A general equilibrium approach to public utility pricing: determining prices for a water authority

A Genera ic ity Pricing: Determining Prices for a Witer Authority Peter B. Dixon, University of Melbourne

1. INTRODUCTION This article shows hc..w a general equilibrium model can be constructed for determining an appropriate pricing policy for a public utility. In section 2, our general equilibrium method is explained by means of a simplified model. Then in section 3, a detailed model is set ouu for determining optimal prices in an urban water authority. Results for the water authority of Melbourne are reported in section 4. Concluding remarks are in section 5. 2. A SIMPLIFIED EXAMPLE OF A GENERAL EQUILIBRIUM APPROACH TO PUBLIC UTILITY PRICING For our simplified example, we assume that the household sector consumes two goods: water and other goods, water being produced by a public utility. Our problem is to determine the price that should be charged by the public utility for water. We assume that the hotizehold sector chooses its level of consumption of water (C,) and other goods (C,) to maximize a utility function (u) subject to a budget constraint. Equivalently, we assume there exists a value for p, the marginal utility of expenditure, such that Address correspondence to Peter B. Dixon, institute of Applied Economic and Social Research, University of Melbourne, Parkville WC 3052, Australia

This article is based on a reportby Dixon and Norman (1989) preparedfor the waterauihorities of Melbourne, Sydney, and Perth. Many people in those authorities gave helpful advice and assisted us with data. We are particularlyindebted to Tony Ryan, Howard Rose, Kaz Grzanowski, and Miss A. W. Johns of the Melbourne and Metropolitan Board of Works; Sergio Bogeholz and Raj Goyal of the Sydney WaterBoard and Lloyd Werner, Pietro Guj, Keith Cadee, Geoffrey Mauger, and Chris Pollett of the WaterAuthority of Western Australia. Neither these individuals nor their employers should be held responsible for the views expressed in the original reportand in this article. Through the course of the research reported here, David Baker did an excellent job in solving numerous challenging computing problems and MargaretMead provided valuable librarianservices. Received January 1990; accepted September 1990. Journal of Policy Mod. ling 12(4):745-767

0 Society for Policy Modeling, 1990

(1990)

745 OI61-8938/90/$3.50

P. B. Dixon U,(C,,C,) = BP,, i = 1, 2

(2.1)

and P,G, = E,

(2.2)

i=l

where Ui is the marginal utility of good i (or the partial derivative of U with respect to Ci), Pi is the price of good i, and E is the household expenditure level. Next we assume that inputs (X,) to the public utility (water authority) and the output of other goods (X2)are supplied by firms that maximize revenue subject to an economywide production possibilities function (g). That is, we assume there exists a value for A, the shadow value of a unit of the economy’s scarce resources, such that hg,(X,., X2) = Pi, i = w and

2

(2.3)

and gK,

m =z

(2.4)

where 2 is a parameter reflecting the size of the economy’s resource base and Equation 2.4 defines a production possibilities frontier of the usual shape (concave to the origin). In practical work on public utility pricing of the type reported in sections 3 and 4 of this article, the main effort is in the specification of the utility’s production technology. Here we simply define the technoiogy for the water authority by

x, = W,.),

(2.5)

where $ is a function having a positive first derivative. We assume that the market clears for both water and other goods, that is, Ci = Xi, i = 1,2.

(2.6)

P, = 1.

(2.7)

We also assume that

This last equation specifies good 2 as the numeraire. A variable that is normally of interest to people concerned with public utilities is the level of surplus. This is given in the present model bY

DETERMMNG

PRICES FOR A WATER AUTHORITY

SW= P,C, - PJ,,.

747

(2.8)

Using (2.2) and (2.6) in (2.8) we see that E = PJw

+ P&,

+ S,.

(2.9)

Hence, built into the system (2.1) through (2.8) is the assumption that households spend their entire factor income (P,X, + P *X2)plus the surplus of the water authority. If SWis positive: then we can think of the surplus being returned to the household sector in thr: form of a dividend. On the other hand, if SWis negative, then we should think of the water authority’s deficit as being financed by a levy paid by the household sector. In total our simplified model consists of 11 equations with 12 endogenous variables. A solution is a set of values for the variables in the list

a = {X,;

X2;X,.; C,; Cz;I’,; P,; P,; f3; A; E; Sm.}

that satisfy (2.1) through (2.8). With the number of endogenous variables exceeding the number of equations, we can expect there to be multiple solutions. The optimal pricing problem for the water authority can be thought of as the selection of one of these solutions. This selection can be made in a variety of ways. For example, the authority might look for the value of X,,, that leads to the particular solution, E(optimal), in which U(C, , C2) is as large as possible. Equivalently, the authority can be thought of as choosing either P,Vor P, to maximize U. Although in this model the authority has three instruments (Xw,P,, and P,), it has only one degree of freedom. Once the value of one of the instruments is chosen, the values of the other two, as well as the values of all the other variables, are determined by equations (2.1) through (2.8). If it is accepted that the water authority should aim to facilitate utility maximization for the households, then a convenient way of finding the authority’s optimal strategy is by solving the problem of choosing C, CZ, X,, X,, and X, to maximize WC,9Cd,

