1. Renewable Resources
Chapter 17

2. Renewable flow resources
Such as solar, wave, wind and geothermal energy.
These energy flow resources are non-depletable.

3. Renewable stock resources
living organisms: fish, cattle and forests, with a natural capacity for growth
inanimate systems (such as water and atmospheric systems): reproduced through time by physical or chemical processes
arable and grazing lands as renewable resources: reproduction by biological processes (such as the recycling of organic nutrients) and physical processes (irrigation, exposure to wind etc.).
Are capable of being fully exhausted.

4. Renewable Resources Key points of interest
private property rights do not exist for many forms of renewable resource
the resource stocks are often subject to open access; tend to be overexploited
policy instruments: relative merits

5. Renewable Resources Biological growth processes Gt = St+1 - St
where S = stock size and G is amount of growth
This is called density dependent growth.
In continuous time notation:
G = G(S)

6. Renewable Resources Biological growth processes G = G(S)
An example: (simple) logistic growth –
Where g is the intrinsic growth rate (birth rate minus mortality rate) of the population.
Now turn to the Excel file logistic.xls
Biological growth processes
In order to investigate the economics of a renewable resource, it is first necessary to describe the pattern of biological (or other) growth of the resource. To fix ideas, we consider the growth function for a population of some species of fish. This is conventionally called a fishery.
We suppose that this fishery has an intrinsic (or potential) growth rate denoted by g. This is the proportional rate at which the fish stock would grow when its size is small relative to the carrying capacity of the fishery, and so the fish face no significant environmental constraints on their reproduction and survival. The intrinsic growth rate g may be thought of as the difference between the population’s birth and natural mortality rate (again, where the population size is small relative to carrying capacity). Suppose that the population stock is S and it grows at a fixed growth rate g. In the absence of human predation the population grows exponentially over time at the rate g and without bounds. This is only plausible over a short span of time.
Any population exists in a particular environmental milieu, with a finite carrying capacity, which sets bounds on the population’s growth possibilities.
A simple way of representing this effect is by making the actual (as opposed to the potential) growth rate depend on the stock size. Then we have what is called density-dependent growth.
Now let us suppose that under a given set of environmental conditions there is a finite upper bound on the size to which the population can grow (its carrying capacity). We will denote this as SMAX.
A commonly used functional form which has the properties of “compensation” and a maximum stock size is the simple logistic function, in which the constant parameter g > 0 is what we have called the intrinsic or potential growth rate of the population.

7. MSY
SMAX
SMSY

9. Figure 17.2 Steady-state harvests
G, H
Much of our attention will be devoted to steady-state harvests.
Consider a period of time in which the amount of the stock being harvested (H) is equal to the amount of net natural growth of the resource (G).
Suppose also that these magnitudes remain constant over a sequence of consecutive periods.
We call this steady-state harvesting, and refer to the (constant) amount being harvested each period as a sustainable yield.
With steady-state harvesting the resource stock remains constant over time.
What kinds of steady states are feasible? To answer this, look at Figure There is one particular stock size (SMSY) at which the quantity of net natural growth is at a maximum (GMSY). If at a stock of SMSY harvest is set at the constant rate HMSY, we obtain a maximum sustainable yield (MSY) steady state.
A resource management programme could be devised which takes this MSY in perpetuity.
It is sometimes thought to be self-evident that a fishery, forest or other renewable resource should be managed so as to produce its maximum sustainable yield.
Economic theory does not, in general, support this proposition.
HMSY is not the only possible steady-state harvest. Indeed, Figure 17.2 shows that any harvest level represented by a point on the growth curve is a feasible steady-state harvest, and that any stock between zero and SMAX can support steady-state harvesting. For example, H1 is a feasible steady-state harvest if the stock size is maintained at either S1L or S1U. Which of these two stock sizes would be more appropriate for attaining a harvest level of H1 is also a matter we shall investigate later.

10. Steady-state harvests
Let H = harvest and G = net natural growth (G)
In steady-state harvesting G = H and so = dS/dt = 0 (the resource stock size remains constant over time).
There are many possible steady states or sustainable yields.
There is one particular stock size at which the quantity of net natural growth is at a maximum. This is a maximum sustainable yield (MSY) steady state.
Many people believe that a renewable resource should be managed to produce its maximum sustainable yield.
Economists do not agree that this is always sensible (but it may be in some circumstances).

11. A model of a commercial fishery
See Excel file logistic.xls
and also
Word file exploit6.xls

12. Economic models of a fishery
Open access fishery
Closed access (private property) fishery

13. Economic models of a fishery
Open access fishery
enter if profit per boat is positive
leave if profit per boat is negative
in steady-state, R = C (revenue = costs)
Closed access (private property) fishery
Static (single period) model: select E to maximise Profit = R – C
(maths derivation of solution given in Box 17.3, page 579)
Dynamic (intertemporal) model
Commercial fisheries
Open access vs Restricted access fisheries
What is an open access fishery?
Consequences of open access: entry continues until all rents are dissipated (profit per boat = zero).
Stock sizes will tend to be lower, and harvest rates will tend to be higher (but may not always be) compared with a restricted access fishery.
Extinction is more likely, but will not necessarily happen.

14. Closed access (private property) fishery
Static (single period) model
Choose effort to maximise profits in any ‘representative’ single period
Dynamic (intertemporal) present-value maximising model
Choose a path of effort over time to maximise the wealth of the firm (maximise the present value of the fishery)

15. An open-access fishery
Our first model of renewable resource exploitation is an open-access fishery model.
It is important to be clear about what an open-access fishery is taken to mean in the environmental economics literature.
The open-access fishery model shares two of the characteristics of the standard perfect competition model.
First, if the fishery is commercially exploited, it is assumed that this is done by a large number of independent fishing ‘firms’. Therefore, each firm takes the market price of landed fish as given.
Second, there are no impediments to entry into and exit from the fishery.
But the free entry assumption has an additional implication in the open-access fishery, one which is not present in the standard perfect competition model.
In a conventional perfect competition model, each firm has enforceable property rights to its resources and to the fruits of its production and investment choices. However, in an open-access fishery, while owners have individual property rights to their fishing capital and to any fish that they have actually caught, they have no enforceable property rights to the in situ fishery resources, including the fish in the water. On the contrary, any vessel is entitled or is able (or both) to fish wherever and whenever its owner likes. Moreover, if any boat operator chooses to leave some fish in the water in order that future stocks will grow, that owner has no enforceable rights to the fruits of that investment. It is as if a generalised ‘what one finds one can keep’ rule applies to fishery resources. We shall see in a moment what this state of affairs leads to. First, though, we need to set up the open-access fishery model algebraically.
[This lack of de facto enforceability may derive from the property that the fish are spatially mobile. Or it may derive from the property that where fish stocks are spatially immobile no state or any other entity, has jurisdiction, or does not or can not exercise its jurisdiction. ]

16. Open access equilibrium
The open-access model has two components:
a biological sub-model, describing the natural growth process of the fishery;
an economic sub-model, describing the economic behaviour of the fishing boat owners.
The model is laid out in full in Table 17.1.
We shall be looking for two kinds of ‘solutions’ to the open-access model.
The first is its equilibrium (or steady-state) solution. This consists of a set of circumstances in which the resource stock size is unchanging over time (a biological equilibrium) and the fishing fleet is constant with no net inflow or outflow of vessels (an economic equilibrium). Because the steady-state equilibrium is a joint biological–economic equilibrium, it is often referred to as bioeconomic equilibrium.
The second kind of solution we shall be looking for is the adjustment path towards the equilibrium, or from one equilibrium to another as conditions change. In other words, our interest also lies in the dynamics of renewable resource harvesting. This has important implications for whether a fish population may be driven to exhaustion, and indeed whether the resource itself could become extinct.

