Long Run Timber Supply: Price Elasticity, Inventory Elasticity, and the Capital-Output Ratio

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

LONG KUN TMBER

SUPPLY:

PRICE W C l T Y . INVENTORY ELATICITY.

AND THE

CAPITAL-OUTPUT RATIO

Clark S. Binkley

February 1985 WP-85-10

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 FCR QUOTATION WITHOLT PERMISSIOR' OF THE AUTHOR

LONG

RUN TIMBER

SUPPLY: PRICE ELASTICITY, INYENTORY ELATICITY,

AND

THE CAPITAL-OUTPUT RATIO

Clark S . Binkley

F e b r u a r y 1985 WP-85-10

Associate P r o f e s s o r of Forestry

School of F o r e s t r y and Environmental Studies Yale University

205 P r o s p e c t S t r e e t New Haven, CT 065U USA

and

R e s e a r c h S c h o l a r IIASA

A-2361 Laxenburg AUSTRIA

Working Papers are interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have received only lim- ited review. Views o r opinions e x p r e s s e d h e r e i n d o not neces- s a r i l y r e p r e s e n t those of t h e Institute o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

FOREWORD

The objective of t h e Forest S e c t o r P r o j e c t at IIASA is t o study long- term development alternatives f o r t h e f o r e s t s e c t o r on a global basis. The emphasis in t h e P r o j e c t i s 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 f o r f o r e s t policy, f o r e s t industrial s t r a t e g y , and r e l a t e d t r a d e policies.

The key elements of s t r u c t u r a l change in t h e f o r e s t industry are r e l a t e d t o a variety of issues concerning demand, supply, and international t r a d e in wood products. Such issues include t h e growth of t h e global econ- omy and population, development of new wood products and of substitute f o r wood products, f u t u r e supply of roundwood and alternative fiber sources.

development of new technologies f o r f o r e s t r y and industry, pollution regu- lations, cost competitiveness, t a r i f f s and non-tariff t r a d e b a r r i e r s , etc.

The aim of t h e P r o j e c t i s t o analyze t h e consequence of f u t u r e expectations and assumptions concerning such substantive issues.

This a r t i c l e r e p r e s e n t s a n equilibrium analysis of timber supply. Based on a single model of timber production, t h e c h a r a c t e r i s t i c s of supply func- tion have been studied in detail. Such an analysis provides foundations f o r t h e P r o j e c t ' s e f f o r t s t o study long-term development of f o r e s t r e s o u r c e s in market economies.

Markku Kallio Leader

Forest S e c t o r P r o j e c t

ABSTRACT

Timber production r e q u i r e s substantially more capital p e r unit output than d o most economic e n t e r p r i s e s . The quantity of capital deployed depends primarily on t h e rotation length and t h e output p r i c e of stumpage.

In a long r u n timber supply model this gives r i s e t o a "backward bending"

supply curve. This p a p e r summarizes a long r u n model of timber supply, and computes t h e associated p r i c e and inventory elasticities. The role of capi- t a l in timber production i s explored through a continuous time formulation of t h e usual Faustmann point input/point output model. The t h e o r e t i c a l r e s u l t s are illustrated through a n example based on loblolly pine yields.

CONTENTS

INTRODUCTION

1. LONG R U N S U P P L Y , P R I C E ELASTICITY AND MARKET INSTABILITY L a n d o w n e r B e h a v i o r

T i m b e r O u t p u t L o n g R u n S u p p l y M a r k e t I n s t a b i l i t y 2. INVENTORY ELASTICITY 3. CAPITAL : OUTPUT RATIO

4. AN EXAMPLE: S I 80 LOBLOLLY P I N E 5. CONCLUSION

R E F E R E N C E S

-

^{v i i}

-

LONG RUN T m B E R

SUPPLY:

PRICE ELASTICITY.

lHVJ3NTORY ELATICITY.

AND

THE CAPITAL-OUTPUT RATIO

Clark S. Binkley

INTRODUCTION

This p a p e r analyses t h e long r u n supply of timber, with p a r t i c u l a r attention paid t o t h e r o l e of capital. The model is long r u n in t h e s e n s e t h a t t h e time period of t h e analysis i s adequate f o r t h e capital stock t o adjust t o t h e economically optimal steady state level. The period of time n e c e s s a r y f o r t h i s condition t o b e m e t depends on t h e initial a g e s t r u c t u r e of t h e f o r e s t , t h e level of demand and t h e underlying biological productivity of t h e f o r e s t ; i t might r a n g e from l e s s than a decade t o more than a century.

