# The Stable Equilibrium Analysis of Nonlinear Renewable Resource Stock Evolution

Authors: Gökhan Güven; Selim İnançli
Journal: Journal of Mathematics
 (2023)

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## Abstract

This paper provides the formal proofs of propositions concerning the stability analysis of steady-state equilibrium in autarky with open-access renewable resources. We assume three different resource environments in which excessive harvesting, pollution, and simultaneous coexistence of these interacting pressures on resource stocks are considered. The propositions mentioned on J.A. Brander and M.S. Taylor,Open access renewable resources: trade and trade policy in a two-country model, Journal of International Economics, 44 (2), 181–209, 1998; B.R. Copeland and M.S. Taylor, Trade, spatial separation, and the environment, Journal of International Economics, 47 (1), 137–168, 1999; and H. A. Rus, Renewable resources, pollution and trade, Review of International Economics, 24 (2), 364–391, 2016 have been shown only on the diagrams and also implicitly explained. This paper aims to contribute the renewable open-access resources literature by providing formal solutions of stable equilibrium analysis in a closed economy within the long run.


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## Body

## 1. Introduction

The open-access resource problem and steady-state equilibrium in the closed economy structure have been discussed in various studies in the literature. It is known that weakly defined property rights pertaining to resource extraction cause the resource stock to suffer from depletion and degradation problems because of overharvesting by economic agents [1].Brander and Taylor [2] analyzed the implications of trade liberalization in a Ricardian general equilibrium framework in which two countries produce two different goods, one of which uses nationally owned open-access natural resources. Brander and Taylor [3] constructed a model in which international trade led to a decline in the utility levels of an open economy with weaker property rights, exporting goods that are produced using natural resources. Brander and Taylor [4] extended this consequence to the case of a country with an ill-defined environment-controlling system (with a consumer country having laxer environmental regulations) importing a resource-intensive good in a steady state. Chichilnisky [5] carried out a dynamic analysis by indicating that a country imposing relatively weak-defined management regimes on resources exported resource good in the steady-state equilibrium. Chichilnisky [6] also demonstrated the features of North-South trade, given a certain assumption where two countries share the same initial parameters except for property rights regimes and national resource stock levels. Building on Brander and Taylor’s models, Hannesson [7] modified their model, changing the constant returns to scale assumption in the less resource-intensive sector with diminishing returns in the polluter industry. This changing initial condition diversified the post-trade welfare, resulting in a steady-state equilibrium. Yanase and Dong [8] developed a small-open economy model where intermediate goods (resource goods) were used to produce final goods, which is also an open-access externality problem related to natural resources. Under this assumption, the extent to which the resource stocks were depleted depended on the labor allocation and the natural growth function of the stock of environmental resources.Benchekroun et al. [9] gave a more recent theoretical analysis of the open-access problem on common property goods. In this regard, they concluded that more restricted environmental control and monitoring must be imposed to alleviate the degradation of natural resources stemming from the combination of trade openness and laxer property rights. Upmann and Gromov [10] aimed to demonstrate the structure of optimal efforts for harvesting in the open-access problem under different scenarios. Takarada et al. [11] examined the open-access resources with shared renewable resource assumption under no resource management. This paper characterized that natural resources were not only accessible by local residents, implying that resources were internationally common. By classifying nations into high- and low-income groups, Dube and Quaas [12] also looked at the effects of trade on utility in the context of restricted and unrestricted access to natural resources. According to their research, high-income countries can gain more from international trade than low-income ones due to the less stringent exertion of resource management controls. Eisenbarth [13] sheds light on the root causes of overfishing in the open-access problem by discussing the increasing demand for resource goods after trade liberalization and presenting novel empirical results.Another negative externality with harmful pressure on resource stock is the pollution generated by industrial activities. This pollution externality damages the environmentally sensitive sectors by creating interindustry detrimental influences on available resource stock to reduce productivity and lead to separated incompatible industries [14]. Copeland and Taylor [15] analyzed that economic growth might be the primary factor of excessive industrial pollution compared to the agglomeration of polluter industries in a specific region. Burrows [16] studied the impacts of nonconvexities in the social production set concerning pollution externality in the form of external costs