1 – Introduction
1In models of Classical, Sraffian or longterm inspiration, natural resources are usually confined to “Ricardian” lands, i.e. resources of which the quality and quantity remain unchanged over time, no matter how they are used. When they become scarce, a positive price must be paid for their use. The theory of rent examines all the intricacies to which this may give rise; a few of the main contributions to this literature are: Sraffa (1960, ch. 11), Quadrio Curzio (1966), Schefold (1971), Montani (1975), Kurz (1978), AbrahamFrois and Berrebi (1980), Steedman (1982), D’Agata (1983), Salvadori (1983), Woods (1987), Bidard and Woods (1989), Erreygers (1990), and Bidard (2010). Considerably less work has been done on either exhaustible or renewable resources. Exhaustible resources have been treated by Parrinello (1983, 2004), Kurz and Salvadori (1997, 2000), and Bidard and Erreygers (2001a, 2001b); this literature is reviewed by Ravagnani (2008). The cornguano model presented by Bidard and Erreygers has been subject to debate (see, among others, Kurz and Salvadori, 2001; Parrinello, 2001; and Schefold, 2001), and applied to the problem of waste treatment (Hosoda, 2001).
2As far as renewable resources are concerned, there is very little besides the salmon model of Kurz and Salvadori (1995, pp. 35157). In their model they conceived of the renewable resource—salmon—as a nonbasic commodity. They examined whether longperiod equilibria exist for a given demand for salmon, and arrived at the conclusion that in some circumstances both stable and unstable equilibria may exist. They also showed that in some cases the population of wild salmon is doomed.
3My purpose here is to construct a simple model for the study of renewable resources in a longterm framework. My model is similar, yet distinct from the model presented by Kurz and Salvadori. The main difference is that my assumptions about the fishing technology are less general than those of Kurz and Salvadori, but that allows me to discuss the effects of specific policy interventions. Since I draw my inspiration from existing simple longterm models, I begin with a brief recapitulation of the corn model and natural resources (Section 2). The core of the paper consists of the presentation, analysis and discussion of the corntuna model. The characteristics of the corntuna model are meant to reflect salient features of the use of renewable resources (Section 3). I start the analysis of the model with a comparison of two “pure” systems of production (Section 4) before considering the “mixed” system (Section 5). This leads to the examination of two different government policies (Section 6). I then explore what can be learned from this model, and point out some of its limitations and weaknesses (Section 7).
2 – The Corn Model and Natural Resources
4Students of classical political theory are undoubtedly familiar with the notion of a “corn model.” In his introduction to The Works and Correspondence of David Ricardo, Sraffa (1951, pp. xxxxxxiii) implied that Ricardo must have had such a model in mind as the foundation for the cornratio theory of profits. In its most simple form (Bidard, 2004, chapter 1), the corn model assumes there exists only one produced commodity, corn, which is produced by means of itself and labour. In a more developed version, land is introduced explicitly. The basic character of Ricardian land is that it is indestructible: its quality and quantity remain unchanged, both when it is used for the production of corn and when it lies fallow. Land can therefore be seen as a joint product of corn production. The theory of rent (extensive, intensive, and other types of rent) studies the properties of systems of production using land.
5The introduction of an exhaustible resource leads to serious complications. In contrast to land, an exhaustible resource is “used up” when it is used for corn production, while its quality and quantity remain unchanged when it is left untouched. This implies that in a competitive environment the price of an exhaustible resource must change over time. At any moment of time t a resource owner faces the choice between selling his resource immediately, in which case he obtains a price of p_{g,t} for every unit of resource g, or letting it lie idle and selling it at some future date t’ at the price p_{g,t} per unit. If one of the two options were more profitable than the other, all resource owners would follow the most profitable course of action, which would mean that either the whole supply of the resource would be exploited at time t, or that none of it would. In general this is impossible: in any period of time, except the one in which the resource is depleted, part of the supply is exploited and part of it is conserved. This means that in equilibrium the two options must be equally profitable. If the first option allows you to obtain a rate of return r when you invest the proceeds of selling the resource, then so must the second. Hence we must have:
7Equation (1) implies that the price of the exhaustible resource, i.e. its royalty, rises at a rate equal to the ruling rate of profit. This is the result known as the Hotelling rule, after the seminal contribution of Harold Hotelling (1931). Since exhaustible resources are typically goods which are used for the production of other goods, the Hotelling rule implies that one can expect that the prices of all goods will change over time.
8The cornguano model (Bidard and Erreygers, 2001a, 2001b) has been conceived as a methodological tool to shed light on the dynamics, especially with regard to prices, of models with exhaustible resources. In the cornguano model there is only one produced commodity, corn, and one exhaustible resource, guano. Table 1 describes the available processes. The first, or “guano process,” specifies how corn can be produced by means of guano. The second, or “backstop process,” is an alternative corn production method; the name is suggested by the property that it will be used after the exhaustion of the stock of guano. [2] The third process expresses the fact that unused quantities of guano are transferred without loss of quality to the next period. In order to unravel the dynamics of prices in the cornguano model, assumptions must be made about the expected exhaustion date of guano and about the standard of value of those who earn profits.
