File: Thirdch17.doc
Date: 24 July 2002
Words: 34,500
Notes to Editor:
The pictures for this chapter are collected together in two files:
(a) Picturesforchapter17.ppt (for Figures 17.8, 17.9, 17.10 and 17.11)
(b) Chapter 17 Pictures.doc (for all other Figures in the chapter)
Chapter 17 Renewable resources
It will appear, I hope, that most of the problems associated with the words
‘conservation’ or ‘depletion’ or ‘overexploitation’ in the fishery are, in reality, manifestations of the fact that the natural resources of the sea yield no economic rent. Fishery resources are unusual in the fact of their common-property nature; but they are not unique, and similar problems are encountered in other cases of common-property resource industries, such as petroleum production, hunting and trapping, etc.
Learning Objectives
Gordon (1954)
After studying this chapter, the reader should be able to
 understand the biological growth function of a renewable resource, and the notions of compensation and depensation in growth processes
 interpret the simple logistic growth model, and some of its variants including models with critical depensation
 understand the idea of a sustainable yield and the maximum sustainable yield
 distinguish between steady state outcomes and dynamic adjustment processes that may (or may not) lead to a steady state outcome
1

 specify, and solve for its bioeconomic equilibrium outcome, an open access fishery, a static private property fishery, and a present value (PV) maximising fishery
 undertake comparative statics analysis and simple dynamic analysis for open access and private property models
 explain under what conditions the stock, effort and harvesting outcomes of private fisheries will not be socially efficient
 describe conditions which increase the likelihood of severe resource depletion or species extinction
 understand the workings, and relative advantages, of a variety of policy instruments that are designed to conserve renewable resource stocks and/or promote socially efficient harvesting
Introduction
Environmental resources are described as renewable when they have a capacity for reproduction and growth. The class of renewable resources is diverse. It includes populations of biological organisms such as fisheries and forests which have a natural capacity for growth, and water and atmospheric systems which are reproduced by physical or chemical processes. While the latter do not possess biological growth capacity, they do have some ability to assimilate pollution inputs (thereby maintaining their quality) and, at least in the case of water resources, can self-replenish as stocks are run down (thereby maintaining their quantity).
It is also conventional to classify arable and grazing lands as renewable resources. In these cases reproduction and growth take place by a combination of biological processes
(such as the recycling of organic nutrients) and physical processes (irrigation, exposure to wind etc.). Fertility levels can naturally regenerate so long as the demands made on the soil are not excessive. We may also consider more broadly-defined environmental systems
2

(such as wilderness areas or tropical moist forests) as being sets of interrelated renewable resources.
The categories just described are renewable stock resources. A broad concept of renewables would also include flow resources such as solar, wave, wind and geothermal energy. These share with biological stock resources the property that current harnessing of the flow does not mean that the total magnitude of the future flow will necessarily be smaller.
Indeed, many forms of energy flow resources are, for all practical purposes, non-depletable.
Given this diversity of resource types, it will be necessary to restrict what will, and will not, be discussed here. Most of the literature on the economics of renewable resources is about two things: the harvesting of animal species (‘hunting and fishing’) and the economics of forestry. This chapter is largely concerned with the former; forestry economics is the subject of the following chapter. Agriculture could also be thought of as a branch of renewable resource harvesting. But agriculture – particularly in its more developed forms - differs fundamentally from other forms of renewable resource exploitation in that the environmental medium in which it takes place is designed and controlled. The growing medium is manipulated through the use of inputs such as fertilisers, pesticides, herbicides; temperatures may be controlled by the use of greenhouses and the like; and plant stocks are selected or even genetically modified. In that sense, there is little to differentiate a study of (developed) agricultural economics from the economics of manufacturing. For this reason, we do not survey the huge literature that is ‘agricultural economics’ in this text, although some of the environmental consequences of agricultural activity are discussed in Agriculture.doc
in the Additional Materials . For reasons of space, we also do not cover the economics of renewable flow resources. Again, a brief outline of some of the main issues is given in Renewables.doc
in the Additional
Materials .
It is important to distinguish between stocks and flows of the renewable resource. The stock is a measure of the quantity of the resource existing at a point in time, measured either as the aggregate mass of the biological material (the biomass) in question (such as the total
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weight of fish of particular age classes or the cubic metres of standing timber), or in terms of population numbers. The flow is the change in the stock over an interval of time, where the change results either from biological factors, such as ‘recruitment’ of new fish into the population through birth or exit due to natural death, or from harvesting activity.
One similarity between renewable and non-renewable resources is that both are capable of being fully exhausted (that is, the stock being driven to zero) if excessive and prolonged harvesting or extraction activity is carried out. In the case of non-renewable resources, exhaustibility is a consequence of the finiteness of the stock. For renewable resources, although the stock can grow, it can also be driven to zero if conditions interfere with the reproductive capability of the renewable resource, or if rates of harvesting continually exceed net natural growth.
It is evident that enforceable private property rights do not exist for many forms of renewable resource. In the absence of regulation or collective control over harvesting behaviour, the resource stocks are subject to open access. We will demonstrate that openaccess resources tend to be overexploited in both a biological and an economic sense, and that the likelihood of the resource being harvested to the point of exhaustion is higher than where private property rights are established and access to harvesting can be restricted.
As we have said, this chapter is principally about the harvesting of animal resources.
Our exposition focuses on marine fishing. With some modifications, the fishery economics modelling framework can be used to analyse most forms of renewable resource exploitation. We begin by setting out a simple model of the biological growth of a fish population. Then the properties of commercial fisheries are examined under two sets of institutional arrangements: an open access fishery and a profit-maximising fishery in which enforceable private property rights exist.
For the case of the private property fishery, the analysis proceeds in two steps. First we examine what is usually known as the static private property fishery. The analysis is kept simple by abstracting from the need to deal explicitly with the passage of time. We do this
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by focussing attention on steady state (or equilibrium) outcomes in which variables are taken to be unchanging over time. Some unspecified interval of time is chosen to be representative of all periods in that steady state. The equilibrium is found by solving the model for its profit maximising solution. By construction that equilibrium would apply to every time period, provided that economic and biological conditions remain unchanged.
The second step involves a generalisation in which the passage of time is modelled explicitly. In this case we investigate a private property fishery that is managed so as to maximise its present value over an infinite lifetime. All nominal value flows are discounted at some positive rate to convert to present value equivalents. Describing the second variant as a generalisation of the first is appropriate because – as we shall show – the static private fishery turns out to be a special case of the present value maximising fishery in which owners adopt a zero discount rate. Where discounting takes place at some positive rate, the outcomes of the two models differ.
In common with the practice throughout this text, we also examine the outcomes of the various commercial fishery regimes against the benchmark of a socially efficient fishery.
We demonstrate that under some conditions the harvesting programme of a competitive fishery where private property rights to the resource stocks are established and enforceable will be socially efficient. However, actual resource harvesting regimes are typically not socially efficient, even where attempts have been made to introduce private property rights. Among the reasons why they are not is the existence of various kinds of externalities. Open access regimes will almost certainly generate inefficient outcomes. The chapter concludes by examining a set of policy instruments that could be introduced in an attempt to move harvesting behaviour closer to that which is socially efficient.
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17.1 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 the population in 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 rate g. Then in the absence of human predation the rate of change of the population over time is given by
1 dS

S
 gS dt
(17.1)
By integrating this equation, we obtain an expression for the stock level at any point in time:
S t

S
0 e gt in which S
0
is the initial stock level. In other words, for a positive value of g, 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 of fish exists in a particular environmental milieu, with a finite carrying capacity, which sets bounds on the population's growth possibilities.
1
Be careful not to confuse a rate of change with a rate of growth. A rate of change refers to how much extra is produced in some interval of time. A rate of growth is that rate of change divided by its current size (to measure the change in proportionate terms). Note that we shall sometimes refer to an ‘amount of growth’ (as opposed to a growth rate); this should be read as the population size change over some interval of time.
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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. Using the symbol

to denote the actual growth rate, the growth function can then be written as
  
( S ) S where

(S) states that

is a function of S, and shows the dependence of the actual growth rate on the stock size. If this function has the property that the proportionate growth rate of the stock ( / S ) declines as the stock size rises then the function is said to have the property of compensation.
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 S
MAX
. A commonly used functional form for

(S) which has the properties of compensation and a maximum stock size is the simple logistic function:

( S )
 g

 1

S
S
MAX

 in which the constant parameter g > 0 is what we have called the intrinsic or potential growth rate of the population. Where the logistic function determines the actual population growth rate, we may therefore write the biological growth function as
 dS dt

 g

1

S
S
MAX

 S (17.2)
The changes taking place in the fish population that we have been referring to so far are
‘natural’ changes. But as we want to use the notation
S and dS dt in the rest of this chapter to refer to the net effect of natural changes and human predation, we shall use the alternative symbol G to refer to stock changes due only to natural causes. (More completely, we shall use the notation G(S) to make it clear that G depends on S.) With this change the logistic biological growth function is
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G ( S )

 g

1

S
S
MAX

 S (17.3)
The logistic form is a good approximation to the natural growth processes of many fish, animal and bird populations (and indeed to some physical systems such as the quantity of fresh water in an underground reservoir). Some additional information on the simple logistic growth model, and alternative forms of logistic growth, is given in Box 17.1.
Problem 17.1 also explores the logistic model a little further, and invites you to explore another commonly used equation for biological growth, the Gompertz function.
Insert Box 17.1 near here
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Box 17.1 The logistic form of the biological growth function
Logistic growth is one example of density-dependent growth: processes where the growth rate of a population depends on the population size. It was first applied to fisheries by
Schaeffer (1957). The equation for logistic growth of a renewable resource population was given by Equation 17.3.
Insert Figures 17.1a, 17.1b, 17.1c and 17.1d near here (all stacked below one another as one four-part diagram if possible) Editor: These are to be found in the second edition as the four parts of Figure 9.1 on page 217.
Figure 17.1a
Caption: Simple logistic growth.
Figure 17.1b
Caption: Logistic growth with a minimum population threshold.
Figure 17.1c
Caption: Logistic growth with depensation.
Figure 17.1d
Caption: Logistic growth with critical depensation.
Simple logistic growth is illustrated in Figure 17.1(a), which represents the relationship between the stock size and the associated rate of change of the population due to biological growth. Three properties should be noted by inspection of that diagram.
(a) S
MAX
is the maximum stock size that can be supported in the environmental milieu.
This value is, of course, conditional on the particular environment circumstances that happen to prevail, and would change if any of those circumstances (such as ocean temperature or stocks of nutrients) change.
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(b) By multiplication through of the terms on the right-hand side of Equation 17.3, it is clear that the amount of growth, G, is a quadratic function of the resource stock size, S.
For the simple logistic function, the maximum amount of growth (S
MSY
) will occur when the stock size is equal to half of S
MAX
.
(c) The amount of biological growth G is zero only at a stock size of zero and a stock size of S
MAX
. For all intermediate values, growth is positive.
This last property may appear to be obviously true, but it turns out to be seriously in error in many cases. It implies that for any population size greater than zero natural growth will lead to a population increase if the system is left undisturbed. In other words, the population does not possess any positive lower threshold level.
However, suppose there is some positive population threshold level, S
MIN
, such that the population would inevitably and permanently decline to zero if the actual population were ever to fall below that threshold. A simple generalisation of the logistic growth function that has this property is:
G(S) = g(S S
MIN
)

 1 -
S
S
MAX

 (17.4)
Note that if S
MIN
= 0, Equation 17.4 collapses to the special case of Equation 17.3. The generalisation given by Equation 17.4 is illustrated in Figure 17.1(b). Several other generalisations of the logistic growth model exist. For example, the modified logistic model:
G ( S )
 gS


 1

S
S
MAX

 has the property that for

> 1 there is, at low stock levels, depensation , which is a situation where the proportionate growth rate (G(S)/S) is an increasing function of the stock size, as opposed to being a decreasing function (compensation) in the simple logistic case where

= 1. A biological growth function exhibiting depensation at stock levels below S
D
(and compensation thereafter) is shown in Figure 17.1(c).
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Finally, the generalised logistic function
G ( S )

 g

S
S
MIN

1 




 1

S
S
MAX

 S exhibits what is known as critical depensation . As with Equation 17.4, the stock falls irreversibly to zero if the stock ever falls below S
MIN
. This function is represented in Figure
17.1(d). It should be evident that if a growth process does exhibit critical depensation, then the probability of the stock being harvested to complete depletion is increased, and increased considerably if S
MIN
is a large proportion of S
MAX
.
END OF BOX 17.1
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17.1.1 The status and role of logistic growth models
The logistic growth model is a stylised representation of the population dynamics of renewable resources. The model is most suited to non-migratory species at particular locations. Among fish species, demersal or bottom-feeding populations of fish such as cod and haddock are well characterised by this model. The populations of pelagic or surfacefeeding fish, such as mackerel, are less well explained by the logistic function, as these species exhibit significant migratory behaviour. Logistic growth does not only fit biological growth processes. Brown and McGuire (1967) argue that the logistic growth model can also be used to represent the behaviour of a freshwater stock in an underground reservoir.
However, a number of factors which influence actual growth patterns are ignored in this model, including the age structure of the resource (which is particularly important when considering long-lived species such as trees or whale stocks) and random or chance influences. At best, therefore, it can only be a good approximation to the true population dynamics.
Judging the logistic model on whether it is the best available at representing any particular renewable resource would be an inappropriate for our present purposes. One would not expect to find that biological or ecological modellers would use simple logistic growth functions. They will use more complex growth models designed specifically for particular species in particular contexts. But the needs of the environmental economist differ from those of ecological modellers. The former are willing to trade-off some realism to gain simple, tractable models that are reasonably good approximations. It is for this reason that much economic analysis makes use of some version of the logistic growth function
Of more concern, perhaps, is the issue of whether it is appropriate to describe a biological growth process by any purely deterministic equation such those given in Box 17.1. Ecological models typically specify growth as being stochastic, and linked in complex ways to various
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other processes taking place in more broadly defined ecosystems and sub-systems. We shall briefly explore these matters later in the chapter.
17.2 Steady-state harvests
In this chapter, much of our attention will be devoted to steady state harvests. Here we briefly explain the concept. 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 as a sustainable yield.
Defining S as the actual rate of change of the renewable resource stock, with

G

H , it follows that in steady-state harvesting S = 0 and so the resource stock remains constant over time. What kinds of steady states are feasible? To answer this, look at Figure 17.2. There is one particular stock size (S
MSY
) at which the quantity of net natural growth is at a maximum
(G
MSY
). If at a stock of S
MSY
harvest is set a the constant rate H
MSY
, 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. We shall see later that economic theory does not, in general, support this proposition.
Insert Figure 17.2 here (See the pictures file “Chapter 17 Pictures.doc”).
Caption: Steady-state harvests.
H
MSY
is not the only possible steady state harvest. Indeed, Figure 17.2 shows that any harvest level between zero and H
MSY is a feasible steady-state harvest, and that any stock between zero and S
MAX
can support steady state harvesting. For example, H
1
is a feasible steady-state harvest if the stock size is maintained at either S
1L
or S
1U
. Which of these two
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stock sizes would be more appropriate for attaining a harvest level of H
1
is also a matter we shall investigate later.
Before moving on, it is important to understand that the concept of a steady state is a heuristic device: useful as a way of organising ideas and structuring analysis. But, like all heuristic devices, a steady state is a mental construct and using it uncritically can be inappropriate or misleading. Like the broader concept of equilibrium, fisheries and other resource stocks are rarely, if ever, in steady states. Conditions are constantly changing, and the ‘real world’ is likely to be characterised by a more-or-less permanent state of disequilibrium. For some problems of renewable resource exploitation the analysis of transition processes is more important or insightful than information about steady states.
We shall examine some of these ‘dynamic’ matters later in the chapter. Nevertheless, we will proceed on the assumption that looking at steady states is useful, and next investigate their properties under various institutional circumstances.
17.3 An open access fishery
Our first model of renewable resource exploitation is an open access fishery model. In conformity with the rest of this book, we study this using continuous time notation.
However, the equations which constitute the discrete time equivalent of this continuous time model (and others examined later in the chapter) are listed in full in Appendix 17.1.
These will be used at various places in the chapter to give numerical illustrations of the arguments. The numerical shown in our illustrative examples are computed using an Excel spreadsheet. Should the reader wish to verify that the values shown are correct, or to see how they would change under alternative assumptions, we have made the two spreadsheets used in our computations available to the reader in the Additional Materials : Comparative statics.xls and Fishery dynamics.xls
. While we hope that some readers (or instructors) will find them useful, they are not necessary for an understanding of the contents of this chapter.
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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.
2
On the contrary, any vessel is entitled or is able (or both) to fish wherever 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.
The open access model has two components:
(1) A biological sub-model, describing the natural growth process of the fishery.
(2) An economic sub-model, describing the economic behaviour of the fishing boat owners.
The model is laid out in full in Table 17.1. Subsequent parts of this section will take you through each of the elements described there. 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
2 This lack of de facto enforceability may derive from the fact that the fish are spatially mobile, or from the fact that boats are spatially mobile (or both).
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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 turns out to have important implications for whether a fish population may be driven to exhaustion, and indeed whether the resource itself could become extinct. The properties of such adjustment paths are examined in Section 17.4.
Insert Table 17.1 near here (a new table)
Caption: The open access fishery model.
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Table 17.1 The open access fishery model.
BIOLOGICAL SUB-MODEL:
General specification
Biological growth
(17.5, 17.3) dS/dt = G(S)
Specific forms assumed
G ( S )

 g

1

S
S
MAX

 S
ECONOMIC SUB-MODEL:
Fishery production function
(17.6, 17.7)
H = H(E, S)
Net growth of fish stock
(17.8) dS/dt = G(S) – H
H = eES dS / dt

 g

1

S
S
MAX

 S - H
C = wE Fishery costs
(17.9, 17.10)
C = C(E)
Fishery revenue
(17.11)
B = PH, P constant
Fishery profit
(17.12)
Fishing effort dynamics: open access entry rule (17.13)
NB = B - C dE/dt =

NB
BIOECONOMIC
EQUILIBRIUM CONDITIONS:
Biological equilibrium
(17.14)
G = H
Economic equilibrium
(17.15)
E = E* at which NB = 0
B = PH, P constant
NB = B - C = PeES -wE dE/dt =
G = H

(PeES -wE)
E = E* at which NB = 0
Note: Numbers in parentheses refer to the appropriate equation number in the text.
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17.3.1 The model described
17.3.1.1 Biological sub-model
In the absence of harvesting and other human interference, the rate of change of the stock depends on the prevailing stock size dS/dt = G(S) (17.5)
For our worked numerical example, we assume that the particular form taken by this growth function is the simple logistic growth model given by Equation 17.3.
17.3.1.2 Economic sub-model
17.3.1.2.1 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.
3
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)
3
See Discussion Question 17.4 for more on this matter and the notion of schooling and nonschooling fisheries.
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This relationship can take a variety of particular forms. One very simple form appears to be a good approximation to actual relationships (see Schaeffer, 1954 and Munro, 1981,
1982), and is given by
H = eES (17.7) where e is a constant number, often called the catch coefficient.
4 Dividing each side by E, we have
H
 eS
E which says 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)
17.3.1.2.2 The costs, benefits and profits 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.
5
4
The use of a constant catch coefficient parameter is a simplification that may be unreasonable, and is often dropped in more richly specified models. 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 ))
5
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
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Let B denote the gross benefit from harvesting some quantity of fish. The gross benefit will depend on the quantity harvested, so we have
B = B(H)
In a commercial fishery, the appropriate measure of gross benefits is the total revenue that accrues to firms. Assuming that fish are sold in a competitive market, each firm takes the market price P as given and so the revenue obtained from a harvest H is given by
6
B

