Estimation of Forest Products Demand as an Intermediary Function

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

December 1984 WP-84-9 1

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

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

ESlTMATION OF

FOREST

PRODUCTS DEMAND AS AN INTEmmDLARY FUNCTION

Kke E. Anaersson

December 1984 WP-84-9 1

Working Papers a r e interim reports on work of t h e International Institute for Applied Systems Analysis and have received oniy limited review. Views or opinions expressed herein do not necessarily represent those oi t h e Institute or of its National Mem ber Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

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

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

The key elements of structural change in the forest industry a r e related to a variety of issues concerning demand, supply, and interna- tional trade of wood products. Such issues include t h e development of the 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 trade barriers, etc. The aim of the Project is to analyze the consequences of future expectations and assumptions concerning such substantive issues.

In this article t h e problem of demand forecasting is discussed from a quantitative point of view. It is shown that an intermediate demand approach is preferable to t h e common final demand procedures of forest product demand studies.

Markku Kallio Project Leader

Forest Sector Project

CONTENTS

METHODS FOR ANALYZING INTERMEDIATE DEMAND CHANGING INPUT DEMAND STRUCTURE

THE INFLUENCE OF PRICES ON CHOICE OF TECHNIQUES AND I N T E R M E D W FOREST PRODUCT DEMAND

AN A L T E R N A W APPROACH TO INTERMEDIATE DEMAND ANALYSIS ESTIMATION WHEN DATA ARE SCARCE

CONCLUSIONS REFERENCES

ESlTlMTION OF FDREfX PRODUCTS DEMAND A S AN INTEEtWDIAR'Y FUNCTION

b e E. Andersson

Forest sector products are intermediary commodities. Less than 15 percent of total output is delivered t o households for final consumption.

A study of seven OECD countries shows a marked decline of the house- hold share in paper purchases in the period 1965-1975. A projection for the U S indicates that the household purchases of paper will drop below 10 percent of total output in the 1980s. Any development of improved methods for demand forecasting must consequently address the issue of intermediate demand and its determinants.

METHODS FOR

ANALYZING

INTEEWEDIATE DEMAND

The classical approach to intermediate demand analysis is input-output modeling. An input-output model is based on a n

accounting framework of the following type:

8. Total input

1

X,

1 x2 1 xs x4 ¹ x5 ! ^x8 ! ^x7 1

Consistency requires that the value of total input

xqj = 3 = xqj.

i i

Total output 8 X 1 x2

x~

x4

X5

4 3 x7

Deliveries from

I

Deliveries to

This means, for instance, t h a t t h e total value of all sales from paper pro- cessing should equal t h e value of all purchases of raw materials, intermediary commodities (Xzs

+

X3,

+

X,,

+

X,,), plus the cost of house-

%,

Xz5 X3, X45 Xi55

x85

I

.-

/

^{t i 2 3 4 5}

C1

C2

' C3

C4 C5

..

c

hold deliveries of labor, capital, and land services. Governmental costs

6 7

G 1

G2 G, G4 Gs

x87

,

..

a r e often treated as delivery charges from industry even if there is no 1. Forestry

2. Wood processing 3. Paper processing

measurabie delivery of government services ( t h e lack of consistency in

X13

X2, Xss

this respect is represented by double dots in the table).

XI, X2, Xs4 4. Other industries

5. S r v i c e s 6. Households 7. Government

& I

Xzl

x3,

In t h e classical input-output studies by Leontief (1951) and many

X12

Xz2 Xs2

others it has often been assumed that t h e interdependencies between X4i

x51 x81

. .

sectors can be roughly represented by a fixed coefficient assumption.

Thus t h e degree of interdependency is represented by the coefficients X42

'

^X43

¹

^Xu

T -

atj

= 4

e.g.. t h e interdependency between forestry (sector 1) and wood

4

Xs2

x82

. .

processing (sector 2) would be represented by two coefficients:

X53 XIS

.

^.

X54

x64

. .

'

cost of delivery of forestry products t o wood processing value of output of wood processing

l

a n d

n-- = cost of delivery of wood processing products t o forestry

1 ,

-"I -

L

value of output of forestry products

J

C, a n d Gi would normally be t r e a t e d as exogenous p a r a m e t e r s a n d t h e forecasting problem is t h u s a question of solving t h e following set of linear equations (in which all relative prices are assumed c o n s t a n t for t h e forecasting period).

o r in matrix form

z

=

Ax

+

c

+

g where

The solution c a n be computed a s

where ( c

+

^{g )} is forecasted independently.

he

expression ( I

-

^A)-'

⁼

^I

⁺

^A

+ - . +

A'

-'

which indicates t h a t t h e interdependen- cies cascahe though t h e whole economic system. E.g., a dollar worth of services implies less t h a n a dollar worth of paper, which implies a need for wood raw material, which implies a n e e d for chemicals, which implies a need for

...

a n d s o o n a d infinitum

In any productive. economy the sum of all direct and indirect needs is bounded, producible and compatible with a general equilibrium of the economy.

