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Forest Economics © Peter Berck 2003 Type of Site, j Many “birthdays” First is –M. h j (t,s) t is calendar time s is birthday of stand h is acres harvested D j (t-s) is volume per acre Warning: See article to get > and >= correct. Simple Forest Planning Problem cont… v(t) is cut at t

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forest economics

Forest Economics

© Peter Berck 2003

simple forest planning
Type of Site, j

Many “birthdays”

First is –M.


t is calendar time

s is birthday of stand

h is acres harvested

Dj(t-s) is volume per acre

Warning: See article to get > and >= correct.

Simple Forest Planning
problem cont
Problem cont…
  • v(t) is cut at t
  • v=js>-M Dj(t-s) hj(t, s)
  • Max present value
    • of P times V
  • s.t. biology
  • v(t+1)  v(t) non declining flow
  • t-s > CMAI or h = 0
Initial Acres = Cut over all time

Aj(s) = t>s hj(t,s)

Cut acres regrow and are recut

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 Johnson and Scheurman, Model II.

  • W is what is left standing
    • w(time, birthday)
    • wj(z,s) = Aj(s) - t<Z hj(t,s)
      • For stands born before time zero s< 0
    • wj(z,s) =t<s hj(s,t) - a<z hj(a,s)
      • For stands born s>0. first sum is total acreage in stand regenerated in time s
      • Second sum is amount cut in times prior to z from stands regenerated at time s.
expanded objective function
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 wj(s,t) Ej(s,t)
  • Max present value of Y(t).
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

constraint on w

More treatments

commercial thin

pre-commercial thin

More meaning to the model
  • 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
  • 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.
  • 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 bare land one period older next period.
valuing stock
Valuing stock
  • Easy: Just add terms to the objective function of the form
  • W S
  • Where W is the stock and S is the valuebv
  • Dual now includes added term in S (big S if held and little S if not, one presumes)
  • This formulation takes care of carbon sequestration.
turning js into a estimating model
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.
how do we get 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
  • 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, v.
  • 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.
redwood national park
Redwood National Park
  • Oldgrowth Redwood stands have zero net growth.
  • Can use exhaustible resource framework
hotelling s model 3 equations
Hotelling’s Model: 3 Equations
  • 1. Capital Market Equilibrium
  • 2. Feasibility
  • 3. Flow Market Equilibria.
price goes up a rate of interest
Price Goes Up a Rate of Interest
  • Hotelling’s Rule
  • Rate of change in price is capital gain
    • No uncertainty
    • Must equal sure rate r
    • dp/dt = r p where t is time
  • (1 ) p = p0ert, where p0 is initial price
use no more than there is
Use no more than there is
  • Second, the sum of the stumpage cut, q(t), over time equals the original stock of stumpage,
  • (2 )
flow market equilibrium
Flow Market Equilibrium
  • c is the cost of converting resource stock to resource flow:
    • example: standing trees into lumber (or other semi-processed product).
  • Thus, s = p + c is the price of lumber
  • Let h be variables, such as housing starts, that shift the demand for lumber
more on the flow
More on the Flow
  • Q*(s, h) demand for (flow) lumber.
        • Assume that it takes x units of stumpage to make one unit of lumber. Then, the derived demand for stumpage is Q(p + c, h) = xQ*(s, h).
  • Q(p+c, h) is (stock) stumpage demand
  • (3) q(t) = Q(p(t) + c, h).
solving the model
Solving the Model
  • (1 )p = p0ert, where p0 is initial price
  • (3) q(t) = Q(p(t) + c, h).
  • SO
  • q(t) = Q(p0ert+ c, h).
what to estimate
What to estimate
  • P as function of stock, housing starts, interest rate etc
  • Demand function
  • Was able to show that the cross equation constraints in the two equations were not violated when one chooses a flexible enough form for P(x,h,r…)
taking of the redwood park
Taking of the Redwood Park
  • In 1968 and again in 1978 the US took a total of 3.1 billion bd ft of standing timber from private companies to form the Redwood National Park
  • The amount by which the price of Redwood went up as a result of the take is called enhancement
  • Amount by which the price goes up when the private timber is taken into the park
  • Enhance = p(xafter take) – p(xbefore take)
enhancement lowering x 0
Enhancement: Lowering X(0)


price path is

result of new X(0)

Arrow shows size

of enhancement

P0 ert






450 line

folded diagram model
Folded Diagram Model
  • p(x(t)) = p0ert
    • price as function of stock is same as price as function of time
  • price in year t + 1 is just p(x(t) – q(t)) which is also p0er(t+1)
  • p(x(t) – ) = p0er(t+n)
  • price after n years of cutting equals the price at
  • time t (p0ert) times the interest factor for n years (er n).
  • Choose n so that the Park taking equals
enhancement years method
Enhancement: Years Method


X(0) is again red area.

Arrow shows number of

years need to wait to

find equivalent

P0 ert






450 line

value of enhancement
Value of Enhancement
  • The 1978 Park taking was 1.4 billion board feet, which is the equivalent of 2.26 years of cutting.
  • price 1978, was $311 per MBF.
  • real interest rate—7 percent
  • 2.26 years at 7 percent real per year or 17 percent of price
  • Gov’t paid $689 million for second take
  • enhancement was $583 million
    • estimated by reduced form method
  • Therefore the US paid nearly twice for the park
forest area deforestation
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
na ve
  • 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
    • Anna’s work on Indonesia was done because of deforestation
  • 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
  • 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 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
  • 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
  • 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
  • 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
Other subjects
  • Employment
  • Trade (and the Lumber Wars)