PH (17.11)
Fishing profit is given by
NB = B – C (17.12)
17.3.1.2.3 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.
7
Conversely, individuals or firms will leave the 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). We return to this matter later.
6 We could justify this assumption either by saying that the harvesting industry being examined is a small part of a larger overall market, or by arguing that the resource market is competitive, in which case each firm acts as if the market price is fixed (even though price will actually depend ex post on the realised total market supply.
7 The terms rent, economic rent, royalties and net price are used as alternatives to economic profit. They all refer to a surplus of revenue over total costs, where costs include a proper allowance for the opportunity of capital tied up in the fishing fleet.
20

fishery if revenues are insufficient to cover the costs of fishing. A simple way of represent this algebraically is by means of the equation dE/dt =

NB (17.13) 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

. Although the true nature of the relationship is unlikely to be of the simple, linear form in Equation 17.13, this suffices to capture what is essential.
17.3.1.2.4 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 (17.14)
Economic equilibrium requires that the amount of fishing effort be constant through time.
Such an 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 = 0 (17.15) 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*.
17.3.2 Open access steady state equilibrium
We can envisage an open access fishery steady state equilibrium by means of what is known as the fishery’s yield-effort relationship. To obtain this, first note that in a
21

biological equilibrium H = G. Then, by substituting the assumed functions for H and G from Equations 17.7 and 17.3 respectively we obtain:
 gS

1 -
S
S
MAX

 = eES (17.16) which can be rearranged to give
S

S
MAX

 1 e g
E


(17.17)
Equation 17.17 is one equation in two endogenous variables, E and S (with parameters g, e and S
MAX
). It implies a unique equilibrium stock at each level of effort.
8
Next substitute
Equation 17.17 into Equation 17.7 (H = eES) giving
H
 eES

MAX

1 e g
E


(17.18)
In an open access economic equilibrium, profit is zero, so
PH = wE (17.19)
Equations 17.18 and 17.19 constitute two equations in two unknowns (H and E); these can be solved for the equilibrium values of the two unknowns as functions of the parameters alone. The steady state stock solution can then be obtained by substituting the expressions for H and E into Equation 17.7. We list these steady state solutions in Box 17.2, together with the numerical values of E, H and S under the particular set of assumptions about numerical values of the parameters given in Table 17.2.
This solution method 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 Ushape 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 E
1
, then the unique intersection of the harvest-stock relationship and biological growth function determines a steady state
8 This uniqueness follows from the assumption that G(S) is a simple logistic function; it may not be true for other biological growth models.
22

harvest level H
1
at stock S
1
. The lower effort level E
2
determines a second steady state equilibrium (the pair {H2, S2}). We leave the reader to deduce why the label E
MSY
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.
The points of intersection in Figure 17.3 not only satisfy Equation 17.17 but they also satisfy Equation 17.18. 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 17.4. 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 17.18.
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 17.4.
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 {E
OA
, H
OA
}is the open access effort-yield equilibrium.
Insert Figure 17.3 near here (See the pictures file “Chapter 17 Pictures.doc”).
Caption: Steady state equilibrium fish harvests and stocks at various effort levels.
Insert Figure 17.4 near here (See the pictures file “Chapter 17 Pictures.doc”).
Caption: Steady state equilibrium yield-effort relationship.
23

Box 17.2 Analytical expressions for the open-access steady state equilibrium and numerical solutions under baseline parameter assumption.
The analytical expressions for E*, S* and H* (where an asterisk denotes the equilibrium value of the variable in question) as functions of the model parameters alone are:
E *
 g e

 1
 w
PeS
MAX

 (17.20)
S *
 w
Pe
H *
 gw
Pe

 1
 w
PeS
MAX


(17.21)
(17.22)
Derivations of expressions 17.20-17.22 are given in full in Appendix 17.2. Throughout this chapter, we shall be illustrating our arguments with results drawn from fishery models using the parameter values shown in Table 17.2. At various points in the chapter we shall refer to these as the ‘baseline’ parameter values. It can be easily verified that for this set of parameter values, the steady state solution is given by E* = 8.0, S* = 0.2 and H* = 0.024.
Insert Table 17.2 here
Caption: Parameter value assumptions for the illustrative numerical example.
Table 17.2 Parameter value assumptions for the illustrative numerical example.
Parameter g
S
MAX e

P w
End of Box 17.2
24
Assumed numerical value
0.15
1
0.015
0.4
200
0.6

17.4 The dynamics of renewable resource harvesting
Our discussion so far has been exclusively on steady states: equilibrium outcomes which, once 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.
A complete description of fishery dynamics is beyond the scope of this book. But some important insights can be obtained relatively simply. In this section, we undertake some dynamics analysis for the open access model of Section 17.3. Suppose a mature fishery exits that has not previously been commercially exploited. The stock size is, therefore, at its carrying capacity. The fishery 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. Have in mind Equations 17.7, 17.10, 17.11 and 17.12 when thinking this through.
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 17.13. 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: the logistic growth function has the property that biological growth is near zero when the stock is near its maximum carrying capacity (Equation 17.3). This process of increasing E and decreasing S will persist for some time, but it 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 rise, and fewer fish are caught.
25

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 processes we are describing here are a little more subtle than this. The changes do not occur as discrete switches but instead are continuous and gradual. We also find that stocks and effort (and also harvest levels) have oscillatory cycles with the stock cycles slightly leading the effort cycles. In some circumstances, these oscillations dampen down as time passes, and the system eventually settles to a steady state outcome such as that described in the previous section. We illustrate such a transition process in Figure 17.5, where parameter values are given by those shown in
Table 17.2. Note that the oscillations shown in this diagram are particularly acute and have massive amplitude; for other combinations of parameter values, the cycles may be far less pronounced. In this case, you should be able to discern that, given enough time, the levels of
S and E will settle down to the steady state values S
*
= 0.2 and E
*
= 8.0.
Insert Figure 17.5 near here (See the pictures file “Chapter 17 Pictures.doc”).
Caption: Stock and effort dynamic paths for the illustrative model.
The oscillations exhibited by the dynamic adjustment path in Figure 17.5 – with the variables repeatedly over and under-shoot equilibrium outcomes – is a characteristic feature of open access fisheries. In this case, the steady state outcome obtained by solving Equations
17.20-22 is eventually achieved (albeit very slowly). But that is not necessarily true. It is evident that if the oscillations were even larger, the population may be driven down to a level from which it cannot recover, and the fishery is driven out of existence, irrespective of whether or not the equilibrium equations imply that a steady state exists with positive stock and effort. For example, if price had been £270 rather than £200, the steady state solution gives, to two decimal places, S = 0.15 and E = 8.52. Verify this by substituting the new parameter values in Equations 17.21 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 to such a huge level that the stock is completely harvested in a short finite span of time.)
26

Other things being equal, the probability that effort oscillations might cause a population to collapse before it can attain a steady state solution is increased when
(a) the growth function has S
MIN
> 0 , and
(b) the growth function or environmental conditions are stochastic.
The first of these should be self-evident. A positive minimum population size – found when the growth function exhibits critical depensation, for example – implies that low populations that would otherwise recover through natural growth may collapse irretrievably to zero. The second is also relatively straightforward. If in practice there is a stochastic component to growth then the achieved growth in any period may be smaller or larger than its underlying mean value as determined by the growth equation 17.3. In some cases, the stochastic component may be such that growth is actually negative. If the population were at a very 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.
Insert Figure 17.6 near here (See the pictures file “Chapter 17 Pictures.doc”).
Caption: Phase-plane analysis of stock and effort dynamic paths for the illustrative model.
Another way of describing this is in terms of a so-called phase-plane diagram. This is shown in Figure 17.6. The diagram defines 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 curved 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 curved 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.
27

But 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 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 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.
Although the calculations required to carry out the kind of dynamic analyses described in this section could be done by hand, this would be extremely tedious to do. It would be far easier to use some mathematical package (such as Maple or Mathematica) or a spreadsheet programme (such as Excel). For the reader interested in seeing how a spreadsheet package can be used for dynamic simulation (and also to calculate numerical values for steady-state stock, harvest and effort) we have provided an annotated Excel workbook Fishery dynamics.xls
in the Additional Materials . 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 by plotting successive values of H, E and S and observing at which numerical values the variables eventually settle down.
28

The Excel file implements the calculations used to obtain Figure 17.5, 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.
Trying to generate a phase-plane diagram by pen-and-paper calculations would be immensely time-consuming. Diagram 17.6 was obtained by setting up the open access model within the Maple package. We provide, in the Additional Materials , file Phase.doc
, a more complete account of phase-plane analysis and an explanation of how Maple - an easy to use and very powerful package - can generate the graphical output of the form in Figure 17.6. The
Maple file used to generate the picture is included as Chapter 17.mws
.
In our analysis to date, we have presumed that environmental conditions remain unchanged over time. However in practice these conditions do change, usually unpredictably, and sometimes very rapidly. When open-access to a resource is accompanied by worsening environmental conditions, and when these changes are either rapid or unforeseen or both, harvesting outcomes can be catastrophic. The recent history of the Peruvian anchovy fishery, described in Box 17.3, illustrates such an outcome. Another facet of open-access is also shown in Box 17.3. Overfishing does not usually happen because of ignorance; it tends to result from the forces of competition in conditions where access is poorly regulated. The New
England fisheries demonstrates that self-regulation on the part of fishermen may do little to overcome the consequences of open access.
17.5 Some more reflections on open access fisheries
The absence of enforceable private property rights does not necessarily imply that renewable resources will fail to be harvested in a rational or conservationist manner. In this respect it is important to distinguish between ‘common property’ and ‘open access’ resources. In the former private property rights are not vested in individuals but rather in communities. Here, relationships of trust and social norms may be sufficiently well
29

developed and entrenched to create patterns of resource exploitation that are both sustainable and rational for the group as a whole (see for example Bromley, 1999). In contrast, open access is usually taken to mean the absence of any binding such norms with a tendency for exploitation to take place under conditions of ‘free-for-all’ individualistic competition.
It is possible that any resource stock could be harvested to exhaustion, or a species driven to extinction, under open access. Indeed, we show later that this is true under almost any regime, including those with enforceable private property rights. Nevertheless, it remains true that open access conditions increase the probabilities of those outcomes occurring. Why is this? The main reason is that in these circumstances there is no collectively rational management of harvesting taking place. Even where what should be done is evident, an institutional mechanism to bring this about is missing. When you compare open access outcomes with those of private property regimes in the sections that follow, you will see that harvest rates are typically higher under conditions of open access; other things being equal, the greater are harvesting rates, the higher is the likelihood of extinction.
Another way of thinking about this is in terms of stock externality effects and economic inefficiency. Open-access harvesting programmes are inefficient because the resource harvesters are unable to appropriate the benefits of investment in the resource. If a single fisher were to defer the harvesting of some fish until a later period, all fishers would benefit from this activity. It would be in the interests of all if a bargain were made to reduce fishing effort by the industry. However, the conditions under which such a bargain could be made and not reneged upon are very unlikely to exist. Each potential bargainer has an incentive to free ride once a bargain has been struck, by increasing his or her harvest while others reduce theirs. Moreover, even if all existing parties were to agree among themselves, the open-access conditions imply that others could enter the market as soon as rents became positive. Open-access resources thus have one of the properties of a public good - non-excludability - and this alone is sufficient to make it likely that markets will fail to reach efficient outcomes.
30

Insert Box 17.3 near here.
17.6 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’ and the stock level may (but will not necessarily) be driven down to biologically dangerous levels.
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.
We shall now discuss a particular set of institutional arrangements that could overcome some of these dilemmas. These arrangements could be described as the private property fishery. This kind of fishery – and several variants of it – have been explored in depth in the fishery economics literature. However, discussions of the private property fishery rarely make explicit the institutional assumptions that lie behind it. It is important to do that, however, and so we shall now describe what the private property fishery is usually taken to mean (and the sense in which we shall be using it).
31

The private property fishery has the following three characteristics:
1.
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.
2.
Each firm is profit (or wealth) maximising.
3.
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 property rights? Within the literature there are several (sometimes implicit) answers to this question. We shall outline two of them. One view regards ‘the fishery’ as an aggregate of a large number of smaller individual fisheries.
Each of these sub-fisheries is privately owned by one firm that has property rights to the fish which are there 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. A second view regards 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.
Neither of these accounts is satisfactory as a statement of what actually does exist, nor what might realistically exist. The first faces problems in deciding how to specify ownership rights to migratory fish. Moreover, it could only be descriptively accurate if the fishery in question is a huge, highly spatially-aggregated, fishery. The researcher not want to study at this level of aggregation. The second concept – the coordinated fishery - seems problematic in that we rarely, if ever, find examples of such internally coordinated fisheries (except in the case of
9 The owners of any fishing firm may, of course, lease or sell their property rights to another set of individuals.
32

fish farming and the like). And even if one were to find examples, it is difficult to imagine that they would operate as competitive fisheries rather than as monopolies or cartels.
10
But to label one or both of these views as descriptively unrealistic is to miss the point somewhat. They should be thought of in “as if” terms. That is, we want a specification such that the industry behaves as if each firm has its own ‘patch’ of fishery that others are not permitted to exploit or as if it were coordinated in the way mentioned above. Given either of these as if assumptions, the researcher can then reasonably assume that owners undertake economically rational management decisions, and are in a position to make investment decisions confident in the belief that the returns on any investment made can be individually appropriated. This is what distinguishes a private property fishery fundamentally from an open access fishery.
An important benefit from thinking about property rights carefully in this way is the help it gives in developing public policy towards fishery regulation and management. If we are confident that a particular property right structure would bring about socially efficient (or otherwise desirable) outcomes, then policy instruments can be designed to mimic that structure. We will argue below that an Individual Transferable Quota (ITQ) fishing permit system can be thought of in this way.
17.6.1 The static profit maximising private property fishery model
As we explained in the Introduction, our analysis of the private property fishery proceeds in two steps. The first, covered in this section, develops a simple static model of a private property fishery in which the passage of time is not explicitly dealt with. In effect, the
10
In fact, two other variant of the private property fishery sometimes discussed in the literature are actually these: the monopoly fishery and the cartel fishery. However, given the fact that they are so uncommon in practice, we do not deal with those models in this text, except for a brief reference to monopoly fishery in Appendix 17.3.
33

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 in any arbitrarily chosen period 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. We shall demonstrate later that this analytical approach only generates wealth maximising outcomes if fishermen do not discount future cash flows. More specifically, the static private property fishery turns out to be a special case of a multi-period fishery model; the special case in which owners use a zero discount rate.
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; the open access entry rule
(dE/dt =

NB), which in turn implies a zero profit economic equilibrium, no longer applies.
Instead, owners choose effort to maximise economic profit from the fishery. This can be visualised with the help of Figure 17.4. As we did earlier, multiply both functions by the market price of fish. The inverted U shape yield-effort equation then becomes a revenueeffort equation. And the ray emerging from the origin now becomes PH = wE, with the righthand 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 E
PP
being parallel to the slope of the H = (w/P)E line.
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 17.4. 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
= 0.0360. To facilitate comparison, the numerical values of the steady-state equilibrium stock, harvest and effort under our baseline parameter assumptions, for both open access and static private property fishery, are
34

reproduced in Table 17.3. Under the assumptions we have made about functional forms, the static private property equilibrium will always lead to a higher resource stock level and a lower effort level than that which prevails under open-access. This is confirmed for our particular parameter assumptions, with the private property stock being three times higher and effort only half as large as in open access.
The steady state harvest may be higher, lower or identical. This is evident from inspection of Figure 17.4. For the particular set of parameter values used in our illustrative example, private property harvest is larger than open access harvest, as shown in the diagram. But it will not always be true that private property harvests exceed those under open access. For example, if P were 80 (rather than 200) and all other parameters values were those specified in the baseline set (listed in Table 17.2) then an open access fishery would produce H = 0.0375, the Maximum Sustainable Yield of the fishery! In contrast, a private property fishery would in those circumstances yield only H = 0.0281.
The source of this indeterminacy follows from the inverted U shaped of the yield-effort relationship. Although stocks will be higher under private property than open access, the quadratic form of the stock-harvest relationship implies that harvests will not necessarily be higher with higher stocks.
Insert Table 17.3 near here
Caption: Steady state solutions under baseline parameter value assumptions.
35

Table 17.3 Steady state solutions under baseline parameter value assumptions.
Steady state solutions:
Open access
Stock
Effort
0.200
8.000
Harvest 0.024
Static Private property
0.600
4.000
0.036
36

Box 17.4 Derivation of the static private property steady state equilibrium for our assumed functional forms
The derivation initially follows exactly that given in Section 17.3.2, with Equations 17.16 to
17.18 remaining valid here.. However, the zero profit condition (Equation 17.19) is no longer valid, being replaced by the profit maximisation condition: maximise NB = PH – wE (17.23)
Remembering that H = eES, and treating E as the instrument variable, this yields the necessary first order condition

(PeES)/

E =

(wE)/

E (17.24)
Substituting Equation 17.17 into 17.24 we have

(PeE S
M AX

 1
 e g
E

 )/

E =

(wE)/

E from which we obtain after differentiation
PeS
MAX
– 2PE S
M AX e
2 g
= w (17.25)
That is, the marginal revenue of effort is equal to the marginal cost of effort. This can be solved for E
*
PP
(the subscript denoting ‘private property’) to give
E
*
PP

1
2 g e

 1
 w
PeS
MAX

 (17.26)
Substitution of E
*
PP
into 17.17 gives
S
*
PP

1
2
PeS
MAX
 w
Pe and then using H = eES we obtain
11
H
*
PP

1
4 g w
2
S
M AX

P
2 e
2
S
M AX
(17.27)
(17.28)
11 In the Excel spreadsheets, an alternative (but exactly equivalent) version of this expression has been used to generate the Excel formulas, namely
H
*
PP

1
4 g ( PeS
M AX
 w )( PeS
P
2 e
2
S
M AX
M AX
 w )
37

End of Box 17.4
38

Table 17.4 Steady state open access and static private property equilibria compared.
Stock
Effort
Harvest
Open access
S *
 w
Pe
E *
 g e