of manufacturing firms. Carrier and Krippl [17] studied the influence of sulfur extraction (pollution emission) in forest areas in European countries (renewable resource stocks) in economic terms. Tahvonen and Kuuluvainen [18] considered both industrial pollution’s detrimental and favorable effects. They analyzed the relationship between the undesirable costs of externalized pollution emissions and the distortion of environmental quality. Li and Yanase [19] presented a model in which excessive harvesting problems induced by open-access externality and industrial pollution took into consideration simultaneously. Based on the relative impact of these two kinds of twin pressures, countries were categorized into two different types, and they concluded that in the case that industrial pollution dominated the resource extraction in damaging terms on resource industry productivity, the production possibilities frontier was drawn in convex-shaped to demonstrate the dominancy of industrial pollution externality. Wang [20] considered not only industrial pollution but also agricultural pollution and incorporated them into the general equilibrium model to figure out the implications of pollution externalities on resource and labor-intensive goods. Karp and Paul [21] also investigated the cross-sectoral pollution externalities between industrial industry and agricultural activities in the context of external economies where manufacturing good was assumed to be produced by generating deteriorating side effect on agricultural outputs (see also [22] for water pollution in the similar context). Li et al. [23] presented a research paper by constructing a general equilibrium model to analyze the importance of agricultural pollution on agricultural production, the development of the rural economy, and wage inequality. Depending on their numerical stimulation, the production process of resource-intensive goods could cause agricultural pollution because of excessive usage of chemical pesticides and fertilizer that damage natural resources (see also [24] for a detailed discussion about industrial pollution and its impacts on economic development). Intraindustry externality problems (excessive harvesting of resource stock) and pollution have appeared simultaneously in the real-life framework. Thus, it is required to consider these two twin pressures together to get closer to real-life examples [25]. In this benchmark paper, the author assumes that the renewable resource stock is simultaneously subject to these two interrelated damaging activities.The articles by [2, 3, 25], respectively, have investigated the open-access resource problem in different settings, and also, the authors have solved the steady-state equilibrium in a closed economy perspective. However, these papers do not provide formal proof of stability analysis of steady-state equilibrium. As in [25], they prefer to analyze the stable equilibrium notion on the graph.The purpose of this paper is to provide formal proofs of the stability analysis of steady-state equilibriums based on the models constructed by these three articles. We restrict our attention to the autarkic steady-state stability analysis within an economy. The authors of [2, 3] have provided a general proof notion; however, they did not analyze the stable, steady-state equilibrium notion as we have done in this paper. Compared to [25], in which he did only provide a graphical explanation of the steady-state equilibria, we extend the stability analysis by using algebraic methods, particularly the features of nonlinear differential equations.Our analytical results are consistent with the real application of the stability analysis of the steady-state equilibrium. The authors of [26] documented that the environmental stability of an ecological system requires a stability parameter of more than 0.67 (value of Ces. st). The south taiga forest-steppe conditions are considered to have ecological stability (Ces. st. was 0.69), which satisfies the steady-state stability. The authors of [27] examined 16 biophysical indicators to understand a “steady-state economy.” This paper implied that a few countries could achieve a defined steady-state economy, which identified under the condition ecological constraints take place. The authors of [28] investigated the relationship between natural resource extraction and labor recruitment and estimated that high-fecundity steady-state equilibrium given optimal harvesting, which provides stability in the long run. The authors of [29] showed that optimal-stock policies could guarantee that during a defined optimal path over time, the economic system would converge to a patterned optimal steady-state equilibrium from any beginning production set. The authors of [30] questioned the possibility of the collapse of open-access natural resources by opening up to trade. This study predicted that the mismanagement of environmental regimes with trade could lead to a decline in resource extinction. Following an exogenous shock, the economy could approach the steady-state stock level and welfare outcomes within specific harvesting conditions (see also [31, 32]).The rest of the paper is organized as follows. Section2 sets up a framework in which only overexploitation of resource stock is assumed as a negative externality. Section 3 analyzes the open-access resources under the pollution externality. Section 4 investigates these two interacting effects simultaneously, and in each section, we provide a formal explanation of stability analysis of steady-state equilibria.