17. Price determination in our simple fishery model
Exogenous price: Market price = P
This is not the only possible way of proceeding. Could have market price determined via a demand curve, and so varying with the harvest rate.

18. The harvest function (or fishery production function)
Many factors determine the size of the harvest, H, in any given period. Our model considers two of these. First, the harvest will depend on the amount of resources devoted to fishing. In the case of marine fishing, these include the number of boats deployed and their efficiency, the number of days when fishing is undertaken and so on. For simplicity, assume that all the different dimensions of harvesting activity can be aggregated into one magnitude called effort, E.
Second, except for schooling fisheries, it is probable that the harvest will depend on the size of the resource stock. Other things being equal, the larger the stock the greater the harvest for any given level of effort. Hence, abstracting from other determinants of harvest size, including random influences, we may take harvest to depend upon the effort applied and the stock size. That is
H = H(E, S) (17.6)
This relationship can take a variety of particular forms. One very simple form appears to be a good approximation to actual relationships, and is given by
H = eES (17.7)
where e is a constant number, often called the catch coefficient. [The use of a constant catch coefficient parameter is a simplification that may be unreasonable, and is often dropped in more richly specified models.]
This implies that the quantity harvested per unit effort is equal to some multiple (e) of the stock size. We have already defined the fish-stock growth function with human predation as the biological growth function less the quantity harvested. That is,
= G(S) – H (17.8)
Note also that equation 17.7 can be regarded as a special case of the more general form H = eES in which the exponents need not be equal to unity. In empirical modelling exercises, this more general form may be more appropriate. Another form of the harvest equation sometimes used is the exponential model H = S(1 – exp(–eE)).
The costs of fishing
The total cost of harvesting, C, depends on the amount of effort being expended
C = C(E) (17.9)
For simplicity, harvesting costs are taken to be a linear function of effort,
C = wE (17.10)
where w is the cost per unit of harvesting effort, taken to be a constant. The equation C = wE imposes the assumption that harvesting costs are linearly related to fishing effort. However, Clark et al. (1979) explain that this assumption will be incorrect if capital costs are sunk (unrecoverable); moreover, they show that even in a private-property fishery (to be discussed later in the chapter), it can then be privately optimal to have severely depleted fish stocks as the fishery approaches its steady-state equilibrium (although the steady-state equilibrium itself is not affected by whether or not costs are sunk).

19. Entry into and exit from the fishery
To complete our description of the economic sub-model, it is necessary to describe how fishing effort is determined under conditions of open access.
A crucial role is played here by the level of economic profit prevailing in the fishery.
Economic profit is the difference between the total revenue from the sale of harvested resources and the total cost incurred in resource harvesting.
Given that there is no method of excluding incomers into the industry, nor is there any way in which existing firms can be prevented from changing their level of harvesting effort, effort applied will continue to increase as long as it is possible to earn positive economic profit.
Conversely, individuals or firms will leave the fishery if revenues are insufficient to cover the costs of fishing.
A simple way of representing this algebraically is by means of the equation dE/dt = ·NB where  is a positive parameter indicating the responsiveness of industry size to industry profitability. When economic profit (NB) is positive, firms will enter the industry; and when it is negative they will leave. The magnitude of that response, for any given level of profit or loss, will be determined by .

20. Bioeconomic equilibrium
We close our model with two equilibrium conditions that must be satisfied jointly.
Biological equilibrium occurs where the resource stock is constant through time (that is, it is in a steady state). This requires that the amount being harvested equals the amount of net natural growth: G = H
Economic equilibrium is only possible in open-access fisheries when rents have been driven to zero, so that there is no longer an incentive for entry into or exit from the industry, nor for the fishing effort on the part of existing fishermen to change. We express this by the equation NB = B – C = which implies (under our assumptions) that PH = wE.
Notice that when this condition is satisfied, dE/dt = 0 and so effort is constant at its equilibrium (or steady-state) level E = E*.

23. Figure 17.3 Steady-state equilibrium fish harvests and stocks at various effort levels
H2 = eE2S
HMSY = eEMSYS
H1 = eE1S H1
Stock, S
SMSY
=SMAX/2 S1 S2 H2
G(S)
HMSY=(gSMAX)/4
This solution can also be represented graphically, as shown in Figures 17.3 and 17.4.
Figure 17.3 shows equilibrium relationships in stock–harvest space.
The inverted U-shape curve is the logistic growth function for the resource. Three rays emanating from the origin portray the harvest–stock relationships (from the function H = eES) for three different levels of effort.
If effort were at the constant level E1, then the unique intersection of the harvest–stock relationship and biological growth function determines a steady-state harvest level H1 at stock S1.
The lower effort level E2 determines a second steady-state equilibrium (the pair {H2, S2}). We leave the reader to deduce why the label EMSY has been attached to the third harvest–stock relationship. The various points of intersection satisfy equation 17.17, being equilibrium values of S for particular levels of E. Clearly there is an infinite quantity of possible equilibria, depending on what constant level of fishing effort is being applied.

24. Figure 17.4 Steady-state equilibrium yield-effort relationship
H=(w/P)E
HPP
HOA
Effort, E
EPP
EOA
g/e
The points of intersection in Figure 17.3 not only satisfy equation but they also satisfy equation Put another way, the equilibrium {E, S} combinations also map into equilibrium {E, H} combinations. The result of this mapping from {E, S} space into {E, H} space is shown in Figure The inverted U-shape curve here portrays the steady-state harvests that correspond to each possible effort level. It describes what is often called the fishery’s yield–effort relationship. Mathematically, it is a plot of equation
Figure 17.4 Steady-state equilibrium yield-effort relationship
The particular point on this yield–effort curve that corresponds to an open-access equilibrium will be the one that generates zero economic profit. How do we find this? The zero economic profit equilibrium condition PH = wE can be written as H = (w/P)E. For given values of P and w, this plots as a ray from the origin with slope w/P in Figure The intersection of this ray with the yield–effort curve locates the unique open-access equilibrium outcome.
Alternatively, multiplying both functions in Figure 17.4 by the market price of fish, P, we find that the intersection point corresponds to PH = wE. This is, of course, the zero profit condition, and confirms that {EOA, HOA} is the open-access effort–yield equilibrium.

25. Comparative statics for the static open access fishery model
It is instructive to inspect how the steady state equilibrium magnitudes of effort, stock and harvest in an open access fishery regime change in response to changes in the parameters or exogenous variables of that model.
Table 17.2 showed that for a particular illustrative (or baseline) set of parameter values, the steady state equilibrium magnitudes of stock, effort and harvest were S* = 0.2, E* = 8.0 and H* = respectively. But we would also like to know how the steady state equilibrium values would change in response to changes in the model parameters. This can be established using comparative static analysis.
Comparative static analysis is undertaken by obtaining the first-order partial derivatives of each of these three equations with respect to a parameter or exogenous variable of interest and inspecting the resulting partial derivative.
The full set of partial derivatives is listed and derived in Appendix What is of particular interest in comparative static analysis is to ascertain whether an unambiguous sign can be attached to a partial derivative, and if so to establish whether that sign is positive or negative. The magnitude of the derivative will also be of interest.
For example, how will S* change as P or w changes? Inspection of the parametric expression for S* shows that open-access S* will increase if w increases and will decrease if P increases. More formally, the two partial derivatives are given by and . As w, P and e are necessarily positive numbers, it follows that S* will fall as P rises (or will rise as P falls) and that S* will rise as w rises (or will fall as P falls). In these two cases, the sign of the effect is unambiguous. Sometimes the direction of an effect cannot be signed unambiguously; this should usually be evident by inspection of the partial derivative. Changes in the steady state value of H*, for example, cannot be unambiguously signed with respect to changes in P, w or e, as can be seen from Table 17.3.

