Conservation of forest based on a fuelwood substitute as well as considering the cultural and spiritual values: an optimal fuelwood harvest model

Excessive fuelwood harvest is a major cause of deforestation in the developing countries. To mitigate 4 this, various preventive measures have been introduced in different countries. Availability of affordable 5 substitute to the community dependent on the forest for domestic energy consumption may prevent 6 further forest degradation. A stock dependent optimal control model of fuelwood harvest from a natural 7 forest is presented here and comparative statics has been used to show that the presence of a fuelwood 8 substitute will reduce its harvest and increase the forest stock. The model indicates that availability of 9 cheaper and high energy content alternative for fuelwood can substantially reduce fuelwood extraction 10 from a forest. Also, a lower discount rate and higher cultural and spiritual values (CSV) will keep the 11 optimal forest stock close to its carrying capacity and reduce fuelwood harvest. The model reveals that 12 the maximum sustainable yield of forest stock and the ratio of energy content per unit mass of fuel play 13 a central role in the fate of forest stock and level of fuelwood harvest. Empirical example of the 14 Southeast Asian forest growth model along with Liquid Petroleum Gas (LPG) as substitute has been 15 used to illustrate the results. The outcomes of this study can incorporated into forest conservation 16 polices. 17


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Forest plays a central role in meeting the social and economic needs of the people living in its vicinity.

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Along with timber, forest provides Non-Timber Forest Products (NTFP) like fuelwood, fodder and from fuelwood to relatively cleaner and convenient energy options like LPG and Biogas, provided such 11 options are readily available (Pandey, 2002). Studies in developing countries have shown that 12 appropriate fuelwood substitutes or efficient use of fuelwood can prevent deforestation (Adhikari, 2002; 13 Agarwala et al., 2017;Roy, 2008). Thereby, understanding the role of fuelwood extraction on the forest factors and the interaction of these factors on the forest regime have been analysed using optimal control 26 theory (Kant, 2000). Impacts of non-timber valuation on the forest stock and timber harvest have been 27 analysed using optimal control theory and comparative statics (Gan, Kolison, & Colletti, 2001).
28 Discrete optimal control model has been used to evaluate the role of fuelwood burning on the climate 29 (Lyon, 2004). Dynamic optimization techniques have been used to model the non-timber forest 30 extraction in spatio-temporal context (Robinson, Albers, & Williams, 2008). Dynamic system 31 modelling based study indicates that fuelwood harvesting causes forest degradation, forest fire, 32 institutional failure and socioecological problems to the forest-dependent communities (Ranjan, 2018).

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Majority of these studies consider fuelwood under the broad heading of non-timber benefits and do not 34 exclusively analyze the impact of fuelwood extraction on the forest stock and harvest decision. Also, 35 role of consumer's choice between fuelwood and its substitute on the forest stock and harvest decision 36 has not been studied using optimal control theory and comparative statics. Moreover, conventionally 37 forest growth is considered as the function of time. Such considerations are appropriate when the forest 38 is private property and objective of the owner is to harvest timber at an optimal rotation period (Amacher 39 et al., 2009). Such an approach is not appropriate for open access resources like a large natural forest 40 with uneven age classes and multiple uses. Thereby, a more appropriate determinant for harvest decision 41 should be forest stock size rather than time.
function, marginal utility of fuelwood (MUF) and marginal utility of substitute (MUS) on the marginal 1 forest growth. Also, comparative statics is used to evaluate the impact of marginal change in discount 2 rate and CSV, marginal utility of fuelwood and substitute on the optimal forest stock and optimal 3 fuelwood harvest. An empirical example of the impacts of fuelwood harvest in the presence of a 4 substitute on the biomass stock of the Southeast Asian forest has been used to illustrate the theoretical 5 results. The outcomes of this model can be generalised to all forms of forest wood products and forest 6 values other than CSV.

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The objective of the consumer is to maximize his utility, (. ) and the CSV, (. ) by selecting the 17 optimal rate of fuelwood harvest, ℎ( ) subject to various constraints given below:

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Subject to: 26 Where, the forest stock, ( ) follows a quasiconcave growth function, ( ( )) such as logistic growth 21 Where = , is the ratio of energy content per unit mass of the substitute to that of fuelwood. The 22 first order condition of (. ), as given in equation (8), is given by:

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Rearranging equations (10) and (11) give: 28 Equation (12) suggest that at optimal utility level the shadow price of the forest stock is equal to the 29 difference between, th-times the MUS from the MUF. Equations (2), (3), (12) and (13) along with 6 inequations (4) to (6), constitute a simultaneous equation system. By setting = ℎ = 0, we can solve 1 for the optimal steady state solution, ( * , ℎ * ). If the forest stock is not at its optimal stock level, then 2 the fuelwood harvest decision can follow any of the two optimal paths. These are the Asymptotic 3 Approach Path or the Most Rapid Approach Path (MRAP), to reach the optimal forest stock (Clark, 4 1990). As per the MRAP or 'Bang-Bang' control approach, the optimal harvest, ℎ * is: 6 Equation (14) suggests that the optimal harvest is equal to the maximum harvest rate whenever the 7 forest stock is above the optimal forest stock. At sub-optimal forest stock level, fuelwood harvest is not 8 appropriate. Lastly, under optimal forest stock condition fuelwood harvest rate equals the natural 9 growth rate of the forest.  Equation (17) implies that, in the absence of CSV, in order to maximize the fuelwood harvest under 20 steady state condition, the marginal forest growth should be equal to the discount rate. Since, ( ( )) 21 is a quasiconcave function, there exists a relation between optimal forest stock and MSY: 23 Equation (18) implies that, depending on the nature of the marginal forest growth function, the optimal 24 forest stock will be below, equal to or above the MSY of the forest (Gan et al., 2001). Moreover, as:

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The limiting condition (19) implies that, as the discount rate approaches zero, the marginal forest growth 27 also approaches zero. From equation (18), this change suggests that as the forest growth rate approaches 28 zero, the optimal forest stock approaches MSY stock size.
In the absence of MUF, equation (16) is expressed as: 2 Equation (20) implies that in the absence of MUF the discount rate is less than the marginal forest 3 growth by a factor equal to the ratio of marginal change in CSV to -times the MUS. In the absence of 4 fuelwood harvest, the lowered discount rate will help in conserving the forest. In the absence of MUS, 5 = 0, equation (16) is expressed as (Gan et al., 2001): 7 Equation (21) implies that in the absence of MUS the discount rate is more than the marginal forest 8 growth by a fraction equal to the ratio of marginal growth of CSV to the MUF. Comparing equation 9 (21) with equation (17) we observe that, in the absence of a substitute, even after considering CSV, the 10 MUF pushes the discount rate above the marginal forest growth, leading to exploitation of the forest for 11 fuelwood, timber and NTFP. Rearranging equation (16) gives: 13 Equation (23) suggests that at equilibrium the marginal forest growth, could be positive, negative or 14 zero.

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Case I: if = 0 or < ℎ − , then < 0, indicating that the optimal forest stock has exceeded the 16 MSY stock size. Such a situation will prevail if ℎ > . Under such circumstances, the optimal forest 17 stock will be above MSY stock. This will cause the utility maximizing consumer to harvest fuelwood 18 using the MRAP strategy as given in equation (14).

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Case II: if = ℎ − then = 0, indicating that the optimal forest stock is equal to the MSY stock 20 size and harvest rate is equal to MSY.

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Case III: if > ℎ − or ℎ < then > 0. Under such circumstances, the optimal forest stock will 22 be below MSY stock size. This will cause the consumer to harvest fuelwood using the MRAP strategy 23 as given in equation (14) and will discourage further fuelwood harvest.

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At equilibrium, let us consider that , ℎ and approaches certain steady state values. Hence, let = 26 , ℎ = and = . Then equation (16) is accordingly modified to:

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Comparative statics is used to analyze the effect of changes in discount rate alone on the optimal forest 3 stock and optimal harvest, by letting = = = 0 in equation (26). Also, let (26) is solved using Cremer's rule: 9 Considering assumption (1), equation (28) suggests that, increase in discount rate will encourage 10 deforestation. On the other hand, equation (29) implies that the optimal harvest increases with increase 11 in discount rate when forest stock is below MSY stock, while it decreases when forest stock is above 12 MSY stock and optimal harvest is not affected by the change in discount rate when forest stock equals 13 MSY stock. 19 From equation (30) it is observed that the optimal forest stock decreases with increase in CSV when the 20 MRS is below the ratio of energy value per unit mass of fuelwood to its substitute ( ). In contrast, 21 optimal forest stock increases when MRS is above . The condition is unknown when MRS is equal to 22 .  (26) and solving:

Effect of marginal utility of fuelwood
And,   , it causes optimal forest stock to approach MSY stock. On the other hand, − − ≥ causes 8 optimal forest stock approach to the carrying capacity and optimal harvest to zero. For instance, with a 9 non-zero CSV and MRS at 0.667, leads to optimal forest stock reach the carrying capacity of Southeast 10 Asian forest, while optimal harvest to zero. These conditions are prevalent at higher CSV. optimal 11 forest stock and optimal harvest will remain constant for any combination ( , , ) as long as 12 ( − − ) remain unchanged.
13 Figure 2 indicates that with an increase in CSV, the optimal forest stock is progressively maintained at 14 its carrying capacity, even at a higher MRS and discount rate. At a higher CSV, a high MRS does not 15 lead to decline in the optimal forest stock. Similarly, the effect of a progressive increase in discount rate 16 is nullified by CSV of forest stock and maintains the forest stock at its carrying capacity. On the other 17 hand, at a lower CSV, a combination of high discount rate and high MRS ratio facilitated decline of the 18 forest stock. A CSV of 12 and above, the forest is protected from deforestation. Regarding optimal 19 harvest, an increase in CSV delays the harvest for a higher MRS value. Also, optimal harvest reaches 20 MSY at a still higher MRS with the increase in CSV. A higher discount rate facilitates harvest at a lower higher MRS.
1 Figure 3a and 3b, gives a cross-section of optimal forest stock and optimal harvest of Southeast Asian 2 forest. Figure 3a indicates that at a fixed discount rate and CSV, a rise in MRS causes a decline in 3 optimal forest stock. In case of optimal harvest, the initial increase in optimal harvest quickly declines 4 due to fall in the optimal forest stock ( Figure 3b). An increase in the discount rate, causes a further and 5 steeper decline in optimal forest stock over MRS. A proportionate amount of optimal harvest also 6 increases with the rise in discount rate. Figure 4a, illustrates the effect of discount rate on optimal forest 7 stock. Optimal forest stock declines over discount rate. The effect gets pronounced with the rise in 8 MRS. optimal harvest initially rises over discount rate, followed by a sharp decline due to fall in the 9 optimal forest stock ( Figure 4b). A rise in MRS shifts the optimal harvest towards lower discount rate.

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A rise in CSV promotes higher optimal forest stock and keeps the forest stock close to the carrying 11 capacity (Figure 5a). A rise in discount rate delays this process to a higher CSV. optimal harvest is 12 higher at lower CSV. But with a rise in the discount rate, fuelwood harvest continues at a higher CSV, 13 though at a lower intensity (Figure 5b).  Gan et al. (2001). The marginal rate of change of optimal forest stock to discount rate was negative 36 (Gan et al., 2001). The marginal rate of change of optimal fuelwood harvest to discount rate was 37 positive, zero or negative whenever the forest stock was below, equal to or above the MSY stock 38 respectively. These outcomes indicate that forest stock has critical role to play in the decision of the 39 consumers (Clark, 1990).

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The marginal rate of change of optimal forest stock to CSV was negative or positive depending on 41 whether the MRS was less than or greater than the κ value. This indicates that a low energy-yielding substitute will promote deforestation even when forest is valued through CSV. On the other hand, a 1 high energy substitute complemented with high CSV will promote forest conservation. There was no 2 solution for marginal rate of change of optimal forest stock to CSV when MRS was equal to the κ value.

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Similarly, the marginal change in optimal harvest to CSV is rather ambiguous. It depended on whether 4 the MRS was greater than, lesser than or equal to the κ value and the sign of marginal change of forest

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The model discussed here is essentially classic and deterministic in nature. Moreover, the analysis is 43 based on the comparative statics of steady state conditions of the forest stock. These conditions were adopted for the ease of analysis, at the cost of ignoring the dynamic and stochastic nature of the forest 1 system. Consideration of steady state conditions is appropriate for long term equilibrium and sustainable 2 forestry. However, in many cases, a forest may not be or is not intended to be in steady state condition.

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The theoretical model and empirical example as discussed here provide relevant insights into the role 4 of discount rate, CSV and MRS on the optimal forest stock and optimal harvesting of fuelwood. It 5 showed that the MSY stock and ratio of energy values per unit mass of fuelwood to its substitute play 6 a critical role in the fate of forest and level of fuelwood extraction. Furthermore, it was observed that 7 by providing a relevant and subsidised energy option like LPG to the households can substantially 8 reduce fuelwood extraction and maintain the forest stock close to its carrying capacity. Also, high CSV 9 of the forest to the community can significantly reduce exploitation of the forest. Thereby, a state may 10 develop welfare schemes to provide subsidised and better substitutes like LPG, biogas or energy 11 plantation to the forest-dependent communities to protect the forest from further exploitation.

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Moreover, conservation agencies can encourage the CSV of the forest through festivals and folk culture 13 that promotes forest conservation.

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The present study provides certain insights into the relation of fuelwood substitute, CSV and forest 16 conditions. The optimal forest stock and optimal harvest critically depend on the ratio of the energy 17 value of the substitute to that of the fuelwood. In addition to that, level of CSV of the consumers' The author contributed to the conceptualization, construction, analysis and interpretation of the model.