Forest Economics
© Peter Berck 2008,2013,
2014
Forest class notes on Linear
Programming and Forests
The Basic Model of Forestry


The control variable is a treatment, a
complete harvest is the one we will
examine.
Other treatments include



Partial harvest or thinning
Fertilizing (pays in Sweden)
Clearing understory with fire
Objective


We examine Net Present Value
Foresters are trained to ask about the
(usually multiple) objectives of owners.
Examples:



Estate tax payment so as to allow a long
term ownership to continue. (Sweden)
Environmental or Aesthetic
Water, Wildlife (hunting), visual buffer,
etc.
Time conventions and cut


Type of Site, j: soil, tree species, altitude, etc
Many “birthdays”


hj(t,s)




First is –M. : That is today is day zero.
t is calendar time
s is birthday of stand; t-s is therefore age
h is acres harvested
Dj(t-s) is volume per acre
Present value of cut




v(t) is cut at t from all stands and ages
V(t) =j s>=-M Dj(t-s) hj(t, s)
P(t) is price of standing trees at time t,
called the stumpage price
Present value is
PV (t )   P ( s )V ( s ) 1  r 
s t
 ( s t )
Conservation of acres

Initial Acres =
Acres cut over all
time

Aj(s) = t>s hj(t,s)





s is birthday
Recall: h(current time,
birthday)
A(s) are the initial
conditions
Cut acres at t regrow and are recut
at a > t
s hj(t,s)


= a hj(a,t)
Cut at t from all birthdays
(s<=t) is what is reborn
at t and therefore cut in
times a>t.
This is where bio
differs from fish.
This is Johnson and Scheurman, Model II.
Full Problem
Choose h >= 0 to maximize PV
s.t. the three constraints and A
given.
PV (t )   P(s)V ( s) 1  r 
 ( s t )
s t
V(s) =j z>=-M Dj(s-z) hj(s, z)
Aj(s) = t>s hj(t,s)
s hj(t,s) = a hj(a,t)

Comment: This
version generates
the most compact
linear programming
code.
Easier to understand and
program

Add another
variable, x(t,s) the
number of acres
standing at time t of
land regenerated at
time s.
x is what is left standing at z
from stands born in s <= z

x(time, birthday)

xj(z,s) = Aj(s)

For stands born before time zero s< 0, A(s) is given.

For stands born at zero or later.
Let Aj(s) = a hj(s,a)


- t<Z hj(t,s)
What is left at time z from what was born
at time s = what there was to start, A,
less what was cut up till now=time z.
Compact Set Up

The same problem explicitly as an LP
in matrix form.

We reduce the number of stands to
one, which is fine until there are
constraints between stands.
Easier to use x as a vector





Originally we had x(t,s)
We find it easier to use
Xt=(x(t,t),x(t,t-1),….x(t,t-n-1))’
That is X is now a vector of acres by
age class. We also make n-1 the
oldest possible acres.
In what follows A will NOT be the A
from before—sorry!
A LP One Stand Example
0   x 1   h 1    1 1 1 1 1   h 1 
 x1  0
     
  
 


0   .   .    0
0 . 
 .  1 0
 .   1
  .    .    
 . 
  
      
 
1
. 
  .   .   
 . 
  x   h    0
 h 
x  
1
1
0
  n  t  n  t  
  n t
 n t 1 
X(t+1) =< A(X(t) – h(t)) +Bh(t)= Ax(t) +(B-A) h(t)
All in acres.
Stack em: 3 periods
 I

A I

A I

  x 1 


   x 2    x 1 * 

 x  
B  A

0

   3  
 B  A    h 1   0 
 h 
 2
(x)t is the col. Vector of (x1,x2,…)t.
Let the giant matrix be F. This is
F ( (x) ,h )’ < ( x1,0,0)’.
Stack em: 3 periods
 I

