File: Fishery.doc
Date: 22 February 2003
ADDITIONAL MATERIALS FOR CHAPTER 17 ^{1}
Several sets of additional materials are associated with Chapter 17. We have classified these materials into a small number of subgroups, the contents which are briefly described below.
(1) APPENDICES TO CHAPTER 17
Two of the Appendices to Chapter 17 were omitted from the text for reasons of space. Appendices 2 and 3 can be downloaded here.
(2) COMPARATIVE STATICS OF FISHING
The file Comparative statics.xls uses Excel spreadsheets to illustrate with numerical examples the properties of the various fishery models examined in Chapter 17. It calculates  from analytical formulae given in the text  the steady states that emerge under the three regimes examined in the chapter:

an open access fishery;

a static private property fishery model; singleperiod profits are maximised;

a dynamic private property fishery model; in which private owners maximise the present value of the fishery (at some given discount rate) over an infinite horizon.
We compute and compare the steady state equilibrium fish stock, fishing effort and harvest in these three regimes, and explore the consequences of changing various parameters. What is being done here is essentially comparative statics. This set of materials is not concerned with the time paths (the dynamics) of how the fishery might get to these steady state equilibria.
The Excel file Comparative statics.xls carries out appropriate computations, and is set up so that you can easily study the long run impacts of changing biological or economic parameters.
A second file associated with this topic is the Maple file Comparative statics.mws . This shows how some of the more tedious calculations needed to obtain the formulae for steady state solutions can be quickly carried out.
Aspects to note from the Comparative statics.xls spreadsheet
(i) Present value maximising outcomes when the discount rate is zero.
The simulations on the “Steady states” sheet demonstrate that the presentvalue maximising model is identical to the static profit maximising model when the discount rate is zero. Try and explain why this is so (and so why the stock level is higher under zero discounting than under positive discounting).
(ii) Present value maximising outcomes when the discount rate is infinity.
We cannot set a value to infinity in an Excel spreadsheet. But we can approximate it closely with a very large number. By making i =1000 in "Steady states", you will see that the present value outcome is more or less identical to that which emerges under open access.
The "Steady states" sheet has been used to calculate the steady state values of S, E and H for interest rates from 0 to 1, and also for i = 10, 100 and 1000. This information is transcribed to Sheet 1, on which we have charted the relationship between S* and i and H* and i for discount rates up to i =1 (100%). Note that as i rises, S falls.
S converges to a level of 0.50 (the open access level) as the discount rate goes to infinity. Why should this be the case? (See the text for some intuition.)
(iii) Present value maximising outcomes when harvest costs do NOT depend on the stock size
The general condition for present value maximisation (in steady state) was shown in the text to be:
(17.33)
However, if H = eE (rather than H = eES) then C = wE/e. Then costs are independent of stock size (C/S = 0) and so (17.33) simplifies to
(17.33)
For our assumed logistic growth model, we have already established that dG/dS = g  (2gS)/S_{MAX}. Equation 17.33 therefore becomes i = g  (2gS)/S_{MAX} and so S* = (g  iS_{MAX})/(2g). For S_{MAX} = 1, g = 0.15, and i = 0.1, this gives S = 0.1667, a much smaller stock size than found in any of our Excel simulations. What is going on here? With a 10% (continuous) discount rate, equilibrium here requires the “growth rate” of the stock to be 10%. But that can only happen in this model with a small stock size. (Note that we must be careful about the phrase “growth rate” in this context. Here we use it to mean dG/dS, which is the slope of the biological growth curve as shown in Figure 17.2 at some particular stock level. However, there are obviously other meanings that could be given to that phrase, which is why confusion might arise.).
In the logistic growth equation, the term g defines the socalled “intrinsic growth rate" of the stock, here assumed to be g = 0.15 = 15%. That is the rate at which the fish stock would grow with respect to the stock when the stock size is very small, and so the fish face no environmental constraints on their reproduction and survival. As the stock size gets larger, these environmental constraints bite more and more strongly, births slow down and deaths increase, and so the growth rate slows down below its “intrinsic” rate of 15%. That happens here by the point at which S = 0.1667.
Question: 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 parameter value assumptions. At what stock size is dG/dS = 0 (the maximum sustainable yield harvest level)? Explain and comment on your findings.
Steady state solution for our assumed functions using Maple
A substantial amount of algebraic manipulation is required to obtain expressions for S*, E* and H* for the NPV maximising fishery. This algebra is shown in the Maple worksheet Chapter 17.mws. Even if you do not know this package, it should be possible to understand what is being done. It may also give you an incentive to learn how to use a mathematical package such as Maple. From this file we obtain expressions for S*, E* and H* in terms of the parameters. Substituting the value i = 0 (that is, no discounting) gives the equivalent expressions for the static (single period) private fishery model.
(3) FISHING DYNAMICS
A third set of materials also uses Excel spreadsheets to illustrate the properties of two fishery models examined in the chapter with numerical examples. Here we are concerned with comparative dynamics. The Excel file fishery dynamics.xls sets up the models on a spreadsheet, and allows you to compare the dynamic behaviour under each regime. Specifically, we examine:

an open access fishery;

a static model of a fishery in private ownership which maximises fishery profits.
The Excel workbook examines how the fishing stock, fishing effort and harvest evolve over time in the two regimes from some arbitrary (nonequilibrium) starting point, and explore how the paths of these variables, and their equilibrium values, alter as relevant parameters are changed. The files also show how dynamic adjustment processes might lead to unstable outcomes, or even to extinction of the fishery.
The files associated with this set of materials are:

An Excel file: Fishery dynamics.xls

A Maple file: Chapter 17.mws
Fishery dynamics.xls
Solution lists the notation used. It then, for the open access fishery model explored in Section 17.3 of the textbook, gives solution expressions for S, E and H in terms of the parameters. These expressions are programmed as formulas in the subsequent Excel worksheets. (The solution expressions for the private fisheries model are also listed for comparison.)
At the foot of this worksheet, we list the parameters used in our baseline model runs, and obtain its steadystate solutions for that set of parameters. You should point to the cells in which those solutions are given, and look at the formula bar at the top of the worksheet to see the Excel formulas, and check that they correspond to the analytical expressions.
Base uses Excel to simulate how, in the open access fishery in which the stock level is initially at its biological maximum (S = SMAX = 1), fishing effort evolves over time from a particular, arbitrary starting level, E = 1. (A maximum stock size of 1 is certainly computationally “convenient” but it may strike you as being rather unrealistic. But note that we can define the stock size in whatever units we desire, and so can without any loss of generality normalise the maximum stock to 1 by an appropriate choice of units. Realism will, of course, require that other parameters be defined reasonably relative to that normalisation).
Note from the two charts on the worksheet how effort (E) and the stock level (S) gradually evolve over time towards stable equilibria at their steadystate values of E* = 8.333 and S* = 0.5. These variables adjust via damped, cyclical processes to their steady states.
A technical Excel point: although neither S nor E become negative at any point in time in this simulation, they could do so for different parameter values. As we intend the reader to see what happens when parameters are changed, we have programmed the Excel formulae to prevent S and E becoming negative (which would be meaningless). To do this we have used:

The Max(0, formula) function to generate S [that is, Excel always returns the greater value of zero or that generated by the formula]

