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2.1
A preliminary statement of the problem
2.2
The problem of open access
2.3
The problem of common property
2.4
Co-ordination failure under common property
2.5
The conjectural variations approach
2.6 The lindahl equilibrium
2.1 A preliminary statement of the problem
Though, as demonstrated in the preceding chapter, high rates of resource exploitation in LDCs may be part of a welfare-maximizing programme, it is often claimed that the rates currently observed in those countries are exceedingly high and that the resulting pattern of management of natural resources is patently inefficient. This observation is the starting-point of an abundant literature which identifies the main source of inefficiency in the management of natural resources with the absence of well-defined property rights and the regime of open access which characterizes them. As early as 1833, William Foster Lloyd (who was concerned with the cheek on population growth imposed by limited employment opportunities) identified the problem which later on came to be known as 'the tragedy of the commons':
Why are the cattle in a common so puny and stunted? Why is the common itself so bare-worn and cropped differently from the adjoining enclosures? No inequality, in respect of natural or acquired fertility, will account for the phenomenon. The difference depends on the difference of the way in which an increase of stock in the two cases affects the circumstances of the author of the increase. If a person puts more cattle into his own field, the amount of subsistence which they consume will be deducted from that which was at the command of his original stock; and, if, before, there was no more than a sufficiency of pasture, he reaps no benefits from the additional cattle, what is gained in one way being lost in another. But if he puts more cattle on a common, the food which they consume forms a deduction which is shared between all the cattle, as well as that of others as his own, in proportion to their number, and only a small part of it is taken from his own cattle. In an enclosed pasture, there is a point of saturation, if I may so call it, (by which, I mean a barrier depending on considerations of interest), beyond which no prudent man will add to his stock. In a common, also, there is in like manner a point of saturation. But the position of the point in the two cases is obviously different. Were a number of adjoining pastures, already fully stocked, to be at once thrown open, and converted into one vast common, the position of the point of saturation would immediately be changed. The stock would be increased, and would be made to press much more forcibly against the means of subsistence. Lloyd 1833, in Hardin and Baden, 1977: 11)
In a more recent neo-Malthusian pamphlet, this position has been restated by Garett Hardin under the expression 'the tragedy of the commons' in the following way:
The tragedy of the commons develops in the following way. Picture a pasture open to all. It is to be expected that each herdsman will try to keep as many cattle as possible on the commons. Such an arrangement may work reasonably satisfactorily for centuries because tribal wars, poaching and disease keep the numbers of both man and beast well below the carrying capacity of the land. Finally, however, comes the day of reckoning, that is, the day when the long desired goal of social stability becomes a reality. At this point, the inherent logic of the commons remorselessly generates tragedy. As a rational being, each herdsman seeks to maximize his gain. Explicitly or implicitly, more or less consciously, he asks: 'What is the utility to me of adding one more animal to my herd?' This utility has one negative and one positive component.
Adding together the component partial utilities, the rational herdsman concludes that the only sensible course for him to pursue is to add another animal to his herd. And another.... But this is the conclusion reached by each and every rational herdsman sharing a commons. Therein is the tragedy. Each man is locked into a system that compels him to increase his herd without limit—in a world that is limited. Ruin is the destination toward which all men rush, each pursuing his own best interest in a society that believes in the freedom of the commons. Freedom in a commons brings ruin to all. (Hardin, 1968: 20)
2.2 The problem of open access
Two different sets of issues must be distinguished in the above statements: the problem of open access and the problem of common property. Let us first analyse the question of open access. When a given resource is in open access, the agents have to decide whether or not they should 'enter' and start exploiting the resource. Their choice is based on the comparison between the price of entry which they have to bear and the expected income they will get. As long as the net expected benefit is positive, they decide to enter and exploit the resource. The problem is that their private evaluation of the expected benefits does not take into account the fall in the others' incomes which is caused by their entry: by their action, they impose an externality on the other agents. Since they do not allow for it, the resulting situation will be typically inefficient.