(2.10)

subject to (2.4j, (2.5) and (2.6). We refer to this as the water authority optimization problem (WAOP). It can be shown that the solution to this problem, togeth::r

P. B. Dixon

748

with the associated Langrangian multipliers, reveals the particular solution to the model (2.1 through 2.8) in which U& m_axi@ze_d. ’ To demonstrate this, we start by noting that if C,, CZ, X,, XZ, and zw, are-a ~olt$ion_to the WAOP, then there exist Lagrangian multipliers, A, g, P,, P2, associated with the constraints (2.4), (2.3, and (2.6) such that Ili(cl. ?z) = Pi* &,(~w, a

i = 1,2,

(2.11) (2.12)

= 5 WKJ,

T;&(X, %) = F*,

(2.13)

P, = 5.

(2.14)

In addition, C,, c2, X,, X, and X, satisfy the constraints. Now we let

By substituting from (2.15) into the left- and right-hand sides of (2.1) through (2.8) we can establish that s is a solution to our model. For example, under (2.15)) we have LHS(2.1, i= 1) = U,(C,, Ez); RHS(2.1, i=l)

= F@,/FJ

= P,.

Equation 2.11 ensures that LHS(2.1, i = 1) = RHS(2.1, i = I). By continuing these substitutions, we can eventually complete the proof that the WAOP reveals a solution to our model (2.1) through (2.8). That 0

= = Z(optimum)

is then clear from the nature of the WAOP. The WAOP generates consumption levels, c,, c2, giving the largest value of U compatible with the technological and resource constraints under which the economy operates. ‘The idea

of solving general

equilibrium models

by means of an associated mathematical

programming problem has been developed by several authors including Dixon (1975) and Ginsburgh and Waelbroeck (1981).

DETERMINING PRICES FGR A WATER AUTHORlTY

749

3. A GENERAL EQUILIB

MODEL FOR DETERMINING PRICES FOR AN URBAN WATER AUTHORITY In this section we specify a general equilibrium model suitable for determining the optimal pricing strategy of an urban water authority. We assume that the authoric produces two products, outside water and inside water. Inside water is the water used inside buildings. Outside water is what goes on the garden, on washing the car, and filling the swimming pool. From the point of view of pricing, the main distinction between inside and outside water is the costs of wastewater disposal. Unlike outside water, nearly all inside water enters the sewerage system and needs costly treatment before it can be discharged. We assume that the water authority has available three pricing instruments: l l l

a volumetric charge for outside water; a volumetric charge for inside water; and an annual fee (service charge) to all house owners.

Our problem is to determine how each of these prices should be set. To solve this problem convincingly, we need to expand the simple model set out in the previous section. We need to allow for capital accumulation and time. Water authority capital is made up largely of dams, water-treatment plants, and main sewers. These tend to be indivisible items requiring considerable time for construction. In deciding how water prices should be set, it is important to consider the availability of capital capacity in the water authority and the costs and lags in the capital .formation process. Because of the connection between water authority capital formation and growth in the housing stock, it is also desirable to include housing variables explicitly in the model. 3A. Household Behavior As in our simple model, we start by specifying our assumptions concerning household behavior. We assume that households choose the paths over the period 1-H of the housing stock (K,(s)), the purchase of new houses (&&)) , and the consumption of four commodities, Ci( S) 9 i= l-4, to maximize

subject to

P. B. Dixon

750 K,(s)

s

Ii-&

-

I) +

C,(s)SK&),s

Q,(s), s = I, . . . , H, = 1,. . . ,N,

(3.2) (3.3)

and

where C,(S) C,(S) C,(S) C,(S) N(S) P,(S), P&) Q&) P,(S) K,(S)

is the consumption of outside water in year S; is the consumption of inside water in year S; is the consumption of housing services in year S; is the consumption of other goods in year S; is the number of households in year S; for i = 1, 2 and 4, is the price of commodity i in year S; is the price paid for the construction of a new house in year S; is number of houses built in year S; is the annual service charge levied by the water authority on each house; is the number of houses in year S, 9(S) is the time preference discount factor for year S; is the interest discount factor for year s; PW E is the present value of .tousehold expenditure for the period I+?, and U and V are concave utility functions.