Processes in the CornGuano Model
Processes in the CornGuano Model
3 – Renewable Resources
3.1 – The Character of Renewable Resources
9It is typical for exhaustible resources that nature has stopped producing them, or produces them at such an extremely slow rate that humans do not notice it. Renewable resources, by contrast, are goods which are produced by natural processes, although sometimes only if a number of critical conditions are satisfied. Think of pelagic fish: as long as the stock of fish swimming in the sea is higher than the critical mass necessary for reproduction and lower than the maximum carrying capacity of the environment, the stock of fish normally increases, which means that fish is produced. One can then harvest fish without endangering the survival of the species, but only if one does not catch too much. Sadly, there are plenty of examples of overfishing culminating in the extinction or nearextinction of fish species.
10In some cases it may also be possible to produce renewable resources by manmade processes. Once again, fish is a good example: various forms of fish farming have become increasingly important in the last decades. This contrasts sharply with the essentially static level of capture fisheries over the last two decades. In its most recent twoyearly report on the state of world fisheries and aquaculture, the FAO wrote:
Global fish production continues to outpace world population growth, and aquaculture remains one of the fastestgrowing food producing sectors. In 2012, aquaculture set another alltime production high and now provides almost half of all fish for human food. This share is projected to rise to 62 percent by 2030 as catches from wild capture fisheries level off and demand from an emerging global middle class substantially increases. If responsibly developed and practised, aquaculture can generate lasting benefits for global food security and economic growth.
12In the following model I try to incorporate some of these features. I assume there exists a renewable resource—tuna—which can be produced by nature and by man. Making some strong assumptions, I will explore the competition between the natural method and the manmade method and examine whether and when these methods can coexist. Let me recall that the exhaustible resource we considered in the previous section, guano, was a means of production, not a consumption good. In the model we are about to present, the renewable resource is a consumption good. So we move from a model with one consumption good to a model with two consumption goods.
3.2 – Catching Fish in the Wild versus Fish Farming
13First of all we have to clarify the interplay of biological and economic factors involved in the production of fish (see, e.g., Schaefer 1954, 1957). In the absence of fishing, the population P tends to grow according to its natural increase, which can be seen as a function of the population level, i.e. dP / dt = ƒ(P). Let us assume there exists a maximum population level, L, at which the population of fish stops growing, i.e. ƒ(L) = 0. If there is fishing, the population changes with the difference between the natural increase and the amount of fish caught in the sea. The catch c depends basically upon two factors, the population of fish P and the effort level e, so that we can assume that we have a functional relationship c = g(P, e). For a given effort level, a larger population normally leads to a higher catch (∂c/∂P > 0): the more fish are swimming in the sea, the easier it is to catch them. Likewise, for a given population, a higher effort level leads to a higher catch (∂c/∂e > 0): the more boats are out on the seas, the more fish is caught. To simplify the analysis, I take the following natural increase and catch functions:
15where k_{1} and k_{2} are positive constants (Schaefer, 1954, pp. 2931). The natural increase function (2) determines a relationship between the population and its natural increase which has an inverted Ushape (see Figure 1). The bilinear catch function (3) implies that the catch per effort is directly proportional to the population.
Natural rate of increase of a population which grows according to the VerhulstPearl logistic
Natural rate of increase of a population which grows according to the VerhulstPearl logistic
16In a longterm perspective, we are especially interested in sustainable catch levels. A catch is sustainable if the effort level ensures that the population of fish remains constant. This means that, given the population of fish, the effort level is chosen in such a way that the catch coincides with the natural increase. The peak of the natural increase curve represents the maximum sustainable yield, which for (2) is equal to k_{1}(L / 2)^{2}. It is easy to see that to any population 0 ≤ P ≤ L we can associate a sustainable catch c*(P) = ƒ(P). Given the effort function we can also determine a sustainable effort level e*(P), where c*(P) = g(P, e*(P)). For the natural increase and effort functions (2) and (3) we have c*(P) = k_{1}P(L−P) and e*(P) = [k_{1}(L − P)]/k_{2}. The relationship between c*(P) and e*(P) has typically also an inverted Ushape. For instance, for our specific catch and effort functions, we have c*(P) = {k_{2}e*(P)[k_{1}L−k_{2}e*(P)]}/k_{1}. This reflects the fact that any catch below the maximum sustainable yield can be sustained either by a large population (and hence a low effort level) or by a small population (and hence a high effort level). For society it seems best that a largepopulation equilibrium is obtained, i.e. an equilibrium on the righthand side of Figure 1. The first reason is that a largepopulation equilibrium is stable: if there is a small but temporary deviation of the effort level from its sustainable level, the population will tend to return to its equilibrium value. The opposite holds for a smallpopulation equilibrium, which means that there is a substantial risk of overfishing and dwindling populations. The second reason is purely economic: in a smallpopulation equilibrium the cost of producing the same amount of fish as in a comparable largepopulation equilibrium is much higher.