Let us take the example of a fishery, and assume that the entire stock of fish, S. could be caught by one fisherman alone (we abstract momentarily from the kind of dynamic considerations which we have made in Chapter I regarding the management of a renewable resource). The fishing technology is such that the entire stock is divided equally among n active fishermen. Consider the choice facing the nth fisherman. The cost of renting a boat is given and equal top. He chooses between not to hire a boat and catch nothing, or to hire a boat at a price p and catch S/n units of fish. As we assume that all other costs are negligible, he will decide to hire a boat iff: p < S/n (the price of fish is assumed to be unity). Since all agents have access to the fishing activity (anybody has a right to fish) and there are many agents in this economy, they will enter as long as the average product of fishing exceeds the price of entry, i.e. the rental price of a boat. The equilibrium number of fishermen, n0, is such that: n0 = S/p. In equilibrium, the rent is totally dissipated. If a second fisherman enters the fishery, the first fisherman's profit will be reduced from (S - p) to (S/2 - p) and, if there are n0 fishermen eventually active in the fishery, his profit, like that of all other operators, will become nil (S/n0 - p = 0). It is clear that such an equilibrium will typically be inefficient (for instance, in our example, efficiency requires only one fisherman on the fishing-ground). The problem addressed here is actually analogous to that of sharing a pie where, to have access to a share of the pie, agents have to pay an entry fee at some positive cost. The source of inefficiency in open-access situations clearly lies in the fact that people must mobilize non-free production factors (factors carrying a positive opportunity cost) where they are not required for production purposes.
Let us now consider another example of a fishery where the total amount of catches depends on the number of fishing boats. Let Y stand for total catches and n for the number of boats operated. For the sake of simplicity, let us assume that, in the relevant range, Y= F(n) can be approximated by the following functional form: Y = an - bn2. In this situation, the openaccess equilibrium will be such that Y/n, the average catch per boat, is equal to p, the rental price of a boat. In other words, open access is characterized by n0 boats, with n0 such that: n0 = (a - p)/b.
n0 = (a-p)/b
On the other hand, if the whole fishery was efficiently managed so as to maximize profits, the marginal catch per boat would be equal to the boat's rental price. Standard calculus yields: n* = (a - p)/2b with n* standing for the profit-maximizing number of boats. It is immediately evident that n* < n0: as a matter of fact, n0 = 2n*. In other words, in this example, the impact of open access is to double the number of boats operating in the fishery, thereby leading to the complete dissipation of the total rent. The mechanics of determination of both the open access and the efficient equilibria is portrayed graphically in Figure 2.1 (where it happens that n0 is much more than twice as high as n*).
That the magnitude of inefficiencies thus generated in fisheries can be considerable is attested by the following recent estimates: Iceland and the EEC could cut their fleets by 40 per cent, Norway by two-thirds, and all three would still catch as much fish as they do today (The Economist, 19 March 1994: 23).
Fig. 2.1. Impact of open access on the equilibrium number of boats
We have so far disregarded the dynamic considerations which were at the centre of the model of fishery presented in the preceding chapter. In the simple case where there are no costs of production and where the price of fish is constant, the optimal catch policy obtains when the stock of fish grows at the same rate as the rate of interest. This rule is widely known in the literature as the Hotelling rule. When there are costs of production, one can show that, in most cases, intertemporal profit maximization entails a long-run stationary catch programme as well as a long-run equilibrium level in the stock of fish. To make matters more complicated, there is in fact a multiplicity of such optimal programmes (for more details, see Spence and Starett, 1977). What bears emphasis is that this equilibrium level does not in general correspond to the maximum biological yield.
