Journal of Economics

, Volume 110, Issue 3, pp 203–224

Time consuming resource extraction in an overlapping generations economy with capital

Authors

    • Department of EconomicsUniversity of Graz
  • Karl Farmer
    • Department of EconomicsUniversity of Graz
Article

DOI: 10.1007/s00712-012-0318-0

Cite this article as:
Bednar-Friedl, B. & Farmer, K. J Econ (2013) 110: 203. doi:10.1007/s00712-012-0318-0
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Abstract

For worldwide fisheries production, two major trends emerge for the next decades: a significantly larger role for aquaculture and reduced output due to climate change impacts. While the former leads to an increase in cost, the latter affects natural regeneration. To address both impacts, we investigate the relevance of resource extraction costs for a private property fishery in an intertemporal general equilibrium model with capital accumulation, commodity production and a labor market. We show how the extraction cost parameter—in addition to time preference, technology and natural regeneration—matters for the existence (and stability) of an economically feasible, nontrivial stationary state. Higher extraction costs increase the equilibrium resource stock, while a reduced regeneration rate (e.g. due to climate change) decreases the stock. Moreover, resource extraction overshoots its new equilibrium value after the cost shock while after a regeneration shock extraction levels adjust monotonically towards the new equilibrium.

Keywords

Renewable resourcesTime consuming extraction costsOverlapping generationsResource shocks

JEL Classification

C62D90Q20

1 Introduction

World fisheries production has grown from 20 million tons in 1950 to more than 140 million tons in 2008, and fisheries are thus an important source of income in both developing and industrialized countries around the world (OECD and FAO 2011). In the years to come, worldwide production is projected to stabilize, but the share of aquaculture will increase and the share of wild capture will decrease (OECD and FAO 2011). This shift to aquaculture leads to higher costs of fishing, e.g. due to feeding cost and rising energy prices. In addition, climatic changes as well as disease outbreaks are likely to have consequences for the availability of fish (OECD and FAO 2011). One may suggest that market competition, high costs, and negative natural shocks endanger the sustainability of resource stocks which is however essential for countries rich in natural resources (fish). In this paper, we thus explore this new fisheries sustainability problem in an intertemporal general market equilibrium model.

One consequence of the growing importance of aquaculture is that this leads to an increase in costs to fisheries. To be able to investigate this cost shock, it is essential to integrate resource extraction cost, a key elements of partial equilibrium fishery models, into a general equilibrium framework. In partial equilibrium models, there is indeed a long tradition in specifying resource harvest cost as dependent on effort (e.g. Neher 1974; Heaps and Neher 1979) and on the available fish stock, following the general wisdom ‘the more fish the easier to catch’ (Smith 1968; Tahvonen and Kuuluvainen 2000). In general equilibrium models, however, with a few exemptions harvest costs are generally neglected (Krutilla and Reuveny 2004; Elíasson and Turnovsky 2004). We argue that this omission is problematic for two major concerns. First, the omission of extraction cost in general equilibrium models would only be acceptable, when unit extraction costs were constant—but as argued above, this assumption is manifestly unrealistic in the case of fisheries. Second, while in partial equilibrium models the price of effort can be assumed to be exogenously given, in general equilibrium models this price is determined by supply and demand. If for instance, the price of labor is high because of high demand for labor in other sectors, resource extraction cost can exceed the sales price of resource extraction, and the price of the resource stock would turn negative. Thus, positivity of the resource stock price needs to be proven in general equilibrium models.

In addition to the increased importance of extraction cost, aquaculture has a second implication for fishery resource modelling: fisheries are increasingly managed under private property instead of open access. Within our model, individuals thus face a choice between preserving a natural resource stock or using resource rents for physical capital accumulation. This decision is complicated by the fact that resource extraction requires effort which competes for labor with commodity production, and that extraction costs are low, but also the resource rent is small, when the stock is high. The intertemporal general equilibrium framework enables us to investigate the existence and stability of a sustainable (i.e. stationary) resource stock and how cost and regeneration shocks affect this stock.

Regarding the general equilibrium approach, we argue that the overlapping generations (OLG) type is more useful than the ILA (infinitely lived agent) type to analyze fisheries resource models. OLG models are well equipped to analyze the store of value function of resources (intergenerational asset allocation problem) since the resource stock can be traded among subsequent generations while in ILA models the resource stock is held by one representative generation forever. Going back to Samuelson (1958) and Diamond (1965), it is well-known that resource stock trade is driven by the demographic structure of finitely lived individuals with an old generation selling the resource stock and a young generation buying. Due to this demographic structure, an endogenous resource stock price dynamics results in OLG but not in ILA models. It is therefore not astonishing, that OLG models have been used frequently to address non-renewable and renewable resource problems (e.g. Mourmouras 1991; Olson and Knapp 1997; Babu et al. 1997; Krautkraemer and Batina 1999; Koskela et al. 2002; Agnani et al. 2005). Going beyond the common assumption in resource-based OLG models where the resource stock serves as the only store of value (e.g. Krautkraemer and Batina 1999; Koskela et al. 2002), we treat the resource stock and manmade capital as alternative assets (as in Mourmouras 1991; Farmer 2000).

Building on Farmer’s (2000) OLG model in which commodity production depends on the inputs of the extraction of a renewable resource, manmade capital, and labor, the conceptual novelty of the present paper is thus the integration of resource extraction cost. While extraction cost in resource-based ILA models are either found at the side of the atemporal production sector (Elíasson and Turnovsky 2004) or included directly in resource dynamics (Krutilla and Reuveny 2004), we assume that the younger household as the resource owner bears the cost of resource extraction. With regard to resource extraction cost, we follow the partial equilibrium literature (for an overview, see Clark 1990; Neher 1990; Conrad 1999; Brown 2000) by assuming that effort depends linearly on the harvest volume and inversely on the resource stock.