26. One should be aware, though, of the limitations of comparative static analysis. Take for example the parameter g, the intrinsic growth rate of the population of interest. Table 17.3 shows that the magnitude of the growth rate has no impact, other things remaining equal, on the steady state equilibrium level of the resource stock. One might be tempted to infer from this that the likelihood of species extinction (or population collapse) is unaffected by how fast or slow the population grows. But that would be an inappropriate conclusion. The steady-state equilibrium value is unaffected; but the adjustment dynamics may be affected by the rate of g, and there is good reason to suppose that extinction is more likely for slow growing species because of circumstances that arise along the dynamic adjustment paths that arise from parameter changes, or as a result of shocks or disturbances to the population of interest. It is to this matter that we now turn.

27. The dynamics of renewable resource harvesting
Discussion so far has been exclusively about steady states: equilibrium outcomes which, if achieved, would remain unchanged provided that relevant economic or biological conditions remain constant.
However, we may also be interested in the dynamics of resource harvesting.
This would consider questions such as how a system would get to a steady state if it were not already in one, or whether getting to a steady state is even possible.
In other words, dynamics is about transition or adjustment paths.
Dynamic analysis might also give us information about how a fishery would respond, through time, to various kinds of shocks and disturbances.
The dynamics of the open access fishery model are driven by two differential equations, determining the instantaneous rates of change of S and E:
When the fishery is in a steady-state equilibrium, by definition both dS/dt and dE/dt are zero.
When the fishery is not in equilibrium the adjustment of S and E over time is governed by these two differential equations.
The properties of these dynamic adjustment processes can be explored by mathematical analysis.
As H = eES, it follows that the two differential equations also determine the dynamic behaviour of the harvest rate, H.
In discrete time modelling the dynamics of the open access fishery model are driven by two counterpart difference equations, determining the period to period rates of change of S and E.

28. Figure 17.5 Stock and effort dynamic paths for the open access model
To help fix ideas, suppose that a stock of fish exists which has not previously been commercially exploited. The stock size is, therefore, at its carrying capacity. This stock now becomes available for unregulated, open-access commercial exploitation. If the market price of fish, P, is reasonably high and fishing cost (per unit of effort), w, is reasonably low, the fact that stocks are high (and so easy to catch) implies that the fishery will be, at least initially, profitable for those who enter it.
If a typical fishing boat can make positive economic profit then further entry will take place. How quickly new capacity is built up depends on the magnitude of the parameter  in equation In this early phase, effort is rising over time as new boats are attracted in, and stocks are falling. Stocks fall because harvesting is taking place while new recruitment to stocks is low: recall that the logistic growth function has the property that biological growth is near zero when the stock is near its maximum carrying capacity. This process of increasing E and decreasing S will persist for some time, but cannot last indefinitely. As stocks become lower, fish become harder to catch and so the cost per fish caught rises. Profits are squeezed from two directions: harvesting cost per fish rises, and fewer fish are caught.
Eventually, this profit squeeze will mean that a typical boat makes a loss rather than a profit, and so the process we have just described goes into reverse, with stocks rising and effort falling. In fact, for the model we are examining, the changes do not occur as discrete switches but instead are continuous and gradual. We find that stock and effort (and also harvest) levels have oscillatory cycles with the stock cycles slightly leading the effort cycles. These oscillations dampen down as time passes, and the system eventually settles to its steady-state equilibrium.
We illustrate such a transition process in Figures 17.5 and 17.6.
These were generated using the Maple mathematical package, with parameter values as given in Table 17.2, and with initial values E0 =0.1 and S0 = 1. Figure 17.5 shows the convergent oscillatory behaviour of S and E as they move from their initial values to their steady-state equilibrium levels of S* = 0.2 and E* = 8.0. It is important to understand that the frequency and amplitude of the oscillations in Figure 17.5 are obtained for one particular set of parameter values. For other combinations of parameter values, the cyclical behaviour of stock and effort will be different, and in some cases variability will be much less pronounced. Problem 7 at the end of this chapter invites you to use Maple or Excel (or both) to explore the properties of the open access fishery model as values of parameters are varied.
The stable but oscillatory behaviour exhibited by the dynamic adjustment paths shown in Figure 17.5 – with E and S repeatedly over- and under-shooting equilibrium outcomes – is a characteristic feature of open-access fisheries. For the ‘continuous time’ specification of the open access model, the steady-state equilibrium outcomes implied by solving equations 17.20–22 will be achieved, albeit slowly. Furthermore, the continuous time open access model exhibits oscillatory but stable and convergent behaviour for any admissible sets of parameter values.
SMAX has been normalised to unity (one) for convenience. So the fact that the initial value of S is 1 indicates that the fishery initially contains a stock at its environmental carrying capacity.
The Maple file implementing the simulations from which Figures 17.5 and 17.6 are taken (Chapter17.mw) is available from the Companion Web Site. For readers unable to read Maple files, a RTF file, readable via Word, is also available.)
This is NOT true, however, in a discrete time counterpart. If the model as specified in Table 17.1 is written in discrete time form (using difference rather than differential equations for the changes in E and S over time) the resulting model dynamics can be entirely different! In particular, in discrete time formulations, some combinations of parameters may induce oscillations so large as to drive the population down to a level from which it cannot recover, and so the fishery is driven out of existence. This can happen even though the parameters imply that a steady state exists with positive stock and effort. For example, in an Excel simulation explored in Problem 7 at the end of this chapter, a fish price of £270 rather than £200 (along with other parameter values as given in Table 17.2) suggests a steady-state solution of S* = 0.15 and E* = (This can be verified by substituting parameter values in equations and 17.20). However, successive iterations of the discrete-time equations from initial values of S = 1 and E = 1 show that the stock collapses to zero after only 7 periods! (It does so because effort explodes so rapidly and to such a huge level that the stock is completely harvested in a short finite span of time.) This matter is explored a little further in Section 17.5.

29. Figure 17.5 (Alternative) Overlaying stock and effort dynamic paths to illustrate their relative phasing.

30. Figure 17.6 Phase-plane analysis of stock and effort dynamic paths for the illustrative model.
Another way of describing the information given in Figure 17.5 is in terms of a ‘phase-plane’ diagram. An example is shown in Figure 17.6.
The axes define a space consisting of pairs of values for S and E.
The steady-state equilibrium (S* = 0.2 and E* = 8.0) lies at the intersection of two straight lines the meaning of which will be explained in a moment.
The adjustment path through time is shown by the line which converges through a series of diminishing cycles on the steady-state equilibrium.
In the story we have just told, the stock is initially at 1 and effort begins at a small value (just larger than zero). Hence we begin at the top left point of the indicated adjustment path. As time passes, stock falls and effort rises – the adjustment path heads south-eastwards. After some time, the stock continues to fall but so too does effort – the adjustment path follows a south-west direction.

31. Phase-plane diagrams
Another way of describing the information given in Figure 17.5 is in terms of a ‘phase-plane’ diagram. An example is shown in Figure 17.6.
The axes define a space consisting of pairs of values for S and E.
The steady-state equilibrium (S* = 0.2 and E* = 8.0) lies at the intersection of two straight lines the meaning of which will be explained in a moment.
The adjustment path through time is shown by the line which converges through a series of diminishing cycles on the steady-state equilibrium.
In the story we have just told, the stock is initially at 1 and effort begins at a small value (just larger than zero). Hence we begin at the top left point of the indicated adjustment path. As time passes, stock falls and effort rises – the adjustment path heads south-eastwards. After some time, the stock continues to fall but so too does effort – the adjustment path follows a south-west direction.
Comparing Figures 17.5 and 17.6, you will see that the latter presents the same information as the former but in a different way.

