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Optimal policies in the delivery of human services
OPTIMAL
POLICIES IN THE DELIVERY HUMAN SERVICES I. BROSHt
Florlda International
and
OF
E. TURBAN
University, Tamiani Trail, Miami, FL 33144. U.S.A.
and E. SHLIFER Technion-Israel
Institute of Technology,
Israel
(Received 13 June 1911; revised 29 August 1977) Abstract-The delivery of services to people is exemplified in two models and analyzed from two different points of view: the provider and the customer. Results indicate that optimal decisions which are made by providers, based on cost minimization, coincide with optimal choices of the customers which are based on the minimization of the total regret of the recipients.
INTRODUCTION
MODEL 1
The delivery of services to humans in our society is continuously increasing as society becomes more and more affluent. Such services range from supply of utilities to the provision of education and public transportation. The delivery of these services is usually characterized by a single provider in a community. This provider can be the state government, the local government, a non-profit corpor,ation or a rate regulated company. In such delivery systems an attempt is usually made to serve all customers on an equal basis; i.e. the services are provided free, or an equal fee per unit of service (or per customer) is ,charged to all customers. In many systems the customers have little say regarding the planning, operation, and management of the services. The customers are seldom served according to their own preferences. In many instances such services are notorious for their inefficiencies and in some cases they are even ineffective. The end result is that the customers are dissatisfied and it is their belief that they can run these systems, if given the chance, much more efficiently. In recent years there have been numerous attempts to apply management science/operations research to the analysis of the delivery of such services (see bibliography). The purpose of this paper is to identify and evaluate optimal policies in the provision of human services. Specifically, two models will be examined. Model I: Free or equal fee services are supplied to all c&omen up to a certain service level. Any customer requiring further services will have a choice of paying higher fees to the provider so he (or she) will get the service at his (her) desired level. Alternatively, this customer will be able to receive the service at a predetermined level free or at a standard rate, and then extend it, at his own expense, to the level of his choice. Model ZZ: Free or equal-fee services are supplied to some customers. The remaining customers will have a choice of a “do-it-yourself” scheme or paying higher fees, if they are connected to the community’s system. These two basic models can be extended to cover any type of service delivered to people in our society.
Of all the services that can fit this model we elected to analyze the supply of water because it is easier to envision such a system as well as to define a service level. However, many other services will fit this model: for example, postal services, garbage collection and public transportation. A water supply system is comprised of a network of pipes, pumping stations, reservoirs and other’ facilities. The cost of providing water is largely dependent upon the location of the consumers (distance and elevation). To simplify the analysis one may view any water network system as being equivalent to a water system in a high rise building. In such a case the cost of providing water depends mainly on the elevation of the residents. The higher the customer resides, the costlier it is to supply him (or her) with water. In general the elevation of each customer will be denoted by h, where h is the elevation above sea level (or water source), and 0 5 h I H; H = max(h). As a service level of the system we can envision the point to which water will be supplied by the provider.
10n leave from Tel-Aviv University
SEPS Vol
12. No 2-A
The provider’s point of view
The provider in this case is presented as a decision maker whose objective is to supply water to all customers (but not necessarily to their residences) such that the total system’s cost is minimized. The decision variable in this case if the level of service, h,, where 0 I h, I H, to which water will be provided in sufficient quantities for the entire population. The provider adopts the following policy: (a) Any customer residing at level h, or lower, will be connected by the provider to the system and will pay a standard fee. (b) Any customer residing at a level higher than h, will have a choice: (1) Request the provider to connect him to the system. In such a case the customer will pay higher fees for the water, as a function of the difference between h and h,, or; (2) The customer may recieve water at level h, and pay the standard fee. From this point on it is his business (and expense) how to transfer the water to his residence.
56
I. BROWeta/.
Formulation of the provider’s problem Let n(h) dh = number of customers residing in the interval [h, h + dh] C,(h,) = total annual cost of supplying water up to level h, C,(h - h,) = average annual cost per customer residing at level h, due to extending the supply from level h, to h, where h z h, C,(h,) = total annual cost of the entire water
system The problem is, thus, formulated as follows: Find h, such that the total annual cost is minimized. In mathematical terms the total cost function can be expressed as:
C,(k) = C,(k) +
I
H
C,(h - h,)n(h) dh.
