Th~ u~e
of hard and soft money budgets, and prices
to lImIt demand for centralized eomputer facility
by SEYMOUR SMIDT
Cornell University
Ithaca, N ew York
charge is a self-policing device and there is no reason to
limit such user's. computer demand by an arbitrary
budget constraint.
However, many users have go~ls whose achievement
cannot easily be measured in mon~tary terms. This is
the usual situation in universities and governmental
organizations, and it also obtains in many research or
staff activities in business organizations. In this case
a charge for using computers will be effective only if the
user is subject to some effective budget constraint.
This paper deals with the problem of effectively
limiting demand for computers in a decentralized organizations in which the computer users are attempting
to maximize some goal that cannot be expressed in monetary terms. The users' goals are assumed to be consistent with the goals of the organization. The users seek
to maximize their goals subject to some budget constraints, in circumstances in which they are charged for
their use of the computer. The user's budget is provided
by the central organization (which is itself subject to a
budget constraint) or by an outside organization. In any
case, the user treats the amount of his budget as an externally determined factor that is outsfde of his control. The size of a user's budget reflects a judgment by
the central organization or an outside agency, about
the relative importance of the goals the user is attempting to achieve.
This paper is concerned with two related issues that
INTRODUCTION
A fundamental problem in any large organization is how
to decentralize decision-making, and still insure that the
decision-makers will act in a manner that is consistent
with the goal of the larger organization. The advantages
of decentralization are well-known, and will not be discussed. Two possible disadvantages of decentralization
are relevant here.
First, the decentralized decision-makers may have
g.oals that conflict with the goals of the larger organizatIOn. Second the actions of a decentralized decisionmaker ~~y /jffect other parts of the organization; and
the deCISIOn-maker may either be unaware of these
effects, or be unable to estimate their significance. This
paper deals with situations in which the goals of the
individual decision-makers are consistent with the
goals of the larger organization, so that goal conflicts are
not a problem. The paper concentrate on ways of overcoming the second type of disadvantage of decentralization.
A centralized computer facility is a good example of a
situation in which the second disadvantage mentioned
above may be a problem. Computers are subject to
e~onomies of scale, so it is often logical for an organizatIO~ to encourage its decentralized decision-making
unIts to share the use of a common computer facility.
In the rest of the paper the word user will mean a decentralized decision-making unit that is, at least potentially, a user of this shared computer facility.
Since the use of the computer by one user imposes
C?sts on the other users and/or on the central organizatIOn, some means of limiting or controlling demand for
computers is essential.
_9ne. means of limiting demand is to charge users for
tne useor-t"hecomnute ""-"rrtn----.-.-..
-.~········--··:t+-·-·· .....
..._.__._._~_~-.~- .....-".".:.r;;:,.-,~.J~.
e user IS a prolllrcenter,
and the computer charge is part of his costs the user
h~s ~n incentive to use the computer only in ~ays that
WIll Increase the profits of his unit. In this case the
*The single price for computer services respresents the average
, price paid by users and credited to the central computer facility
during the relevant planning horizon. This average price may be
t he result of a rather complex pricing system in which the amount
charged for a particular job depends on the characteristics of the
job, the priority assigned to the job, the load on the system at the
time the job is processed, and other relevant considerations. A
number of authors have considered the characteristics that should
be incorporated in a pricing system for computer services. See
bibliography.
499
From the collection of the Computer History Museum (www.computerhistory.org)
500
Fall Joint Computer Conference, 1968
arise in this context. The first issue is what basis should
be used in establishing prices (or charges) for computer
usage. The second is whether or not users should be
- given a separate budget that is applicable only to their
computer usage.
The conclusions reached in this paper are based On a
series of mathematical models, that are described in an
appendix. Readers interested in the details of the logic
should consult this appendix. Only the main assumptions made, and the conclusions that result are summarized in the paper.·
Decision variables available to the organization
The central organization has several sets of decision
variables it can manipulate to achieve its objectives.
However, these decision variables are interrelated. This
section will attempt to describe these interrelations. One
set of decision variables relates to the amount and kind
of computing facilties that will be provided, including
both hardware and software. These decision variables
can be thought of as determining a supply schedule of
computer services. A second set of variables relates to
user budgets. The organization can determine how large
a budget it will allocate to each user. It can also specify,
if it wishes, that a certain amount of the user's budget
can be spent on computers, but not on other goods and
services. Given the characteristics of users, these decisions can be thought of as determining the demand
schedule for computer services. A third set of variables
that the organization may specify relates to the terms
on which computer services are made available to users.