Under open access, however, the dynamic consequences of the current catch policy are completely disregarded. What matters for potential entrants is the average catch per boat and its rental price: there is no way in which, by refraining from fishing today, the agents can be assured that, in the next period, they will receive the amount of fish they have left untouched, augmented by its natural growth. As a result, in an open-access situation, every fisherman is forced to follow the myopic rule (by comparing average instantaneous returns to the rental price of a boat) even though he may well be aware that he thereby contributes to reduce the future stock. The problem is simply that there is no way in which he can reap the future benefits of restraint in the present: the total amount of fish currently caught is independent of the individual decisions taken by the participants. As should be clear from Chapter 1, a sequence of openaccess equilibria may lead to a stationary long-run stock, depending on the particular form of the cost curve and of the natural growth law.
It is in the light of the perverse logic of open access (since fisheries' management has miserably failed in Europe and America, and threatens to fail in developing countries, too) that the following distressing facts must be understood. Almost all the world's 200 fisheries monitored by the FAO are today fully exploited. One in three is depleted or heavily overexploited, almost all in the developed countries. On the other hand, American fishery managers estimate that the US catch is only half as valuable as it could be if fish stocks in federal waters were allowed to recover. The FAO has estimated the annual loss world-wide at $15-30 billion worth of fish (The Economist, 19 March 1994:23 4; FAO, 1993: 31-2).
It is also a well-documented fact that the open-access status of many tropical forests in Latin America has contributed significantly to deforestation under demographic pressure. Since no one has a clear right to the income associated with forest conservation (land-clearing is actually a prerequisite for formal tenure), everyone has a strong incentive to respond immediately to any opportunity to capture the rents generated by land-use conversion (Southgate, 1990; Southgate, Sierra, and Brown, 1991). Binswanger has noted that, in the case of Brazil, incentives to degrade forests are even more perverse since 'a claimant is allocated two to three times the amount of land cleared of forests'. Many people therefore clear land 'simply for purposes of solidifying land claims and increasing the size of allocations' (Binswanger, 1991: 827-8).
2.3 The problem of common property
Common property and open access have often been confused in the literature, as is attested by the following quotation from an article by Comes and Sandler: 'Traditionally common property analyses demonstrate the overexploitation of the scarce fixed resource; the average product of the variable input, not its marginal product, is equated to the input's rental rate when access is free and the number of exploiters is large' (Comes and Sandler, 1983: 787). These two situations are however essentially different in so far as, in a common property, the community has the right to exclude nonmembers from the use of the resource. Under common property, the right of exclusion is assigned to a well-defined group. Under open access, a right of inclusion is granted to anyone who wants to use the resource. As a consequence, common property and open access are also analytically distinct: because the very concept of common property supposes the existence of a well-defined group, the agents are now allowed to interact strategically with each other. In other words, the agents do not any more think that the final outcome is independent of their own individual decisions, as was the case under open access. They actually expect that their action will induce a particular reaction from the other agents and, thereby, affect the collective result.
The problems raised by common property are usually represented by the formal framework of the popular 'prisoner's dilemma'. Let us consider two herdsmen who must decide on the number of animals to let pasture on a 'common' land (belonging to both). To further simplify the presentation, let us assume that the choice facing each herdsman is between letting one or two animals on the common land. If each herdsman chooses to have one animal each, each of them gain $5. If, however, they both choose to have two animals each on the common land, these animals will be underfed and will lose much of their economic value. As a result, the total gain each herdsman may expect for having two animals pasturing is $4. Finally, if one herdsman has only one animal on the common field, and the other has two, their total gains are $3 and $6 respectively. This situation can be summarized by entering the different gains, also called payoffs, in a double-entry matrix, called the payoff matrix, as in Figure 2.2. Note that the first number in each cell is the payoff accruing to the row player, while the second number refers to the column player.
FIG. 2.2. The herdsmen game as a prisoner's dilemma game
A player's (pure) strategy is 'a statement that specifies an action at each of the decision nodes at which it would be player i's duty to make a decision if that node were actually reached' (Binmore, 1992: 30). In the herdsmen game presented above, there is only one decision node for each player, and two possible strategies. Comparing the payoffs associated with each strategy, it is easy to see that each herdsman will choose the strategy 'put two animals'. Such a strategy is called a dominant strategy, since the optimal action for one player does not depend on the strategy followed by the other player: whatever the other does, the action chosen is the one which maximizes his payoff.