32. The phase-plane diagram also presents some additional information not shown in Figure 17.5
First, the arrows denote the directions of adjustment of S and E whenever the system is not in steady state, from any arbitrary starting point.
It is evident that the continuous-time version of the open-access model we are examining here in conjunction with the particular baselines parameter values being assumed has strong stability properties: irrespective of where stock and effort happen to be the adjustment paths will lead to the unique steady- state equilibrium, albeit through a damped, oscillatory adjustment process.
Second, the phase-plane diagram explicitly shows the steady-state solution. As noted above this lies at the intersection of the two straight lines.
The horizontal line is a locus of all economic equilibrium points at which profit per boat is zero, and so effort is unchanging (dE/dt = 0).
The downward sloping straight line is a locus of all biological equilibrium points at which G = H, and so stock is unchanging (dS/dt = 0). Clearly, a bioeconomic equilibrium occurs at their intersection.
Figures 17.5 and 17.6 were generated using the Maple package to set up the continuous time version of the open-access model. We provide, on the Companion Web Site, two files that readers may find useful. Phase.doc contains an elementary level account of phase-plane analysis, plus an explanation of how Maple – an easy-to-use and very powerful package – can generate the graphical output shown in Figure The Maple file used to generate the picture is included as Chapter 17.mw, and a pdf version of that Maple file is also available.

33. Should one use a continuous time model or a discrete time model of the open access fishery?
The open access fishery model developed in this chapter is specified in terms of ‘continuous time’.
This is evident by inspecting equations 17.8 and
These are differential equations, describing the instantaneous rate of change of the net stock of fish and of fishing effort respectively.
However, one could easily rewrite the bioeconomic model in discrete time terms, replacing equations 17.8 and with their difference equation counterparts (giving amounts of change between successive discrete time periods).

34. Which of these should one use?
Often choice is made on the basis of analytical or computational convenience.
If these two approaches yielded results that were exactly (or very closely) equivalent, then convenience would be an appropriate choice criterion.
However, they do not. In some circumstances discrete time modelling generates results very similar to those obtained from a continuous time model.
But this is not generally true. Indeed, the two approaches may differ not only quantitatively but also qualitatively. That is, discrete time and continuous time models of the same phenomenon will often yield results with very different properties.
For example, a reader familiar with spreadsheet software, such as Microsoft Excel, might find it more convenient to carry out model analysis in its discrete time version; someone with fluency in a symbolic mathematical package such as Mathematica or Maple might have a preference for exploring a model in its continuous time version.

35. Which of these should one use?
This suggests that the choice should be made in terms of which approach best represents the phenomena being modelled.
Let us first think about the biological component of a renewable resource model. Gurney and Nisbet (1998, page 61) take a rather particular view on this matter and argue that
“The starting point for selecting the appropriate formalism must be therefore the recognition that real ecological processes operate in continuous time. Discrete-time models make some approximation to the outcome of these processes over a finite time interval, and should thus be interpreted with care. This caution is particularly important as difference equations are intuitively appealing and computationally simple. However, this simplicity is an illusion … . …. incautious empirical modelling with difference equations can have surprising (adverse) consequences.”
It is undoubtedly the case that much ecological modelling is done in continuous time.
But the Gurney and Nisbet implication that all ecological processes are properly modelled in continuous time would not be accepted by most ecologists.
Flora and fauna where generations are discrete are always modelled in discrete time. Some ecologists, and others, would argue that discrete time is appropriate in other situations as well.

36. Which of these should one use?
When it comes to the economic parts of bioeconomic models of renewable resources, it is also far from clear a priori whether continuous or discrete time modelling is more appropriate.
Economists often have a predilection for continuous time modelling, but that may often have a lot to do with physics emulation and mathematical convenience.
Discrete time modelling is appropriate for quite a range of economic problems; and when we get to the data, except for financial data, it’s a discrete time world.
All that can really be said on this matter is that we should select between continuous and discrete time modelling according to the purposes of our analysis and the characteristics of the processes being modelled.
For example, where one judges that a process being modelled involves discrete, step-like changes, and that capturing those changes is of importance to our research objectives then the use of discrete-time difference equations would be appropriate.
For the reader interested in seeing how the Excel spreadsheet package can be used for dynamic simulation of the discrete time counterpart of the open access model used in this chapter, we have provided an annotated Excel workbook Fishery dynamics.xls on the Companion Web Site. This Excel file implements the calculations used to obtain the discrete-time counterpart to Figures 17.5 and 17.6, carries out some additional dynamic simulations, and also illustrates other fishery models examined in this chapter. The workbook is set up so that the reader can easily carry out other simulations, and contains instructions for using the spreadsheet and an explanation of the simulation technique. A few minutes spent doing simulations in this discrete time model counterpart will make it clear that dynamics are qualitatively very different from those shown in Figures 17.5 and 17.6.
In essence, dynamic simulations are done in Excel in the following way. First choose initial values for E and S. Then using the discrete-time versions of our model equations (listed in Appendix 17.1) calculate the values of H, E and S the next time period. Next use the Fill Down function in Excel to copy the formulae which embody these model equations down the spreadsheet. In this way, Excel will recursively calculate levels of the variables over any chosen span of successive time periods. The steady-state solution can be found either analytically by using the parametric solution equations given in Box 17.2 or numerically by plotting successive values of H, E and S and observing at which numerical values the variables eventually settle down.

37. Alternative forms of biological growth function in which there is a positive minimum viable population size
One extension to the simple open access model is to deal with biological growth processes in which there is a positive minimum viable population size.  
Box 17.1 introduced several alternative forms of logistic growth. Of particular interest are those shown in panels (b) and (d) of Figure 17.1, in which there is a threshold stock (or population) size SMIN such that if the stock were to fall below that level the stock would necessarily fall towards zero thereafter. There would be complete population collapse.
In these circumstances, steady-state equilibria become irrelevant, as the fishery collapse prevents one from being obtained. Even though parameter values may suggest the presence of a steady-state equilibrium at positive stock, harvest and effort levels, if the stock were ever to fall below SMIN during its dynamic adjustment process, the stock would irreversibly collapse to zero.
As a result, where a renewable resource has some positive threshold population size the likelihood of the stock of that resource collapsing to zero is higher than it is in the case of simple logistic growth, other things being equal.
Moreover, the likelihood of stock collapse becomes greater the larger is SMIN. These results are to a degree self-evident. A positive minimum viable population size implies that low populations that would otherwise recover through natural growth may collapse irretrievably to zero.

38. Alternative forms of biological growth function in which there is a positive minimum viable population size
But matters are actually more subtle than this.
In particular, where the renewable resource growth function exhibits critical depensation, the dynamics no longer exhibit the stable convergent properties shown in Figures 17.4 and 17.5.
Indeed, over a large domain of the possible parameter configurations, the dynamics become unstable and divergent.
In other words, the adjustment dynamics no longer take one to a steady-state equilibrium, but instead lead to population collapse. The stock can only be sustained over time by managing effort and harvest rates; but the management required for sustainability is, by definition, not possible in an open access fishery.
This illustrates two general points about renewable resources. First, under conditions of open access, the specific properties of the biological growth function for that resource have fundamental implications for its sustainability. Second, open access conditions are likely to be incompatible with resource sustainability where biological growth is characterised by critical depensation.

