lecture 10
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Lecture 10. The efficient and optimal use of renewable resources. What are renewable resources?. Examples: Renewable stock resources : fishery, forestry, water, wildlife, arable land, grazing land, soil, anti-biotic and pesticide resistance - PowerPoint PPT PresentationTRANSCRIPT
Lecture 10
The efficient and optimal use of renewable resources
What are renewable resources?
• Examples:
• Renewable stock resources: fishery, forestry, water, wildlife, arable land, grazing land, soil, anti-biotic and pesticide resistance
• Renewable flow resources: solar, wave, wind, and geothermal energy
• We consider only renewable stock resources
Biological growth processes for renewable resources
• Change of population over time:
• Stock level at any point in time:
dss gs
dt
0gt
ts s e
Integration of the growth process
gSSdT
dS
gS
dTdS
Integrating both sides: gdTdTSdT
dS 1
a) 1cslnS
dS 0s (applying the substitution and log rule)
b) 2cgTgdT
Integration of the growth process
1 2ln s c gT c taking the antilog of ln s and combining the two constants to c
cgTs ee ln se sln
gTcgT Aeees
ceA
If the initial stock level is positive, then ss and we get gTAeTs )( where T is an arbitrary constant. If we set 0T we get 0(0)s Ae A . We can therefore write:
gTesTs )0()( where )0(s denotes the initial stock level.
Growth Function
Important: limits to growth of the biological stock = > carrying capacity
A simple way of including carrying capacity is by making growth rate dependent on the stock rise rather assuming ds/dt being constant: s)s(s
Assume there is a finite upper bound on the rise of population. This will be denoted by maxs .
Commonly used functional form for (s)
maxss
1g)s( Where g>0 and g is a constant parameter
Using s)s(s
maxss
1g)s( ss
s1gs
max
s indicates the net-effect after natural changes and human intervention
Or ss
s1g)s(G
max
Logistic biological growth function
Where is the maximum growth rate?
S)S
S1(g)S(G
max1
or
max
2
1 S
gSgSSG
1
max
20
G gSg
S S
maxS
gS2g
S2Smax
S2
Smax
SMAX
SMSY
0
G(S)
S
Logistic biological growth function
Logistic biological growth function with threshold level
max
min s
s1ssg)s(G
s
G(s)
Logistic biological growth function with depensation
max
( ) 1s
G s gss
s
G(s)
Logistic biological growth function with critical depensation
ss
s11
s
sg)s(G
maxmin
s
G(s)
Steady-State Harvesting
H – h a r v e s tG – n e t n a t u r a l g r o w t h
S t e a d y S t a t e : s t o c k b e i n g h a r v e s t e d ( H ) = n e t n a t u r a l g r o w t h ( G ) H a n d G a r e e q u a l a n d c o n s t a n t o v e r t i m e
I n s t e a d y s t a t e : 0HGs
P o s s i b l e s t e a d y - s t a t e h a r v e s t ?
Figure 17.2 Steady-state harvests (Perman et al.: page 560)
SMAX S1U S
1L
SMSY
GMSY = H MSY
G1 = H 1
0
Open access resource: fishery as a case
Biological Model:
dS/dt = G(S)
Economic Model:
),( SEHH H – harvest E – harvest effort S – resource stock
(harvest depends on stock of fish and effort)
Static analysis of a renewable resource harvesting
Simple harvest function H – harvest E – harvest effort S – resource stock e – catch coefficient
eESH eSE
H
Stock growth including harvest
eESSG
HSGS
)(
)(
The quantity harvested per unit of effort is equal to some multiple (e) of the stock size
Costs and benefits of harvestingLinear cost function ( )C C E wE where w is the constant cost per unit of harvesting effort
Gross benefits or revenue PHHBB )( will depend on quantity harvested H and market price P
PHV
VHB )( PHHB )( simple model )(HPP ; 0H
P
Fishing Profit CBNB
Entry and Exit NBdtdE where - parameter indicating responsiveness of industry size to industry profitability
Bio-economic equilibrium
(1) biological equilibrium: the resource stock is constant
through time, as G(S) = H (2) economic equilibrium: the industry is in an open-
access zero rent equilibrium, as V=C NB = 0
=> PH –wE = 0 => PH = wE (3) bio-economic equilibrium: (1) + (2)
The equilibrium is unique and stable
Open-access steady-state equilibriumH = G
=> eESs
sgS
max
1 =>
E
g
eSS MAX 1
Substituting S in H = eES
E
g
eeESH MAX 1 still biological equilibrium
PH = wE still economic equilibrium There are three equations with three endogenous variables: H, E and S
bio-economic equilibrium
Open-access steady-state equilibrium
