Natural Resource Economics
Academic year: 2015-2016
Prof. Luca Salvatici
luca.salvatici@uniroma3.it
Lesson 8: Forestry
Outline
1.
2.
3.
4.
Trasversality conditions
Forestry
Forest resources: the optimal single rotation
The optimal single rotation: Excel example
Transversality conditions
We consider a variety of alternative ending
conditions for continuous-time dynamic
optimization problems:
• the state variable at the terminal time or
• the ending time are flexible
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Transversality conditions:
taxonomy
STATE
TIME
Fix
Flexible
Fix
Flexible
graph 1
graph 3
graph 2
steady-state
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Vertical or Free-endpoint problems: graph 2
• By vertical end point, we mean
that T is fixed and X(T) can take on
any value.
• This would be appropriate if you
are managing an asset or set of
assets over a fixed horizon and it
doesn't matter what condition the
assets are in when you reach T.
• When looked at from the
perspective of the beginning of
the planning horizon, the value
that X takes on at T is free and,
moreover, it has no effect on what
happens in the future.
• So it is a fully free variable and we
would maximize V over X(T).
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Trasversality: optimal (final) stock
• Relevant for renewable resources
• Example: payment according to the level of stock at the end of
the period
• Instead of X(T) = a
m(T) = 0
• Rationale: if the shadow price of the stock was positive, the
level could not be optimal (by definition)
• B(X(T)) = salvage value
If B(.) is positive, how does transversality condition change?
dB
m (T ) 
dX (T )
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Horizontal terminal line or fixedendpoint problem: graph 3
• In this case there is no fixed endpoint, but the ending state variables
must have a given level.
• For example, you can keep an asset as long as you wish, but at the end
of your use it must be in a certain state.
Economia delle
risorse
naturali
- a.a. 2008-09
Natural
Resource
Economics
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7
Trasversality: optimal period of
duration
Relevant for non-renewable resources
mT *1
F  X T *   YT *   0
p  X T * , YT *  
1 
It adds both the contributions of the state
variable to the objective function. i.e. the "
direct" (p) as well as the "indirect" one (value of
the stock increase)
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The current state of world forest resources
•
•
Comprehensive assessments of the state of the world’s forest resources are
published every five years by the Food and Agriculture Organisation (FAO) of the
United Nations in its ‘Global Forest Resources Assessment’ series:
http://www.fao.org/forest-resources-assessment/en/
One can partition the earth’s total land area into four categories:
1.
2.
3.
4.
•
•
forest land with a high density of tree cover;
other wooded land, that is extensively wooded, but where the density or extensiveness of trees is insufficient to warrant
description by the word ‘forest’;
other land with tree cover, which consists of land that has substantial wooded coverage, but where the density of that coverage
is fairly low or the trees are relatively small;
other land (that does not have forest or tree coverage); this includes agricultural land, meadows and pastures, built-up areas,
barren land, and land incapable of supporting large trees or trees at anything other than very low density.
The earth’s total land area is a little larger than 13 thousand million hectares.
Of this, approximately 40%) consists of forest, other wooded land, or other land
with tree cover.
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Characteristics of forest resources
1. While fisheries typically provide a single service, forests are multi-functional.
2. Woodlands are capital assets that are intrinsically productive.
3. Trees typically exhibit very long lags between the date at which they are
planted and the date at which they attain biological maturity.
4. Unlike fisheries, tree harvesting does not involve a regular cut of the
incremental growth. Forests, or parts of forests, are usually felled in their
entirety.
5. Plantation forestry is intrinsically more controllable than commercial marine
fishing. Tree populations do not migrate spatially, and population growth
dynamics are simpler, with less interdependence among species and less
dependence on relatively subtle changes in environmental conditions.
6. Trees occupy potentially valuable land. The land taken up in forestry often has
an opportunity cost.
7. The growth in volume or mass of a single stand of timber, planted at one
point in time, resembles
that illustrated for fish populations
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Definitions
• Even-aged forest: a forest consisting of trees of the same
species and age
• Q = Q(t) => volume of merchantable timber
• Average annual yield: Q(T)/T
• Maximum Annual Increment (MAI): d[(Q(T)/T]/dT = 0
MAI = Q(T)/T = Q’(T)
Q’(T)/T – Q(T)/T2 = 0 => TQ’(T) – Q = 0
Example: forestry1.xls (downloadable from the website)
Is it the efficient solution?
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Commercial plantation forestry
•
An economist derives the criterion for an efficient forest management and
felling programme by trying to answer the following question:
What harvest programme is required in order that the present value
of the profits from the stand of timber is maximised?
•
•
•
The particular aspect of this question that has most preoccupied forestry
economists is the appropriate time after planting at which the forest should
be felled.
As always in economic analysis, the answer one gets to any question depends
on what model is being used.
We begin with one of the most simple forest models, the single-rotation
commercial forest mdel.
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