39. Stochastic fishery models
In practice the environment in which the resource is situated may change in an unpredictable way.
When open access to a resource is accompanied by volatile environmental conditions, and where that volatility is fast-acting, unpredictable and with large variance, outcomes in terms of stock and/or harvesting can be catastrophic.
Simple intuition suggests why this is the case.
Suppose that environmental stochasticity leads to there being a random component in the biological growth process. That is, the intrinsic growth rate contains a fixed component, g, plus a random component u, where the random component may be either positive or negative at some instant in time. Then the achieved growth in any period may be smaller or larger than its underlying mean value as determined by the growth equation When the stochastic component happens to be negative, and is of sufficient magnitude to dominate the positive fixed component, actual growth is negative. If the population were at a low level already, then a random change inducing lower than normal, or negative, net growth could push the resource stock below the point from which biological recovery is possible. This will, of course, be more likely the further above zero is SMIN.

40. The private-property fishery
In an open-access fishery, firms exploit available stocks as long as positive profit is available. While this condition persists, each fishing vessel has an incentive to maximise its catch.
But there is a dilemma here, both for the individual fishermen and for society as a whole.
From the perspective of the fishermen, the fishery is perceived as being overfished. Despite each boat owner pursuing maximum profit, the collective efforts of all drive profits down to zero.
From a social perspective, the fishery will be economically ‘overfished’ (this is a result that is established in Section 17.11). Moreover, the stock level may be driven down to levels that are considered to be dangerous on biological or sustainability grounds.
What is the underlying cause of this state of affairs? Although reducing the total catch today may be in the collective interest of all (by allowing fish stocks to recover and grow), it is not rational for any fisherman to individually restrict fishing effort.
There can be no guarantee that he or she will receive any of the rewards that this may generate in terms of higher catches later. Indeed, there may not be any stock available in the future. In such circumstances, each firm will exploit the fishery today to its maximum potential, subject only to the constraint that its revenues must at least cover its costs.

41. Private-property fishery: specification
A private-property fishery has the following three characteristics:
There is a large number of fishing firms, each behaving as a price-taker and so regarding price as being equal to marginal revenue. It is for this reason that the industry is often described as being competitive.
Each firm is wealth maximising.
There is a particular structure of well-defined and enforceable property rights to the fishery, such that owners can control access to the fishery and appropriate any rents that it is capable of delivering.
What exactly is this particular structure of private property rights? Within the literature there are several (sometimes implicit) answers to this question. We shall outline two of them.
A private property fishery refers to the harvesting of a single species from lots of different biological fisheries, fishing grounds. Each fishing ground is owned by a fishing firm. That fishing firm has private property rights to the fish which are on that fishing ground currently and at all points in time in the future.9 All harvested fish, however, sell in one aggregate market at a single market price.
The fishery as being managed by a single entity which controls access to the fishery and coordinates the activity of individual operators to maximise total fishery profits (or wealth). Nevertheless, harvesting and pricing behaviour are competitive rather than monopolistic.
9 The owners of any fishing firm may lease or sell their property rights to another set of individuals.

42. Two versions of the static profit-maximising private-property fishery model
Our analysis of the private-property fishery proceeds in two steps.
The first develops a simple static model of a private-property fishery in which the passage of time is not explicitly dealt with.
The second develops a multi-period intertemporal present-value maximising fishery model
The first case turns out to be a special case of the second in which owners use a zero discount rate.

43. The static profit-maximising private-property fishery model
In the simple static model of a private-property fishery, the passage of time (and so time-value of money) is not explicitly dealt with.
In effect, the analysis supposes that biological and economic conditions remain constant over some span of time.
It then investigates what aggregate level of effort, stock and harvest would result if each individual owner (with enforceable property rights) managed affairs so as to maximise profits over any arbitrarily chosen interval of time.
This way of dealing with time – in effect, abstracting from it, and looking at decisions in only one time period but which are replicated over successive periods – leads to its description as a static fishery model.

44. Static private property fishery: model specification
The biological and economic equations of the static private-property fishery model are identical to those of the open-access fishery in all respects but one respect.
The difference lies in that the open-access entry rule , dE/dt = ·NB, which implies a zero-profit economic equilibrium, no longer applies.
Instead, owners choose effort to maximise economic profit from the fishery. This can be visualised by looking back to Figure 17.4.
An algebraic derivation of the steady-state solution to this problem – showing stock, effort and harvest as functions of the parameters – is given in Box It is easy to verify from the solution equations given there that the steady-state values of E, S and H are given by E*pp = 4.0, S*pp = 0.60 and H*pp = To facilitate comparison, parametric solution equations for the steady-state equilibrium stock, harvest and effort, and their numerical values under baseline parameter assumptions, for both open-access and static private-property fishery, are reproduced in Table 17.4.

45. Figure 17.4 Steady-state equilibrium yield-effort relationship
H=(w/P)E
HPP
HOA
Effort, E
EPP
EOA
g/e
Instead, owners choose effort to maximise economic profit from the fishery. This can be visualised by looking back to Figure Multiply both functions by the market price of fish. The inverted U-shape yield–effort equation then becomes a revenue–effort equation. And the ray emerging from the origin now becomes PH = wE, with the right-hand side thereby denoting fishing costs. Profit is maximised at the effort level which maximises the surplus of revenue over costs. Diagrammatically, this occurs where the slopes of the total cost and total revenue curves are equal. This in indicated in Figure 17.4 by the tangent to the yield–effort function at EPP being parallel to the slope of the H = (w/P)E line.
Figure 17.4 Steady-state equilibrium yield-effort relationship
The particular point on this yield–effort curve that corresponds to an open-access equilibrium will be the one that generates zero economic profit. How do we find this? The zero economic profit equilibrium condition PH = wE can be written as H = (w/P)E. For given values of P and w, this plots as a ray from the origin with slope w/P in Figure The intersection of this ray with the yield–effort curve locates the unique open-access equilibrium outcome.
Alternatively, multiplying both functions in Figure 17.4 by the market price of fish, P, we find that the intersection point corresponds to PH = wE. This is, of course, the zero profit condition, and confirms that {EOA, HOA} is the open-access effort–yield equilibrium.
Static Private Property: owners choose effort to maximise economic profit from the fishery. Multiply both functions by the market price of fish. The inverted U-shape yield–effort equation then becomes a revenue–effort equation. And the ray emerging from the origin now becomes PH = wE, with the right-hand side thereby denoting fishing costs. Profit is maximised at the effort level which maximises the surplus of revenue over costs. Diagrammatically, this occurs where the slopes of the total cost and total revenue curves are equal. This in indicated in by the tangent to the yield–effort function at EPP being parallel to the slope of the H = (w/P)E line.

48. It is evident from Table 17
It is evident from Table 17.5 is that, in terms of the signs of the appropriate partial derivatives, open access and static private property are identical for the steady-state equilibrium stock and effort. However, for the steady state harvest level, three partial derivatives that cannot be signed under open access conditions show unambiguous directions of response in the (static) private property case.
Be careful here. The signs of the partial derivatives are identical, but their mathematical expressions and numerical values are not.

49. The present-value-maximising fishery model
The present-value-maximising fishery model generalises the model of the static private-property fishery, and in doing so provides us with a sounder theoretical basis and a richer set of results.
The essence of this model is that a rational private-property fishery will organise its harvesting activity so as to maximise the discounted present value (PV) of the fishery.
This section specifies the present-value-maximising fishery model and describes and interprets its main results. Full derivations have been placed in Appendix 17.3.
The individual components of our model are very similar to those of the static private fishery model. However, we now bring time explicitly into the analysis by using an intertemporal optimisation framework.
Initially we shall develop results using general functional forms. Later in this section, solutions are obtained for the specific functions and baseline parameter values assumed earlier.

51. Necessary conditions for maximum wealth
The necessary conditions for maximum wealth include
It is very important to distinguish between upper-case P and lower-case p in equation 17.29, and in many of the equations that follow.
P is the market, or landed, or gross price of fish; it is, therefore, an observable quantity. As the market price is being treated here as an exogenously given fixed number, no time subscript is required on P.
In contrast, lower-case pt is a shadow price, which measures the contribution to wealth made by an additional unit of fish stock at the wealth-maximising optimum. It is also known as the net price of fish, and also as unit rent.