Solving for H, E, S provides the following steady-state conditions:
MAXPeS
w
e
gE 1*
Pe
wS *
MAXPeS
w
Pe
gwH 1*
C: total cost of harvesting, linear function of effort, C = wE E: effort or level of harvesting technology H: amount of resource being harvested P: price per unit of the resource being harvested S: stock of natural resource e: catch coefficient, explaining harvest per unit of effort from stock (H/E = e S) g: fixed growth rate of the resource w: cost per unit of harvesting effort E
Steady-state equilibrium fish harvests and stocks at various effort levels
H2 = eE2S
HMSY = eEMSYSH1 = eE1S
H1
SSMSY
=SMAX/2
S1 S2
H2
H=(w/P)EHPP
HOA
EEOA
Steady-state equilibrium yield-effort relationship zero economic profit
equilibrium
E
g
eeESH MAX 1
Steady-state harvesting: private property
0
Max , itt t tPH C H S e dt
);( ttt SHC , 0HC , 0SC
subject to 0, 0t t
dSG S H S S
dt
P – gross price of the resource C – harvesting costs i – Opportunity costs of the resource owner’s capital pt – shadow price of resource Necessary conditions for maximum of wealth include:
,t
t
C H Sp P
H
,t
t tt t
dG S C H Sdpip p
dt dS S
Steady-state harvesting: private property
in steady state dp/dt = 0
ip
,C H Sp P
H
difference between market price and marginal costs of an incremental unit of harvested fish
profit foregone not harvesting)
marginal cost of investment marginal benefit of investment
,C H S dG Sp
S dS
reduction in total harvesting costs, by having one more unit of resource,
additional fish grows by dG/dS value at the net-price
Steady-state harvesting: private property
ip
S
C
pdS
dG
Opportunity costs
Reduced harvest costs
Value from additional growth
pdS
dG
S
C
dt
dpip
increase the stock
pdS
dG
S
C
dt
dpip
decrease the stock
Steady-state harvesting: private property
fundamental equation of renewable resources
i CSp
G
S
opportunity costs
natural rate of growth in the stock from a marginal change in
stock size
the value of the reduction in
harvesting costs that arises from a marginal increase
in the resource stock
Social efficient resource harvesting
Tt
t
rt
ttt dteSHCSHBSMaxPV0
,
subject to
tt HSGdt
dS ; S(0) = S0
Steady – state: tt HSG = dt
dS 0 ;
S
SHCS +
Sd
SdG i =
dt
d
tttt
t
,
0
Same result as under private property, if r = i, BS = PH, and CS = C.
Reasons for difference between social and private maximum: 1. Externalities in the benefits function
Tt
t
rt
tttt dteSHCSSHBSMaxPV0
,,
subject to t tdS G S Hdt
- social benefits depend on stock size (number of species, by-catch)
Solving provides the following steady-state result:
C BdGS S
i = - + p p dS
Reasons for difference between social and private maximum: 1. Externalities in the benefits function
C BdGS S
i = - + p p dS
This reformulation shows that it is Hotelling's rule of efficient resource use,
albeit in a modified form. The left-hand side is the rate of return that can be
obtained by investing in assets elsewhere in the economy. The right-hand side is
the rate of return that is obtained from the renewable resource. This is made up of
three elements:
the proportionate reduction in harvesting costs that arises from a
marginal increase in the resource stock
the proportionate increase in existence benefits
the natural rate of growth in the stock from a marginal change in the
stock size.
Other reasons for difference between social and private maximum: 2. Externalities in the fishery production function - e.g. ploughing the sea bottom 3. Difference between social and private discount rate 4. Monopolistic fisheries
Renewable Resource Policies
1. Command and Control
• reducing fishing effort• restriction on fishing gear• spatial restrictions• fleet size reduction• quantity restrictions on catch
2. Incentive-based instruments - landing tax under open access
- open access condition: ,
0t
C H S p P
H
- present – value – maximizing condition private property:
,
,dG S C H S
pdG S C H S dS S ip p p dS S i i
,
0
user cost
H
dG S C H Sp
dS SP C i i
CH: marginal costs of harvesting one additional unit of fish
2. Incentive-based instruments - present – value - maximizing
,
0
user cost
H
dG S C H Sp
dS SP C i i
taxing at user costs:
,dG S C H Sp
dS St i i
post-tax open-access equilibrium open-access taxed at user costs:
,
0H H
dG S C H Sp
dS SP t C P Ci i
=> equals present - value – maximizing fishing rule Why are optimal landing taxes not used in reality?