## 2. The Open-Access Renewable Resource Model with Excessive Harvesting

In this section, the renewable resources (forest or fish species) are subject to weakly defined property rights, and also, no resource management regime has been enforced. We analyze the small closed economy with an open resource problem. It is well-known in the literature that open-access problems lead to overexploitation of renewable resource stock because economic agents are not forced to bear all the costs incurred throughout the production process, which implies that agents produce more than the optimal level in the absence of such an externality.Let us define the basic structure and functions of the model. The economy produces and consumes two goods.H is the resource good from harvesting of resource stock. M is the manufacturing goods, which can be considered other goods representing the whole goods except for H. In addition to renewable resource stock, the economy also has another input factor used in both production processes; labor denoted by L. Considering the resource growth, the renewable resource stock level at time t is identified by St. The natural growth rate of a renewable resource, denoted by G, is a logistic function depending on the resource stock:(1)GSt=rS1−SK,in which r is the intrinsic growth rate of resource stock and K refers to the maximum carrying capacity an economy can reach. The change in resource stock level in time t is defined as follows:(2)dSdt=S˙=GSt−Ht,where Ht is the harvesting activity detrimentally influencing the existing resource stock. We assume that the harvesting industry produces its output based on Schaefer production function, which is(3)HS=qSLH,q is the productivity parameter and LH is the amount of labor devoted to the production of resource goods. Manufacturers use only labor as an input factor, and its production process shows the feature of the constant returns to scale:(4)MS=LM,where LH+LM=L. From the zero-profit condition in the manufacturing sector, it is found that pM=w=1. Labor is the only input factor that has to be paid for under open-access conditions. As the free entry condition holds here and labor is mobile between two sectors, which provides unitary wage for the homogenous labor, the relative price of the resource goods pH=p in autarky must be derived with the equation ΠH=0=pH−wLH. Here, w=pqS is found, or(5)p=1qS,w=1.From the demand side of the economy, a representative agent’s utility function is characterized by the Cobb–Douglas utility function as follows:(6)u=hβm1−β,where β is the taste parameter for resource good and the budget constraint is w=ph+m. Maximizing equation (5) subject to equation (6), we obtain the demand functions as follows:(7)HD=βwpL,MD=w1−βL.The full employment condition provides the material balance, implying thatHS=HD in Ricardian temporary equilibrium. By using equation (5) and equation (6) together, it yields the temporary harvesting level for a constantly renewable resource stock as follows:(8)Ht=qβLSt.Substituting (1) and (8) into (2), we find the dS/dt as an open form:(9)dSdt=GSt−Ht=rS1−SK−qβLS=0.At steady state, the harvesting given by (8) must equal to natural resource stock given by (2), yielding steady-state resource stock levels as follows:(10)S1A=0,S2A=K1−qβLr.Proposition 1.
In autarky, larger rootS2A=K1−qβL/r is a globally stable and unique steady-state equilibrium. After any small perturbations leading to equilibrium deviating from steady state, the economy certainly gets back its beginning steady state in the long run.Proof.
The dynamic evolution of resource stock is governed by substituting (1) and (8) into (2) at steady state, as seen in Figure 1. Recalling the evolution of t:(11)dSdt=GSt−Ht=rS1−SK−qβLS,which is a nonlinear differential equation. Rearranging equation (11) to transform into a linear equation, firstly rewrite equation (11) as follows:(12)dSdt+SqβL−r=−rKS2,in which dS/dt=dy/dt, SqβL−r=Ry, and −r/KS2=Tym. Equation (12) is arranged in the form of dy/dt+Ry=Tym. As ym=S2, implying that m=2. Readjusting equation (12) by dividing both sides with S2, it yields that(13)dSdtS−2+qβL−rS−1=−rK.
By using the change of variables method, we obtain thatz=y1−m=S−1, which results in that −dz/dt=dS/dtS−2. By parallel reasoning, it can be easily seen that qβL−rS−1=qβL−rz. Substituting the last two equations into equation (13),(14)−dzdt+qβL−rz=−rK.
We need to transform equation (14) as an exact differential equation for simplicity. Thus, it is required to multiply both sides of equation (14) by 1−mdt=−dt, which results in as follows:(15)dz+r−qβLz−rKdt=0.
Equation (15) does not satisfy the exact differential conditions yet. To achieve it, let us define an integral factor denoted by I. Using the integral factor in equation (15), we obtain that Idz+Ir−qβLz−r/Kdt=0 in which M=∂Fz,t/∂z=I and N=∂Fz,t/∂t=Ir−qβLz−r/K from Young's theorem. That is, ∂M/∂t=∂I/∂t and ∂N/∂z=Ir−qβL are the partial derivatives written in open form. It is a well-known fact that ∂M/∂t=∂2Fz,t/∂t=∂N/∂z=∂2Fz,t/∂z must be satisfied to define equation (15) as an exact differential equation, which yields(16)∂I∂t=Ir−qβLimplying that∂I/∂tI=r−qβL,that is a constant. This means that the integral factor is growing up in a constant rate. Rearranging equation (16), we find the integral factor as follows:(17)It=Ae∫r−qβLdt=e∫r−qβLdt.
Substituting equation (17) into equation (15) yields(18)e∫r−qβLdtdz+e∫r−qβLdtr−qβLz−rKdt=0.
It is known that∂Fz,t/∂z=M. Taking integral of both sides to reach the simplified version of Fz,t, it yields Fz,t=∫Mdz+Ψt in which M=e∫r−qβLdt. Rearranging the equality, it is seen that(19)Fz,t=ze∫r−qβLdt+Ψt,where Ψt represents all terms not involving the term z. At the same time, we can use the ∂Fz,t/∂t=N, then yielding(20)zr−qβLe∫r−qβLdt+∂Ψt∂t=e∫r−qβLdtzr−qβL−rKe∫r−qβLdt.
We need to identify theΨt from the expression (20). Taking integral of both sides again,(21)Ψt=−∫rKe∫r−qβLdtdt.
Substituting equation (21) into equation (18), which is written as follows:(22)Fz,t=ze∫r−qβLdt+−∫rKe∫r−qβLdtdt=c.
Now,zt can be found by rearranging equation (22) as follows:(23)zt=Ae−∫r−qβLdt+rKr−qβL.
In the beginning, we have assumed thatzt=St−1. Thus, we can switch to St and(24)St=1Ae−r−qβLt+r/Kr−qβL.
Let us redefine equation (24) by considering starting stock level at t=0, which yields(25)A=1St=0−rKr−qβL,in which St=0=S0,r/K=b and r−qβL=a can be represented acknowledged. Thus, this rearrangement provides A=1/S0−b/a. Now, substituting the last equality into equation (24) yields(26)St=11/S0−b/ae−r−qβLt+b/a,1S0−ba=c.
Rearranging equation (26) gives the general form of evolution of resource stock as St=1/b/a+ce−at. For stability analysis of the steady-state equilibrium, firstly rewrite the St without using any abbreviations. Thus, we obtain(27)St=1r/Kr−qβL+1/S0−r/Kr−qβLe−r−qβLt,in which SA=K/rr−qβL can be used to rearrange equation (27), implying that(28)St=S0SAS0+SA−S0e−r−qβLt=S0SAS0+SA−S0e−r−qβLt−1.
The stability analysis examines whether the economy starting from any point in timet=0 will converge to the steady-state resource stock level over time. For this reason, we need to focus on the change of equation (28) over time. Thus,(29)∂St∂t=S0SA⏟>0S0+SA−S0e−r−qβLt−2⏟>0SA−S0r−qβL⏟>0.
Therefore, the sign of∂St/∂t depends on the sign of SA−S0. If(1)
S0<SA results in ∂St/∂t>0, implying that when the initial stock level is lower than the steady-state resource stock, renewable resource stock monotonically converges to steady-state stock(2)
S0>SA results in ∂St/∂t<0, implying that when the initial stock level is higher than the steady-state resource stock, renewable resource stock monotonically converges to steady-state stock again
As a result, the economy deviated from its steady-state stock equilibrium because of any external shock can get back its steady-state equilibrium in the long run.
The proof is completed.Figure 1
The stability analysis of steady-state equilibriums.