The function V is included in the objective function to prevent (3.1) through (3.4) from implying unrealistically low levels for housing construction (Q&)) in years s close to H. Constraint (3.2), with K,(O) taken as given, describes the accumulation of the housing stock. In constraint (3.3), we specify that the housing stock limits the consumption of housing services. The household sector’s budget constraint is specified by (3.4) user Behavior Our specification of producer behavior is very similar to that in the simple model. We assume that in each year S, s = I-H, producers choose the level of output of other goods (X4(s)), the supply of new houses (X5(s)), and the supply of inputs to the water authority (X,(s)) to maximize

X

pi(s)xi(s)

i = 4.5.7

subject to a production possibilities set given by

(3.5)

DETERMINING PRICES FOR A WATER AUTHORITY

gl&W. X,(s), Ws)l

s ad.

751 (3.6)

where P,(S) is the price paid by the water authority per unit of inputs purchased in year s; and Z(s) is a measure of the economy’s overall productive capacity in year s. 3C. Water Authority Technology As out earlie! main effort specifying a utility pricing is in the utility’s technology. For water authority need to accumulation of harvesting and capacity, treatment and sewer We also to include running costs maintenance. We assume that the quantity of water supplied in year s is limited as follows: WATER

HARVESTING

AND STORAGE.

X,(S)+ X*(s)5 b,[K,(s)]T, S = I,

. . . , H,

(3.7)

where X,(s) and X,(s) are the supplies of outside and inside water; b, and yI are positive parameters, the value of y, being less than or equal to unity; and K,(s), dam capital, is a measure of accumulated investment in dams to year s. With y, < 1, we allow for diminishing returns to investment in dams. Dam capital can be expanded according to K,(s) I K,(s- 1) + k,(s)cq,s = I,. . . ,H,

(3.8)

where we refer to k,(s) as the number of new dams becoming available in year s. 01~is a parameter reflecting the size of dams. One way to recognize lumpiness is by restricting k,(s) to being a nonnegative integer. If (Y,is small, then the lumpiness aspect is not important. On the other hand, if
which is currently under investigation, is to specify X,(S) + X*(S) d Wl.7):

(9

P. B. Dixon

752

The creation of new dams requires an investment program. We assume that +(I)

f,(s)

z

x a,,(s)k,(s +

i),

t=l

s = 1, .

. , H,

(3.9)

where I,(S) is the number of units (base period dollars worth, say) of investment activity in dams in year S. a,,(s),a,&), . . . , a,,&) are parameters such that a,,(s) is the number of units of investment activity required in year s to support the completion of a dam in year s + 1. Under (3.9) we allow no endogenous flexibility in construction times. In view of this, we treat k,(s), s = 1, . . . , ~(1) as predetermined variables. The values of these variables have been determined by plans made before our initial year. WASTEWATER

TREATMENT. The specification we adopt for waste-

water treatment is mathematically similar to that adopted for water harvesting and storage. We assume that X,(S)5 b,[K&)l’rZ, S = 1, K&j

S K,(s - 1) + k,(s)a,,

MS) 2

c aJs)k,(s + 0, ,=I

. . . , H,

s = 1, . . . , H,

s = 1, . . . * H.

(3.10) (3.11) (3.12)

Here, KZ(s)refers to the capital of the wastewater treatment facilities. Lumpiness in the construction of these facilities can be accommodated by restricting k*(s) to integer values or by allowing it to be nonzero only in selected years. Diminishing returns can be handled by setting y2 < 1. The variables kZ(s), s = 1,. . . , a(2) are assumed to be predetermined.

W(s) S a[KI(s)lY’;

(ii)

W(s) s qW(s - I) + R(s - 1) - X,(s - I) - Xz(s - 1);

(iii)

R(s) I @[K,(S)]“.

(iv)

In this specific&ion, W(s)is the maximum amount of water that could be consumed in year S. This is limited in constraint (ii) by dam capacity. It is also limited in constraint (iii) by an accumulation relationship. The first term on the right-hand side of (iii) is the amount of water available in the previous year, W(s - I), multiplied by an evaporation factor (q = 0.9, for example). The second term, R(s - I), is the amount of water captured from the environment in year s - I. The third and fourth terms are the sales of water in year s - 1. Constraint (iv) relates the amount of water that can be captured to dam capacity.

DETERMINING PRICES FOR A WATER AUTHORITY

753

MAIN SEWERS.In the basic specification of the model, we assume that all houses must be sewered and that the number of houses is limited by the capacity of a possibly lumpy capital item, main sewers (&(s)). Following the same pattern as in the previous two subsections, we assume that .K”O) I b,[K,(s)]‘“,

&WbK,(s-

f,(s)

s = 1, . . . , H,

I)+-k,(s)a,,

z 2

a,;(s)k,(s + i).

s=

l,...,H,

s = I, . . . , H.

(3.13) (3.14) (3.15)

i=t

k3(s),s = l,...,

~(3) is assumed to be predetermined.

