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The optimal allocation of the groundwater resource over time by the use of the user cost Alexander, Esther 1970

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THE OPTIMAL ALLOCATION OF THE GROUNDWATER RESOURCE OVER TIME BY THE USE OF THE USER COST by ESTHER ALEXANDER A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A. i n the Department of Economics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S tudy. I f u r t h e r a g ree tha t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . It i s unde r s t ood t h a t c o p y i n g or p u b l i c a t i o n o f t h i s thes,is f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date i i ABSTRACT A mathematical model of maximizing the present-value of the ground-water reservoir i s developed. According to i t the optimal condition of the water use i s : to rat i o n the water i n a way which makes the marginal net rent to the water equal to the marginal user cost each year- The model i s v a l i d for every amount of recharge. The rent function i s derived from the a g r i c u l t u r a l production function and i t i s based on the main assumption that diminishing return to the water e x i s t s . Using the rent function the present-value expression f o r the reservoir has been established. I t i s d i f -ferentiated i n respect to the water used - the chosen independent variable of the rent function. In t h i s way the r e s u l t mentioned above has been achieved. The optimal sequence of the water use - which maximizes the present-value of the reservoir - has been calculated by salving the set of equa-tions of the optimal conditions of each year. Both the optimal sequence of the'water use and the user cost have been calculated e x p l i c i t l y and expressed i n formulas by measurable parameters. I t i s argued that only a p r o f i t maximizing sole owner f o r whom the user cost i s meaningful would use the water i n the optimal sequence. I f the reservoir i s a common property resource a non—optimal water use would be practized. This non-optimal sequence of the water use i s calculated by solving the set of equations of the-non-optimal condition of the water use. of each year. Considering the user cost zero the condition of the water use became: to use the quantity of water every year which makes the marginal net rent equal zero. The optimal condition of the water use developed i n t h i s work, which says marginal rent should be equal marginal i i i user cost made possible to establish the non-optimal condition of the water use by turning the user cost to zero. The non-optimal water sequence i s expressed too i n an e x p l i c i t formula by measurable parameters only. I t i s compared to the optimal water se-quence. Also l i f e - t i m e of the reservoir and i t s f i n a l depth are compared f o r the cases of optimal and non-optimal water use. In the case of o p t i -mal water use (sole ownership) the water use i s more gradual, the econo-mical l i f e of the reservoir i s longer and i t s f i n a l depth i s less deep than i n the case of the non-optimal water use. A numerical example at the end of t h i s work shows that - i n these circumstance - the user cost has a s i g n i f i c a n t magnitude. i v TABLE OF CONTENTS CHAPTER Page I INTRODUCTION 1 1. Maximizing the Present-Value of the Groundwater Reservoir 1 2. The Nature of the Reservoir and the Concept of the Model 4 a) The Diminishing Returns to the Water and i t s Role i n the Use of Water 4 b) The Cost of Water - Pumping Cost and User Cost and t h e i r Role i n the Use of Water 7 ( i ) The Pumping Cost 7 ( i i ) The User Cost 9 3. Sole Ownership and Monopoly 12 4. The Advantages of Expressing the Optimal Condition of Water Use i n Terms of the User Cost 15 a) The User Cost as a C r i t e r i o n of Choice 15 b) Comparison of Sole Ownership and Common Property way of Water Use 16 c) The User Cost and Water P o l i c y Decision Making 16 5. The Role of the Recharge i n the Model 18 I I REVIEW OF THE LITERATURE - COMPARISON OF MODELS 22 1. The Kelso Model 22 2. , The Renshaw Model 26 3. The Burt Model 29 V CHAPTER Page I I I THE MATHEMATICAL MODEL 31 1. Def i n i t i o n s and Assumptions 31 2. The Derivation of the Rent Function from the Production Function and the Derivation of the Demand for the Water from the Rent Function 35 3. The Condition of the Optimum Water Use 41 IV THE OPTIMAL SEQUENCE OF WATER USE ( A ) 47 1. General Form of the Solution 47 2. The Calculation of the Constant R and the Calcu-l a t i o n of the User Cost 51. V THE GROUNDWATER AS COMMON PROPERTY RESOURCE 53 1. The Condition of the Water Use 53 2. Calculation of the Common Property Sequence (y^t) 55 3. The Calculation of the Constant Z 57 VI COMPARISON OF THE USE OF THE WATER IN THE CASE OF THE SOLE OWNER AND IN THE CASE OF COMMON PROPERTY 58 1. The Comparison of R and Z 58 2. The Comparison of A and M. and the L i f e Time of the Reservoir 60 3. Comparison of the F i n a l Depths of the Reservoir 64 i -a) The Case when there i s no Recharge 64 b) The Case when Recharge does E x i s t 66 ( i ) The Cose of Common Property 66 ( i i ) The Case of Sole Ownership 67 VII NUMERICAL EXAMPLE 75. BIBLIOGRAPHY B2 v i APPENDIX I The Condition of the Optimal Water Use. A Different Way of Calculation I I Solving the Set of Equations (20) to get the Optimal Sequence of Water Use I I I The Way of Calculation of the Constant R and the User Cost vii LIST OF FIGURES Figure Page 1 Graphical Exposition of the Procedure f o r Determining the Rent Function to X^ from a Multivariates Production Function 38 2 The Rent Function 39 3 The Demand Curve f o r Water 40 4 Graphical Exposition of the Optimal Condition of the Water Use 45 5 Graphical Exposition of the Optimal Sequence of Water Use 50 6 Comparison of the Common Property Sequence of Water Use (*n(c)) and the Optimal Sequence (X n (s)) when there i s no Recharge 63 7 Same as Figure 6 when there i s Recharge (w) 68 8 The Depth of the Reservoir and i t s Economic L i f e i n the Case of Sole Ownership and i n the Case of Common Property 71 9 The Depth of Using the Recharge Only i n the Case of Sole Ownership and i n the Case of Common Property 72 10 Comparison of the Sequence of Water Use assumed by Kelso with those Computed for the Same Numerical Para-meters from our Equations 80 v i i i ACKNOWLEDGMENT I wish to express my appreciation to Prof. Anthony Scott and Dr. P. G. Bradley f o r t h e i r valuable suggestions and help i n wri t i n g t h i s work. CHAPTER I INTRODUCTION 1. Maximizing the Present-Value of the Groundwater Reservoir The subject of the present work i s the groundwater resource. I t deals with the question how to use the resource i n order to achieve the maximum revenue attainable from a certain reservoir; - i n other words, how to maxi-mize i t ' s present-value. This i s an i n t r i g u i n g t h e o r e t i c a l question but i t has p r a c t i c a l applications too. In many areas i n the world groundwater i s the scarce factor i n agriculture so that to economize i t has f i r s t rate importance. Besides the way of use of the reservoir which a p r o f i t maximizing sole-owner would follow, which maximizes the present-value, we are, able to show i n t h i s work the way of water use when the reservoir i s a common property (1) resource. ••' This comparison, which i s - as f a r as we know - new i n the l i t e r a t u r e on groundwater - i s done by the use of the user-cost, which we have defined f o r the groundwater resource ( i n equation 1) and whose value we have calculated i n equation (30). This i s also new.in the present work. The calculation of the quantities of water to be used each year which maximize the present-value of the reservoir i s not new. These r e s u l t s have been derived by Burt (1967). The analysis i n t h i s work leans heavily on his solution f o r the optimal a l l o c a t i o n of water over time. The r e s u l t of our calculation of the optimal sequence of water use - which maximizes the present-value of the reservoir - i s the same as h i s . What i s new i n the present work - i s the way of the cal c u l a t i o n and the advantages gained by (1) For further discussion of the sole-ownership and common property resource see page 12. that (see page 1,5"). Burt (1967) has assumed - and we have done the same i n equation (5) - a quadratic objective function which represents the rent to (2) the water i n a given year. 1 The present-value of the reservoir i s the the discounted sum of the net rent f o r a l l years (equation 15 i n our case). To f i n d the maximum Burt d i f f e r e n t i a t e s the present-value expression with  respect to the quantities of water retained i n the stock i n a certain year. These quantities are dif f e r e n t from year to year as the reservoir i s de-pleted. Thus the water reserves l e f t i n the ground are the chosen: inde-pendent variables of the Burt function (1967). The condition f o r the maxi-mum - which i s the condition f o r the optimal use of the water - which he gets i n t h i s way i s a purely mathematical expression. I t has the advantage that i t i s convenient to calculate from i t the values of the variables -namely the quantity of water l e f t i n the stock each year. I t has no econo-mic meaning by i t s e l f and does not provide a " c r i t e r i o n of choice" f o r the user of the water. His r e s u l t s do, of course, t e l l the user how much water he should use each year to maximize the present-value but the maximum condi-t i o n does not provide any economic explanation why these p a r t i c u l a r quanti-t i e s of water would do that. In our objective function, which we c a l l the rent function (equation 5) the chosen independent variables are the quantities of water used each  year and not the quantities retained i n stock. We define the rent to the water according to Kelso (1961): "This i s the amount above operating outlays, current r e a l estate taxes and oppor-tuni t y charges f o r management, operator and family labor and interest on (2) For further discussion of the rent to the water see page 35. - 3 -non-real estate investment". In other words the rent to a unit of water i s the revenue received from the crop associated with t h i s unit of water after a l l the other expenses - except f o r the pumping costs of the water - have been deducted. Our rent function shows how the rent to the water changes, f 3) i n a given year, when we change the quantity of water used. 1 Our present-value expression (equation 15) i s the discounted sum of the net rent (that i s the rent minus the pumping costs) over the l i f e t i m e of the reservoir. I t assumes - l i k e Burt's expression (Burt, 1967) - constant factor and pro-duct prices and a constant discount rate over time. Thus i n our expression f o r the present value (equations 15 and 17) the variables are the quantities of water used each year. To maximize the present-value we d i f f e r e n t i a t e the expression with respect to the quanti-t i e s of water used (equations 18-19) instead of the amount of water l e f t i n stock. In t h i s way we get that the conditions that maximize the present-value  are: Each year the marginal net revenue should be equal to the marginal user cost (equation 22) I t i s an optimum condition with a c r u c i a l economic meaning by i t s e l f . I t also provides a " c r i t e r i o n of choice" f o r the user of the water which t e l l s him when to stop the use of water i n a certain period i f he wants to maximize the present-value of the reservoir. A. Scott (1955, 1967) has proved that the above r e s u l t - marginal net revenue equals marginal user-cost - i s the marginal condition to maximize (3) For a more detailed discussion of the nature of the rent function see pages 35 et seq. ( 4 ) For the e x p l i c i t d e f i n i t i o n of the user-cost i n the case of ground-water see page 9. - 4 -the present-value of every exhaustible (stock) resource. Here we have proved i t i n a mathematical way by d i r e c t d i f f e r e n t i a t i o n . As f a r as we know t h i s i s new i n the l i t e r a t u r e on natural resources. 2. The Nature of the Resource and the Concept of the Model a) The Diminishing Return to the Water and i t s Role i n the Use of  Water In the preceding paragraph we have shown how we get i n a mathematical way the optimizing p r i n c i p l e MNR = MC u f o r the groundwater resource where MNR = Marginal Net Revenue, MCu = Mar-g i n a l User Cost. This equality should hold f o r the water used each year. Thus we have an equation MNR = MCu f o r each year and therefore a set of equations which have to be solved simultaneously (equation 20). I f we solve f o r the quanti-t i e s of water used each year we get the r e s u l t that we maximize the present-value of the reservoir i f we ra t i o n the water over time and use a certain  d e f i n i t e amount every year, and not by emptying i t i n one period. The question arises i f these r e s u l t s are acceptable. Why not use up the whole reservoir i n the present when a positive discount rate e x i s t s i n the economy - which we c e r t a i n l y assume? Mathematically we get the r e s u l t of rationing water over the years because our rent function has a negative quadratic term (equation 5) which represents diminishing returns to the water (see graphical exposition i n figure 2 and figure 3). I f we maximize the present value expression (equa«* ti o n 15) based on a rent function which assumes diminishing returns we get the r e s u l t that we have to r a t i o n the water i n order to maximize the present-value of the resource. The question which arises now i s i f the assumption of diminishing r e -turns to the water i s a r e a l i s t i c one and how i t a c t u a l l y stops the water user from using up the reservoir i n one year. Before we answer the question we want to c l a r i f y the concept of dimi-nishing returns i n t h i s context. In general diminishing returns means that when one increases the production of a product the t o t a l net revenue i n -creases with a diminishing rate. We may also say, which i s the same, that the net revenue to an additional unit of product decreases as the produc-t i o n i s increased. In connection with diminishing returns we always t a l k about the net revenue to a unit of product as Marshall (1920, page 167) says: "... the returns of which we speak i n the law of diminishing return i s a net y i e l d " . The phenomenon of diminishing returns e x i s t s i n ore mining too. Scott (1967), who reaches by diff e r e n t tools the same conclusion of rationing of ore production over time, i n order to maximize the present—value of the mine, also bases t h i s r e s u l t on the assumption of diminishing returns to mining. There i s however a difference between the nature