## 3. The Renewable Resource Model with Pollution

We now consider the renewable resource in which the only detrimental pressure on resource stock is pollution. The industrial pollution, denoted byZ, is generated by manufacturing facilities situated adjacent to environmental resource stock, which creates a flow of pollution influencing the level of resource stock, S, and impairs productivity in resource-based sectors. In the case of damaging the productivity of the resource industry, it can also create a possibility of depleting resource stocks’ long run sustainability. As noted previously, there are two primary factors: labor L and resource stock S. The stock level can be depleted or enhanced over time based on the flow of pollutants and the intrinsic growth rate of the stock. Assume that the change in resource stock is defined as follows:(30)dSdt=S˙=GSt−Zt.It is also assumed thatγ is the pollution intensity parameter, implying that one unit of manufactures yields γ unit of pollution. We know that M=LM, and then, Z=γM is found. The amount of pollution can be thought of as an average detrimental polluter generated by industries on forestry or fish and is also unattached from available resource stock. The environmentally sensitive industry has a production process using renewable resource stock, which is considered harvesting activity defined in the previous model such as forestry. Hence,(31)H=FSLH,where FS=Sϵ with 0<ε<1. We differentiate the harvesting supply function by decreasing returns to scale to resource stock. GSt is also linear growth function as defined as follows:(32)GSt=rK−S.The production and demand sides of the model in this section are similar to the previous model written previously. In an autarkic economy, sustainable production frontier can be obtained by substituting equation (30) into equation (32) with manufacturing production equilibrium as follows:(33)dSdt=S˙=rK−S−γM.A steady state corresponds todS/dt=0. Thus, steady-state equilibrium in autarky is(34)SA=K−γMM,γr=γM.This tells us the steady-state relationship between the manufacturing sector yielding pollutants negatively affecting the stock level and renewable resource stock. And we have the followings:Proposition 2.
In autarky,SA=K−γ/rM is a globally stable and unique steady-state equilibrium. The deviated resource stock can get back its steady-state equilibrium point in the long run.Proof.
The evolution of stock within time is governed by equation (33). We will show that the steady state observed at a point in which the supply and demand of manufacturing goods are equal to each other is stable and unique. Manufacturing demand is found in equation (7) as MD=1−βL. Substituting MD into equation (33), we must have(35)dsdt=rK−S−γ1−βL.
Rearranging equation (35) yields that dS/dt+rSt=rK−γ1−βL, a differential equation indicating the evolution of the resource stock over time, starting from any initial stock levels. The general form of the differential equation is dy/dt+ay=b. The first-order nonhomogeneous differential equation has consisted of a combination of two different solution notions St=Stc+Stp:(1)
dS/dt+rSt=0, implying that dS1/S=−rdt. Taking the integral of both sides yields lnS+c1=−rt+c2 in which c=c2−c1. Having rearranged the terms for simplifying, we find S=e−rtA where A=ec. Thus, Stc=Ae−rt is the general solution corresponding to the dS/dt+rSt=0.(2
)We now focus on the particular solution of the first-order nonhomogeneous differential equation. Assume that the resource stock is defined in the simplest way such asSt=z in which z is a constant. We know that St=z provides the steady-state equilibrium in which dS/dt=0, so we have known that z=K−γ/rM. This is the particular solution that needs to be converged in the long run, so Stp=K−yMM.
St is the general solution identified as follows:(36)St=Stc+Stp=Ae−rt+K−yMM.
Before solving equation (36), A must be obtained at time t=0 in which the steady-state equilibrium is S0=K−yMM. Hence, St=0=Ae0+S0 can be transformed into A=St=0−S0. Substituting the last equation into equation (36), we have(37)St=S0+St=0−S0e−rt.
Stability analysis of equation (37) yields that resource stock growth at the steady-state equilibrium is equal to zero, that is, ∂St/∂t=0. S0 is the resource stock level to be converged after deviating the equilibrium, and St=0−S0e−rt is the deviation term emerged after an external shock hits the economy in the autarky. It is found that as >0,(38)limt⟶∞Stc=limt⟶∞Ae−rt=0.
Consequently,St converges to the steady-state equilibrium value S0 over time. The steady-state equilibrium is stable. Regardless of our initial stock level, the stock level will converge towards equilibrium in the long run.
The proof is completed.Proof.
St=S0+St=0−S0e−rt is defined above as a solution of the first-order nonhomogeneous differential equation. Considering the change of this equation over time, we have obtained that ∂St/∂t=−rSt=0−S0e−rt.(1)
IfSt=0>S0, which means that ∂St/∂t<0 will be obtained. Therefore, when the initial stock level is above steady-state equilibrium, convergence towards the equilibrium takes place as a result of negative stock growth over time.(2)
IfSt=0<S0, implying that ∂St/∂t>0 will be result in. As a result, when the initial stock level is below steady-state equilibrium, convergence towards the equilibrium occurs as a result of positive stock growth over time.
The proof is completed.