A pair of strategies is a Nash equilibrium if each player's strategy is optimal, i.e. maximizes his payoffs, given the strategy of the other player. In the present context, each player has a dominant strategy so that the Nash equilibrium of the game comes out naturally as the one where each player chooses to put two animals on the common field. Here lies the tragedy of the commons: even though it would be better for both herdsmen to put only one animal on the commons, it is individually rational for each of them to put two animals, and the Paretodominated outcome obtains. In the words of Roemer: 'Everyone's welfare can be improved by exercising a restraint that no one has any interest to exercise in the state of nature' (Roemer, 1988: 2).
The representation of the commons problem as a prisoner's dilemma (PD) game (See, 1967; Dasgupta and Heal, 1979) is not without problem, however. Indeed, if the prisoner's dilemma game may be useful as a first approximation of the problem at work in the 'tragedy of the commons', it is not strictly speaking an appropriate representation of the commons problem. Indeed, such a problem is in general not characterized by the existence of dominant strategies for each agent. To show this, let us return to the example of the fishery presented above, where Y stands for total catches, ni for the number of boats operated by player i, and p, for the initial price of a boat. Assume furthermore that, within the relevant range:
Let us consider the case where access to the lake is restricted to two fishermen. Their individual profit function can be written as:
Given ni, the profit-maximizing number of boats for fisherman i is:
His optimal strategy, njN, depends on the number of boats the other fisherman operates. He has therefore no dominant strategy. The Nash equilibrium of this game is characterized by each fisherman operating (a - p)/(3b) boats. The total number of boats operated in this case is equal to two-thirds of the number of boats operated under open access (i.e. (a - p)/b) and to four-thirds the socially optimal number (i.e. (a-p)/2b). It is interesting to note that, if more fishermen are admitted on the common fishery, the total number of boats as well as the total catches will increase and gradually approach the open-access situation. As a matter of fact, if we let N stand for the number of fishermen allowed into the fishery, it is easy to show that the ratio of the total number of boats operated to the social optimum, n*, increases with N as follows:
FIG. 2.3. Common property, open access and social optimum in a fishery
The open-access situation obtains
when 
and the
value of the above ratio then equals 2. This is portrayed in
Figure 2.3. As is illustrated, the degree of inefficiency
resulting from common ownership depends crucially on the number
of agents operating on the commons: the lower the number of
agents, the greater the extent to which they can take into
account the negative consequences of their actions on the
productivity of the common property.
2.4 Co-ordination failure under common property
The Pareto-inefficieney of the Nash equilibrium under common property also holds when the production function exhibits increasing returns to scale (and not only under decreasing returns to scale). In that case indeed, the agents will typically underexploit the common property, leaving productive opportunities unused in the absence of any commitment or contractual arrangement. One may even find situations where two Nash equilibria coexist, one supporting the collectively rational outcome and the other yielding the increasing-returns-to scale equivalent of the tragedy of the commons.
Fig. 2.4. Co-ordination failure under common property
When such a situation obtains, the problem is not so much that the collectively rational outcome cannot be supported by utility-maximizing agents without a change in their economic environment but that it will not materialize because the agents fail to communicate and coordinate their actions.
Let us at this stage present a very simple example derived from that model with a view to highlighting what is meant by 'co-ordination failure'. Suppose a CPR exploited by two agents, agent A and agent B. They have to choose between two different techniques, X and Y. where X is inefficient compared to Y but where, for technique Y to be advantageous, it must be used by both agents simultaneously. In other words, the two techniques are interdependent in such a way that Y appears as a superior technique to an agent only if the other already uses it. Otherwise, X is chosen. By co-ordination failure, we mean the case where both agents do not choose the superior technique Y because they fail to communicate and co-ordinate. They are both stuck into an inferior Nash equilibrium, while another Pareto-dominating Nash equilibrium exists.