17Fish farming, by contrast, resembles any other industrial or agricultural activity. Fish farms are very much the same as pig farms or cattle farms.
3.3 – The CornTuna Model
18While fish farming can be integrated into a Sraffian model without any problem, catching fish in the wild poses a real challenge. One problem is that for a given population of fish, the natural method of producing fish is not necessarily characterized by constant returns to scale: doubling the effort level may not double the catch of fish. Moreover, as we have just seen, in a sustainable (and hence longterm) equilibrium the effort level is determined by the population of fish. This means we are obliged to introduce some biological elements into the model.
19The first consumption good in our model, corn, is produced by means of itself and land. The second consumption good, tuna, which is the model’s renewable resource, can either be caught in the sea by means of boats (this is the natural process) or be produced artificially in aquaculture ponds (the manmade process). In the first case boats are needed, and to make the model a bit more realistic I assume that every year a fraction of the existing fleet must be replaced. With regard to the second case, I assume that ponds also need to be renewed from time to time, so that each year new ponds have to be constructed. Ignoring land and sea, the model therefore consists of 4 goods (corn, tuna, boats and ponds) and 5 processes. I stress that I make the probably unrealistic assumption that wild and farmed tuna are indistinguishable commodities. Consumers are indifferent between the two: they have a demand for tuna, not for wild or farmed tuna.
20The available processes are described in Table 2. The first process is the single corn production process of the economy. The second and fourth processes are the two available methods to produce tuna: the former using boats to catch fish in the wild, the latter using ponds in which fish are farmed. The third process is the construction process for boats, and the fifth that of ponds. Since boats are used only to catch wild tuna, processes 2 and 3 will always be operated jointly; hence {2,3} is the wild tuna technique. The same holds with respect to processes 4 and 5, and so {4,5} constitutes the farmed tuna technique.
Processes in the CornTuna Model
Processes in the CornTuna Model
21As mentioned, some caution is needed with regard to the second process. The amount of tuna caught per boat, b_{22,t}, is contextdependent and hence it has a time index. Recall that the total catch of tuna in a given period is a function of the existing population of tuna and of the effort level. While the population at time t will be denoted by P_{t} the effort level in the period between time t and time t + 1 will be measured by the size of the fishing fleet active in this period, i.e. y_{2,t}. [3] The catch function can therefore be written as g(P_{t},y_{2,t}). Since the total catch is equal to the catch per boat b_{22,t} multiplied by the number of boats y_{2,t}, we have b_{22,t} = g(P_{t},y_{2,t})/y_{2,t}. Given the effort function (3) the average tuna catch per boat b_{22,t} is a simple linear function of the population P_{t}:
23As far as the b_{23} coefficient is concerned, I assume it is smaller than 1. This expresses the fact each year some boats break down or perish, so that new boats have to be constructed in order to keep the fleet constant. The number 1 − b_{23} measures the fraction of the fleet which must be replaced each year in order to keep the fleet constant. A similar reasoning underpins the assumption that the b_{44} coefficient is smaller than 1: ponds also have to be replaced, and the replacement rate is equal to 1 − b_{44}.
24Please note that in order to simplify the analysis, I assume that tuna is not used as a means of production, except in aquaculture. In this respect the model presented here is slightly different from the salmon model of Kurz and Salvadori (1995, p. 352). Given that overfishing can lead to the exhaustion of fish stocks, the wild tuna technique is to a certain extent comparable to the guano process, whereas the farmed tuna technique resembles the backstop process in the cornguano model.
25Finally, when I talk about the sea, I always understand that this is not a privately owned good. Whereas the amount of fish in the sea may be large or small, the sea itself is assumed to be vastly large. The sea therefore does not have a price, and no rent will ever be paid for its use. Nevertheless, it may occur that boat owners are in a situation to capture the advantage of using the sea as a productive agent. The “sea rent” must therefore be understood as a kind of extraprofit earned by boat owners. Although land is (usually) privately owned, I assume here that it is so plentiful that its owners never earn rent. This means that we can drop the land from both the quantity and the price equations.
3.4 – The Quantity Side
26The quantity equations specify that the activity levels are such that the system is capable of satisfying the net demands for corn and tuna of the current period, and of producing everything which is needed to continue production in the next period. This may require an expansion of the fishing fleet and of the number of ponds.
27Let the net demands for corn and tuna in period t be equal to d_{1,t} and d_{2,t}. The activity level of process i in period t is represented by y_{i,t}. My choice of units implies that y_{i,t} stands for the area of land used for corn production, y_{2,t} for the effective number of boats used to catch tuna, y_{3,t} for the number of new boats constructed, y_{4,t} for the effective number of ponds used to produce tuna, and y_{5,t} for the number of new ponds constructed. The quantity equations can be written as follows:
29The first equation specifies that the output of corn must be equal to the sum of the aggregate input of corn in the next period and the net demand for corn in the current period. The second equation stipulates the same with respect to the output of tuna. The third equation expresses that the size of the fleet of period t + 1 is equal to the sum of what remains of the fleet used in period t and the number of new vessels produced in period t. The fourth equation states the same with regard to the number of ponds.