 1
 w
PeS
MAX


H *
 gw
Pe

 1
 w
PeS
MAX


Static Private property
S
E
*
PP
*
PP


1
2
1
2
PeS g e
H
*
PP

1
4 g
MAX

 1

Pe
 w w
PeS
MAX

 w
2
S
M AX

P
2 e
2
S
M AX
39

17.6.2 Comparative statics
For convenience, we list in Table 17.4 the expressions obtained in earlier sections for the steady state equilibria of E, H and S. We can use these expressions to make qualitative predictions about the effects of changing a particular parameter on the equilibrium levels of
S*, E* and H*. Doing this is known as comparative statics. For example, how will S* change as w rises or as P increases? Inspection of the formula in the top left cell shows that open access S* will increase if w increases and will decrease if P increases. This is also true in the case of the static private property steady state, as can be seen by inspection of the top righthand side cell.
Insert Table 17.4 near here
Caption: Steady state open access and static private property equilibria compared.
Where the sign of a relationship can not easily be found by inspection, we may be able to obtain it from the appropriate partial derivative. For example, although it is easy in this case to confirm by inspection that a rise in w will increase open access S*, this inference is corroborated by the fact that the partial derivative of S* with respect to w is 1/Pe. As P and e are both positive numbers, the partial derivative 1/Pe is also positive. Sometimes, of course, the direction of an effect can not be signed unambiguously; this should usually be evident by inspection of the partial derivative.
Table 17.5 lists the signs of these effects from the appropriate partial derivatives. A + sign means that the derivative is positive, a - sign means that the derivative is negative, 0 means that the derivative is zero, and ? means that no sign can be unambiguously assigned to the derivative (and so we cannot say what the direction of the effect will be without knowing the actual values of the parameters that enter the partial derivative in question). Note that variations in

have no effect on any steady-state outcome (although it does effect how fast, if indeed at all, such an outcome may be achieved).
40

Insert Table 17.5 near here
Caption: Comparative static results.
41

Table 17.5 Comparative static results.
OPEN ACCESS
S*
E*
H*
STATIC PRIVATE PROPERTY
S
*
PP
E
*
PP
H
*
PP
P
-
-
+
?
+
+ w
+
-
?
+
-
-
E
-
-
?
?
?
+ g
0
+
+
0
+
+
0
0

0
0
0
0
42

17.6.3 The present value maximising fishery model
The present value maximising fishery model generalises the model of the static private property fishery. In doing so it formulates a model that has a more sound theoretical basis and generates 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 value of the fishery. We shall refer to this value as the present value (PV) of the fishery. In this section, we outline how a model of a present-value maximising fishery can be set up, state the main results, and provide interpretations of them. 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 in this chapter.
As in previous sections of the chapter, we assume that the market price of fish is a constant, exogenously given number. Moreover, as before, the market is taken to be competitive. However, Appendix 17.3 will also go through the more general case in which the market price of fish varies with the size of total industry catch, and will briefly examine a monopolistic fishery.
It will be convenient to regard harvest levels as the instrument (control) variable. To facilitate this, we specify fishing costs as a function of the quantity harvested and the size of the fish stock. Moreover, it is assumed that costs depend positively on the amount harvested and negatively on the size of the stock.
12, 13
That is
12
The reader may be confused about our formulation of the harvest cost function. In an earlier section, we wrote C = C(E), Equation 17.9. But note that we have also assumed that H
= H(E,S), Equation 17.6. If 17.6 is written as E in terms of H and S, and that expression is then substituted into 17.9, we obtain C = C(H,S). It is largely a matter of convenience
43

C t

C ( H t
, S t
) C
H

0 , C
S

0
The initial population of fish is S
0
, the natural growth of which is determined by the function G(S). The fishery owners select a harvest rate for each period over the relevant time horizon (here taken to be infinity) to maximise the present value (or wealth) of the fishery, given an interest rate i. Algebraically, we express this as
Max


0

PH t
C( H t
, S t
)
 e
it dt subject to dS
= G( dt
S t
) H t and initial stock level S(0) = S
0
. The necessary conditions for maximum wealth include p t

P


C ( H , S )

H t dp t dt
 ip t
 p t dG ( S )
 dS t

C ( H , S )

S t
(17.29)
(17.30) whether we express costs in terms of effort or in terms of harvest and stock. In our discussion of open access equilibrium, we chose to regard fishing effort as a variable of interest and did not make that substitution. In this section, our interest lies more in the variable H and so it is convenient to make the substitution. But the results of either approach can be found from the other.
13 There is another issue here that we should mention. The costs of fishing should include a proper allowance for all the opportunity costs involved. For land-based resources, the land itself is likely to have alternative uses, and so its use in any one activity will have a land opportunity cost. For fisheries however, there is rarely an alternative commercial use of the oceans, and so this kind of opportunity cost is not relevant. However, from a social point of view there may be important alternative uses of the oceans (for example, as conserved sources of biodiversity). Hence a difference can exist between costs as seen from a social and a private point of view.
44

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, 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 p t
is a shadow price, which measures the contribution to wealth made by an additional unit of fish stock at the wealth-maximising optimum. We call it the net price of fish. Two properties of this 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 a steady state equilibrium). In general, therefore, it is necessary to attach a time label to this net price.
Equation 17.29 defines the net price of the resource (p t
) as the difference between the market price and marginal cost of an incremental unit of harvested fish. Equation 17.30 governs the behaviour over time of the net price, and implicitly determines the harvest rate in each period. This equation is interpreted in the discussions below.
17.6.3.1 Steady state equilibrium in the present value maximising fishery
Let us investigate the properties of a PV maximizing fishery steady-state equilibrium. An explanation of how such a state may be achieved is left to Appendix 17.3. In a steady state all variables are unchanging with respect to time, so dp/dt = 0. We have G(S) = H and the optimizing conditions 17.29 and 17.30 collapse to the simpler forms p

P


C ( H , S )

H ip
 p dG ( S ) dS


C ( H , S )

S
(17.31)
(17.32)
17.6.3.2 Interpretation of Equation 17.32
How is the present value of profits maximised? The key to understanding profitmaximising 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
45

to consider the owner’s decision about whether to marginally change the amount of fish harvesting currently being undertaken.
Choosing not to harvest some fish 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.
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 choosing not to harvest an incremental unit, the fisher incurs an opportunity cost in holding a stock of unharvested fish. Holding these units sacrifices an available return. The marginal cost of the investment is ip. How is this obtained? Sale of the harvested fish would have led to revenue which can be invested at the prevailing rate of return on capital, i. The foregone revenue is equal to the net price of the resource, p. Note that this is actually a net revenue as p = P - C
H
, where P is the market price and C
H
=

C/

H is the marginal cost of one unit of the harvested resource. However, 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 the resource investment (not harvesting the incremental unit). There are two categories of benefit:
1.
As a consequence of an additional unit of stock being added, total harvesting costs will be reduced by the quantity C
S
=

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.
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. That is:
46

ip < dG dS p -

C

S
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: ip > dG dS p -

C

S
These imply the asset equilibrium condition, Equation 17.32. 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 17.32 by the net price p to give i = dG dS
-

C

S p
(17.33)
Equation 17.33 is one (steady-state) 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
Equation 17.33 is an implicit equation for the unknown PV maximising equilibrium value of S. Solving 17.33 for S gives an analytical solution for the equilibrium stock. (This solution is given in Appendix 17.3.) Given that, expressions for the PV optimal solutions
47

for H and E can be obtained using the biological growth function and the fishery production function, Equation 17.6.
We have used the spreadsheet Comparative ststics.xls
to calculate 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. These equilibria are listed in Table 17.6. 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 latter two models, equilibrium values do not vary with the interest rate.
Three important results are evident from an inspection of the data in Table 17.6:
1.
In the special case where the interest rate is zero, the steady state equilibria of a static private property fishery and an PV maximising fishery are identical. For all other interest rates they differ.
2.
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.
3.
As the interest rate becomes arbitrarily large, the PV maximising outcome converges to that of an open access fishery.
At the moment we just note these results. We shall explain them later.
Insert Table 17.6 near here (a new Table)
Caption: Table 17.6 Steady state equilibrium outcomes from the illustrative spreadsheet model for baseline parameter values in (a) a static private property fishery, (b) open access fishery and (c) a PV maximising fishery with various interest rates.
48

0.1
0.2
0.3
0.4
0.5
Table 17.6 Steady state equilibrium outcomes from the illustrative spreadsheet model for baseline parameter values in (a) a static private property fishery, (b) open access fishery and
(c) a PV maximising fishery with various interest rates.
Static private fishery

PV maximising fishery Open access fishery
 i
0.0
S
0.6
E
4.0
H
0.036
S E H S
0.6000 4.0000 0.0360 0.20
E
8.0
H
0.024
0.6
0.6
0.6
0.6
0.6
4.0
4.0
4.0
4.0
4.0
0.036
0.036
0.036
0.036
0.036
0.4239 5.7607 0.0366 0.20
0.3333 6.6667 0.0333 0.20
0.2899 7.1010 0.0309 0.20
0.2677 7.3333 0.0293 0.20
0.2527 7.4734 0.0283 0.20
8.0
8.0
8.0
8.0
8.0
0.024
0.024
0.024
0.024
0.024
0.6
0.7
0.8
0.9
1.0
0.6
0.6
0.6
0.6
0.6
4.0
4.0
4.0
4.0
4.0
0.036 0.2434 7.5660 0.0276 0.20 8.0
0.036 0.2369 7.6314 0.0271 0.20 8.0
0.036 0.2320 7.6798 0.0267 0.20 8.0
0.036 0.2283 7.7171 0.0264 0.20 8.0
0.036 0.2253 7.7467 0.0260 0.20 8.0
0.024
0.024
0.024
0.024
0.024
10.0 0.6 4.0 0.036 0.2024 7.9759 0.0242 0.20 8.0 0.024
100.0 0.6 4.0 0.036 0.2002 7.9976 0.0240 0.20 8.0 0.024

Static private fishery and open access fishery steady state equilibrium values do not vary with the interest rate. These equilibrium values have been copied into every row for purposes of comparison.
Source of data: Comparative statics.xls
49

17.7.1 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. But in such cases (which are rather unlikely case in general), 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 is an interesting result. It 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. At the present value-maximising resource stock size (which we denote by S
PV
), i = dG/dS. It is clear from this diagram that as the interest rate rises, the profit-maximising stock size will fall.
Furthermore, it is also clear from inspection of Figure 17.7 that if i > 0 and

C/

S =0, then S
PV
is less than the maximum sustainable yield stock, S
MSY
. 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.
A further deduction can be made from Equation 17.32, and also illustrated with the help of Figure 17.7. If harvesting costs do not depend on the stock size, and decision makers use a zero discount rate, 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 which leads to the
50

maximum sustainable yield, S
MSY
. (In Figure 17.7, the straight line is horizontal and is tangent to the growth function at S
MSY
.) 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. This result is of little practical relevance in commercial fishing, however, as it is not conceivable that owners would select a zero discount rate.
Insert Figure 17.7 here (See the pictures file “Chapter 17 Pictures.doc”).
Caption: Present value maximising fish stocks with and without the dependence of costs on stock size, and for zero and positive interest rates.
However, for many marine fisheries, it is likely to be the case in practice that harvesting costs will rise as the stock size falls. This property is built into the equations used in the illustrative example in this chapter. We have found that when costs depend negatively on the stock size, the present value-maximising stock is higher than it would be otherwise. This 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 (S
PV
) the dependence of costs on the stock size.
Note from Equation 17.32 that as i rises, it will be necessary for dG/dS to fall to maintain the equality. That requires a smaller stock size. Hence we find that i and S will be negatively related for a PV maximising fishery, whether or not costs depend on the stock size. This is consistent with our findings in Table 17.6.
Finally, note that in the general case where a positive discount rate is used and

C/

S is non-zero, 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
51

relative magnitudes of i and (

C/

S)P. Indeed inspection of Figure 17.7 shows the following: i
 
(

C /

S ) P

S
*
PV i
 
(

C /

S ) P

S
*
PV i
 
(

C /

S ) P

S
*
PV

S
M SY

S
M SY

S
M SY
This confirms an observation made earlier in this chapter: a maximum sustainable yield
(or a stock level S
MSY
) 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
17.7.2 Dynamics in the PV maximising fishery
Our analysis of the open access fishery showed complex patterns dynamic behaviour. 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, 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).
These various dynamic processes reflect the unplanned, uncoordinated and myopic 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. Owners in private fisheries are either assumed to coordinate their behaviour in a rational maximising way, or the
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 a MSY solution.
52

individual private property assumption 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 17.3. 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). This is characterised by a simple rule
H t
 
0
H
M AX when when
S
S t t

S
*
PV

S
*
PV
That is, 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
Matters are more complicated where the price is endogenous, depending on the harvest rate. Relevant results, obtained using phase-plane analysis, are presented below in Appendix 17.3.
17.8 Bringing things together: the open access fishery, static private property fishery and PV maximising fishery models compared
17.8.1 Steady state characteristics
For pedagogical reasons, we have used a variety of methods of deriving results for the three fishery models investigated in this chapter. In addition, a variety of graphical techniques have been employed. It turns out that all of these can be related to one another
15 Our Excel workbook does not exhibit this MRAP form of adjustment for a private (nor for a PV maximising) fishery. Instead, the workbook uses a gradual 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.
53

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.
Let us look first at the difference between the two private property fishery models.
Earlier remarks have – without proof - stated that the static private property fishery is nothing more than a special case of the PV fishery – the special case being that 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. One way of verifying this assertion would be to substitute 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 17.4.
To understand intuitively why this happens, return to the dynamic asset-equilibrium condition given by Equation 17.32: ip
 p dG dS


C

S
When i = 0, this collapses to dG dS p


C

S
The left-hand side of this expression is the marginal revenue (with respect to stock changes) and the right-hand side is the marginal cost (with respect to stock changes). Profitmaximising equilibrium (in the static private property fishery model without discounting) requires that these be equal. This is, of course, the standard result for any static profitmaximising model. Indeed, if the reader obtains the derivatives dG/dS and

C/

S for our particular model equations, substitutes these into the expression above, and then solves for the equilibrium level of stock, she will find that the result is identical to that derived in
Section 17.4 (and listed in the column labelled ‘Static Private Property’ in Table 17.4). We recommend that you try this exercise; it can be checked against our solution, given in
Appendix 17.2.
54

Finally, is there any way in which we can bring the open access fishery model 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. Hence it is appropriate to modify a statement made earlier. 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.
55

Box 17.3 A story of two fish populations
One species of fish - the Peruvian anchovy - and one group of commercial fish - New
England groundfish – provide us with case studies of the mismanagement and economic inefficiency which often characterise the world's commercial fisheries. In this box, we summarise reviews of the recent historical experiences of these two fisheries; the reviews are to be found in WR(1994), chapter 10.
Peruvian anchovy are to be found in the Humboldt upswelling off the west coast of South
America. Upswellings of cold, nutrient-rich water create conditions for rich commercial fish catches. During the 1960s and 1970s, this fishery provided nearly 20% of the world's fish landings. Until 1950, Peruvian anchovy were harvested on a small scale, predominantly for local human consumption, but in the following two decades the fishery increased in scale dramatically as the market for fishmeal grew. The maximum sustainable yield (MSY) was estimated as 9.5 million tonnes per year, but that figure was being exceeded by 1970, with harvests beyond 12 million tonnes. In 1972, the catch plummeted.
This fall was partially accounted for by a cyclical natural phenomenon, the arrival of the El
Nino current. However, it is now recognised that the primary cause of the fishery collapse
(with the catch down to just over 1 million tonnes in the 1980s) was the conjunction of overharvesting with the natural change associated with El Nino. Harvesting at rates above the MSY can lead to dramatic stock collapses that can persist for decades, and may be irreversible (although, in this case, anchovy populations do now show signs of recovery).
The seas off the New England coast have been among the most productive, the most intensively studied and the most heavily overfished in the world since 1960. The most important species in commercial terms have been floor-living species including Atlantic cod, haddock, redfish, hake, pollock and flounder. Populations of each are now near record low levels. Although overfishing is not the only contributory factor, it has almost certainly been the principal cause of stock collapses. The New England fisheries are not unusual in this; what is most interesting about this case is the way in which regulatory schemes have failed to achieve their stated goals. In effect, self-regulation has been practised in these
56

fisheries and, not surprisingly perhaps, regulations have turned out to avoid burdening current harvesters. This is a classic example of what is sometimes called ‘institutional capture’; institutions which were intended to regulate the behaviour of firms within an industry, to conform with some yardstick of ‘the common good’, have in effect been taken over by those who were intended to be regulated, who then design administrative arrangements in their own interest. The regulations have, in the final analysis, been abysmal failures when measured against the criterion of reducing the effective quantity of fishing effort applied to the New England ground fisheries.
Long-term solutions to overfishing will require strict quantity controls over fishing effort, either by direct controls over the effort or techniques of individual boats, or through systems of transferable, marketable quotas. We investigated some of these instruments in
Chapter 7 and do so further later in this chapter.
END OF BOX 17.3
57

17.9 Socially efficient resource harvesting
As with all resource allocation decisions, there can be no guarantee that privately maximising decisions will be socially efficient (let alone socially optimal). In this section, we review some of the reasons why divergences may take place. First of all, though, we establish the conditions that must be satisfied for socially-efficient harvesting.
Let r be the social consumption discount rate. Socially efficient harvesting emerges as the solution to the following maximisation problem
Max


0

BS( H t
) CS( S t
, H t
)
 e
rt dt subject to dS
= G( dt
S t
) H t and initial stock level S(0) = S
0
. In this expression BS(H) denotes the social benefits as a function of quantity of fish landed, and CS(S, H) denotes the social costs as a function of stock size and harvest rate. The current-value Hamiltonian, L, for this problem is
L ( H , S ) t

BS ( H t
)

CS ( S t
, H t
)
  t
( G ( S t
)

H t
) where
 t
is the (social) shadow net price of a unit of the stock of the renewable resource.
The necessary conditions for a maximum include
( i )

L

H t t

0
 dBS
 dH t

CS

H t
  t
(ii) d
 dt t
= r
 t
-
 t dG d S t
+

CS

S t
(17.34)
(17.35) and the resource net growth equation
(iii) dS dt
= G( S t
) R t
58

We continue to assume that the market price of fish is exogenously fixed at P. Now consider also the following additional conditions:
(i) The market price, P, correctly reflects all social benefits (which then, together with the assumption of exogenously given prices, implies dBS/dH = P)
(ii) There are no fishing externalities on the cost side so that CS(S,H) = C(S,H).
(iii) The private and social consumption discount rates are identical, i = r.
If these additional conditions are satisfied, and are imposed on Equations 17.34 and 17.35, the first-order conditions for social efficiency are identical to those for the PV maximising fishery (Equations 17.29 and 17.30) except for the fact that the shadow price is denoted as

in the former and p in the latter. But since shadow prices are themselves solutions to optimising problems, and the problems are in this case identical, it follows that p and
 must also be identical. Hence the PV maximising fishery is socially-efficient under the set of conditions we have just described. But, of course, it also follows that if one or more of those conditions is not satisfied, private fishing will not be socially efficient. We investigate next some reasons why such a divergence might arise.
17.9.1 Externalities in the benefits function
The first case is that in which social benefits depend not only on the size of the resource harvest but also on the level of the resource stock. For many biological species that are harvested, intentionally or unintentionally (as bycatch), it is evident that society does place a value on the existence of these species and is concerned about the number of them that do exist. This is clearly true for many large animals such as big cats, whales, and apes, and surely extends much more widely than that. In this case, the social welfare maximising problem should be generalised to
Max