AS
(I)
The optimal level, hi, can be found by equating the hrst partial derivative of eqn (l), to zero. Thus, the condition for optimal h, is:
C(k) ah,
act(k) ;
n(h) dh = 0.
ah,
(2)
Any provider who deviates from hf is considered inefficient and the result is a larger total cost to the entire community. Let us now see how the customers may view the management of such a system. The customer’s point of view
A customer may consider two alternative courses of action: (1) he may request the provider to supply the water to his residence but not further (since the function C,(h) is monotonically increasing with h). His average annual cost in such a case is:
C,(h) N where C,(h) is the total annual cost of providing water up to level h; N = total number of customers in the community. (2) In the second alternative the customer may prefer that water be provided not to his residence, but rather to a lower level h,. From this level on he will build an extension up to his residence, or transport the water in containers to his residence. The average annual cost per customer in such an event will be: y+C,(h-hJ
where h, = level, selected by the customer, to which he would like the provider’s system to be extended; C,(h,) = total annual cost of supplying water up to level h,; C,(h - h,) = average annual cost, per customer, required to deliver water from level h, to level h. The customer’s decision
If the customer is interested in minimizing his own tExceptionsare public hearingson rate increasesrequestedby utility companiesin some countries. However, very seldom is the public asked directly about the level of service to be provided.
cost, without considering the rest of the community, then his choice between the two alternatives is simple: If the value of eqn (3) is larger than that of eqn (4), he selects the second alternative (the system extends up to h, plus extra delivery). If, on the other hand the value of eqn (3) is smaller than that of eqn (4), he selects the first alternative, namely he requests that the provider supply the water up to his residence. The customer’s choice, when compared with the provider’s optimal choice, hf, indicates a potential conflict of interest between the provider and the customer. From the customer’s point of view, h, should be either at his residence, h, when the value of eqn (3) is smaller than that of eqn (4), or at level h, otherwise. In either case the customer will prefer to see the system not exceeding his residence, h. Such an approach is in direct contlict with the cost minimization objective of the provider. Further, there is no way that the provider can satisfy all customers’ preferences. This, in part, explains the policies used by most providers: do not ask for customer’s preferences;t deliver services to satisfy all customers’ needs, and charge equal fees. It will be shown next that the provider-customer conflict may be resolved if the customer’s attitude is viewed differently. To do so, the concept of the conscientious customer is introduced. The conscientious customer
This customer is characterized by his concern for the community as a whole and his willingness to accept a policy in which he (or she) as an individual, may pay a little but more, if the end result is a system which is efficient and equitable from a societal point of view. It is suggested that the conscientious customer is ready to settle for a service level h, which minimizes the total regret of all customers, although h, is not necessarily his preferred service level as an individual. Regret is defined as the cost difference incurred by the customer when the optimal value of h, does not coincide with the customer’s preferred service level. The total regret function C,(h,) for all customers is derived in Appendix 1. The result of this derivation is shown in eqn (5) CdW =
C,(k) +
C,(h - h,)n(h) dh + A.
(5)
To find the level at which the total regret is minimized we differentiate eqn (5) with respect to h, The term A is not a function of h, The remaining two terms of the right hand side of the equation correspond to eqn (I), with h, replacing h,. Differentiating eqn (5) to find the optimal service level which minimizes total regret, yields an equation corresponding to eqn (2) with h, replacing h,. Therefore, hz equals hr. The implication of this interesting result will be discussed after examining Model II. MODELIi
In the preceding model it was assumed that all residents of the community share equally the cost of water supply, up to a certain level (h,, h,, or h,). Those who live above these levels incur additional expenses. In the present model, another type of service will be investigated, where cost sharing is confined to the direct recipients of the service, while part of the population has the choice of satisfying its needs independently of the
Optimal policies in the delivery of human services
rest of the population, hence, not being involved in the redistribution of the cost of service among the total population. An example of such a contingency is a sewer system. Home owners outside the sewer network, using septic tanks constructed and maintained at their own cost, are exempted from the sewer tax or fees. As in Model I, here, too, it is assumed that all customers must have some sewer services. Other examples of services that fit this model are the self-generation of electricity by some farms, the digging of private water wells (common in Florida), and the most recent use of solar energy for heating water rather than the use of gas or electricity provided by a utility company. The provider’s p&t of view The provider in this case is a decision maker whose objective is to minimize the total cost of the system. The provider constructs a central sewer system up to a point (or level) h,, which can be visualized similarly to that of Model I. He adopts the following policy: (a) Any customer residing at level h, or closer (lower), must be connected to the sewer system, paying a standard fee. (b) Any customer residing at a level further (higher) than h, will have ,a choice: (1) Build his own septic tank (or sewer plant) at his own expense, operate it independently and pay no sewer taxes or sewage fees. Or; (2) Request the provider to extend the central sewer system to his residence, which is located at level h, and pay higher fees than those residents residing at levels closer (lower) than h,. Formulation of the provider’s problem
Let h = distance (or level) of the customer’s house from a reference point (sewage collector, treatment plant, etc.) n(h) dh = number. of customers in the interval [h, h + dh]. : C,(h,) = Total annual cost of providing a sewer network for all customers up to h,. C, = The customer’s annual average cost of owning and maintaining a septic tank. C&h,) = the system’s total annual cost. The provider’s problem is to find the level h, so that the total system’s cost will be minimized. This total cost is expressed in eqn (6). H C.&J = C,(h,)+
I4
C,n(h) dh
(6)
By differentiating eqn (6) and equating it to zero (as shown in eqn (7) the optimal value, hf, can be computed. (7) Let us examine now the situation from the customer’s point of view. The customer’s decision situation
The customer in this case is visualized as an individual decision maker residing at level h, his decision alternatives are: (a) Stay independent. (b) Request that the sewer network be extended to his home.
