Forest Economics Peter Berck 2008 The Basic Model
Forest Economics © Peter Berck 2008
The Basic Model of Forestry
When to cut? n n How much old timber to hold? Costs of not profit max policies ¡ ¡ n Like don’t cut till CMAI (top of growth curve) Like hold onto oldgrowth for a while Next: the basic optimizing model
Simple Forest Planning n n Type of Site, j Many “birthdays” ¡ n hj(t, s) ¡ ¡ ¡ n First is –M. 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
Problem cont… n n n v(t) is cut at t v= j s>=-M Dj(t-s) hj(t, s) Max present value ¡ n n n of P times V s. t. biology v(t+1) v(t) non declining flow t-s >= CMAI or h = 0
Biology—conservation of acres n Initial Acres = Cut over all time ¡ Aj(s) = t>s hj(t, s) n Cut acres at t regrow and are recut at a > t ¡ s hj(t, s) n = 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.
x is what is left standing at z from stands born in s <= z ¡ x(time, birthday) ¡ xj(z, s) = Aj(s) n ¡ - t<Z hj(t, s) For stands born before time zero s< 0 xj(z, s) = t <s hj(s, t) - a<z hj(a, s) n n For stands born s>0. first sum is total acreage in stand regenerated at time s from birthdays t which are < s. Second sum is amount cut in times prior to z from stands regenerated at time s.
Expanded Objective Function n n Let E(s, t) be value of wildlife, etc Y(t) = j s>=-M Dj(t-s) hj(t, s)P(t) + ¡ n j s>=-M xj(s, t) Ej(s, t) Max present value of Y(t).
More meaning to the model n Types of sites, j ¡ ¡ ¡ n different species site classes critical locations n n near streams visual buffers More Constraints ¡ ¡ n Don’t cut type j Keep N% of forest at age, t-s, > 100 More treatments ¡ ¡ commercial thin pre-commercial thin
Biology n Could use stand table. ¡ n n Mc. Ardle 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
n n This JS model was basis for huge planning exercise for national forests. Required by RPA ¡ n n (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.
Compact Set Up n The same problem explicitly as an LP in matrix form.
A LP One Stand Example X(t+1) =< A(X(t) – h(t)) +Bh(t)= A(x(t) +(B-A) h(t) All in acres.
Objective
Stack em (x)t is the col. Vector of (x 1, x 2, …)t
LP n n n Last 2 slides were the LP for one stand in standard form. Let F be the giant matrix and c be the vector of E’s and G’s Let z = (x, h) Max c’z s. t Fz =< (x(1), 0… 0)’ primal Min 1 x(1) s. t. F >= c’ dual You can solve the dual recursively
Steps n n 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
Dual n n 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.
Lagrange n n n This problem can also be solved using the Kuhn Tucker conditions from the constrained optimization problem. Let M(t, s) be the shadow value at t of wood born in s. M(t+1, s) = Max of (M(t, s) and ¡ P D(t, s)/(1+r)t +M(t, t)
Stochastic n n 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 n n n 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.
Turning JS into a estimating model n n 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 n n Model of previous section has value function J(P 1, …, 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 n n Demand is estimated from time series data. Price and housing starts are most important variables in demand Forest stock identifies the demand equation.
n n n 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
n 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 n US: Virgin forest to today: less forest ¡ ¡ n n 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 n 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 n Gross forest statistics like US ¡ ¡ n Area goes down Then up Why? ¡ ¡ Market stories require property rights— FR implicitly assume such. Demand forest products goes up, forests should go up. n n Long run, true Short run could go other way. Not so obvious
FR n n n 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 ¡ ¡ ¡ n 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 forest, don’t have plausible that more income = more forest
Carbon n n Carbon sinks include soil and trees From Sohngen and Mendelson ¡ 10% more carbon could be sequestered in forests n n ¡ ¡ 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 n n To decide what to do need to know the value of carbon sequestration by time period. S-M model ¡ ¡ Damage function of carbon stock d. Stock/dt = emissions – abatement Reducing emissions and abatement are costly Minimize present value of costs
soln n n 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 n Melds forest and climate model ¡ ¡ Gets price for emissions abatement Finds how sequestration changes land forest prices Finds equilibrium with higher prices forest land (bid up because of sequestration) Sequestration makes sense, but is less profitable than with no price rise
Other subjects n n Employment Trade (and the Lumber Wars)
- Slides: 35