30Given that the tuna catch per boat depends on the population of tuna (cf. (4)), the activity levels can be determined only if we know how the demands for corn and tuna as well as the population of tuna change over time. One could try to derive the dynamics of the system by means of these equations and inequalities making assumptions about the level of demand in each period, the evolution of the population of tuna, and the initially available quantities of the different goods. But since a choice has to be made among different processes, we also need to consider the price side.
3.5 – The Price Side
31As usual in Classical models, I assume that none of the available methods makes extraprofits, and that all methods which are actually operated break even. I take corn to be the numéraire good, and the rate of profit r to be given, constant over time, and nonnegative.
32Let the prices of the four goods in period t be denoted as p_{i,t}, i = 1,2,3,4, the wage as w_{t} and the “sea rent per boat” as z_{t}. Since corn is the numéraire, we have p_{1,t} = 1 in every period. The prices, wage and rent must be such that the following conditions hold:
34where the variable between square brackets is zero if the corresponding inequality is strict.
35Boat owners are in a position to capture rent only if they can produce tuna more cheaply than tuna farmers, and if they are unable to satisfy the demand for tuna. We will be more specific about that in what follows.
3.6 – Stationary Conditions
36In the following sections I focus on situations which correspond to the longterm positions usually studied in Classical theory. Let us make the strong assumption that the demand for corn and tuna remain constant over time. I will explore whether there is room for longterm equilibria characterized by constant prices and activity levels, and where the catch of wild tuna is sustainable. This implies that I will consider only situations where the population of tuna remains constant. I will try to find out whether it is possible that the same set of processes is used for an extended period of time, with the prices and activity levels equal to their longterm equilibrium levels. [4]
4 – Wild versus Farmed Tuna
4.1 – Two Pure Systems
37It seems useful to begin the analysis with a comparison of two “pure” systems of production. We will look at the possible combination of the two at a later stage.
38The first pure system is the one in which tuna is caught in the wild only. Two conditions are vital here: it must be cheaper to catch tuna in the wild than to farm it, so that there is no incentive to switch to aquaculture, and the catch must be both sufficient to satisfy demand and sustainable, so that the existing population of tuna remains the same. This will be called the Wild Tuna System. In formal terms, only processes {1,2,3} are operated, and the activity levels and prices are determined by the following systems of equations (since the activity levels and prices are assumed to be constant, we can drop the time index from the quantity and price equations):
Wild Tuna System
Wild Tuna System
40We already know that the catch per boat is b_{22} = k_{2}P. The existing population P must be such that the supply and demand of tuna coincide with the sustainable catch (y_{22}b_{22} = d_{2} = c*(P)), and the size of the fleet with the corresponding effort level (y_{2} = e*(P)). For the natural increase and catch functions (2) and (3) this means that the fleet size is equal to y_{2} = [k_{1}(L − P)]/k_{2}.
41The second pure system is the one in which tuna is produced in aquaculture ponds only. The crucial condition here is that it must be cheaper to farm tuna than to catch it in the wild. But there is another aspect which must be taken into account: if no tuna is caught in the wild and the population of tuna is below its maximum (but not zero), the population will tend to increase and it might be that at some future date catching tuna in the wild becomes more profitable than farming tuna. It must therefore be cheaper to farm tuna than to catch it in the wild for all possible population levels. In this system, which I will call the Farmed Tuna System, only processes {1,4,5} are operated, and the activity levels and prices are determined by the following systems of equations:
Farmed Tuna System
Farmed Tuna System
4.2 – The Activity Levels
43Let us begin by exploring how the activity levels are determined if either the Wild Tuna System or the Farmed Tuna System is operated. If only wild tuna is to be produced, the demand for tuna cannot be higher than the maximum sustainable yield. Hence we must have:
45We already know that the catch per boat is b_{22} = k_{2}P and that a sustainable catch requires a fishing fleet equal to y_{2} = [k_{1}(L − P)]/k_{2}. Since the total catch is then y_{2}b_{22} = k_{1}(L − P)P, it follows that the population P must be such that:
47As can be seen from Figure 1, when the demand for tuna is lower than the maximum sustainable yield, there exist two population levels which are able to sustain the demand for tuna: the smallpopulation equilibrium and the largepopulation equilibrium . Once the population P and the fleet size y_{2} are known, the third equation of (W.Q) determines the required number of new boats y_{3}, and the first equation determines the activity level of corn production y_{1}.
48By contrast, if the supply of tuna comes only from farmed tuna, the second equation of (F.Q) determines the required number of ponds y_{4}, the third the required number of new ponds y_{5}, and the first the activity level of corn production y_{1}.