CURRENT ACTIVITIES OFTHEWATERAUTHORITY. Current activities include maintenance, pumping, and the use of chemicals and other materials in wastewater treatment. We specify the level of current activities (measured in base period dollars, for example) as 4s) =

x

i=l.Z

6,X,(s)+ i

snXi(~),

s=

l,...,H.

(3.16)

i=o

In this equation, the water authority is viewed as having current cost components that move proportionately with the fluid flows X,(s) and X,(s). Use of resources in maintenance is assumed to be made up of terms proportional to the accumulated investments in water storage (K,(s)), treatment capacity (K*(s)), and main sewer capacity (&(s)). We also include a maintenance term proportional to the number of houses, K,(s). This allows for maintenance of the pipes connecting each house to the water authority services. Installation of these pipes is not a water authority responsibility in the version of our model described here. However, the authority is responsible for their maintenance.

We assume that only good 4 is traded outside the urban economy. Where M&) is the level of net imports of good 4, the market-clearing equations in our model are

P. B. Dixon

754 X,(s) = C,(s),

i = 1 and 2, s = 1, . . . , H,

X,(s) + M,(s) = Cd(S), s = 1, . . . , H, X,(s) = Q,,(s),

s = 1, . . . , H,

X,(s) = A(s) + e Us), ;= I

s = 1,. . . ,H.

(3 .‘1?) (3.18) (3.19) (3.20)

Equation 3.17 ensures that supply equals demand for outside water (i = 1) and inside water (i = 2). In (3.18), the supply of good 4 available to the local market is X4(s) (production) plus &(s) (net imports). This is equated to demand, C&). In (3.19), the number of new houses produced, X,(s), is equated to the number purchased, &(s). Finally, in (3.20), we assume that the supply of inputs to the water authority, X,(s), is equal to the demand for ther I arising from current activities (A(s)) and from investment in dam capacity (I,(s)), treatment capacity (I&)), and main sewer capacity (b,(s)). 3E. Balance of Trade Constraint and Surpluses and Deficits of the Water Authority and the Households We assume that

This equation restricts the present value of the urban society’s balanceof-trade deficits to the amount B, possibly zero. We assume that P&j is exogenously given (i.e., determined outside the urban society). Similarly, we assume that the interest discount factor p(s) is exogenous. The cash surplus of the water authority in year s is given by

The first term on the right-hand side of (3.23) is household expenditure in year s, the second term is the earnings of the household sector from production, and the third term is the baiance of trade deficit. On taking present values we find that

DETERMINING PRICES FOR A WATER AUTHORITY

s,, = -s, -

755

B,

(3.24)

where S, is the present value of the cash surpluses and deficits of the water authority and S,, is the present WIUCof the stream of household saving and dissaving. Thus, our model implies that any deficit of the water authoritymust be financed either by household savings (which include forced savings, i.e., taxes) or by borrowing from outside the urban economy. 3F. The Water Authority Optimization Problem A solution to the model is a list, 8, of variable values satisfying the optimizing behavior, market equilibrium conditions, and technological relationshipsspecified in (3.1) through(3.24). Many variables in our model are treated as exogenously given. These are 4(s), N(s), p(s), Z(s), P4(s) for all s; kits) for i = 1,2,3 and s = 1 through I; B; and &(O) for i = O-3. Nevertheless, at this stage there are many more free variables in our model than restrictions. This is because we have not yet tied down the behavior of the water authority. To close the model, we assume that the water authority sorts through the possible solutions E, looking for the solution E(optimum). This is the solution that gives the highest value of the household utility function, (3.1), compatible with terminal ccnditions of the form K,(H)Z z,(H),

i =

(3.25)

i,2,3,

and ki(H + q) 2 &(H + q),

i = 1,2,3; q = 1, . . . , T(i),

(3.26)

where the values for K,(H) and ki(H + q) are set exogenously. In imposing (3.25) ?nd (3.26), we can insist that the authority bequeath to the generations iiving beyond year H an adequate water infrastructure. Although the details are too space-consuming for inclusion here, it is not difficult to show that &optimum) can be revealed by solving the water authority optimization problem (WAOP) of choosing for s = l,..., H the values of Ci(S), i = I, . . . , 4;

K,(S), i = 0, . . . , 3;

ki[S + T(i)], i = 1,2,3,;

Q&j,

Ii(S), i = 1,2,3;

X,(S), i = !,2,4,5,7;

A(s);

and M.,(S),

where all variables except M,(s) are restricted to being nonnegative, so as to maximize (3.1), subject to

P. B. Dixon

756 (3.2) through (3.3), (3X-0, (3.7) through (3.21), (3.25). and (3.26).