2. Incentive-based instruments - property rights and transferable harvesting quota - 200 mile fishing zones - individual transferable quota (ITQ) (total allowable catch
(TAC) are distributed among fishermen
Where the forests go
Community ownership and administration of forests
Forestry Funds
A possible global forest situation: 2050
A single rotation forest model• Stand of timber of uniform type and age
• All trees are planted at the same time• All trees are to be cut at the same time• Once felled the forest will not be replanted• The land has no alternative uses: • All costs and prices are constant• The forest generates only timber as a value, other
possible values are ignored• Felling of the forest has no external effects
A single rotation forest modelP gross price of timber
c harvesting costs
p net price of harvested timber
TS volume of timber at time T
i private discount rate opportunity costs of capital
k planting costs
Maximisation of profit present value T h e p r e s e n t v a l u e o f p r o f i t s i s m a x i m i z e d a t t h a t v a l u e o f T w h i c h g i v e st h e NPVmax :
kepSkeScPNPV iT
T
iT
T )(max
0
T
peSpe
T
S
T
NPV iT
T
iTT
P r o d u c t r u l e
T
epe
T
p
T
pe iTiT
iT
C h a i n r u l e
})({0 1 TiT eeip TiT eipe )1(
TTiT eipe
TTiTipe iTipe
Maximisation of profit present value
0)(
iT
T
iT pieSSpeT
NPV
T
iTiT SipeSpe
TiSS or iS
S
T
or TS
Si
The NPV is maximized when the growth rate of the resource stock is equal to the private discount rate
Example
S = 40t + 3.1 t2 – 0.016 t3
P = 10; market price per cubic foot of felled timber;
k = 5000; total planting costs;
c = 2; harvesting costs per cubic foot, incurred at whatever time theforest is felled
Figure 18.1 (a) The volume of timber in a single stand over time (Perman et al., page 603)
0.0
5000.0
10000.0
15000.0
20000.0
25000.00 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
105
110
115
120
125
130
135
140
145
Years after planting
Vo
lum
e o
f ti
mb
er
S = 40t + 3.1 t2 – 0.016 t3
Figure 18.2 Present values of net benefits at i = 0.00 (NB1) and i = 0.03 (NB2)(Perman et al.: page 606)
-20
0
20
40
60
80
100
120
140
160
180
200
0 4 8
12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96
100
104
108
112
116
120
124
128
132
136
140
144
PV
(tho
usan
ds)
Years after planting
NB1
NB2
Slope = i = 3%
t = 50 t = 135
Figure 18.3 Variation of the optimal felling age with the interest rate, for a single-rotation forest (Perman et al.: page 607)
0
1
2
3
4
5
6
0 20 40 60 80 100 120 140 160
T
Infinite rotation forestry modelsWhat happens if we allow planting another forest after theclear cutting of the existing forest?
Assumptions: Stand of timber of uniform type
and age All trees are planted at the same
time All trees are to be cut at the same
time All costs and prices are constant The forest generates only timber
as a value
Felling of the forest has noexternal effects
The forest will be immediatelyreplanted for the next cycle afterclear-felling
First planting starts at 00 t Rotation of the forest continues
infinitely
Infinite rotation forestry models: NPV
The NPV from the first rotation will be: kepSNPV tti
tttt
)(
)(/01
0110 The NPV of profits over an infinite sequence is:
kepS tti
tt
)(
)(01
01 kepSe tti
tt
tti
)(
)(
)( 12
12
01
kepSe tti
tt
tti
)(
)(
)( 23
23
02 kepSe tti
tt
tti
)(
)(
)( 34
34
03
... Conditions remain the same. The optimal length in the rotation model remains the same. We can write for the length of rotation
Ttttttt ...231201
Infinite rotation forestry models: NPVSimplification:
kepS iT
T kepSe iT
T
iT kepSe iT
T
Ti 2
kepSe iT
T
Ti 3 ... Factorising out iTe gives:
kepSekepSekepSekepS iT
T
TiiT
T
iTiT
T
iTiT
T 2 The term in the round braces on the RHS equals .