In this paper a single price variable will be assumed to
represent these terms. *
Because the three sets of variables are interrelated,
the organization cannot arbitrarily pick levels for all
three variables independently. Rather, if levels are set
for any two of the three, a level for the third variable is
implied. For example, suppose that the demand variables and the supply variables have been determined.
Then the price variable must be left free to adjust demand and supply. If the central organization attempts
to fix the price variable as well, then some other aspect
of the terms on which computer services are available
will perform the adjustment. For example, if the price
for computer services is not free to rise when demand
exceeds supply, then turnaround times will increase,
which in turn will impose costs and inconveniences on
users. The effects are similar, if not quite identical, to
thos!=l that would result from an increase in the price a
user m~st pay to achieve a given rate of turnaround.
Alternatively if the organization has determined the
demand schedule by fixing user budgets, and wishes to
maintain a certain price level without compensating
changes in other non-price terms, it must be prepared
to supply the amount and kind of computing facilities
that will be demanded. Finally, the organization may
decide on the amount and kind of computer facility it
is willing to provide, and the terms on which computer
services will be made available, and then try to adjust
the demand variables in a manner that is consistent
with the IEwels of the other two variables.
In practice, an organization cannot accurately predict what price level for computer services will result
from a given set of decisions about the supply and demand variables. If the price level that results is not
what was desired, some adjustments will be necessary.
Either the demand or supply variables, or both, should
be adjusted to move toward the desired equilibrium.
Whether the initial response to a disequilibrium price
takes place by adjusting the demand variables or the
supply variables depends on how long it takes to adjust
one or the other of these variables. If user budgets take
a long time to change, but changes in hardware and software can be made relatively quickly, then the appropriate initial response to a disequilibrium is to adjust the
supply variables, "or" taking into account the fact that
the demand variables will not be changed for some time.
In other circumstances it may be that hardware and
software changes take a long time but budget changes
can be made relatively quickly.
The conclusion from these comments is that for
long-range planning one should consider all three sets of
variables as subject to control by the decision-maker,
while for short-run adjustments it may be necessary
to treat either the demand variables or the supply variables as being outside of the control of the decisionmaker. The first three models considered in this paper
take a long-range viewpoint. The fourth model assumes
that the supply conditions have been fixed, but that
user budgets are subject to control.
General assumptions
The class of models considered in this paper have the
following characteristics. Each user receives a budget
allocation from the centralorga;nization. * The user
tries to purchase amounts of computer services and of
non-computer inputs that maximize his utility subject
to his budget constraint. The users treat the prices of
all inputs as given.
The central organization must pay for the costs of the
computing facility, and for any non-computer inputs
purchased by users. It has available, to meet these payments, a predetermined quantity of its own resources
plus any amounts that users have added to their budget
*One model allows for the possibility that a user may receive
budget allocations from other sources as well as from the university.
From the collection of the Computer History Museum (www.computerhistory.org)
Use of Hard and Soft Money Budgets
allocations from outside sources. The only resourcs
allocation decisions made directly by the administration are those concerning the type and capacity of the
computing facility. The organization allocates budgets
to users who make the detailed decisions about the use
of the computing facility, and the amounts of noncomputer inputs purchased. The organization indirectly
controls the behavior of users by setting the pricing
schedule for computer services, by determining the
budget allocation assigned to each user, and possibly
by requiring that a certain amount of the user's budget allocation can be spent only on computers. It is assumed that the pricing system used does not discriminate among users.
The paper assumes that there is no conflict between
the goals of users and the goals of the central organization. Specifically, this means that if a user can increase
his utility without reducing the utility level, of any
other user, then the level of utility achieved by the organization will also increase.
In describing the various models, the terms hard and
soft will be applied to monetary amOl.~nts that constrain
a decision-maker. Specifically, hard money is money
that can be spent for any purpose, while soft money is
money that can be used only in some limited way. The
adjectives hard and soft are also applied to budgetary
R.llocations.