52. Necessary conditions – other aspects
Three properties of the net price deserve mention.
First, it is typically an unobservable quantity.
Second, like all shadow prices, it will vary over time, unless the fishery is in steady-state equilibrium. In general, therefore, it is necessary to attach a time label to net price.
The third property concerns an equivalence between the net price equations for the renewable and non-renewable resource cases.
Equation defines the net price of the resource (pt) as the difference between the market price and marginal cost of an incremental unit of harvested fish.
Differential equation governs the behaviour over time of the net price, and implicitly determines the harvest rate in each period.
Note that if growth is set to zero (as would be the case for a non-renewable resource) and the stock term in the cost function is not present (so that the size of the resource stock has no impact on harvest costs) then the net price equation for renewable resources (equation above) collapses to a special case that is identical to its counterpart for a non-renewable resource
See equation 14.6 in Chapter 14 or the expressions for net price in section 15.3 in chapter 15).
Thus the differential equation for the net price of a non-renewable resources turns out to be a special case of a more general differential equation expression for the net price of a renewable resource.

53. Interpretation of equation 17.32   
A decision about whether to defer some harvesting until the next period is made by comparing the marginal costs and benefits of adding additional units to the resource stock. By not harvesting an incremental unit this period, the fisher incurs an opportunity cost that consists of the foregone return by holding a stock of unharvested fish. The marginal cost of the investment is ip, as sale of one unit of the harvested fish would have led to a revenue net of harvesting costs (given by the net price of the resource, p) that could have earned the prevailing rate of return on capital, i. Since we are considering a decision to defer this revenue by one period, the present value of this sacrificed return is ip.
The owner compares this marginal cost with the marginal benefit obtained by not harvesting the incremental unit this period. There are two categories of marginal benefit:
1. As an additional unit of stock is being added, total harvesting costs will be reduced by the quantity CS = ∂C/∂S (note that ∂C/∂S < 0).
2. The additional unit of stock will grow by the amount dG/dS. The value of this additional growth is the amount of growth valued at the net price of the resource.
How is the present value of profits maximised? The key to understanding profit-maximising behaviour when access to the resource can be regulated lies in capital theory.
A renewable resource is a capital asset.
To fix ideas, think about a fishery with a single owner who can control access to the resource and appropriate all returns from it. We wish to consider the owner’s decision about whether to marginally change the amount of fish harvesting currently being undertaken. Specifically, our interest lies in the marginal cost and the marginal benefit of changing the harvest rate for one period only.
A decision about whether to defer some harvesting until the next period is made by comparing the marginal costs and benefits of adding additional units to the resource stock. What is the marginal cost of deferring the harvesting of one unit of fish from the current period until the following period? Choosing not to harvest now is equivalent to a capital investment. The uncaught fish will be there next period; moreover, biological growth will mean that there is an additional increment to the stock next period, over and above the quantity of fish left unharvested. This amounts to saying that the asset – in this case the fishery – is productive. By not harvesting an incremental unit this period, the fisher incurs an opportunity cost that consists of the foregone return by holding a stock of unharvested fish.
A present-value-maximising owner will add units of resource to the stock provided the marginal cost of doing so is less than the marginal benefit.
This states that a unit will be added to stock provided its ‘holding cost’ (ip) is less than the sum of its harvesting cost reduction and value-of-growth benefits. Conversely, a present-value-maximising owner will harvest additional units of the stock if marginal costs exceed marginal benefits.
These imply the asset equilibrium condition, equation When this is satisfied, the rate of return the resource owner obtains from the fishery is equal to i, the rate of return that could be obtained by investment elsewhere in the economy. This is one of the intertemporal efficiency conditions we identified in Chapter 11. To confirm that this equality exists, divide both sides of equation by the net price p to give
(17.33)
Equation is a version of what is sometimes called the ‘fundamental equation’ of renewable resources. The left-hand side is the rate of return that can be obtained by investing in assets elsewhere in the economy. The right-hand side is the rate of return that is obtained from the renewable resource. This is made up of two elements:
the natural rate of growth in the stock from a marginal change in the stock size;
the value of the reduction in harvesting costs that arises from a marginal increase in the resource stock.

54. Some intuition lying behind Equation 17.32

56. Three important results are evident from an inspection of the data in Table 17.6:
In the special case where the interest rate is zero, the steady-state equilibria of a static private-property fishery and a PV-maximising fishery are identical. For all other rates they differ.
For non-zero interest rates, the steady-state fish stock (fishing effort) in the PV profit-maximising fishery is lower (higher) than that in the static private fishery, and becomes increasingly lower (higher) the higher is the interest rate.
As the interest rate becomes arbitrarily large, the PV-maximising outcome converges to that of an open-access fishery.
Equation (see slide above) is an implicit equation for the unknown PV-maximising equilibrium value of S. Solving it for S gives an analytical expression for the equilibrium stock. Given that, expressions for the PV-optimal solutions for H and E can be obtained using the biological growth function and the fishery production function, equation 17.6.
Table 17.6 lists the equilibrium values of S, E and H for interest rates between zero and 100 per cent (and two higher values), conditional on our assumed functional forms and other baseline parameter values. For purposes of comparison we also show the equilibrium values for a static private-property fishery and an open-access fishery. Note that for these last two models, equilibrium values do not vary with the interest rate.
The solution is derived in Appendix 17.3.
These calculations are implemented in both the Excel spreadsheet Comparative statics.xls and in the Maple file Chapter 17.mw.

57. The consequence of a dependence of harvest costs on the stock size
The specific functions used in this chapter for harvest quantity and harvest costs are H = eES and C = wE respectively. Substituting the former into the latter and rearranging yields C = wE/eS.
It is evident from this that, under our assumptions, harvest costs depend on both effort and stock. However, suppose that the harvest equation were of the simpler form H = eE. In that case we obtain C = wE/e and so costs are independent of stock.
In such (rather unlikely) cases, when harvesting costs do not depend on the stock size, ∂C/∂S = 0 and so the steady-state PV-maximising condition (17.32) simplifies to i = dG/dS. 
This interesting result tells us that a private PV-maximising steady-state equilibrium, where access can be controlled and costs do not depend upon the stock size, will be one in which the resource stock is maintained at a level where the rate of biological growth (dG/dS) equals the market rate of return on investment (i) – exactly what standard capital theory suggests should happen.
This is illustrated in Figure 17.7.

58. Figure 17.7 Present value maximising fish stocks with and without dependence of costs on stock size, and for zero and positive interest rates
Slope = i - [-(C/S)/p]
Slope = i S
SPV*
SPV
G(S)
Where costs do NOT depend on stock size, at the present-value-maximising resource stock size (which we denote by SPV), i = dG/dS.
It is clear from this diagram that as the interest rate rises, the profit-maximising stock size will fall.