4.3 – The Costs of Producing Wild and Farmed Tuna
49The choice between the Wild and Farmed Tuna Systems depends crucially upon which of the two systems produces tuna in the cheapest way. At first sight the comparison of the two systems seems impossible because they do not produce the same set of goods: in the Wild Tuna System ponds do not occur, and in the Farmed Tuna System boats are inexistent. But the problem is only apparent. In fact, in both systems the first process determines the wage rate, and this is all that is needed to calculate the shadow prices of boats and ponds, even if they are not actually produced. Since the wage is the same in both systems, the (shadow) prices of boats and ponds are also the same in both systems. Hence the choice is determined exclusively by the price of production of tuna; the system with the lowest tuna price is the most efficient.
50Let us formalize this reasoning. The wage w is determined exclusively by the first process. Given the wage, the (shadow) price of boats is determined by the third process. Likewise, the (shadow) price of ponds is determined by the fifth process. Given the wage and the price of boats, the price of tuna in the Wild Tuna System is determined by the second process. This gives us the price of production of wild tuna p^{W}_{2}:
52In the Farmed Tuna System, the price of tuna is determined by the fourth process. This gives us the price of production of farmed tuna p^{F}_{2}:
54For a given rate of profit, what can we say about the prices of tuna in the two systems? From (17) it follows that the catch per boat b_{22} has a crucial influence on the wild tuna price. When the catch per boat is relatively high, the wild tuna price is low and probably aquaculture will not be able to produce tuna more cheaply. But when the catch per boat is relatively low, the wild tuna price is high and may very well be above the farmed tuna price. In that case, traditional fishery loses the competition with aquaculture, which turns out to be the cheapest tuna producer. The threshold level of the catch per boat can be calculated with precision. The transition occurs for the value of b_{22} for which we have p^{W}_{2} = p^{F}_{2}; in other terms, aquaculture is more efficient than traditional fishery when the catch per boat sinks below the critical value:
56In view of (4) it is possible to define a corresponding critical population level:
4.4 – Wild versus Farmed Tuna
58We can now be more specific in our conclusions about the comparison between the two systems. The Wild Tuna System will dominate the Farmed Tuna System if the demand for tuna does not exceed the maximum sustainable yield, and if in addition wild tuna can be produced more cheaply than farmed tuna. Nevertheless, there seem to be some Wild Tuna System equilibria which are very unlikely to occur. This is due to their unstable character.
59When the first condition holds, we have seen there are two possible outcomes: a smallpopulation equilibrium P^{S} and a largepopulation equilibrium P^{L}. The smallpopulation equilibrium is clearly an unstable equilibrium: any deviation of the catch from its sustainable level c*(P^{S}) = k_{1}(L − P^{S})P^{S} will entail that the population moves away from P^{S}. Although it is in principle possible that the economy remains in the smallpopulation equilibrium, it is hard to see how the economy could ever reach it. The largepopulation equilibrium, on the other hand, is stable. It seems therefore that we must discard the smallpopulation equilibria and concentrate on the largepopulation equilibria.
60When we look at the second condition, it turns out that the probability that it is satisfied is always higher in a largepopulation equilibrium than in a smallpopulation equilibrium. Since more boats are required in the smallpopulation equilibrium than in the largepopulation equilibrium, the catch per boat is lower in the smallpopulation equilibrium than in the largepopulation equilibrium (b^{S}_{22} < b^{L}_{22}). In the smallpopulation equilibrium the price of production of wild tuna is therefore always higher than in the largepopulation equilibrium; the ratio of the two prices is equal to b^{L}_{22}/b^{S}_{22} = P^{L}/P^{S}.
61To check whether the second condition is verified, we have to look at the critical value of the catch per boat (cf. (20)) or, equivalently, at the critical population level P̃ (cf. (21). Three cases can be distinguished. If P̃ < P^{L}, catching tuna in the wild is more efficient than farming tuna for a tuna population equal (or close) to P^{L}. Aquaculture will not get off the ground: it is simply too expensive. If P^{L} < P̃, catching tuna in the wild is less efficient than farming tuna for a tuna population equal (or close) to P^{L}. Hence, the fishing fleet will remain idle and the demand for tuna will be satisfied by tuna farmers. But if no wild tuna is caught, the population of tuna will increase, and after some time the point may be reached where it is once again more efficient to catch tuna at sea. We therefore have to distinguish two subcases: either a population level exists for which this point is reached, or no such population level exists. More precisely, the former arises if P̃ < L, and the latter if L ≤ P̃. In the first case, the likely outcome is a mixture of both systems of production—this will be examined in the next section. In the second case, no wild tuna will ever be caught: it is much too expensive to do so.
62It is also interesting to note that it may depend upon the rate of profit which of the two systems is the most profitable one. It can occur that the Wild Tuna System is more profitable than the Farmed Tuna System for low rates of profit, but less profitable for high rates of profit. We may even see reswitching for very high rates of profit, as shown in the following example.
