Derived Demand and Substitution for Forest Products Based on Cobb-Douglas and CES Production Functions

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W O R K I I V G P A P E R

DERIVED D m

AND

-ON FOR F O B PRODUCE B t W 3 ON COBEDOUGLAS AND

CES

PRODUCTION P'LJNCTIONS

Markku Kallio Runar B r k n l u n d Esko Uutela

December 1984 WP-84- 104

I n t e r n a t i o n a l I n s t i t u t e for Applied Systems Analysis

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NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

DEFWED

DEMAND

AND SUBSITUTION M)R FOREST PRODUCTS

BASED

ON COBBDOUGLAS AND CES PRODUCTION FUNCTIONS

Markku Kallio Runar Brannlund Esko Uutela

December 1984 WP-84- 104

W o r h g Papers a r e interim reports on work of the International Institute for Applied Systems Analysis a n d have received only limited review. Views or opinions expressed herein do not necessarily r e p r e s e n t those of t h e Institute or of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

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The objective of t h e Forest Sector Project a t IIASA is t o study long- t e r m development alternatives for t h e forest sector on a global basis.

The emphasis in t h e Project is on issues of major relevance t o industrial and governmental policy makers in different regions of t h e world who a r e responsible for forestry policy, forest industrial strategy, a n d related trade policies.

The key elements of s t r u c t u r a l change in t h e forest industry a r e related t o a variety of issues concerning demand, supply, and international t r a d e of wood products. Such issues include t h e development of t h e global economy and population, new wood products and substitution for wood products, future supply of roundwood and alternative fiber sources, technology development for forestry and industry, pollution regulations, cost competitiveness, tariffs and non-tariff t r a d e barriers, etc. The a i m of t h e Project is t o analyze t h e consequences of future expectations and assumptions concerning such substantive issues.

The r e s e a r c h program of t h e Project includes an aggregated analysis of long-term development of international trade in wood products, and thereby analysis of t h e development of wood resources, forest industrial production and demand in different world regions. This article analyzes t h e robustness of derived demand with respect t o assumptions concerning underlying production functions. As a case, t h e wood consumption in t h e Canadian construction sector is chosen.

Markku Kallio Project Leader

Forest Sector Project

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S t a n d a r d production t h e o r y with Cobb-Douglas a n d CES production functions i s applied t o derive d e m a n d functions for forest products.

Time-series d a t a for 1961-1978 from Canadian c o n s t r u c t i o n s e c t o r is employed for estimation, a n d sensitivity of t h e demand f o r e c a s t i s t e s t e d with r e s p e c t t o t h e choice of t h e production function.

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1. INTRODUCTION 2. ASSUMPTIONS

3. THE COBB-DOUGLAS CASE 4. THE CES CASE

5. APPLICATION TO THE CONSTRUCTION SECTOR IN CANADA 6. CONCLUSION

REFERENCES

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DERIYED DEMAND AND SUBSTITUTION FOR FORESI' PRODUCTS BASED ON COBEDOUGLAS AND CES PRODUCTlON FUNCTIONS

Markku Kallio, Runar Brannlund a n d Esko Uutela

1. INTRODUCTION

The purpose of this n o t e is t o study demand (and thereby also s u b stitution) of various production i n p u t s in a given production sector. We a r e particularly i n t e r e s t e d in s u c h derived demand a n d substitution for forest products, and therefore, t h e construction s e c t o r which is a major c o n s u m e r of mechanical wood products was selected for this study.

Using s t a n d a r d production theory, we first derive t h e demand func- t i o n s when t h e production s e c t o r is described by CobbDouglas or Con- s t a n t Elasticity of Substitution (CES) production functions. These func- t i o n s will be e s t i m a t e d for t h e construction sector in Canada. for each of t h e years between 1961 a n d 1978. Finally, estimates will be provided for t h e demand function p a r a m e t e r s in 1990. Testing will be c a r r i e d out concerning t h e sensitivity of demand with respect to t h e assumptions on t h e production function.