59. The consequence of a dependence of harvest costs on the stock size: other results
If i = 0 and ∂C/∂S = 0 (that is decision makers use a zero discount rate and harvesting costs do not depend on the stock size) then the stock rate of growth (dG/dS) should be zero, and so the present-value-maximising steady-state stock level would be the one generating the maximum sustainable yield, SMSY.
In Figure 17.7, the straight line labelled ‘Slope = i’ would be horizontal and so tangent to the growth function at SMSY. Intuitively, it makes sense to pick the stock level that gives the highest yield in perpetuity if costs are unaffected by stock size and the discount rate is zero. However, this result is of little practical relevance in commercial fishing, however, as it is not conceivable that owners would select a zero discount rate.
If i > 0 and ∂C/∂S = 0 (that is decision makers use a positive discount rate and harvesting costs do not depend on the stock size) then SPV is less than the maximum sustainable yield stock, SMSY.
When total costs of harvesting, C, depend negatively on the stock size, the present-value-maximising stock is higher than it would be otherwise.
Result 2 follows from the fact that the equality i = dG/dS can only be satisfied for positive i where dG/dS is positive, which requires a stock size below SMSY. Intuitively, with positive discounting and no stock effect on costs, the stock is drawn down below the maximum sustainable yield as future losses in income from higher harvests are discounted and there is no penalty from harvest cost increases. However, for many marine fisheries, it is likely to be the case in practice that harvesting costs will rise as the stock size falls, and so this result is unlikely to be of wide applicability.
Result 3 follows from the result that the discount rate i must be equated with dG/dS – {∂C/∂S}/p), rather than with dG/dS alone. Given that {∂C/∂S}/p is a negative quantity, this implies that dG/dS must be lower for any given interest rate, and so the equilibrium stock size must be greater than where there is no stock effect on costs. The intuition behind this is that benefits can be gained by allowing the stock size to rise (which causes harvesting costs to fall). This is also illustrated in Figure 17.7, which compares the present-value optimal stock levels with (S*PV) and without (SPV) the dependence of costs on the stock size. This result is consistent with our findings in Table 17.6.

60. The consequence of a dependence of harvest costs on the stock size: other results
In the general case where a positive discount rate is used and ∂C/∂S is negative, the steady-state present-value-maximising stock level may be less than, equal to, or greater than the maximum sustainable yield stock.
Which one applies depends on the relative magnitudes of i and (∂C/∂S)p. Inspection of Figure 17.7 shows the following:
i = –(∂C/∂S)p  S*PV = SMSY
i > –(∂C/∂S)p  S*PV < SMSY
i < –(∂C/∂S)p  S*PV > SMSY
This confirms an observation made earlier in this chapter: a maximum sustainable yield (or a stock level SMSY) is not in general economically efficient, and will only be so in special circumstances. For our illustrative numerical example, using baseline parameter values, it turns out that this special case is that in which i = 0.05; only at that interest rate does the PV-maximising model yield the MSY outcome.14
Result 2 follows from the fact that the equality i = dG/dS can only be satisfied for positive i where dG/dS is positive, which requires a stock size below SMSY. Intuitively, with positive discounting and no stock effect on costs, the stock is drawn down below the maximum sustainable yield as future losses in income from higher harvests are discounted and there is no penalty from harvest cost increases. However, for many marine fisheries, it is likely to be the case in practice that harvesting costs will rise as the stock size falls, and so this result is unlikely to be of wide applicability.
Result 3 follows from the result that the discount rate i must be equated with dG/dS – {∂C/∂S}/p), rather than with dG/dS alone. Given that {∂C/∂S}/p is a negative quantity, this implies that dG/dS must be lower for any given interest rate, and so the equilibrium stock size must be greater than where there is no stock effect on costs. The intuition behind this is that benefits can be gained by allowing the stock size to rise (which causes harvesting costs to fall). This is also illustrated in Figure 17.7, which compares the present-value optimal stock levels with (S*PV) and without (SPV) the dependence of costs on the stock size. This result is consistent with our findings in Table 17.6.
14 You can verify this yourself using the spreadsheet Comparative Statics.xls. Note also that if P were chosen differently, then another value of i would be required to bring about an MSY solution.

61. Dynamics in the PV-maximising fishery
Our analysis of the open-access fishery showed complex patterns of dynamic behaviour.
These various dynamic processes reflect the unplanned and uncoordinated behaviour of fishing effort in an open-access fishery.
Given the assumptions that have been made in this chapter about the PV-maximising fishery (and about the static private-property fishery) we would not expect to find those dynamic processes there. Even where there is a large number of profit maximising owners acting as if they were in a competitive market, the individual private property assumption is sufficient for owners to somehow or other coordinate their behaviour to be collectively rational. This rules out the intertemporal stock externality effect (whereby each boat’s harvest reduces the future stock available for others) that exists in an open-access fishery.
A brief description of adjustment dynamics in PV-maximising private-property fisheries is given in Appendix We suggest there that the efficient (wealth-maximising) paths to steady-state equilibrium outcomes will typically not be oscillatory. In fact, for the model examined in this section where the market price of fish is exogenously fixed, adjustment takes a particularly simple form, known as the most rapid approach path (MRAP).
With respect to point 1, out of equilibrium, the processes of adjustment over time were (under baseline assumptions) characterised by oscillatory paths for stock, effort and harvest, with levels of these variables repeatedly over-shooting and under-shooting their steady-state values. In the case investigated with simple logistic growth, the dynamic adjustment processes were dynamically stable, eventually converging to equilibrium levels. For other sets of parameter values, other patterns of dynamic behaviour can be found, including fishery collapse and chaotic behaviour. (See Clark, 1990, and Conrad and Clark, 1987 for further details.)
With respect to point 4, a simple rule emerges: do no harvesting when stocks are less than the equilibrium value, and harvest at the maximum possible rate when stocks exceed the PV equilibrium.15 Spence and Starrett, 1975, identify the conditions on the Hamiltonian which make MRAP optimal. Matters are more complicated where the price is endogenous, depending on the harvest rate. Relevant results, obtained using phase-plane analysis, are given in Appendix 17.3.
15 Discrete time simulations, shown in the Excel workbook that accompanies this chapter, do not exhibit this MRAP form of adjustment for a private (nor for a PV-maximising) fishery. Instead, the workbook uses a discrete period-by-period adjustment rule in which effort increases (decreases) when marginal profit of effort is positive (negative). This rule generates a correct steady-state outcome, but the adjustment to it is slower than optimal, and does show some oscillatory behaviour. This corroborates an earlier comment that even for equivalent parameterisations discrete time and continuous time models can have qualitatively very different dynamic properties.

62. Encompassing the open-access, static private-property and PV-maximising fishery models in a single framework
 All three fishery models investigated here can be related to one another and so what might appear to be three fundamentally different sets of models and results can all be reconciled with one another and encompassed within a single framework.
Consider first the two private-property fishery models. The static private-property fishery is a special case of the PV fishery – the case in which the interest rate is zero. Put another way, the steady-state PV-maximising fishery results collapse to those of the simple private fishery model when the interest (discount) rate is zero.
The open-access fishery model can also be brought into this encompassing framework. We can do so by observing that an open-access fishery can be thought of as one in which the absence of enforceable property rights means that fishing boat owners have an infinitely high discount rate. The interest rate of boat owners in an open-access fishery is, in effect, infinity, irrespective of what level the prevailing interest rate in the rest of the economy happens to be. We have already demonstrated that as increasingly large values of i are plugged into the PV-maximising solution expression, outcomes converge to those derived from the open-access model.
With respect to point 2, this can be verified by substitution of the value i = 0 into the expressions for PV-maximising stock, effort and harvest levels (shown in Appendix 17.3). It will be found that they collapse to the simpler expressions for static private-property equilibria in Table In that case, the marginal revenue (with respect to stock changes) and the marginal cost (with respect to stock changes) are equal, as is required by profit-maximising equilibrium in the static private-property fishery model without discounting. This is, of course, the standard result for any static profit-maximising model.

63. Socially efficient resource harvesting
If we go through the process (as in section 17.11) of identifying the necessary conditions for socially optimal resource allocation decisions, we find that privately maximising decisions (as with the NPV maximising private property fishery model) will be socially efficient provided that a number of conditions are satisfied.
These conditions include:
 The market in which fish are sold is a competitive market.
The (gross) market price, P, correctly reflects all social benefits (i.e. there are no externalities on the demand side).
There are no fishing externalities on the cost side.
The private and social consumption discount rates are identical, i = r.
If these conditions are satisfied, the first-order conditions for social efficiency are identical to those for the PV-maximising fishery. Hence the PV-maximising fishery would be socially efficient.
But, of course, it also follows that if one or more of those conditions is not satisfied, private fishing will not be socially efficient.