(3.27)

with (2.10), the WAQP in the simple model, problem (3.27) involves no endogenous prices either in its objective function or its constraints. The values of the price variables in &optimum) are deduced from the Lagrangian multipliers associated with the solution. In particular, the prices that should be charged by the water authority relative to the price of good 4 can be calculated for s = 1, ...,H as follows: (3.28) p,(s) = V%(s)+ M%~)l/k(S)l As

p*(s) = U%) + KM

+

~z&(s)l~~xs),

(3.29)

and l%(s)= IkW + &?dA(S)~~~4~S),

(3.30)

where p,(s) is the volumetric charge for outside water relative to the price of other goods; p*(s) is the relative volumetric charge for inside water; p,(s) is the relative annual fee to all house owners; P,(s), p,,,(s), and p,,(s) are the Lagrangian multipliers assoriated with the dam, treatment, and main sewer capacity constraints, (3.7), (3. lo), and (3.13), in year s; and P,(s), P,(s) are the Lagrangianmultipliers associated with the market-clearingequations for good 4 and for water authority inputs, (3.18) and (3.20) in year s.

Equation 3.28 implies that the felative price of outside water, p,(s), should reflect the rental value, P,(s), of darn capacity and the costs (e.g. pumping) di_rectlyassociated with supplying a marginal unit of outside water (6&(s)). EZquation3.29 implies that the relative price of inside water, p*(s), should reflect the rental values of dam capacity and treatment facilities, Pg(s) + P ,0(s), and the costs (e.g. pumping and chemicals) that are directly associated with an extra unit of inside water (8&(s)). Ekptation 3.30 implies that the relative annual fee, p&), should reflect the rental value of main sewer capacity, p,,(s), and _the annual maintenance costs per house of connecting pipes (k?lo6(s)). AUT

TTY PRICING MODEL

Applications of the model set out in section 3 have been made for Sydney, Perth, and Melbourne. The model has proved flexible enough

DETERMINING FRICES FOR A WATER AUTHORITY

757

to capture special features of the water supply situation in each of these cities. For Sydney, where there is currently a problem of beach pollution arising from the discharge into the sea of insufficiently treated wastewater, we allowed for improved treatment by assuming a decline over the next few years in b2 in (3.10) and an increase in a2 in (3.16). In Perth there is a severe shortage of freshwater sources, each new source involving higher costs than the last. We allowed for this by adopting a low value for y , in (3 7). In Melbourne, the special feature is the recent completion of a very large dam, the Thomson. The associated reservoir will be full by the end of 1992. By then it will have increased the annual amount of water that can be made available to Melbourne residents by about 50 percent. To allow for this, we set high values in (3.8) and (3.9) for the increments, k,(s), in dam capacity in early years. We also set low values for the appropriate U,,(S) parameters in (3.9), thereby recognizing that the increments in dam capacity associated with the filling of the Thomson reservoir will require very little further investment. Full reports on these applications are given in Dixon and Norman (1989). Here we provide a brief description of the Melbourne application. 4A. Setting the Parameters and Exogenous Variables in the Melbourne Application In this application, the initial year (s = 1) is 1988 and the final year is 2037, implying H = 50. Data for 1987, year 0, are given in Table 1. The utility function U in (3.1) was assumed to be of the form H

u =

r, c/l-” i

X=1

1=I

pi ln[C,(s)lN(s) -

&I.

(4.1)

The pi are nonnegative parameters summing to 1, and 4 is a parameter with value between 0 and 1. The “subsistence” parameters hi are unrestricted in sign. Under utility maximizing behavior, (4.1) gives the well-known linear expenditure system CJs)lN(s) = H + lP,R(s)l lE(s)lN(s)- C,cLS,(~)l~

(4.2)

where E(s) is the expenditure level in year s. After setting N(O) at unity and after making assumptions about expenditure and price elasticities in year 0, we combined (4.2) with the data for consu and prices from Table 1 to deduce values for the parameters

P. B. Dixon

758 Table 1: Data for Melbourne (1987) Consumption (Quantities) Outside water, C,(O), millions of kl Inside water, C,(O), millions of kl Housing services, C,(O), billion Other goods, C,(O), $ billion

86 276 50 50

Capital stocks ($ billion) Housing, KJO) Dams and large water mains, K,(O) Treatment plants, K,(O) Main sewers, KS(O) Reticulation capital, K,(O)

50 3.419

0.972 2.189 2.295

Outlays by water authority ($ billion) Water delivery expenses, S,[X,(S) + X,(O)] Inside water treatment, (S, - 6&(O) Maintenance of reticulation system, &,,,,&,(O) Maintenance of dams, 6,&,(O) Maintenance of treatment facilities, S,,,&(O) Maintenance of main sewers, S,,&O) Total expenses, A(0) Capital expansion outlays, I,(O) + I,(O) + Z3(0) Total outlays, X,(O)