As a consequence of time-consuming extraction cost, a nontrivial stationary state in which all prices (in particular of the resource stock) are positive might not exist. Since this case cannot emerge in a model without harvest cost, the second contribution of the paper is the rigorous analysis of the existence (and stability) of a nontrivial stationary state equilibrium which is also economically feasible. In particular, we show that it is important to compare the regeneration capacity of the resource stock to the resource demand in commodity production, taking account of resource harvest cost.

Finally, this paper investigates the economic impacts of two types of resource shocks: a shock to the regeneration ability of natural resources and a push in harvest costs. Potential sources of the former shocks are climatic changes on reproduction such as El Niño, infectious diseases or invasive species that displace native ones. A push in extraction costs can be caused by an increase of feeding costs (e.g. of agricultural products or fish meal) or of energy prices (crude oil). While a negative regeneration shock leads to a lower resource stock, a cost shock increases the stock due to the inverse stock dependence of harvest cost. Contrary to first thoughts, that the type of the resource shock is irrelevant for the characteristics of the harvest dynamics, we find that the harvest cost shock generates harvest transition dynamics arising qualitatively different from the dynamics in response to shocks to resource regeneration.

The paper is structured as follows. The following section contains a description of the model and derives the intertemporal equilibrium dynamics. In Sect. 3, existence of a stationary state and its stability properties are analyzed. Section 4 analyzes the comparative statics and transitional dynamics of extraction cost and regeneration shocks and compares them to each other. The final section discusses the results and provides suggestions for extensions of the analysis.

2 Model description

We extend the OLG model of Farmer (2000) with manmade capital and a renewable resource stock by incorporating resource extraction cost costs.

As in Diamond’s seminal work (Diamond 1965), the economy is inhabited by identical consumers and each generation lives for two periods, one working and one retirement period. For the sake of analytical tractability, population of each generation is held constant and normalized to one.

Private property rights for the natural resource stock (and also for the physical capital stock) are fully specified and enforced without cost.1 The resource stock is thus an asset which the young household uses to transfer income into the retirement period, and similarly for physical capital. As necessary in OLG models (de la Croix and Michel 2002), it is assumed that the initially old generation is endowed with the stock of the renewable resource and physical capital.

Since the resource stock is in private property of the households, households are faced with the decision on how to allocate time between employment in commodity production and resource harvest effort. For expositional purposes we shall identify the resource stock as an aquaculture fishery.

