJ :tion s 
 
 ising: ty 
 
 the ; DS ut 
 
 on ( 
 
 e. 
 
 
 L 
 
utl 
 
 ure 
 
 ■ 
 c is in equi 
 to 40 hours 
 
 oposs he is 
 him $60 a wet 
 
 50 per cent Ing the guarantee as earned .is 
 
 plan has akevCn" at point B (greater t. treflpo'. 
 
 ied income of $120 ft; to the 1 point he receives 
 
 payr e neg< 
 
 onvent s (in 
 
 ■ 
 
 lay re 
 
 ex a 
 
 ■ 
 
 a 2 
 
 of Is. 
 

 TBO 
 
 L 
 C 
 
 60 20 
 
 ours of Work 
 
 40 60 
 
 of Leisure 
 
 Figure 4: Rf 
 
was s 
 
 ■ 
 
 . 
 
 
 .e. , 
 
 ■I . 
 
 l a 2/ 
 
 hen proceeds along an expansion path wil pe (cu However, if 
 before the point F is r< he encounters his budget constraint ABG, I 
 must then stop at a point E on indifference curve I. which corresponds I 
 no change in income with the entire - of the negative income tax p2 
 manifest*- eased leisure (i.e., a -eduction in laf .). 
 
 Intuitively, the explanation is as f s. The redu. >£ the 
 effective wage as a result of the >s that the pric .u»-e hr 
 
 Ly speaking, dropped. Hence, he will consume more leisure, 
 eve , MRS - (- — ) , f i . 
 
 (th 
 
 - 
 
Follow 
 
 bee 
 
 utilii 
 
 • y wafi ippeareri to 
 
 .action of the level framework of a CDS p: ^on 
 
 fur. :r equal 1 oo would h chat tl 
 
 returns to scale were g fun< els o: 
 
 ■ght of re ,'orld developments and the "energy crisis/' t> 
 
 question of whether there g returns to scale 
 
 electricity takes on added imp. creasing 
 
 relevant ranges of output i lie utilities have important impj ns 
 
 in terms z subsidies and investment policies in such an industr. 
 
 ^rder to investigate the returns to scale, Nerlove sugges 
 
 r'irms can be viewed as taking their output and i i3 as 
 
 exogeneous, and the c total cost 
 
 Cobb-Douglax pr e data use 
 
 was based on Nerlove [9 ) and 
 
 sour 
 
 Suppr. 
 pre jrm 
 
 (l- Q - e ± l° 
 
 Pa <:'s [ 9 , >] OWT. gS CO' 
 
 questic uction timet io,. .rothet 
 
 •wed m a an i 
 
ency . 
 are the c< Unlike 
 
 I be at ' ec ? ua 
 
 F , or 6 F . T i fuel knot, f 1 - 4 (nri.lli< 
 
 3tu's), is admittedly somewhat arbitrary. Hopex in more elaborate 
 
 applications the choice of 7., or for that fact, labor and capital kr rill 
 
 reflect technological considerations of the production process. The chc 
 here is no more arbitra n the five, group breakdown used by Nerlove, ai 
 
 it corresponds to approximately the 33rd percentile of the firms' fuel input 
 level. 
 
 Following Nerlove [9, FP» 171-175] it can be showed that cost 
 minimization subject to (15) and exogenous output and input prices implies 
 the cost function 
 
 M 1 ! ^InQ+^Ur, lnP F ) + ^ ( laP K -lr,P F ) 
 
 where w ost, P L is Zhe 
 
 labor, P is the price of capital, P , is the price L, and 
 
 
 - In v . In 6. 
 
 Proceeding as i assumes ti 
 
 nplies 
 
 this I 
 be ways to hand . knots 
 
 Whi 
 
 at lea >ry be ap; more comp 
 
 6 ], a: for do 
 

 ible 1 
 
 Coe 
 
 
 
 
 
 Coefficient 
 
 
 ■ 
 
 dare 
 
 a 
 
 
 
 33 
 
 5 X 
 
 .5905** 
 
 
 40 
 
 S 2 
 
 .5591** 
 
 
 .1244 
 
 n 
 
 .01574 
 
 
 .2394 
 
 
 
 1823. 
 