0

BS( H t
, S t
) CS( H t
, S t
)
 e
rt dt subject to dS
= G( dt
S t
) H t
59

Note that S now enters the benefits function (in which it had not appeared previously).
Intuition suggest that the optimal solution to this model will be different to that obtained from the model where benefits only depend on the harvest rate, H. This intuition is correct.
Assuming that utility is an increasing function of S, the solution will in general be one in which the optimal stock level is higher, reflecting the positive utility that the resource stock generates. We invite the reader to work through the maths to obtain this result. To check your analysis, we have placed such a derivation in a separate document in the
Additional Materials , Stock Dependent Utility.doc
.
More generally, society is likely to have multiple objectives which are not well represented by the private harvester’s own objective function (which will tend to be dominated by catch quantity considerations). This is very important for many terrestrial resources, particularly woodlands and forests as we shall see in the next chapter. But is also applies to marine resources. For example, society may have an interest in the maintenance of population diversity or genetic diversity; it may be willing to pay a larger risk premium to ensure high resistance to disease among marine organisms; or it may prefer to maintain stock levels much higher than would private harvesters – at a safe minimum standard – in response to uncertainty and the threats of catastrophic change. All these could be thought of as additional arguments that would appear in the social objective function (but which would not usually enter private profit functions).
Alternatively, we might choose to model some or all of externalities in terms of costs.
That is, private owners are likely to fail to make adequate provision for the full social opportunity costs of their activities, and so externalities are not being internalised. This leads us to the next category of sources of inefficiency.
17.9.2 Externalities in the fishery production function
A second source of social inefficiency arises from externalities operating through the fishery production function. There are two important types of harvesting externality. First, it often happens that resource harvesting inadvertently destroys other species. Beam
60

trawling, for example (in which a net is weighed down to the sea bed by heavy beams and is trawled along the sea bed to catch bottom-feeding fish such as cod), causes immense damage to other seabed creatures, and can cause those populations to collapse. This is a classic externality problem and it is clear that outcomes are most unlikely to be efficient in such cases. Some form of regulation of fishing practices seems to be appropriate here. We return to this matter later.
A second kind of externality is often known as a ‘crowding’ diseconomy. Suppose that each boat’s harvest depends on its effort and on the effort of others. Then each boat's catch imposes a contemporaneous external cost on every other boat. In effect, boats are getting in each others’ ways. When any boat is fishing, the costs of harvesting a given quantity of fish become higher for all other boats. This externality drives the average costs of fishing for the fleet as a whole above the marginal costs of an individual fisher.
If a crowding effect of this kind exists, the function C(H,S) from the point of view of an individual boat operator may differ from the function CS(H,S) from the social point of view. Whether it does or not depends on what institutional conditions apply. If the private
(or PV maximising) fishery is one in which effort is somehow or other coordinated in the common interest, then it would be sensible to assume that the size of the fishing fleet as a whole (and the spatial patterns of fishing) would be optimally chosen. The optimal size of fleet would balance the additional benefits of extra boats against the additional external costs of extra boats. The crowding diseconomies become internalised in this way, leading to efficient outcomes. Alternatively, if the fishery were in private property ownership but were carefully and effectively regulated , then it is conceivable that such regulation might also internalise the externality.
Under conditions of open access, there is virtually no possibility that crowding effects would be internalised by the actions of fishermen alone. The kind of coordination we referred to above cannot happen in the competitive struggle to grab fish. Almost certainly, the unregulated industry would consist of more harvesters and more harvesting capital
61

than is economically efficient. However, it is still possible that some form of regulation might achieve the required coordination. We discuss this further below.
It is important to note that the crowding diseconomy we have just referred to is entirely different from the ‘stock externality’ effect which arises in open access. Crowding externalities are contemporaneous; stock externalities are intertemporal. The latter exist when the taking of fish today imposes additional costs in the future by virtue of the reduced future stock size. We have already remarked that this kind of externality fundamentally distinguishes the open access and private (and PV maximising) fishery cases. Realistically or not, modellers typically assume that the stock externality will not be internalised in the former case but will be in the latter case. The first part of this assumption is surely true. Indeed, there is ample evidence that fishing effort is massively excessive and inefficient in many open-access fisheries throughout the world. We give some evidence on this later in Box 17.7.
It is of interest to note that the Schaeffer form of fishery production function we have used in this chapter (H = eES) rules out the existence of crowding externalities. For a given stock size, H will change in proportion to changes in E. In other words, the marginal product of effort is constant, precluding crowding effects.
17.9.3 Difference between private and social discount rates
Maximised social and private net benefits also diverge when social and private discount rates differ. We have already remarked that this is one way of explaining the excessive harvesting that takes place in open access fisheries, but it should be clear that the problem is more wide-reaching than that case alone.
17.9.4 Monopolistic fisheries
The existence of monopoly ownership of a fishery may also generate inefficient outcomes.
A resource market is monopolistic if there is one single price-making harvester. It is known from standard microeconomic theory that marginal revenue exceeds marginal cost
62

at a monopolistic market equilibrium.
16
A monopoly owner would tend to harvest less each period, and sell the resource at a higher market price, than is socially efficient.
Therefore, if a renewable resource were harvested under monopolistic rather than competitive conditions, an economically inefficient harvesting level may result. The qualifier ‘may’ in the previous sentence arises from second-best considerations. In a world where there are other market failures pushing harvest rates to excessive levels, monopoly harvests may be closer to the second-best efficient allocation than those from
‘competitive’ PV maximising fisheries.
17.10 A safe minimum standard of conservation
Our discussion of the ‘best’ level of renewable resource harvesting has focussed 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 primary forests) or jeopardise an environmental system itself (such as a wildlife reserve containing extensive biodiversity) then the criterion of efficiency may be insufficient or inappropriate.
We observed earlier that 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.
16 Similar conclusions also apply to imperfectly competitive markets in which a small number of harvesters dominate the industry.
63

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 recent 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 17.8. The function shows the resource stock level that will be available in period t + 1 (S t+1
) for any level of stock that is conserved in period t (S t
). 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
Figure 17.8 Insert near here (located in Picturesforchapter17.ppt
)
Caption: A safe minimum standard of conservation.
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
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.
64

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
0
line (labelled 'slope = 1') constitute sustainable stocks. The minimum sustainable level of stock conservation is labelled S
MIN
.
The efficient level of stock conservation is S t
*
. 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 S t
* 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 17.7.1. That is, it is the form that Equation 17.32 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 R t
*
.
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 17.8. 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 S
~
M S is conserved in each period, the condition for perpetual sustainability of the stock will be maintained. We might regard S
~
M S 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 S
MIN
is an appropriate minimum stock in the absence of uncertainty, S
~
M S takes account of uncertainty in a particular way.
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.
65

In fact, S
~
M S 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
R
MIN
is judged to be that minimum required level of resource consumption. Then Randall and Farmer's safe minimum standard of conservation is S M S . If the stock in period t is kept from falling below this level then, even in the worst possible case, R
MIN
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 S M S . 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.
17.11 Models of biological interaction among species
So far in this chapter, our bio-economic models have considered just one population of a single biological species interacting with its natural environment (the biological component) and human harvesting of that population (the economic component). There are two circumstances in which it is sensible to develop a bio-economic model in this way. First, if the biological growth behaviour of a single population of interest is independent of that of any
66

other population or species. In that case, there is no useful information to be brought into the modelling exercise by jointly studying the biological growth of this species and any other. It is most unlikely that any population will be strictly independent of all others, but it may be approximately so in some cases; as a practical matter, this might justify its analysis in isolation.
Second, we may accept that the behaviour of the population of interest is affected in many ways by many other populations or species, but with no single relationship being particularly dominant. In that case, a researcher might choose to regard all other populations/species as constituting (part of) that population’s environment. The modeller would then give greater attention to modelling that environment – perhaps by treating it as being stochastic or uncertain in some way – but would not develop models of interaction between specific species or populations.
But there are circumstances where proceeding in this way is not appropriate. Of most importance are those cases where particular species interact in important ways. These relationships may be as predators and prey, as with big cats and herbivore mammals; they may involve parasitism in which one species inhibits the growth of another; they may exhibit mutualism (or symbiosis) where each species requires the other for its survival (such as some bacteria and hosts); or they may involve one of several other types of relationship.
19
Wherever the relationship between two or more populations is significant, the researcher is throwing away important relevant information by ignoring that relationship. In this section we investigate some models of biological interaction between populations, and then briefly show
19
Ecologists recognise at least four other forms of biological interaction (see, for example,
Dajoz, 1977): competition , where species are in competition for scarce resources; cooperation , where symbiotic relationships are chosen for common purposes, such as security against predation; commensalism , where one species benefits but the other neither suffers nor benefits; and amensalism , where one species is unaffected but the other suffers from a relationship.
67

how such models can be used to shed some light on the issues of biodiversity and sustainability.
20
It will be helpful to begin by outlining a classification of models of biological interaction developed by Shone (1997). We shall also investigate some of the examples he considers.
Suppose that there are two populations of different species, labelled H and S. Let h denote the net contribution of a ‘representative’ individual in H to the magnitude of population H, and let s denote the net contribution of one individual in S to the size of population S. We specify h and s in the following, relatively general, ways. h
 α  β H t
 χ S t s
 δ  ε
S t
 φ
H t
(17.36)
(17.37)
In Equations 17.36 and 17.37, the parameters

and

are population-specific natural growth coefficients. They correspond to the parameter g that we called the intrinsic growth rate of a fish population in the logistic growth model examined earlier. Parameters

and

are population-specific crowding (or self-limiting) coefficients, which can be interpreted in the following ways:

If

and

are both negative, both species are subject to (intra population) crowding effects, and so there will be limits to which the population size could grow, even in the absence of any limits imposed by relationships with other species.

If

and

are both positive, the fertility of each population increases the larger is the size of that population. There is a kind of mutualism within a population.
Mixed cases are, of course, also possible. The parameters

and

relate to the interaction between different species or populations. In particular:
20 You may have noticed that the text sometimes refers to species interaction and sometimes to population interaction. Generally, it is the former which is relevant, provided we take care to note that the populations are of different species. In some special cases, the populations of interest may equate with entire species, in which case either is appropriate.
68

 
,

both negative implies inter species competition
 
,

both positive implies mutually beneficial interaction (mutualism or cooperation)
 
,

of opposite sign implies a predator- prey relationship (the population with the positive parameter being the predator, and with the negative being the prey)
In the absence of migration, the growth of each population is given by the product of the net contribution of a representative individual and the number of individuals in the population.
That is
H
 hH t
S
 sS t
By imposing restrictions on this general specification, various particular models of biological interaction are generated. We consider three of these.
17.11.1. (a) Competition between species with no crowding or self-limitation
This model is obtained from the general specification (17.36 and 17.37) by imposing the restrictions {

> 0,

= 0,

< 0} and {

> 0,

= 0,

< 0}. The population growth models may then be written as

 a
 bS t

H t
(a, b > 0)

 c
 dH t

S t
(c, d > 0)
(17.38)
(17.39)
What do these equations tell us? First, both intrinsic growth rate coefficients (here a and c) are positive, so in isolation each population becomes larger over time. Indeed, the absence of any self-limitation (strictly speaking, the property that

= 0 and

= 0) means that in isolation the populations would grow without bounds. However, the populations are not in isolation; the negative coefficients in the terms -bS and -dH show that each species is in competition with the other for scarce resources. In Equation 17.38, for example, the term –bS t
means that the net contribution to the H-population of one individual member of H is negatively related to the size of the other population, S. As this relationship is true for both populations, we have a ‘species competition’ model.
69

There are three possible outcomes to this interactive relationship. It may be helpful to think of the example of two competing garden plants (or perhaps a chosen plant and a weed) to visualise these. Two of these are equilibrium outcomes. The first – a trivial (uninteresting) solution - is that the equilibrium stock of each is zero.
21
A second equilibrium outcome is that there is a positive stock of both, in which the competitive force that each exerts on the other is completely balanced. But such an equilibrium will be a ‘knife-edge’, or unstable, equilibrium.
Any event leading one population to become larger than its equilibrium level will precipitate a chain reaction sequence driving the other to zero. This points to the third kind of outcome: one species will become increasingly dominant, and the stock of the other will be driven towards zero. 22
It will be helpful to give a numerical example. We use Example 12.4 in Shone (1997) for which Equations 17.38 and 17.39 take the particular forms


4

3 S t

H t


3

H t

S t
(17.40)
(17.41)
By definition, an equilibrium occurs where the two population levels are simultaneously constant, that is, H

0 and S

0 . Imposing these equilibrium conditions on 17.40 and 17.41 gives
0


4

3 S t

H t
0


3

H t

S t
21 Perhaps the gardener pulls out all members of both in exasperation!
22 Mathematically, this kind of outcome is a not an equilibrium outcome in this model, as no fixed point will ever be reached. It is worth noting that this property comes from the fact that this model has no (intra-population) crowding or self-limitation; hence there is no fixed point to which it can grow.
70

which yields the pair of solutions {H = 0, S = 0} and {H = 3, S = 4/3}. These are shown in
Figure 17.9 by the intersection of the two straight lines representing H

0 and S

0 . The arrows show the directions in which the two populations will move (defined by Equations
17.40 and 17.41) from any arbitrarily-chosen starting point. If sequences of these directional arrows are connected together, we obtain dynamic time paths for the two populations. Several such paths are shown in the diagram by the heavily-drawn curved lines.
23
It is evident from that diagram that our previous conclusions are valid. A steady state equilibrium with positive values of H and S does exist. However, this is an unstable knife edge equilibrium; any deviation from this equilibrium will lead to the populations levels diverging even further, with one of the species becoming increasingly dominant and the other becoming ever-closer to zero with the passage of time. Moreover, almost all initial starting points fail to find that equilibrium and so – in the absence of deliberate management – the equilibrium is most unlikely to be achieved.
Insert Figure 17.9 near here. (See the pictures file “Picturesforchapter17.ppt”).
Caption: Equilibrium and dynamics of the competition with no crowding model.
17.11.1. (b) Predator-prey model with no population-specific crowding or self-limitation
Suppose we wish to examine biological interaction between populations of a predator and a prey, but still retain the property that neither population is subject to crowding or self-
23
Figure 17.9 is an example of what is called phase plane analysis, a technique that is explained in the file Phase.doc
in the Additional Materials . Also to be found there is the
Maple file interaction.mws
that was used to generate Figure 17.9. A discrete time counterpart to the continuous time specification used for this model is given by:
H t

1

H t
S t

1

S t
 
4

3 S t

H t


3

H t

S t
The file Interaction.xls
(worksheet COM), also in Additional Materials , shows how this discrete time model can be analysed by means of an Excel spreadsheet.
71

limitation. This can be done by imposing the restrictions {

> 0,

= 0,

< 0} and {

< 0,

=
0,

> 0}on the general specification (equations 17.36 and 17.37). The resulting model – first developed by Lotka (1925) and Volterra (1931) – takes the form

 a
 bS t

H t
(a, b > 0)


 c
 dH t

S t
(c, d > 0)
In this predator-prey model, H is the prey population and S is the predator population. In the differential equation for H, the positive intrinsic growth rate a implies that in the absence of predation the prey population would increase through time. Moreover, the restriction

= 0 implies the absence of any self-limiting factor, and so this growth process would be without bounds: the prey population would expand indefinitely.
However, the prey population is constrained by its relationship with the predators. The term – bHS, found by multiplying out the terms within and outside brackets, shows that the prey population falls because of predation, and that this effect is larger the greater is the predator population.
In contrast, the negative intrinsic growth coefficient for the predator implies that the predator depends on the prey population for its existence: in the absence of H, the population
S would collapse to zero. However, the presence of prey acts to increase the predator population; this effect is represented by the interaction term dHS.
What kind of outcome would one expect in this situation? Intuition suggests that we might find a balance between predator and prey populations. To see why, note that an equilibrium will, by definition, exist where both predator and prey populations are constant. The prey population will be constant where net recruitment (aH) equals net losses due to predation
(bHS). The predator population will be constant when natural population loss (cS) is just balanced by growth associated with presence of the prey population (dHS). This gives us two equations; knowing values of the parameters a, b, c and d, these two equations could be
72

solved for the two unknowns, the equilibrium levels of H and S. By way of example, we take the parameter values used in Example 12.5 of Shone (1997), which are a = 2, b = 1/100, c = 2 and d = 1/50. The two equilibrium equations are then
2 H

( 1 / 100 ) HS and
2 S

( 1 / 50 ) HS which yield the equilibrium solution H
*
= 100 and S
*
= 200.
24
However, this intuition and ‘steady-state algebra’ both fail to reveal one of the important features of this model. Except fortuitously, or through deliberate management, the equilibrium outcome H
*
= 100 and S
*
= 200 will never be realised. Instead, what will happen is that populations of both H and S will continually fluctuate, cycling above and below those steady state levels but not actually converging to them.
25 This can be observed by examining
Figure 17.10.
26
Insert here Figure 17.10 (See the pictures file “Picturesforchapter17.ppt”).
Caption: Equilibrium and dynamics in the LV predator-prey model.
17.11.1. (c) Predator-prey model with population-specific crowding
24
There is also a second, trivial, solution H = 0 and S = 0.
25
Strictly speaking, this is only true in continuous time models. If a discrete time counterpart model is examined, cyclical behaviour will also be observed, but it will be explosive, with cycles of increasing amplitude (until the system collapses). This can be seen by examining sheet LV in the Excel file Interaction.xls
in the Additional Materials .
26 Figure 17.10 was generated using the Maple file Interaction.mws. A discrete time version can be found in the Excel file Interaction.xls
(sheet = LV), found in the Additional Materials to Chapter 17.
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The cycling, non-convergent dynamic behaviour of the model examined in the last subsection is a mathematical property of the set of differential equations (and the associated parameter restrictions) which underlies the L-V predator-prey model. Many biologists regard these dynamic properties as being inconsistent with the observed evidence, or feel that they are overly restrictive. Various generalisations to the predator-prey model have been developed.
One such generalisation involves introducing upper limits to the population sizes of the predators and prey, much as we did earlier when looking at biological models of fisheries.
This can be implemented by making the parameters

and

in the general specification each be negative (rather than zero). A predator-prey model with logistic-like population specific upper size limits thus can be specified with the following general structure
  a
 bS t
 uH t

H t

 c
 dH t
 vS t

S t
(a, b, u > 0)
(c, d, v > 0)
These equations imply that each population has a component that corresponds to a logistic form of biological growth function.
27, 28
To see that these equations do indeed contain logistic-like growth functions, consider the H population. Multiplying out terms and rewriting, the resultant equation gives
27 For the generalisation that we are discussing here to make sense, it is usual to specify that the parameter c in the predator equation is positive, unlike in the LV model without crowding effects (where it was negative).
28
Note that as H
MAX