of diminishing r e -turns f o r ore mining and f o r water mining. In Scott's model (A. Scott, 1967) the diminishing returns e x i s t because of the increasing marginal cost of mining when the production i s increased. In the case of the groundwater resource the cause of the diminishing returns i s the decrease of the rent to an additional unit of water when the water use i s i n -creased. This appears i n Burt's model (Burt, 1967) and i n our model too. 6 In Hotelling's model ( H o t e l l i n g , 1931) the cause of diminishing returns to mining are the f a l l i n g prices of the products with increasing supply. The diminishing returns to the water are mainly due to the differences i n the qua l i t y of land. There i s land of high q u a l i t y and of lower wuality. Burt (1967) expresses i t as follows: "In i r r i g a t e d agriculture marginal net output w i l l tend to decrease with rate of use because of decreasing returns to the water on a given acreage and the l i m i t to p o t e n t i a l l y i r r i g a b l e land within the basin. The most productive land, and:that having the le a s t development costs w i l l come under i r r i g a t i o n f i r s t which w i l l also tend to give decreasing returns to the water." Certain kins of land provide higher rent to a unit of water than other kinds of land a f t e r each kind has received the most proper crop and c u l t i v a -(5) t i o n - as i s assumed i n the rent function. v ' The r a t i o n a l farmer - we assume r a t i o n a l i t y as the basic assumption of human economic behavior - would i r r i g a t e the land which gives high rent to the water f i r s t and gradually go on to lower quality land. Thus the margi-nal rent to the water i s diminishing while the water use i s expanding. At the point where the marginal net rent vanishes the r a t i o n a l user stops the use of water. He w i l l s t a r t again i n the following year when the higher q u a l i t y lands are free again. Another reason f o r the diminishing return to the water i s the nature of the crop i n agriculture. There i s an optimum quantity of water f o r each crop. More than that decreases the returns. We have seen here what we mean by diminishing returns to the water and (5) For further discussion of the rent function see page 35. - 7 -how i t stops the use of water i n the present. We have also seen that i n the case of groundwater f o r i r r i g a t i o n one period of use means a year. In the same way i n the case of ore mining the l i m i t i n g factor which brings about diminishing returns to mining i s the increasing cost of mining which brings about the stopping of the mine's exhaustion i n the present per-iod (Scott, 1967). b) The Cost of Water. Pumping Cost and User Cost and t h e i r Role i n  the Use of Water. < ( i ) The Pumping Costs The di r e c t cost of the water used i s the pumping cost. A unit of pump-ing cost i s defined as the cost of l i f t i n g one acre foot of water on foot. In our model i t i s assumed constant, independent of the t o t a l depth. The marginal pumping cost - the cost of l i f t i n g out an additional unit of water -i s then a function of the depth of the water l e v e l i n the reservoir (equa-ti o n 2a). As a r e s u l t i t depends on the t o t a l amount of water previously pumped out. Thus there i s a cumulative cost which depends on the u t i l i z a -t i o n of the reservoir i n the past and not on the current rate of output. The marginal pumping cost increases from year to year - because the depth of the reservoir increases - i f any production i s taking place. This w i l l be true whether the annual rate of use i s constant, increases or decreases i n time. (Unlike the non-cumulative marginal production costs which increase, decrease are constant, according to the annual rate of output.) For t h i s reason we cannot say that the pumping cost i s a l i m i t i n g factor which causes the rationing of water over time - i n spite of the fact that the costs are increasing with the production. The r a t i o n a l farmer would not gain anything i n pumping costs by l i m i t i n g the production i n the present period because the next unit of water would have the same pumping costs whether i t i s (6) pumped i n t h i s period or i n the next period. ' There i s thus a cumulative cost, as we have emphasized already, quite d i f f e r e n t from the cost of pro-duction i n an industry or mine. We have seen why the pumping costs cannot be the " l i m i t i n g f a c t or" and l o g i c a l l y the law of diminishing returns does not apply to them. As Scott (1967, page 28) has pointed Out "This law should be stated as a function of the rate of output, not of the remaining reserves". Marshall (1920, page 167) uses the example of pumping out of a reservoir to show that there i s no diminishing return to mining. Scott (1967) shows that t h i s conclusion was wrong and there are diminishing returns to mining. They r e s u l t however from the increasing of cost with the rate of production and not from the costs which stem from the deepening of the mine. The l a t t e r are cumulative and equivalent to our pumping cost. To t h i s type of cost the law of dimi-nishing returns i s just not applicable, ( i t i s v a l i d only f o r production costs which increase per unit of production when we apply more and more factors i n the same place and at the same time.) Nevertheless the pumping costs do play an important role i n determin-ing the way the reservoir i s used. They appear i n the net rent with a n e -gative sign: This means that they make the rent to the water smaller than i t would be otherwise. Obviously the net rent to the water decreases fast e r than the rent to the water. Thus the user of the water reaches the (6) In our discussion below we neglect the increase i n pumping costs within each period. This i s j u s t i f i e d when the depletion of the reservoir i n any one period i s small but would of course be wrong i f the whole reservoir or a large part of i t would be used i n any one period. - 9 -point where the net rent equals zero i n a given period - and stops the use of water - a f t e r using a smaller amount of water than he would use without pumping costs. In other words the r a t i o n a l user stops the use of water when the marginal gross rent equals the marginal pumping costs and not when i t equals zero. Thus the eff e c t of the pumping costs i n water production i s to decrease the production i n each period. This r e s u l t i s brough about by the mere existence of the pumping cost. The cumulative property of the pumping costs - the increase of the marginal pumping cost from year to year - therefore causes a decrease i n the use of water each year. The marginal gross rent has to be equal to the marginal pumping costs - and therefore also increases from year to year. Because of the diminishing returns to the water t h i s can be achieved only by using less water each year than i n the preceding year. ( i i ) The User Cost In the process of production the resource i s depreciated. This i s also a cost of production. We c a l l i t the user cost. As the u t i l i z a t i o n i s advancing ;the reservoir i s getting deeper and deeper and the water becomes more and more expensive because of increasing pumping costs. Thus we can see that the true cost of a unit of water taken out i s composed of two parts: f i r s t there i s the straight pumping cost and second| by pumping i t out we have increased the pumping cost of a l l the water l e f t i n stock. The second cost i s the user cost. We define the marginal user cost; i t i s the present-value of the increase i n the pumping  cost of a l l the water which w i l l be used i n the future because of the pump- ing out of one additional unit of water i n the present (equation 1). Scott (1967, page 34) defines the user cost as follows: "... user cost i s de-fined as the maximum increment to the mine's present-value that could be gained by a decision to allocate the unit of output to a future period." - 10 -We can see that our d e f i n i t i o n f i t s that of Scott (1967). The p r o f i t maximizing sole owner of the reservoir would not take out an additional unit of water i f i t s contribution to the present value of the reservoir (the mar-g i n a l net rent) i s less than i t s deduction from the present value (the user cost as defined above). He w i l l allocate the water to a future period when  the net rent to i t i s lower than i t s user cost. Thus the p r o f i t maximizing sole owner stops the use of water i n a given period when the marginal net rent equals the user cost. This i s exactly the re s u l t we get by d i f f e r e n t i a t i n g the expression f o r the present-value with respect to the quantity of water used (equations 19 and 22). In other words  we can interpret the expression we get by d i f f e r e n t i a t i o n as the user cost. The user maximizes the present-value of the reservoir i f he determines the quantity of water he uses each year so that the marginal rent becomes equal to the user cost i n that year. The user cost i s recognizable only i n the economic considerations of a sole—owner of the reservoir. A sole-owner i s the only person - mostly a public authority i n the case of the groundwater resource - who has the r i g h t to use the reservoir. For the sole owner there i s a value attached to the water l e f t i n stock. This value i s the present-value of a l l future pumping costs saved by leaving t h i s ' u n i t of water i n the a q u i f i e r . The sole owner i s able to consider t h i s and to allocate the marginal unit of water i n the most prof i t a b l e way - that i s i n a way which adds the maximum to the present value of the resource. When the user i s not a sole-owner but one of many who have the r i g h t to use the reservoir - i n other words, i f the reservoir i s a common pro-perty resource - the water l e f t i n the stock has no value f o r him. For a - 11 -common property user only the water pumped out - which earns rent - has value. There i s no use f o r him to leave water i n the stock i n order to pre-vent the deepening of the reservoir, because someone else w i l l take the water out. Scott and Christy (1965, page 6) define the common property natural resource: No single user has exclusive use righ t s to the resource nor can he prevent others from sharing i n i t s e x p l o i t a -t i o n . An increase i n the number of users affects each user's enjoyment of the resource." Even though there i s no value to the water l e f t i n stock f o r the com-mon property user, the diminishing returns to the water force him - as we have shown above - to stop the use of water each year at the point where the marginal gross rent equals the marginal pumping cost. Kelso (1960) shows how the common property user stops the use of water by himself be-cause of r i s i n g pumping costs. He does not assume diminishing returns to the water e x p l i c i t l y but i t i s c l e a r l y i m p l i c i t i n his model. The sole owner also stops the use of the reservoir i n the present because of the diminishing returns to the water. Because he takes the user cost into account h i s stopping point i s however dif f e r e n t from that of the common property user. He stops the use of water i n each period when the marginal rent to the water equals the marginal pumping cost plus the marginal user cost. We can see here the r o l e of the user cost i n determining the way the groundwater reservoir i s used. While the diminishing returns to the ' water are responsible f o r the fact that the use of water i n each period i s li m i t e d , the user cost determines the optimal quantity of water to be used - 12 -which maximizes the present value of the reservoir We can also conclude that only the sole owner, for whom the user cost ; i s a r e a l cost, i s able to take t h i s into consideration. Thus the sole ownership i s the e f f i c i e n t form of ownership which provides the way of water use which maximizes the present-value of the reservoir. 3. Sole Ownership and Monopoly I t would be a misunderstanding to confuse sole ownership with monopoly and common property use with perfect competition and to conclude that our argument implies that monopoly i s more e f f i c i e n t than perfect competition. The i n a b i l i t y of the common property user to recognize the user cost i s a market imperfection. I t r e s u l t s i n excessive use of the resource and therefore i n i t s "exploitation". This means that the rent to the resource f o r i t s users decreases and i n some cases disappears completely. Common ownership or free use of a natural resource i s e f f e c t i v e l y equi*-' valent to no ownership of a production factor. I t i s quite di f f e r e n t from free or perfect competition which does assume sole (or private) ownership of the production factors. Scott (1967, page 33) compares the exhaustible natural resource to a machine to which we have to apply other production factors i n order to pro-duce and which produces a f i n i t e amount of product during i t s l i f e t i m e . C l e a r l y free access to the use of the machine i s not what we mean by (7) In Scott's model of mining (Scott, 1967) the user cost plays the same ro l e . The increasing cost of mining and the positive discount rate are not s u f f i c i e n t to determine the rate of output which maximizes the present-value of the mine. For that the user cost must, be taken into account. perfect competition. In that case no user would consider the depreciation of the machine and t h i s would c l e a r l y lead to misallocation of resources. Establishing of sole-ownership of a groundwater reservoir - or of any other exhaustible common property natural resource - comes to remove the market imperfection described above. I t s purpose i s to stop the free access to the use of the resource and to provide a way of use which maxi-mizes i t s present-value and provides the f u l l rent to the users. We have seen above that only the sole owner for whom the depreciation of the r e -source i s a r e a l cost, i s able to achieve t h i s . A regular monopoly ex i s t s independently of any kind of market imper-fe c t i o n . I t does not come to correct any. As economic science has proved i t causes misallocation of resources. In the l i t e r a t u r e (Smith, 1967) sole ownership i s sometimes just a way of use which a hypothetical sole-owner would follow. A public authority - which i s usually the sole owner of a former com-mon property natural resource, may enforce optimal use by different kinds of regulations and taxation without a c t u a l l y establishing ownership. Such regulations are, f o r example, widely used i n the f i s h e r i e s (Scott and Christy, 1965). Sole ownership of a reservoir also does not assume that i t i