These adjectives are often used in informal discussions of computer budgeting problems. Their formal
use is justified by analogy to their usage in international
trade theory. A hard currency is acceptable as a medium
of exchange in one country, and is easily convertible
into the currencies of other countries. Thus hard currency is effectively usable anywhere. A soft currency
is acceptable as a medium of exchange !n one country,
but is not (easily) convertible into the currencies of
other countries. Thus soft currency can be spent only
on a limited range of goods.
Efficient pricing 'with and without external
financing of U8er8
The first model to be considered assumes that all
users receive their budget allocations entirely from the
central organization, and that the organization does
not impose any restrictions on how the user can allocate
the budget assigned to him. Furthermore this model assumes that the organization sets a price level for computer services, and then purchases (or rents) whatever
amount and kind of computing facility is necessary to
satisfy the user's demands at the given price.
Under these condition the central administration has
two independent decisio.ns to make. It can decide how
large a buget to al10cate to each user, an!i it can decide
501
the price it will charge for computer services. To maximize its utility the administration should assign budgets to each user so that at the margin the increase in
satisfaction the organization derives from each additional dollar allocated to a partiCUlar user~s budget is
the same for all users. In practice, it is ass:Umed that the
ordinary budget-setting procedures approximate this
formal requirement. Second, the administration should
price computer services at their marginal. cost to' the
organization. * This second condition is correct provided that the organization would not be better off
with no computer facility. ** The results from this set of
assumptions are hardly surprising; but they provide a
benchmark for comparing the results from other sets of
assumptions.
The next model to be considered continues the assumptions that only hard money budgets are used, and
that computer capacity is adjusted to demand. However, this model allows for the possibility that some
users receive at least part of their buget allocations
from sources outside the central organization. t
Budget allocations that a user receives from outSIde
soUrces are assumed to be hard money from the point of
view of the user, but soft money from the point of view
of the organization. Specifically the organization is assumed free to reduce the budget it allocates to a user if
the user receives funds from outside. However, the organization cannot take funds a user receives from outside, and re-allocate them to other users. Thus the total
budget allocation of a user (from all sources) must be at
least as large as the allocation he receives from outside.
The fact that a particular user receives some budget
allocations from outside sources (and that the organization receives a corresponding amount of funds) may
or may not change any of the conclusions of the previous model. If, when a particular user receives some
funds from outside, the organization reduces the
amount it allocates to that user by a corresponding
amount, and reallocates those funds among all users,
then the effect is exactly the same if the funds had been
given directly to the orga,nization for its unrestricted
*If some flexible pricing system is used", this means that the
average charges earned per day by the computer facility should
equal the incremental costs of expanding capacity (including
operating costs).
**This need not mean that users have no computer services
available to them. It might mean that higher priced computer
services are purchased from outside sources.
t Examples would be a university in which some professors
receive research support from non-university sources, or a local
government in which some programs are supported in part by
grants from the federal government.
From the collection of the Computer History Museum (www.computerhistory.org)
502
Fall Joint Computer Conference, 1968
use. Under these conditions the formal results of the
previous model are equally applicable here.
However, if some users who are financed from outside sources receive larger budget allocations than they
would have received from the organization, then it no
longer follows that computer services should be priced
at marginal cost. In general, under these circumstances, the optimal price for computer services will be
somewhere between marginal cost and the price at
which marginal cost equals marginal revenue. The extreme case in which the optimal price is set so that marginal cost equals marginal revenue would occur only if
every user who actually used computer services was
financed from outside, and if this outside financing was
so generous that the organization would gain no additional utility from an additional dollar allocated to the
budget of such a user.
Significance of marginal cost pricing
The importance of the possibi1ity that the optimum
price may exceed marginal cost will be clearer if the
cost structure of the computing center is considered.
It is commonly believed that computers are subject to
economies of scale. That is, a one per cent increase in the
amount spent on owning and operating a computer
leads to more than a one per cent increase in the quantity of computing that is possible. Under these cost
conditions, if prices are set equal to marginal cost, the
total revenue of the computing center will be less than
its total cost. That is, the computing center will operate at a deficit.
By contrast, setting prices so. that marginal costs
equal marginal revenue is the rule to follow if one wants
to maximize the profits (or minimize the deficits) of the
computing center.
To· the extent that some users receive larger budget
allocations from outside than they would have received
from internal sources, raising prices above marginal
cost becomes advisable. (The higher prices would apply
to all users.) By paying more than marginal cost, outside financed users tend to reduce the deficit of the computing center and thus increase the amount the organization has available for internally financed users. In
effect, by setting prices above marginal cost the organization uses the computing center as an indirect means
of re-allocating funds from externally financed users to
internally financed users.