## 4. The Open-Access Renewable Resource Model with Overharvesting and Pollution

In this section, we integrate both detrimental influences defined in previous models together. It is more logical to assume that renewable resource stocks are subject to two different interacting problems simultaneously in the real world. As noted aboveHt, harvesting activity level is related to the renewable resource stock. As known in the literature, when property rights are weakly defined, agents tend to harvest the stock excessively, which is the first negative externality observed in the model. Zt, pollution level shows up as a result of the manufacturing production, which is analyzed in Section 2 in detail. As [25] mentioned, combining these two kinds of pressures has a negative degradation and depletion influences on resource stocks. These harmful activities on resource stocks commonly coexist together, so they must be considered jointly in the model setup.Depending on the assumptions, we identify the stock of renewable resources according to(39)dSdt=S˙=GSt−Ht−Zt,where GSt and Ht are, respectively, similar to equation (1) and equation (3). The flow of pollution, Zt, produced in the manufacturing process is defined as in Section 2. We also assume that production and demand conditions are analogous to Sections 2 and 3. For this reason, we do not redefine the functional forms here. Given the Cobb–Douglas utility preferences, zero profit conditions, and free labor mobile between sectors, Ricardian temporary harvesting and pollution levels in the short-run autarky equilibrium are as follows:(40)Ht+Zt=qβLS+γ1−βL.The open-access steady-state condition is defined under the condition thatdS/dt=S˙=0. As can be seen from equation (39), by substituting equations (1) and (40), we have(41)dSdt=S˙=rS1−SK−qβLS+γ1−βL=0,which yields two possible autarkic steady-state renewable stock levels. The larger root is S1A=Kr−qβL+qβL−r2−4r/Kγ1−βL/2r, and the other one is identified as S2A=Kr−qβL−qβL−r2−4r/Kγ1−βL/2r. (According to [3], it is necessary to define that the two real roots are interior solutions that the pollution intensity is not too high, or γ<Kr−qβ4r1−βLKr−qβL−qβL−r2−4r1−βL/Kγ1−βL/2r as identified in the reference model). The steady states are pointed in Figure 2 as follows.Figure 2
The stability analysis of steady states in autarky.Proposition 3.
In autarky, the larger rootS1A=Kr−qβL+qβL−r2−4r/Kγ1−βL/2r is a locally and asymptotically stable and unique steady-state equilibrium. The other root S2A=Kr−qβL−qβL−r2−4r/Kγ1−βL/2r not locally and asymptotically stable, leading to extinction or the larger root S1A when the external shock hits the equilibrium.Proof.
To demonstrate all the algebra details, we do not have enough space here. So, we only identify the basic steps of the proof. As known,dSt/dt=rS1−S/K−qβLSt−γ1−βL=0 is a nonlinear differential equation. Rearranging the equation, dSt/dt+qβLS−rS1−S/K+γ1−βL=0 is found. We need to identify the equation as an exact differential equation for simplicity. For this reason,(42)1dSt+qβLSt−rS1−SK+γ1−βLdt=0.
∂FS,t/∂S=1=TS,t and ∂FS,t/∂t=qβLSt−rS1−S/K+γ1−βL=RS,t are the functions to be used to associate equation (42) to exact differential equation form. However, their cross partial derivatives are not equal to each other ∂TS,t/∂t≠∂RS,t/∂t. Hence, integration factor ∐S must be used. Rearranging the (4.4),(43)1∐SdSt+qβLSt−rS1−SK+γ1−βL∐Sdt=0.
To describe equation (43) as the exact differential equation,(44)TS,t∐SdSt+RS,t∐Sdt=0,must be satisfied. So, the condition for the exact differential equation:(45)∂∂tTS,t∐S=∂∂SRS,t∐S.
We need to determine the integration factor in structural form by using equation (44). We know that(46)∂∂tTS,t∐S=0.(47)∂∂SRS,t∐S=∂∐S∂SqβLS+γ1−βL−rS1−SK+∐SqβL+2rSK−r.
By equating equations (46) and (47) each other, ∂∐S/∂SqβLS+γ1−βL−rS1−S/K+∐SqβL+rS/K−r1−S/K=0 is identified. We need to identify the integral factor first. Defining the integral factor as a growth function, which yields(48)∂∐S/∂S∐S=−βLq+rS/K−r1−S/KβLqS+γ1−βL−rS1−S/K=−βLKq+Kr−2rSKLγ1−β+SqβLK−Kr+rS.
By taking integral of both sides to obtain integral factor in simplified form, we have(49)∫∂∐S/∂S∐SdS=ln∐S+c1.(50)∫−βLKq+Kr−2rSKLγ1−β+SqβLK−Kr+rSdS=−lnβKLqS+KLγ1−β+rSS−K.
Equating equations (49) and (50) to each other,(51)Ln∐S=−lnβKLqS+KLγ1−β+rSS−K.
After making some arrangements in that equation,(52)∐S=βKLqS+KLγ1−β+rSS−K−1,is obtained. Equation (52) is substituted into equation (44) RS,t∐S+∂St/∂t∐STS,t=0:(53)1βKLqS+KLγ1−β+rSS−K⏟∐SqβLS+γ1−βL−rS1−SK⏟RS,t+∂St∂t1⏟TS,t1βKLqS+KLγ1−β+rSS−K⏟∐S=0.