2.1 Household and firm optimization

The representative consumer’s intertemporal utility depends on consumption during the working period, \(C_{t}^{1}\), and consumption during the retirement period, \(C_{t+1}^{2}\), \(0\!<\!\beta \!<\!1\) denoting the total time discount factor, i.e. for a generational length of 25 years.2 For simplicity, the representative household’s preferences are represented by a log-linear intertemporal utility function:
$$\begin{aligned} u=u(C^{1}_{t},C^{2}_{t+1})=\ ln\ C^{1}_{t}+\beta \ ln\ C^{2}_{t+1}. \end{aligned}$$
(1)
In maximizing intertemporal utility (1), the young household faces a budget constraint in each period of life. Assuming that resource harvest requires labor (or effort) as input (see e.g. Krutilla and Reuveny 2004; Elíasson and Turnovsky 2004), the household splits total working time (normalized to one) on employment in the commodity production sector and on resource harvesting effort \(E_t\). Assuming moreover that effort is inversely related to the resource stock, harvest effort is represented by3
$$\begin{aligned} E_t=\lambda \frac{X_t}{R_t^d}, \end{aligned}$$
(2)
where \(\lambda \) is the extraction cost parameter. The budget constraint in the working period is thus:
$$\begin{aligned} p_{t}\ R_{t}^d+C_{t}^{1}+K_{t+1}=w_{t}(1-E_t)+q_{t} X_{t}, \end{aligned}$$
(3)
where \(w_{t}\) denotes real wage, \(q_{t}\) the price of resource harvest, \(p_{t}\) the price of the resource stock demanded, and the price of the consumption good in period \(t\) serves as the numeraire. In addition to wage income, the young household gains income from selling of the resource harvest \(X_{t}\). Income from wage and selling of the resource harvest is spent on consumption \(C_{t}^{1}\). For transferring income to the retirement period, the household saves in terms of manmade capital \(K_{t+1}\) and in terms of the natural resource stock \(R_{t}^d\). As a consequence of exclusive private property rights (as for e.g. a fish pond or a fishing ground), the younger household acquires the resource stock from the older household in the competitive resource stock market.4 Due to exclusive private property rights, the younger household can also appropriate any rents delivered by the stock.5
From saving, the household gains income in the retirement period, where \(z_{t+1}\) denotes the interest factor of manmade capital (i.e. one plus the interest rate). Thus, revenues when old derive from sale of the capital stock and of the resource stock to the younger generation and are spent on consumption \(C_{t+1}^{2}\), forming the second period budget constraint:
$$\begin{aligned} C^{2}_{t+1}=z_{t+1} K_{t+1}+p_{t+1} R_{t+1}. \end{aligned}$$
(4)
As argued above, the resource stock is held in private property of the household and therefore the dynamics of the aggregate resource stock form the third constraint to household optimization:6
$$\begin{aligned} R_{t+1}=R_{t}^d+\text{ g}(R_{t}^d)-X_{t}. \end{aligned}$$
(5)
As in other OLG models with a renewable resource (e.g. Krautkraemer and Batina 1999; Farmer 2000; Koskela et al. 2002), we assume that the aggregate resource stock evolves according to a concave growth function \(\text{ g}(R_{t}^d)\), \(R_t^d \in [0,R_{max}]\), where \(\text{ g}^{\prime \prime }<0\) and \(\text{ g}^{\prime }(0)>0\).7 In the following analysis, we assume that resource regeneration is governed by the logistic function \(\text{ g}(R_{t}^d)= r\left[R_{t}^d-({R_{t}^d})^{2}/{R_{\mathrm{max}}}\right]\), where \(r>0\) denotes the regeneration rate and \(R_{\mathrm{max}}\) the carrying capacity. Alternatively, we will focus on the case of constant regeneration in which resource growth is independent of the stock and thus \(\text{ g}(R_{t}^d)=r\) (as e.g. in Mourmouras 1991; Kemp and Van Long 1979).
The representative household thus chooses \(C_{t}^{1}\), \(C_{t+1}^{2}\), \(R_t^d\), \(K_{t+1}\) and \(X_t\) to maximize (1) with respect to  (3) (taking account of (2)), (4), and (5). This yields the following first order condition for intertemporal optimality in consumption:
$$\begin{aligned} \frac{C_{t+1}^{2}}{\beta \; C_{t}^{1}}= z_{t+1}, \end{aligned}$$
(6)
which requires that the intertemporal marginal rate of substitution between consumption when young and consumption when old equals the interest factor.
The second condition equates the net return on resource harvest (i.e. the revenue on selling an incremental additional unit, net of the marginal harvest cost) to the discounted price of the resource stock:
$$\begin{aligned} q_{t} -w_t\frac{\lambda }{R_t^d}=\frac{p_{t+1}}{z_{t+1}}. \end{aligned}$$
(7)
In (7), there is a remarkable difference to partial equilibrium models: marginal harvest cost, \(w_t \lambda /R_t^d\), depends on the endogenous wage rate (which is determined in the labor market) and on the resource stock demanded. For a positive harvest level (an interior solution), it is thus necessary that the left hand side of (7) is positive for the general equilibrium solution values of all endogenous variables.
A modified Hotelling rule holds which requires that the factors of return on manmade capital and on the renewable resource stock are balanced:
$$\begin{aligned} p_{t}=\frac{p_{t+1}}{z_{t+1}} \left(1+\text{ g}^{\prime }(R_{t}^d)\right)+w_t \frac{\lambda X_t}{{R_t^d}^2}. \end{aligned}$$
(8)
The resource owner thus compares the marginal costs of investment in the resource stock (left hand side of (8)), to the marginal benefits of this investment (right hand side of (8)). The marginal benefits consist of the discounted market value of the resource stock not harvested times its growth factor (first term on right hand side of (8)) and the reduction of harvest cost because of an increase in the stock (second term). This latter effect would vanish if harvest cost depended only on the harvest volume but not the stock.8
Finally, the intertemporal budget constraint follows from (3) and (4):
$$\begin{aligned}&C_{t}^{1}+\frac{C_{t}^{2}}{z_{t+1}}= w_{t}\left[1-\lambda \frac{X_t}{R_t^d}-\lambda \Phi (R_t^d)\right]+q_{t}\Phi (R_t^d)R_t^d, \nonumber \\&\quad \Phi (R_{t})\equiv \frac{\text{ g}(R_{t}^d)}{R_{t}^d}-\text{ g}^{\prime }(R_{t}^d). \end{aligned}$$
(9)
The firm is assumed to behave competitively and to maximize profits given output and input prices. Output \(Y_{t}\) is produced according to a constant-returns-to-scale Cobb–Douglas production function \(Y_{t}=(X_{t}^d)^{\alpha _{1}}(N_{t}^d)^{\alpha _{2}}(K_{t}^d)^{\alpha _{3}}\), \(\alpha _{1}+\alpha _{2}+\alpha _{3}=1\), with labor \(N_{t}^d\), capital services \(K_{t}^d\), and resource harvest \(X_{t}^d\) as inputs and \(0<\alpha _{i}<1\) denoting the constant production elasticities of the resource harvest, labor, and capital services, respectively. Acknowledging (2), the firm’s first order conditions read as follows:
$$\begin{aligned} q_{t}X_{t}^d =\alpha _{1}Y_{t}, \quad w_{t}N_{t}^d = \alpha _{2}Y_{t}, \quad z_{t}K_{t}^d = \alpha _{3}Y_{t}. \end{aligned}$$
(10)
All markets are assumed to clear every period, i.e. the markets for the resource stock (\(R_t^d=R_t, \forall t\)), for resource harvest (\(X_t^d=X_t, \forall t\)), for labor (\(N_{t}^d=1-\lambda X_t/R_t^d, \forall t\)), and for (manmade) capital (\(K_t^d=K_t, \forall t\)). Moreover, capital is assumed to depreciate fully every period. Finally, commodity market clearing coincides with Walras’ Law and is therefore redundant:
$$\begin{aligned} (X_{t})^{\alpha _{1}}(1-\lambda X_t/R_t)^{\alpha _{2}}(K_{t})^{\alpha _{3}}=C_{t}^{1}+C_{t}^{2}+K_{t+1}. \end{aligned}$$
(11)

