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 near-extinction of fish species.

10In some cases it may also be possible to produce renewable resources by man-made 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 two-yearly report on the state of world fisheries and aquaculture, the FAO wrote:

11

Global fish production continues to outpace world population growth, and aquaculture remains one of the fastest-growing food producing sectors. In 2012, aquaculture set another all-time 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 man-made 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:

14

15where k₁ and k₂ are positive constants (Schaefer, 1954, pp. 29-31). The natural increase function (2) determines a relationship between the population and its natural increase which has an inverted U-shape (see Figure 1). The bilinear catch function (3) implies that the catch per effort is directly proportional to the population.

Figure 1

Natural rate of increase of a population which grows according to the Verhulst-Pearl logistic

Natural rate of increase of a population which grows according to the Verhulst-Pearl logistic

16In a long-term 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 k₁(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) = k₁P(L−P) and e*(P) = [k₁(L − P)]/k₂. The relationship between c*(P) and e*(P) has typically also an inverted U-shape. For instance, for our specific catch and effort functions, we have c*(P) = {k₂e*(P)[k₁L−k₂e*(P)]}/k₁. 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 large-population equilibrium is obtained, i.e. an equilibrium on the right-hand side of Figure 1. The first reason is that a large-population 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 small-population equilibrium, which means that there is a substantial risk of overfishing and dwindling populations. The second reason is purely economic: in a small-population equilibrium the cost of producing the same amount of fish as in a comparable large-population 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 Corn-Tuna 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 long-term) 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 man-made 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.

Table 2

Processes in the Corn-Tuna Model

Inputs Outputs Corn Tuna Boat Pond Labour Corn Tuna Boat Pond a11 0 0 0 l1 → b11 0 0 0 a21 0 1 0 l2 → 0 b22,t b23 0 a31 0 0 0 l3 → 0 0 1 0 a41 a42 0 1 l4 → 0 b42 0 b44 a51 0 0 0 l5 → 0 0 0 1

Processes in the Corn-Tuna Model

21As mentioned, some caution is needed with regard to the second process. The amount of tuna caught per boat, b_22,t, is context-dependent 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:

22

23As far as the b₂₃ 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 − b₂₃ 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 b₄₄ coefficient is smaller than 1: ponds also have to be replaced, and the replacement rate is equal to 1 − b₄₄.

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 corn-guano 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 extra-profit 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:

28

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 extra-profits, 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:

33

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 long-term 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 long-term 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 long-term equilibrium levels. [4]

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):

39

Wild Tuna System

Wild Tuna System

40We already know that the catch per boat is b₂₂ = k₂P. The existing population P must be such that the supply and demand of tuna coincide with the sustainable catch (y₂₂b₂₂ = d₂ = c*(P)), and the size of the fleet with the corresponding effort level (y₂ = e*(P)). For the natural increase and catch functions (2) and (3) this means that the fleet size is equal to y₂ = [k₁(L − P)]/k₂.

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:

42

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:

44

45We already know that the catch per boat is b₂₂ = k₂P and that a sustainable catch requires a fishing fleet equal to y₂ = [k₁(L − P)]/k₂. Since the total catch is then y₂b₂₂ = k₁(L − P)P, it follows that the population P must be such that:

46

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 small-population equilibrium and the large-population equilibrium . Once the population P and the fleet size y₂ are known, the third equation of (W.Q) determines the required number of new boats y₃, and the first equation determines the activity level of corn production y₁.

48By contrast, if the supply of tuna comes only from farmed tuna, the second equation of (F.Q) determines the required number of ponds y₄, the third the required number of new ponds y₅, and the first the activity level of corn production y₁.