∐SRS,t can be reorganized as follows:(54)1βKLqS+KLγ1−β+rSS−KqβLS+γ1−βL−rS1−SK=1K.
Plugging (54) into equation (53), we obtain 1/K+∂St/∂t1/βKLqS+KLγ1−β+rSS−K=0. Rearranging that expression,(55)dt1K+dSt1KLγ1−β+βKLqSt+rStSt−K=0,where 1/K=PS,t and 1/KLγ1−β+βKLqSt+rStSt−K=QS,t.
According to Young's theorem,∂PS,t/∂S=∂QS,t/∂t=0 condition is already satisfied. Equation (55) is treated as an exact differential equation, and the solution will be followed by using equation (55). It is known that FS,t=∂FS,t/∂tdt+∂FS,t/∂SdS=0 must be held and here assuming that ∂FS,t/∂t=PS,t,∂FS,t/∂S=QS,t, so FS,t=c must be constant. By taking the integral of ∂FS,t/∂t=PS,t=1/K in both sides,(56)∫∂FS,t∂tdt=∫1Kdt=FS,t=t1K+gS,where gS is a closed functional form referring to all terms not including S.(57)t1K=βLqStβKLqS+KLγ1−β+rSS−K−rS1−S/KtβKLqS+KLγ1−β+rSS−K+Lγ1−βtβKLqS+KLγ1−β+rSS−K,can be defined in more simplified form, and characterizing the right-hand side of the equation as ΩS,t, equation (57) is being transformed to(58)FS,t=ΩS,t+gS,where gS can be clarified by taking the derivative of equation (58) and using the ∂FS,t/∂S=QS,t equality. That is,(59)∂FS,t∂S=∂∂SβLqStβKLqS+KLγ1−β+rSS−K−rS1−S/KtβKLqS+KLγ1−β+rSS−K+Lγ1−βtβKLqS+KLγ1−β+rSS−K+gS.
The partial derivatives of ordered terms in equation (59) can be shown separately as follows:(60)∂∂SβLqStβKLqS+KLγ1−β+rSS−K=−βLqSβKLq+rS+rS−KtβKLqS+KLγ1−β+rSS−K2+βLqtβKLqS+KLγ1−β+rSS−K.(61)∂∂SrS1−S−KtβKLqS+KLγ1−β+rSS−K=rS1−S/KβKLq+rS+rS−KβKLqS+KLγ1−β+rSS−K2+rStKβKLqS+KLγ1−β+rSS−K+r1−S−KtKLqS+KLγ1−β+rSS−K.(62)∂∂SLγ1−βtβKLqS+KLγ1−β+rSS−K=Lγ1−ββKLq+rS+rS−KtβKLqS+KLγ1−β+rSS−K2.
Substituting the last three equations into equation (59) and using dgS/dS, we have(63)∂FS,t∂S=QS,t=−βLqSβKLq+rS+rS−KtβKLqS+KLγ1−β+rSS−K2+βLqtβKLqS+KLγ1−β+rSS−K−rS1−S/KβKLq+rS+rS−KtβKLqS+KLγ1−β+rSS−K2−rStKβKLqS+KLγ1−β+rSS−K−r1−S/KtKLqS+KLγ1−β+rSS−K−Lγ1−ββKLq+rS+rS−KtβKLqS+KLγ1−β+rSS−K2+dgSdS=1KLγ1−β+βKLqSt+rStSt−K.
Simplifying the expression equation (63) and find that(64)dgSdS=1KLγ1−β+SβKLq−Kr+rS.
Taking both sides integration to obtain a simplified version of equation (64),(65)∫dgSdSdS=gS=1KLγ1−β+SβKLq−Kr+rS.
Rearranging equation (65) gives us ∫1/KLγ1−β+SβKLq−Kr+rS2dS. In this phase of the proof, the denominator is being completed the square, and changing the variable method is used to simplify the integration. Rewriting the denominator as the form of ax2+bx+c, which is defined as follows:(66)rS+βKLq−Kr2r2−βKLq−Kr24r−KLγ1−β=ax2+bx+c.
Simplifying equation (66) to be more apparent,(67)∫1rS+βKLq−Kr/2r2−βKLq−Kr2/4r−KLγ1−βdS.
It is time to implement a change in the variable method. Assume that∐=rS+βKLq−Kr/2r, we have d∐=rdS and dS=1/rd∐. Redefining equation (67),(68)∫1∐2−βKLq−Kr2/4r−KLγ1−β1rd∐=1r∫1∐2−βKLq−Kr2/4r−KLγ1−βd∐.
Putting the common denominator parentheses by usingβKLq−Kr2/4r−KLγ1−β and making some arrangements in the denominator for simplification, we have(1)
1/r∫1/βKLq−Kr2/4r−KLγ1−β4r∐2/βKLq−Kr−4rKLγ1−β−1d∐(2)
1/r∫1/βKLq−Kr2−4rKLγ1−β/4r4r∐2/βKLq−Kr2−4rKLγ1−β−1d∐(3)
1/r∫4r/βKLq−Kr2−4rKLγ1−β4r∐2/βKLq−Kr2−4rKLγ1−β−1d∐
Taken out all terms not dependent on∐ of integral,(69)1r4rβKLq−Kr2−4rKLγ1−β∫14r∐2/βKLq−Kr2−4rKLγ1−β−1d∐.
Using the change in variable method again by definings=2r∐/βKLq−Kr2−4rKLγ1−β, and we obtain s=2r/βKLq−Kr2−4rKLγ1−βd∐. It is important to note that d∐ is obtained by putting all the terms taken out of the integral back in integral. Now, rewrite equation (69)(70)∫2rβKLq−Kr2−4rKLγ1−βd∐⏟″equalstods.″2βKLq−Kr2−4rKLγ1−β1s2−1.
Equation (69) can be reformed, which holds that(71)2rβKLq−Kr2−4rKLγ1−β∫1s2−1ds=−2βKLq−Kr2−4rKLγ1−β∫11−s2ds.
It is a well-known fact that the general form of hyperbolic functions is−∫1/1−s2ds=−tanh−1s+c. Therefore, rearranging equation (71) based on hyperbolic function, we get rid of integration as follows:(72)gS=−2tanh−1SβKLq−Kr2−4rKLγ1−β+c.
Now, let us put back the expressions we found in the variable change phase. We know thats=2r∐/βKLq−Kr2−4rKLγ1−β and substituting it into equation (72) gives us:(73)−2tanh−12rL/βKLq−Kr2−4rKLγ1−ββKLq−Kr2−4rKLγ1−β=gS
And now, let us substitute the∐=rS+βKLq−Kr/2r assumed before into equation (73), gS can be obtained as open form:(74)gS=−2tanh−12rrS+βKLq−Kr/2r/βKLq−Kr2−4rKLγ1−ββKLq−Kr2−4rKLγ1−β=−2tanh−1βKLq−Kr+2rS/K¯Kr−βLq2−4rLγ1−βK−4rLγ1−β+Kr−βLq2.