64. . . Regeneration function St+1 Slope = 1 + r Slope = 1
Slope (variable) = 1 + 
A safe minimum standard of conservation
Our discussion of the ‘best’ level of renewable-resource harvesting has focused almost exclusively on the criterion of economic efficiency. However, if harvesting rates pose threats to the sustainability of some renewable resource (such as North Atlantic fisheries or tropical primary forests) or jeopardise an environmental system itself (such as a wildlife reserve containing extensive biodiversity) then the criterion of efficiency may be inappropriate.
Even in a deterministic world in which population growth rates are known with certainty the pursuit of an efficiency criterion is not sufficient to guarantee the survival of a renewable resource stock or an environmental system in perpetuity, particularly when resource prices are high, harvesting costs are low, or discount rates are high. Where biological systems are stochastic, or where uncertainty is pervasive, threats to sustainability are even more pronounced.
Many writers – some economists, but particularly non-economists – argue that correcting market failure and eliminating efficiency losses should be given secondary importance to the pursuit of sustainability. This would suggest that policy be targeted to the prevention of species extinction or the loss of biological diversity whenever that is reasonably practical. Efficiency objectives can be pursued within this general constraint.
Such considerations brings us back to the principle that policy be oriented around the criterion of a safe minimum standard of conservation (SMS), an idea examined earlier in Chapter 13. How does this idea apply to renewable resource policy? A strict version of SMS would involve imposing constraints on resource harvesting and use so that all risks to the survival of a renewable resource are eliminated. This is unlikely to be of much practical relevance. Virtually all human behaviour entails some risks to species survival, and so a strict SMS would prohibit virtually all economic activity.
In order to make the concept usable, it is necessary to impose weaker constraints, so that the adoption of an SMS approach will entail that, under reasonable allowances for uncertainty, threats to survival of valuable resource systems are eliminated, provided that this does not entail excessive cost. For decisions to be made that are consistent with that weaker criterion, judgements will be necessary particularly about what constitutes ‘reasonable uncertainty’ and ‘excessive cost’, and which resources are deemed ‘sufficiently valuable’ for application of the SMS criterion.
Let us explore the concept of an SMS in this context by following the exposition in a paper by Randall and Farmer (1995). Suppose there is some renewable resource the expected growth of which over time is illustrated by the curve labelled ‘Regeneration function’ in Figure The function shows the resource stock level that will be available in period t + 1 (St+1) for any level of stock that is conserved in period t (St). Notice that the greater is the level of stock conservation in period t, the higher will be the stock level available in the following period.17
Randall and Farmer restrict their attention to sustainable resource use, interpreting sustainability to mean a sequence of states in which the resource stock does not decline over time. Therefore, only those levels of stock in period t corresponding to segments of the regeneration function that lie on or above the 45° line (labelled ‘slope = 1’) constitute sustainable stocks. The minimum sustainable level of stock conservation is labelled SMIN.
The efficient level of stock conservation is St*. To see this, construct a tangent to the regeneration function with a slope of 1 + r, where r is a (consumption) social discount rate, and let  denote the rate of growth of the renewable resource.18 At any point on the regeneration function, a tangent to the function will have a slope of 1 + . For the particular stock level St* we have
1 + r = 1 +  or r = 
and so the rate of growth of the renewable resource, , is equal to the social discount rate, r. This is a condition for economically efficient steady-state renewable resource harvesting that we have met earlier in Section That is, it is the form that equation takes in steady-state equilibrium where harvest costs are independent of stock size. Note also that at the efficient stock level, the amount of harvest that can be taken in perpetuity is Rt*. By harvesting at this rate, the post-harvest stock in period t + 1 is equal to that in period t, thus satisfying the sustainability requirement.
Now suppose that the regeneration function is subject to random variation. For simplicity, assume that the worst possible outcome is indicated by the dotted regeneration function in Figure At any current stock, the worst that can happen is that the available future stock falls short of the expected quantity by an amount equal to the vertical distance between the solid and dotted functions. Now even in the worst outcome, if is conserved in each period, the condition for perpetual sustainability of the stock will be maintained. We might regard as a stock level that incorporates the safe minimum standard of conservation, reflecting the uncertainty due to random variability in the regeneration function. Put another way, whereas SMIN is an appropriate minimum stock in the absence of uncertainty, takes account of uncertainty in a particular way.
In fact, is not what Randall and Farmer propose as a safe minimum standard of conservation. They argue that any sustainable path over time must involve some positive, non-declining level of resource harvesting and consumption in every period. Suppose that RMIN is judged to be that minimum required level of resource consumption. Then Randall and Farmer’s safe minimum standard of conservation is . If the stock in period t is kept from falling below this level then, even in the worst possible case, RMIN can be harvested without interference with sustainability.
The SMS principle implies maintaining a renewable resource stock at or above some safe minimum level such as . In Figure 17.8, there is no conflict between the conservation and efficiency criteria. The safe minimum standard of conservation actually implies a lower target for the resource stock than that implied by economic efficiency. This will not always be true, however, and one can easily imagine circumstances where an SMS criterion implies more cautious behaviour than the economically efficient outcome.
Finally, what can be said about the qualification that the SMS should be pursued only where it does not entail excessive cost? Not surprisingly, it is difficult to make much headway here, as it is not clear how one might decide what constitutes excessive cost. Randall and Farmer suggest that no society can reasonably be expected to decimate itself. Therefore, if the SMS conflicted with the survival of human society, that would certainly entail excessive cost. But most people are likely to regard costs far less than this – such as extreme deprivation – as being excessive. Ultimately, the political process must generate views as to what constitutes excessive costs.
17 The non-linear relationship shown here is plausible for many types of biological resource, but will not be a good representation for all such resources.
18 Strictly speaking,  is the derivative of the amount of resource growth with respect to the stock size; it is what we have earlier written as dG/dS. St
Figure 17.8 A safe minimum standard of conservation

65. EXCESSIVE HARVESTING AND SPECIES EXTINCTION
There are many reasons why human behaviour may cause population levels to fall dramatically or, in extreme cases, cause species extinction. These include:
Even under restricted private ownership, it may be ‘optimal’ to the owner to harvest a resource to extinction. Clark (1990) demonstrates, however, that this is highly improbable.
Ignorance of or uncertainty about current and/or future conditions results in unintended collapse or extinction of the population.
Shocks or disturbances to the system push populations below minimum threshold population survival levels.

66. WHATEVER THE REGIME, SPECIES EXTINCTION IS MORE LIKELY:
the higher is the market (gross) resource price of the resource
the lower is the cost of harvesting a given quantity of the resource
the more that market price rises as the catch costs rise or as harvest quantities fall
the lower the natural growth rate of the stock, and the lower the extent to which marginal extraction costs rise as the stock size diminishes
the higher is the discount rate
the larger is the critical minimum threshold population size relative to the maximum population size.

67. Forestry
As far as forests as sources of timber are concerned, much of the previous analysis applies.
But what is also important here is the multiple service functions of much forestry.

68. Renewable resources policy
Command-and-control:
Quantity restrictions on catches (EU Total Allowable Catches)
Fishing season regulations
Technical restrictions on the equipment used - for example, restrictions on fishing gear, mesh or net size, or boat size.
Incentive-based policies:
Restrictions on open access/property rights
Fiscal incentives
Establishment of forward or futures markets
Marketable permits (‘individual transferable quotas’, ITQ)