2.2 Intertemporal equilibrium dynamics

A competitive intertemporal equilibrium is a positive sequence of all prices and all quantitities such that household’s and firm’s first order conditions, (6)–(10), and the market clearing conditions hold. By using the goods market clearing condition and the first order conditions, the intertemporal equilibrium dynamics can be reduced to a two-dimensional system in \(R_{t}\) and \(X_{t}\). The equation of motion of the resource stock is identical to the stated natural growth function:
$$\begin{aligned} R_{t+1}=R_{t}+\text{ g}(R_{t})-X_{t}, \quad \text{ for} \text{ given} \ R_{0}>0. \end{aligned}$$
(12)
Regarding the intertemporal equilibrium path of the resource harvest, we derive first two relationships for \(K_{t+1}\) and equate. For that, we have to ensure that along the intertemporal equilibrium path \(K_{t+1}>0\). The first relationship results from (11) by taking account of (4), (6)–(10):
$$\begin{aligned} \frac{K_{t+1}}{Y_{t}}&= \frac{\text{ A}(R_{t},X_{t})}{\left[R_t-\lambda X_{t}\right]X_t}, \end{aligned}$$
(13)
with
$$\begin{aligned} \text{ A}(R_{t},X_{t})&\equiv (\alpha _{1}+\alpha _{2}\sigma ) X_{t}\left[R_{t}-\lambda X_{t}\right]-\alpha _2 \lambda X_t^2-\text{ B}(R_t,X_t)\big [(1-\sigma )\Phi (R_t)\nonumber \\&+1\!+\!\text{ g}^{\prime }(R_t)\!\big ]R_t, \\ \sigma&\equiv \frac{\beta }{(1\!+\!\beta )},\quad \text{ B}(R_{t},X_{t})\equiv \alpha _1 R_t\!-\!(\alpha _1+\alpha _2)\lambda X_t.\nonumber \end{aligned}$$
According to (13), \(K_{t+1}>0\) if \(\text{ A}(R_{t},X_{t})>0\) and \(1-E_t=1-\lambda X_t/R_t>0\).
Another relationship for \(K_{t+1}\) can be derived from (7), (8), and (10):
$$\begin{aligned} \frac{K_{t+1}}{Y_{t}}=\frac{\alpha _3 X_{t+1} R_{t+1}\left[R_{t+1}-\lambda X_{t+1}\right]\text{ B}(R_{t},X_{t})}{ \left[R_t-\lambda X_{t}\right]X_t\left\{ \left[1+\text{ g}^{\prime }(R_{t+1})\right]R_{t+1}\text{ B}(R_{t+1},X_{t+1})+\alpha _2 \lambda X_{t+1}^2\right\} }, \end{aligned}$$
(14)
where \(K_{t+1}>0\) if \(\text{ B}(R_t,X_t)>0\) and \(1-E_t>0\), for all \(t\). But using (10) in (7) shows that the discounted price of the resource stock in \(t+1\) is positive when \(\text{ B}(R_t,X_t)>0\) and \(1-E_t>0\):
$$\begin{aligned} \frac{p_{t+1}}{z_{t+1}}=\frac{\text{ B}(R_t,X_t)}{\left[R_t-\lambda X_{t}\right]X_t} Y_t, \end{aligned}$$
(15)
and hence according to (8) also the price in period \(t\) is positive, such that \(K_{t+1}>0, \forall t\) in (14).9 As a consequence, we can equate (13) and (14), which yields the equation of motion for resource harvest:
$$\begin{aligned} \alpha _3 X_{t+1} R_{t+1}\left[R_{t+1}-\lambda X_{t+1}\right]&= \Gamma (R_{t},X_{t}) \left\{ \left[1+\text{ g}^{\prime }(R_{t+1})\right]R_{t+1}\text{ B}(R_{t+1},X_{t+1})\right.\nonumber \\&\qquad \qquad \qquad \left.+\alpha _2 \lambda (X_{t+1})^2\right\} , \nonumber \\ \Gamma (R_{t},X_{t})&\equiv \text{ A}(R_{t},X_{t})/\text{ B}(R_t,X_t). \end{aligned}$$
(16)
While the derivation of (16) involves the capital stock implicitly, the explicit dynamics of the capital stock follow from (14) for an initially given, positive capital stock.10 Proposition 1 ensures the existence of a strictly positive intertemporal equilibrium sequence.

Proposition 1

(Existence of intertemporal equilibrium) Let \(\tilde{\text{ A}}(R_t,X_t)=\{(R_{t},X_{t})|\)\(\text{ A}(R_t,X_t)=\eta >0,R_t \in [0,R_{max}]\}\) and \(\tilde{\text{ B}}(R_t,X_t)=\{(R_{t},X_{t})|\text{ B}(R_t,X_t)=\varepsilon >0, R_t \in [0,R_{max}]\}\). If \(\alpha _2\ge \alpha _1\) and \(2(\alpha _2-\alpha _1)\alpha _3\Gamma (R_t,X_t)\left[1+\text{ g}^{\prime }(R_{t+1})\right]\)\(\lambda R_{t+1}+(\alpha _3)^2 R_{t+1}> \Gamma (R_t,X_t)^2\lambda \left[1+\text{ g}^{\prime }(R_{t+1})\right]\left\{ 4 \alpha _1\alpha _2-(\alpha _1+\alpha _2)^2\right.\)\(\left.\lambda \left[1+\text{ g}^{\prime }(R_{t+1})\right]\right\} \), with \(R_{t+1}=R_{t}+\text{ g}(R_{t})-X_{t}\) and all \((R_{t},X_{t})\in \tilde{\text{ B}}(R_t,X_t)\) and \((R_{t},X_{t})\in \tilde{\text{ A}}(R_t,X_t)\), then an intertemporal equilibrium sequence \(\{(R_{t+1},X_{t+1},K_{t+1},C_t,^1,C_t^2,p_t,z_t,\)\(w_t, q_t)_{t=0}^{\infty }\gg 0\}\) such that (16) holds exist.

 

Proof

See Appendix A.1.