57
The costs under these options are: --If a customer is independent, (alternative a) his average annual cost is C,. -If a customer is connected to the central system (alternative b) his cost is:
C,(h)
I
h
0
n(h) dh
The customer’s preferred service level depends on the relationship between eqn (8) and C,. If Cz is smaller than the value of eqn (8) the customer will prefer to be independent. Otherwise he would like to be part of the system which extends up to his house (but not further!). The implication is exactly as in Model I, namely, a conflict of interest exists between the provider and the customer. This conflict may be resolved considering the concept of the conscientious customer, as in Model I. The conscientious customer
Assuming a conscientious customer as described in Model I, the total regret function for this model is derived (see appendix 2) and is expressed in eqn (9).
GA) = C,(k) +
C&h) dh - GWJ FHp
Jh,
(9)
Differentiating eqn (9) with respect to h, and equating it to zero, yields eqn (10)
Gdhr) ah,
-----ah,-
aC,(h,)
c
n(h
2
)
r
=
o .
W-0
From this equation the optimal service level, hr, can be computed. Though the objective function of the provider, eqn (6), is different from the objective function expressing the total regret of the customers, eqn (9); the optimal service levels derived from eqns (7) and (lo), are the same with the substitution of h, for h,. Therefore, h3 = h*,.
58
1. BROSH et al.
the existing “free services”, or “equal fee”, since it discriminates against the few. In our society there is an ever increasing trend to improve the quality of life and the standard of living. Typical to this trend is the flight from the urban areas to the suburbs. The expansion of services to the suburbs may, at least in the short run, be regarded as disadvantageous to the city dwellers, if they pay an equal fee for such services as water or sewage. The irony is that the more affluent people are usually the ones who leave the cities; therefore, existing equal fee policies may benefit the rich rather than the poor.t Implementation
The two basic models discussed in the paper can be extended to cover almost any type of service delivered to people. The optimal policies proposed can be utilized by planners of the delivery of human services. If indeed providers would be allowed to set discriminative rates and deliver services only up to an optimal level h,, they would be able to minimize the total cost of delivering services, thus minimizing the societal expenses and regret for the desired services. In implementing the proposed policy several issues should be investigated: (a) How to convince customers (or the community) that the provider’s decision is indeed the best, i.e. how to convince even a conscientious customer that the community’s total regret will indeed by the lowest? (b) How to deal with politics, politicians, pressure groups, commissioners, rate setters and alike, that today dominate the delivery of human services? (c) How to calculate the fee for the extended services. (d) What to do if the poor are those who reside at the outskirts of the communities? (e) What to do in cases where the government is interested in the redistribution of the population. Despite these and probably other questions, the proposed policies may be so superior over existing policies that at least a trial run must be considered. In an attempt to assess some of the attitudes of the public with regard to rate setting in the supply of water we conducted a small scale survey of 95 residents of Dade and Broward Counties in South Florida. The results of the survey indicated that: (1) 40% of those surveyed knew very little about water supply systems and rate structure. (2) Of the 60% who said they do understand the system, 67% believe that the providers of water try to minimize the total system’s cost. (3) 50% of those surveyed paid monthly fees for water. (4) Out of those who pay for water, 75% think that they pay equal rates.4 Most interesting were the results reflecting the attitudes of the subjects toward the paper’s proposals of rate discrimination and/or the system’s structure. 74 per cent were against the idea of not providing water to all customers at their residences and only 26% agreed to the idea of having to extend individually the water system. tobviously the opposite may be true; in some cases the poor the ones who dwell further from the center of the community, but this is the exception rather than the rule. $It is interesting to note that one of the largest water supply companies in Florida is using rate discrimination on their sewage .. disposal bill, but not for their water bills, charging all the custome& residing outside the city of Miami 25% more. are
61% of all the subjects are against rate discrimination structure. The results, even though based on a small sample, are enlightening. We found no significant difference in the attitudes of people regarding our proposal, with respect to the subjects’ sex, area of residence and type of dwelling unit. The results of the survey might be