, H t
/H
MAX

0, and so the logistic component disappears. Hence the linear form used in the L-V model can be regarded as a special form of logistic in which the quadratic collapses to a linear by virtue of there being an infinitely large upper limit to the population size.
74

t

 a
 bS t
 uH t

H t
 aH t
 uH
2 t
 bS t
H t
 aH t


1 where H
MAX
= a/u. uH t a


 bS t
H t
 aH t


1

H t
H
MAX


 bS t
H t
It is evident that there are here two limiting influences on the population of H:
1.
H
MAX
is the maximum carrying capacity of the population in the absence of the population S; as H rises from low levels, members of the H-population face increasingly intense “self-competition” given the environmental milieu in which they are located.
2.
The multiplicative term – bS t
H t
implies that the prey population, H, is negatively related to the size of the predator population, S.
For a numerical example we consider the following specific forms:
 aH t


1

H t
H
MAX


 bH t
S t

0 .
5 H t


1

H t
100



0 .
01 H t
S t
(17.42)
 cS t


1

S t
S
MAX


 dH t
S t
 
0 .
2 S t


1

S t
250



0 .
00001 H t
S t
(17.43)
Setting H and S
 equal to zero, and solving these two equations for H and S gives the equilibrium solution H
*
= 4000 and S
*
= 300.
29
The dynamics of this model are shown in
Figure 17.11. It is evident from looking at the directional arrows and the examples of dynamic adjustment paths (shown by the heavy continuous lines) that H * = 4000 and S * = 300 is an equilibrium solution that will eventually be achieved (dynamic adjustment paths from any arbitrary position all lead to it) and that it is a stable equilibrium (a disturbance would only knock the system out of equilibrium temporarily as dynamic adjustments will restore the equilibrium).
A useful exercise for you to do at this point would be to set up a discrete time counterpart to
Equations 17.42 and 17.43 yourself in a spreadsheet, and to verify the solution we have just described. Try a series of alternative starting values of H and S (ideally not too far away from
29
Mathematically there are also three other solutions. Can you deduce what these are?
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the equilibrium values) and observe what happens. You might also like to see what happens as you change the parameter values of the model. If you wish to verify that you have set up your spreadsheet correctly, an example is provided in the file Interaction.xls
in the Additional
Materials .
Insert Figure 17.11 near here (See the pictures file “Picturesforchapter17.ppt”).
Caption: Equilibrium and dynamics in the predator-prey model with crowding.
17.11.2 Economic Policy
The various models of species interaction that we have examined in this section can be used to generate a number of policy implications. One way of doing so is to add, as an extra component to the model, a social welfare function (SWF). This could be specified in various ways. It might contain H and S as arguments, and so reflect society’s relative valuations over the possible combinations of H and S populations. Alternatively, the arguments of the SWF might be the costs and benefits of harvesting each population at various rates. If the in situ stock size of H and/or S also contributes to utility, those benefits should also enter as arguments of the SWF.
Given a SWF, an optimisation exercise can then be undertaken, maximising social welfare subject to the constraints of the differential equations that are thought to be appropriate for H and S, and for given initial values of H and S. Note that an optimisation analysis of this kind will only be useful for policy purposes if populations of H and/or S can be controlled by human intervention. Hartwick and Olewiler (1998) do an analysis of this kind for two interacting species, sharks and tuna, and demonstrate how interesting policy inferences can be drawn from such an exercise.
An alternative way of deriving policy insight is to bring human intervention explicitly into the model being considered. One way of doing so is to introduce yet another species into the model, one which is bred and maintained for human benefit, yet which also interacts with one or more species of our model. In most cases of interest, this third species will be some kind of
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domesticated livestock animal or farm crop. Conrad (1999, pages 173-182) investigates a three species model of grass-herbivore-predator interactions. This is a useful basis for studying policy implications of farming or agriculture, and so we shall examine a slightly modified version of his model, given by Equations 17.44-17.46:
30
Grass biomass (G) dynamics:
 gG t


1

G t
G
MAX


 α
1
H t
(17.44)
Herbivore (H) dynamics:
 hH t


1

H t
θ
G t


 β
H t
S t
Predator (S) dynamics:
 sS t


1

S t
S
MAX


 χ H t
S t
(17.45)
(17.46)
Examining these three equations, you will see that the first component on the right-hand side of each equation constitutes a logistic growth process, with the qualification that a fixed maximum herbivore population has been replaced by the term

G t
, indicating that the upper limit to which the herbivore population can grow is a fixed multiple of grass biomass.
The remaining terms on the right-hand side further specify the form of biological interactions. Grass is consumed by the herbivore population at the rate

per individual, and so at the rate

H t
by the population at time t. The term -

H t
S t shows that herbivores are consumed by predators at a rate given by a fixed multiple of the product of the herbivore and the predator populations. In contrast, the predator population is positively related to that product. Once again, given knowledge of the parameter values and initial values for the
30 The major variation concerns Equation 17.46 below. Unlike in our equation, Conrad specifies the predator population to be directly related to the herbivore population, and does not contain a logistic component in the form given by 17.46.
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variables G, H and S, one could identify steady state (equilibrium) populations, and simulate the dynamic evolution of these variables through time.
Next we follow Conrad by introducing a fourth population, domestic cattle, into the model.
This is done by assuming that cattle consume grass at the rate

2
per head. Equation 17.44 is then amended to become 31
 gG t


1

G t
G
MAX


 α
1
H t
 α
2
C (17.47)
Note that in Equation 17.47, the variable C is not time-subscripted. That is, we are treating the cattle stock as a fixed constant, presumably predetermined by economic agents.
At this stage of development, the model has become quite rich, and can be used in various ways. The researcher could leave the various equations of the model in general parametric form (as in 17.45 to 17.47 above) and could undertake qualitative analysis. Alternatively, particular parameter values and initial values of the variables G, H, S and C could be chosen and quantitative analysis be done. In either case, it would be possible to address the following types of questions:
(1) Given particular choices of cattle stock, C, what are the steady-state equilibrium values of G, H and S. How do those equilibrium values change as C is changed? values of C
(2) Using dynamic simulation, how do the paths of G, H and S vary over time in response to changes in the stock of domestic cattle? Do adjustment paths converge on a new steady state equilibrium?
(3) What, if any, are the limits to which cattle stocks can be raised before the biological system breaks down (with one or more of the other stocks being driven to zero, or below some critical threshold)?
31 Conrad (1997) actually uses (in his Equation 8.7) a discrete time counterpart to our continuous time specification 17.47.
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(4) What level of cattle stocks maximise some appropriately specified social welfare function?
Conrad takes this kind of model one stage further by making the grass growth function stochastic rather than deterministic. This can be done by replacing the deterministic growth rate g in Equation 17.47 by g t
, a random variable with some suitably chosen distribution. It should be evident that provided that the expected value of g t is g steady state equilibrium solutions will not be changed. However, the dynamic adjustments paths will now exhibit more variability. Moreover, if the variance of g t
is sufficiently large, these dynamic time paths may well breach biological threshold points leading to population collapses. In a stochastic environment, therefore, it is likely to be the case that human impacts on the system
(measured in this case by the size of cattle stocks) will need to be smaller to avoid possible ecosystem collapse. This is one way of modelling the idea of a safe minimum standard of conservation that we described earlier. Alternatively, it is a useful way of modelling sustainability questions in a relatively simple bio-economic model framework.
This brief account of a multiple species bio-economic model has been given primarily as a pointer to how you might go about doing this kind of modelling yourself. We take it no further in this chapter. If you are interested in seeing how some of the issues here could be operationalised, a sensible option might be to follow the exposition in Conrad (1999),
Chapter 8. Alternatively, we have provided in the Additional Materials to this chapter an
Excel file ( Interaction.xls
) which simulates a stochastic version the ‘three species + cattle’ model that has just been described, and which is used to explore the questions raised in this sub-section.
17.12 Resource harvesting, population collapses and the extinction of species
It will be useful to draw together some of the results obtained in this chapter that relate to the possibilities of major population declines or the extinction of species. Human activity can have adverse effects on biological resources in a variety of ways. Two general classes of effects can be distinguished. One operates on particular species (or local populations of
79

a species) that are the direct targets of harvesting activity. In this category we shall also include effects on related species or populations that are strongly dependent on or interrelated with the targets. The second class concerns more widely diffused, indirect impacts on systems of biological resources, induced in the main by disruptions of ecosystems. Much of this generalised impact on biological resources is brought together under the rubric of decline of biological diversity. We consider the causes of biodiversity decline in a separate document in the Additional Materials .
As we have seen, property rights have important consequences for patterns of resource exploitation. In general, harvesting effort will be higher and stock sizes lower where a renewable resource is open access than when enforceable property rights exist. But stronger claims are often made. In particular, it is sometimes argued that populations of renewable resources will inevitably be degraded, suffer serious collapse, or even be driven to zero under open access conditions. The inevitability of these events finds no support in the models we have examined so far. Indeed, in examining the steady state equilibrium of an open access fishery, our numerical example generated a maximum sustainable yield stock size. And for higher values of harvesting costs or lower prices the stock would be even higher, possibly as large as S
MAX
, the environmental carrying capacity of the resource.
To confirm this, look again at Figures 17.3 and 17.4. Consider, for example, the consequence of an increase in w. The ray which plots H = (w/P)E in Figure 17.4 shifts anticlockwise, and so leads to lower effort. Lower effort rotates the eES curve in Figure
17.3 clockwise, leading to a higher steady-state stock. For a sufficiently high value of w, no positive steady state fishing effort will be profitable, and the population will rise to its environmental carrying capacity. Clearly, there is nothing inevitable about population collapses in open access conditions. Much depends on economic factors that may be favourable or unfavourable to large population sizes. Nevertheless, the possibility that a population may be driven to zero is greater
 when the resource is harvested under conditions of open access than where enforceable property rights prevail
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

 the higher is the market price of the harvested resource the lower is the cost of harvesting a given quantity of the resource when prices are endogenous, the more that market price rises as catch costs rise or as harvest quantities fall

 the lower the natural growth rate of the stock the lower the extent to which marginal extraction costs rise as the stock size diminishes.
 the higher is the discount rate
Even under private property conditions an optimal harvesting programme may drive a fish stock to zero. This is most likely where the prey are simple to catch even when the stock approaches a critical minimum threshold level, and where the harvested resource is very valuable. In this case, the optimal harvest level could exceed biological growth rates at all levels of stock.
As the analysis in this chapter have been derived from a simple logistic biological growth model one should be wary about claiming too much generality for our results.
Clearly, other biological growth functions could have generated different results. Three matters are noteworthy here:

The biological growth function may have a positive (and possibly quite large) minimum sustainable population size, as in Figures 17.1(b) and (c) . If for any reason the stock falls below this level, the population cannot survive. Many large mammal species appear to be cases in point.

The growth process may be stochastic (have a random component); stochastic shocks or disturbances might lead to population collapses.

Populations are often inter-dependent in ecosystems. Changes in other populations may lead to the collapse of some population in which we are interested.
But there is another set of reasons why we should also be cautious. Our comments so far in this section have been largely about steady state or equilibrium outcomes. But systems are not always (or perhaps not even very often) in such a state. Indeed, our Excel spreadsheet
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simulations of an open access fishery show that the fishery may be in disequilibrium over very long periods of time even where parameters and environmental conditions are unchanging. Where those factors are frequently changing – as we would expect to find in practice – disequilibrium states will prevail more-or-less indefinitely, and it is very difficult to predict outcomes. It is clearly possible that harvest rates may be persistently above natural population growth rates. Also, ignorance, uncertainty or institutional failure could lead to a population falling below its minimum threshold size even where that event is ( ex post ) irrational.
Notice in particular that the existence of a steady state equilibrium is no guarantee that such a state may be attained. Even if we restrict attention to the adjustment paths along which effort and stocks levels travel in the open access model of Section 17.3, the Excel simulations described in Fishery dynamics.xls
demonstrate that the population might collapse to its minimum threshold size (and never recover) even though a positive steady state equilibrium stock level exists. Non-targeted species may be casualties in this process too. Many forms of resource harvesting, particularly marine fishing, directly or indirectly reduce stocks of other plants or animals that happen to be in the neighbourhood, or which have some biologically complementary relationship with the target resource.
Extinction of species is more likely, other things being equal, where the critical minimum threshold population size is relatively large. In this case, a greater proportion of possible harvesting programmes will lead to harvest rates that cannot be maintained over time. The existence of uncertainty also plays a very important role. Uncertainty may relate to the size of the minimum threshold population, to the actual stock size, or to the current and forecast harvesting rates. If errors are made in estimating any of these magnitudes, then it is clear that the likelihood of stock extinction is increased.
Another influence is the role of chance or random factors. Our presentation has assumed that all functions are deterministic - for given values of explanatory variables, there is a unique value of the explained variable. But many biological and economic processes are stochastic, with chance factors playing a role in shaping outcomes. In these circumstances,
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there will be a distribution of possible outcomes rather than one single outcome for given values of explanatory variables. We discussed some consequences of risk and uncertainty in Chapter 13.
Elementary theories of commercial resource harvesting tend to ignore these possibilities, as we did in much of the exposition given to this point in this chapter. Steady-state fishing involves a balance between regeneration and harvesting rates, and so precludes population collapses. Even sophisticated dynamic models of commercial resource harvesting suggest that species extinction, while being possible in principle, is likely only under very special circumstances. For example, Clark (1990) shows that a privately optimal harvesting programme (where access to the resource is controlled) may be one in which it is ‘optimal’ to harvest a resource to extinction, but also demonstrates that this is highly improbable.
But even a casual inspection of the evidence suggests that much resource harvesting does not fit comfortably with the theoretical models we have outlined, nor does it appear to have the consequences that those models predict. Reading the sections on the state of renewable resources in recent issues of World Resources makes it clear that the world is experiencing extensive losses of many renewable resource population stocks, and unprecedented rates of species extinction. Some examples of these phenomena, looking particularly at cases where harvesting rates of target animal populations have been high relative to natural rates of regeneration, are presented in the file Population Collapses in the Additional Materials .
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Claims that marine resources are currently being, or have been, overfished are commonly made. But what constitutes ‘overfishing’? To answer this, we need some reference point of
‘optimal’ fishing. Here we find some important differences of opinion.
An Economic Perspective
To an economist, optimal fishing is usually taken to mean ‘efficient fishing’. That is, the level of fishing harvest (and the corresponding stock level) which maximises economic net benefits, looked at from a social point of view. This is the criterion used throughout most of this chapter, and was derived analytically for a private property fishery with enforceable property rights. An economist would define optimal fishing capacity as the cost-minimising level of effort required to attain that efficient harvest.
Biological Perspective
The economic notion of optimal fishing has had little practical impact (at least until recent years) on fishery authorities and regulators. Instead, optimal fishing has usually been defined by a particular biological criterion: Maximum Sustainable Yield (MSY). And, correspondingly, optimal fishing capacity has commonly been taken to be that level of fishing effort required to harvest the fishery’s MSY.
As Figure 17.7 demonstrates, the economically efficient stock (or harvest) may be less than, equal to, or greater than the biologically optimal stock (or harvest). Whether the tangency point between G(S) and {i + (

C/

S)/P} lies to the left or right of S
MSY evidently depends on the interest rate, market price of fish, and parameters of the fishery production function.
When is a fishery overfished?
The answer to this question will obviously depend on whether one takes a biological or economic view of ‘optimal’ fishing. Most discussions about overfishing by international
84

agencies (such as the United Nations FAO) and by national and regional regulatory agencies in fact use a biological criterion.
If we were, for the sake or argument, to take MSY as a criterion of optimal fishing, then we can get some insight into whether a fishery is overfished by looking again at Figure 17.2. We can think about this diagram in terms of various stock categories:

When stock is at S = S
MAX
, the fishery is unexploited (and in effect has been for some considerable span of recent time).

When stock is above S
MSY
(but below S
MAX
) the fishery could be described as being in a developmental phase. Specifically, we might say that the fishery is ‘under exploited’ if stocks are close to S
MAX
so that the fishery is capable of producing a great deal more under increased fishing pressure, or as ‘moderately exploited’ if S is closer to S
MSY
so that the fishery is capable of producing some more under increased fishing pressure.

When S is in the (close) neighbourhood of S
MSY
the fishery is ‘fully exploited’ (and the fishery is producing close to its MSY).