0.0984 0.0886 0.0337 0.0502 0.0143 0.0321 0.3173 0.0695 0.3868

1.4 Base period prices ($ billion) Outside water, P,(O) Inside water, PJO) Housing service, P,(O) Other goods, P,(O) New houses, PJO) Water authority inputs, P,(O)

0.00015 0.00030 0.05 1.0 1.0 1.0

1.5 Outputs ($ billion) Other goods, X,(O) New houses, X,(O) Expenditure by water authority on inputs, X,(O)

50.0 1.0 0.3868

Data for water authority production, capital stocks, prices, and outlays were extracted from the 1988 Annual Report of the Melbourne water authority. Other items were estimated from national data after taking account of the population of Melbourne relative to that of Australia. The algebraicexpressions in the table indicate the definition of each data item in relation to our model. The only item that has not alreadybeen defined in section 3 is P,(O), the price of housing services. Units are chosen so that the quantity of housing services consumed in year 0 is ‘$50 billion. We estimakd that the amount paid for these housing services (i.e., the rental value of houses) was $2.5 billion, implying a price, P,(O), of $0.05 billion per unit of housing services. Furtherdetails on the data are given in Dixon and Norman (1989).

DETERMINING PRICES FOR A WATER AUTHORITY

759

pi. In our base case for Melbourne, we assumed that the expenditure elasticities of emand in year 0 for goods 1, 2, and 3 were 1.0, 0.2, and 0.25 and that the “subsistence” share, Ij,P,(O)pJE(O), in total expenditure was 0.5. Under these assumptions, the implied absolute values for the own price elasticities for outside and inside wster are 0.5 and 0.1 .3 The value used for the households’ time preference factor,
i-

1) -

pJ.

(4.3)

The

subsistence parameter, b, was set at the base period housing stock per household, that is, &(0)/N(O). In setting the value for PO,we made sure that the value implied by our model for a unit of housing stock in year H was plausible in light of values implied for earlier years. The interest rate discount factor, p(s), appearing in (3.4) and (3.21) was set according to p(s) = (1.04)‘?

(4.4)

For the price of good 4, we assumed that Pa(s) =

1 frx all s.

(4.5)

Under (4.4) and (4.5) we assume a zero rate of inflation and a 4 percent real rate of interest. The assumption of 4 percent for the real interest rate is often used by the Melbourne water authority for long-term planning purposes. The zero inffation assumption has no important implications. The results from our model for real variables and relative prices are unaffected by our inflation assumption. Consisteimtwith population projections supplied by the water authority, we set N(s) according to h’(s) =

N(O)( 1.OlSy.

(4.6)

The

production possibilities frontier (3.6) was specified in the GET form (Powell and Gruen, 1968) as these values were chosen with reference to unpublished material for Penh supplied by the Water Authority of Western Australia. Sensitivity analysis suggeststhat our results for optimal prices are not very sensitive to the values used for the price elasticities. As can be seen from the example reported in this article, optimal prices depend mainly on costs. not demand.

P. B. Dixon

760

(4.7)

where the gk are positive parameters summing to 1, and 4 is a parameter with value less than - 1. Initially, we set q at - 3. In most solutions of the model, there is a considerable decline through time in the production of water authority inputs, X,(s), relative to the production of housing, X5(s), and other goods, X,(S). With q set at - 3, this produces rather sharp dec!ines in &(s) relative to P,(S) and Pa(s). After some experimentation, we decided that - 1.5 was a more suitable value for q, implying a less curved production possibilities frontier and less movement in relative prices. Having set a value for q, we deduced gk, k = 4,5,7 and Z(0) by assuming that the production and price data in Table 1 are consistent with the producer behavior specified in (3.5) and (3.6). The growth factor on the right-hand side of (4.7) allows the economy to expand at 2.5 percent per year, 1.5 percent reflecting population growth and one percent reflecting technological improvements. In (3.7), (3. lo), and (3.13) we set y,, y2, and y3 at 1. Consequently, we assumed constant returns to scale in the creation of water authority capital capacity. We were then able to deduce b,, b2, and b3 from the data in Table 1 on K,(O), i = 0, . . . , 3 and Xi(O), i = 1,2. In making these deductions, we assumed that (3.7) (3.10), and (3.13) held as equalities in year 0. After discussions with water authority officials, we set the gestation periods for dams, treatment plants, and main sewers, 7(l), r(2), and 7(3), at 10 years. The QS in (3.8), (3.1 I), and (3.14) were set at 1 and a,,(S) and axi in (3.12) and (3.15) were set at 0.1 for all S. Thus,

for treatment plants and main sewers, we assumed that a tenth of the expenditure associated with any increment to capacity in year s takes place in each of the preceding 10 years. We also used 0.1 for most of the ali in (3.7). However, we set LZ,~(S), s = l-4, and i = 1 through S) to values close to zero. This allowed us to specify large (5 increments in dam capacity over the period 1988-1992 associated with the filling of the Thomson reservoir without requiring further significant investment expenditures. The filling of the Thomson reservoir is expected to increase capacity by about 9 percent per year for 5 years. Consequently, we set k,(s) = O.O9K,(s -

and

I), s = I-5,

DETERMINING PRICES FOR A WATER AUTHORITY

761

k,(s) = 0, s = 6-10.