According to Proposition 1, a positive, or economically feasible, equilibrium sequence requires that \(\text{ A}(R_t,X_t)\) and \(\text{ B}(R_t,X_t)\) are positive and hence the capital stock in the subsequent period is positive. The interpretation of \(\text{ A}(R_t,X_t)>0\) is that the production remaining after consumption of younger and older households needs to be positive to ensure \(K_{t+1}>0\). \(\text{ B}(R_t,X_t)>0\) is equivalent to the requirement that \(\lambda X_t/R_t=E_t<\alpha _1/(\alpha _1+\alpha _2)\). Accordingly, the effort devoted to resource extraction needs to be less than the relative production share of resource input. If this condition is ensured the net price of the resource stock, i.e. after acknowledging resource extraction costs, is positive. There is an important difference here to the cases of linear harvest cost or a model without harvest cost because there \(\text{ B}(R_t,X_t)>0\) and \(1-E_t>0\) are trivially fulfilled.

Finally, note that the resulting equilibrium dynamic for the resource harvest (16) is a second order polynomial, but since the larger root is not compatible with \(R_{t+1}>0\) (see (21) in the Proof to Proposition 1) only the smaller root, defined as \(X_{t+1}=\phi (R_{t+1},X_t)\), is feasible.

3 Existence and stability of stationary states

Having ensured that the intertemporal equilibrium sequence of the resource stock and the resource harvest are economically feasible (i.e. prices do not turn negative), the next step is to analyze whether strictly positive resource prices, a strictly positive resource harvest, and a strictly positive resource stock can be ensured in the long run (stationary resource stock which is sustainable by definition). For that purpose, we analyze the existence and stability of nontrivial stationary states in our system.

By setting \(R_{t+1}=R_{t}=R\) and \(X_{t+1}=X_{t}=X\), for all \(t\), a stationary state to the dynamic system (12) and (16) is defined as follows:
$$\begin{aligned}&\displaystyle X=\text{ g}(R),\end{aligned}$$
(17)
$$\begin{aligned}&\displaystyle \frac{\alpha _3 X R\left[R-\lambda X\right]}{\text{ A}(R,X)}= \frac{\left\{ \alpha _1 \left[1+\text{ g}^{\prime }(R)\right]R\left[R-\lambda X\right]-\alpha _2 \lambda X\left[\left[1+\text{ g}^{\prime }(R)\right]R-X\right]\right\} }{\text{ B}(R,X)}.\nonumber \\ \end{aligned}$$
(18)
Inserting (17) into (18), we can define the resulting left hand side by \(\text{ LHS}(R)\equiv \alpha _3 \text{ g}(R) R \left[R-\lambda \text{ g}(R)\right]/\text{ A}(R)\) and the right hand side by \(\text{ RHS}(R)\equiv \left\{ \left[1+\text{ g}^{\prime }(R)\right]\right.+\)\(\left.\text{ B}(R)R+\alpha _2 \lambda \text{ g}(R)^2\right\} /\text{ B}(R)\). Proposition 2 provides sufficient conditions for the existence of a unique nontrivial stationary state.

Proposition 2

(Existence of nontrivial stationary state) If \(\lambda <\alpha _1/[(\alpha _1+\alpha _2)r]\) and moreover \(\lim _{R\rightarrow 0^+}\text{ LHS}^{\prime }(R)<\lim _{R\rightarrow 0^+}\text{ RHS}^{\prime }(R)\) or if \(\lambda >\alpha _1/[(\alpha _1+\alpha _2)r]\), then a unique nontrivial stationary state \(0<R<R_{max}\) satisfying (17), (18), \((R,X)\in \tilde{\text{ B}}(R,X)\) and \((R,X)\in \tilde{\text{ A}}(R,X)\), exists.

 

Proof

See Appendix A.2.

According to Proposition 2 an economically feasible stationary state requires that in the nontrivial stationary state \(\text{ A}(R,X)\) and \(\text{ B}(R,X)\) are positive.11 Inspecting (18) reveals that \(\text{ LHS}(R)\) has a pole \(\hat{R}\) at \(\text{ A}(\hat{R})=0\) and \(\text{ RHS}(R)\) has a pole \(\tilde{R}\) at \(\text{ B}(\tilde{R})=0\). Regarding the second pole, we have to distinguish two cases according to whether \((p/z)\) is positive for rather small \(R\) or for sufficiently large \(R\) only (see (15)). In case 1 (see Fig. 1), \(\text{ B}(0)=0\) and moreover \(\text{ B}^{\prime }(0)=\alpha _1-(\alpha _1+\alpha _2)\lambda r>0\) which implies that \((p/z)>0\) in the interval \((0,\hat{R})\).12 In other words, because \(\lambda \) is small, economically feasible solutions are possible over the whole range \((0,\hat{R})\). On the other hand, since \(\lambda \) is relatively small in comparison to \(\alpha _1/\left[(\alpha _1+\alpha _2)r\right]\) and \(\text{ LHS}^{\prime }(0)\) increases while \(\text{ RHS}^{\prime }(0)\) decreases with smaller \(\lambda \), \(\lambda \) must not be too small to ensure that \(\text{ LHS}^{\prime }(0)\) is larger than \(\text{ RHS}^{\prime }(0)\), and hence a nontrivial stationary state \(R\) exists.
https://static-content.springer.com/image/art%3A10.1007%2Fs00712-012-0318-0/MediaObjects/712_2012_318_Fig1_HTML.gif
Fig. 1

Existence of a unique stationary state (for \(\lambda <\alpha _1/[(\alpha _1+\alpha _2)r]\))