explained by the fact that water is relatively a cheap commodity (the average monthly bill in our sample was $17.1) as well as the fact that the population is accustomed to get water at their residence. In addition, water is considered to be an essential commodity. The results of the survey indicate that should the optimal system’s structure, as proposed in this paper, require an extension, it would have to be done by the provider and not necessarily by the customer. As for the rate structure, it should probably be structured as a step function, where customers in the same community pay the same rate while those who live in furtheraway communities pay higher rates than those who live in close-by communities. CONCLUSIONS
In this paper it was shown that an optimal level of human services can be computed by a provider and that this level coincides with the optimal level desired by conscientious customers. Two types of models were selected. These models can be adapted to the delivery of almost any human services, ranging from garbage collection to education and transportation. The implementz)tion of the proposed policy, which at first glance may seem to be unfair, may indeed be the best from a societal point of view and therefore should be considered by all planners and providers of human delivery services. It takes far more than the discovery of a solution on paper to eliminate any major public policy problem or misconception. Among many tasks there is the one of communicating the results to all involved and convincing them that it is a valid solution. Communication of ideas to the public is very often difficult especially in matters that are likely to have strong emotional views, such as in cases of safety, comfort and out of pocket outlays. Nevertheless to solve social problems one must find the way to induce social change, and to persuade the individuals that their personal goals are well taken into consideration within the social structure. REFERENCES
1. D. S. Alberts, A Plan for Measuring the Performance
or Social Programs: The Applications of Operations Research Methodology. Prauger, New York (1970). 2. K. J. Arrow,Social Choice and Individual Values. Wiley, New
York (1951). ~ --,3. Jack Byrd, Operations Research Models for Public Administration, Lexington Books, Lexington, Mass (1975). 4. Ida R. Hoos, Systems analysis in public policy: A critique. University of California Press, Berkeley (1972). 5. A. Y. Lewin and M. F. Shaken, Policy Sciences: Methodology and Cases. Pergamon, Oxford (1976). 6. Philip M. Morse, Operations Research for Public Systems. MIT. Press, Cambridge, Mass. (1967). 7. E. S. Quade, Analysis for Public Decisions, Elsevier Publishing Co.. 1975. g. Jerome Rothenberg, The economics of congestion and pollution: an integrated view. Am. &on. Reu. 60, 114-121(1970). 9. T. L. Saaty, The future of operations research in the government. fnterfaces (the Bulletin of the Institute of Management “. ~__^,_. .^__ Xlences) 2, I-Y (Feb. EJ72).
Optimal policies in the delivery of human services
59
APPENDIXI
APPENDIX2
The regret function for model type I A customer chooses between h and h,, based on the cost comparison of eqns (3) and (4). The conscientious customer is willing to accept a service level h, which minimizes a total regret function, C,(h,). Let G[.] be the customer’s regret function if he resides at level h. It is a function of: [(what he has to pay should the system extend to h,) -(what he would have paid if he could have had his preferred choice)]. Formally:
The regret function
for model type II A customer residing at level h chooses either to be connected
to the sewer system or to stay independent, based on the cost comparison of eqn (8) and the cost C,. The conscientious customer is willing to accept a service level h, which minimizes a total regret function C&h,). Let G[.] be the customer’s regret function if he resides at level h. It is a function of: [(what he has to pay should the system extend to h,) - (what he would have paid if he could have had his preferred choice)]. HP = maximum h for which a customer would have chosen to be connected to the system. Formally:
Cl.1 =*
ifOchs:h,
I
if h, s h c H.
if h, I h I HP
G[.]=
(15)
If the customer’s regret is a linear function of the cost differences in eqn (11) namely G[.] = [.I, the total regret function is: if HP 5 h.
C,(h,)=
If the customer’s regret is a linear function of the cost differences in eqn (IS), namely if G[*] = [.I, the total regret function is:
C,(hA
C,(h,) =
h,
-7
n(h) dh
Equation (12) can be expressed as
C,(h) n(h) dh
1
n(h)dh
” C,(h,) = C,(h,) +
I h,
C,(h - h,)n(h) dh
t A.
(13)
t
(16)
Where A equals: Equation (16) can be expressed as: A=-
%j!d+C,(h_h) hpC1(h)n(h)dh_
n(h)&
N
and A is not a function of h,.