 
 34* 
 
 "*" denotes significance at the 57. level. 
 "**" denotes significance at the 1% level. 
 

 w. In. C - 6, + In a P f + n In P r - In e 
 
 + H In P^. Note that isturbancc term in 
 
 to be homoscedaatic when returns to scale vary (i.e., <*>.. i 6 ? ), (16) 
 multiplied through by w . 
 
 ier the assumption that the residuals in (17) are independent, homo- 
 scedastic, and normally dit,r ed, the likelihood function corresponding 
 (17) can be maximized subject to the continuity constraint In 6- " In 0. + 
 (6. - 6„) In P . The resulting maximum likelihood estimators a, d. , 6 ? , n» ana 
 
 .ill asymptotically be consistent, normally distributed, and efficient. Since 
 the likelihood function is nonlinear in these parameters, it is necessary to 
 employ a technique such as Marquardt's method of steepest descent to obtain 
 the maximum. 
 
 The actual maximum likelihood estimators (together with their standard 
 errors) are given in Table 1. The results are similiar to those of Nerlcve* 
 the output elasticities of labor and fuel highly significant, and the output 
 elasticity of capital is insignificant (as it often was in Nerlove's study). 
 Of course the result of principal concern is the estimated change 6. - 6~ m 
 .03140 in the on lastieity o! at F . The asymptotically normal test 
 
 i. 
 
 statistic for testing the significance of this change is 
 
 112 
 
 (Est. Var ...-•.- 
 
 which is highly significant. Thu^ .e there are increasing raturns to 
 veryvhere, they appear to undergo a significant drop along F . 
 

 
 [I] Co! :ion, -i& 
 Supp. 1. (1928), 139-165. 
 
 [2] Fu Jayne red Polynomials as Approximating Functions," 
 Australian Journal ■ rlcultural Econon^ [ (June, 1969), 
 35-46. 
 
 [3] Gallant, A. R. and Fuller, Wayne A., "Fitting Segmented Polynomial 
 Regression Models Whose Join Points Have to be Estimate imal 
 of the American Statistical Association , LXVIII (March, 1973), 144-147, 
 
 [4] Greene, William H. , "Factor Substitutions and Returns to Scale in 
 Electrical Supply," unpublished manuscript, 1974. 
 
 [5] Halpern, Elkan F. , "Bayesian Spline Regression when the Number of 
 Knots is Unknown," Jou rnal of the Royal Statistical Society , 
 (1973), 347-360. 
 
 [6] Hinkley, David V., "Inference in Two-Phase Regressirn," Journal ..; 
 
 the American Statistical Association , LXVI (December, 1971), 736-743. 
 
 [7] Horner, David, "The Impact of Negative Taxes on the Labor Supply 
 Low- Income Male Family Heads," in Final Report of the New Jersey 
 Graduated Work Incentive Experiment (Madison: Institute for 
 Research on Poverty, University of Wisconsin, 1973), Chapter B-II 
 
 dson, Derek J., "Fitting Segmented Curves Wh 'n Points Have 
 to Be Estimated," Journal of the American Statistical Association , 
 I (December, 1966), 1097-11. 
 
 [9] in 
 
 Measuremen t In St. (Stanford: 
 
 Stanford Dnlvex 1963) , 
 
 [10] Poirier, Dale J na in Economics 
 
 ted dissertatio isconw 9 
 
 (II] , ewiee Regres sing Cubic Splineb 
 
 Journal of the Amer^ atibtlcal Association , LXVIII (Septemb 
 1973), 5J 
 
 [12] , "On the Use of Bilinear Splines 
 
 ming. 
 

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