s the only reservoir i n the country or that there i s no other form of water production which would be assumed f o r a monopoly. On the contrary, i n the l i t e r a t u r e and also i n our model - sole ownership i s related to one d e f i n i t e reservoir which i s one out of many, and the farmers of i t s basin have no control over the Crop.prices i n the. market. They are given to them. - 14 -This i s an underlying assumption of our production function and i s not i n contradiction to sole ownership but i s obviously inconsistent with mono-poly. ' In summary we can say that 'sole-ownership" i s a careful formulation. I t does not say anything about the position of the resource i n the economy. I t may be one of many or an only resource. I t does not assume, but also does not exclude monopoly. I t does not say i f the sole owner i s private or a public authority. I t could be either. Sole ownership only assumes a single owner who i s interested and knows how to maximize the present-value of the resource, as opposed to many common property owners who are not able to do t h i s . In certain circumstances sole ownership of a resource may be a monopoly. Even i n t h i s case i t may be j u s t i f i e d on e f f i c i e n c y grounds l i k e other mono-polies which come into existence because of di f f e r e n t kinds of market im-perfections such as "natural monopolies". This i s however a more complicated question which involves the problems of the desirable form of ownership and • of public intervention i n natural resources. This i s not the place to d i s -cuss these problems. We only want to emphasize that the concept of sole-ownership i s d i f f e r -ent from the concept of monopoly. (8) For a discussion of t h i s subject see A. Scott, "Resourcefulness and Responsibility. - 15 -4. The Advantages of Expressing the Optimal Condition of Water-use i n  Terms of the User Cost At the beginning of t h i s chapter we said that Burt (1967) has calculated a sequence of water a l l o c a t i o n over time which maximizes the present value of a ground water res e r v o i r , without expressing the user cost i n his ca l c u l a t i o n . We have obtained the same sequence i n a d i f f e r e n t way which introduces the user cost e x p l i c i t l y and obtains an expression f o r i t . What have we gained by doing this? a) The User Cost as a C r i t e r i o n of Choice F i r s t , by d i f f e r e n t i a t i n g the present value expression with respect to the quantity of water used (equation 19) we get an expression (the r i g h t hand side of equation 22) which we are j u s t i f i e d i n c a l l i n g the user cost of the water used. Thus we have proven — and as we have mentioned t h i s i s new i n the l i t e r a t u r e - by d i r e c t c a l c u l a t i o n , that i n order to maximize the pre-sent value of an exhaustible resource marginal net revenue should, i n each period, equal marginal user cost. Thus by obtaining an e x p l i c i t expression f o r the user cost (equation 22) we have provided not only a c i r t e r i o n of choice, the optimal stopping point of water use, which Burt (1967) has also provided. We are also able to t e l l why t h i s amount of water i s the optimal one: namely because the marginal net rent to i t i s equal to the marginal user cost. The expression of the user cost gives us more insight into the use of the resource. The user cost i t s e l f can be a p r a c t i c a l c r i t e r i o n of choice too, because we have obtained a formula which expresses i t i n terms of known measurable parameters (equation 30). With the help of t h i s formula -which i s new i n the l i t e r a t u r e as f a r as we know - the value of the user 16 -cost can be calculated and the user of the water can get i t each year as a de f i n i t e positive number. Thus i f the marginal net rent reaches t h i s num-ber i n a certain year, the p r o f i t maximizing sole owner w i l l stop the use of water. b) Comparison of Sole-Ownership and Common Property Way of Water Use Second, the expression of the user cost also enables us to compare the way of use of a reservoir when i t i s under sole ownership with i t s use as a common property resource. After d i f f e r e n t i a t i n g the present-value expression we get the optimal conditions of water use f o r each year. This i s a set of equations which say: marginal net rent equals marginal user cost i n each year. When we c a l -culate the quantities of water to be used each year from t h i s set of equa-tions we get the optimal sequence of water use, which would be realized under sole ownership. Now i f we equate the marginal net revenue i n a l l these equations to zero, instead of equating them to the user cost (equation 31) and use these equations to calculate the quantities of water used each year, we get the sequence of water use when the reservoir i s a common property resource -(see page 56 ). This sequence i s , of course, not optimal. This comparison - and the cal c u l a t i o n of the non-optimal water use of a common property reservoir - i s also new i n the l i t e r a t u r e as f a r as we know. I t i s made possible only by having the optimum condition expressed i n terms of the user cost. In Burt's (1967) method of calculation i t cannot be done. c) The User Cost and Water P o l i c y Decision Making Third. The expression f o r the user cost ( r i g h t side of equation 22) and especially i t s e x p l i c i t formulation (equation 30) enable an interested - 17 -authority to calculate the value of the user cost and to use i t f o r pricing the water. Thus the user cost i s a very useful number f o r water poli c y de-c i s i o n making. I f the public authority i s a sole-owner of a reservoir, which s e l l s the water, and i t s p o l i c y i s the most e f f i c i e n t water use - which i s not necessarily the only possible po l i c y - i t can achieve i t s aim by equat-ing the price of a unit of water to the sum of the marginal pumping cost and the marginal user cost. The pumping cost can be determined d i r e c t l y and the user cost from our expressions. The farmers w i l l then stop the use of water, each year, when the marginal rent to i t reaches t h i s price. In t h i s way, as we have seen, they w i l l use the amount of water i n each period which maxi-mizes the present value of the reservoir. I f the authorities f i n d the user cost negligible and the cost of i n t e r -vention higher than the gain they may decide to leave the reservoir f o r a..,-mon use. In t h i s case - as we have seen - the users w i l l stop the water use each year when the marginal rent to i t reaches the pumping cost (Kelso 1961). The water authorities may f i n d t h i s perfectly s a t i s f a c t o r y on e f f i c i e n c y grounds from the society's point of view. I f they are the sole owners they may s e l l the water i n t h i s case at i t s pumping cost - which has the same r e s u l t . But they could not reach t h i s decision i n a r a t i o n a l way without knowing the user cost. I f the public authorities have other preferences than maximizing the rent to the reservoir, and are therefore interested i n subsidizing or over-pr i c i n g the water, knowledge of the f u l l e f f i c i e n c y price may serve as a good measure of the r e a l magnitude of the subsidy or overpricing. I f an authority prefers to ensure the use of the rent maximizing t o t a l amount of water each year by d i s t r i b u t i n g water r i g h t s instead of pric i n g - 18 -the water i t can do t h i s without using the user cost and without using the ef f i c i e n c y price of the water. Burt's model (Burt, 1967) which determines the optimal amount of water to be used each year i s perfectly enough for t h i s purpose. I f they want to enforce t h i s d i s t r i b u t i o n by taxing the water used above the " r i g h t s " the user cost w i l l serve as a good basis f o r the taxes. I f the taxes are set at the l e v e l of the user cost and are therefore added to the farmer's own pumping cost he w i l l use the r i g h t amount of water. Thus - as we have shown above - the user cost can serve as a very use-f u l instrument i n the water pol i c y decision making process. Our formula (equation 30) makes i t possible to calculate i t s value i n terms of measur-able parameters. 5. The Role of the Recharge i n the Model So f a r we have discussed the ground water reservoir as a special kind of mine. This i s r i g h t i f there i s no recharge or i f the recharge i s i n s i g -n i f i c a n t . What happens however i f there i s s i g n i f i c a n t recharge? Can we s t i l l treat the reservoir as an exhaustible resource? When the stock of water i s exhausted and the users use the recharge only the groundwater i s no longer a stock resource l i k e the mine. As long as the stock i s not exhausted there i s however no reason not to use the stock water too - that i s to mine the reservoir together with the use of the recharge - even i f i t i s s i g n i f i c a n t . Leaving usable water i n the ground unused i s not less a waste than the use of expensive water f o r cheap crops which we wanted to eliminate. Both contradict the maximization of the present-value of the reservoir. When the recharge i s s i g n i f i c a n t our - 19 -problem i s to determine how much to use out of the stock above the recharge, i n each year i n order to maximise the present value of the reservoir, i n -stead of determining the quantity to be used out of stock, when there i s no recharge, f o r the same purpose. As we see the two problems are e s s e n t i a l l y the same and we can put the recharge into our expression f o r the present value (equation 17) without changing i t s c h a r a c t e r i s t i c s . Thus as long as the reservoir i s not exhausted and the use of the stock i s essential to the present-value maximization i t s use remains a mining problem regardless of the magnitude of the annual recharge. The recharge can vary from zero to r e a l significance and the way of solution remains the same. j We use t h i s concept of the resource f o r our model. I t i s a r e a l i s t i c concept and i t has the great advantage that i t enables us to use the same model f o r every amount of recharge. Burt's model (Burt, 1967) has the same property. Thus i t i s not necessary to establish separate models for ground-water mining when there i s no or very l i t t l e recharge and f o r the case when the recharge i s large - as Renshaw has done (Renshaw, 1963). In our model the exhaustion of the reservoir has physical as well as economical meaning. A reservoir i s exhausted physically when there i s no more stock water l e f t i n i t . But i t i s exhausted economically when the water platform gets so deep and the pumping cost so high that i t i s no longer pr o f i t a b l e to pump the water out. We do not use the assumption of homoge-neous reserves - l i k e Scott (1967) uses i n his model of mining. This would be u n r e a l i s t i c i n the case of groundwater. Instead we assume - as we have seen already - that the water close to the surface i s cheaper than thB water l y i n g deeper down because the pumping cost increases with depth. When the reservoir reaches a depth where there cannot be any more net revenue to the - 20 -water from mining the reservoir we consider the reservoir exhausted. From that time on the users use the recharge only and there i s no more mining. In the model, f o r mathematical convenience - i n order to make i t pos-s i b l e to work with i n f i n i t e series the t o t a l exhaustion of the reservoir happens only at i n f i n i t e time. The water use i s diminishing from year to year and assimptotically approaches through time the quantity of the r e -charge (see figure 4). I m p l i c i t l y t h i s assumes that the reservoir i s never exhausted physically and that the recharge i s s u f f i c i e n t l y large so that farming at the recharge l e v e l of water supply i s s t i l l p r o f i t a b l e . The model can be extended to the more general case, as indicated very b r i e f l y by Burt (196?), but t h i s considerably complicates the mathematics and i s not discussed here. I t also can be shown to have a r e l a t i v e l y small effect on the r e s u l t s except very close to the physical exhaustion or the point where agriculture i s stopped. The recharge i n our model i s an annual average and i t i s constant. I t i s assumed constant f o r the purpose of the calculation l i k e any other para-meters which are assumed constant i n time - f o r example the pumping cost from a given depth - f o r the same purpose. I f we are going to use the model fo r water pol i c y purposes we can use the a c t u a l l y measured water l e v e l -which depends on the actual (random) recharge i n the past - i n the equa-tions at the beginning of each year together with the best value of the average recharge and thus calculate each year the optimal way of use and the current user cost. This means that we determine the use of the reser-v o i r each year as though t h i s were the f i r s t year of use. This i s the way we should work with the model. I t i s a way of substituting r e a l measured parameters at each stage f o r our constant ones which were instruments to work out the optimal way of use of the reservoir. Burt writes about the use of the model (Burt, 196?): "The nature of the poli c y or strategy stemming from the use of a sequential decision r u l e may need some elabora-t i o n . The po l i c y i s a conditional decision r u l e that s p e c i f i e s the quantity of water to be withdrawn from storage f o r a given stock of water ... The (actual) rates of use i n future years are not determined u n t i l storage i s observed at the beginning of the year i n question ... Thus a great deal more information i s u t i l i z e d than would be possible i f rate of use were specified f o r more than a year i n advance." - 22 -CHAPTER I I REVIEW OF THE LITERATURE: COMPARISON OF MODELS (9) 1. The Kelso Model 1 J The data used i n the model are related to the stock water resource of Central Arizona. The model emphasizes the importance of the pumping cost which increases with the depth of the aquifer - or i n other words: with the use of the water - as a natural regulating factor of the water use over time. The assumptions the model i s based on are: (1) The aquifer i s a common property. The rule of "capture" i s v a l i d and i t " e n t i t l e s " every farmer to use the water below his land. There i s no central regulation whatsoever which would r e s t r i c t the farmer to us the water as he. wants, even the "ownership" of the resource being shared by the other farmers. (2) The in d i v i d u a l farmers are p r o f i t maximizing. They want to maxi-mize the net rent to the water. Kelso defines t h i s as the