The increments, &(s) and k,(s), in treatment and main sewer capacity for s = 1- 10 were set to achieve annual growth rates of 1.5 percent in K2 and KS.This rate of expansion is consistent with the water authority’s capital expenditure plans. The 6 coefficients in (3.16) were set on the basis of the data in Table 1 for water authority outlays, capital stocks, and output levels. In (3.21) we set B equal to zero. Finally, in (3.25) and (3.26) we set the terminal conditions for water authority capital and investment by an iterative procedure. Each of the Ki(50) were adjusted to values consistent with extrapolations from earlier points on the Ki time paths. For example, we ensured that Km)

= K,(45)([K,(45)/K,(35)]“-

y

(4.8)

Similarly, the ki(H + q) were adjusted to ensure that the implied time paths for water authority capital stocks beyond the terminal year were consistent with extrapolations based on the time paths up to the terminal year. Having set the values for the parameters and exogenous variables, we solved the WAOP (3.27), using a linear programming package. The numerous nonlinearities in the WAOP were handled by piecewise linear approximations. The method was similar to that described in Dixon (1978). 4B. Some Results for Melbourne Figures 1 and 2 give some results from the Melbourne version of the model. The dominant fact in the Melbourne data input is the recent completion of the Thomson dam. The filling of the Thomson over the next few years is expected to increase the amount of water that can be supplied to Melbourne from 362 million kiloliters per year in 1987 to about 560 million kiloliters per year in 1992. In 1987, the price paid by most Melbourne families for their last kiloliter of water was 15 cents. Our model indicates that the prices of water should be raised immediately to cover short-run marginal costs. The price for outside water should be about 27 cents per kiloliter and the price for inside water should be about 59 cents. Twenty=seven cents is the cost of “pumping” water to households. (Table 1 indicates that in 1987 the cost of delivering 362 million kiloliters of water was $0.0984 billion, i.e., 27 cents per kiloliter.) The cost in 1987 of treating a kilolitel of inside water was 32 cents. (Table 1 indicates that 276

762

t

763

P. B. Dixon

744

million kiloliters were treated at a cost of $0.0886 billion.) Thus, 59 cents is the pumping and treating cost per kiloliter of inside water. With the prices of water set to cover short-run marginal costs, the model indicates that demand for water will be well below dam capacity. This can be seen in Figure 1, where demand in the first period of the model (1988) is about 330 million kiloliters and annual dam capacity is about 4.00 million kiloliters. In Figure 2, we see that there should be very little change in the real price of outside water before 2018. Until then, this price should continue to reflect just the short-run marginal costs of supplying water. The gradual decline to 2018 in the price of outside water is a general equilibrium effect. Our model implies that there will be a gradual decline in the opportunity cost of resources used by the water authority. This reflects a gradual decline in the amount of resources used by the water authority relative to the size of the urban economy. For inside water, the mo&! indicates that there should be a jump in the price in the year 2000. L@ to that stage. tFf sewage treatment facilities are not fully used. Cctisequenr~y, to the year B300, the optimal price of inside water includes ZIGclaw, me %r the costs of these facilities. After 2000, the facilities are fully used. From 2002 the facilities are expanded in line with demand. (No indivisibilities are allowed in treatment facilities in this version of the model.) Consequently, from the year 2002, the optimal price of inside water includes the capital costs of treatment facilities. Notice that the jump in the price of inside water between the years 2000 and 2002 is about 19 cents. This can be rationalized in terms of the data in Table 1 as follows: Annualcapital costs per kl of treatment

0.972 x ZQ9 = 276 x r$

..J-

0.0143 0.972 ) t maintenance cost per dollar of treatment capital

= 19 cents.