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2. A S U M I T O N S

Consider a production s e c t o r (e.g., t h e construction s e c t o r ) whose production (quantity) y is c h a r a c t e r i z e d by a production function

with q being t h e i n p u t (quantity) of production f a c t o r i, _for i=1.2, ^* . . .n. Let pi be t h e price of factor i a n d y

=

a given level of o u t p u t . Assume t h a t t h e input mix (qi) r e s u l t s from cost minimizing:

s.t. y * = y ( q i ) . (3)

The optimality conditions for t h e Lagrangian function L are:

a~ a~

- = 0 a n d -

=

⁰ for all i ,

BP aqi

where p i s t h e dual multiplier for c o n s t r a i n t (3).

3.

THE

COBB-DOUGLAS CASE

Let t h e production function be specified a s

where a a n d

pi

a r e p a r a m e t e r s . Condition (4) yields

multiplying by q i , s u m m i n g over i , a n d observing (3) we obtain f r o m (6)

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By strong duality, x p i q i = pyO, a n d therefore

z g i = l .

Equation (6) yields also

pigi Bi

- = -

for all i, j . P j g j Bj

Relations (8)-(9) shall be used below for e s t i m a t i o n of p a r a m e t e r s

&.

Thereafter (5) may be employed for estimation of p a r a m e t e r a.

Solving q i from (9), for all i # j , a n d s u b s t i t u t i n g in (3). we obtain

Observing (8) we obtain f r o m ( 1 0 )

This d e t e r m i n e s t h e derived d e m a n d for all j. Equivalently, if is t h e base year consumption, 71 t h e production volume index a n d Xi t h e price index for i n p u t i, for all i, t h e n

4. THECESCASF, Assume now t h a t

where b

>

^0,yi r 0, z y i

=

1 a n d a 5 1. Condition (4) yields

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This implies

for all i , j .

Solving q i for all i f j and substituting into (3) ylelds

5. AF'PLlCATlON TO THE CON!RRUCl'ION SEClWR IN CANADA

Time-series d a t a for 1961-70 from t h e Canadian input-output statis- tics was employed t o study t h e derived demand in t h e construction sector.

For t h e Cobb-Douglas case, t h e output elasticity parameters

pi

and t h e scale factor a were estimated for each 18 years and tabulated in Table 1, which also indicates t h e input categories employed. Also esti- m a t e s for t h e year 1990 a r e given. The development of parameters

pi

over t i m e is illustrated in Figure 1.

The CEScase was studied in two cases: first with t h e value for l / ( l - a ) , t h e elasticity of substitution equal t o -5, and t h e second with t h e elasticity of substitution equal to 0.1. Tables 2 and 3 p r e s e n t t h e values of t h e distribution p a r a m e t e r s yi and of t h e scale parameter b over 1961-78 a s well a s two sets of estimates for 1990 for both cases. The first set i s a conservative estimate assuming t h e distribution p a r a m e t e r s t o stay a t t h e level of year 1978. The second s e t assumes a radical t r e n d of the 1970s t o continue until 1990. Forecasts a r e also presented in graphical form in Figure 2.

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0.10 -,

Cement 0.086

0.08 = -

,.---- _-L. ,&.,

^,^,^,^,^-

---

^0.069

0.06-

--

^{- a} ^\.em*-e^N-

we<:/* - - - ---

^0.056

0.04

--

^J^W^K^K^O

0.02

-.

0.00 ^I^I I I 1 1

1

1961 1965 1970 1975 1980 1985 1990

Other Inputs

J

Figure 1. Elasticity parameters /Ii for a Cobb-Douglas production function for the Canadian construction sector in 1961-78 and forecasts for 1990.