t o t a l revenue to the i r r i g a t e d land, minus a l l the payments to a l l the productive factors except the water, l i k e : land rent, rent to machines, wages, payment to the owner, etc. The net revenue l e f t i s the payment to the water. This d e f i n i -t i o n f i t s the d e f i n i t i o n of rent. This i s the gross rent to the water. Net rent to the water i s gross rent minus pumping cost. For a p r o f i t maximizing i n d i v i d u a l user of a common property reservoir the water l e f t i n the ground has no value, because, i f he leaves i t there, (9) Kelso (1961). This paper i s referred to as Kelso i n t h i s chapter. - 23 -somebody else may take i t out. For him the stock water value i s zero, be-cause he cannot "save" i t . So he w i l l take out a l l the water which gives him any net rent above zero. (3) The area under i r r i g a t i o n i s constant. The number of farms and scale of farms are constant. (4) The composition of crops i n the farm does not change. (5) Prices and technology do not change during the years. (6) The pumping cost of l i f t i n g one acrefoot water one foot higher i s the same i n every depth of the aquifer. Here i t i s $0.04. Under these circumstances what i s going to happen to the water use over time? We assume that an acre of land gives $100 of gross return to the water i f i t i s i r r i g a t e d by 51/2, and $75 gross return i f we use 3 2/3 acrefeet of 1/ acrefeet water per acre of land per year. I f the farmer continues to use 5/2 — — — acre water, the net rent to the water - gross revenue ($100) minus pumping cost (5/^2 x 0.04 x depth) - decreases faster per year than i n the case of con-tinuous use of 3 2/3 a c ^ ^ e t water - $75 - (3 2/3 x 0.04 x depth). The acrefeet reason i s f that the pumping cost i n the case of 3 2/3 — — — water use i n -acre creasing much more slowly per year because the deepening per year i s less and the amount of water to l i f t i s l e s s , than i n the former case. Thus even thoug the gross revenue to the water i n the former case i s higher, than i n the l a t t e r case, there should come a year when the net revenue to the water per acre i n both of the cases w i l l be the same, ( i n the year 24 and $25 r e -acrefeet spectively). From t h i s year on the use of 3 2/3 — — — — water w i l l be the acre profitable one. - 24 -Thus i t i s proved, that the increasing pumping cost with the increasing use causes diminishing water use over time without any intervention of autho-r i t i e s . They w i l l stop to pump out water from the ground i n the year, when because of the high pumping cost the net t o t a l rent to i t w i l l be zero. This t terminates the economic l i f e of the reservoir. This i s shown i n the Kelso diagram below. Figure 1 Kelso's Diagram (10) The l i n e A shows the case of continuous water use of 5/2 . The acre l i n e B the use of 3 2/3 a c r e f s e t 0 f w a t e r after the year 24. The l i n e B(1) acre i s the t h e o r e t i c a l l i n e (our addition) shows the net rent to the water i n the case of 3 2/3 — T T T T " " of water use from the depth we get by using 5/2 acre acrefeet acre water i n each year. We see, that t h i s crosses and, l i e s above the Line A a f t e r the year 24, as described before. (10) I t i s an interesting question, i f the Kelso model assumes diminishing returns of water. I t does even i f i t does not state i t e x p l i c i t l y . We see i n the model, that 51/2 acrefeet w a t e r yields $100 gross rent, and 3 2/3 of the water yields a c r s 3/4 of the revenue. I t means, i f we add 1/3 more water, we get only 1/4 more revenue. - 25 -Line C i s the net rent to the water i f an authority intervenes and a acrefset water-use r e s t r i c t i o n p o l i c y makes farmers use 3 2/3 — — — — — water from acre the beginning. The question, which Kelso r a i s e s , i f i t was worthwhile? How to see i f i t has increased at the present-value of the groundwater stock or not? The answer i s , i f the present-value of the tri a n g l e UVW - the loss - at a " f a i r " discount rate i s less than the present value of the area WXYZ - the gain -the pol i c y was j u s t i f i e d . . r , I The model makes hints about the problem of maximizing present-value of the resource, and about a need for "choice c r i t e r i o n " of the water use i n each year but does not specify them. I t does not specify a " c r i t e r i o n of choice" and does inot show a way to f i n d the present-value maximizing quanti-t i e s of water to use each year, ( i t does not claim that the l i n e C does "maximize" present-value). But he states, that i f we f i n d a way to define the quantities of water to be used each year i n order to maximize present-value of the, ^resource, we have to compare i t to the "way of use foregone" -the water use without intervention - and so to decide i f i t w i l l be worth-while to enforce such a p o l i c y , or the difference between the two i s not big enough to j u s t i f y i t . In t h i s work we are going to develop a " c r i t e r i o n of choice" which t e l l s whether to use an additional unit of water or not to use i t . And on t h i s basis to calculate the optimal amount of water to use each year. Furthermore we w i l l compare the way of use i n the case of maximizing pre-sent value and i n the case Df common property use of the resource. The formulas developed i n t h i s work could show us what are the differences and how large they are i n s p e c i f i c s i t u a t i o n of a certain reservoir. Then the authorities could decide what i s the best pol i c y to follow. - 26 In the l a s t chapter of t h i s work the data of the Kelso model are used fo r numerical example of the above mentioned general formulas. (11) 2. The Renshaw Model1- J The model i s composed to two parts. The f i r s t part deals with the case, when there i s no recharge e x i s t i n g or i t i s i n s i g n i f i c a n t i n respect to the amount of water withdrawn each year. This i s the case of the Texas High Plains where water withdrawal i n 195? was 5,000,000 acrefeet and the r e -charge only 30,000 acrefeet. The second part deals with the case recharge i s not i n s i g n i f i c a n t . The model assumes common property f o r the ground water resource. Thus f o r the in d i v i d u a l users the water i n the stock has zero value. They w i l l pump i t out as f a r as i t gives any revenue above zero. The f i r s t part of the model shows us, that i t i s a very wasteful way of using water when o recharge e x i s t s , and what we need i s optimal mining. The model assumes e x p l i c i t l y l i n e a r diminishing return to water i n the agriculture - not l i k e the Kelso model - and on t h i s basis - according to the law of the triangles - i t shows us, that cutting the water use by half i n a certain year, w i l l y i e l d only one fourth less of the t o t a l revenue to the water. (What i s the revenue to the water i s not defined c l e a r l y l i k e i n Kelso's model.) This i s expressed i n the diagram below: (11) Renshaw (1963). This paper i s referred to as Renshaw i n t h i s chapter. - 27 $ Marginal net rent Thus we can double the economic l i f e time of the reservoir and gain by that. This suggestion may be an attempt to maximize present-value, but i t r e a l l y does not. I t i s assumed that people w i l l use every year the same amount of water. The pumping cost i s neglected. I t i s not assumed to regulate water use likis i n the Kelso model. Here i t i s stated that only 2/3 of the water i n the ground can be p r o f i t a b l y removed f o r purposes of i r r i g a t i o n - probably be-cause of the increasing pumping cost, but i t i s not stated. I t i s assumed, i f there i s no intervention, people w i l l use the same amount of water every year - and what i s more important - w i l l earn the same amount of net revenue - $13,500 per year i n the farms of the Texas High Plains f o r 30 more years. There w i l l not be any more economically pro f i t a b l e water and i n lack of any i r r i g a t i o n the revenue per farm w i l l drop suddenly to $6,000 per year. The alter n a t i v e conduct i s , the model states, to r e s t r i c t water use and have a $10,240 per year per farm revenue f o r 60 more years. We do not think i t i s conceivable, that the net revenue stays on the same high l e v e l a l l through the years and then suddenly they have to stop pumping, because i t does not give any more revenue. The model somehow lacks time dimension but makes a case f o r saving water by emphasizing the diminishing returns to i t . - 28 -The second part of the model deals with case when s i g n i f i c a n t recharge does e x i s t . In t h i s case overdraft of the groundwater - using more than the recharge - makes the recharge more expensive, because from t h i s year on we w i l l have to pump i t out from greater depth. So i f we leave the stock water i n the ground i t w i l l be a substitution f o r the pumping cost. The annual 1 savings w i l l be: C—Y $>. C i s the pumping cost to l i f t one acrefoot water one foot high. I t i s constant and does not depend on the depth - l i k e i n Kelso's model. (Here i t i s $0.05.) s i s the s p e c i f i c y i e l d , a natural c h a r a c t e r i s t i c of the reservoir i t -1 s e l f . — i s the deepening of the reservoir by pumping out one acrefoot of s water. Y i s the net expected annual recharge expressed i n acrefeet per acre overlying the reservoir. 1 O-Y has to be c a p i t a l i z e d ^ d i v i d e d by an acceptable discount rat e , to s get the c a p i t a l i z e d value of groundwater l e f t i n the stock. The capita-1 l i z e d value of the groundwater - as a function of — and Y i s sometimes con-siderably high. I t may be higher, than the revenue of using i t . The pol i c y implication of the model: t h i s waste of water could be avoided by assigning water r i g h t s or by a pric i n g system which would take into account the c a p i t a l i z e d value of the ground water as a substitute for pumping cost. 1 The expression 0-Y i s nothing but the marginal user cost i n the case s r when a sole owner uses only the recharge, as i t i s defined i n the present work. In Renshaw*s model i t i s cal l e d the substitution of water retained i n the ground f o r pumping cost. - 29 -In t h i s thesis a more general concept of the user cost i s developed, a concept which i s v a l i d f o r every quantity of water used annually net the r e -charge only. In Renshaw's model, when there i s no recharge, there i s no user cost. The concept of user cost developed i n t h i s thesis states that every unit of water taken out has a user cost regardless i f i t i s the r e -charge or mining case, furthermore i t i s used as a "choice c r i t e r i o n " to the present value maximizing sole owner when he stops the water use each year. Renshaw suggests to use the user cost f o r p o l i c y purposes, f o r pri c i n g the water i n such a way that w i l l avoid overdraft above the recharge. I t i s a very s i m i l a r p o l i c y implication of the user cost used i n t h i s thesis. The difference i s , that the general concept of user cost enables us to de-sign a p r i c i n g policy which leads to a certain pattern of water use over time which maximizes the present—value of the resource - not only to avoid the overdraft. 3. The Burt Model^ 1 2^ The previous models of Kelso and Renshaw assume common property nature of the reservoir and do not provide a way of maximizing i t s present-value. The Burt model - as we have seen i n Chapter I - assumes sole ownership and calculates the present-value maximizing water use. I t s p o l i c y implication i s the prorating of the water according to his solution. The nature of the model, i t s role i n our work and the differences of our model from i t , are discussed i n Chapter I. (12) Burt (1967). This paper i s referred to as Burt i n t h i s chapter. The model developed i n t h i s thesis incorporates many of the properties of the above described three models. I t deals with the case of sole owner-ship, l i k e the Burt's model assumption. But i t also deals with the ground-water resource as common property, l i k e the assumption of the Kelso and Ren-shaw models. I t has adopted the "rent to the water" d e f i n i t i o n of Kelso, the significance of the pumping cost i n defining net rent, of Kelso and Burt and the concept of diminishing return to the water of Renshaw and Burt. CHAPTER I I I THE MATHEMATICAL MODEL 1. Definitions and Assumptions The d e f i n i t i o n of the marginal user cost: S n =» marginal user cost C m «a the t o t a l pumping cost i n year m X n = the t o t a l amount of water taken out i n year n r = discount rate n = the year's number The marginal user cost i s the increase i n the future t o t a l pumping cost because using one addition unit of water today. The t o t a l pumping cost: (2) C (X ,d )=o(X d v ' xv xv nJ n n d n = the depth of "the reservoir i n year n Q( = the pumping cost of one acrefoot of water to l i f t one foot The t o t a l pumping cost i s a function of the quantity of water used and of the depth of the reservoir. We get i t i f we multiply the 0( X n d n as we see i n equation (2). dC (2)a j = marginal pumping cost n There i s a r e l a t i o n between the X and the d . I f we use more water. n n ' the depth of the reservoir increases. - 32 -The r e l a t i o n between the quantity of water used and the depth: n-1 v X -w (3) d =d + > — — m<> v J n o ^ — s >• m=o w a the amount of the recharge i n a year. In t h i s model i t i s a constant. An average of many years. 