When S is less than S
MSY
we might choose to apply one of three labels depending on how much lower is S than S
MSY and in which direction the fishery is moving:
1.
The fishery is ‘overfished’: stocks at lower level than S
MSY
; and the catches in recent years have been showing a downward trend (so current catches less than recent historical high).
2.
The fishery is ‘depleted’ – a more extreme version of ‘overfished’ in which stocks are very far below S
MSY
.
3.
A ‘recovering’ fishery is one in which stocks are very low relative to historical maximum levels, but in which harvest levels are trending upwards (the fishery is moving from left to right towards S
MSY
in terms of
Figure 17.2).
In fact, the notation we have just used is that currently employed by the FAO and by most policy-related discussions of fishery. The reader should note that these terms would not, in general, be appropriate from an economics perspective. However, it is not difficult to see why
85

they are attractive to policy makers. This form of classification has low information requirements, as it does not require any knowledge of economic parameters (prices, costs, fishery production function parameters). Indeed, at a pinch all that is required are time series data on harvests. To see this, note that given a sufficiently long run of harvest data alone, the current position of any fishery could be ascribed with a reasonable degree of confidence to the appropriate one of the categories just listed.
END OF BOX 17.5
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17.13 The current state of marine fisheries
Information about marine fisheries globally has been collected by the FAO since 1970, and published regularly since then in the publication The State of World Fishery Resources,
Marine Fisheries . Much of this data is now publicly available; sources are provided in the
Further Reading at the end of this chapter.
Working at the highest level of aggregation, we might begin by asking what is the uppermost limit to the global fish catch, and where we are now relative to that. A commonly quoted answer to the first part of this question is that provided by Gullard (1971) who estimated the global sustainable yield upper limit to be 100 million tonnes per annum. Figures for the global fisheries catch per annum over a long time span until the present are shown in
Figure 17.12. Global annual production of marine fisheries has is estimated to have increased from 19 million tonnes in 1950 to about 80 million tonnes in the mid-1980s. If these figures are trustworthy, there does not appear to be much scope left for sustained fisheries expansion globally.
Insert Figure 17.12 near here. (See the pictures file “Chapter 17 Pictures.doc”).
Caption: Trends in global marine fisheries production.
Source: Garcia and de Leiva Moreno (2001), page 5. [Permission required]
Figure 17.13 shows an estimate by Garcia and de Leiva Moreno (2001) of the proportions of total fisheries in each of the states classed in Box 17.5. Trends in these proportions are shown
– in summary form - in Figure 17.14. Since 1974 the proportion of fully or moderately exploited stocks has decreased steadily from 95% to just over 70%; the proportion of overexploited, depleted or recovering stocks has increased from about 10% in the mid-1970s to nearly to 30% in the late 1990s.
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Insert near here Figure 17.13 (See the pictures file “Chapter 17 Pictures.doc”).
Caption: State of world stocks in 1999.
Source: Garcia and de Leiva Moreno (2001), page 4. [Permission required]
Insert near here Figure 17.14 (See the pictures file “Chapter 17 Pictures.doc”).
Caption: Global Trends in the state of world stocks since 1974.
Source: Garcia and de Leiva Moreno (2001), page 5. [Permission required]
17.13.1 Are global fisheries in a good or bad state?
If we continue to work with a largely biological criterion of the state of a fishery, then it would be reasonable to conclude from Figure 17.13 that 25% of fisheries are in a good state
(the top two categories) and 28% are in a poor state (the bottom three categories). This begs the question of how we should treat the 47% classed as Fully Fished. This is a moot point, and one which has profound implications for the conclusion one reaches about the state of marine fisheries, and for public policy towards fisheries.
A conventional view of fisheries management is that the stock which corresponds to MSY is a appropriate policy target . This view is reflected, for example, in the United Nations
Conference on the Law of the Seas, which espouses the objective of managing stocks at or above the MSY level of abundance. Using this criterion, Figure 17.13 suggests that 72% of the stocks are in a good state.
More recently, these statistics have been interpreted rather differently. This has coincided with the interest in the use of Sustainability Indicators, and the Precautionary Principle and
Safe Minimum Standard of Conservation concepts. In this case, S
MSY
is regarded as an appropriate limit or threshold . Stocks which are currently exploited near to S
MSY
are potentially endangered by future harvesting behaviour. This changes the way in which the
‘fully fished’ category is treated, and implies that 75% of fishery stocks are in a poor state.
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Such assessments are of course crude, and need much qualification. We have already mentioned in this chapter that an economic appraisal would very often yield entirely different estimates of appropriate targets (or limits) to that of S
MSY
. Ideally, an assessment would also include qualitative information (rather than data of catch or stock quantities alone). This would include such considerations as spawning stock biomass, magnitude of by-catch and discards, and the species composition and age structures of catches and marine stocks. Most current assessments of the state of marine fisheries worsen when these factors are taken into account.
17.14 Renewable resources policy
What goals might one reasonably expect of governmental policy towards the use of renewable resources? Several seem relevant, including efficiency, biological and ecological sustainability, and regional employment and cultural support.
Efficiency goals have been the main focus of this chapter. When the use of resources is economically inefficient, there are potential welfare benefits to the community from policy which leads to efficiency gains. This suggests that policy may be directed towards removing externalities, improving information, developing property rights, removing monopolist industrial structures, and using direct controls or fiscal incentives to alter rates of harvesting whenever there is reason to believe that harvesting programmes are inefficient. Efficiency gains, in the form of improved intertemporal resource extraction programmes, may also be obtained if government assists in the establishment of forward or futures markets. As we saw in Chapter 5, efficient outcomes are not possible in general unless all relevant markets exist. The absence of forward markets for most natural resources suggests that it is most unlikely that extraction and harvesting programmes will be intertemporally efficient. Chapter 7 examined the policy instruments available to attain environmental pollution targets, and much of what was written there is relevant for our present discussion. We will examine two particular instances of incentive-based policy instruments later in this section.
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The provision of information matters not only to the pursuit of efficiency but also to biological and ecological sustainability. Given that uncertainty is so great in matters relating to natural resources, the government's role in the provision of information is likely to be crucial. In the case of commercial fisheries, for example, individual fishermen will not be in a position to know, in quantitative terms, how previous and current behaviour has affected and is likely to affect the population levels of relevant species. The consequences of cyclical natural phenomenon, such as the El Nino current mentioned in Box 17.3, will similarly be largely unpredictable by individual agents. Obtaining this kind of information requires a significant monitoring and research effort which is unlikely to be undertaken by the industry itself. Even if it were obtained privately, the dissemination of such information would probably be sub optimal, as those who devote resources to collecting the information may well seek to limit its availability to others.
17.14.1 Command-and-control regulations
Throughout the world most fisheries have been, and remain, in essence open access fisheries. By far the most common way of regulating fisheries is some form of commandand-control approach. The large set of restrictions and regulations that fall under this heading can be loosely classified in the following way:
1.
Regulations aimed at reducing fishing effort. Examples include restrictions on the boat size or other capital equipment used by fishermen, closed fishery seasons, limits on days of fishing permitted per boat.
2.
Restrictions on fishing gear and mesh or net size, aimed at controlling the qualitatitive nature of the catch. Particular targets have included protection of juvenile fish, reduction of by catch and catch discards, and reducing environmental damage associated with harvesting.
3.
Spatial restrictions on harvesting activity, aimed at reducing conflict among fishing operators.
4.
Fleet size reductions.
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5.
Quantity restrictions on catches. This is the centrepiece of fishing regulation in the
European Community, for example, in the Total Allowable Catch system.
A difficulty with the first three of these command-and-control approaches is that objectives are reached at the cost of decreased harvesting efficiency (in effect, a reduction in the catch coefficient e in Equation 17.7) and/or large financial costs to fishing operators.
These controls deliberately impose economic inefficiency on the industry, in an effort to reduce harvest sizes, and so cannot be cost-efficient methods of attaining harvest reduction targets.
The fourth approach – fleet size reduction – may avoid cost-inefficiencies if the fishing industry capital stock is reduced to its optimal (cost-minimising) level. Governments have tried try to attain this by incentives for firms to leave the industry. But there are two serious limitations at work here. First, capital reductions in some fisheries (particularly in those of the industrialised economies) result in older vessels being sold cheaply to fisheries elsewhere; the effective capacity is not actually cut. Second, such controls are useless in a state of generalised over capacity when reduced effort by one set of operators is just matched by increased effort from other firms.
The fifth approach, quantity of catch restrictions, may be attractive if enforceable, but its record of achievement is extremely poor. Box 17.6 shows that even one of the few examples of quota schemes that was thought to be successful turned out not to be so on closer examination. The widespread failure of catch quota instruments is partly the consequence of the mobile and geographically dispersed nature of the industry and its prey. That is being overcome by modern electronic-based methods of Monitoring, Control and Surveillance. However, in the final analysis, this instrument – like others in the list – are flawed focus on proximate causes and fail to tackle the fundamental causes of the excess fishing capacity problem. This has its roots, as we have seen, in the absence of well-defined and enforceable property rights, such that many fisheries are de facto open access resources. Where property rights do not exist, perverse incentive structures result.
Individual fishing firms have incentives to buy larger boats, or more boats, or install more
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sophisticated harvesting technology in order to win a larger share of the fixed quota for themselves. These things happen at the same time as government is trying to buy boats out of fishing. The failure to effectively limit capacity (and attain quota targets) then results in regulators imposing shorter fishing seasons.
32
As a result of processes like this, the larger capital stock either lies idle for even longer periods, or boats turn to exploit other, less tightly regulated, fisheries, thereby imposing stock depletion problems elsewhere.
Insert Box 17.6 near here
Caption: Quota restrictions in the Pacific halibut fishery
Note to Editor: Please try to locate footnote 15 inside (but at the foot of) Box 17.6.
32 For example, the season for eastern Pacific yellowfin tuna fell from nine months prior to catch quotas being introduced to just three months after a quota restriction was imposed.
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Box 17.6 Quota restrictions in the Pacific halibut fishery
Gordon (1954) provides an interesting account of the use of quota restrictions in the
Pacific halibut fishery. During the 1930s, Canada and the USA agreed to fixed-catch limits. For many years, the scheme was hailed as an outstanding success, with catch per unit effort quantities rising over two decades, one of the few quota schemes to have achieved this goal.
However, Gordon shows that the improvements were not the result of quotas, but of a natural cyclical improvement in Pacific halibut stocks. Catches rose rather than fell during the period when quotas were introduced. Even then, the total catch taken was only a small fraction of the estimated population reduction prior to the introduction of regulation.
Furthermore, the efficiency loss of the regulations was enormous, with the fishing season before quotas were met falling from six months in 1933 to between one and two months in
1952. Despite their widespread use, these quantitative restrictions on either effort or catch have very little justification in either economic or biological terms.
End of Box 17.6
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17.14.2 Incentive-based instruments: a landing tax.
Compliance with regulations is often so poor, and illegal, unregulated and unreported
(IUU) fishing so widespread, that command-and-control techniques may simply fail to reach their objectives at all. Not surprisingly this has led to a search for alternative policy instruments. We have seen in several earlier parts of this book that a tax instrument can sometimes internalise an externality so as to bring about an efficient solution. The
‘problems’ associated with open access fisheries can also be interpreted as externality problems. In this section we show how – at least in principle - an open access fishing industry might be induced to harvest in a socially-efficient manner through the use of a tax instrument.
Specifically, the tax being looked at here is a fish landings tax. It is levied at a fixed rate per unit of the fish resource landed. How should the level of such a tax be set? Consider the steady state present-value maximising condition 17.32: ip = p dG dS
-

C

S
Remember that p in this equation is the net price of a unit of the fish resource. Rewrite this equation in the form p
 dG p dS  i

C

S i
(17.49)
How can we interpret this equation? Consider a decision to not harvest an additional unit of the resource. The left-hand side of this expression is the net price of the fish; it is, therefore, the rent foregone by not harvesting that unit. As that rent would be obtained immediately if the fish were harvested, the left-hand side is already in present value terms. The right-hand side of 17.49 consists of the present value of the benefits obtained by not harvesting that unit of fish. This benefit has two components:
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 dG p dS is the present value of the extra biological growth that would result from leaving the i fish unharvested (including any new growth of fish that could be attributed to it)
 

C

S i
is the present value of the reduced fishing costs that would result from leaving the fish unharvested (noting that the costs per unit harvest will generally be smaller the larger is the fish stock level).
How do we know that these two terms are in present values? It follows from the expression for the present value of an infinite duration annuity. The present value of an infinite annuity of A at an interest rate i is A/i. As we are considering steady states only here, the numerator term in each of the two components is an infinitely repeated benefit or cost. Dividing one repetition of either by the interest rate gives us its present value, therefore.
The right-hand side of Equation 17.49 is often known as the user cost of fishing. We note that the user cost of fishing is taken into account in optimal (present value maximising) harvesting choices. However, in an open access fishery, even though each fisherman may well be aware of these two components of costs, the user cost of fishing will not be taken into account in choices about effort or harvesting. Why not? The open access property gives the fisherman no incentive to leave fish in the sea (to encourage more growth in the future). Others would simply harvest any fish left by one fisherman. The fisherman would be unable to appropriate the fruits of his investment in the future. Each fisherman in effect sees the present value of these future benefits as zero (even though collectively they would not be zero if all abstained together).
Recall that in open access conditions, fishing effort expands until a point is reached at which the net price (or rent) from an additional unit of effort is zero. That is, in equilibrium p = P -

C/

H = 0 where p is the net price of the fish resource. However, the condition that must be satisfied by optimal (present value maximising) fishing is, using Equation 17.49:
95

p




 dG p dS i


C
 i
S



 
0 and so
( P

C
H
)
 

 dG p dS i


C

S i




0
(17.50) where C
H
=

C/

H is the marginal cost of each fish harvested for an additional unit of effort.
But now suppose that the open access fisherman is required to pay a tax (t) on each unit of fish landed equal to the user cost of fishing dG p dS i


C

S i
.
Then, the post-tax open access equilibrium condition would be amended so that post tax net price is zero. That is
P - t - C
H
= (P




 dG p dS i


C

S i 



0 which is identical to the present value maximising fishing rule. A tax on landed fish equal in value to the user cost of fishing would therefore bring about an efficient outcome.
In fact, there may be an additional tax adjustment required to bring about a sociallyefficient outcome. In the models discussed in this chapter, we have adopted simplified functions which impose equality of average and marginal fishing costs, and rule out the existence of crowding diseconomies. Where crowding diseconomies do exist, these warrant a second component to the optimal tax; that component would be set at a value that internalises crowding externalities. Overall, therefore an optimal landing tax will have two components: one part corrects for the fact that fishermen in open access take no account of the future benefits to be obtained by refraining from harvesting; the second part internalises crowding
96

diseconomies. Despite its theoretical attractiveness, tax systems of this kind are very uncommon (or perhaps non-existent).
17.14.3 Incentive-based instruments: property rights and transferable harvesting quotas
A central theme of this chapter has been the key role played by property rights – or their absence - in renewable resource exploitation. Where enforceable property rights do not exist, the resource is said to be subject to open access. An obvious way of trying to achieve efficient resource harvesting is to define and allocate exclusive property rights to it. Where resources are exclusively owned, and where that ownership generates for its owner or owners the full income flow attributable to that resources, owners have incentives to maximise its present value and so to take whatever investment or conservation decisions are consistent with that goal.
Many nations have extended the limits of their national jurisdictions over marine resources to 200 miles over their coastal waters. Is this sufficient to meet the conditions we have just described for exclusive and enforceable property rights? The answer is, in most cases, that by itself it is not sufficient. Even if a government were willing and able to prevent all access to its fisheries by foreign nationals, there may still be de facto open access to fishing boats of the nation in question. The problem is not resolved, and clearly something additional is required.
One way of thinking what additional instrument is required is to follow up on a device we used when introducing the notion of a ‘private property fishery’. There we presented two
‘scenarios’. The first envisaged a fishery consisting of a large number of sub-fisheries, each exclusively owned by one operator. The second envisaged a fishery in which enforceable property rights are collectively vested in a particular large set of fishing firms and in which the behaviour of those individual owners was somehow or other coordinated to be collectively rational. We did not specify, though, what either of these mechanisms might be in practice.
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There is a large number of ways in which one or other of these scenarios could be achieved or, rather, mimicked. We shall investigate just one in this section. It involves the state or regulator acting as coordinator, and allocating exclusive property rights to particular quantities of harvested fish to a particular set of fishing operators. Note that this will involve more than merely allocating quotas or allowable catches to the fleet as a whole; the allocations are to individuals (or possibly to some small collective units that can be relied upon to internally coordinate their actions).
Moreover, if these exclusive property rights are to have their full worth to those to whom they are allocated, they must – like all property rights – be transferable or marketable.
Bearing in mind results we obtained in Chapter 7, the reader will notice that this marketability characteristic will be consistent with the objective of achieving whatever target is sought at least cost. This corresponds with the requirement we are looking for that the system in some way mimics a coordinated present-value maximising fishery.
One scheme that has these properties is the ‘individual transferable quota’ (ITQ) system.
This operates (approximately) in the following way for some particular stock of a controlled species. Scientists assess current and potential stock levels, and determine a maximum total allowable catch (TAC). The TAC is then divided among fishers. Each fisher can catch and land up to the amount of the quota it holds. Alternatively, some or all of the ITQs that an operator holds can be sold to others. No entitlement exists to harvest fish in the absence of holding ITQs.
To see how the ITQ system can result in the harvest of a given target quantity of fish in a cost-efficient manner, consider the following hypothetical example in which, for arithmetical simplicity only, the industry consists of just two fishing firms. The two firms differ in terms of harvesting costs, one being low cost ($2 per tonne) and the other high cost ($4 per tonne). A tonne of fish can be sold for $10. Each firm has historically caught and sold 100 tonnes of fish each period. Now consider what will happen if the government imposes an industry TAC of 100 tonnes. Suppose that a non-transferable quota of 50 tonnes is imposed on each firm. The total catch will be 100 tonnes, at a total cost of $300
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(that is, 50

4 + 50

2). Next, suppose that a transferable quota of 50 tonnes is allocated to each fisher; what will happen in this case? Given that the low-cost fisher makes a profit
(net price) of $8 per tonne, while high-cost fisher makes a profit of $6 per tonne, a mutually advantageous trade opportunity arises. Suppose, for example, that an ITQ price is agreed at $7 per tonne of landed fish. The high-cost producer will sell quotas at this price, obtaining a higher value per sold quota ($7) than the profit foregone on fish it could otherwise catch ($6). The low-cost producer will purchase ITQs at $7, as this is lower than the marginal profit ($8) it can make from the additional catch that is permitted by possession of an ITQ. A Pareto gain takes place, relative to the case where the quotas are non-transferable. This gain is a gain for the economy as a whole as can be seen by noting the total costs after ITQ trading. In this case, all 50 ITQs will be transferred, and so 100 tonnes will be harvested by the low-cost fisher, at a total cost of $200.
This example is, of course, unrealistically simple. We really want our story to apply to large numbers, not just two. And the idea that the industry consists of two sets of fishers, each with a constant (but different) set of harvesting costs, will not hold in practice.
Nevertheless, the underlying principle is correct and applies generally to marketable permit or quota systems in a wide set of circumstances. Transferability ensures that a market will develop in the quotas. In this market, high-cost producers will sell entitlements to harvest, and low-cost producers will purchase rights to harvest. The market price will be set at some level intermediate between the net prices or profits of the different producers.
(We demonstrated in Chapter 7 that this efficiency property is also shared by a tax system; indeed, a tax rate of $7 per tonne of harvested fish would bring about an identical outcome to that described above. Why is this so?)
The transferable quota system has been used successfully in several fisheries, including some in Canada and New Zealand. Some aspects of its practical experience are examined in Box 17.8.
In this section we have focussed on one specific form of property rights - exclusive individual private property rights - and one particular form of policy instrument –
99