Equation (58) can be redefined explicitly by putting equation (74), and it is given(75)FS,t=c1=−2tanh−1βKLq−Kr+2rS/KKr−βLq2−4rLγ1−βK−4rLγ1−β+Kr−βLq2+βLqStβKLqS+KLγ1−β+rSS−K−rS1−S/KtβKLqS+KLγ1−β+rSS−K+Lγ1−βtβKLqS+KLγ1−β+rSS−K.
From equation (75), the expression St defined in the model as the resource stock level on which we apply the stability analysis. By skipping intermediate steps for simplicity, St is given as(76)St=12r1Yβ2L2K3/2tanh12Zq2−2βLrK3/2tanh12Zq+r2K3/2tanh12Z−4LrKγ1−βtanh12Z+12rr−qβLK,where(77)Z=−β2L2K3/2c1q2Y+β2L2tKq2Y−2βLrKqtY+2βLrK3/2c1qY−r2K3/2c1Y+4LrKγ1−βc1Y−4Lrtγ1−βKY+r2tKY,Y=r−βLq2K−4Lrγ1−β.
In this step, before proceeding further, let us show how to express the steady-state resource stock level in terms ofY. The larger root of steady-state equilibrium S1A can be redefined as(78)S1A=12rKr−qβL+1KY,which will be involved our analysis in further steps. The simple version of equation (76) can be given(79)St=12r1Yβ2L2K3/2q2−2βLrK3/2q+r2K3/2−4rLKγ1−βtanh12Z+12rr−qβLK.
The expression in the square brackets is the coefficient oftanh1/2Z. Editing the coefficient and converting it into a function dependent on Y, we have(80)β2L2K3/2q2−2βLrK3/2q+r2K3/2−4rLKγ1−β=Kβ2L2Kq2−2βLrKq+r2K−4Lrγ1−β=YK.
Based on these expressions, the equation showing the evolution of the resource stock in the time (equation (79))is to be written in a simpler form:(81)St=12r1Ytanh12ZYK+12rr−qβLK.
Now, let us get back to equation (77) and rearrange it to obtain a more comprehensible functional form:(82)ZYY=−β2L2K3/2c1q2Y+β2L2tKq2Y−2βLrKqtY+2βLrK3/2c1qY−r2K3/2c1Y+4LrKγ1−βc1Y.−4Lrtγ1−βKY+r2tKY
We focus on the numerator of equation (82). We can express the numerator as the sum of two different expressions, such as Znumerator=Z1+Z2(83)Z1=−β2L2K3/2c1q2+2βLrK3/2c1q−r2K3/2c1+4LrKγ1−βc1=−c1β2L2K3/2q2−2βLrK3/2q+r2K3/2−4LrKγ1−β=−c1YK,(84)Z2=β2L2tKq2−2βLrKqt−4Lrtγ1−βK+r2tK=tr2K−2βLrKq+β2L2Kq2−4Lrγ1−βK=tYK.
We know thatZ is given as =Z1+Z2/Y; thus,(85)Z=tY/K−c1YKY=Yt−c1K.
Equation (81) can be written by using the simpler form of Z found in equation (85)(86)St=12r1Ytanh12Yt−c1KYK+12rr−qβLK.
The terms situated outside of the square brackets1/2rr−qβLK can be arranged to emulate expression in equation (78). Firstly, add and subtract ±1/2rYK into equation (86), we have(87)St=12r1Ytanh12Yt−c1KYK+12rr−qβLK+12rYK−12rYK.
Rearranging equation (87) based on equation (78) as the following expression:(88)St=12r1Ytanh12Yt−c1KYK+S1A−12rYK.
The constant termc1 needs to be converted to a specific and unique value. Solve equation (87) for time t=0 as follows:(89)St=0=12r1Ytanh12Y−c1KYK+SA−12rYK.where SA=S1A and also, S0=St=0. Rearranging equation (89), we obtain(90)−S0−SA2rY+KYYK=tanh12Yc1K.
Redefining both sides of equation (90) by using tanh−1, we obtain −tanh−1S0−SA2rY/YK+1=1/2YKc1. Now, a specific and unique c1 is as follows:(91)c1=A=2YK−tanh−1S0−SA2rYYK+1.
Substituting equation (91) into stock evolution equation (88), the renewable stock level of the economy t:(92)St=12r1Ytanh12Yt+2YKtanh−1S0−SA2rYYK+1KYK+S1A−12rYK,where S0 is the level that occurred at time t=0. To understand how the reaction will appear when the small perturbation hits the larger root S1A equation (92) needs to be analyzed by considering time t approaches infinity. It can be easily shown that limt⟶∞tanht=1. Then,(93)limt⟶∞St=limt⟶∞12r1Ytanh12Yt+2YKtanh−1S0−SA2rYYK+1KYK+S1A−12rYK,where limt⟶∞tanh1/2Yt+2/YKtanh−1S0−SA2rY/YK+1K=1. Rearranging Equation (91), finally, the St is found as follows:(94)St=12r1YYK+S1A−12rYK,implying thatSt=S1A.
Equation (94) refers to the fact that in case of any deviation from the steady-state equilibrium point S1A, the economy can converge back to equilibrium in the long run. As a result, S1A is globally stable and unique.
For proving the other steady-state equilibriumS2A is not stable, the same logic will be applied. Until the point St=1/2r1/Y2tanh1/2Yt−c1KYK+1/2rr−qβLK+1/2rYK−1/2rYK. is obtained, mathematical steps are the same. After this stage, S2A=1/2rKr−qβL+1/KY is plugged into the above equation and is given that(95)St=12r1Ytanh12Yt−c1KYK+12rKr−qβL+1KY+12rYK.
It is known that the intermediate steps are being implemented as done previously, and the steady-state equilibrium is defined as time proceeds to infinity:(96)St=12r1YYK+S2A+12rYK,implyingSt=S2A+1rYK.
Equation (96) implies the fact that in case of any deviation from the steady-state equilibrium point S2A, the economy cannot turn back the equilibrium in the long run. Equilibrium is deviated as 1/rYK. Eventually, S2A is an unstable equilibrium.
The proof is completed.


