Forest Economics © Peter Berck 2003
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… • 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. Biology This 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).
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
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 • 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 • Oldgrowth Redwood stands have zero net growth. • Can use exhaustible resource framework
Hotelling’s Model: 3 Equations • 1. Capital Market Equilibrium • 2. Feasibility • 3. Flow Market Equilibria.
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 • Second, the sum of the stumpage cut, q(t), over time equals the original stock of stumpage, • (2 )
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 • 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 • (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 • 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 • 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 • 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) 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 • 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 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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 • Employment • Trade (and the Lumber Wars)