individual transferable quotas. However, we do not wish to imply that these are the only ways of achieving efficient outcomes. In principle, many forms of property rights regime, and many forms of policy instrument, can deliver efficient outcomes. The suggestions for further reading at end of this chapter point you to, and elaborate on, some other possibilities.
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Box 17. 8 The Individual Transferable Quota system in New Zealand fisheries
In 1986, New Zealand introduced an individual transferable quota (ITQ) system for its major fisheries. The ITQ management system operates in the way we described earlier.
Government scientists annually assess fish stock levels, and determine maximum total allowable catches (TAC) for controlled species. New Zealand legislation requires that the
TAC levels are consistent with the stock levels that can deliver maximum sustainable yields. The TAC is then divided among fishers, with the shares being allocated on the basis of individual catches in recent years. Each fisher can catch fish up to the amount of the quota it holds, or the quotas can be sold or otherwise traded.
The ITQ system has a number of desirable properties. First, fishermen know at the start of each season the quantity of fish they are entitled to catch; this allows effort to be directed in a cost-minimising efficient manner, avoiding the mad dash for catches that characterises free-access fishery. Secondly, as a market exists in ITQs, resources should be allocated in such a way that firms with low harvesting costs undertake fishing. The reasoning behind this assertion is explained in the main text.
The ITQ system operates in conjunction with strictly enforced exclusion from the fishery of those not in possession of quotas. This access restriction generates appropriate dynamic incentives to conserve fish stocks for the future whenever the net returns of such
‘investments’ are sufficiently high.
The evidence of the ITQ system in operation suggests that, in comparison with alternative management regimes that might have been implemented, it has been successful both as a conservation tool and in terms of reducing the size of the uneconomically large fleets. The
ITQ system has not eliminated all problems, however. The fishing industry creates continuous pressure to push TAC levels upwards, and great uncertainty remains as to the levels at which the TAC can be set without jeopardising population numbers. The ITQ system has failed to find a clear solution to the problems of bycatch - the netting of
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unwanted, untargeted species - and highgrading - the discarding of less valuable species or smaller-sized fish, in order to maximise the value of quotas set in terms of fish quantities.
The ITQ system now operates, to varying extents, in the fisheries of Australia, Canada,
Iceland and the United States.
Source: WR(1994), chapter 10.
END OF BOX 17.8
Box 17.7 Overcapacity in fishing fleets and fishery profitability
FAO data suggests that the tonnage of the world fleet has almost doubled between 1970 and
1999 (from 13.5 to 25 million tons) increasing nominal fishing capacity at about 3% per year.
However, a more appropriate measure would be effective fishing capacity, taking account of technological progress. Garcia and de Leiva Moreno (2001) construct such an index and suggest that with a correcting factor being applied the increase in fishing capacity appears to be from 9 to 40 million tons, at about 12% per year.
This average rate of increase of effective fishing capacity is at the very least worrying, not only because it outstrips the growth of catches but also as it suggests that fishing effort may be highly cost-inefficient.
33
Previous editions of this textbook reported evidence (for example, Huppert, 1990; FAO, 1992, and WR, 1994) which suggested that, taken as a whole, the costs of the global fishing fleet exceeded its revenues by a substantial amount.
33
It should be noted, however, that the global fleet capacity appears to have been relatively constant during the last decade, suggesting that the trend has been broken. Note also that technology improvements do have important, positive effects too. Not only can fish be caught at lower real effort but technology change has also improved catch preservation and quality. It also improves the potential effectiveness of Monitoring, Control and Surveillance (MCS) measures.
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This appraisal was supported by estimates made by the Food and Agriculture Organisation
(FAO) of the United Nations during the early 1990’s (see FAO, 2002).
However, this assessment did not fit well with the observed evidence that most individual fisheries were commercially viable, a fact confirmed by detailed regional studies undertaken by the FAO during the period 1995-97, and by the EC (for individual European countries). The only cases of loss-making fisheries appeared to be at extreme ends of the size scale: small scale gill netting, and large industrial deep-sea trawling. Losses in the former case appear to be due to an inability to compete with technologically superior capture techniques (such as purse seiners and coastal trawlers). In the latter case, losses are attributable to excess fishing capacity and the associated high operational and capital costs per unit of harvested fish. Also of importance here is that a substantial part of deep sea trawling occurs outside national territorial limits, and so is less heavily regulated. It is these kinds of fisheries that most closely resemble the pure open access fishery model examined in this chapter.
It is interesting to note that many of the commercially most profitable fisheries are found in developing rather than industrialised countries. This is explained, at least to some degree by three sets of conditions that relatively favour developing countries: lower operational and labour costs; lower capital costs (because of the use of older vessels, many of which are sold at very low costs to developing countries from the surplus fleets of the industrialised nations); and less overfished marine resources.
Two factors go some way to account for the apparent paradox that the global fleet as a whole is loss-making while individual fisheries are typically commercial. First, global estimates include both active and non-active vessels in cost calculations, whereas individual fishery studies include only active vessels. This is thought to account for much of the difference. Second, the global estimates do not include subsidies and other transfers to fisheries, while the individual fishery studies do. Data on subsidy payments suggests that these transfers are often extremely large, with the result that true economic loss making firms are able to achieve financial viability.
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Thus, our conclusion in previous editions – that there is chronic overcapitalisation in the fishing industry, with far more fishing capacity available than is necessary to catch at current levels, as a result of which the industry is massively inefficient – might still be valid. Latest evidence does confirm this for several fisheries, including deep sea trawling fisheries (as noted above) and long-line fleets exploiting tuna in temperate waters, in which excess capacity is estimated to be in the order of 20 to 30% (FAO, 10/05/2002).
We also noted in previous editions a linkage between overcapacity and subsidy payments that remains valid. Overcapacity is partly a result of responses to fluctuations in fish populations. As fish stocks rise, perhaps as a result of unusually beneficial environmental conditions, the industry installs new capital, often with government support. Subsequently, as stocks fall, either because of natural changes in environments or because of excessive harvesting, the industry is left heavily overcapitalised. Normal market mechanisms, driving firms out of loss-making industries, commonly work slowly or not at all, as government protective subsidies are introduced to maintain employment in areas where fishing activity is heavily concentrated. This linkage is likely to be weakened to the extent that international pressures against the use of subsidies for protection of ‘nationalfavourites’ intensifies.
End of Box 17.7
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Summary

Simple models, such as the logistic growth equation, can be used to describe the biological growth properties of renewable resources. However, for some species – those which exhibit critical depensation – there will be some positive level of population size below which the stock cannot be sustained (and so will eventually collapse to zero).

The notion of a sustainable yield is a useful heuristic device for analysing resource harvesting. Resource owners are sometimes recommended to manage stocks so as to extract a maximum sustainable yield. This is only economically efficient under special circumstances.

Open access fisheries are characterised by conditions of individualistic competition combined with an inability of each boat owner to appropriate the gains of her investment in fish stocks. An open access fishery is likely to be characterised by an economically excessive amount of fishing effort.

It is important – particularly when conditions of open access prevail – to distinguish between steady state equilibrium outcomes and dynamic adjustment paths for a renewable resource stock. In some cases equilibrium outcomes are unobtainable (and so irrelevant) because stocks are pushed beyond critical threshold points during adjustment processes.

Open access conditions do not necessarily imply stock collapses. But economic
(and biological) over-harvesting are more likely to occur where the stock is exploited under conditions of open access than where access can be regulated and enforceable property rights exist.

Comparative statics analysis (for both open access and private property models) suggests that steady state stocks will be higher (fishing effort will be lower) when resource prices fall and when harvesting costs rise. Increases in the efficiency of fishing effort will reduce stocks. The effects of parameter changes on harvest quantities can not be unambiguously signed, as harvest quantities depend on the
105

particular configuration of stock and effort changes (which may be in opposite directions).

Where resource owners have a positive discount rate, and aim to maximise the resource net present value ( PV ), resource stocks will typically be kept at smaller levels than where no discounting of cash flows is undertaken. As the discount rate becomes arbitrarily large, a PV maximising fishery model converges in its outcomes to that of an open access fishery.

There are several reasons why privately rational resource exploitation decisions are not socially efficient or optimal. These include poorly defined or unenforceable property rights, a failure to internalise the value of the resource stock size (and various other kinds of harvesting external effects) in the objective function, and a divergence between private and social discount rates.

Resource harvesting may, under certain circumstances, lead to the biological exhaustion of stocks or even extinction of species. Sometimes the stocks in question are target species; but they may also be incidental (untargeted) sufferers of harvesting behaviour.

Outcomes in which fishery stocks are economically depleted are common in practice, as are outcomes in which stocks are being harvested beyond safe biological limits. Occasionally these processes have led to species extinction.

Fisheries regulatory instruments have typically taken the form of some kind of total allowable catch license system, combined with controls over effort, such as fishing seasons, closure of fisheries, and boat or gear restrictions. Controls have also been targeted at controlling catch ‘quality’ or composition (such as minimum mesh sizes). It is difficult to find any evidence that such instruments are effective in achieving their stated goals.

Market based instruments, particularly transferable fishing licenses, are theoretically more attractive and have shown much promise where used to date.
However, they are not a panacea for attaining fishery objectives.

As with all environmental assets, renewable resources typically provide a multiplicity of valuable services. Their value for supply of food or materials – which drives private harvesting behaviour – is only a partial reflection of total
106

resource values. It is not surprising, therefore, that socially optimal resource exploitation will need to use several instruments, and/or be subject to a variety of constraints.
107

General reviews : Excellent reviews of the state of various renewable resources in the world economy, and experiences with various management regimes are contained in the biannual editions of World Resources . See, in particular, the sections in World Resources 1994-95 on biodiversity (chapter 8) and marine fishing (chapter 10). Various editions of the United
Nations Environment Programme, Environmental Data Report also provide good empirical accounts.
Renewable resources : Clark (1990), Conrad and Clark (1987) and Dasgupta (1982) provide graduate-level accounts, in quite mathematical form, of the theory of renewable resource depletion, as do Wilen (1985), Conrad (1995) and Rettig (1995). Good undergraduate accounts are to be found in Fisher (1981, chapter 3), the survey paper by
Peterson and Fisher (1977), Neher (1990), Hartwick and Olewiler (1998) and Tietenberg
(1992, update). Gordon (1954) is a classic paper developing the idea of open-access resources. Bromley (1999) examines common property resource regimes, and argues that many fisheries have transformed to open access with population growth. Ostrom et al
(2002) provides a multi-disciplinary reappraisal of the role of the notion of the commons as an explanation of human activity. Other references on open access are Ostrom (1998) and Baland and Platteau (1996), the latter of which includes a game-theoretic approach.
Fisheries
: Comprehensive accounts of the world’s fisheries can be found from the UN FAO at www.fao.org
. In particular, see the FAO’s publication
State of World Fisheries and
Aquaculture , updated every few years and available online. FAO (2002) gives a good account of recent thinking about sustainability, property rights, and the precautionary principle in fisheries management. Works which focus specifically on fisheries include the early classics by Munro (1981, 1982); Hannesson (1993); and Cunningham, Dunn and Whitmarsh (1985).
Van Kooten and Bulte (2000) contains an excellent survey of the economics of fishing, covering with great care harvesting under various forms of uncertainty. Iudicello, Weber and
Wieland examine the economics of overfishing. L. Anderson (1986) and Waugh (199?) deal
108

with fisheries management. NRC (19??) examines individual fishing quotas. More advanced discussions of fisheries management can be found in Conrad and Clark (1987), Conrad
(1991), Harris (1998) which discusses ‘high grading’, Neher (1990), Graves et al (1994),
Salvanes and Squires (1995), Sutinen and Anderson (1985), Ludwig et al (1993), Tahvonen and Kuuluvainen (1995), and L.G. Anderson (1977, 1981, 1995), the last of which examines gear restrictions and the ITQ system. L.G.Anderson (1985, chapter 7) gives a very thorough and readable analysis of policy instruments that seek to attain efficient harvesting of fish stocks, using evidence from Canadian experience. That book also provides a good account of models of fluctuating fish populations, an issue of immense practical importance. Rettig
(1995) considers the problem of management with migratory and transboundary populations.
There are now several very useful web based sources of information on fishery problems and fishery management. One such source is the website of the International Institute of Fisheries
Economics and Trade (IIFET) at http://osu.orst.edu/Dept/IIFET (search via ‘Resources’) which contains conference proceedings and contributions on ITQ management and many other fisheries and resource-related topics.
Empirical models include Wilen (1976) on north Pacific fur seal; Amundsen et al (1995) on North Atlantic minke whale; Bjorndal and Conrad (1987) on North Sea herring;
Henderson and Tugwell (1979) on a lobster fishery; and Huppert (1990) on Alaskan groundfish.
Ecological considerations: Ecologists have fundamentally different notion of population viability to that held by many environmental economists. In particular, they typically reject the largely deterministic bioeconomic models examined in this chapter, and would argue that ecological complexities cannot be properly dealt with by a fixed number for S
MIN
, nor by the simple inclusion of a stochastic factor in an otherwise deterministic growth function. It is, therefore, sensible to read some expositions of issues relating to resource exploitation and conservation, and the nature of stochasticity in biological systems written from an ecological perspective. Krebs (1972, 1985) contain good expositions of ecology.
MacArthur and Wilson (1967) consider spatial aspects of biological population dynamics
109

(island biogeography; meta-populations); Lande et al (1994) discuss extinction risk in fluctuating populations.
May (1994)
Ludwig et al (1993) uncertainty and conservation
Hilborn et al (1995)
Sethi and Somanathan (1996) discuss the use of evolutionary game theory to examine ecological systems.
Species extinction and biodiversity decline.
Several references to the meaning, measurement and statistics/estimates regarding biodiversity decline were given at the end of Chapter 6. The various causes of biodiversity loss are surveyed in a separate document in the Additional Materials , What is causing the loss of biodiversity.doc
. A simple, non-technical account of species loss arising from harvesting and human predation is given in Conrad (1995). For a more rigorous and complete account, see Clark (1990). Barbier et al. (1990b) examine elephants and the ivory trade from an economics perspective. In addition to those, important ecological accounts of the issue are
Lovelock (1989, developing the Gaia principle), Ehrenfeld (1988) and Ehrlich and Ehrlich
(1981, 1992). The classic book in this field is Wilson (1988). More recent evidence is found in Groombridge (1992), Hawksworth (1995), Jeffries (1997) and UNEP (1995).
Perrings (1995), Jansson et al (1994) and Perrings et al (1995) examine biodiversity loss from an integrated ecology-economy perspective. Several other ecological accounts, together with more conventional economic analyses of the causes, are found in Swanson (1995a,b) and
OECD (1996). Repetto and Gillis (1988) examine biodiversity in connection with forest resource use. The economics of biodiversity is covered at an introductory level in Pearce and
Moran (1994), McNeely (1988) and Barbier et al (1994)..
Other economics-oriented discussions are Simon and Wildavsky (1993). There is also a journal devoted to this topic, Biodiversity and Conservation . Articles concerning biodiversity are regularly published in the journal Ecological Economics . Policy options for conserving
110

biodiversity are covered in OECD (1996). Common and Norton (1992) study conservation in
Australia.
Conservation : A large literature now exists examining the economics of wilderness conservation, including Porter (1982), Krutilla (1967) and Krutilla and Fisher (1975).
Excellent accounts of the notion of a Safe Minimum Standard of conservation are to be found in Randall and Farmer (1995) and Bishop (1978). Other good references in this area include Ciriacy-Wantrup (1968), Norgaard (1984, 1988), Ehrenfeld (1988) and Common
(1995).
Water : Water may, of course, be regarded as a renewable resource. Good discussions of the valuation of water quality improvements are found in Desvousges et al. (1987) and
Mitchell and Carson (1984) (which both emphasises valuation issues), Ecstein (1958) and
Maass et al. (1962), both of which focus on CBA. Water quality management is considered by Johnson and Brown (1976), Kneese (1984) and Kneese and Bower (1968).
The USEPA website contains much useful information about ground water (and drinking water) resources (web links include http://www.epa.gov/safewater/protect/sources.html
and http://www.epa.gov/OGWDW/ It may also be helpful to check out the site of the
Groundwater Foundation (a non-profit organization) at http://www.groundwater.org/GWBasics/gwbasics.htm
and the US Geological Survey
(USGS) ground water information websites (http://ga.water.usgs.gov/).
Dams : Large dam construction and operations can have dramatic effects on human and biological populations. A large literature on this topic has been spawned by the World Bank
Operations Evaluation Department and the IUCN-The World Conservation Union. Follow their respective web sites. See also the World Commission on Dams (2000) with website at http://www.damsreport.org/ .
Land use policy : El-Swaify and Yakowitz (1998) is a general survey.
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Discussion questions
1. Would the extension of territorial limits for fishing beyond 200 miles from coastlines offer the prospect of significant improvements in the efficiency of commercial fishing?
2. Discuss the implications for the harvest rate and possible exhaustion of a renewable resource under circumstances where access to the resource is open, and property rights are not well defined.
3. To what extent do environmental ‘problems’ arise from the absence (or unclearly defined assignation) of property rights?
4.
Fish species are sometimes classified as ‘schooling’(such as herring, anchovies and tuna) or ’searching’(non-schooling) classes, with the former being defined by the tendency to ‘school’ in large numbers. In the text we specified fishery harvest by the Equation H =
H(E,S). For some species, the level of stocks has a much less important effect on harvest, and so (as an approximation) we may write H = H(E). Is this more plausible for schooling or searching species, and why?
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Problems
Problems marked with an asterisk * require that the reader either construct his or her own spreadsheet programme, or adapt the file exploit5.xls.
1.
(a) The simple logistic growth model given as Equation 17.3 in the text
G ( S )

 g

1

S
S
MAX


S gives the amount of biological growth, G, as a function of the resource stock size, S. This equation can be easily solved for S = S(t), that is the resource stock as a function of time, t.
The solution may be written in the form
S ( t )

1
S
MAX
 ke
 gt where k

S
MAX
S o

S
0 and S
0
is the initial stock size (see Clark, 1990, page 11 for details of the solution). Sketch the relationship between S(t) and t for:
(i)
S
0
> S
MAX
(ii) S
0
< S
MAX
* (b) An alternative form of biological growth function is the Gompertz function
G ( S )
 gS ln


S
M AX
S


Use a spreadsheet programme to compare – for given parameters g and S
MAX
– the growth behaviour of a population under the logistic and Gompertz growth models.
2.
A simple model of bioeconomic (that is, biological and economic) equilibrium in an open-access fishery in which resource growth is logistic is given by
G ( S )

 g

1

S
S
MAX


S - eES and
V - C = PeES - wE = 0
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with all variables and parameters defined as in text of the chapter.
(i) Demonstrate that the equilibrium fishing effort and equilibrium stock can be written as
E
 g e

 1
 w
PeS
MAX

 and S
 w
Pe
(ii) Using these expressions, show what happens to fishing effort and the stock size as the 'cost-price ratio' w/P changes. In particular, what happens to effort as this ratio becomes very large? Explain your results intuitively.
3. In what circumstances would it be plausible to assume that, as a first approximation, harvest costs do NOT depend on stock size?
4.
(i) The results of this chapter have shown that the outcomes (for S, E and H) are identical in what has been called the PV maximising model and the static private fishery profit maximising model when the discount rate is zero. Explain why this is so. Also explain why the stock level is higher under zero discounting than under positive discounting.
(ii) It has also been shown in this chapter that as the interest rate becomes arbitrarily large, the PV maximising outcome converges to that found under open access Why should this be the case? (If you are using the exploit5.xls spreadsheet, this result can be quickly verified. In the worksheet ‘Steady states
(2)’, note that at i =1000, the present value outcome is more or less identical to that which emerges under open access.)
5. Calculate the ‘growth rate’, dG/dS, at which the fish population is growing in the open access equilibrium, the static private property equilibrium, and the present value maximising equilibrium with costs dependent on stock size and i = 0.1, for the baseline assumptions given in Table 17.3. At what stock size is dG/dS = 0 (the maximum sustainable yield harvest level)?
Explain and comment on your findings.
114

6. Demonstrate that open access is not cost-minimising.
115

Appendix 17.1 The discrete time analogue of the continuous time fishery models examined in Chapter 17.
The analysis in the chapter text uses continuous time notation. However, when a spreadsheet is used for dynamic simulation (as in exploit5.xls) we necessarily adopt discrete time , as spreadsheets are set up to do calculation recursively in discrete time intervals. Tables 17.8 and 17.9 present discrete time analogues of the continuous time equations (with the latter reproduced from Table 17.1 for convenience).
Table 17.8 The fishery models: general function specification
Biological growth
Continuous Time Model dS/dt = G(S)
Fishery production function H = H(E, S)
Net growth of fish stock dS/dt = G(S) – H
Fishery costs C = C(E)
Fishery revenue B = PH, P constant
Fishery profit NB = B – C
FISHING EFFORT DYNAMICS
Open access entry rule dE/dt =

