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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

© 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 PlanningProblem cont…

- v(t) is cut at t
- v=js>-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.

BiologyThis is Johnson and Scheurman, Model II.

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

- Let E(s,t) be value of wildlife, etc
- Y(t) = js>-M Dj(t-s) hj(t, s)P(t) +
- js>-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 modelBiology

- 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

- 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.

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

- 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

- 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

- 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

- 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 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(p+c, h) is (stock) stumpage demand (3) q(t) = Q(p(t) + c, h).

- 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).

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

- 10% more carbon could be sequestered in forests

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)

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