A I

A I

  x 1 


   x 2    x 1 * 

 x  
B  A

0

   3  
 B  A    h 1   0 
 h 
 2
F
z
≤b
Expanded Objective Function


Let E(s,t) be value of wildlife, etc
Y(t) = j s>=-M Dj(t-s) hj(t, s)P(t) +


j s>=-M xj(s,t) Ej(s,t)
Max present value of Y(t).
Objective
Git  Di Pt / (1  r )t where D is the volume in age class i
E it is the value of stock in class i at time t, e.g. for
carbon or spotted owls
G is the vec(G it ) and E is vec  Eit 
max E ' x  G ' h subject to constraints on
previous slide and x,h  0
LP




Let F be the giant matrix and c be the
vector of E’s and G’s Let z = (x,h) and
b= (x*(1),0…0)’ ( a ‘ is a transpose)
Max c’z s.t Fz ≤ b (x(1),0…0)’ primal
A result from LP is that an equivalent
problem is
Min b’  s.t. F’  ≥ c dual
Lagrange and dual
max z c ' z s.t. Fz  b
min  0 max z L()  c ' z + '( b  Fz )
min  0 max z L()   ' b  (c '  ' F ) z = b '   z '(c  F '  )
min  b '  s.t. F'  c


The dimension of b is the number of constraints, say m. So 
is also a m-column vector. Dimension of z is the number of
states plus controls, say n. So F has m rows and n columns.
Recall that ’F >= c’ is just fine for the dual constraint.
Writing out our probelm




Lets cut this down to size: 3 periods
and 3 age classes. So x is just born,
one year old, two (or more) years old.
Write out problem and dual.
Solve dual for a simple rule.
Sketch below
Steps


Write out A and B for 3 age classes.
What are the dimensions of A and B?
What are the dimensions of F?

I count 9 constraints, one for each age class in
each time period. Group your  so that ()1=(age=1,
time=1,  a=2, t=1,  a=3, time=1).

Group your c’s the same way. Recall that the first 3 c’s are the value of
standing stock and the last two the value of harvest.
Using your neat new notation


Write out ’F >= c’ in terms of the A’s
and B’s. You should have 5 matrix
equations.
.
Dual


One can show that the dual to the
simple problem is:
Max( value of cutting, value of leaving
alone)


Cutting is just Dj(t-s) P + shadow of bare
land at t.
Leaving stand is shadow of land one
period older next period.
Expanding the model
When to cut with more
constraints?


How much old timber to hold?
Costs of not profit max policies




Like don’t cut till CMAI (top of growth
curve)
Like hold onto oldgrowth for a while
Like have a nondecling flow of timber
National Forest Management Act.
More meaning to the model

Types of sites, j




different species
site classes
critical locations


near streams
visual buffers
More Constraints



Don’t cut type j
Keep N% of forest
at age, t-s, > 100
More treatments


commercial thin
pre-commercial thin
Biology

Could use stand table.



McArdle Bruce Meyer tables for doug fir
Could use stand simulator and then
table the results
Must handle changes in stand
discretely—possibly as stand with new
growth function—eg. After fertilize


This JS model was basis for huge
planning exercise for national forests.
Required by RPA



(Resources Planning Act of 1974)
Hampered by large numbers of
additional constraints on types of land
that could be cut and when.
Eventually died of its own weight.
Stochastic


Can be turned into stochastic program.
Dixon and Howitt do this by taking
linear quadratic approximations and
solving them. (AJAE)
Fire, insects, make stochastic
advisable if planning is objective.
Valuing stock





Easy: Just add terms to the objective
function of the form
XE
Where X is stock and E is value
Dual now includes added term in E
This formulation takes care of carbon
sequestration.
Carbon



Imagine adding 1t co2e to the stock in the
DICE model and finding the change in
value. That is the price of carbon for each
time.
Now imagine cutting a tree and having it
release carbon over time while its
replacement is growing.
There will be a delta for each time. Got to
multiply by the shadow value.
Example:





Price of co2e: 10, 15, 20 in 3
decades.
Cut 1 ton co2e tree in first decade,
20% goes back to atmosphere each
decade.
Growth is 2., .3 , .4 tons
Net is 0, .1, .2 tons; Hence +$ 5.5
Need to do this over a very long time
to get it right.
Example 2





Plant a forest that burns after 30 years.
P = 10, 20, 30.
Growth .2, .3, -.5
Does this make you wonder about
what the prices must really mean?
Perhaps clearer if rental rate: take out
one unit co2e in yr 1 and put it back in
year 2.
Turning JS into a estimating
model


Want to know if private and public
forest were managed differently and if
so what was “optimal” or what the
shadow losses were of public
management.
Need to estimate future prices and
appropriate interest rate.

Hotelling: Derivative of profit function
is supply.

CS ( P )   Q( z )dz where Q is demand curve.
p
How do we get P




Model of previous section has value
function J(P1,…,Pn, r) where P are the
prices in the n periods and r is the
interest rate.
Let CS(Pi) be consumer surplus of i
Consider functional Z(P,r) = J + S
CS(Pi)
Function takes a minimum where
supply = demand (2nd deriv positive)
Demand


Demand is estimated from time series
data. Price and housing starts are
most important variables in demand
Forest stock identifies the demand
equation.





Now– for each choice of r, using the
rule that P mins Z we can find P(r)
Given the Prices, the planning part of
the model gives the cut, h.
Residual is predicted less actual cut
Min sum sq. resids by varying r
This estimates the model

Given the r and the P’s it is a simple
matter to value the losses to cmai
(small) and to oldgrowth retention,
large.
Forest Area/Deforestation

US: Virgin forest to today: less forest




However NE and S. both regrew
Large parts of rural US are going back to
forest
General trend is for less forest
Foster and Rosenzweig look at India
Naïve

Many LDC’s have insufficient land
ownership to protect forests


Marcos denuded the Phillipines for profit
Nepal has problems with marginal ag
taking over forest regions
India

Gross forest statistics like US



Area goes down
Then up
Why?


Market stories require property rights—
FR implicitly assume such.
Demand for forest products goes up,
forests should go up.


Long run, true
Short run could go other way. Not so obvious
FR



Interest is the in the matched dataset of
sattelite imagery (historical forest cover) to
village surveys.
Find that increased population or
expenditure on forest products leads to
more forest land.
Wages, ag land prices insignificant




New England can be told with wages or time to
regenerate
Need relative ag land/ forest land price to do this
in the normal way
Also need the product price for forest, don’t have
plausible that more income = more forest
Carbon


Carbon sinks include soil and trees
From Sohngen and Mendelson

10% more carbon could be sequestered in
forests




Either more land
Or more intensive management
Unclear how one would keep it tied up in soil or
trees
$1-150 per ton are estimates for sequestration
Optimal


To decide what to do need to know the
value of carbon sequestration by time
period.
S-M model




Damage function of carbon stock
dStock/dt = emissions – abatement
Reducing emissions and abatement are
costly
Minimize present value of costs
soln


There is a shadow price of carbon, the
marginal value of reducing the stock
by one unit. Marginal costs = that
Problem: forestry stores the carbon
for a while. Uses rental rate for carbon



Interest on value less
Price increase
Worth investigating==might not be right
Empirical

Melds forest and climate model




Gets price for emissions abatement
Finds how sequestration changes land
and forest prices
Finds equilibrium with higher prices for
forest land (bid up because of
sequestration)
Sequestration makes sense, but is less
profitable than with no price rise
Other subjects





Employment
Trade (and the Lumber Wars)
Private non industrial supply
Mean reversion of prices
Diversity and the need for large
untouched blocks vs. diversity in cut
areas
Steps




1. write out the dual constraints, all 5
of them.
2. What is lambda(3)?
3. Now work backwards and get the
other two
Remember x and lamda are nonnegative. Also need to make lamda
small says the objective function