We have used an If(S>0 then formula, else 0) formula for E so that the expression is used to calculate E only when S was positive in the previous period.
QUESTIONS:
Verify that the steady state stock level is at the level that generates the maximum sustainable yield for this fishery. What is the size of that yield?
What happens to fishery profits (and why) over the simulation period shown? How are changes in profits associated with changes in fishing effort?
Critical depensation model
Uses the equation of logistic growth with critical depensation given in Box 17.1 to generate an Excel chart in the form of Figure 17.1d
Random
Presents a simulation of the dynamic adjustment path comparable to that shown in the sheet “Base” but with the growth parameter g a random variable rather than a constant value. (g ~ N(0.15, 0.9)).
PP Model
Calculates the private property fishing equilibrium. The difference between this model and the open access model lies in Equation (3), the capital dynamics equation. Under open access, the change in effort is proportional to the level of total profit. So effort increases (decreases) above (below) its current level when total profit is positive (negative). But under private property, the change in effort is proportional to the level of marginal profit. Therefore, effort will increase if marginal profit is positive, and will only reach a constant level when marginal profit is zero (and so total profit has been maximised). The derivation of the marginal profit function can be found in Chapter 17 of the text.
Charts display the dynamic adjustment path to steady state equilibrium. Note that in this sheet (unlike several others in the workbook) nonnegativity constraints have not been built into the Excel formulae for stock and effort. Computes and displays the steady state equilibria under open access and static private property. Note that other things being equal, steady state stock is higher under private property and effort is lower, as we know from the analysis in Chapter 17. .
QUESTION: Explain why the private property stock level is higher than under open access. Can we unambiguously establish whether the harvest level will be always higher (or always lower) under private property compared with open access, for any assumed set of parameters?
Some additional simulation exercises the reader may wish to do.
(1) Increase the effort adjustment parameter, d, from 0.4 to 1.45 (so that the rate of effort applied changes over three and a half times more quickly in response to fishery profit levels in this simulation than it did in the base case). Leave all other parameters at their base values. The stock chart that results appears to show an undamped “two period cycle”, with the stock repeatedly under shooting and then over shooting its steady state value of 0.5. Note that although the steady state value is unchanged at 0.5, the fishery does not seem to converge to that equilibrium. It certainly does not over the time horizon we have shown.
Use the fill down command to double the number of periods over which the simulation is run. Examine the maximum and minimum levels of stock in each repetition of the cycle to see whether an equilibrium would eventually be achieved. (It does in fact converge, but very slowly).
Why does a change in d alter none of the steady state solution values? If it does not alter these, what does it alter?
Construct a chart for fishing effort. What kind of behaviour does it exhibit over time, and how is this related to stock size?
What happens if d is increased to 3.0?
A CONCLUSION: A high value for the adjustment parameter d can bring about the possibility of species extinction. Effort increases so quickly that stock collapses to zero and so the stock is unable to recover. (e.g. This happens when d = 3).
(2) Increase the gross (market) price from 200 to 350. All other parameters are left at their base values. The stock chart shows a damped cycle, with the stock repeatedly under shooting and then over shooting, but eventually converging to its steady state value of 0.2857.
Examine other possible market prices. How does the outcome vary as the price is changed?
If P is increased in steps (of 50, say) from 350 to 450, and then from 450 to 500, we observe the following. (Try to explain why). As price rises above about 490, the fishery collapses because the adjustment path pushes the stock to zero.
(3) Increase the cost of fishing effort, w, from 0.9 to 1.3 (with all other parameters left at their original “base” values). The stock chart initially shows a smooth convergence to equilibrium. What happens if w is increased to 1.9? Or decreased to 0.1? And why?
(4) To think about an efficient fish landing tax, using trial and error, adjust the market price of fish (P) from its baseline value of 200 until the same steady state fish stock is found in the open access model as under net present value maximising fishery with i = 0.1. [You can find this stock size from the text or from the Excel file Comparative Statics.xls.]
What is the required price, and so what is the implied tax rate?
How does the optimal tax rate change as the interest rate increases?
What do these findings imply about the market equilibrium price of transferable permits to catch fish?
A CONCLUSION: A high value for the adjustment parameter d can bring about the possibility of species extinction. Effort increases so quickly that stock collapses to zero and so the stock is unable to recover. (e.g. This happens when d = 3).
(5) Examine the effects of varying the intrinsic growth rate parameter, g. For example, observe the consequences of g being increased 10 fold from 0.15 to 1.5. In this case, the stock adjusts fairly rapidly, and without any overshooting, to its steady state value, which is unchanged (relative to baseline parameters) at 0.5. What happens to the effort level (and why)?
You may like to investigate this more systematically. For example, increase g in steps as follows: 0.15, 0.3, 1.5, 3, 3.2, 3.3. Record what happens. Observe that for the final value of g in this series (g = 3.3) the stock collapses to zero between periods 11 and 12, and so the fishery goes to complete collapse.
A CONCLUSION: A high value for the growth parameter g can bring about the possibility of species extinction. Effort increases so quickly that stock collapses to zero and so the stock is unable to recover. (e.g. This happens when g = 3.3).
SPECIES COLLAPSE MAY OCCUR BECAUSE OF THE DYNAMICS OF THE ADJUSTMENT PATH EVEN IN CASES WHERE THE STEADY STATE STOCK IS POSITIVE (BUT IS NEVER ACTUALLY REACHED).
(6) Observe the consequences of varying the harvesting efficiency parameter, e. For example, if e is decreased from 0.009 to 0.005 (with all other parameters left at their base values) we observe a non oscillating smooth convergence to a steadystate stock, which has risen from 0.5 to 0.9. Lower harvesting efficiency makes this fishery less attractive; steady state effort falls from 8.33 to 3. What happens if the efficiency parameter decreases a little further to 0.0045 (half its original level), and why?
(7) Plots the G(S) against S for the logistic growth function with our baseline parameter values.
QUESTION: Change g or SMAX and observe how this chart changes. (To do this will you will need to alter the formula itself that generates the stock level in cell B7 and then use the Fill down command to copy this updated formula down the whole column.)
To instructors: A suggested student exercise
To undertake this exercise, use the two Excel files Fishery dynamics.xls and Comparative statics.xls as appropriate.
(a) Consider each of the following parameters of the open access version of fishery model specified in the Excel files:

The price of landed fish.

The cost per unit of fishing effort.

The responsiveness of entry into or exit from the fishing effort to changes in average fishing profitability.

The intrinsic growth rate of the fish stock.
How do variations in each of the those parameters affect

the equilibrium fish stock and harvest;

the likelihood that the fishery will be driven to a state of complete stock exhaustion? (Hint: to answer this, ensure that you consider not only the steady state outcome but also, and more importantly, the dynamic adjustment path towards a new equilibrium.)
(b) For the fishery models examined in Chapter 17, and for identical price, cost and biological growth parameters, how do the outcomes of open access and private property (restricted entry) conditions in the fishery differ in terms of

Steady state (equilibrium) levels of fish stock, fishing effort and harvest;

The likelihood of complete collapse of the fishery?
(4) MODELS OF BIOLOGICAL INTERACTION
The Word file Biological Interaction.doc outlines and explains some alternatives to the singlespecies fisheries models used in Chapter 17. This material was intended to be included there, but was omitted for reasons of spaceshortage. The models examined here involve interactions between two or more species, and so substantially extend the scope of our modelling of renewable resource economics. Pictures associated with this document are to be found in the file Picturesforbiolint.ppt.
We intend later – but have not yet done this – to add further materials which extend, or are alternatives to, the Schaefer approach to biological growth modelling, including: Gompertz growth functions; the BevertonHolt approach (which models separate cohorts (ageclasses) through time, as for example in Flaaten and Kolsvik, 1996 (for a fishery) and van Kooten et al (for wildlife ungulates); and other discrete time fishery models, including the stock recruitment model.
(4) DYNAMICS: PHASE PLANE ANALYSIS
The Maple file Chapter 17.mws and the associated Word file Phase.doc outline, explain, and illustrate the use of phase plane analysis to examine the nature of dynamic adjustment processes.
(5) THE STATE OF MARINE FISHERIES
This Word file contains a summary of the state of Marine Fisheries as assessed by the UNFAO at the year 2000.
(6) BIOLOGICAL DIVERSITY
The file What is causing the loss of biological diversity.doc discusses the various causes of loss of biological diversity.
(7) AGRICULTURE
The Word file Agriculture.doc contains a miscellany of items relating to the topic of agriculture. It includes an investigatation into forms of environmental pollution associated with developed agricultural systems; a case study on the costs of reducing pollution on a Swedish island; and consequences of modern agricultural practices in developing countries.
(8) RENEWABLE ENERGY
The file Renewables.doc provides an outline of some of the main issues arising in the exploitation of renewable sources of energy.
(9) STOCK DEPENDENT UTILITY
The file Stock Dependent Utility.doc generalises the benefits function so that utility depends not only on the harvest level of the resource but also on its current stock level. This file derives the optimising conditions for such a model.
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