4.5 – A Numerical Example (I)
63Consider an economy characterized by the following data (for the moment we leave the value of the catch per boat undetermined):
64Under the Wild Tuna System the prices and wage will be:
66and the shadow price of ponds
68Under the Farmed Tuna System we obtain:
70and the shadow price of boats
72The catch per boat for which the wild and farmed tuna prices coincide is equal to:
74For instance, when r = 1/15≈6.67%, we have p^{W}_{2} = 47/(3b_{22}), p^{F}_{2} = 1.5, p_{3} = 40.4, p_{4} = 26.8 and w = 118/15≈7.87. It follows that the critical value is equal to 94/9≈10.44.
75To find out which of the two systems is the most efficient, we have to determine the value of the average catch per boat and check whether it is smaller or larger than the critical value . Let us assume that the parameters of the natural increase and catch functions are k_{1} = 1/126900 and k_{2} = 1/1000, and the maximum population L = 20772. The average catch per boat is therefore b_{22} = P/1000, which means that every boat is able to catch 0.1% of the existing population. The maximum sustainable yield of wild tuna is slightly higher than 850.
76Suppose that the demand for tuna is equal to d_{2} = 832.23, which is below the maximum sustainable yield. The two population levels which are able to sustain this demand are P^{S} = 8883 and P^{L} = 11889. We can safely discard the lowpopulation equilibrium. For the highpopulation equilibrium we have b_{22} = 11.889>10.44≈b_{22}, which implies that the Wild Tuna System dominates the Farmed Tuna System when r = 1/15. The required size of the fishing fleet is y_{2} = 70, which means that in every period 7 new ships have to be built. The price of production of wild tuna is p^{w}_{2} = 47000/35667≈1.32.
77In order to show the influence of the rate of profit on the comparison between the two systems, let us suppose that the demand for tuna is equal to d_{2} = 846. This can be sustained by a tuna population P^{L} = 10575, with a catch per boat equal to b_{22} = 10.575 and a fleet size of y_{2} = 80 boats. The wild tuna price is therefore equal to p^{w}_{2} = (−360r^{2} + 1564r + 524)/423. From this we can infer that the Wild Tuna System dominates the Farmed Tuna System for r < 1/10 = 10%, the Farmed Tuna System dominates the Wild Tuna System for 1/10 < r < 259/360, and the Wild Tuna System again dominates the Farmed Tuna System for r > 259/360≈71.94%.
5 – The Mixed Tuna System
5.1 – The Mixed Tuna System
78At the end of our comparison of the Wild and Farmed Tuna Systems it transpired that there may be circumstances in which the demand for tuna will be satisfied by both wild and farmed tuna. For this to happen, the Wild Tuna System must dominate the Farmed Tuna System for large populations; in formal terms, the critical population P̃ must be such that:
80If the price of production of wild tuna were lower than that of farmed tuna, boat owners would be in a position to earn rents. These boat rents constitute extra profits, and act as incentives to increase the number of boats. Hence, the fleet size would increase, more wild tuna would be caught, and the tuna population would decrease. Obviously, the process comes to an end only if the difference between the two costs of production disappears. An equilibrium is reached when the fishing fleet is expanded to its economic limit, i.e. up to the point where the price of production of wild tuna becomes equal to that of farmed tuna.
81In the Mixed Tuna System all processes {1,2,3,4,5} are operated and the following systems of equations hold:
Mixed Tuna System
Mixed Tuna System
83Moreover, as we have just seen, the population of tuna must be equal to P̃.
5.2 – The Activity Levels and Prices
84Once we know the equilibrium population level, it is straightforward to calculate the activity levels. Since the catch per boat is b_{22} = k_{2}P̃ and the size of the fleet y_{2} = [k_{1}(L – P̃] / k_{2}, the supply of wild tuna is equal to the sustainable catch k_{1}(L – P̃)P̃. It follows that the net amount of tuna which needs to be supplied by tuna farmers is equal to d_{2} – k_{1}(L – P̃)P̃. Once the supplies of wild and farmed tuna are known, the other activity levels are determined very easily.
85As far as the wage and prices are concerned, they are the same as in the Farmed Tuna System.
5.3 – A Numerical Example (II)
86Let us return to our previous example, and assume that the rate of profits is r = 10%. The critical value of the catch per boat is . This implies that P̃ = 10575, with the corresponding fleet size equal to y_{2} = 80 boats and the sustainable catch of wild tuna equal to . Since we have 10386<10575<20772, condition (22) is verified.