1 - a i t i s the deepening of the reservoir because pumping out one acrefoot water. I t depends on the natural circumstances of 1 aquifer. I f i t i s narrow and deep, the — i s large, i f i t i s 5 1 wide and shallow, the — i s small. I t also depends on the qual i t y of the s o i l i n which the water l i e s , s i s c a l l e d the " s p e c i f i c y i e l d " of the s o i l . I t i s the engineering factor. d(d ) m The depth i n a certain year i s the sum of the i n i t i a l depth plus the sum of a l l the deepening caused by a l l the units of water taken out up to t h i s year. (Equation 3). Now we can write down the d e f i n i t i o n of the user cost (equation 1) i n a more e x p l i c i t way. z dC m dX n n . a r» }m—n m=n+1 (1+rJ dC dC(X ,d ) dC(x d ) d(d ) dX = dX dd ' dX n n m n d ( d j ^ = — (Equation 3a) n - 33 -dC dC(X ,d ) . m v nr nr 1 dX = dd m Now we substitute into equation (1) S dC(X ,d ) v nr nr dd m n = m«n+1 [ 1 + r J m _ n Now: C(X m,d m) = o U d m m mm dC(X ,d ) v m' nr i v dd <*Xm m We substitute again into equation ( l ) a : (l)b According to equation ( l ) b , the marginal user cost i s a function of the amount of water used i n every future year. Thus, i f we want to calculate the marginal user cost, we have to know the water use each year a f t e r the unit of water i n question was taken out. The e x p l i c i t expression of the user cost, we have got i n equation ( l ) b i s a l o g i c a l one. ck i s the increase i n the marginal pumping cost, a f t e r an additional unit of water was taken out. This additional unit of water has deepend the water platform by — feet. To l i f t one acrefoot 1 foot costscJL » s 1 , 1 1 to l i f t one acrefoot — costs oL— d o l l a r . To l i f t X acrefeet water — , w i l l s ^ s n s ' 1 j 1 cost X cA— . X Ok— i s the increase i n the t o t a l pumping cost i n the year n, n s n s because the use of-one unit of water sometime before the year n. ( i n t h i s model we do not count the deepening effect within one year.) The discounted 34 -sum of t h i s increase i n the t o t a l cost i n each future year i s the marginal user cost, as we have defined i t , and t h i s i s expressed i n t h i s e x p l i c i t way i n equation (1)b. In chapter I we have seen, that the optimal condition of using water, as a sole owner would use i t i s as follows: So f a r we have got the marginal user cost and also the condition f o r the optimal use of the water by d e f i n i t i o n only. We are going to show that we can have i t by calculation. When we establish the expression for the present-value of the water resource and d i f f e r e n t i a t e i t by the water used, we get the optimal condition of the water use and the marginal user cost as i t was defined above. The present-value of the water resource (PV) i s the discounted sum of the net rent to the water every year (equation 15). The rent to the water (F ) i s the net revenue from the crop associated to that water (equation 5). The net rent to the water (G n) i s the rent minus the pumping cost (equation 14). MNR = MC (U) n n v 1 Marginal net revenue should be equal marginal user cost. MNR n - 35 -2. The Derivation of the Rent Function from the Production Function and  the Derivation of the Demand for the Water from the Rent Function We assume quadratic rent function l i k e Burt (1967) which expresses demi-nishing returns to the water used: (5) AX - ^BX 2 - C = F v 1 n 2T n n 1 2 AX - -r-BX expresses the gross rent to the water. n <L u C stands f o r the fixed cost. 1 2 - TJBX . The sign minus here means that there i s diminsihing r e -[ i 3 l turns to the water. v 1 • Equation (5) i s the rent function. I t expresses, how the rent to the water changes, i f we change the water used on the land. I t assumes, that a l l the other productive factors are optimized f o r a certain amount of water used. (They are a l l used i n proportions which maximize the produc-tion function f o r a certain amount of water.) The rent function i s different from the a g r i c u l t u r a l production func-t i o n . The production function expresses how much revenue expect f o r a (14) certain combination of productive factors and t h e i r prices. (13) = A - B X N i s the marginal rent. I t i s a l i n e a r diminishing func-n of X. The slope of the l i n e i s -43. (14) S t r i c t l y speaking, the production function should be i n r e a l terms, and express the various possible combinations of factors to have a certain product. The production function here i s r e a l l y a revenue function, the product i s multiplied by i t ' s price. In t h i s way we get the product i n r e a l term. We need i t because the rent i s defined i n r e a l terms. - 36 -The production function i s : (6) VP = VP(X 1 f X g f X 3, X n) The X's here stand f o r different productive factors. V f o r the price of the product. Kelso (1961) defines the rent to the water but does not establish a rent function. Burt (1967) does not d i f f e r e n t i a t e between rent and production function, even though we can interpret his objective function as rent func-t i o n . We suppose that X^ i s the factor - water - for which we want to estab-l i s h a rent function. For every X^ we want to use the other factors i n such proportions which maximize the production function P. (7) *1 = $ i s a c o n s ^ a n ^ number dP dP dP dX 2 dX 3 dX ( 8 ) P r i c e ( x J " P r i c e ( x J = " ' = Price ( x j = C o n s t a n t i r The above equations are the well known conditions f o r maximizing the production function: Marginal Physical Product of X 2 Marginal Physical Product of Price of X g " Price of X 3 " ~ e t c ° " We have n-1 equations and n-1 unknowns so we can calculate a l l the X's and substitute them into the production function, and calculate the optimum product, when X^ i s constant and a l l the other factors are used i n t h e i r optimum proportion. We can repeat the same process f o r another value of X^. This way we can get f o r every value of a certain value of net product, - 37 -which we c a l l the rent to which changes - as we have seen above, by changing independently'' the value of X^. (9) Rent = B[XA) =MaxP (x., = J , X 2, X 3, X 4) For every X,, and X 2 < A more general form, which expresses that X^ i s the independent v a r i -able i n the rent function, looks l i k e that: (10) x 1 - I (11) F = F {.jf) - MaxP (X 1 =jf , X 2, X 3, ... X p) In our case i t takes the form of equation (5) (12) F = AX - |BX 2 - C I f we d i f f e r e n t i a t e the rent function i n respect to X we get the func-tion of the demand curve fo r the water: (13) ~ = A - BX This i s the value of the marginal physical product of the water, the demand for the water. I t decreases while X i s increasing. Figure 1 represents the production function. We take X^, the water as constant. I f i t i s equal to a certain amount (A), and we combine i t with the variable factors ( l _ ) , we get the t o t a l gross revenue curve I. The t o t a l cost i s represented by C, which i s a straight l i n e . The maximum rent we have, where the difference between the t o t a l cost and t o t a l revenue i s maximum. There marginal cost equal marginal revenue. I t i s at the point of the other factors when the water (X,j) equal the amount A. • When X^ = B, the maximum rent we get, i f we combine i t with L^t and when X. = C, the maximum rent i s at Figure 1 Graphical exposition of the procedure f o r determining the rent function to X^ from a multivariable produc-tion functions - 39 -Figure 2 The rent function - 40 -Figure 3 The demand curve f o r water - 41 -In Figure 2 we change the axes and we take the X 1 as the independent variable, and t h e r e n t to i t , as i t s function. Then we copy from Figure 1 the height of the rent to every into Figure 2. The other factors are optimally combined - which maximizes the rent f o r every X^. The slope of the rent function i s the value of the marginal physical product of the water, which i s nothing but the demand f o r the water, which i s represented i n Figure 3. To forecast future demand f o r water i n the agriculture i s the most im-portant thing i n making a water policy. Here we can do that, i f we estimate r i g h t the production function and we derive the rent function from i t i n the way i t has been shown. The demand for the water depends on the rent to i t , l i k e the demand f o r any other resource of perfectly i n e l a s t i c supply. 3. The Condition of the Optimum Water Use We get the net rent (G n) i f we subtract the pumping costs (C n) from the gross rent (F ). The t o t a l pumping cost i s according to equation (2): C =0(X d n n n The net rent function i s therefore: (14) G = F -C = AX - ^ X 2 - C - o ( X d v ' n n n . n «Tn ^ n n We see that G i s a function of X as w e l l as of d . X and d are the n n n n n independent variables. Now the present-value of the reservoir (PV) can be written: - 42 -(15) oO r G (X ,d ) n n n (1+r)" n=0 - ( A X - §, * - c - * x ^ ) ' • ^ , - » i 2 - _ ° - ^ A ) + O O O O 1+r ( A X 2 - | x 2 2 - c - d x 2 d 2 ) (1+r) 2 " + . ( A X _ | x - c - d x d ) (1+r)" + . We want to express the d R i n terms of;the X N and to have only one type of independent variable i n the present value function, ( i n fact there are many variables because X i s di f f e r e n t every year.) We know from equation (3) X - w d = d + -2 1 o s where d^ i s the depth of the reservoir i n year 1, d i s the i n i t i a l depth, X i s the quantity of water used i n the year o, and w i s the average annual ! X - w recharge. The expression, t e l l s us how much the water taken out i n year n above the recharge has deepened the reservoir. I f we add t h i s number to the i n i t i a l depth, we get the depth i n the next year. We can continue: X -w X,-w (16) d 0 = d + -2— + —!— • 2 o s s This leads to equation (3): d = d + *-~r. m n o m=0 s Now we are going to substitute the X f o r the d i n the equation of the present value: (Equation 15). B w 2 X -w o R ? (AX. - "EX/, - C - o(,x.d -dx,-—--) P V = ( A X -|< 2 - c - o t x d ) + — 3 — 2 ~ ! , 1 ° o 2 a ; o o J 11+rJ 43 -(1+r) 2. + 5 (l+r)3 ~ ( A X - | x 2 - c - d x d - o U ^ — ) f 1 0i , . n 2 n n o N n m53 s J , V ' ' J + » • • + ——————————————————————•—•—•—••——•»«——-••»——«-—• + ... We see i n equation (17) that the net present-value i s a function of a series of X ' s , that i s of the di f f e r e n t quantities of water taken out every year and year. We are going to take the p a r t i a l derivative of the present-value with respect to X q . ( i t t e l l s us how much the PV^ne^.j would change i f we change o dPV °^X1 <*X2 <*Xn a A - BX -r^d - . - - ... dX 0 ^ 0 s(1+r) /•., -.2 r. -.n . -o v 1 s(1+rj s(1+rj 00 ( 1 8) ffi = A - BX v J dX 0 0 sm=1 o (1+rJ Si m i l a r l y f o r the water used i n any year (X^: A - BX - o l d M X dPV _ n ^ n _ 2" mdX ( . ->n s m=n+1 /• . , -im n (.1+r) l 1+^J which we can also write as: ^ X ( 1 9) (l+r) n®> » A - B X -Wd-2^2- _ - J E v ' v ' dX n n s uion+1 n (1+r) The l a s t term on the R.H.S. of t h i s equation r e f l e c t s the fact that the - 44 -pumping costs i n a l l years subsequent to n (m}n) depend on X n through i t s effect on the depth of the reservoir (d ). m The condition that the present-value be a maximum i s the set of equations: (20) f U n n f o r a l l X . n E x p l i c i t l y t h i s can be written f o r the f i r s t year (n = o): oo (21) A - BX - c*d w '" ~' o and i n general: o s n-1 ( l 4 r ) " i (22) X m A - B X - o l d = — n n s m=n+1 r a \m—n (1+r) We have defined above the user cost S n i n equation (1): S =2i 2 , 2 n s m=n+1 r. >m—n (1+r) The l e f t side of equation (22) i s the marginal net rent and the ri g h t side of i t , according to equation (1b), i s nothing else but the marginal user cost (S ). Thus we have got by calculation the optimal condition of the water use,  which says that marginal net revenue should be equal to marginal user cost. Figure 4 shows graphically the diminishing return to the water. The point R represents the optimal condition of the water use, when marginal rent equals marginal pumping cost plus marginal user cost. At t h i s point: X n(s) the sole owner would stop to use the water. - 45 -Marginal $ (gross) rent and marginal costs Figure 4 Graphical exposition of the optimal condition of the water use - 46 -The common property user stops at point 2 when marginal revenue equals marginal pumping cost and X n • X n ^ j . More than i n the case of the sole owner use. The marginal user cost within a certain year i s constant f o r every unit of water because we do not count the deepening effect within a year. The optimal condition: MNR » ^ ( u ) c a n ^ B d e r i v / B C ' * n a m o r e general form. In that case we get: dC m dX n MNR m=sn+1 r A -ifn—n (1+r) which i s according to the d e f i n i t i o n i n equation ( l ) . This derivation i s shown i n Appendix I. - 4? -CHAPTER IV THE OPTIMAL SEQUENCE OF WATER USE 1. General Form of the Solution We have the conditions f o r the optimum use of water: MNR = S , which i s expressed i n equation (22.): A - B X n - d d n = f , ^ + 1 - -In order to use the reservoir at an optimal rat e , the above expression should be true i n every year. So we are looking f o r the X n's which s a t i s f y t h i s equation i n every year. In other words we are looking f o r the optimal sequence of water used from the reservoir, the sequence which would maximize the present value of the resource. We can see immediately i n equation (22), that the X which s a t i s f i e d i t i n the year n, does not do i t i n the n + 1. ( i n the year n+1 the d has changed and also the discount factors and the X m*s on the r i g h t side, so the same X cannot s a t i s f y i t . ) Every year we have another X, which solves the equation. We have got here a system of equations and we are looking for the X's which solve i t . We show i n Appendix I I that the solution of t h i s set of equations (equa-ti o n 22) has the form (23) where: (24) A + - 1 "+ P / 2 + y ( l + | ) 2 - (1 + r ) - 48 -and we have defined: (25) P . rB/(B - d/s) As defined i n chapter I I I r i s the interest r a t e , B i s the c o e f f i c i e n t of diminishing returns i n the rent function and i n the pumping cost per unit of water and s i s the s p e c i f i c y i e l d of the reservoir. A l l are measurable parameters. Burt has got an equivalent r e s u l t . In equation (23) R + and R_ are constants which we are going to determine below. We saw i n Appendix I I (equation I I 20) that the A are roots of a quadra-t i c equation. This i s the reason we have A and A . As mentioned i n A: .an-dix I I and shown i n d e t a i l by Burt (1967) ^  + >^ 1 and o < A _ < 1 i n a l l s ituations where the model i s meaningful. The nature of the water use from the resource requires us to consider only the A which i s smaller than one. X i s t n e number which t e l l s us how we go from X to X ... What f r a c t i o n of the water used i n the year n we are n riT I going to use i n the year n+1 ? We are certain that we are not going to use more water i n the year n+1 than i n the year n. For t h i s reason we cannot use the A which i s larger than one. To use /\ larger than one would mean exactly that the best pattern of water use i s using more water every subsequent year. This cannot be true so we only use the /\ < 1 i n our solution. I t i s the A i n t h i s calculation. The solution, which means a r i s i n g use of water from year to year i s not feasible economically. I t does not make sense to use more water next year than t h i s year - c e f e r i s paribus - because r - the d i s -count rate - i s positive. The fa c t of the increasing pumping cost works i n - 49 -the same dir e c t i o n too. For example, the s i x t h unit of water i s cheaper t h i s year, than i n the next year, when i t comes af t e r the t o t a l ten units of water. Thus the net revenue to the s i x t h unit of water next year i s smaller, than t h i s year, f o r two reasons. F i r s t , because i t s pumping cost i s higher than t h i s year, and second, because we have to discount i t to get