This model of timber supply d a t e s a t l e a s t t o Vaux's (1954) analysis of timber production in California, and h a s been used many times since: f o r t h e Douglas f i r region of t h e United S t a t e s by t h e USDA Forest S e r v i c e (1963) and Hyde (1980). f o r pine in t h e s o u t h e r n US by Robinson (1980) and f o r t h e spruce-fir r e s o u r c e in Maine by Binkley (1983). Jackson (1980) and Hyde (1980) discuss some of t h e t h e o r e t i c a l a s p e c t s of t h e model.

The p r e s e n t analysis i s both n a r r o w e r and d e e p e r than t h e s e o t h e r efforts. The r o t a t i o n a g e i s t h e only decision v a r i a b l e considered. To a g r e a t e x t e n t , t h e r o t a t i o n a g e determines t h e quantity of c a p i t a l used in timber production, s o a n analysis of c a p i t a l logically focuses on t h i s vari- able. In addition, t h e r o t a t i o n length i s p e r h a p s t h e single most important v a r i a b l e determining t h e level of output from a f o r e s t (Davis, 1976). This analysis omits any formal discussion of management intensity o r t h e amount of land devoted t o timber production, although t h e concluding section com- ments on how v a r i a b l e s would influence t h e p r e s e n t r e s u l t s

.

Similarly t h e nontimber p r o d u c t s of t h e f o r e s t ( r e c r e a t i o n , water, environmental ser- vices) which may a f f e c t t h e optimal h a r v e s t period are excluded from

consideration ( s e e Hartman, 1976 and Bowes, Krutilla, and Sherman, 1984 f o r discussions of how t h e p r e s e n c e of valuable nontimber forest p r o d u c t s affects t h e analysis). This narrow p e r s p e c t i v e i s adopted to focus on t h e r o l e of c a p i t a l in timber production, and i s consistent with, for example, Samuelson's (1976) t r e a t m e n t of t h e problem.

The r e m a i n d e r of t h e p a p e r is in f o u r p a r t s and a concluding comment.

The f i r s t section d e t a i l s t h e long r u n supply model. This section shows t h a t t h e p r i c e elasticity of supply may b e negative, indicating a backward bend- ing supply c u r v e f o r timber. Instability may a r i s e in t h e long r u n market equilibrium because of t h i s unusual supply behavior.

Many s h o r t r u n models of timber supply use t h e level of timber inven- t o r y t o explain h a r v e s t levels. Section 2 below compares t h e inventory elasticity implied by t h e long r u n model with t h e r e s u l t s f r o m t h e s e studies.

The f o u r t h section examines t h e capital/output r a t i o for t h e steady state forest, and c l a r i f i e s t h e importance of capital in timber production

The fifth section u s e s yield information for loblolly pine grown in t h e s o u t h e a s t e r n United States to demonstrate t h e p r a c t i c a l significance of t h e t h e o r e t i c a l r e s u l t s .

1. LONG RUN SUPPLY. PRICE ELASTICITY AND -T INSTABILITY

The long r u n timber supply model h a s two p a r t s . The first d e s c r i b e s t h e r o t a t i o n decisions of f o r e s t owners as a function of timber p r i c e and o t h e r r e l e v a n t p a r a m e t e r s . This decision determines t h e level of timber production. The second explains precisely how t h e amount of timber pro- duced depends on t h e r o t a t i o n a g e , and t h e r e f o r e on p r i c e .

Landowner Behavior

The f o r e s t owner s e l e c t s t h e r o t a t i o n a g e which maximizes t h e n e t p r e s e n t value n of timber r e c e i p t summed o v e r a n infinite planning horizon.

Capital m a r k e t s are p e r f e c t s o t h e f o r e s t owner c a n lend and borrow at a constant, known i n t e r e s t rate i. (Equivalently, land markets p e r f e c t l y r e f l e c t t h e p r e s e n t value of partially grown stands). Timber yield v p e r unit area is a known function of s t a n d a g e t ; t h e yield function does not change o v e r time. Regenerating a stand costs c p e r unit a r e a , a n amount which i s constant through time. Lastly, t h e even-aged f o r e s t is r e g e n e r a t e d promptly a f t e r clearcutting if i t i s p r o f i t a b l e to d o so.