Figure 2 indicates that after 2018 the price of outside water should begin to climb quite steeply. This causes a similar increase in the price of inside water. The increase in dam capacity associated with the Thomson project is so large that despite income and population growth it will take until 2018 for the total usage of water to catch up to the

DETERMINING PRICES FOR A WATER AUTHORITY

765

potential supply. From 2018 to 2025 dam capacity is fully used but is not expanded. During this period, increases in demand are chocked off by price increases for both types of water. As can be seen from Figure 1, consumption of outside water falls. Consumption of inside water, which is comparatively price-inelastic, continues to rise. Total consumption of water is fixed. By the year 2025, households are, at last, willing to buy the available water supply at prices sufficiently high to cover full costs, including the capital cost of dams. At this stage, dam capacity can be expanded. With the expansion in capacity, prices are stabilized at levels that cover full costs. Notice that the incrtiases in the prices of the two types of water between *he years 2018 and 2025 are both about 52 cents. This can be unders,ood in terms of the data in Table 1 as follows: 3.419 x IO’ (0.04 + yg) . = 362 x 10”

Annual capital costs per kl of dams

= 52 cents.

In 1987, the annual fee (service charge) levied by the Melbourne water authority averaged about $600 per house. Our model implied that this should be reduced to about $154. This would cover the share of each house in the annual capital costs of the main sewers and in the maintenance costs of the reticulation system. There are about 1 million houses in Melbourne. Thus, from the data in Table 1, we can rationalize the model’s result for the service fee as follows: Service fee =

2.189 x 10’ (0.04 + !z?-& lo”

+ O*033;o~ lo9

= $154.

In summary, the results for Melbourne suggested that the prices of outside and inside water should be increased, eventually reaching about $0.80 and $1.30 per kiloliter. At the same time, nonvolumetric charges (service fees) should be reduced. In the past, the Melbourne water authority has relied mainly on nonvolumetric charges for its revenue. Volumetric charges (prices) have been set very low. This may have led to overinvestment in water resources with the costs of producing marginal units of water exceeding the benefits associated with their consumption. 5.

CONCLUDING REMARKS

The model described in this article could be extended and improved in several ways. The current work program at the IAESR includes the

P. 8. Dixon

xi6

implementation of the improved specification of water accumulation set out in endnote 2. After that, we are hoping that the water authorities will sponsor research aimed at extending the model to allow analysis of peak load problems. This will involve disaggregation of the water products in the model by the time of the day and the season of the year in which they are supplied. From a methodological point of view, perhaps the most novel feature of the work reported here is the general equilibrium approach. Previous analyses of public utility pricing adopt a partial equilibrium approach.4 Individual public utilities are not normally a very large part of the economy. For example, the water authority studied in this article uses inputs worth only about 0.8 percent of the GDP of its urban economy. Consequently, for looking at public utility pricing policies, it could be argued that a partial equilibrium approach, ignoring the effects of the utility’s policies on prices outside the authority, is adequate. However, a general equilibrium approach is theoreti&y superior. A partial equilibrium approach can be justified only if it is significantly less costly to implement in terms of research and computational effort. For the study reported in this article, our general equilibrium method proved quite straightforward and produced interpretable and interesting results. The computational burden was probably greater than if we had used a partial equilibrium method of comparable sophistication, Nevertheless, the computational costs (programmer time and computer time) were less than a quarter of the costs of the study. Thus, the extra cost of the general equilibrium approach is likely to have been quite small. FERENCES Bos, D. (1986) Public Enterprise Economics Amsterdam: North-Holland. Brown, S. J.. and Sibley, D. S. The Theory of Public Utility Pricing Cambridge: Cambridge University Press. Dixon, P. B. (1975) The Theory of Joint Marimizution. Amsterdam: North-Holland. Dixon, P. B. (1978) The Computation of Economic Equilbria: A Joint Maximization Approach Merroeconomica 28 (December; January): f73- 185. Dixon, P. B., and Norman, P. M. (1989) A Model of water Pricing for Melbourne, Sydney and Perth. Report preparedat the IAESR for the water authoritiesof Melbourne, Sydney and Perth. Ginsburgh, V. A., and Waelbroeck, J. L. (1981) Activity Analysis and General Equilibrium Modelling. Amsterdam: North-Holland.

4For a very helpful theoretical discussionin a partial equilibrium framework, see Ng (1987). O!her modem theoretical treatments include Bos (1986) and Brown and Sibley (1966). For important earlier theoretical and applied work, see Turvey (1968, 1971) and Webb (1973).

DETERMINING PRICES FOR A WATER AUTHORITY

767

Ng. Y. K. (1987) Equity, Efficiency and Financial Viability: Public-Utility Pricing with Special Reference to Water Supply. Ausfralian Economic Review, Third Quarter. pp. 21-35. Powell, A. A., and Gruen, F. H. (1969) The Constant Elasticity of Transformation Frontier and Linear Supply System. Infernafiona! Economic Review 9(Gctober): 315-328. Turvey, R. (1968) Optimal Pricing and Investment in Electricity Supply. London: George Allen and Unwin. Turvey, R. ( 197I) Economic Analysisand Public Enterprises. London: George Allen and Unwin. Webb, M. G. (1973) The Economics of Nationalized Industries. London: Thomas Nelson and Sons.