i t s present-value. So i f there i s any sense to use a s i x t h unit of water, we w i l l use i t t h i s year and not i n the next year. So a solution which indicates a r i s -ing use of water from year to year i s out of question. As we have already stated - according to Burt(1967) -A + > 1 I f we are going to use ^ only, we have to rewrite equation (23), Thus the solution of the system of equations MNRn = S n which we want has the form: (26) X - w + R X " n - x-This solution means that the quantity of water used each year i s decreas-ing - A < 1 and n, the years are increasing - and getting closer and closer - asymptotically to w. In figure 5 we show how the water use over time approaches the amount of recharge w. - 50 -X n Quantity of water 0 1 2 Number of years Figure 5 Graphical exposition of the optimal sequence of water use (equation 26) 51 2. The Calculation of R and the Calculation of the User Cost In the preceding chapter we have seen that the solution of the set of equations (equation 26): X » w + R \ n n — — Equation (24) gives us the expression f o r A . Now we have to know the R i n order to calculate X . n R i s nothing else but the quantity of water used from the stock i n the very f i r s t year of the use of the reservoir. We have X = R A " + (26) (27) w I f n = o, then A " c A ° 88 1 s o t n a t X = R + w o -Now we want to calculate the R and to get i t by the given parameters l i k e we have the ^ As shown i n Appendix I I I we can get R by substituting the expression (27) f o r X q i n equation (21,). This gives (aquation I I I 11): (23) A - ( B ^ ) w -o(d c B + We also got the expression f o r the user cost (equations I I I 8 and I I I 9): (29) i c*R \ o = s.r s( 1 +r- ^  __) - 52 -(30) 8 < * « * R - A - n + 1 n s.r s( 1+r-^ _ J Using our previous expression f o r )\ (equation 24) we can therefore express R_ and the user cost (S n) i n terms of the measurable parameters A, B,cl , s, d Q , w and r. Having the equation f o r the optimal sequence of using the water (26) and the expressions f o r \ (equation 24) and R (equation 28) expressed by the parameters we can calculate the quantity of water to use each year. We are able to know the optimal amount of water to be pumped out from the stock during the f i r s t year (R +W) and we know what i t should be i n every subsequent year i n order to maximize the present-value of the resource. A l l these we can learn from Burt's a r t i c l e (Burt, 1967) too, even though he uses a d i f f e r e n t notation. What we have new here i s that we have also calculated the user cost, which does not appear i n Burt's way of c a l c u l a -t i o n . . We have got an expression f o r the user cost (29, 30) expressed i n terms of known parameters, so we can get i t numerically. We have already pointed out that t h i s i s extremely useful i n determining a r e a l i s t i c water policy. Tjie expression of the optimum condition by the user cost (equation 22) enables us to formulate the problem of water use i n the case when -the reservoir being a common property - the users of the water do not consider user cost. In what sequence they would use the r e s e r v o i r , and i f i t ' ' d i f f e r s s i g n i f i c a n t l y from the optimal use? This i s a question we are gibing to answer i n the next chapter.; - 53 -CHAPTER V THE GROUNDWATER AS COMMON PROPERTY RESOURCE I f there i s no l e g i s l a t i o n that the ground water belongs to a cen-t r a l authority, then naturally i t belongs to the owner or owners of the land which i s the surface of the reservoir. This way i t can belong to as many users as there are landowners above the water and i t becomes th e i r common property. The common property water user does not consider the user cost i n his use of water because f o r him there i s no gain i f he leaves a unit of water i n the ground. He cannot maintain the desirable depth of the reservoir by leaving the water i n - and i n t h i s way eliminate undesirable future cost increase, when t h i s cost increase would be higher than the net revenue to the same unit of water i f i t i s used. I f he leaves the water i n the ground, others would take i t out and would deepen the reservoir j u s t the same. One user's action effects every user's cost. Thus the best r a t i o n a l p o l i c y f o r the in d i v i d u a l user of the common water property i s to take the water out as long as the net revenue to i t i s above zero. We can summarize the ef f e c t of the common property by saying that the user would stop to use the water from the reservoir, when the net revenue to i t i s equal to zero and not to the user cost, as i t i s i n the case of the sole ownership. 1. The Condition of the Water Use The condition of stopping the water use i n the common property case i s : dG n - 54 -where i s the net revenue factor i n *bhe year n. Equation (31) replaces fo r the common property owner equation (22) which would be used by the sole owner to determine the amount of water he uses. Using our model f o r G n (equation 14) equation (31) becomes: (32) A - B X - d d = 0 n n Equation (32) represents a set of equations s i m i l a r to equatlons(19) which we have to solve f o r every Xn« We have seen already that equation (32) i s not the optimal condition, which maximizes the present-value of the resource. The optimal condition i s expressed i n equation (22): A - B X - d d = S n n n which i s the condition of stopping the water use by a sole owner. The question we have to ask here i s what i s the timepath of the water use i f the reservoir i s a common property? How much does i t d i f f e r from the optimal timepath we have already calculated? I f the difference i s s i g -n i f i c a n t , which requires public intervention, or not? Kelso (1961) i n his a r t i c l e deals with t h i s case of common property. He points out, that the water use w i l l decrease every year because the increasing depth increases the pumping cost. Thus the average net r e -venue to the water used i n a certain year decreases with the years. In the following we are going to calculate how f a s t the water use decreases over time, when there i s no intervention of the authorities. The r a t i o which shows i t we w i l l c a l l ^4. the equivalent of the A i n the former case of optimum water use. - 55 -We write down again equation (32): A - B X - dv d = • n n This i s true each year, while the d n i s increasing from year to year, so the X R has to adjust i t s e l f i n order that equation (32) be true each year. The question i s - as i n the former case - what are the X n's which solve the whole set of equations f o r a l l the years of the l i f e of the r e -servoir? How X „ comes out of X ? n+1 n 2. Calculation of the Common Property Sequence We have for the year n: (33)a A - B X - ok d =• n n and s i m i l a r l y f o r the year n+1 (33) b A - B X p + 1 - d d n + 1 =0 subtracting (33)b from (33)a gives (34) -BX + BX . — Ckd + o U . - 0 V n n+1 n n+1 As i n Appendix I I i t i s now convenient to define q = X - w n n and from equation 3 d , = d + -lq n+1 n s -n 1 because i s the deepening of the water during the year n because of pumping out the quantity q n from the stock. We substitute into equation 34. 56 (34) -e(w+qj + B(w+q n + 1) - d d„ + cU^H-r) - 0 n s w cancels, and o l d cancels too, thus: n ck -Bq + Bq ,, + — q = 0 n n^+1 s n (35) B ! q = ( B _ 0L)q Ti+1 v s / n Equation (35) t e l l s us what proportion i s the water taken out from the stock i n the year n+1, of the water taken out i n the year n. This should be true i n respect of the years n+2 and n+1 and so on, because we can go through the above calculation f o r any two years and get the same r e s u l t . So the sequence of the sue of the water by i n d i v i d u a l farmers i s : (36) - - *n = X - w n (3?) X r , = w + Z/4. n (commonJ / This i s s i m i l a r to the r e s u l t i n the case of optimal use of a sole owner, which i s i n equation (26). (21) X = w + R A n The Z i n equation (36) i s l i k e the R_ i n equation (21), the amount of stock water used i n the f i r s t year by the t o t a l of the i n d i v i d u a l farmers. - 57 -( i t should be d i f f e r e n t from R_ and we w i l l calculate i t l a t e r . ) TheyH.11 i n the same equation t e l l s us what proportion of Z i s used each year. 1 J Equation (35) shows, that yU. i s less than one as long as — CA i s a positive number and smaller than B. 3. The Calculation of Z For the purpose of the calculation of the Z we use the equation (32) v a l i d f o r the f i r s t year: A - BX - o l d =0 o o (38) X = w + z o A A A Z = B(w+Z) - did =0 o Bw - BZ - Okd =0 o Bw - old = BZ o A - Bw - ok d o B (39) A - ol d £ B - w - 58 -CHAPTER VI COMPARISON OF THE USE OF THE WATER IN THE CASE OF SOLE OWNER  AND IN THE CASE OF COMMON PROPERTY 1. The Comparison of B-and Z I t i s d i f f i c u l t to compare the expressions f o r R_ (equation 28) and Z (equation 39) and to t e l l which one i s larger, or to compare \ (equa-t i o n 24) with (equation 35) and to t e l l which one i s smaller, i n d i -cating faster or slower use of the water. (Of course we can do that i f we have numbers to substitute them f o r the parameters.) There i s no r e a l need to do t h i s , because we can t e l l much about the way of using the r e -servoir i n both of the cases ( j u s t by looking at the basic equations of the water use.) X m (22) A - B X - o l d = — ^—-s m=n+1 r. ->m-n n (1+r) (32) A - B X -old =0 ' n n Equation (22) can be rewritten: A - B X = d d + — ^ -i „ m n ^ n s m«sn+1 — t A \m—n (1+r) and equation (32) rewritten A - B X = Ol d n n Equations(22) and (32) t e l l us what i s happening with the water use i n a certain year - i n the case of a sole owner and common property r e -spectively. We imagine one reservoir, and we want to know how the very - 59 -same reservoir w i l l be used under central management and under common property s i t u a t i o n . According to equation (22), the sole owner w i l l use i t i n a certain year, as f a r as the net revenue to a unit of water taken out i s equal to i t s user cost. In our model the user cost i s the same for every unit of water taken out i n the same year. The net revenue to a unit of water taken out within the same year i s not the same. I t d i f f e r s from unit to uni t , because of the diminishing return to the water. The higher net revenue goes to the f i r s t unit of water and i t decreases as we go on with the water use. When i t reaches the amount of the user cost, the sole owner stops the water use i n the concerned year. A common property user stops the water use i n a certain year, when the net revenue to the l a s t unit i s equal to zero. The diminishing return to the water means that we have to use more water to achieve that the marginal net revenue be zero, than we have to use for the marginal net revenue to be equal to the user cost, which i s a positive number. Thus i n the year, when the depth of the reservoir - and so d ( d n , the pumping cost - i s the same for both of the cases, the common property users w i l l take out more water than the sole owner. For t h i s very same reason, only i n the f i r s t year of the use the reservoir has the same depth. In the second year, the common property reservoir w i l l be deeper - d n larger - because more water has been taken out i n the previous year, when i t had the same depth. Thus we are able to conclude, that i n the f i r s t year the common property user w i l l use more water. Z i s larger than R . (40) - 60 -2. The Comparison of A - and f\ , and the Lifetime of the Reservoir Now the question i s , what i s going to happen i n the following years? More s p e c i f i c a l l y , i f the A * s larger or lower, than the/H. ? Again, as i n the case of the comparisons of Z and R , we are not going to compare the expressions f o r A and fory*t , which we already know, but we are going to answer the question by comparing again our two basic equations (22) and (32). Equation (22) rewritten t e l l s us that the sole owner w i l l use the water of the reservoir each year as f a r as the marginal gross revenue equal to the marginal pumping cost plus the user cost. Equation (32) rewritten t e l l s us that the common property users w i l l use the water as f a r as the marginal gross revenue equal to the marginal pumping cost. Because the diminishing return to the water, mentioned above, the following sentence i s true: In every year, when the pumping cost i n the case of common property ( d d r \] i s lower than the pumping cost of 1 v n(common)7 a the sole owner plust the user cost, ( (old /• , -\)+S), the . ' v v ^ n(sole ownerj-' nJ' common property users w i l l use more water out of the reservoir, than the sole owner would do. I f : (41) o i d r A ^ „i d /• , i + S K J ^ n(. common J \ Ck n^sole ownerj n than: (42) Xn(common) ^ X n ( s o l e owner This i s surely the case i n the f i r s t year of the use of the reser-v o i r , as we already have proved, and presumably i n t h e ' f i r s t few years. - 61 A l l these we know straight from equations (22) and (32). We even are able to conclude: i f d ( •> i s growing so f a s t , that the pumping cost n^common J o l d f -v associated with i t w i l l pass i n a certain year the pumping n\common J cost plus user cost of the sole owner (ckd r , -v + S ), then from v n^sole ownerj x\'' t h i s year and on the sole owner w i l l use more water than the common pro-perty user. I f : (43) o l d r , S o(df i ^ + S v ' n(common) / ^ n(sole ownerj n Then: (44) X ( , < X r . -v v nlcommon) x n^sole owner) We w i l l show i n the following that i f t h i s s i t u a t i o n described i n the in e q u a l i t i e s (43) (44) w i l l be true i n a certain year, then \ i s larger than . We w i l l also show that t h i s must be the case. We cannot see t h i s d i r e c t l y from our basic equation (22) and (32). But, i f we assume t e n t a t i v e l y , that i n both cases the f i n a l depth of the reservoir w i l l be the same, or i n other words: we assume, that the t o t a l amount of water taken out from the reservoir stock i s the same i n the case of common property as i n the case of sole ownership, then there must come a year, when the common property users s t a r t to take out less water from the reservoir than the sole owner each subsequent year. The s i t u a t i o n described i n i n e q u a l i t i e s (43) and (44) have to come into existence and the two time paths of the use should cross each other, as we see i n figure 6. 