P
1
(,4)
G(k)
= C,(h,)+ C2/” n(h)dh - C,(H,). h,
(17)
POLICIES IN THE DELIVERY HUMAN SERVICES I. BROSHt
Florlda International
and
OF
E. TURBAN
University, Tamiani Trail, Miami, FL 33144. U.S.A.
and E. SHLIFER Technion-Israel
Institute of Technology,
Israel
(Received 13 June 1911; revised 29 August 1977) Abstract-The delivery of services to people is exemplified in two models and analyzed from two different points of view: the provider and the customer. Results indicate that optimal decisions which are made by providers, based on cost minimization, coincide with optimal choices of the customers which are based on the minimization of the total regret of the recipients.
INTRODUCTION
MODEL 1
The delivery of services to humans in our society is continuously increasing as society becomes more and more affluent. Such services range from supply of utilities to the provision of education and public transportation. The delivery of these services is usually characterized by a single provider in a community. This provider can be the state government, the local government, a non-profit corpor,ation or a rate regulated company. In such delivery systems an attempt is usually made to serve all customers on an equal basis; i.e. the services are provided free, or an equal fee per unit of service (or per customer) is ,charged to all customers. In many systems the customers have little say regarding the planning, operation, and management of the services. The customers are seldom served according to their own preferences. In many instances such services are notorious for their inefficiencies and in some cases they are even ineffective. The end result is that the customers are dissatisfied and it is their belief that they can run these systems, if given the chance, much more efficiently. In recent years there have been numerous attempts to apply management science/operations research to the analysis of the delivery of such services (see bibliography). The purpose of this paper is to identify and evaluate optimal policies in the provision of human services. Specifically, two models will be examined. Model I: Free or equal fee services are supplied to all c&omen up to a certain service level. Any customer requiring further services will have a choice of paying higher fees to the provider so he (or she) will get the service at his (her) desired level. Alternatively, this customer will be able to receive the service at a predetermined level free or at a standard rate, and then extend it, at his own expense, to the level of his choice. Model ZZ: Free or equal-fee services are supplied to some customers. The remaining customers will have a choice of a “do-it-yourself” scheme or paying higher fees, if they are connected to the community’s system. These two basic models can be extended to cover any type of service delivered to people in our society.
Of all the services that can fit this model we elected to analyze the supply of water because it is easier to envision such a system as well as to define a service level. However, many other services will fit this model: for example, postal services, garbage collection and public transportation. A water supply system is comprised of a network of pipes, pumping stations, reservoirs and other’ facilities. The cost of providing water is largely dependent upon the location of the consumers (distance and elevation). To simplify the analysis one may view any water network system as being equivalent to a water system in a high rise building. In such a case the cost of providing water depends mainly on the elevation of the residents. The higher the customer resides, the costlier it is to supply him (or her) with water. In general the elevation of each customer will be denoted by h, where h is the elevation above sea level (or water source), and 0 5 h I H; H = max(h). As a service level of the system we can envision the point to which water will be supplied by the provider.
10n leave from Tel-Aviv University
SEPS Vol
12. No 2-A
The provider’s point of view
The provider in this case is presented as a decision maker whose objective is to supply water to all customers (but not necessarily to their residences) such that the total system’s cost is minimized. The decision variable in this case if the level of service, h,, where 0 I h, I H, to which water will be provided in sufficient quantities for the entire population. The provider adopts the following policy: (a) Any customer residing at level h, or lower, will be connected by the provider to the system and will pay a standard fee. (b) Any customer residing at a level higher than h, will have a choice: (1) Request the provider to connect him to the system. In such a case the customer will pay higher fees for the water, as a function of the difference between h and h,, or; (2) The customer may recieve water at level h, and pay the standard fee. From this point on it is his business (and expense) how to transfer the water to his residence.
56
I. BROWeta/.
Formulation of the provider’s problem Let n(h) dh = number of customers residing in the interval [h, h + dh] C,(h,) = total annual cost of supplying water up to level h, C,(h - h,) = average annual cost per customer residing at level h, due to extending the supply from level h, to h, where h z h, C,(h,) = total annual cost of the entire water
system The problem is, thus, formulated as follows: Find h, such that the total annual cost is minimized. In mathematical terms the total cost function can be expressed as:
C,(k) = C,(k) +
I
H
C,(h - h,)n(h) dh.