In case 2 (see Fig. 2), also a pole of \(\text{ RHS}(R)\) emerges at \(\text{ B}(\tilde{R})=0\). To the left of \(\tilde{R}\), \((p/z)<0\) because \(\lambda \) is relatively large (for given \(r\)) and thus the stationary resource stock \(R\) has to be larger than \(\tilde{R}\). As for case 1, \(R\) has to be smaller than \(\hat{R}\) to ensure that \(\text{ A}(R)>0\) and thus \(R\) has to be in the interval \(\tilde{R}<R<\hat{R}\) (see gray shaded area in Fig. 2).13 However, \(\text{ B}^{\prime }(0)<0\) is equivalent to \(r>\alpha _1/[(\alpha _1+\alpha _2)\lambda ]\) and moreover \(\text{ B}(R)>0\) is equivalent to requiring that effort becomes not too large: \(E<\alpha _1/(\alpha _1+\alpha _2)\). As a consequence, \(r>\alpha _1/ \left[(\alpha _1+\alpha _2)\lambda \right]>E/\lambda =X/R\), i.e. the regeneration rate at the origin is larger than stationary resource extraction used in production relative to the resource stock. Hence, in contrast to case 1, there is is no need for an additional parameter restriction for the slopes of \(\text{ LHS}\) and \(\text{ RHS}\).
https://static-content.springer.com/image/art%3A10.1007%2Fs00712-012-0318-0/MediaObjects/712_2012_318_Fig2_HTML.gif
Fig. 2

Existence of a unique stationary state (for \(\lambda >\alpha _1/[(\alpha _1+\alpha _2)r]\))

Regarding dynamic stability and comparative stationary state analysis, we will focus our analysis on the special case of constant regeneration, i.e. \(\text{ g}(R)=r\), implying that the extraction dynamics (16) simplifies to:
$$\begin{aligned} X_{t+1}=\frac{\alpha _1 \Gamma (R_t,X_t)R_{t+1}}{\alpha _2 \Gamma (R_t,X_t)+\alpha _3 R_{t+1}}. \end{aligned}$$
(16′)
It is important to note that for constant regeneration only case 2 is possible (\(\tilde{R}<R<\hat{R}\)) but not case 1 since \(\text{ B}(R)>0\) if and only if \(R>(\alpha _1+\alpha _2)\lambda r=\tilde{R}\). However, since case 2 (large \(\lambda \)) is economically more interesting, there is no loss of generality when we focus on constant regeneration and unitary extraction costs (\(\lambda =1\)).

Regarding dynamic stability of the stationary state, Proposition 3 claims that the unique stationary state is a stable saddle point.

Proposition 3

(Stability of stationary states) Let \(g(R)=r\) (constant regeneration), \(r<1\) and \(\lambda =1\) hold. Then, the stationary state \((R,r)\) is a stable saddle point.

 

Proof

See Appendix A.3.

For the case of linear regeneration, saddle path stability of the stationary state is indicated by the arrows of motion in Fig. 3 where \(XX\) and \(RR\) are the loci which result from setting \(X_{t+1}=X_t\) and \(R_{t+1}=R_t\) in (16′) and (12). While we cannot prove analytically that the same pattern emerges also for the more general case of logistic regeneration, intensive numerical investigation reveals that a stable saddle point emerges as in the former case of constant regeneration, and this is illustrated in Fig. 4.
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Fig. 3

Local stability around stationary state \((R,r)\) (saddle point) in model with constant regeneration

 
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Fig. 4

Local stability around stationary state \((R,X)\) (saddle point) in model with logistic regeneration

 

4 Comparative statics and transitional dynamics of an extraction cost push and a regeneration shock

As outlined in the introduction, there are two trends in fisheries: a growing importance of aquaculture which induces higher costs, and shocks to regeneration due to climatic changes or disease outbreaks.

In the following, we first derive analytically the stationary state (long-run) and transitional dynamics effect of the two types of shocks under the assumption of constant regeneration and generalize these results numerically for the logistic case. To be able to study the transitional dynamics following a shock, the type and the timing of the shock must be specified. To keep the analysis as simple as possible we assume that the shock is permanent, unannounced and occurs at the beginning of the transition period.

4.1 An extraction cost push: a rise in the cost parameter \(\lambda \)

In this section, we analyze the consequences of an extraction cost push. Taking the example of fisheries, such a harvest cost push may arise when with the switch from capture fishery to aquaculture a higher unit effort is needed or when in case of aquaculture an increase in feed costs emerges (for fish meal and grain, rising food prices are projected according to OECD and FAO 2011, pp. 148–158). In our model, an extraction cost push is specified as an exogenous change in the extraction cost parameter \(\lambda \), i.e. an increase in the effort necessary to harvest one unit of the resource stock.

Focusing again on the special case of constant regeneration, we find that an extraction cost push leads to an increase in the stationary state resource stock while the harvest volume is not affected.

Proposition 4

(Comparative statics of an extraction cost push) Let \(\text{ g}(R)=r\) and \(\lambda =1\) and hence \(X=r\). Then, \(\text{ d}R/\text{ d}\lambda >0\) and \(\text{ d} X/\text{ d} \lambda =0\).

 

Proof

See Appendix A.4.

Figure 6 illustrates our numerical results for the general case of logistic resource regeneration. As for stability, the case of constant regeneration is generic for this more general case: for a broad range of admissible parameter values, an extraction cost push leads to a larger stationary state resource stock. However, the main difference to constant regeneration is that stationary state resource harvest is no longer fixed. Whether stationary state resource harvest falls or rises is dependent on where the initial stationary state lies. For the case depicted in Fig. 6, the stationary state lies to the right of maximum sustainable yield and thus stationary state resource harvest is reduced, while to the left harvest would be increased.