For the sake of illustration, we may have the payoff matrix (note the difference with a PD game structure) depicted in Figure 2.4.
Both agents may choose the inferior equilibrium (X,X), with a payoff (2,2) and have no interest in a unilateral change while they could be much better off if they both chose (Y,Y) with a payoff (3,3). In the game structure given above, we cannot tell a priori whether an equilibrium will be chosen, nor which one. What is interesting, however, is to consider that X is the old technique used since ages by both agents, and Y is a new technique which has just been made available. What the example above then shows is that it is likely that, by 'failing to co ordinate', both agents will not switch to this new opportunity but will choose to keep the old technique at work and thus maintain the status quo.
Such a problem would of course not appear were the resource privately owned, since, in that case, the private owner would immediately choose the more efficient technique. In other words, a private owner, by being alone to exploit a given share of the resource, would internalize the externalities involved in his own choice of techniques.
2.5 The conjectural variations approach
We have so far identified two sources of inefficiency for the unregulated common: first, in a common property game, where technology is such that the production function exhibits decreasing or increasing returns to the variable factor, the Nash equilibrium of the game is Pareto-inefficient, and second, people may fail to coordinate their action. In recent years, the very idea of applying the concept of Nash equilibrium—which implies that, when deciding on their actions, the agents assume that the other players don't change their decisions—to analyse the former type of issues has been questioned by R. Comes and T. Sandler in a paper entitled 'Commons and Tragedies'. They propose that, instead of applying the concept of Nash equilibrium, we should allow players of a 'common ownership game' to hold 'non-zero conjectural variations regarding what one exploiter thinks will be the effect of his exploitation on the exploitation efforts of the others' (Comes and Sandler, 1983: 787). In other words, when maximizing his payoffs, the agent should anticipate that the others will 'react' to his own action, and take into account these reactions in his private calculus. It is clear that, when the conjectural variations are nil, that is when the player does not expect the other players to react, the Nash equilibrium obtains. Similarly, the Pareto-optimal outcome results when each player expects the others to behave exactly in the same way he does (in the case of identical players).
Since there is virtually an infinity of conjectural variations which may be assigned to the players, it is desirable to restrict the set of possible conjectures by requiring that they should have a particular property, namely that they are consistent: 'a conjectural variation is consistent if it is identical to the optimal (actual) response of the other agent(s) in a neighborhood of the equilibrium based upon this conjecture . . . at the equilibrium, the slope of each firm's reaction path equals the corresponding conjectural variation held by the other firms.' In other words, 'conjectures are consistent when they conform to reality' (Comes and Sandler, 1983: 791).
In the context of a simple model of a fishery (similar to that presented above), the authors show that, in general, there is no consistent conjectural equilibrium. It is only in the particular case where the industry's profits are equal to zero that a consistent conjectural equilibrium exists. The required conjecture is that 'an individual firm expects the rest of the industry to reduce its input by one unit in response to a one unit increase in the firm's input, so that industry output will remain uncharged' (Comes and Sandler, 1983: 792).
Our problem with conjectural variations is that, in general, they correspond to ad hoc theorizing. Furthermore, economic theory provides a much more rigorous alternative to that approach under the form of repeated games: indeed, in the latter framework, the analyst is forced to clearly specify the structure of the game, and by doing so and making use of the appropriate equilibrium concept, he (she) is led to investigate how players interact in a dynamic setting, why players can reasonably expect a particular behaviour from the other players, and how the system evolves from its initial to its final position . . . Irrationality is not called for in the latter setting. Let us see more precisely what kind of irrationality underlies the conjectural variation approach.