## 5. Conclusion

A central goal of the paper is to outline the formal proofs of the propositions about the stability of autarkic economy steady-state equilibriums identified in seminal papers [2, 15, 25]. Motivated by two considerations, to date, the environmental economics literature has often investigated the autarkic steady-state analysis by using a graphical display, and that global and unique steady-state equilibrium in autarky has abstracted from formal and general proof. This paper provided a mathematical proving method for three different well-known renewable resource models. It is shown, as regards the stability and sustainability of renewable resource stocks, that the larger equilibrium resource level in an autarkic economy tends to be globally unique and stable and also ensures long-run resource level to be reached by economic agents. Moreover, in the case of two interacting detrimental activities affecting renewable resource stock, the lower autarkic steady-state equilibrium is inclined either to extinction or to the larger steady-state equilibrium following the small perturbations hit the economy; this means that policymakers should direct the economy to the larger resource stock equilibrium in autarky in order to sustainable and effective growth in the long run. It can be suggested that stronger and well-defined property rights, as well as stricter pollution management regimes on polluter industries, are able to help maintain the renewable resource stock by deterring excessive harvesting and also the flow of pollutants into the environment.To carry out the analysis, our paper shows a theoretical framework of an open-based economy by providing proof of autarkic steady-state equilibriums in three different frameworks in terms of stability. The strength of the model is its ability to fill a mathematical area that has not been emphasized and analyzed in detail in the literature. In addition, we provide a general proof notion for similar renewable resource models.Possible extension of this research may include investigating the stability of autarkic steady-state equilibrium by incorporating more detrimental effects on resource stocks, allowing for a more realistic framework to figure out happens in autarky steady-state equilibriums in the sense of stability.



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*Source: 9215883-2023-03-22.xml*