These assumptions imply t h e r o t a t i o n problem is s t a t i o n a r y , so t h e f o r e s t owner solves

max n ( t )

=

--c + p v ( t ) e 4 :

+

n ( t ) e 4 : 1.1

t

The stumpage p r i c e p is endogenous. The optimal r o t a t i o n a g e t*(p), and t h e r e f o r e t h e level of output, v a r i e s with p r i c e level.

The f i r s t o r d e r optimality conditions f o r t* (p) can easily b e found (see d n

Jackson, 1980; Hyde, 1980; o r Chang, 1983) by solving

- =

0.

d t

d v

where v

= -

d t

Timber Output

Given t h e optimal r o t a t i o n a g e , how much timber i s produced? In t h e long r u n , c a p i t a l c a n a d j u s t to t h e economically d e s i r a b l e level. F o r timber production t h i s implies t h a t t h e f o r e s t h a s no timber o l d e r t h a n t*

0).

and e a c h y e a r all timber r e a c h i n g t h i s a g e i s harvested. Averaged o v e r a r o t a - tion, t h e annual output of a f o r e s t of area A i s Av It* @ ) ] / t* @). Without loss of generality t h i s p a p e r t a k e s A = 1 , s o t h e supply function i s

A "fully regulated" f o r e s t produces identically t h e long r u n a v e r a g e annual output e a c h y e a r . This i s achieved by a n a r r a n g e m e n t of a g e c l a s s e s s o t h a t e a c h occupies a n area equal t o A / t*

.

Before continuing, i t will b e helpful l a t e r in t h e analysis t o note t h a t supply i s maximized at t h e r o t a t i o n a g e which satisfies

Equation 1.4 i s simply a r e s t a t e m e n t of t h e f o r e s t e r ' s familiar maxim t h a t a v e r a g e annual output i s maximized when c u r r e n t annual increment ( v ) equals mean annual increment (v /

t ) .

I t i s n a t u r a l t o call t h e r o t a t i o n which satisfies 1.4 t h e maximum sustained yield (MSY) rotation. F o r typical timber yield functions, r o t a t i o n s are l e s s t h a n MSY if a n only if v / v

>

^{1/ t}

.

Long Run Supply

The supply model is illustrated in Figure 1. The t o p panel d e p i c t s t h e optimal r o t a t i o n condition 1.2. The lower panel shows forest growth/steady state supply, equation 1.3. Given all t h e p a r a m e t e r s of t h e model e x c e p t p r i c e , t h e supply c u r v e i s defined by t h e following algorithm. F o r a given p , 1.2 i s solved ( u p p e r panel of Figure 1) to find t h e optimal r o t a t i o n age.

Long r u n supply t h e n c a n b e found by 1.3 ( t h e lower panel of Figure 1). The p r o c e s s i s i t e r a t e d for v a r i o u s p r i c e levels until t h e supply c u r v e is identi- fied with suitable precision.

The supply function c a n a l s o b e developed in a n o t h e r way

.

F i r s t solve 1.2 for p t o link p r i c e d i r e c t l y with t h e optimal r o t a t i o n a g e

F'IGUm I. Optimal r o t a t i o n and t i m b e r s u p p l y .

Given a yield model v ( t ) , c o s t and i n t e r e s t rate, t h e supply c u r v e c a n b e constructed by examining a s e r i e s of r o t a t i o n a g e s . F o r e a c h r o t a t i o n a g e , 1.5 gives t h e p r i c e which makes t h a t r o t a t i o n optimal, and 1.3 gives t h e sup- ply at t h a t p r i c e .

The p r o c e d u r e i s illustrated in Figure 2. The N W q u a d r a n t labelled t* (p ) plots 1.5. The S W q u a d r a n t plots 1.3. The S E q u a d r a n t contains a 45"

t r a n s f e r line to map t h e a v e r a g e output from t h e S W q u a d r a n t onto t h e quantity a x i s of t h e supply c u r v e , s (p), which i s shown in t h e N E quadrant.

Figure 2 shows t h e construction of t h e supply c u r v e for t h r e e impor- t a n t cases. In case (a), t h e p r i c e i s s o low t h a t -rr

=

0, and no long-run pro- duction t a k e s place. Case (b) o c c u r s at MSY. Binkley (1985) h a s shown t h a t t h e MSY r o t a t i o n o c c u r s at a p r i c e of

FIGURE 2 Long run timber supply.