AS
(I)
The optimal level, hi, can be found by equating the hrst partial derivative of eqn (l), to zero. Thus, the condition for optimal h, is:
C(k) ah,
act(k) ;
n(h) dh = 0.
ah,
(2)
Any provider who deviates from hf is considered inefficient and the result is a larger total cost to the entire community. Let us now see how the customers may view the management of such a system. The customer’s point of view
A customer may consider two alternative courses of action: (1) he may request the provider to supply the water to his residence but not further (since the function C,(h) is monotonically increasing with h). His average annual cost in such a case is:
C,(h) N where C,(h) is the total annual cost of providing water up to level h; N = total number of customers in the community. (2) In the second alternative the customer may prefer that water be provided not to his residence, but rather to a lower level h,. From this level on he will build an extension up to his residence, or transport the water in containers to his residence. The average annual cost per customer in such an event will be: y+C,(h-hJ
where h, = level, selected by the customer, to which he would like the provider’s system to be extended; C,(h,) = total annual cost of supplying water up to level h,; C,(h - h,) = average annual cost, per customer, required to deliver water from level h, to level h. The customer’s decision
If the customer is interested in minimizing his own tExceptionsare public hearingson rate increasesrequestedby utility companiesin some countries. However, very seldom is the public asked directly about the level of service to be provided.
cost, without considering the rest of the community, then his choice between the two alternatives is simple: If the value of eqn (3) is larger than that of eqn (4), he selects the second alternative (the system extends up to h, plus extra delivery). If, on the other hand the value of eqn (3) is smaller than that of eqn (4), he selects the first alternative, namely he requests that the provider supply the water up to his residence. The customer’s choice, when compared with the provider’s optimal choice, hf, indicates a potential conflict of interest between the provider and the customer. From the customer’s point of view, h, should be either at his residence, h, when the value of eqn (3) is smaller than that of eqn (4), or at level h, otherwise. In either case the customer will prefer to see the system not exceeding his residence, h. Such an approach is in direct contlict with the cost minimization objective of the provider. Further, there is no way that the provider can satisfy all customers’ preferences. This, in part, explains the policies used by most providers: do not ask for customer’s preferences;t deliver services to satisfy all customers’ needs, and charge equal fees. It will be shown next that the provider-customer conflict may be resolved if the customer’s attitude is viewed differently. To do so, the concept of the conscientious customer is introduced. The conscientious customer
This customer is characterized by his concern for the community as a whole and his willingness to accept a policy in which he (or she) as an individual, may pay a little but more, if the end result is a system which is efficient and equitable from a societal point of view. It is suggested that the conscientious customer is ready to settle for a service level h, which minimizes the total regret of all customers, although h, is not necessarily his preferred service level as an individual. Regret is defined as the cost difference incurred by the customer when the optimal value of h, does not coincide with the customer’s preferred service level. The total regret function C,(h,) for all customers is derived in Appendix 1. The result of this derivation is shown in eqn (5) CdW =
C,(k) +
C,(h - h,)n(h) dh + A.
(5)
To find the level at which the total regret is minimized we differentiate eqn (5) with respect to h, The term A is not a function of h, The remaining two terms of the right hand side of the equation correspond to eqn (I), with h, replacing h,. Differentiating eqn (5) to find the optimal service level which minimizes total regret, yields an equation corresponding to eqn (2) with h, replacing h,. Therefore, hz equals hr. The implication of this interesting result will be discussed after examining Model II. MODELIi
In the preceding model it was assumed that all residents of the community share equally the cost of water supply, up to a certain level (h,, h,, or h,). Those who live above these levels incur additional expenses. In the present model, another type of service will be investigated, where cost sharing is confined to the direct recipients of the service, while part of the population has the choice of satisfying its needs independently of the
Optimal policies in the delivery of human services
rest of the population, hence, not being involved in the redistribution of the cost of service among the total population. An example of such a contingency is a sewer system. Home owners outside the sewer network, using septic tanks constructed and maintained at their own cost, are exempted from the sewer tax or fees. As in Model I, here, too, it is assumed that all customers must have some sewer services. Other examples of services that fit this model are the self-generation of electricity by some farms, the digging of private water wells (common in Florida), and the most recent use of solar energy for heating water rather than the use of gas or electricity provided by a utility company. The provider’s p&t of view The provider in this case is a decision maker whose objective is to minimize the total cost of the system. The provider constructs a central sewer system up to a point (or level) h,, which can be visualized similarly to that of Model I. He adopts the following policy: (a) Any customer residing at level h, or closer (lower), must be connected to the sewer system, paying a standard fee. (b) Any customer residing at a level further (higher) than h, will have ,a choice: (1) Build his own septic tank (or sewer plant) at his own expense, operate it independently and pay no sewer taxes or sewage fees. Or; (2) Request the provider to extend the central sewer system to his residence, which is located at level h, and pay higher fees than those residents residing at levels closer (lower) than h,. Formulation of the provider’s problem
Let h = distance (or level) of the customer’s house from a reference point (sewage collector, treatment plant, etc.) n(h) dh = number. of customers in the interval [h, h + dh]. : C,(h,) = Total annual cost of providing a sewer network for all customers up to h,. C, = The customer’s annual average cost of owning and maintaining a septic tank. C&h,) = the system’s total annual cost. The provider’s problem is to find the level h, so that the total system’s cost will be minimized. This total cost is expressed in eqn (6). H C.&J = C,(h,)+
I4
C,n(h) dh
(6)
By differentiating eqn (6) and equating it to zero (as shown in eqn (7) the optimal value, hf, can be computed. (7) Let us examine now the situation from the customer’s point of view. The customer’s decision situation
The customer in this case is visualized as an individual decision maker residing at level h, his decision alternatives are: (a) Stay independent. (b) Request that the sewer network be extended to his home.