The use of forest sector products in different sectors of the economy is highly skewed. Unlike labor or energy these a r e inputs that seem to be reducible from most types of production. The connections to other sectors of the economy a r e highly structured a s can be seen from Table 1. The table shows that for t h e mechanical wood processing two sectors stand out as important users

-

Construction and Fbmifu7.e.

Different types of paper used by a limited group of industries a r e of special importance:

-

Printing (printing and writing, newsprint)

-

Food (packaging)

-

Offices (printing and writing)

-

Domestic trade (packaging)

-

Chemicals (printing and writing)

-

Hotels and Restaurants (household and sanitary)

A demand study should focus on the choice of technique of production in these user sectors.

CHANGING INPUT DEXAND STRUCTURE

it has been suggested that t h e constant price coefficients aij should be made functions of time in order to reflect the technological develop ment of production processes. This implies no further problem in solv-

TAJ3I.E 1. The s t r u c t u r e of intermediate deliveries in t h e Canadian forest s e c t o r 1971 (and 1978) (% of total output). Source:

Statistics Canada, Input-Output Tables in Constant Prices.

Delivery Fabricated Furniture Paper Constr. Food Printing Dom. Services Accommod. Chemicals Offices Rest

Delivery to wood products t r a d e food

from products service

Forestry

Lumber &

timber Veneer &

plywood Other wood

fabricated materials Pulp

Newsprint

& o t h e r paper stock Paper pro-

d u c t s

ing t h e input-output system of equations.

An example of the development of relative forest sector inputs is given in Table 2. In this table the value coefficient development is given relative to the value coefficient for other inputs into the construction sector

-

a major user of forest sector products.

The table indicates an average decline of 2 percent per year in t h e relative importance of wood deliveries to t h e construction sector. This cieveiopment is, however, subject to one major qualification: what t h e table shows is the development of the ratio ( 4 u b / s b , i.e., the value input-output coefficient for wood into buildings relative to t h e non-wood coefficient.

It can be shown that these figures can be decomposed into a sum of percentage relative price change and percentage relative quantity

T A B U 2. Forest sector construction input divided by other builhng material construction input. (Sources: UN-ECE Standardized Input-Output Tables of ECE Countries, 1865 and 1875. United Nations, 1977 and 1882, New York).

1965 1975 Average annual

Country change (7.)

Australia Austria Canada

Czechoslovakia Denmark West Germ any Italy

Japan

Netherlands Norway Portugal Spain Sweden

United Kingdom

change. Taking this into account the quantitative decline is still there and amounts to approximately one half to one percent per year.

The demand for inputs is rigidly fixed to the scale of operations in most input-output models. In reality choice of techniques is adaptable to prices through substitution. The problem of input substitution is therefore an important subject also in intermediary commodity demand studies.

THE INFLUENCE OF F'EUCES ON CHOICE OF TECHNIQUES AND INJXEZMEDLAKY FOREST PRODUCT DE3MNIl

I have argued above that the demand for forest sector products should be based on intermediary dernand theory.

One approach to this problem is to assume that t h e physical inputs per unit of physical output is determined by profit maximization a t the level of the plant.

We can further assume that each plant is producing an output fiow qj per unit oi time, being constrained by a production function q j = q j ( q l j t . . ~ q = j *

. . .

,qnj). For simplicity we assume this function to be of the CES form, i.e.,

where

qij

=

physical Aow of input i into output j , A , aij, p j

=

parameters representing technology.

A short run maximization of profits, given input prices, p , corresponds to solving the problem

where

5,

a r e t h e observed prices (or expected price, reflecting t h e observed prices).

The conditions of a profit maximum a r e

This implies t h a t t h e input quantities should be adjusted so that:

1 a k

where ckij

= -

^In

-

^; and 1 + p . ₁ %

where

As a special case of this technology representation we have t h e Cobb- Douglas production function. Such a C-D function is a good approxima- tion if an estimated 6 is not statistically significantly different from 1. En that case values of input-output coefficients that remain constant over time is also a necessary consequence (Klein 1974).

There a r e consequently t h r e e types of tests that a r e possible (pro- vided we can make t h e necessary assumptions of a statistical n a t u r e for the estimations). We can test for a) dj t 0; _b) bj # 1; and c)

aj

for one

f

pair of input being different from bj for another pair of inputs.

dj # 0 indicates t h a t price variation could be of importance for t h e choice of technique within the technology available (and portrayed by t h e production function chosen). b j # 1 indicates that we have to rule

out the Cobb-Douglas representation.