Rewrite: iTiT
T ekepS ; kepSe iT
T
iT 1 ;
iT
iT
T
e
kepS
1
Infinite rotation forestry models: NPV
Maximizing by selecting the rotation length T that provides the highest NPV
iT
iT
T
e
kepS
1
T
011
1
TekepS
e
pieSSpe iTiT
TiT
iT
T
iT
T
e
Te
iTiT
111
1 iTiT iee 211 2
1 iT
iT
e
ie
Infinite rotation forestry models: NPV
0
11 2
iT
iTiT
TiT
iT
T
iT
e
iekepS
e
pieSSpe
T
0 iTiT
T
iT iepieSSpe
0 ipiSSp T
ipiSSp T
Faustmann rule
iipSSp T
Sp (additional value if the forests grows for one more year)
TipS(interest earned from the value of trees if trees are cut immediatelyafter one period)
i (interest earned for one period if land would be sold after the treeshave been felled on the price received for the land)
Hotelling dynamic efficiency condition The Faustmann rule is a Hotelling dynamic efficiency condition for the harvesting of timber. Rewrite the Faustmann rule:
TT pS
ii
S
S
The Hotelling rule states that the growth rate of the shadow price of the resource (that is, its own rate of return) should equal the social utility discount rate:
pP
P
T
T
or the opportunity costs
Observations: A positive value of the land increases the opportunity costs and hence the
rotation rate will be shorter The land value can also be derived by looking for the other opportunity costs of
land than the forest
Multiple use forestry
Assumptions: the same as for the infinite rotation forestry model
tNT are the non-timber benefits in yeart after the forest has beenplanted. These are amenity values, but also include non-timber forestproducts like game, fruits and others. Another example is the capacity ofthe forest to store CO2, protect watersheds or reduce soil erosion
The tNT can be seen in the same way as the increase in timber. The valueof non-timber benefits accumulated over the rotation period is
NT = T
iT
t dteNT0
Multiple use forestry: NPV
The first rotation period NPV can be written as: 1 T TiTNPV pS N e k
The infinite rotation model including non-timber benefits, indicated by a * is:
..
*
3
2
keNpSe
keNpSe
keNpSe
keNepS
iT
TT
iT
iT
TT
iT
iT
TT
iT
iT
T
iT
T
Multiple use forestry: NPVFactorizing out iTe gives:
...
*3
2
keNpSe
keNpSe
keNpSe
keNpS
ekeNpSiT
TT
iT
iT
TT
iT
iT
TT
iT
iT
TT
iTiT
TT
** iTiT
TT ekeNpS
iT
iT
T
iT
T
e
keNepS
1
*
Optimal length of rotation
T h e o p t i m a l l e n g t h o f r o t a t i o n c a n b e d e r i v e d b y g e t t i n g t h e f i r s td e r i v a t i v e o f w i t h r e s p e c t t o T w h i c h i s s e t e q u a l t o z e r o
011
*2
iT
iTiT
TTiT
iT
T
iTiT
T
iT
e
iekeNpS
e
eiNNepieSSpe
T
0 iiNNpiSSp TT
iipSiNNSp TT
Optimal length of rotation
iipSiNNSp TT
Sp additional timber value for one year N additional non-timber benefits for one year
TiN opportunity costs of non-timber benefits that have accumulated over time (if they could be sold)
TipS opportunity costs of not harvesting the trees immediately (one period)
i opportunity costs of land including the non-timber values, where is the
value of land (one period)
Decision about the optimal rotation rate The left-hand-side (LHS) describes the benefits from leaving the forest for one more unit
of time, say one year, unfelled. The benefits are the change in volume of saleable timber (which could be negative) and the value of non-timber benefits
The right-hand-side (RHS) describes the income that the forest owner would receive over
one period if the forest would be cut. The first term expresses the benefits from selling non-timber benefits. These benefits are zero for most cases. The can be negative in the case of releasing fixed carbon. They can be positive in the case where game is harvested when the trees are felled
The second term on the RHS expresses the returns the forest owner would get over one
time period, if he cuts the trees immediately and invests the revenue at a rate of return of i The third term is the return if the forest land is sold at price * and the revenue invested
at a rate of return of i The non-timber benefits increase the value of waiting to fell the trees and increase the
rotation interval other things being equal The non-timber benefits increase the value of the land, * , as well. The increase in * ,
other things being equal, reduces the rotation interval
Conclusions• Non-timber benefits can shorten, lengthen or leave
unchanged the optimal rotation length
• If the flow of non-timber benefits is constant over theforest cycle, the optimal rotation length will be shortenedcompared to a situation that excludes non-timber benefits,other things being equal
• If non-timber benefits increase at an increasing rate withthe age of the forest, then the optimal rotation length willincrease
• In extreme cases the magnitude and timing of non-timberbenefits could result in no felling being justified