57
The costs under these options are: --If a customer is independent, (alternative a) his average annual cost is C,. -If a customer is connected to the central system (alternative b) his cost is:
C,(h)
I
h
0
n(h) dh
The customer’s preferred service level depends on the relationship between eqn (8) and C,. If Cz is smaller than the value of eqn (8) the customer will prefer to be independent. Otherwise he would like to be part of the system which extends up to his house (but not further!). The implication is exactly as in Model I, namely, a conflict of interest exists between the provider and the customer. This conflict may be resolved considering the concept of the conscientious customer, as in Model I. The conscientious customer
Assuming a conscientious customer as described in Model I, the total regret function for this model is derived (see appendix 2) and is expressed in eqn (9).
GA) = C,(k) +
C&h) dh - GWJ FHp
Jh,
(9)
Differentiating eqn (9) with respect to h, and equating it to zero, yields eqn (10)
Gdhr) ah,
-----ah,-
aC,(h,)
c
n(h
2
)
r
=
o .
W-0
From this equation the optimal service level, hr, can be computed. Though the objective function of the provider, eqn (6), is different from the objective function expressing the total regret of the customers, eqn (9); the optimal service levels derived from eqns (7) and (lo), are the same with the substitution of h, for h,. Therefore, h3 = h*,.
58
1. BROSH et al.
the existing “free services”, or “equal fee”, since it discriminates against the few. In our society there is an ever increasing trend to improve the quality of life and the standard of living. Typical to this trend is the flight from the urban areas to the suburbs. The expansion of services to the suburbs may, at least in the short run, be regarded as disadvantageous to the city dwellers, if they pay an equal fee for such services as water or sewage. The irony is that the more affluent people are usually the ones who leave the cities; therefore, existing equal fee policies may benefit the rich rather than the poor.t Implementation
The two basic models discussed in the paper can be extended to cover almost any type of service delivered to people. The optimal policies proposed can be utilized by planners of the delivery of human services. If indeed providers would be allowed to set discriminative rates and deliver services only up to an optimal level h,, they would be able to minimize the total cost of delivering services, thus minimizing the societal expenses and regret for the desired services. In implementing the proposed policy several issues should be investigated: (a) How to convince customers (or the community) that the provider’s decision is indeed the best, i.e. how to convince even a conscientious customer that the community’s total regret will indeed by the lowest? (b) How to deal with politics, politicians, pressure groups, commissioners, rate setters and alike, that today dominate the delivery of human services? (c) How to calculate the fee for the extended services. (d) What to do if the poor are those who reside at the outskirts of the communities? (e) What to do in cases where the government is interested in the redistribution of the population. Despite these and probably other questions, the proposed policies may be so superior over existing policies that at least a trial run must be considered. In an attempt to assess some of the attitudes of the public with regard to rate setting in the supply of water we conducted a small scale survey of 95 residents of Dade and Broward Counties in South Florida. The results of the survey indicated that: (1) 40% of those surveyed knew very little about water supply systems and rate structure. (2) Of the 60% who said they do understand the system, 67% believe that the providers of water try to minimize the total system’s cost. (3) 50% of those surveyed paid monthly fees for water. (4) Out of those who pay for water, 75% think that they pay equal rates.4 Most interesting were the results reflecting the attitudes of the subjects toward the paper’s proposals of rate discrimination and/or the system’s structure. 74 per cent were against the idea of not providing water to all customers at their residences and only 26% agreed to the idea of having to extend individually the water system. tobviously the opposite may be true; in some cases the poor the ones who dwell further from the center of the community, but this is the exception rather than the rule. $It is interesting to note that one of the largest water supply companies in Florida is using rate discrimination on their sewage .. disposal bill, but not for their water bills, charging all the custome& residing outside the city of Miami 25% more. are
61% of all the subjects are against rate discrimination structure. The results, even though based on a small sample, are enlightening. We found no significant difference in the attitudes of people regarding our proposal, with respect to the subjects’ sex, area of residence and type of dwelling unit. The results of the survey might be