62 -The reason i s , that i f the Z i s larger, than R , so the common pro-perty user i n the f i r s t years use more water than the sole owner, and the t o t a l water taken out of the reservoir i s the same - by assumption - i n both cases, so there should be years, when the common property users would use less than the sole owner would do. i t., We see t h i s i n figure 6, that i f we assume that the area under the time path l i n e of x n ( c ) ^ s equal to the area under the path l i n e of x n^gj» then the two l i n e s should cross each other - the in e q u a l i t i e s of (43) and (44) are true - and \ i s larger thany^ . (We can add here, that i n t h i s case, the reservoir i n sole ownership has a longer l i f e than i t had as common property. (See figure 6).) C stands f o r common property S stands f o r sole owner Proved that: Z > R _ I f r\=&o n= o° j~z*nd - y~~n A n d , £ 0 / n n' Then: (46) A - > / * a n d "(sole owner) > f (common property) The area under the l i n e s represents the quantities of water used from the stock. (See figures 4 and 5). - 63 -Water used Z - X f -i R = X o(s) (s) and (c) X f i * Z >H n(c) n years Figure 6 Comparison of the common property sequence of water use ( X n ( c p and the optimal sequence (X n ( s ) ) when there i s no recharge - 64 -3. The Comparison of the F i n a l Depth of the Reservoir a) The case when there i s no recharge We want to show here, that the areas under the two l i n e s i n figure 6 are equal, when w = 0. In other words: The t o t a l quantities of water pumped out from the reservoir i n the case of sole ownership and i n the case of common property are equal, i f there i s no recharge. I f i t i s tajue, then i n t h i s case X should be greater t h a n a s figure 6 shows us. But i f X ^>in the case when w = 0, then^\ w i l l ..always be greater thanJh., because none of them i s a function of w. (See equa-tions 24 and 35). Instead of showing that the t o t a l quantities of water used are equal i n case of w = 0, we w i l l show that the f i n a l depth of the reservoir i s the same i n the case of sole ownership as i n the case of common property, i f there i s no,,recharge. I t does not make any difference, because the depth could be the same only i f the quantity of water taken out was the same. We w i l l show i t by using our two basic equations (22) and (32) only. We have to rewrite them again i n the following way: ^ A - B X n ( c ) - o t d n ( c ) =0 <47> -*<n(C) = ^ d n ( C ) - A Equation (47) shows us, i f the increasing marginal pumping cost ^ ^ n f c ) ~ * n **ne c o m r a o n Property case - reaches the magnitude of A -which i s constant - B X n ^ j w i l l be zero, so no more water w i l l be pumped out of the reservoir. (We do not forget that there i s no recharge,) Thus the f i n a l depth (d f) w i l l be: \ - 65 -(48) <* d f( D) = A (49) d. A In the sole owner case: ( 1 3) A " B Xn(S) - ^ d n ( S ) * S n (50) ^ X n ( s ) = ( d d n ( s ) + S n) - A Equation (50) shows us, that the sole owner w i l l stop taking out water from the reservoir altogether i n the year, when the marginal pump-ing cost plus the marginal user cost become equal to A. But i n the l a s t year of using the reservoir - we c a l l i t n.p — no user cost e x i s t s , because there w i l l be no further use of the reservoir, so the deepening of the reservoir during t h i s year does not increase the cost of anything. (3) S = 2T Xm v 1 n s m=n+1 . ( l + r ) m - n I f X =0 f o r m>n- thus S =0 too. m ' f n (In the year n„+1•the reservoir w i l l not be used and i n the S we do v r n not count the eff e c t of deepening i n the same year.) I f S n =0, equation (47) i s the same as equation (50). (51) -BX f i -old - A n f(s) * n f ( s ) I f d d f ( s ) - A Then: A ^ d f ( s ) d - 66 -Equation (52) i s the same as equation (49):-. (S3> df(s) - df(c) So the f i n a l depth of the reservoir w i l l be the same i n the case (s) as i n the case ( c ) , i f there i s no recharge. Thus: ^  ^ i s proved, b) The case when recharge does e x i s t ( i ) The case of common property We have already seen, that the quantity of water used every year (X n) i s decreasing, because of the increasing pumping cost - which i s brought about by the increasing depth of the reservoir. Thus there w i l l come a year, when the quantity of water used w i l l be equal to the quantity of the recharge (w) which i s a constant. From t h i s year and on, the X n -which i s now the w - i s not going to decrease any more and each year, the same quantity - w - w i l l be taken out. This quantity comes back every year, and there i s no more deepening of the water platform. There i s no reason i n any year to take out less than the recharge, because i n t h i s case i n the next year, the reservoir w i l l . b e higher - instead of deeper -and they w i l l take out more than w, making the reservoir deeper again, and take out w i n the following year. I f the optimal X n to the common property user i s just equal w, they w i l l not take out more ei t h e r , be-cause then they w i l l get negative marginal net revenue. Thus i n the case, when w solves the equation (32): X n « w. (54) A - Bw - d d n = 0 - 67 -This i s the equilibrium s i t u a t i o n . There w i l l be no more change i n the water use and t h i s p a r t i c u l a r d n i s the f i n a l depth of the reservoir. (See figure 7) This depth w i l l be: (54) A - Bw - d d n ( c ) " 0 f c c 1 . A-Bw ( 5 5 ) dn(C) = — ( i i ) The case of the sole ownership What i s the f i n a l depth, which i s reached by the sole owner? For the year, when the decreasing X n reaches the magnitude of the w, we can write the equation (22) as follows: (56) A - B w - o C d n ( s ) = S n . X n = w A-Bw-S The marginal user cost (S n) i s not zero here, unlike i n the case, when there i s no recharge. Thus - according to equations (55) and (57) - the reservoir w i l l be deeper i n the case of common property use than i n the case of sole ownership, when they w i l l s t a r t to use the recharge f i n a l l y . The common property users mine more out of the stock. In figure 7 the areas under the two l i n e s , defined by ZT w i n the common property case and by R P w i n the sole owner case, are not equal. These areas represent the depletion of the reservoir (see figure 4), and i t i s not the same f o r both the cases, because each s t a r t s to use the quantity w at a di f f e r e n t depth. 7 (o) (8)" n years Figure 7 Same as figure 6 when there i s recharge (w) - 69 -We conclude that: (59) R _ A ; < ZM." n=o "~ n=o Because as was shown i n equations (58) and (57): d f ( C ) > d f ( s ) The inequality (58) - which means that the common property users mine deeper than the sole owner - i s based on the statement that the marginal user cost i n equation (57) i s not zero. That a user cost does e x i s t , when the sole owner uses only the recharge. One would think, that i n t h i s case no more user cost e x i s t s , because using w only does not deepen the reservoir anymore. So i n what sense the user cost does exist? The user cost does e x i s t even when a sole owner uses only the w. I f we take out one more unit of stock water i t w i l l increase the cost of pumping out the recharge f o r a l l the coming years. The increase of t h i s cost would be: °s s obviously. (This expression appears i n r Renshaw's model. (Renshaw, 1963) (60) s = 2 ^ ' n s.r I t i s equivalent to the general expression n s, m=n+1 (1+rj«n for the case when we use the recharge only. The sole owner w i l l s t a r t to use the recharge only, when mining of one more unit of stock water w i l l increase the pumping cost of the recharge more than the net - 70 -revenue.to t h i s unit of stock water would be. I t indicates, that the sole owner w i l l s t a r t to use recharge only from a lesser depth, than the common property user, who does not consider the cost associated with the deepening of the water platform. Using the w quantity of water i s an equilibrium s i t u a t i o n f o r the sole onwer too. He would never use less of i t , because then the water platform would be higher i n the next year, the pumping cost lower, and he would use more, up to the quantity of w. He would never use more, because i n the next year - after w came back - the reservoir would be deeper, than i n the current year, and he would use l e s s , up to w again. The case i s the same, as of common property user, the only difference i s , that the sole owner would use the quantity of the recharge from a less deep reservoir, than common property user, as has been shown above. This r e s u l t i s natural and can be concluded straight from the two basic equations (22) and (32). (32) A - B X -d, d = 0 v ' n n (22) A - B X - A d = S ' n ^ n n We have to keep i n mind the law of diminishing return to the water when we examine the two equations. Thus we have seen already, that they show us, (15) In our model w i s only the l i m i t to which the water use i s getting close i n i n f i n i t e . t i m e - and so the depth i s , from which i t i s taken out - X n = w + R AJ?, so X n i s never w. But with the years R_A n i s getting very small and i n a certain year we can neglect i t e n t i r e l y , the same i s true f o r X » «wZR n (equation 3?). - 71 -d n Depth of the reservoir n 1 n2 n 3 n4 n y e a * 3 Figure 8 The depth of the reservoir and i t s economic l i f e , 1J without recharge, 2J with recharge, al i n the case of sole owner b) " " " " common property user The depth f o r using the recharge only 1) f o r sole'owner 2) f o r com. prop, user - 73 i f the depths are the same, the water use should be di f f e r e n t - i n order to s a t i s f y the conditions. This was the case i n the f i r s t year of the use. Now we can add, that they also show us, i f the quantities of water used are the same - x n ( c ) " X n ( s ) 8 8 w " t n e n **ne deP^ °f * n e reservoir . should be d i f f e r e n t . In summary and more s p e c i f i c a l l y we can say, that we have proved, that the common property user uses more water i n the f i r s t year than the sole owner, he reaches equilibrium - the use of the recharge only - i n time before the sole owner does - his pattern of depletion i s faster -and he st a r t s to use t h i s recharge at a deeper water l e v e l , than the sole owner. (Mines the reservoir deeper.) These r e s u l t s are represented by figures 8 and 9. Here we see the significance of the formulation of the maximizing : condition of the present-value:of the resource, optimal condition of the water use expressed by the user cost, when we have di f f e r e n t i a t e d the present-value of the resource i n respect to the quantity of the water used. dX n (22) A - B X -odd = S ' n n n This form of optimal condition enabled us to compare the pattern of use of a certain reservoir i f i t i s i n sole ownership - which can be a certain authority - or i f i t i s i n common use. The way of the common property user i s the general pattern: marginal net revenue should be equal to zero. Kelso (1961) says i t e x p l i c i t l y : 74 -MNR - 0 (32) A - BX,,- d d o - 0 I f we have equation (22), so we can compare. The optimal time path (y\ ) can be achieved by d i f f e r e n t i a t i n g the present-value of the resource i n respect to the change of the stock water too, as Burt (1967) di d , but then we cannot make the above comparisons. To be able to compare the outcome of the common property use with that of a sole owner i s important to a water p o l i c y making authority. I t i s quite possible that there i s no economic significance whatsoever of these differences, but we cannot know t h i s without examining them. The calculations above are tools to do i t . After a l l the differences could be s i g n i f i c a n t enough to j u s t i f y intervention or even planning of the water use by an authority. The next chapter i s dedicated to see the differences i n the way of using the water by a sole owner and by a group of in d i v i d u a l s , in. a numerical example. The data used are from the a r t i c l e of M. Kelso (1961). - 75 -CHAPTER VII NUMERICAL EXAMPLE The Kelso model and i t s assumptions are described already i n chap-ter I. We have to keep i n mind mainly the assumption that the land under i r r i g a t i o n i s constant. The reason i s , that most of the data are related to an acre of land - l i k e : water used per acre - then i n case we want the t o t a l amount, we have to multiply i t by the number of acres under i r r i g a -t i o n . So i n order to be able to work with the numbers related to an acre of land, the t o t a l area should be assumed to be constant. {16l Given and I n i t i a l l y Calculated^ ' Pumping cost per acrefoot per foot ( c i ) $0.04 The i n i t i a l depth of the reservoir ( c L ) 183 feet The t o t a l pumping cost i n the year of the beginning (ck d Q) $7.32 The discount rate (r) $ °/o I f the water used on an \acre per year (X q) i s 51/2 acrefeet The deepening of the reservoir i n a year i s 672 feet Thus the displacing of the reservoir per acre- 61/2 13 foot per acre (^ ) 51/2 11 I f the water used on an acrefoot f o r a year i s 51/2 acrefeet Then the gross rent f o r the water per acre $100 The water used per acre i n the year of the beginning (z) 572 acrefeet (16) The i n i t i a l l y calculated numbers have r e a l sign. The o r i g i n a l l y given data have none. - 76 -I f the water used on an acre f o r a year i s 3 2/3 acrefeet Then the gross' rent f o r the water per acre i s $ 75 No recharge e x i s t s . From these data given we want to calculate the pattern of optimal use of the water. How much a p r o f i t maximizing sole owner should take out i n the f i r s t year (R or X ), at what sequence the water use should diminish over time (\") what i s the user cost? Furthermore we want to compare these re s u l t s with t h e i r equivalent i n the common property s i t u a t i o n . A special i n t e r e s t to see, how s i g n i f i c a n t the user cost r e a l l y i s . Assuming our form f o r the production function we can write down the following equations: The f i r s t two are f i r s t developed i n equation (5), the t h i r d one i n equation (32) AX - |U 2 - C = 100 X = 51/2 o 2 o o AX - §X 2 - C = 75 X = 3 2/3 n 2 n n ' A - B X - d d =0 o o From the above three equations we are able to calculate A, B and C. A . 45 B = 6.89 C - 44.5 The amount of water to be withdrawn each year: (24) X = X A n w - 0 K J n o - . 77 -The parameters to calculate i t are given i n the Kelso model, so we can calculate i t . (24) \ _ = 1 + £ + £ _ y ( | + £ ) 2 + 2 6 where £ - * * . = | (P-r) 2 ( B - 4 ) 2 0.08 4? 0.04 2 2(6.89-0.04 ~ ) w , 0 . 0 8 0 . 