Case (c) o c c u r s at t h e quantity asymptote of t h e supply c u r v e . To see t h i s asymptote, refer t o 1.2. Note t h a t as p -, m, t h e r a t i o of c / p a p p r o a c h e s 0. The l e f t hand side of 1.2 a p p r o a c h e s t; / v , and t h e supply c u r v e grows increasingly inelastic at a p r i c e determined by t h e i n t e r e s t r a t e and t h e biological productivity of t h e f o r e s t .

From Figure 2, i t is c l e a r t h a t t h e supply c u r v e c a n have a negative slope. In general, what i s t h e p r i c e elasticity of supply implied by t h i s model? By definition, t h e p r i c e elasticity is

Since

6s 7;t

-

v

- - -

d t t 2 ^{, and}

d t

I t i s well known t h a t

-

is negative, s o t h e sign of t h e long r u n supply elasticity depends on t h e sign of d~ v / v

-

^1/ t

.

If t h e p r i c e level is such t h a t v / V <1/ t , t h e n t h e p r i c e elasticity of supply is positive. Recall t h a t t h i s condition obtains only if t h e optimal r o t a t i o n a g e is g r e a t e r t h a n t h e M S Y rotation. Consequently, only if t h e optimal r o t a t i o n is longer t h a n MSY will t h e long r u n supply c u r v e h a v e t h e usual positive slope. Otherwise, t h e sup- ply c u r v e will have a negative slope. The "backward bending" supply phenomenon h a s been noted by Clark (1976) f o r t h e case of f i s h e r i e s , and in passing by Hyde (1980) f o r t h e case of timber (although his empirical exam- ples d o not r e v e a l t h i s situation).

Before turning t o questions of m a r k e t equilibrium, note t h a t if p r i c e s are s o low t h a t rr

<

0 , no production at all will o c c u r in t h e long r u n . High enough c o s t s on i n t e r e s t rates c a n c l e a r l y lead t o a situations where all production o c c u r s on t h e negatively sloped p a r t of t h e supply c u r v e . Thus, unlike Clark's (1976) f i s h e r i e s example, t h e e n t i r e long r u n timber supply c u r v e might h a v e a negative slope. This o c c u r s because inputs are r e q u i r e d t o p r o d u c e timber, where Clark (1976) t a k e s t h e f i s h e r y t o b e wholly self reproducing.

ARarket Instability

For a n open access f i s h e r y , Clark (1976) points out t h a t t h e backward bending supply c u r v e c a n lead t o unstable m a r k e t equilibria. Figure 3 shows t h e situation f o r competitive timber supply with t h r e e levels of demand. The lowest level of demand, D l , c o r r e s p o n d s t o t h e usual s o r t of m a r k e t equilibrium, a n d t h e stability of E l depends on well-known elasticity and adjustment conditions. A t a slightly h i g h e r level of demand, D 2 , t h r e e m a r k e t equilibria e x i s t , and i t i s e a s y t o see t h a t E t t z i s unstable. S h o r t r u n timber demand i s thought t o b e v e r y inelastic. Long r u n demand i s likely t o b e more e l a s t i c but if i t remains fairly inelastic, t h e n t h e p r i c e level asso- ciated with t h e equilibrium point E"I2 could b e much h i g h e r t h a n t h a t asso- c i a t e d with EIz. The m a r k e t instability implied by t h i s analysis would t h e n t r a n s l a t e into d r a m a t i c p r i c e instability. Finally, if t h e m a r k e t equilibrium settles at E t t t 2 , t h e i n c r e a s e in demand from Dl t o Dz i s accompanied by a d e c r e a s e in consumer's surplus.

FIGURE 5. Instability in long-run timber markets.

2. INYENTORY ELASTICITY

In t h i s supply model, timber supply i s implicitly a function of timber inventory level. Many forest sector m o d e l s use t h e level of timber inven- t o r y as a determinant of timber supply behavior (e.g. Adams and Haynes, 1980; Binkley and Cardellichio, 1985). Consequently i t i s of some i n t e r e s t to examine how supply r e s p o n d s to inventory level in t h i s long r u n model.

The inventory of a unit a r e a , steady s t a t e forest is

This inventory c a n b e i n c r e a s e d in t h r e e g e n e r a l ways. The f i r s t two cases r e f l e c t exogenous changes in t h e inventory, where t h e t h i r d i n c o r p o r a t e s endogenous inventory changes.

The f i r s t way t o alter t h e inventory adds to t h e f o r e s t a n o t h e r unit of land which i s identical to t h a t a l r e a d y in production. Alternatively, t h e yield function c a n b e i n c r e a s e d by a constant f r a c t i o n at all ages. In both cases i t is obvious t h a t t h e inventory elasticity i s 1 because both supply (1.3) and inventory (2.1) are augmented by precisely t h e same amount.

The t h i r d , more interesting, case alters t h e inventory endogenously by changing some p a r a m e t e r

-

^{i ,} c or p

-

s o t h a t t h e optimal r o t a t i o n a g e changes. An "apparent" inventory elasticity of supply c a n b e calculated.

This elasticity i s termed a n "apparent elasticity" because both t h e change in supply and t h e change in inventory are due t o t h e exogenous change in some o t h e r model p a r a m e t e r .

By definition t h e inventory elasticity EI is

which c a n b e r e w r i t t e n

From 2.1.

d s

Substituting t h e value of

-

f r o m 1.8 and r e a r r a n g i n g gives d t

Because

>

^0, v ( t ) t

>

^Jv ( z ) d z f o r all values of t , and t h e numerator of

0

t h e second t e r m in 2.5 i s positive. Thus t h e sign of t h e a p p a r e n t inventory elasticity depends on t h e sign of t h e t e r m in b r a c k e t s . This t e r m i s positive if t*

<

^{W ,} z e r o if t*

=

UTY, and negative if t*

>

UTY. The a p p a r e n t inventory elasticity i s positive for s h o r t r o t a t i o n , falls to z e r o a t MSY and t h e n becomes negative.

A t any point along t h e long r u n supply c u r v e , t h e p r i c e and inventory elasticities will h a v e opposite signs ( e x c e p t at hEY where t h e y are both identically zero). Most supply studies t a k e both elasticities t o b e positive.

Timber supply studies which employ inventory as a n independent vari- a b l e generally use o n e of t w o a p p r o a c h e s t o estimate t h e r e q u i s i t e elasti- city. F i r s t , because time s e r i e s d a t a on timber inventory levels are fre- quently p o o r (and inventory would probably change only gradually o v e r time in any c a s e ) , i t sometimes i s not possible to obtain usuable s t a t i s t i c a l estimates f o r a n inventory term. In such cases t h e supply v a r i a b l e c a n b e recaste as t h e r a t i o of h a r v e s t t o inventory, and t h e inventory v a r i a b l e omitted from t h e independent variables. This specification implicitly con- s t r a i n s t h e inventory elasticity t o b e unity. To see t h i s , consider a supply function specified as

where X i s v e c t o r of nonprice independent v a r i a b l e s thought t o a f f e c t sup- ply. The inventory elasticity of supply in this model is

In some US regions, Adams and Haynes (1980) u s e t h i s specification f o r softwood timber supply. The Data Resources, Inc. FORSIM softwood s e c t o r model uses t h i s specification in all regions as does t h e model of t h e US hardwood lumber s e c t o r developed by Binkley and Cardellichio (1985).

Using a unitary inventory elasticity is consistent with t h e f i r s t two kinds of inventory elasticities discussed above.

F o r some regions, Adams and Haynes (1980) were a b l e t o estimate soft- wood stumpage supply equations with inventory as a n independent variable.

For t h e regions where this w a s possible, they obtained estimates ranging from 0.2 t o 1.46, with p e r p o n d e r a n c e of values n e a r 0.5. A s shown in sec- tion 4, t h e s e r e s u l t s are not empirically inconsistent with t h e t h i r d case examined if timber r o t a t i o n s are l e s s than iUSY.

3.

CAPITAL

: OUTPUT RATIO

Because of t h e long time period involved in f o r e s t production, capital is a c r i t i c a l input. The capital stock r e q u i r e d f o r a steady state f o r e s t c a n b e measured in s e v e r a l ways, and t h e p r e s e n t analysis uses p e r h a p s t h e most conservative definition.

This definition c a n b e aeveloped most easily using a continuous time model of t h e timber production process. In each period t h e n e t income f o r a unit of f o r e s t c a n b e decomposed into t h r e e p a r t s :

pt;

=

g r o s s income 3.la

zpv

=

opportunity cost of t h e growing stock 3 . l b

r

=

land r e n t 3 . l c

In t h i s context, t h e n e t p r e s e n t value function c a n b e written as

t t

n(2)

=

- c

+

~ v ( z ) e 4 " d z - J i p ~ ( z ) e - ( ~ d z

0 0

-

J r e 4 " d z t 0

Before using t h e s e definitions t o d e r i v e t h e capita1:output r a t i o , l e t us show t h a t 3.2 i s equivalent t o 1.1. F i r s t , i n t e g r a t e t h e f i r s t integral in 3.2 by p a r t s

Now substitute 3.3 into 3.2 t o g e t t

~ ( t )

=

^{- c} + p v ( t ) e " + J r e 4 ' d z

0

Efficient land markets imply t h a t r adjusts s o T = 0 (see Samuelson, 1976 on this point). Integrating t h e last t e r m of 3.4 and imposing this condi- tion implies

The term on t h e l e f t hand side of 3.5, r ^{/ t} , is simply t h e capitalized value of land r e n t s , and c o r r e s p o n d s t o t h e economic r e n t w e seek t o maximize in 1.1. Thus 4.2 i s precisely equivalent t o t h e original problem. Casting t h e problem as a continuous input/continuous output problem gives precisely t h e same r e s u l t s as t h e more conventional point input/point output formula- tion.

The continuous formulation i s useful because i t highlights t h e r o l e of capital in timber production. In t h i s context, 4 . l b comprises t h e most lim- ited definition of capital possible. That is, t r e a t regeneration c o s t s as

"labor", and land r e n t a l c o s t s "land" (although land h a s adequate durability t o be viewed as a form of capital).

In t h e steady s t a t e f o r e s t t h e c u r r e n t annual value of t h e a v e r a g e cap- ital deployed k i s

Capitalized o v e r perpetuity at rate i , t h e capital deployed in a steady state f o r e s t becomes

t

k

=

Jv (z )dz 3.7

0

By 1.3, t h e annual income from t h e f o r e s t i s

The c a p i t a l : output r a t i o k / y is then

t

Jv (2 )dz

Suppose t h a t t h e discount r a t e i increases. How does t h e capital/output r a t i o respond? The rotation will d e c r e a s e ( s e e , f o r exam- ple, Chang, 1983), and by 2.4 t h e value of t h e capital embodied in t h e inven- t o r y of growing stock will decline. Output may i n c r e a s e o r d e c r e a s e with t h e change in rotation, s o t h e direction of t h e change in k / y i s ambiguous.

The loblolly pine example developed below shows t h a t f o r most rotations of d- k

i n t e r e s t ,

-

i s positive, s o i n c r e a s e s in i n t e r e s t rates will lead t o reduc- d l

tion in t h e use of c a p i t a l p e r unit output.

4. AN EXAMPLE: SI 80 LOBLOLLY PINE

While t h e foregoing t h e o r e t i c a l analysis provides some definitive r e s u l t s concerning t h e n a t u r e of t h e long r u n supply c u r v e f o r timber. t h e p r a c t i c a l importance of some of t h e theoretical c o n c e r n s is not readily a p p a r e n t . To provide a modest d e g r e e of empirical insight into t h e n a t u r e of this timber supply model, this section p r e s e n t s a n example using yield information f o r S I 80 loblolly pine.

To ease t h e numerical burden of t h e example, t h e cubic f o o t / a c r e yields given by Schumacher and Coile (1960) w e r e f i t t o a two-parameter yield equation

The numbers in p a r e n t h e s e s a r e t h e

t

-statistics f o r t h e null hypothesis t h a t t h e associated coefficient i s z e r o . For t h e purposes of this example, ordi- n a r y l e a s t s q u a r e s r e g r e s s i o n produced an a c c e p t a b l e f i t t o t h e yield table.

For this example, r e g e n e r a t i o n c o s t s $50/acre and t h e i n t e r e s t rate equais 0.025. Binkley (1985) h a s shown t h a t f o r t h e long r u n supply c u r v e t o have any positively sloping portion, t h e i n t e r e s t rate must b e l e s s than t h e inverse of t h e

MW

rotation. Applying 1.4 t o 4.1 shows a

MW

rotation of 23.5 y e a r s , s o t h e i n t e r e s t r a t e must b e l e s s than 1/23.5=0.043 t o show t h e general case of where t h e supply c u r v e h a s f i r s t a positive slope at low p r i c e s then a negative slope at h i g h e r prices. A comparatively low r a t e of i

=

0.025 w a s t a k e n in o r d e r to i l l u s t r a t e t h e g e n e r a l principles involved.

Indeed i t i s of some i n t e r e s t t h a t more r e a l i s t i c i n t e r e s t rates would pro- duce supply c u r v e s which a r e e i t h e r almost perfectly inelastic o r slope backwards throughout t h e i r e n t i r e range.

Figure 4 shows t h e long r u n supply c u r v e f o r t h i s situation. For r o t a - tion a g e s ranging in 1 y e a r increments from 1 to 50, 1.5 w a s solved f o r t h e p r i c e which makes t h a t rotation a g e optimal (for

t <

1 9 , t h e

p ( t * ) <

0).

Supply a t e a c h p r i c e w a s determined from 1.3. Figure 4 plots t h e r e s u l t s of t h e s e calculations. The initial p a r t of t h e c u r v e slopes upward until t h e p r i c e r i s e s t o t h e point t h a t h&SY i s reached. Under t h e assumptions of t h i s example, U m o c c u r s at a p r i c e of 0.082/cf, o r about $6.40/cd. The c u r v e is asymptotic at a quantity of about 103.5 c f / a c / y r , o r at a supply level which is about 98% of t h e

MW

supply. For

p <

0.0360, n

<

O and no long r u n production o c c u r s . This p r i c e c o r r e s p o n d s t o a n optimal r o t a t i o n a g e of 30 y e a r s .

102 103 104 105 106 S(P)

103.5 MSY cf/a/yr

FIGURE 4. Long-run supply, SI 00 Loblolly pine (c = $50/a, t = 0.025).

Figure 5 plots t h e p r i c e elasticity as a function of p r i c e level. This c u r v e w a s d e r i v e d numerically from 1.5 and 1.9. Nowhere i s t h e supply c u r v e v e r y elastic. I t i s p e r f e c t l y inelastic at MSY a n d again at t h e quantity asymptote.

Figure 6 shows t h e "apparent" inventory elasticity as a function of t h e r o t a t i o n age. A t t h e quantity asymptote, t h e a p p a r e n t inventory elasticity i s 2.88, falling t o 0.0 at Urn and t o -5.7 when timber production i s no l o n g e r economic. The empirical r e s u l t s from t h e short-run timber supply studies cited in section 2, above, fall close t o t h e p r e s e n t r e s u l t s n e a r Urn.

Finally, Figure 7 shows t h e c a p i t a l : output r a t i o as a function of t h e r o t a t i o n age. A t Urn, t h e r a t i o i s about 1 0 and at t h e quantity asymptote t h e r a t i o i s about 7. In 1980, t h e r a t i o of r e p r o d u c i b l e fixed assets to value added for all US manufacturing industry w a s 0.34 (UN, 1981, Tables u s 2.15 and u s 4.3). Even using t h e p e r h a p s t h e m o s t r e s t r i c t e d definition of capi- tal possible, timber production r e q u i r e s two o r d e r s of magnitude more capi- t a l p e r unit output t h a n d o e s t h e US industrial s e c t o r t a k e s as a whole.

FIGURE 5. Long-run p r i o e e l a s t i o i t y , Loblolly pine (c = $50, .2 = 0.025).

FIGURE 6. Inventory eiasticitp, Lobioily pine (c = $50/a, i = 0.025).

i s positive. A s one would Throughout the range shown in Figure 7,

-

_{d t}

e x p e c t , increases in capital c o s t s will lead t o fiss capital used p e r unit of output. Until a g e 7 the converse i s true, however.

0 I I

10 i 0

t

^{30 40 t}

MSY

FIGURE 7. Capital : output ratio, SI 80 Loblolly pine (c = 150/a, t = 0.025).

5. CONCLUSION

C ~ p i t a l is a major component of f o r e s t production costs. By determin- ing t h e amount of growing stock inventory, t h e rotation length l a r g e l y d e t e r n i n e s t'ne quantity of capital used in a timber ente-rprise. The r o t a t i o n a g e a l s o strongly influences t h e a v e r a g e output f r o m t h e f o r e s t . High stun?- page p r i c e s imply not only t h a t t h e output from t h e f o r e s t h a s a high value, but a l s o t h a t c a p i t a l in t h e form of growing stock h a s a high opportunity cost. A t high p r i c e s i t i s optimal t o c o n s e r v e on t h e u s e of capital, and t h e r e f o r e t o r e d u c e t h e growing stock inventory by reducing t h e r o t a t i o n age. This kind of c a p i t a l conservation c a n also r e d u c e f o r e s t growth. In t h e long r u n , timber supply c a n t h e r e f o r e fall as a consequence of h i g h e r timber p r i c e s . Higher output p r i c e s inevitably mean h i g h e r c a p i t a l c o s t s , and t h e timber supply c u r v e bends backward as a r e s u l t of t h e necessity t o r e d u c e t h e s e costs. Market instability may r e s u l t from t h i s unusual c o s t s t r u c t u r e f o r timber production.

These conclusions are of c o u r s e s t r i c t l y correct only u n d e r t h e many assumptions n e c e s s a r y f o r such concise and unambiguous r e s u l t s . Two important m a r k e t adjustments have been ignored in t h e analysis: changes in management intensity and changes in t h e area of land devoted t o timber pro- duction. Both kinds of adjustments make timber supply m o r e e l a s t i c t h a n found h e r e , p a r t i c u l a r l y at l o w p r i c e s where t h e r e i s much latitude f o r intensifying silvicultural p r a c t i c e s o r f o r extending t h e intensive and extensive margins of timber production. A s a f o r e s t sector develops, how- e v e r , t h e opportunity f o r t h e s e adjustments will diminish, and capital will tend t o dominate t h e timber production c o s t s t r u c t u r e . The s t r u c t u r a l problems f o r t h e s e c t o r associated with inelastic o r backward bending sup- ply will t h e n a p p e a r .

REFEBENCES

Adams, D., and R. Haynes. 1980. The 1980 Softwood Timber Assessment Market Mode!: S t r u c t u r e , Projections, and Policy Simulations. For.

Sci. Mono 22.

Binkley, C.S. 1983. Long Run Timber Supply f o r t h e Spruce-Fir Region of Maine. P r e p a r e d f o r t h e James Sewall Co., Old Town, ME.

Binkley, C.S. and P.A. Cardellichio. 1985. The US Hardwood Assessment Market Model. d r a f t ms.

Sowes, M.D., J.V. Krutilla, and P.B. Sherman. 1984. F o r e s t Management f o r Increased Timber and Water Yield. Water Resources Research 20:655- 663.

Chang, S. J . 1983. Rotation Age, Management Intensity and t h e Economic F a c t o r s of Timber Production: Do Changes in Stumpage P r i c e , I n t e r e s t Rate, Regeneration Cost and F o r e s t Taxation Matter. Forest Science 29:267-277.

Clark. C.W. 1976. Mathematical Bioeconomics. New York: John Wiley &

Sons.

Davis. K. 1974. Forest Management. New York: McGraw-Hill.

Hantman, R. 1976. Harvesting Decision When a Standing F o r e s t Has Value.

Economic I n q u i r y 14:52-58.

Hyde, W.F. 1980. Timber S u p p l y , L a n d Allocation a n d Economic

m-

c i e n c y . Baltimore: Johns Hogkins P r e s s .

Jackson, D.W. 1980. Microeconomics of t h e Timber I n d u s t r y . Boulder:

Westview P r e s s .

Robinson, V.S., A.A. Montogomery, and J.D. S t r a n g e . 1981. Economic Growth Goal f o r Timber in t h e Southeast. For. R o d . J., 31:69-76.

Samuelson, P.A. 1976. Economics of F o r e s t r y in a n Evolving Society.

Economic I n q u i r y 14:466-492.

Schumachon, F.X. and T.S. Coile. 1960. G r o w t h a n d E e l d of N a t u r a l S t a n d s of t h e S o u t h e r n P i n e s . Durham, NC: T.S. Coile. Inc.

United Nations. 1981. -81 Yearbook of N a t i o n a l A c c o u n t s , v. 11. N e w York: UN Publications.

USDA F o r e s t S e r v i c e . 1963. Timber Trends in Western Oregon and Western Washington. P a c . N. W. F o r e s t & Range Exp. S t a t . Res. Pap. PNW-5.

Vaux, H.J. 1954. Economics of Young-Growth S u g a r P i n e Resources. Univ.

Calif./Berkeley, Div. of Ag. Sci. Bull. 78.