explained by the fact that water is relatively a cheap commodity (the average monthly bill in our sample was $17.1) as well as the fact that the population is accustomed to get water at their residence. In addition, water is considered to be an essential commodity. The results of the survey indicate that should the optimal system’s structure, as proposed in this paper, require an extension, it would have to be done by the provider and not necessarily by the customer. As for the rate structure, it should probably be structured as a step function, where customers in the same community pay the same rate while those who live in furtheraway communities pay higher rates than those who live in close-by communities. CONCLUSIONS
In this paper it was shown that an optimal level of human services can be computed by a provider and that this level coincides with the optimal level desired by conscientious customers. Two types of models were selected. These models can be adapted to the delivery of almost any human services, ranging from garbage collection to education and transportation. The implementz)tion of the proposed policy, which at first glance may seem to be unfair, may indeed be the best from a societal point of view and therefore should be considered by all planners and providers of human delivery services. It takes far more than the discovery of a solution on paper to eliminate any major public policy problem or misconception. Among many tasks there is the one of communicating the results to all involved and convincing them that it is a valid solution. Communication of ideas to the public is very often difficult especially in matters that are likely to have strong emotional views, such as in cases of safety, comfort and out of pocket outlays. Nevertheless to solve social problems one must find the way to induce social change, and to persuade the individuals that their personal goals are well taken into consideration within the social structure. REFERENCES
1. D. S. Alberts, A Plan for Measuring the Performance
or Social Programs: The Applications of Operations Research Methodology. Prauger, New York (1970). 2. K. J. Arrow,Social Choice and Individual Values. Wiley, New
York (1951). ~ --,3. Jack Byrd, Operations Research Models for Public Administration, Lexington Books, Lexington, Mass (1975). 4. Ida R. Hoos, Systems analysis in public policy: A critique. University of California Press, Berkeley (1972). 5. A. Y. Lewin and M. F. Shaken, Policy Sciences: Methodology and Cases. Pergamon, Oxford (1976). 6. Philip M. Morse, Operations Research for Public Systems. MIT. Press, Cambridge, Mass. (1967). 7. E. S. Quade, Analysis for Public Decisions, Elsevier Publishing Co.. 1975. g. Jerome Rothenberg, The economics of congestion and pollution: an integrated view. Am. &on. Reu. 60, 114-121(1970). 9. T. L. Saaty, The future of operations research in the government. fnterfaces (the Bulletin of the Institute of Management “. ~__^,_. .^__ Xlences) 2, I-Y (Feb. EJ72).
Optimal policies in the delivery of human services
59
APPENDIXI
APPENDIX2
The regret function for model type I A customer chooses between h and h,, based on the cost comparison of eqns (3) and (4). The conscientious customer is willing to accept a service level h, which minimizes a total regret function, C,(h,). Let G[.] be the customer’s regret function if he resides at level h. It is a function of: [(what he has to pay should the system extend to h,) -(what he would have paid if he could have had his preferred choice)]. Formally:
The regret function
for model type II A customer residing at level h chooses either to be connected
to the sewer system or to stay independent, based on the cost comparison of eqn (8) and the cost C,. The conscientious customer is willing to accept a service level h, which minimizes a total regret function C&h,). Let G[.] be the customer’s regret function if he resides at level h. It is a function of: [(what he has to pay should the system extend to h,) - (what he would have paid if he could have had his preferred choice)]. HP = maximum h for which a customer would have chosen to be connected to the system. Formally:
Cl.1 =*
ifOchs:h,
I
if h, s h c H.
if h, I h I HP
G[.]=
(15)
If the customer’s regret is a linear function of the cost differences in eqn (11) namely G[.] = [.I, the total regret function is: if HP 5 h.
C,(h,)=
If the customer’s regret is a linear function of the cost differences in eqn (IS), namely if G[*] = [.I, the total regret function is:
C,(hA
C,(h,) =
h,
-7
n(h) dh
Equation (12) can be expressed as
C,(h) n(h) dh
1
n(h)dh
” C,(h,) = C,(h,) +
I h,
C,(h - h,)n(h) dh
t A.
(13)
t
(16)
Where A equals: Equation (16) can be expressed as: A=-
%j!d+C,(h_h) hpC1(h)n(h)dh_
n(h)&
N
and A is not a function of h,.
P
1
(,4)
G(k)
= C,(h,)+ C2/” n(h)dh - C,(H,). h,
(17)