0 6 - ^ 0.04 0.08 ^ 0.04 - / ( 5" + To") + 2 2 2(6.89-0.04 -|f) 2(6.89-0.04 -]f) which gives X = 0.994 This means, that i n the f i r s t year we pump out the portion 0.994 of 1 = 2 the water used i n the year 0. The second year i t i s the portion 0.994 of the water used i n the year 0, and so on. This i s an optimal path of the water use. Now we calculate the water used i n year 0. A - Bw - d d - — (28) X - ° r S v ' o B < l +r-A _ w = 0 x 45 - 7.32 ° 6.89 +'0.04 41 • °- 9 9 4 11 ' 1+0.08+0.994 X = 5 acrefeet o - 78 -This means that i n the year of the beginning (o) the water used ( X Q ) should be 5 acrefeet per acre i n order to maximize present-value of the resource. In Kelso's data which described common property resource, the water used i n the year i s 572 acrefeet per acre land. This difference comes from the fact - which was proved above - that the p r o f i t maximizing sole owner w i l l stop to use the water i n a certain year, when the condition marginal net revenue equal to marginal user cost i s f u l f i l l e d ; while the i n d i v i d u a l user of a common property water r e -source w i l l stop to use the water i n a certain year when the condition marginal net revenue equals marginal user cost, i s achieved. So the user cost, which i s ex i s t i n g f o r the sole owner and does not e x i s t f o r the ind i v i d u a l user, who i s one of many, causes the difference i n the use of water. The formula of the user cost i s : (29) S = — + o ~ sr s " 1+r-^ and when w = 0: S =—d\X „ ; \ o s o 1+r-\ s o , i f o.o4 , 5 , ( o . g a 9 ! 4 ^ - » 3-° S = $ 3 o The user cost of every unit of water pumped out i n the year zero i s $3 per acrefoot of water. The straight cost to pump out one acrefoot of water i n t h i s p a r t i c u l a r reservoir i s $7.32. Thus we see the user cost i s not n e g l i g i b l e , i t i s about 40°/o of the pumping cost. - 79 -An authority which i s interested to maximize the present-value of the reservoir adds the marginal user cost - $3 - to the marginal pumping cost - $7.32 - i n order to make farmers to use less water, less by the amount which would make the present-value maximum. The incorporation of the user cost into the p r i c e of water w i l l ensure t h i s aim. I t i s the r e a l marginal p r i c i n g of the water. I t i s a matter of interest to see what i s the way of use of the water of the i n d i v i d u a l farmers, when there i s no intervention of autho-r i t y . From the model of Kelso (1961) we know they s t a r t with 5 7 2 acrefeet water per acre i n the year zero. He assumes that i n the year 24 they s t a r t to use 3 2/3 acrefeet per acre. But i n r e a l i t y the water use goes down along a curve and not along a straight l i n e . I t i s described i n figure 10. The equation of the curve i s B - ^ ( 3 5 ) yH—E-a -6.89 - 0.04 - i f — 0.993 6.89 ^ m 0.993 This portion of the water used i n the year 0 w i l l the i n d i v i d u a l farmer use i n the f i r s t year. I t i s very s i m i l a r to the X of the sole owner. )\ _ = 0.994 Even so the quantities of the water used each year i n the two di f f e r e n t cases should be s i g n i f i c a n t l y d i f f e r e n t . 80 Acrefeet Figure 10 Comparison of the sequence of water use assumed by Kelso with those computed f o r the same numerical parameters from our equations (X (c) and X (s)) - 81 -X =* X A " (26) n(sole owner) = o ( s ) A -(36) Xn(common property) ~ * o ( c ) ^ X t \ stands f o r the water used i n a certain year i n the case of sole ownership. * n ( c ) ^ n * n e c a s e °? common property. Even though there i s not much difference between }\ ^  and , the differences are growing i n the years between }\ n andy^l n. Furthermore Qnr'st'pR'fc there i s a s i g n i f i c a n t difference betwee X r •>, 5 — — — — and o[s J' acre A/ Qcrs^sst X r •>, 5/2 . Thus we see from the equations above, there should o^ c J acre be s i g n i f i c a n t differences between X and X f \. n(S) n(c) These r e s u l t s f i t the spec i a l circumstances of the Central Arizona water resource. In other circumstances, where the parameters are d i f f e r -ent, the re s u l t s w i l l be d i f f e r e n t . Especially the marginal user cost can be expected to be much higher, when there i s a recharge. The r e -charge component of the marginal user cost i s proportionate with the amount of recharge. The formula of the water use and the user cost developed here can be used i n every circumstance of ground water mining. The d i f f e r -ences i n the possible r e s u l t s are due to the different parameters only. - 82 -BIBLIOGRAPHY Bator, Francis M,, "The Anatomy of Market F a i l u r e " , Quarterly Journal of  Economics, March 1958, 351-379. Brown, Gardner, J r . , and McGuire, C.C., "A S o c i a l l y Optimum P r i c i n g f o r a Water Agency", Water Resources Research, F i r s t Qaurter 1967, 33-34. Burt, Oscar R., "Economics of Conjunctive Use of Ground and Surface Water", HilgaTdia (36), 1964-65, 31-111. Burti Oscar R.i 'Economic Control of Ground Water Reserves", Journal of  Farm Economics (48), 1966, 632-647. Burt, Oscar R., "Ground Water Management under Quadratic C r i t e r i o n Func-tionsy Water Resources, Third Quarter, 1967, 673-682. Castle, Emery N. and Lindebory, K a r l H., "The Economics of Ground Water Allocations: A Case Study", Journal of Farm Economics (42), 1960, 150-160. Dominco, P.A., Anderson, D.V., and Case, CM,, "Optimal Ground Water Mining", Water Resources Research, A p r i l 1963, 247-255. Fox, Irving K., and Herfindhal, Orris C , "E f f i c i e n c y i n the Use of Natural Resources. Attainment of E f f i c i e n c y i n S a t i s f y i n g De-mands f o r Water Resources", American Economic Review, May 1969, 193-206. Gordon, Scott H., "On a Misinterpretation of the Law of Diminishing Re-turns i n Marshall 1s P r i n c i p l e s " , Canadian Journal of Economics  and P o l i t i c a l Science (18), February 1952, 96-98. Groyther, V a l . , "Progress i n the U t i l i s a t i o n of Ground Water i n B r i t i s h Columbia", B.C. Natural Resource Conference, 1953, 64-73. Ho t e l l i n g , Herald, "The Economics of Exhaustible Resources", Journal of P o l i t i c a l Economy (39), A p r i l 1931, 137-175. - 83 -/ Kelso, Maurice M., "The Stock Resource Value of Water", Journal of Farm  Economics (43), 1961, 1112-1129. Marshall, A l f r e d , P r i n c i p l e s of Economics, 8th ed., New York: The Mac-millan Company, 1920. McGauhey, P.H. and E r l i c h , H., Economic Evaluation of Water, Part I . Search f o r C r i t e r i a . University of C a l i f o r n i a , Berkley, 1960. Renshaw, Edward F., "The Management of Ground Water Reservoirs", Journal  of Farm Economics (45), 1963, 285-295. Ruttan, V.W., The Economic Demand f o r Ir r i g a t e d Acreage, Resources f o r the Future, Inc. - The Johns Hopkins Press, Baltimore, Maryland. Scott, Anthony, "Notes on User Cost", Economic Journal (63), June 1953, 368-389. Scott, Anthony, Natural Resources: The Economics of Conservation, The University of Toronto Press, 1955. Scott, Anthony, "Resourcefulness and Responsibility", Canadian Journal  of Economics and P o l i t i c a l Science Scott, Anthony and Christy, F.T., The Common Wealth i n Ocean Fisheries. Some Problems of Growth and Economic A l l o c a t i o n , Baltimore, Published f o r Resources f o r the Future by Johns Hopkins Press, 1965. Scott, Anthony, The Theory of the Mine Under Condition of Certainty. Extractive Resource and Taxation. Edited by Mason Gaffney, University of Wisconsin Press, Milwaukee, 1967, 25-62. Sewell, W.R. Derrick, and Bower, B l a i r T,, Forecasting the Demands f o r Water, P o l i c y and Planning Branch, Department of Energy, Mines and Resources, Ottawa, Ontario, Queen's P r i n t e r j 1968. " Smith, Vernon, L,, On Models of Commercial Fishing", MS, 1967. . ... r Smith, Vernon, L., "Economics of Production from Natural Resources", MS, 1967. Proceedings, Conference on Water Resources Research i n the University of C a l i f o r n i a , 1957, WatBr Resource Center, University of C a l i f o r n i a . Proceedings, Conference on C a l i f o r n i a Ground Water S i t u a t i o n , University of C a l i f o r n i a , 196G. Proceedings, Water P r i c i n g P o l i c y Conference, 1968, University of C a l i -f o r n i a , Water Resources Center. APPENDIX I THE CONDITION OF, THE OPTIMAL WATER USE A DIFFERENT WAY OF CALCULATION The production function f o r net t o t a l output i s : G(x,d) a F ( x ) - c ( x , d ) where F f u i i s the t o t a l revenue from the water used and C/-^  i s the l x J IX,dj t o t a l cost of water used, and again: X i s the quantity of water used and X r stands f o r X i n y e a r n , d i s the depth of the well and d n i s the depth i n year n. r i s the interest rate. The present-value of the resource: ( I 2 ) py a f ~ G ( X n f d n ) = f t l F(Xn) ~ C(Xntdn]2 n=0 ( l + r ) n ndD ( , | + r ) n We maximize the present-value according to the quantity of water used ravery year (X ).. dG dC n - ^ r — m , o w dX > dX r T r i i dPV n / n _ ^ 1.13) -rr- = —• • = 0 m > n dX r. i n m=n+1 r. \va ' n (.1+rJ (.1+rJ The second term of the r i g h t hand of the above equation stands f o r the fact that the pumping costs of a l l the years a f t e r the n year i s a th function of X n» and not the cost i n the n year only. The equation above can be written: - 86 -(14) dG oo dC n _ m dX ST dX n S n fx i n m=n+T r „ >m (1+rJ (1+r) Now we are multiplying by ( l + r ) n . _ dC co m dG S dX u J dX ~ m=n+1 r. m^-n ~ n n (1+r) The l e f t hand of the equation i s the marginal net revenue. The ri g h t hand of the above equation i s the marginal user cost as we have defined i t i n equation (1): the sum of the present-value of the increase i n the pump-ing cost every year after the year n, i n which we pumped out an additional unit of water - instead of leaving i t i n the stock. Here I have got t h i s expression of the marginal user cost straight by d i f f e r e n t i a t i n g the pre-sent-value of the resource to X n < The re s u l t we have got i n t h i s c a l c u l a -t i o n i s : that i n order to maximize present-value of the resource we have i to use every year the quantities of water which equate marginal net reve-nue to marginal user cost. Exactly the r e s u l t which we had before by de f i n i t i o n . - 8 7 -APPENDIX I I SOLVING THE SET OF EQUATIONS f22) TO GET THE OPTIMAL SEQUENCE OF WATER USE We want to solve the set of equations (22) (111) A - BX - d d - S • 1 n n n where we have for the depth of the reservoir (equation 3): n-1 x (112) d = d + Y"~ m *" W m=0 s and the user cost S n i s defined i n equation (1) by (113) S - ^ J T " % n s fe+1 ( 1 + r r We want to solve the equations(ill) f o r a l l years from n=0 to n = N where N i s the l a s t year the reservoir w i l l be used. N can be i n f i n i t e . We notice from equation (lI3) ^ S n = ( 8 „ + 1 + T X n + l ) / t l + r ) which can be checked by comparing the expressions (equation 113 for S n and f o r S ^• This equation(il4) also has a simple direct interpretation. Using equation (EI4) i n equation .(111) we can write, (115) A - BX p - c U p - S n - ( S n + 1 + ^ X n + 1 ) / (1 + r ) but we know: (116) A - B X n + 1 - C ( d n + 1 = S n + 1 . We can now substitute ( l I 6 ) i n (115) (117) A - B X - o l d = ( A - B X - o l d . + — X J /(1+r) v n n n-»-| n+1 s n+1J ' K ' Rearranging terms t h i s can be written: 88 -(118) p = rA - (l+r)BX + (B--^-)X - (l+r)o(d + o(d , = 0 v n v 1 n s 1 n+1 1 , n °* n+1 where we have defined the polynomial P n by the expression to the r i g h t for convenience. Now from equation (112) we have X -w (119) d . = d + - E — v 1 n+1 n s so that: A (X -w) ( . ( 1 + r ) o l d n + o i d n + i . ~ - - o ( r d n •• Substituting t h i s i n equation(118)we get: (1111) P = rA - ~w - /"(1+r) B - — 7x + (B - — )X . - ro(d = 0 v 1 n s *• v •* s—' n v s••' n+1 n and s i m i l a r l y : (H12) P n + 1 - rA - f w - f( 1+r)B - ^ J x ^ + (B - ^ ) X n + 2 - rc^ d R + 1 = 0 We can now subtract P . from P : n+1 n (IH3) P n - P n + 1 - - f(Ur)B - 4 j X n + /-(2+r)B - ^ J x ^ -- ( B - ^ X n + 2 + r c i t d n + 1 - d J = 0 We can now use(119)again to eliminate ^ n +^ and d n to get the f i n a l form of the equations which involves only the X n: ( n i 4 ) - 0 * ) ( B - i ) x n + /-(^r)B - ^ i j x n + 1 - (e - - f ) x n + 2 - £ $ E . 0 We have therefore replaced the o r i g i n a l equations ( i l l ) which involved X < d . and S . and therefore depend on a l l X from m=0 to m<=N. by the new n' rf n' m ' set of equations (ii14) each of which involves only three d i f f e r e n t X . - 89 -I t i s now convenient to defines ( i n s ) X r = w + q n When we substitute t h i s i n (14) we f i n d f o r Q: (1116) - (1*)(B - $ - • „ • r t a « ) B - ^ . 7 % + 1 - (B - 4 ) q n + 2 = 0 One notices however that there are only N-2 equations of the type (1116) instead of the N o r i g i n a l equations ( i l l ) . This r e s u l t s from the fa c t that we have assumed that we stop using the reservoir i n the year N so that X N only obeys equation ( i l l ) f o r n <^  N. As a r e s u l t n+2 i n equations! I114) or (1116) can at most equal N and there are only N-2 equations of t h i s type. Since we have N variables ( X M or a^) we have to add two of the o r i g i n a l equations to the set (1116). I t i s most convenient to use the f i r s t and l a s t equations: (1117) A - Bfw+q ) - o l d = S 0 o o (1118) A - B(w+qn) - o ( d N =0 We can now return to the N-2 equations (I116). These aquations require a solution of the form: (1119) q n = R +A" + R_A" where A + and /\ are the two roots of the quadratic equation; (1120) - ( i + r ) ( B - f ) +/"(2+r)B - ^ - j A " (B - ^ - ) A 2 - 0 so that: (1121) /\ + - 1 * | i | / ( 1 + I ) 2 - (1+r) where (1122) P = rB / (B - ^ ) - 90 -I t i s easy to check that (ill9) i s a solution of the equations ( l I 1 6 ) . One can also show that f o r a l l those values of the parameters B and — f o r which / i s r e a l (and not complex). This i s always the case f o r r e a l i s t i c values and i n any case the model i s c e r t a i n l y not applicable i f A i s Complex. Using the r e s u l t (IHE? which assures the solution of the (N-2) equa-tions (I116) we can now use the two additional equations ( i i 17) and (II18) to determine R + and R . I t should also be emphasized that the solution we get i s i d e n t i c a l to that of Burt (1967). - 91 -APPENDIX. I l l THE WAY OF CALCULATION OF THE CONSTANT R AND THE USER COST We want to calculate R . We have f o r X o (III1) A - BX - d d - S ^ o o a ill* and from equation (27): X = w + R o substituting t h i s i n equation ( l j l l ) We ..have (III3) A - Bw - BR_ - Q( d Q = SQ". We s t i l l have to calculate S Q. .In general we have: £7° (ma) s n . 4 xi Xm m=n+1 (1+r) and we know that (III5) X„ - w + R A m . m — — Substituting t h i s i n ; ( I I I 4 ) gives . V 2 w+fi T m (HIS) S = — *• • 1 n s m*=n+1 r. ->m-n (1+r) ck S 1 ^ «• o A " V ~ c * ^(m-n) m=n+1 (1+rJ m=n+1 Summing the two geometric series we get: d i p ) s „ = 4 w . | + 4 R _ A n - _ A ^ _ n s and i n p a r t i c u l a r ( I I I8) S ,lL2^"\ , v ' p s.r sU+ r-7\_J - 92 -This can now be substituted into equation ( I I I 3 ) : 1 ( I I I 9 ) A - Bw - BR_ - <*do - + s i l ^ J and f i n a l l y solving f o r R__: A - (B + ^ ) w - old ( H U G ) R = R S R _ 2 " R , .<*A-8 + s l l + r - ^ ) / 

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