Regarding transitional adjustment, we write the equilibrium dynamics around the initial stationary state as follows:
$$\begin{aligned}&\displaystyle R_{t+1}=R\left( 1-\eta _{2} \right) +\eta _{2} R_{t} , \quad \text{ given}\, R_{0} >0,\end{aligned}$$
(19)
$$\begin{aligned}&\displaystyle X_{t} =r+\left[1-\eta _{2}\right] \left(R_{t} -R\right), \end{aligned}$$
(20)
where \(0<\eta _{2}<1\) is, according to Appendix A.3, the stable eigenvalue of the Jacobian. Inspecting (20) for \(t=0\) reveals that the extraction cost push triggers an initial drop in resource harvest because of \(\text{ d}R/\text{ d}\lambda >0\) and \(1-\eta _2>0\). Since moreover, due to linear regeneration, the new stationary state harvest level is again equal to the regeneration rate, overshooting of the harvest level occurs (see Fig. 5). Afterwards, both the resource stock and the resource harvest adjust non-oscillatory towards the new stationary state. According to Fig. 5, also in case of logistic regeneration an extraction cost push triggers non-oscillatory adjustment towards the new stationary state, with the initial drop in resource harvest being stronger than the stationary state effect and thus overshooting occurs.
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Fig. 5

Effects of an extraction cost push (\(\lambda ^\prime >\lambda \)) in model with constant regeneration

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Fig. 6

Effects of an extraction cost push (\(\lambda ^\prime >\lambda \)) in model with logistic regeneration

4.2 A regeneration shock: a change to the rate of resource regeneration

Let us now turn to a regeneration shock, namely an exogenous change in the regeneration rate \(r\). To facilitate intuition for this shock, a diminished regeneration rate (i.e., smaller fish) could be the result of a natural variation e.g. due to El Niño phenomenon or adverse effects of climate change potentially resulting in higher water temperature, more extreme weather periods, less run-off water to lakes and rivers, and more diseases (OECD and FAO 2011).

According to Proposition 5, a decline of the regeneration rate (i.e. a negative regeneration shock) reduces stationary state resource harvest and the resource stock.

Proposition 5

(Comparative statics of a regeneration shock) Let \(\text{ g}(R)=r\), \(r<1\), and \(\lambda =1\) and hence \(X=r\). Then, \(\text{ d}R/\text{ d}r>0\) and \(\text{ d} X/\text{ d}r=1\).

 

Proof

See Appendix A.5.

For the model with logistic regeneration, numerical analysis confirms a similar pattern for the resource stock, while the effect on resource harvest depends on whether we are to the left or to the right of the maximum sustainable yield level. When the \(RR\) locus at the stationary states is negatively sloped as illustrated in Fig. 8, then the stationary state harvest level declines in response to a reduction in the regeneration rate (\(\text{ d}X/\text{ d}r<0\)), while an adjustment in opposite direction results when the shock affects stationary states at a point where the \(RR\) locus is positively sloped.

Regarding transition, a look on (20) reveals that in contrast to the cost push no overshooting of the new harvest level occurs since the stationary state resource stock decreases due to a lower \(r\). Again, this result is also confirmed numerically for logistic regeneration (see Fig. 8).
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Fig. 7

Effects of regeneration shock (\(r^\prime <r\)) in model with constant regeneration

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Fig. 8

Effects of regeneration shock (\(r^\prime <r\)) in model with logistic regeneration

 

4.3 A comparison of an extraction cost push and a negative regeneration shock

Geometrically, the differences between an extraction cost push and a negative regeneration shock in case of logistic regeneration can be explained as follows. As can be seen from Fig. 6, the \(XX\) locus shifts downwards in response to an increase in \(\lambda \) because for a fixed level of the resource stock only a smaller level of harvest is feasible. This leads to an increase in the stationary state resource stock. Thus, an extraction cost push leads to a resource recovery due to the substitution of labor and capital for harvest input. This finding is also a confirmation of the result in a partial equilibrium fishery model where inverse stock dependency of harvest costs leads to a preserving effect for the resource stock (Perman et al. 2011).

On the other hand, a negative regeneration shock implies a downward shift of the \(RR\) locus (see Figs. 7, 8). In case of logistic generation, also the XX locus shifts slightly downward (see Fig. 8). Accordingly, both the stationary-state harvest volume and the resource stock decline. On interpreting the shift of the \(XX\) locus, we find that households substitute manmade capital for the resource stock since the resource stock becomes less profitable.

The economic intuition why overshooting occurs for an extraction cost push, but not for a decline in the regeneration rate, is as follows. For the case of the cost shock, there is a direct impact of the cost push on the harvest level, leading to a sharp initial drop in the harvest volume and a corresponding rise in the harvest price. As a result, the price of the resource stock rises in the shock period, too. Over time, the resource stock recovers to a new higher stationary state level and the new stationary state harvest volume is only moderately lower than in the pre-shock equilibrium. Thus, the harvest volume initially overshoots its new stationary state level and so does its price. Moreover, in the new stationary state manmade capital, total output and consumption of both young and old households are lower than in the pre-shock equilibrium.

In contrast, a negative shock in the resource growth rate has a direct impact on the resource stock and only an indirect one on the harvest volume. Thus, in the post-shock period the harvest volume is only moderately reduced, its price increases moderately and the price of the resource stock increases moderately, too. As for the extraction cost push, consumption of the young household declines while consumption of the old household increases in the shock period. Thereafter, however, both the resource stock and the harvest volume degrade gradually to their new lower stationary state levels. As before, in the new stationary state manmade capital, total output and consumption levels of young and old households are lower than in the initial stationary state.

Initial non-overshooting of resource extraction in response to a negative regeneration shock can be seen as a rationale for modest immediate economic responses to natural disasters in contrast to their much larger long-run consequences.

5 Conclusions

In this paper time consuming resource extraction was introduced in an OLG model with a renewable natural resource and with manmade capital. We proved the existence and dynamic stability of a unique stationary state solution. It turns out that for the proof of the existence of the stationary state two cases need to be distinguished: In case 1, while the natural regeneration rate in the origin is not sufficient to cover the demand for resource use in commodity production, the relatively small extraction cost parameter ensures a positive resource stock price over the whole range of \(R\) (economically feasible solution). In this case the existence condition ensures that in spite of low natural regeneration a positive stationary state exists. In case 2, \(\lambda \) is rather large and therefore the price of the resource stock is negative over a broad range of resource stocks while the natural regeneration rate at the origin is relatively large. Hence, natural regeneration allows for a large stationary state resource stock which reduces unit extraction costs such that the resource stock price becomes positive (economically feasible).

