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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, WilliamGreene, 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, -

Oc.oOK,-ih-

K

1

(i,J)'

• * •

i

(i fJ) • • 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 ti"ai+i ) ln

^i 3 ,2 ,. . .,1-1

for all j, and

(3) ln61(J+1)

-li + (V6j+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 Ri 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 '

$51 **i

+5

61K 1'

J-2

2milar comparisons in terms of grafted polynomials , see Fuller (

TOd Gallant and Fuller [J].

wh*

chati

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

9U J

l/6d

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' ithe grid

lines. The iso y convex iff

and

> a1+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- ,JI )

— 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.-riI

Greater

Smaller

Inconclusive

Figure

Li-

te S orisons

I

isoq

dKrtL^

•

' L •* L " 6J

dKdL"

limitT~

Letting w denote the

for all price ratios w/r such t

,d r the price of capita]

(7)dK.v

dKdL~

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 *

dKdL

^.It

(10)5. < - <

Lt

Li

hold;.

.

the ab ted.

K

K

K

1 7La

(a)

K**

K*

/

Li

(b)

K-

K" -

'*».K—'Kx

K" -

)

Figure 3: Expansion P

'11-

>d in

Jgous to the

po • -a at

proceed along either

training p / not d by ( (10) is I

(ID w i+1

J+l K

slope C6xWi

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 :tions

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

exa

a2

of Is.

TBO

LC

60 20ours of Work

40 60of Leisure

Figure 4: Rf

was s

. .e.

,

I .

l

a2/

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, f1

- 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, lnPF) +

^(laPK -lr,P

F)

where w ost, PL

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

5X

.5905** 40

S2

.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 Pf+ n In Pr - 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 testi.

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 PolynomialRegression Models Whose Join Points Have to be Estimate imalof 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," Journal 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 SupplyLow- Income Male Family Heads," in Final Report of the New JerseyGraduated Work Incentive Experiment (Madison: Institute forResearch on Poverty, University of Wisconsin, 1973), Chapter B-II

dson, Derek J., "Fitting Segmented Curves Wh 'n Points Haveto Be Estimated," Journal of the American Statistical Association ,

I (December, 1966), 1097-11.

[9] inMeasuremen t In St. (Stanford:Stanford Dnlvex 1963) ,

[10] Poirier, Dale J na in Economicsted dissertatio isconw 9

(II] , ewiee Regres sing Cubic SplinebJournal of the Amer^ atibtlcal Association , LXVIII (Septemb

1973), 5J

[12] , "On the Use of Bilinear Splinesming.

of Hu: