Long Run Timber Supply: Price
Elasticity, Inventory Elasticity, and
the Capital-Output Ratio
Binkley, C.S.
IIASA Working Paper
WP-85-010
February 1985
Binkley, C.S. (1985) Long Run Timber Supply: Price Elasticity, Inventory Elasticity, and the Capital-Output Ratio. IIASA
Working Paper. IIASA, Laxenburg, Austria, WP-85-010 Copyright © 1985 by the author(s). http://pure.iiasa.ac.at/2690/
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W O R K I N G PAPER
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
International Institute
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
February 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
Research Scholar
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 limited review. Views o r opinions expressed herein do not necessarily 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 longterm 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 economy 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 regulations, 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 function 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 capit 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
Landowner Behavior
Timber Output
Long Run Supply
Market Instability
2.
INVENTORY ELASTICITY
3.
CAPITAL : OUTPUT RATIO
4.
AN EXAMPLE: S I 80 LOBLOLLY P I N E
5.
CONCLUSION
REFERENCES
- vii -
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 sense t h a t
t h e time period of t h e analysis is 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 necessary
f o r this 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 less than a decade t o more than a century.
This model of timber supply dates a t least t o Vaux's (1954) analysis of
timber production in California, and has 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 southern 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 theoretical a s p e c t s of t h e model.
The p r e s e n t analysis i s both narrower and d e e p e r than t h e s e o t h e r
efforts. The rotation a g e is t h e only decision variable considered. To a
g r e a t extent, t h e rotation a g e determines t h e quantity of capital used in
timber production, s o a n analysis of capital logically focuses on t h i s variable. In addition, t h e rotation length is p e r h a p s t h e single most important
variable 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 comments on how variables would influence the present r e s u l t s . Similarly t h e
nontimber products of t h e f o r e s t (recreation, water, environmental services) which may a f f e c t t h e optimal harvest period are excluded from
consideration (see Hartman, 1976 and Bowes, Krutilla, and Sherman, 1984
f o r discussions of how t h e presence of valuable nontimber forest products
affects t h e analysis). This narrow perspective i s adopted to focus on t h e
r o l e of capital in timber production, and i s consistent with, for example,
Samuelson's (1976) treatment of t h e problem.
The remainder 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 details 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 be negative, indicating a backward bending 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 invent 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 clarifies t h e importance of capital in timber production
The fifth section uses yield information for loblolly pine grown in t h e
southeastern United States to demonstrate t h e p r a c t i c a l significance of t h e
theoretical results.
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 describes
t h e rotation 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 relevant parameters. This decision determines t h e level of timber
production. The second explains precisely how t h e amount of timber produced depends on t h e rotation a g e , and t h e r e f o r e on price.
Landowner Behavior
The f o r e s t owner selects t h e rotation 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 markets are p e r f e c t s o t h e f o r e s t owner can lend and borrow at a
constant, known i n t e r e s t rate i. (Equivalently, land markets perfectly
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 stand 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 profitable to do so.
These assumptions imply t h e rotation problem is stationary, so t h e
f o r e s t owner solves
max n ( t )
t
= --c + p v ( t ) e 4 : + n(t)e4:
1.1
The stumpage p r i c e p is endogenous. The optimal rotation 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
dn
Jackson, 1980; Hyde, 1980; o r Chang, 1983) by solving - = 0.
dt
where v
dv
=dt
Timber Output
Given t h e optimal rotation a g e , how much timber i s produced? In t h e
long r u n , capital can adjust to t h e economically desirable level. For 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 than t* 0).and
e a c h y e a r all timber reaching t h i s a g e i s harvested. Averaged o v e r a rotation, 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 this 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 arrangement of a g e classes
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 rotation a g e which satisfies
Equation 1.4 i s simply a restatement 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 ) . It i s natural t o call t h e rotation which
satisfies 1.4 t h e maximum sustained yield (MSY) rotation. For typical
timber yield functions, rotations are less than MSY if a n only if v / v > 1/ t .
Long Run Supply
The supply model is illustrated in Figure 1. The top panel depicts t h e
optimal rotation 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. For a given
p , 1.2 i s solved (upper panel of Figure 1) to find t h e optimal rotation age.
Long r u n supply then can b e found by 1.3 (the lower panel of Figure 1). The
p r o c e s s i s i t e r a t e d for various p r i c e levels until t h e supply c u r v e is identified with suitable precision.
The supply function c a n also 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 directly with t h e optimal rotation age
.
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 ) , cost and i n t e r e s t rate, t h e supply c u r v e can b e
constructed by examining a s e r i e s of rotation ages. F o r each rotation a g e ,
1.5 gives t h e p r i c e which makes t h a t rotation optimal, and 1.3 gives t h e supply 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 quadrant labelled
t* (p ) plots 1.5. The SW quadrant plots 1.3. The S E quadrant 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 SW quadrant 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 import 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 production 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 rotation 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 curve. To see this
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 approaches
0. The left hand side of 1.2 approaches t; / v , and the supply c u r v e grows
increasingly inelastic at a p r i c e determined by t h e interest r a t e and t h e
biological productivity of t h e forest.
From Figure 2, i t is c l e a r t h a t t h e supply c u r v e can have a negative
slope. In general, what is t h e p r i c e elasticity of supply implied by this
model? By definition, t h e p r i c e elasticity is
Since
-6s- dt
7;t
-v
t2
, and
I t i s well known t h a t
dt
is negative, s o t h e sign of t h e long r u n supply
d~
-
elasticity depends on t h e sign of v / v
1/ t . If t h e p r i c e level is such t h a t
v / V <1/t , then t h e p r i c e elasticity of supply is positive. Recall t h a t this
condition obtains only if t h e optimal rotation a g e is g r e a t e r than t h e MSY
rotation. Consequently, only if t h e optimal rotation is longer than MSY will
t h e long r u n supply c u r v e have t h e usual positive slope. Otherwise, t h e supply 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 fisheries, and in
passing by Hyde (1980) f o r t h e case of timber (although his empirical examples do not r e v e a l this situation).
Before turning t o questions of market 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 run. High
enough c o s t s on i n t e r e s t rates c a n clearly 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 curve. Thus,
unlike Clark's (1976) fisheries example, t h e e n t i r e long r u n timber supply
c u r v e might have a negative slope. This o c c u r s because inputs are
r e q u i r e d t o produce timber, where Clark (1976) t a k e s t h e fishery t o b e
wholly self reproducing.
ARarket Instability
For a n open access fishery, Clark (1976) points out t h a t t h e backward
bending supply c u r v e can lead t o unstable market 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 , corresponds t o t h e usual s o r t of
market equilibrium, and t h e stability of E l depends on well-known elasticity
and adjustment conditions. A t a slightly higher level of demand, D2, t h r e e
market equilibria exist, and i t i s easy 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 very inelastic. Long r u n demand i s likely t o
b e more elastic but if i t remains fairly inelastic, then t h e p r i c e level associated with t h e equilibrium point E"I2 could b e much higher than t h a t associated with EIz. The market instability implied by this analysis would t h e n
t r a n s l a t e into dramatic p r i c e instability. Finally, if t h e market 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 this 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 invent 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 responds to inventory level in this long r u n model.
The inventory of a unit a r e a , steady s t a t e forest is
This inventory can b e increased in t h r e e general 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 already in production. Alternatively, t h e
yield function can b e increased by a constant fraction 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
i , c or p - s o t h a t t h e optimal rotation a g e
changing some p a r a m e t e r
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 parameter.
-
By definition t h e inventory elasticity EI is
which can b e rewritten
From 2.1.
Substituting t h e value of
Because
ds
f r o m 1.8 and r e a r r a n g i n g gives
dt
> 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 brackets. 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 rotation, falls to z e r o a t MSY and
then 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 have opposite signs (except at hEY where they 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 varia 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 requisite elasticity. First, because time s e r i e s d a t a on timber inventory levels are frequently p o o r (and inventory would probably change only gradually o v e r
time in any case), i t sometimes i s not possible to obtain usuable statistical
estimates f o r a n inventory term. In such cases t h e supply variable can 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 variable
omitted from t h e independent variables. This specification implicitly cons t r a i n s t h e inventory elasticity t o b e unity. To see this, consider a supply
function specified as
where X is v e c t o r of nonprice independent variables thought t o a f f e c t supply. The inventory elasticity of supply in this model is
In some US regions, Adams and Haynes (1980) use 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.
For some regions, Adams and Haynes (1980) were able t o estimate softwood 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 perponderance of values n e a r 0.5. A s shown in section 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 rotations are less 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 critical input. The capital stock required f o r a steady state f o r e s t can
b e measured in s e v e r a l ways, and t h e present analysis uses p e r h a p s t h e
most conservative definition.
This definition can b e aeveloped most easily using a continuous time
model of t h e timber production process. In each period t h e net income f o r
a unit of f o r e s t can b e decomposed into t h r e e p a r t s :
= gross income
zpv = opportunity cost of t h e growing stock
3.lb
r = land r e n t
3.lc
pt;
3.la
In this context, t h e n e t p r e s e n t value function can b e written as
t
t
n(2) = - c + ~ v ( z ) e 4 " d z - J i p ~ ( z ) e - ( ~ d z
0
0
t
- Jre4"dz
0
Before using t h e s e definitions t o derive t h e capita1:output ratio, l e t us
show t h a t 3.2 is equivalent t o 1.1. First, integrate 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 get
t
~ ( t =) - c
+pv(t)e"
+Jre4'dz
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 condition implies
The term on t h e left hand side of 3.5, r / t , is simply t h e capitalized value of
land r e n t s , and corresponds t o t h e economic r e n t w e seek t o maximize in
1.1. Thus 4.2 is 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 formulation.
The continuous formulation i s useful because i t highlights t h e r o l e of
capital in timber production. In this context, 4.lb comprises t h e most limited definition of capital possible. That is, t r e a t regeneration costs as
"labor", and land r e n t a l costs "land" (although land has 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 capital deployed k is
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
0
By 1.3, t h e annual income from t h e f o r e s t is
The capital : output r a t i o k / y is then
t
Jv (2 )dz
3.7
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 (see, f o r example, Chang, 1983), and by 2.4 t h e value of t h e capital embodied in t h e invent o r y of growing stock will decline. Output may increase 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 is ambiguous.
The loblolly pine example developed below shows t h a t f o r most rotations of
k
di n t e r e s t , -i s positive, s o increases in i n t e r e s t rates will lead t o reducdl
tion in t h e use of capital p e r unit output.
AN EXAMPLE: SI 80 LOBLOLLY PINE
While t h e foregoing theoretical 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 run supply c u r v e f o r timber. t h e
practical importance of some of t h e theoretical concerns is not readily
apparent. 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 presents a n example using yield
information f o r SI 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 fit t o a two-parameter
yield equation
4.
The numbers in parentheses 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 is zero. For t h e purposes of this example, ordinary least squares regression produced an acceptable fit t o t h e yield table.
For this example, regeneration costs $50/acre and the i n t e r e s t rate
equais 0.025. Binkley (1985) has 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 less 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 less than 1/23.5=0.043 t o show the
general case of where t h e supply curve has f i r s t a positive slope at low
p r i c e s then a negative slope at higher prices. A comparatively low r a t e of
i = 0.025 w a s taken in o r d e r to illustrate t h e general principles involved.
Indeed i t is of some i n t e r e s t t h a t more realistic i n t e r e s t rates would produce 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 this situation. For r o t a tion a g e s ranging in 1 y e a r increments from 1t o 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 < 19, t h e p ( t * ) < 0).
Supply a t each 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
these 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, Um 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 cf/ac/yr, 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 run
production occurs. This p r i c e corresponds t o a n optimal rotation a g e of 30
years.
102
103
104
105
103.5
FIGURE 4. Long-run supply, SI 00 Loblolly pine
106
MSY
S(P)
cf/a/yr
(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 derived 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 perfectly inelastic at MSY and again at t h e quantity
asymptote.
Figure 6 shows t h e "apparent" inventory elasticity as a function of t h e
rotation 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
longer 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 capital : output r a t i o as a function of t h e
rotation 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 reproducible fixed assets to value
added for all US manufacturing industry w a s 0.34 (UN, 1981, Tables us 2.15
and us 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 capital possible, timber production r e q u i r e s two o r d e r s of magnitude more capit a l p e r unit output t h a n does 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).
Throughout the range shown in Figure 7,
i s positive.
dt
A s one would
expect, increases in capital costs will lead to fiss capital used p e r unit of
output. Until age 7 the converse is true, however.
0
10
i0
t
I
I
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 lis 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 largely
d e t e r n i n e s t'ne quantity of capital used in a timber ente-rprise. The rotation
a g e also strongly influences t h e a v e r a g e output f r o m t h e forest. 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 also t h a t capital 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 conserve on t h e use 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 rotation
age. This kind of capital conservation can also r e d u c e f o r e s t growth. In
t h e long r u n , timber supply can t h e r e f o r e fall as a consequence of higher
timber prices. Higher output p r i c e s inevitably mean higher capital costs,
and t h e timber supply c u r v e bends backward as a result 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 this unusual cost
s t r u c t u r e f o r timber production.
These conclusions are of course s t r i c t l y correct only under t h e many
assumptions n e c e s s a r y f o r such concise and unambiguous results. Two
important market 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 production. Both kinds of adjustments make timber supply m o r e elastic than
found h e r e , particularly 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, howe 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 cost 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 supply will then a p p e a r .
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