NB
Private property entry rule dE/dt =

(dNB/dE)*E
STEADY STATE CONDITIONS
Biological equilibrium
Economic equilibrium
G = H
E(t) = E*
Discrete Time Model
S
H
S
C
B t+1 t+1 t t t
-S
-S t t
= G(S
= H(E
= G(S
= C(E
NB
= PH t
= B t t t t
, S
) t t t
)
)
) - H t
, P constant
- C t
E t+1
-E t
=

NB t
E t+1
-E t
=

(dNB t
/dE t
)*E
G t
= H t
E t+1
= E t
= E*
116

Table 17.9 The fishery models: with assumed functional forms
Pure biological growth
Fishery production function
Net growth of fish stock
Fishery costs
Fishery revenue
Fishery profit
Continuous Time Model
G ( S )
 g


1

S
S
MAX


S
H = eES
G ( S )
 g


1

S
S
MAX


S - H
Discrete Time Model
S
H t t

1

= eE
S t t
S t

1

S t
S t
 g


1

S t
S
MAX


S t

 g

1

S t
S
MAX


S t

H t
C = wE
B = PH, P constant
C t
= wE t
B t
= PH t
, P constant
NB = B - C = PeES –wE NB t
= B t
- C t
= PeE t
S t
-wE t
FISHING EFFORT
DYNAMICS
Open access entry rule
Private property entry rule dE/dt =

(PeES -wE) dE/dt =

(dNB/dE)*E
E t+1
-E t
=

(PeE t
S t
-wE t
)
E t+1
-E t
=

(dNB t
/dE t
)*E t
117

Appendix 17.2 Derivation of steady-state equilibrium for an open access fishery and for a private property fishery.
It is most natural to regard fishing effort, E, as the choice (or instrument) variable when considering open access fisheries, as we did in our discussion in the text. However, in order to facilitate comparison with other fishery regimes, it will be more convenient to work with the fish stock as the choice variable in this appendix. Identical results would be found by treating any of S, E or H as the choice variable. The reader might like to verify this as an exercise.
The open access fishery
Steady state economic equilibrium in open access will be characterised by the zero economic profit condition

( S )

R ( S )

C ( S )

0
Given that we are assuming that the industry is competitive and so each firm is a price taker), R(S) = PH(S).
Steady state biological equilibrium requires that H(S) = G(S).
A bioeconomic equilibrium requires simultaneous satisfaction of these two equilibrium conditions. Substituting the second into the first gives
PG(S) = C(S)
For our assumed functions
C = C(E) = wE
H = eES

E = H/eS and so
C
 wH eS
(17.51)
Since in steady state H = G and with G(S) given by Equation 17.3 we have
C ( S )
 w eS gS ( 1

S
S
M AX
)
 wg e
( 1

S
S
M AX
) (17.52) giving the total cost as a function of stock.
118

Now we obtain an expression for revenue.
G ( S )
 gS ( 1

S
S
M AX
) hence total revenue is
R(S) = PG ( S )

PgS ( 1

S
S
M AX
) is total revenue as a function of stock.
Hence wg e
( 1

S
S
M AX
)

PgS ( 1

S
S
M AX
)
(17.53) wg

PgS e
S
*  w
Pe
H=G=eES gS ( 1

S
S
M AX
) = eES.
Substitute for S then rearrange g w
Pe
( 1
 w
PeS
M AX
) = eEw/Pe. g w
Pe
( 1
 w
PeS
M AX
) = Ew/P.
E *
 g e

 1
 w
PeS
MAX


(17.54)
(17.55)
Substituting S* and E* into H = eES then yields H *
 gw
Pe

 1
 w
PeS
MAX

 (17.56)
Note that these expressions for open access steady state effort, stock and harvest are those given earlier as Equations 17.20, 17.21 and 17.22, and reproduced in Table 17.4. The outcome can be illustrated diagrammatically, as in Figure 17.15. The total cost and total
119

revenue curves shown in the diagram are Equations x and y respectively. The open access equilibrium is at S
OA
.
Insert Figure 17.15 here (See the pictures file “Chapter 17 Pictures.doc”).
Caption: The static private fishery and open access fishery equilibria compared.
Private property fishery.
Once again, treat S as the choice variable. Objective is to maximise profits,

, by an appropriate choice of S.
Max

( S ) requires dR ( S ) dS


C ( S )

S
In steady state equilibrium we have R(S) = PG(S). Hence dR ( S )
 dS d [ PG ( S )] dS

P dG ( S ) dS
Thus dR ( S ) dS


C ( S )

S

P dG ( S ) dS


C ( S )

S
(17.57)
We therefore need expressions for marginal revenue and marginal cost. Total revenue is
PG ( S )

PgS ( 1

S
S
M AX
)
We note that dG dS
 g

2 g
S
M AX
S
P dG dS

P ( g

2 g
S
M AX
S ) and so marginal revenue is
(17.58)
As in the open access case, fishing costs are given by Equation 17.52
C
 w eS gS ( 1

S
S
M AX
)
 wg e
( 1

S
S
M AX
)
120

Taking the derivative we obtain marginal cost
34

C

S
  wg eS
M AX
(17.59)
Equating marginal revenue and marginal cost gives
P ( g

2 g
S
M AX
S )
  wg eS
M AX which on solving for S gives
S

1
2
( S
M AX
 w eP
) or
S
*
PP

1
2
PeS
M AX
 w
Pe which is the expression given in Table 17.4
(17.60)
A few lines of algebra suffice to obtain expressions for steady state effort and harvest.
Noting again that H=G=eES, we obtain gS ( 1

S
S
M AX
) = eES.
Substitute for S then rearrange g (
PeS
M AX
2 Pe
 w


)

1

PeS
M AX
 w
S
2 Pe
M AX



 eE (
PeS
M AX
2 Pe
 w
)

 g

1

PeS
M AX
 w
S
2 Pe
M AX



 eE
E
*
PP
 g
2 e

 1
 w
PeS
MAX

 (17.61)
34
We continue to use partial derivative notation for consistency with notation in the chapter, despite having substituted H out of this expression.
121

Then by substituting the solutions for E and S into H = eES gives
H
*
PP

1
4 g w
2
S
M AX

P
2 e
2
S
M AX
(17.62)
Diagrammatically, in Figure 17.15, the slopes of the total cost and total revenue curves - that is, marginal cost and marginal revenue, given by Equations 17.59 and 17.58 - are equal at S
PP
, the static private property equilibrium. The open access equilibrium, in comparison, is at S
OA
.
End of Appendix 17.2
122

Appendix 17.3 The dynamics of renewable resource harvesting
In this appendix we derive the optimal solution to the net present value maximising fishery described in Section 17.6 in the text. We begin the analysis with general functional forms for the biological and economic equations underlying the model. Two cases are considered. First the case in which the market price of fish is an exogenously given constant. In the second case, the fishery faces a downward sloping demand curve for fish, and so fish prices are endogenous.
In both cases, the derivation involves specifying the private optimising problem, obtaining the current-value Hamiltonian, deriving necessary first order conditions, and solving those conditions to obtain a set of analytical conditions that constitute the main elements of the model solution. We obtain the steady state solution to the model from these conditions. In addition, an elementary account of system dynamics is given.
This appendix also derives parametric solutions for the particular functional forms used in this chapter. Finally, it solves these parametric solutions for their numerical values for the
‘baseline’ parameters assumed in Box 17.2 and for i = 0.
To assist comparison with the main text, equations in this appendix that appeared previously in the text are given their earlier number, but marked with an asterisk. Other equations are numbered sequentially and are not asterisked. Where notation has been used before in the text, it will not be redefined here.
A.17.3.1 The Optimising Problem
The problem to be solved is
Max


0

R( H t
) C( S t
, H t
)
 e
it dt subject to dS
= G( S t dt
) H t
123

and initial stock level S(0) = S
0
., where R(H) is the total revenue from resource harvesting.
A.17.3.2 Private present value-maximising harvesting in a competitive market with enforceable property rights and constant prices.
With constant price R(H) = PH and so the current-value Hamiltonian is
L ( H , S ) t

PH t

C ( S t
, H t
)
 p t
( G ( S t
)

H t
) with necessary first order conditions
( i )

L

H t t

0

P


C

H t
 p t
(ii) d p dt t
= ip t
p t dG d S t

C
+

S t
(17.63)
(17.30*)
( iii ) dS dt
= G( S t
) H t
Equation 17.30* is the Hotelling efficiency condition for a renewable resource in which costs depend upon the stock level. It applies whether or not the fishery is in steady state equilibrium.
As described in the text, 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. The marginal cost is the foregone current return, the value of which is the net price of the resource, p. However, since we are considering a decision to defer this revenue by one period, the present value of this sacrificed return is ip.
Out of steady state the benefits obtained by the resource investment are threefold:

The unit of stock may appreciate in value by the amount dp/dt.

With an additional unit of stock, total harvesting costs will be reduced by the quantity

C/

S (note that

C/

S < 0).
124


The additional unit of stock will grow by the amount dG/dS, the value of which is this quantity multiplied by the net price of the resource, p.
A present value-maximising owner will add to (subtract from) the resource stock provided the marginal cost of doing so is less (greater) than the marginal benefit. These imply the asset equilibrium condition 17.30*.
Dividing both sides of Equation 17.30* by the net price p (and ignoring time subscripts from now on for simplicity) gives i = dp dt p
-

C

S p

+ dG dS
(17.64)
This reformulation shows that it is Hotelling's rule of efficient resource use, albeit in a modified form. (It is instructive to compare this version of Hotelling's rule with other versions that we have derived previously. See, for example, Equation 7.20c in Chapter 7.)
The left-hand side of Equation 17.64 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 three elements:
 the proportionate growth in net price
 the proportionate reduction in harvesting costs that arises from a marginal increase in the resource stock
 the natural rate of growth in the stock from a marginal change in the stock size.
A.17.3.2.1 Steady state equilibrium.
In a steady state, all variables (including shadow prices) are unchanging with respect to time, so dp/dt = 0. So the three first order conditions collapse to ip = p dG dS
-

C

S
(17.32*)
125

p

P ( H )


C

H
G(S) = H
(17.29*)
(17.14*)
This constitutes a system of three equations in three unknowns S, H and p. Given functional forms for G(S) and C(H,S), we may then be able to obtain the steady state solution for those three unknowns in terms of the model parameters. These are derived later in this section for our assumed functional forms.
If i = 0 the steady-state equation 17.32* simplifies to

C

S
= dG dS p (17.65)
This was the condition 17.57 that we found in Appendix 17.2 for a private property fishery, and demonstrates that the static private property fishery is a special case of the PV maximising fishery when the discount rate is zero
We can also express Equation 17.65 in terms of market price rather than net price. Noting that p = P - C
H
(where C
H
=

C/

H) then

C

S
= dG dS
(P C
H
)

C

S
= dG dS
P dG dS
C
H or

G

S
P = dG dS
C
H
+

C

S
(17.66)
This is the steady-state condition for maximising the net present value of the renewable resource in the special case where the discount rate is zero. The left-hand side can be interpreted as the steady-state marginal revenue of harvesting, the right-hand side the steady-state marginal cost of harvesting, taking account of the dependence of costs on effort, the cost per unit effort and the dependence of cost on the size of the resource stock.
126

A.17.3.2.2 Initial and terminal conditions, and adjustment dynamics.
The problem has a single initial condition, namely that there exists a fixed, positive initial stock level, S
0
. Let {H
T
, S
T
} denote the terminal, steady-state values of the harvest rate and resource stock, such that dH
= 0 and dt dS
= 0 dt
The nature of the adjustment dynamics depends on whether or not the Hamiltonian is linear in the control variable (here taken to be H). Adjustment may be rapid and discontinuous (along a MRAP) or gradual and continuous, depending on the underlying functions. We return to this below.
A.17.3.2.3 Parametric solution for the assumed functional forms, where P is constant
For our assumed functions, the Hamiltonian and first-order conditions are:
L ( H , S )

PH
 wH eS gS
2
 p ( gS

S
M AX

H ) (17.67)
( i )

L

H

0

P
 w eS
 p
(ii)

= ip p ( g

2 gS
S
M AX
)
 wH eS
2
(17.66)
(iii)

= gS
 gS
2
S
M AX
H
In steady state, Equation (17.66) can be rearranged to the form ip
 p ( g

2 gS
S
M AX
)
 wH eS
2
Solving this first order condition equation to obtain an expression for H as a function of p (net price) and S gives:
H
 p ( iS
MAX
 gS
MAX

2 gS ) eS
2 wS
MAX
127

and then substituting for p from the first order condition p = P -

C/

H yields:
H
 w eS
( iS
M AX
 gS
M AX

2 gS ) eS
2 wS
M AX
Given that G(S) = H in steady state, we next substitute the biological growth function, G(S) into the previous equation:
 g

1

S
S
MAX

 S


PeS
 w

( iS
MAX
 gS
MAX

2 gS ) S wS
MAX
After further manipulation and simplification, we can obtain from this last equation an explicit solution for S* in terms of model parameters alone:
S
* 
1
4

PeiS
M AX

PegS
M AX
 wg

P
2 e
2 i
2
S
M AX
2 
2 P
2 e
2 iS
M AX
2 g

6 PegwiS
M AX

P
2 e
2 g
2
S
M AX
2  
2 Peg
2
S
M AX w
 w
2 g
2
Peg which is the rather cumbersome expression that has been used to calculate S
*
in the associated Excel spreadsheets. A simpler form emerging after further manipulation is:
S
* 
1
4
S
M AX


 w
PeS
M AX

1
 i g


 w
PeS
M AX

1
 i g


2

8 wi
PegS
M AX



(17.69)
We may now substitute S* into G(S) to get H*:
1
4 i
H
* 
1
4 g



3
4


 w
PeS
M AX


1

1
4 PeS i g
 w
M AX


 w g
PeS
M AX

1
4

1


 i g


2 w
PeS
M AX


1
 i g
8 wi
PegS
M AX





2



8 wi
PegS
M AX




S

M AX

(17.70) and then finally obtain an expression for steady-state effort, E*, by using E* = H*/(eS*):
128

E
*  g e



3
4

1
4 w
PeS
M AX

1
4 i g

1
4

 w
PeS
M AX

1
 i g


2



8 wi
PegS
M AX





(17.71)
Equations 17.70 and 17.71 are used to obtain the PV maximising steady state values of H and E in the associated Excel spreadsheets. If i = 0 these expressions collapse to those for a static profit maximising model.
A.17.3.2.4 Dynamics
The Hamiltonian (Equation 17.67) is linear in H, the instrument variable. In this case, adjustment is of the bang-bang (most rapid approach path or MRAP) variety. That is, if S
< S
*
then harvesting is zero until S = S
*
, after which H = H
*
. If S > S
*
then harvesting is undertaken at its maximum possible rate until S = S
*
, after which H = H
*
.
129

A.17.3.3
Private present value-maximising harvesting in a competitive market with enforceable property rights and a downward sloping demand curve for fish.
The Hamiltonian is now
L ( H , S ) t

R ( H t
)

C ( S t
, H t
)
 p t
( G ( S t
)

H t
) (17.72)
The necessary conditions for a maximum are now

L

H t t

0
 dR dH t


C

H t
 p t d p dt t
= ip t
p t dG d S t
+


C
S t
(17.73)
(17.30*)
Differentiating 17.73 with respect to t (and omitting time subscripts for simplicity) yields dp
 dt d
2
R dH
2 dH dt

 2
C

H
2 dH dt
Equating this with 17.30* gives ip p dG dS
+

C

S
 d
2
R dH
2 dH dt

 2
C

H
2 dH dt
(17.74) which can be solved to give dH/dt as a function of H and S.
We also have dS dt
= G( S t
) H t
(17.75)
Equations 17.74 and 17.75 constitute a system of two differential equations in two variables H and S which can be solved for the paths and steady state solutions of those two variables.
130

A.17.3.3.1 Worked example for assumed functions and parameter values
We assume that dB/dH = P(H) = a – bH, with a = 250 and b = 2000. As before, the biological growth equation, the fishery production function, and fishing costs are as specified in the right-hand column of Table 17. Therefore C(H,S) = wH/eS. Parameter values are those given in Table 17.2.
The current-value Hamiltonian and the first order conditions (omitting time subscripts for simplicity except where necessary) are
L ( H , S )
  h
0
( a
 bH ) dh
 wH eS gS
2
 p ( gS

S
M AX

H ) (17.76)
( i )

L

H

0
 a
 bH
 w eS
 p
(ii)  = ip p ( g

2 gS
S
M AX
)
 wH eS
2
(17.77)
(17.78)
(iii)

= gS
 gS
2
S
M AX
H
Differentiating 17.77 with respect to time gives
  b
  w eS
2
Substituting 17.80 into 17.78 gives
 b
  w eS
2
 ip p ( g

2 gS
S
M AX
)
 wH eS
2 gives


1 b

 ( i
 g

2 gS
S
MAX
) p
 wH eS
2
 w eS
2


(17.79)
(17.80) which after substituting for S from (17.79), p from (17.77) and simplifying gives
131



1 b

 ( i
 g

2 gS
S
MAX
)[ a
 bH
 w eS
]
 wg
( 1
 eS
S
S
MAX
)

 (17.81)
Equations (17.69) and (17.81) constitute the required system of 2 differential equations in the variables S and H. From these, we can (after parameter values have been substituted in) obtain solution values for S and H. The dynamic trajectories may be shown graphically by means of a phase plane diagram (to be explained below).
The steady-state solutions to the system are found where

= 0 and H

0 . That is, we solve simultaneously
0 = gS
 gS
2
S
M AX
H
0

1
 b

 ( i
 g

2 gS
S
MAX
)[ a
 bH
 w eS
]
 wg
( 1
 eS
S
S
MAX
)

 which, with the baseline parameter values, yield S* = 0.449 and H* = 0.0371. Then P* = a
– bH* = 250 – 2000(0.0371) = 175.8. Computations can be found in the Maple file
Chapter 17.mws
, in which the phase plane diagram showing the system dynamics is generated.
It is of interest to note that if the intercept of the inverse demand function has been a =
273.26 rather than a = 250, the solution yields S* = 0.4239, H* = 0.0366 and P* = 200, identical results to those obtained for the exogenously fixed price model variant with P =
200.
A.17.3.4
Private present value-maximising harvesting in a monopolistic market with enforceable property rights and a downward sloping demand curve for fish.
For a monopolistic industry, R = P(H)H and so
132

dR d dH
= dH

P(H)H

 dP dH
H + P(H) and so dR/dH < P, as dP/dH <0. Hence the monopolistic profit-maximising solution is different from the competitive market (and the socially efficient) solution. To see precisely how it is different, look at the net price first-order condition for the two alternative market structures.
Monopoly:
P(H) =

C

H
+ p
 dP dH
H
Competition:

C
P(H) =

H
+ p
As dP/dH is negative, this implies that the market price is higher in monopolistic than in competitive markets, and so less is harvested and sold. One would expect the stock size to be larger in monopolistic than in competitive markets.
End of Appendix 17.3
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