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# Notes on Forest Models - PowerPoint PPT Presentation

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

© Peter Berck 2003

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

• 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

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.

• 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.

• 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).

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.

• 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.

• 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.

• 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.

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

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

Example Concluded: losses to cmai (small) and to oldgrowth retention, large.Reduced Form

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 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 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) 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 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 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 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 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 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

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 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

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 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 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

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

• Employment

• Trade (and the Lumber Wars)