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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 l.jpg

Forest Economics

© Peter Berck 2003


Simple forest planning l.jpg

Type of Site, j

Many “birthdays”

First is –M.

hj(t,s)

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 l.jpg
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


Biology l.jpg

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.

Biology

This is Johnson and Scheurman, Model II.


Slide5 l.jpg
W

  • 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 l.jpg
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 l.jpg

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


Biology8 l.jpg
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


Stochastic l.jpg
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.


Slide10 l.jpg
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 bare land one period older next period.


Valuing stock l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Demand

  • Demand is estimated from time series data. Price and housing starts are most important variables in demand

  • Forest stock identifies the demand equation.


Slide15 l.jpg


Slide16 l.jpg


Redwood national park l.jpg
Redwood National Park losses to cmai (small) and to oldgrowth retention, large.

  • Oldgrowth Redwood stands have zero net growth.

  • Can use exhaustible resource framework


Hotelling s model 3 equations l.jpg
Hotelling’s Model: 3 Equations losses to cmai (small) and to oldgrowth retention, large.

  • 1. Capital Market Equilibrium

  • 2. Feasibility

  • 3. Flow Market Equilibria.


Price goes up a rate of interest l.jpg
Price Goes Up a Rate of Interest losses to cmai (small) and to oldgrowth retention, large.

  • 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 l.jpg
Use no more than there is losses to cmai (small) and to oldgrowth retention, large.

  • Second, the sum of the stumpage cut, q(t), over time equals the original stock of stumpage,

  • (2 )


Flow market equilibrium l.jpg
Flow Market Equilibrium losses to cmai (small) and to oldgrowth retention, large.

  • 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 l.jpg
More on the Flow losses to cmai (small) and to oldgrowth retention, large.

  • 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 l.jpg
    Solving the Model losses to cmai (small) and to oldgrowth retention, large.

    • (1 )p = p0ert, where p0 is initial price

    • (3) q(t) = Q(p(t) + c, h).

    • SO

    • q(t) = Q(p0ert+ c, h).


    Finishing the solution l.jpg
    Finishing the Solution losses to cmai (small) and to oldgrowth retention, large.


    Example q p c 0 no h l.jpg
    Example: Q = p losses to cmai (small) and to oldgrowth retention, large.- ; c=0;no h


    Example concluded reduced form l.jpg
    Example Concluded: losses to cmai (small) and to oldgrowth retention, large.Reduced Form


    What to estimate l.jpg
    What to estimate losses to cmai (small) and to oldgrowth retention, large.

    • 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 l.jpg
    Taking of the Redwood Park losses to cmai (small) and to oldgrowth retention, large.

    • 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


    Enhancement l.jpg
    Enhancement losses to cmai (small) and to oldgrowth retention, large.

    • 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 l.jpg
    Enhancement: Lowering X(0) losses to cmai (small) and to oldgrowth retention, large.

    p

    price path is

    result of new X(0)

    Arrow shows size

    of enhancement

    P0 ert

    p0

    p0

    q

    t

    q

    450 line


    Folded diagram model l.jpg
    Folded Diagram Model losses to cmai (small) and to oldgrowth retention, large.

    • 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 l.jpg
    Enhancement: Years Method losses to cmai (small) and to oldgrowth retention, large.

    p

    X(0) is again red area.

    Arrow shows number of

    years need to wait to

    find equivalent

    P0 ert

    p0

    p0

    q

    t

    q

    450 line


    Value of enhancement l.jpg
    Value of Enhancement losses to cmai (small) and to oldgrowth retention, large.

    • 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


    Conclusion l.jpg
    Conclusion losses to cmai (small) and to oldgrowth retention, large.

    • 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 l.jpg
    Forest Area/Deforestation losses to cmai (small) and to oldgrowth retention, large.

    • 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 l.jpg
    Naïve losses to cmai (small) and to oldgrowth retention, large.

    • 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


    India l.jpg
    India losses to cmai (small) and to oldgrowth retention, large.

    • 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


    Slide38 l.jpg
    FR losses to cmai (small) and to oldgrowth retention, large.

    • 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 l.jpg
    Carbon losses to cmai (small) and to oldgrowth retention, large.

    • 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 l.jpg
    Optimal losses to cmai (small) and to oldgrowth retention, large.

    • 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


    Slide41 l.jpg
    soln losses to cmai (small) and to oldgrowth retention, large.

    • 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 l.jpg
    Empirical losses to cmai (small) and to oldgrowth retention, large.

    • 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 l.jpg
    Other subjects losses to cmai (small) and to oldgrowth retention, large.

    • Employment

    • Trade (and the Lumber Wars)


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