on: Notes "Private Property Rights and Forest Preservation in Karnataka Western Ghats, India," by M. Bhat and R. Huffaker, AJAE 73(1991), 357387
1. Introduction 1.1. The question addressed in this paper is whether converting a specific publicly-owned open-access forest resource to private ownership would encourage sustainable use of the resource 1.2. The specific resource is government-owned forest land bordering cultivated cropland and orchards in the Western Ghats region of Karnataka 1.2.1. The government has traditionally granted orchard owners exclusive rights to the soppinabeta portion of the forest lands to prune trees for leaves and to sweep dry leaves from the forest floor for farmyard mulch that is used to conserve moisture and retard soil erosion 1.2.2. Orchard owners are not allowed to cut down large trees or sell any of the forest products 1.2.3. The remaining forest lands are reserved for fuel, leafy matter and small timber needs of all villagers, regardless of land/orchard ownership 1.3. It had been observed in prior research that privately-owned orchard land was being used in a more or less sustainable manner, but that the publicly-owned land was being depleted of renewable biomass 1.4. The suggested policy measures were to convert soppinabeta lands to private ownership or to convert those lands into community ownership, where the community would ensure ecological balance in the resource use
1.5. The question addressed in this paper is whether private ownership can lead to sustainable use of the resource 2. Bioeconomic model of areca nut production 2.1. Model setup 2.1.1. Consider a representative areca nut producer who mulches one sixth of an acre of orchard with biomass from a privately-owned 1 acre plot of forest land 2.1.2. Let B(t) be foliage density in kg/acre 2.1.3. R(t) be the harvest (recovery) rate in year t, in kg/acre/t 2.1.4. The dynamics of foliage biomass are governed by
where G(B) represents the average annual growth of foliage biomass in the forest 2.1.4.1. G(B) is characterized by density dependence and a minimum viable biomass stock 2.1.4.1.1. Which means that marginal foliage production increases at low biomass densities and decreases at high densities 2.1.4.1.2. And that a forest defoliated beyond the minimum viable stock will eventually die out 2.1.4.2. G(B) is required then to be strictly concave for 0
2.1.4.3. And there is some B and , which is the minimum viable stock and the carrying capacity, respectively 2.1.4.3.1. Where 0 2.1.4.3.2. Where if tend toward 0 2.1.4.3.3. And where G(B) = =0 B B the stock will tend toward , and if B0 B the stock will
2.1.4.4. The following generalized logistic equation meets the above requirements
2.1.4.4.1. This equation implies that G(B) < 0 at B = 0, so we have to impose the constraint 0 in the optimization process
2.1.5. Let A(R) be the annual areca nut production on one-sixth of an acre of orchard land in kg/ 1 6 acre/t 2.1.5.1. A(R) varies only with the recovered foliage biomass 2.1.5.2. A(R) exhibits positive but diminishing marginal returns to biomass 2.1.5.3. So 0 and 0
2.1.5.4. A(R) has a horizontal asymptote at the maximum periodic production level 2.1.5.5. The Menton-Michaelis functional form meets these requirements 1
where
0 and q is the fraction of recovery made up solely of leaves
2.1.6. Let p denote the price of areca nuts in rupees per kilogram, Rs/kg 2.1.7. C(B) is the cost per unit of biomass recover y in Rs/kg 2.1.7.1. It is assumed that as more biomass is recovered the cost of recovery increases (to reflect the additional labor required to prune limbs of greater height as biomass decreases) 2.1.7.1.1. So, 0
2.1.7.2. Also, unit costs are assumed to approach zero as biomass increases 2.1.7.3. An exponential cost function satisfies these requirements
where c1
0
2.1.8. Periodic profits (Rs/ac/t) are
Where FC represents fixed costs associated with areca nut production
2.1.9. The present value of profits over the interval 0
is
2.2. The producer's problem is to select a function R(t) and a scalar T that maximizes the present value of profits subject to the equation of motion for biomass growth (2.1.4) 2.2.1. The to solution this type of problem (with the specific stated non-negativity constraints) requires following a procedure that is a little different from the standard maximum principle we covered in class 2.2.2. The lagrangian function for this problem is , , , , , _2
where H[B, R] is the standard current-valued hamiltonian 2.2.2.1.1.
2.2.2.2. A solution to the above problem statement must solve: 2.2.2.2.1. This set of three Pontryagin conditions 0
2.2.2.2.2. The non-negativity constraints on R and B 0; 0
0; lim
0
2.2.2.2.3. The free terminal-time transversality condition , 0
2.2.2.2.4. And the non-negativity conditions on the terminal biomass level B(T) if the optimal value of T is finite
lim lim
0 0
2.2.2.3. If the Pontryagin conditions can be satisfied with nonzero values of R and B, then normal 2.2.3. The Pontryagin conditions become 0 0 by the constraints in 2.2.2.2.2, and we can solve the system as
2.2.3.1. The first condition can be rearranged so that
2.2.3.1.1. The above can be differentiated with respect to time to give
2.2.3.2. The equations under 2.2.3.1 and 2.2.3.1.1 can be substituted into the 2nd Pontryagin equation under 2.2.3
2.2.3.2.1. And we can substitute the third Pontryagin condition into the above to get
2.2.3.2.2. Rearrange the above to get
2.2.3.2.3. So we get
2.2.3.2.4. Or in the paper's notation
2.2.3.2.5. Combine the above with
2.2.3.2.6. And the solution is defined in two variables B and R in a two equation dynamical system 2.3. Recovery (harvest) strategies 2.3.1. Recovery strategies are characterized either `continuous' or `abandonment' 2.3.1.1. If T is infinite, recovery continues indefinitely at a steady state of the optimal dynamical system 2.3.1.2. If T is finite, the resource is abandoned after some finite time 2.3.2. An optimal continuous recovery strategy 2.3.2.1. In order to identify the steady-state of the optimal system, we use a phase portrait 2.3.2.1.1. Phase portraits identify the possible trajectories of a dynamical system 2.3.2.2. The authors start with a characterization of the isoclines (sometimes called nullclines) of the system 2.3.2.2.1. An isocline for a dynamic variable, say B, is the set of all B, R combinations for which 0
2.3.2.2.2. A point in a two variable dynamic system where the two isoclines meet is a steady-state
2.3.2.2.3. The isocline for R is found by setting of B 0
0 and solving for R as a function
2.3.2.2.3.1. So, the isocline (set of all R, B combinations where annual growth function of biomass
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