2, +

^6, indicates that the func- tional form chosen is not a valid representation of the true technology.

These tests have been performed on a limited set of user sectors defined above, repair residential and non-residential construction, and a limited set of inputs.

Some results of the Canadian economy a r e presented in Table 3. I t is clear from the table t h a t we cannot accept only one of the three hypotheses to be tested: 6,

+

^0. We have, however, no ground whatso- ever to accept the Cobb-Douglas representation and little reason t o accept t h e CES-form. The analysis of intermediate demand should be built on another foundation.

AN ALTEIRNM"M3 APPROACH TO KlTXlEDIATE DE3UND ANALYSLS

In order to accommodate t h e results of t h e testing of the CES pro- duction function we can follow a track of complicating the production function to accommodate different elasticities of substitution and com- plementarities between inputs. Instead, another route has been fol- lowed. I t has been argued by Diewert (1971) and McFadden & Fuss (1978) that a dual approach to input demand could be used. Their argument essentially is that cost functions a r e t h e dual of production functions, although it is often very hard to reconstruct t h e production function underlying a given (optimized) cost function. In some cases the duality is obvious. A unit cosi function of Cobb-Douglas form in input prices has a Cobb-Douglas function a s a dual production function (Nerlove 1965). A cost function, allowing for complements and differences in substitution

TABLE 3. Results for constant elasticity of substitution regressions for construc- tion sectors of Canade based on data for period 1961-1974.

Forestry prod. _]

₌

_{-9.2 In-}

==

^F

_-

_O.1t

In [ Fabr. wood prod. pw

(4) (3)

Repair constr.

Lumber Timber

1

= -0.73 in

-

PLT

-

^0.08t

In [ Fabr. wood prod. P~

(3) (2.0)

Nan res. constr.

Lumber & Timber P ~ m

]

=

-0.6 ln--

-

^0.03t

In [ Fabr. metal prod. P~ _{- -}

(2) (3.0)

Non res. constr.

Lum Timber ]

=

-0.86 in

-

PLT

-

^0.04t

In [ Cement prod. P~~

(3) (5)

Non res. constr.

Lumber br Timber in [ =

Struct. Met. prod.

+

0.03t

p a

(3.7) (1.3)

Res. constr.

[Panels

In ]

=

-1.84 In- 'P"

+

0.03t Iron & s t e e l prod. P~~~

(4.1) (2.3)

Res. constr.

Fabr. wood prod.

] = -1.4 ln-

ln [ Cement prod. Pm

+

0.05t PCP

(2.3) (3.4) Res. constr.

elasticities between different inputs has been proposed by Diewert. This function is also proposed to be used in our study of forest product input demand.

The idea is quite straightforward and goes back to Hotelling (see Fuss and McFadden 1978) and Shephard (1953). It can be illustrated with a simple diagram (Figure 1). Suppose two inputs are free to be varied and that the goal of a plant manager is to minimize costs subject to an exogenously specified quantitative demand level and given input prices.

It is fairly obvious that a slight variation of the factor price ratio a t a given demand ievel will trigger a predictable response in terms of demand for the two inputs.

Shephard's theorem can be stated in the following simplified fashion (Diewert 197 1). Assume that:

(1) A cost function, C, in input prices, p , and demand level, y . (2) C(y , p ) is positive real valued, defined and finite for all p

>>

⁰

and y

>

0.

(3) C(y;p) is nandecreasing, left continuous and tends to infinity for p

>>

0.

(4) C(y ;p ) is nondecreasing in p .

(5) C(y;p) is linearly homogeneous in p for y

>

0.

(6) C(y ; p ) is concave in p .

(7) C(y ;p) is everywhere differentiable in p . Then

lnput 1 not minimized cost

c-

demand level

7

minimum

I

^cost

^{\ \}

lnput 2

FIGURE 1. Cost minimizetion and factor demand.

A proof is given in Diewert ( 197 1).

This theorem, based on .economically reasonable assumptions, states in short that all input demands a r e uniquely determined if the price vector is given.

One econometrically amenable form suggested by Diewert and oth- e r s is, the Generalized Leontief Cost Function

for pi ² 0 ; y

>

⁰ with bij

=

b j t . h(y) could be any continuously and

monotonically increasing function, for instance, h(y)

=

aoYal with ao.

a ,

>

0.

With ao, a,

=

1 we have

In order to generate the input-output coefficients needed for a full interdependency analysis of the economy we only have to divide

zi

by y , i.e..

This permits the immediate formulation of a dual general equilibrium formulation. The whole set of input-output coefficients, depending on relative prices, can be arranged as a matrix of functions A(p)

=

)qj (p){.