First, in the case where conjectures are non-consistent, players do not anticipate correctly the optimal response of the other players. Irrationality prevails because, knowing that the other players expect you to behave in a certain sub-optimal way, you may, and in general you will, take advantage of this situation by undertaking actions which do not correspond to what the other players expect, nor to the 'optimal' behaviour under conjectural variations. Therefore, adherence by the players to non-consistent conjectural variations is simply irrational. Second, one does not know where the 'conjectures' come from: do they originate in past experience (in which case the learning process and the original conjecture ought to be defined and specified), do they come from the institutional setting, from pre-play discussions between players? All this should be made explicit. Nor can one see why they should be continuous. Third, the use of the consistent conjectural variations hypothesis is of no use in understanding how the consistent equilibrium emerges: what are the forces driving the system towards this particular equilibrium? The problem is particularly serious since, in the example chosen by Comes and Sandler, the consistent conjectural equilibrium corresponds to the zero-profit situation. One cannot find any reason why (nor how) this particular equilibrium will be chosen or attained by the players.
To conclude, we do not consider the conjectural variation approach as a fruitful method to analyse the commons problem: it is not an alternative to the Nash equilibrium concept. As a matter of fact, a well-defined game can be developed to analyse any question raised by the partisans of conjectural variations.
By definition, the concept of externality, of which the tragedy of the commons provides a particular illustration, refers to a missing market. In the words of S. Cheung, 'externalities . . . are thus attributable to the absence of the right to contract' (Cheung, 1970: 50). In the commons problem, if a competitive market could be established for the externality which i imposes on j by putting one more animal on the common field or by putting one more boat at sea, the externality and the resulting inefficiency would simply disappear. This is the basic idea underlying the Lindahl equilibrium.
It can be shown that, by creating competitive markets for 'named goods', the 'good' exchanged being the externality agent j imposes on agent i, the competitive path of exploitation of the resource is optimal. In this respect, a Lindahl equilibrium is a simple generalization of the competitive equilibrium, and the traditional welfare theorems apply. The generalization consists in regarding the externality i imposes on j as a private good which can be exchanged on a competitive market.
Does this efficiency result imply that it is desirable to create and develop markets on which agents sell and buy rights to generate externalities? In the case of the commons, the general answer is 'no' (the discussion provided below is directly based on Dasgupta and Heal, 1976: ch. 3). Indeed, the problem with the Lindahl equilibrium is that one cannot see clearly how to implement it. First, private property rights in externalities may be impossible to define and enforce since, in principle, one must be able to exclude non-buyers from consuming the commodities purchased by someone else. Second, the markets supporting the Lindahl equilibrium typically involve one buyer and one seller: the market structure is that of bilateral monopoly. One cannot see clearly the driving forces leading to a competitive equilibrium in this framework. The problem is made even more complicated if one admits that many externality problems are characterized by private information (this issue will be analysed in more detail in Chapter 8).
Finally, in many commons problems, the Lindahl equilibrium fails to exist. To illustrate this, let us consider, in the fishery problem, the market where agent i buys from agent j the right to generate externalities on j by putting fishing boats to sea. Let us call this commodity xij, and its price, pij. If pij is positive, then j can close down and sell an unlimited quantity of these rights xij, while i asks only for a limited amount of them: demand and supply will not match and pij > 0 is not an equilibrium. But, if pij is negative (or nil), j does not supply any positive amount of xij, while i asks for an infinite (or positive) amount of it: again, supply and demand for the named commodity do not match. As a result, a competitive market for the externality does not exist (the technical reason thereof is that, as long as the resource is common property, the production possibility set is not a convex set). The source of the problem, however, is that property rights so defined are not appropriate: if the fishing ground could be divided into parcels privately appropriated by the fishermen, the inefficiency problem simply would not arise. In other words, the non-convexity of the production possibilities set, in this particular case, originates in the absence of well-defined property rights (this, however, is not always the case: a good illustration of this point, for the case of pollution, is given in Dasgupta and Heal, 1979: 78-92). It is to the definition of such rights that we shall now turn our attention.