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0.6

--

/ ^/^/ ^(0.62) / I

/ I

Other Inputs I -

0 ----,, ,,

0.5.- 0.51

0 -28

0.2- (0.17)

Services (0.13)

0.12

0.0

-

^I ^I ¹

1961 1965 1970 1975 1980 1985 1990

0.08.- 0.06.- 0.04 0.02-

Figure 2. Distribution parameters ^y,for a CES p r o d u c t i o ~ function (elasticity of substitution = 0.5) for t h e Canadian construction sector in 1961-78 and two s e t s of forecasts for 1980.

0.0- Î , Î Î Î

(0.056) 0.046

O-O (OsO 71 0.016(0.018)

--

^Cement

I ---@--=

^-^>^*^u

=----

^-^-^,^,^,

Wopd

r; -

_

--

^--A-.

--- ^: ^- ^. =

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Applying formulas (11) a n d (16). t h e resulting d e m a n d forecasts for wood products for u n i t of output in t h e construction s e c t o r a r e shown i n Table 4. P r i c e indices a n d t h e i r forecasts for 1990 a r e given in Table 5.

The r e s u l t s a r e also compared t o t h e situation where prices would not change a t all, o r t h a t relative prices were c o n s t a n t . The corresponding wood inputs in 1990 for t h e two price scenarios a r e .052 and .053, when t h e Diewert function approach is used (Andersson e t al. 1984).

6. CONCLUSION

The r e s u l t s of Table 5 show t h a t the choice of production function a s s u c h does n o t radically afiect t h e derived demand levels: t h e difference between CD a n d CES r e s u l t s was only 0-4%. depending on t h e case. The r e s u l t s of t h e CES function proved also t o be r a t h e r insensitive with r e s p e c t t o t h e value of elasticity p a r a m e t e r a, as well a s t o different fore- c a s t values of p a r a m e t e r yi. The effects of changes i n relative prices appears t o be m o r e i m p o r t a n t for demand derived from t h e CD function than t h a t from t h e CES function.

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TABLE 1. Scale parameter a and elasticity parameters Bi for t h e Cobb-Douglas production function for the Canadian construction sector.

Year a

wood Metals Cement Labor Services Other products

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TABLE 2. Scale p a r a m e t e r b a n d distribution p a r a m e t e r s yi for t h e CES production function, elasticity of substitution equaling to 0.5, for t h e Canadian coc- s t r u c t i o n s e c t o r .

Year

Ti

b Wood

products Metals Cement Labor Services Other

1990

4.4 .016 .056 .O 18 .28 .12 -51 (conserv.)

1990

4.4 .01B .046 .017 .13 .17 .62 (radical)

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TABLE 3 Scale parameter b and distribution parameters yi for t h e CES production function, elasticity of substitution equaling to 0.1, for t h e Canadian construction sector.

Year b Wood Metals Cement Labor Services Other

products

z z104 Z I O - ~ z lo-2

1990

3.4 .240 .19 .57 .30 .090 .70

(conserv.) 1990

3.4 -175 ,093 -45 .042 .I64 .94 (radical)

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TABLE 4 . Price indices for inputs of construction sector.

Year Wood

products Metals Cement Labor S e r v i c e s Other

1) the original data material included an obvious error (original value for me- tals in 1974 was 0.40) and an estimated value was used. Table 1 to 3 were corrected accordingly.

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TAJ3LE 5. Demand for wood products ic 1990 per unit of construction output

Case Prices from Constant price Difference (%)

Table 4

Cobb-Douglas .056 .066

CES (a = -1, conservative) .054 .060

CES (a

=

-1, radical) .05? .062

CES (a

=

^-9,conservative) .056 .058

CES (a = -9, radical) .055 .056

Diewert .052 .053 -2

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REFERENCES

Andersson,

RE.,

R. Brannlund, and G. Kornai. 1984. The Demand for Forest Sector Products. IT-64-07. Laxenburg, Austria: International Institute for Applied Systems Analysis.