This expression can be used to formulate the general equilibrium equa- tion (in t h e dual space). In the sequel we disregard capital and other fixed inputs.

pizi = x p j a j i ( p ) q

+

uriatiz, ; (1

=

1 , .

. .

, n ) revenue input cost labor cost

where

uri =househoid income per unit of labor input

z, =

total output (and input) of commodity i

z,

can be reduced from the system of equations Ap =PA@) + w a L

If the use of labor would be determined in a substitution process, then relative price of labor would also be determined endogenously and the

system could be reformulated:

AP, = p , A , b , )

where p

,

includes the relative wage rates.

This is a non-linear fixed point eigenvalue problem which has a n equilibrium solution (Nikaido 1968). Once solved for the equilibrium p ; and A; the quantitative equilibrium can be easily calculated using inver- sion of t h e equilibrium matrix, A;.

ESIXIWI"I'0N

WHEN DATA ARE

SCARCE

Statistical data good enough for t h e approach suggested above only exist for Norway and Canada. In these two countries time series of input-output tables a r e actually available. In most other countries only time series of unrelaked input and output series a r e available (see Table 4). A series of construction output and furniture output does, for instance, exist in most countries. Similarly sawnwood and panel (apparent) consumption data does exist for most countries and so does the associated price indexes. it is thus fairly straightforward to form a basis for dual approach intermediate demand estimation.

It is possible t o estimate macro-input-output functions

The size of t h e weight 7 can be determined in a full input-output table containing data on t h e relative importance of the furniture sector as a user of sawnwood.

TABLE 4. Hypothetical data for estimation of macro intermediate demand func- tion.

In order to uphold the symmetry constraint information on the use of building metals, cement and labor has to be organized in a form simi- lar to Table 4. Analogous data tables can be assembled for most of t h e commodities of the GTM and most developed countries.

Apparent Year consumption

of sawnwood O xi ( 0 ) 1 z i ( l >

2 2, (2)

t z i ( t >

N z i ( N ) I

Ordinary Least Square is not a reasonable approach in such a con- strained estimation procedure. In the estimations currently done on Canadian data, t h e "Seemingly Unrelated Regression" method has been chosen as a starting point for the econometric work. This work is to be reported in papers in Andersson e t al. (1984).

Forest sector products a r e intermediate commodities. This implies that input-output models or their generalizations into programming models should be used a s forecasting tools. The recent energy crisis has shown that in periods of substantially changing relative prices,

Construction output

~ ~ ( 0 1 Y c ( l ) t/c(2)

vc(t

1

V C ( N )

Furniture output

~ ~ ( 0 1

~ ~ ( 1 1

~ ~ ( 2 1

Y F ( ~ )

V F ( N )

Price of

sawnwood Pi ( 0 ) Pi(1) Pi ( 2 )

P i ( t )

Pi ( N )

Price of labor Pw(0)

Pw ( 2 )

P u ( t )

Pu ( N ) Price

of

cement P8 ( 0 ) P s ( l ) Ps ( 2 )

Ps(t)

I P8 ( N ) Price of metals

Pm ( 0 )

P m ( l ) P w ( l ) Pm ( 2 )

P m ( t )

Pm ( N )

substitution of inputs must be taken into account (see Bjerkhalt e t al.

1983). I t is argued above that combining interdependency analysis with substituion models of the Generalized Leontief Cost Function category is a reasonable methodology. Indeed, it is a superior method a t least if compared t o classical inpout-output modeling o r substitution/interdependency analysis.

REFERENCES

Andersson,

A.E., R.

Briinnlund, and G. Kornai. 1904. The Demand for Forest Sector Products. WP-84-87. Laxenburg, Austria: International Institute for Applied Systems Analysis.

Diewert, W.E. 1971. An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function. Jotmud of fbliticorl Economy, 79(3).

Frenger, P. 1978. Factor Substitution in the Interindustry Model and the use of Inconsistent Aggregation. Chapter V.2 in Fuss and MacFadden (1970).

Fuss,

M.

and D. McFadden (eds). 1970. Fhxiuction Economics: A Dual Approach to i7aeory and AppLications. Amsterdam: North-Holland.

Klein, L. 1974. A R z t b o o k of Econometrics. Englewood Cliffs, N.J.: Pren- tice Hall.

Leontief, W. 1951. The Shuctum of the Anaerkcm Economy 1919-1929, 2nd ed. New York: Oxford Univ. Press.

Nerlwe, M. 1965. Estimartwn and Msntifrcatwn of C o b b - h u g l a s A.ockrc- t i o n k n c t w n s . Amsterdam: North-Holland.

Nikaido H. 1968. Convez S b u c h r r e s and Economic 7heory. New York Academic Press.

Shephard,

R.W.

1953. Cost d Reduction h n c t w n s . Princeton:

Princeton University Press.