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^W₂:

51

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^F₂:

53

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 b₂₂ 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 b₂₂ for which we have p^W₂ = p^F₂; in other terms, aquaculture is more efficient than traditional fishery when the catch per boat sinks below the critical value:

55

56In view of (4) it is possible to define a corresponding critical population level:

57

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 small-population equilibrium P^S and a large-population equilibrium P^L. The small-population equilibrium is clearly an unstable equilibrium: any deviation of the catch from its sustainable level c*(P^S) = k₁(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 small-population equilibrium, it is hard to see how the economy could ever reach it. The large-population equilibrium, on the other hand, is stable. It seems therefore that we must discard the small-population equilibria and concentrate on the large-population equilibria.

60When we look at the second condition, it turns out that the probability that it is satisfied is always higher in a large-population equilibrium than in a small-population equilibrium. Since more boats are required in the small-population equilibrium than in the large-population equilibrium, the catch per boat is lower in the small-population equilibrium than in the large-population equilibrium (b^S₂₂ < b^L₂₂). In the small-population equilibrium the price of production of wild tuna is therefore always higher than in the large-population equilibrium; the ratio of the two prices is equal to b^L₂₂/b^S₂₂ = 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):

Inputs Outputs Corn Tuna Boat Pond Labour Corn Tuna Boat Pond 2 0 0 0 1 → 1 0 0 0 01 0 1 0 1 → 0 b22 9/10 0 1 0 0 0 5 → 0 0 1 0 1 1 0 1 1 → 0 10 0 9/10 3 0 0 0 3 → 0 0 0 1

64Under the Wild Tuna System the prices and wage will be:

65

66and the shadow price of ponds

67

68Under the Farmed Tuna System we obtain:

69

70and the shadow price of boats

71

72The catch per boat for which the wild and farmed tuna prices coincide is equal to:

73

74For instance, when r = 1/15≈6.67%, we have p^W₂ = 47/(3b₂₂), p^F₂ = 1.5, p₃ = 40.4, p₄ = 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 k₁ = 1/126900 and k₂ = 1/1000, and the maximum population L = 20772. The average catch per boat is therefore b₂₂ = 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 d₂ = 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 low-population equilibrium. For the high-population equilibrium we have b₂₂ = 11.889>10.44≈b₂₂, which implies that the Wild Tuna System dominates the Farmed Tuna System when r = 1/15. The required size of the fishing fleet is y₂ = 70, which means that in every period 7 new ships have to be built. The price of production of wild tuna is p^w₂ = 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 d₂ = 846. This can be sustained by a tuna population P^L = 10575, with a catch per boat equal to b₂₂ = 10.575 and a fleet size of y₂ = 80 boats. The wild tuna price is therefore equal to p^w₂ = (−360r² + 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%.

5.1 – 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:

79

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:

82

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 b₂₂ = k₂P̃ and the size of the fleet y₂ = [k₁(L – P̃] / k₂, the supply of wild tuna is equal to the sustainable catch k₁(L – P̃)P̃. It follows that the net amount of tuna which needs to be supplied by tuna farmers is equal to d₂ – k₁(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 y₂ = 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 d₁ = 356 and d₂ = 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 y₄ = 40 ponds. In each period 8 vessels are replaced, as well as 4 ponds. Summarizing, we have:

88

89With regard to prices and the wage, we have:

90

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 ȳ₂. 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 ȳ₂, the associated population P̄ can be found through the equation ȳ = [k₁(L – P̄)] / k₂. 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:

96

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:

98

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:

102

103Given P, it is easy to determine the size of the fishing fleet y₂ = [k₁(L−P)]/k₂, the catch per boat b₂₂ = k₂P, and the sustainable catch b₂₂y₂ = k₁(L−P)P.

104The systems of equations determining the prices and quantities can now be written as:

105

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 y₂ = 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 ȳ₂ = 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 d₁ = 356 and d₂ = 2052 are equal to:

108

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^w₂ = 1, p^F₂ = 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.

7.1 – Predictions of the Model

111It may be useful to summarize what the corn-tuna 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 corn-tuna 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 corn-guano model, is explained by the profit-maximizing behaviour of the resource owners and occurs even when demand is not growing. In the corn-tuna 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 corn-tuna 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 corn-tuna 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.