In contrast to a partial equilibrium model, in our general equilibrium model we had to explicitly investigate the positivity of the price of the resource stock along the intertemporal equilibrium path and in the stationary state. By distinguishing the case where the extraction cost parameter is relatively small from that where it is relatively large, we were able to prove that in both cases an economically feasible stationary state exists. Thus, the concerns raised in the introduction that sustainability of resource stocks might be endangered by time consuming extraction costs, private property, and market competition, are not warranted.

Our second objective was to study the potentially different impacts of an extraction cost and of a regeneration shock, both in the stationary state as well as along the transition path towards the new stationary state. As regards the stationary state effects of an extraction cost push, we found a positive impact on the resource stock whereas the resource harvest decreases (when the stationary state is to the right of the maximum sustainable yield resource stock). That the stationary state resource harvest declines while the resource stock increases is also the main reason why the resource harvest of the shock period overshoots its new stationary state value. As a consequence, the price of the resource harvest and the price of the resource stock overshoot, too.

A negative shock to the regeneration rate has a negative impact on the main dynamic variables in the new stationary state. In view of the equilibrium manifold leading to the new saddle point the resource harvest does not overshoot its new, lower stationary state value. This result can be seen as a rationale for modest immediate economic responses, observed in real world circumstances, to shocks in nature’s technology in spite of their much larger stationary state impacts.

Two directions for future research are easily identified. First, other functional forms for the effort into resource extraction could be used, e.g. relaxing the assumption of harvest costs linear in the harvest volume. Another option were to replace the inverse impact of the resource stock such that harvest costs increase with the resource stock, a specification suitable e.g. for species-rich ecosystems like tropical forests. Second, rather than focusing on log-linear intertemporal utility functions a more general function (like CES) should be considered. The equilibrium dynamics would then depend on manmade capital in a nontrivial way, and a richer and hence empirically more interesting dynamical system is to be expected.

Footnotes
1

For the consequences of different property right regimes (private property versus open access) on renewable resource exploitation, see Alvarez-Cuadrado and Van Long (2011).

 
2

Because population is normalized to one, \(C_{t}^{1}\) and \(C_{t+1}^{2}\) are either per-capita consumption or aggregate consumption in the working and retirement period, respectively.

 
3

Solving (2) for \(X_t\) yields the well-known Schaefer (1954) harvest function, a functional specification popular in mostly partial equilibrium fishery models (see, e.g. Brown 2000; Conrad 1999; Clark 1990).

 
4

As in Mourmouras (1991) and Farmer (2000), the resource stock is bought at the beginning of the period from the old household and the resource harvest is sold simultaneously at a one-period forward market to the representative firm where it is used as an input to commodity production. Instead of a beginning-of-period market setting, some authors employ an end-of-period notion (Koskela et al. 2002; Bréchet and Lambrecht 2011). However, there is no essential difference between these two specifications as shown by Farmer (2000).

 
5

Despite the resource harvest being generated from different resource sites, all resource harvest sells on a competitive market at a single market clearing price.

 
6

Assuming that each generations consists of identical individuals and that there are no externalities among individual resource stocks, implies that the own rate of return on each individual’s resource stock depends on its resource stock (e.g. the fish in a privately owned pond), and not the aggregate stock. Multiplying the individual resource stock dynamics by the number of individuals yields resource dynamics at the aggregate level as stated in (5). For an alternative specification where individual resource return depends linearly on all other individual resource returns, see Bréchet and Lambrecht (2011).

 
7

The class of concave natural growth functions corresponds to the sigmoid growth function verified for density dependent species populations (see, e.g. Begon et al. 1996; Brown 2000).

 
8

One reviewer rightly remarked that it is important to clarify how this condition can be decentralized for the individual household. For this we assume that there are no externalities between individual resource stocks, and thus both the own rate of return on the resource stock and harvest cost (and hence also rent) depend on the individual resource stock (e.g. the fish in a privately owned pond), and not the aggregate one. Otherwise the individual’s first order condition would read as: \(p_{t}=p_{t+1}/z_{t+1}\).

 
9

We would like to thank Andreas Rainer for pointing us to the fact that the requirement \(p_{t+1}>0\) can be violated in case of harvest cost.

 
10

The main difference of the present model compared to an OLG model without capital and log-linear utility, is that the intertemporal dynamics are described by a planar system instead of a single difference equation as in (Koskela et al. (2002), pp. 509–510) or Krautkraemer and Batina (1999). This dimensional difference is the result of manmade capital being here both an alternative asset to the resource stock and a factor of production.

 
11

Since \(\lambda X/R\) is not defined for \(R=X=0\), a trivial stationary state does not exist in this model—as in an OLG with endogenous labor-leisure choice (Nourry 2001).

 
12

The figures are drawn for illustrative purposes based on the following parameter set: \(\alpha _1=0.1, \alpha _2=0.6, \alpha _3=0.3, \beta =0.44, r=0.55, R_{max}=10\), and \(\lambda =0.25\) (Fig. 1), or \(\lambda =1\) (Fig. 2). Note that the total time discount factor of 0.44 for a typical generational length of 25 corresponds to an annual time discount factor of 0.968.

 
13

The other intersection point of \(\text{ LHS}\) and \(\text{ RHS}\) is not economically feasible. Whether the resulting stationary state resource stock is to the left or to the right of maximum sustainable yield depends on the parameter values of \(\lambda \), \(\alpha _1\), \(\alpha _2\), and \(r\).

 

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© Springer-Verlag Wien 2012