87Suppose the net demands for corn and tuna are equal to d_{1} = 356 and d_{2} = 1206. Given that the supply of wild tuna is equal to 846 units, the net supply of farmed tuna must be 1206−846 = 360 units, which requires an aquacultural park equal to y_{4} = 40 ponds. In each period 8 vessels are replaced, as well as 4 ponds. Summarizing, we have:
89With regard to prices and the wage, we have:
6 – Government Policies
6.1 – An Unstable Situation
91The Mixed Tuna System comes into being when the demand for tuna is too high to be satisfied by wild tuna only, and when the price of production of wild tuna happens to be lower than that of farmed tuna for sufficiently high levels of the population of tuna. Condition (22) is in fact crucial to having a stable equilibrium. Suppose that wild tuna remains cheaper to produce even at relatively low levels of the population of tuna, or more formally, that P̃ < L / 2. As long as the tuna population exceeds P̃, boat owners are in a position to capture rents, and hence incentives exist to increase the fishing fleet. Owning a boat may be a source of considerable wealth, and it can be expected that rent seeking will occur (Krueger, 1974). The expansion of the fishing fleet inevitably leads to capture levels exceeding the natural increase, and hence the tuna population will decrease. In principle, this process comes to an end when the population reaches the equilibrium level P̃, but since this is an unstable equilibrium, there is a very high risk that we end up in a situation where the population of tuna will steadily decline and eventually perish.
92Put differently, if the population level for which wild and farmed tuna are equally costly is too low, mechanisms are at work which will lead to overfishing and extinction of tuna. In comparison to farmed tuna, wild tuna remains too cheap for too long. As long as there is cost advantage for wild tuna, the fishing fleet will expand. But when the cost advantage finally disappears, the remaining population of tuna will be low and fragile.
93Governments or supranational organizations such as the European Union which are concerned about the survival of wild tuna might think about ways of preventing the system from entering the danger zone of overfishing and extinction. In what follows I present two possible policies, one working on quantities, the other on prices.
6.2 – Limiting the Fleet Size
94Since it is the expansion of the fleet size which pushes the system towards overfishing, an obvious policy consists of imposing a limit on the number of boats. Suppose that the government introduces a strict boat licensing system by which the maximum number of boats is fixed at the level ȳ_{2}. Evidently, in order to be effective this maximum number should be small enough to keep the system in the region of stable equilibria.
95The presence of a constant and consistently enforced limitation of the fleet size gives rise to a variant of the Mixed Tuna System, characterized by a positive boat rent z. With the number of boats limited to ȳ_{2}, the associated population P̄ can be found through the equation ȳ = [k_{1}(L – P̄)] / k_{2}. Given P̄, we can determine the catch per boat and hence the sustainable catch . The systems of equations determining the prices and quantities can now be written as:
Mixed Tuna System with Fleet Limitation
Mixed Tuna System with Fleet Limitation
97The rent per boat is determined by the difference between the cost prices of wild and farmed tuna and equal to:
99The selling price of tuna, whether it be wild or farmed, is equal to the cost price of farmed tuna.
6.3 – Taxing Fishing Boats
100The incentive to increase the number of vessels comes from the existence of rents. An alternative policy consists of reducing the rents by taxation. Suppose the government imposes a flat tax per boat equal to T. This obviously raises the cost of production of wild tuna. As in the case of a fleet limitation policy, the tax should be sufficiently high to keep the system out of the region of unstable equilibria.
101Boat owners earn a positive net rent as long as z > T. The incentive to increase the number of vessels therefore disappears when the size of the fishing fleet is such that z = T. In view of (23), the corresponding population level is determined by means of the following equation:
103Given P, it is easy to determine the size of the fishing fleet y_{2} = [k_{1}(L−P)]/k_{2}, the catch per boat b_{22} = k_{2}P, and the sustainable catch b_{22}y_{2} = k_{1}(L−P)P.
104The systems of equations determining the prices and quantities can now be written as:
Mixed Tuna System with Boat Taxation
Mixed Tuna System with Boat Taxation
6.4 – Numerical Example (III)
106Let us once again return to our numerical example, with one change: the maximum population of tuna is now L = 30879. Assume that the rate of profits is r = 10%. The critical value of the catch per boat remains and that of the population of tuna P̃ = 10575. The corresponding fleet size is now y_{2} = 160 boats and the sustainable catch of wild tuna . Since we have 10575 < 15439.5, condition (22) is not verified.
107Assume that the fishing fleet is restricted to ȳ_{2} = 110 vessels. The corresponding tuna population is P̄ = 16920, the catch per boat and the sustainable catch . The activity levels corresponding to the net demands for corn and tuna d_{1} = 356 and d_{2} = 2052 are equal to:
109For r = 10% the prices and wage are the same as in the case without limitation of the fleet size. Since the cost price of wild tuna is significantly different from the cost price of farmed tuna (p^{w}_{2} = 1, p^{F}_{2} = 1.6), the rent per boat is substantial: each boat earns a rent equal to z = 10.152.
110Assume alternatively that a boat tax equal to T = 10.152 is imposed. Then the same solution will be obtained.
7 – Discussion
7.1 – Predictions of the Model
111It may be useful to summarize what the corntuna model predicts under plausible assumptions about the evolution of the net demand for corn and tuna. To make things interesting, let us start the analysis at a point in time at which the world’s oceans contain plenty of wild tuna and the demand is relatively small.
112At first, a small fishing fleet suffices to catch all the wild tuna that is required, at a low per unit cost. Wild tuna therefore dominates the market. As the demand for tuna increases, the fishing fleet becomes larger, the average catch per boat declines, and the price of wild tuna gradually increases. The rate at which the wild tuna price increases is the same as the rate at which the catch per boat decreases. Inevitably, one of two things will happen: either farmed tuna will enter into competition with wild tuna when the population of tuna is relatively large (higher than L/2), or it will do so when the population of tuna is relatively small (lower than L/2). In the former case, a stable equilibrium exists where the market for tuna is divided between wild and farmed tuna. Boat owners will not earn rents, and the risk of overfishing appears to be low. In the latter case, however, the existence of rents will push the system into the danger zone of unstable equilibria, with a considerable risk of overfishing and depletion. Without government intervention, the most probable outcome is that wild tuna will eventually become extinct, and the whole supply of tuna will consist of farmed tuna.
113In the Good Scenario (i.e. the first case) the wild tuna price is initially lower than the farmed tuna price, but not by too much. As the demand for tuna increases, the wild tuna price increases until it reaches the level of the farmed tuna price. From that moment on, the price of tuna remains constant and equal to the farmed tuna price, even though wild tuna continues to be supplied to the market. In the Bad Scenario (i.e. the second case) wild tuna is initially much cheaper than farmed tuna. But the problem is that it remains too cheap for too long: farmed tuna enters into competition with wild tuna only when the stocks of tuna have become very low. It is difficult to say what the evolution of the tuna price will be in this highly unstable environment.
114The corntuna model therefore predicts that in a situation of growing demand the price of the renewable resource will steadily increase, until it reaches a steady equilibrium (the Good Scenario) or becomes unstable (the Bad Scenario). This may be thought of as a kind of Hotelling rule for renewable resources, but it must be kept in mind that the logic behind this rule is fundamentally different from the logic behind the Hotelling rule. The increase of the price of an exhaustible resource, such as guano in the cornguano model, is explained by the profitmaximizing behaviour of the resource owners and occurs even when demand is not growing. In the corntuna model, by contrast, the price increase of the renewable resource is driven by a combination of biological and economic factors and occurs only when demand is growing.
115The model also predicts that in the long run the renewable resource will either have a declining market share (the Good Scenario) or be replaced entirely by an industrially produced ersatz good (the Bad Scenario), unless a policy is implemented to prevent the collapse of the renewable resource industry. Such a policy can take the form of quantitative restrictions or of price incentives.
7.2 – Limitations and Shortcomings of the Model
116It remains to be seen whether the predictions of the corntuna model are in accordance with the stylized facts of the world’s fisheries. It may very well be that this simple model misses out on some essential features of the real world. The model surely cannot be applied to all renewable resources. For instance, it is not always the case that the natural “source” which “breeds” the renewable resource (in the corntuna model the source is the sea) has no owner. Environmental economists know very well that different property regimes may lead to different outcomes (see, e.g., Pearce and Warford, 1993, Chapter 10).
117The model can be improved by establishing explicit links between crucial variables. Feedback effects, positive or negative, are absent. The model also ignores technical change, both in the traditional fishing industry and in aquaculture. These might be powerful enough to counteract the tendency of increasing wild tuna prices, or to bring about a more rapid change to aquaculture.
8 – Conclusion
118In this paper I have examined how renewable resources can be integrated into Classical theory. Starting from the insights which can be obtained from the corn model and the cornguano model, I have developed a corntuna model in which a valuable renewable resource (tuna) can be produced either by nature or by man. In the first case it is harvested by means of boats, in the second it is farmed using aquaculture ponds. Increasing levels of demand will entail the intensified use of the natural process, but eventually the manmade process will become more and more dominant. It could very well be that the natural process is operated for such a long time and at such a high level that the renewable resource becomes overexploited, which will lead to the exhaustion of the resource. Since Classical theory has a longterm perspective, I believe it should try to come to grips with this issue.
Notes

[1]
Department of Economics, University of Antwerp; guido.erreygers@uantwerpen.be

[2]
Both processes obviously also require land. Assuming land to be of uniform quality and always abundantly available, its price will be zero. Hence land plays no role in this model, and therefore it can be dropped from the inputs and outputs.

[3]
To be precise, the effort consists of a combination of boats, corn and labour, but since these are always used in the same proportion, there is no harm in treating boats as a proxy.

[4]
Of course, in reality changes occur all the time. The perspective adopted here is that it makes sense to focus on longterm equilibrium positions if the changes are slow and gradual, and if the price and quantity dynamics occasioned by these changes are stable, so that the prices and activity levels will never deviate very much from their longterm equilibrium levels. However, as we will see below, there may be circumstances in which stability is not guaranteed.