The use of hard and soft money budgets
In the next model, the assumption that all users are
financed from internal sources is reinstated, and the
assumption that ~apacity is adjusted to effective demand is retained. However, in this model, the organ-
ization is permitted to make two types of budget allocations to users ..
The hard money budget allocation can be used for
either computer services or for other inputs. The soft
money budget can be used only for computer service
charges. Under these condition, if the optimal amounts
of hard and soft money are allocated to each user, the
total budget allocation each user receives will be the
same as he would have received if only hard money had
been allocated. Furthermore, the amount of computer
services each user purchases will be the same as in the
hard budget model. Also the optimal price for computer
services is still at marginal cost.
In summary, if all users are financed from inte:tnal
sources, and computer capacity is adjusted to demand,
there is no advantage to be gained from distinguishing
between hard money and soft money in making budget
allocations to users. In practice, distinguishing between
hard money allocations and soft money allocations under these conditions is likely to lead to a less efficient
use of resources, since users have less possibility of adjusting their expenditure patterns as circumstances
change. within the budget period.
The fourth and final model considered is one in which
users receive no outside financing, both hard and soft
money budgets are allowed, and the capacity of the
computing facility is assumed to be fixed. Since computing capacity is fixed, it is assumed that the price charged
for computer services is free to adjust so as to equate
the quantity demanded to the fixed supply. * The only
decisions that remain to be made by the organization
are how much hard money and how much soft money
to allocate to each user. Under these conditions, it may:
be desirable to use both. hard and soft money budget
allocations.
The necessary condition for an optimum under these
assumptions is that the marginal utility the organization derives from an additional dollar of hard or soft
money allocated to a particular user's budget be proportional to the extra cash outlays that will result for
the organization. If computer capacity were variable"
and were priced at marginal cost, each dollar of hard or
soft money allocated to a user would lead to an expenditure of one dollar by the organization. But when
computer capacity is fixed an additional dollar spent by
a user on computer services will cause less than one dollar of additional expenditure by the organization. The
immediate effect of the user's expenditure is simply to
• Although strictly speaking the mathematical model assumes
that pric~s are flexible, in practice if prices were also fixed, demand
and supply would be equated by variations iIi the non-:-price costs
of using the computing facility. For example, if the demand function increased, and prices could Dot rise, turnaround times would
increase, queues would form at remote terminals, etc.
From the collection of the Computer History Museum (www.computerhistory.org)
Use of Hard and Soft Money Budgets
drive up the price of computer services. Additional cash
outlays by the organization occur only to the extent
that the higher price causes some other users who have
been spending hard money budget allocation on computer services to spend less on computer services, and
more on other inputs. **
Suppose a fixed computer capacity is described as
excess (deficit), to the extent that it exceeds (is less
than) the organization would have chosen if capacity
were -variable. In practice, soft-money budgets are
likely to be useful to an organization only during periods
when there is excess computer capacity. Under these
conditions allocating soft money to users encourages
them to make use of the excess capacity, at a lower
dollar cost to the organization than if they had been
allocated hard money. By contrast if there is a deficit
of computer capacity reducing a user's soft-money
budget past a certain point has the same effect on his
computer usage as reducing his hard money budget.
There soft money is not effective in limiting demand
when there is a shortage of computer capacity. t
When soft money budgets are appropriate, their
amounts should be determined by comparing the benefits the user receives from his additional use of the computer, to the real dollar cost that his usage causes the
organization. As explained above, when computer capacity is fixed, the real dollar cost to the organization will
be greater than zero, but less than the soft-dollar expenditure by the user.
user cannot transfer funds to another. Two commodities
(really categories of commodities) are available to users.
Commodity one is purchased from outside the organization at a market price that cannot be changed by the
behavior of the organization, or by users individually or
collectively. Commodity two, representing computer
services, is produced by the organization, and decisions
about the quantity available, and the price at which
it will be sold are made by the organization. However,
given the allocation of users budgets, there is a one-toone relation between the supply of computer services
and the price at which it is sold to users. This is because the organization's policy is to allow prices for
computer services to fluctuate to equate supply and
demand. For users, that price is taken as given. For the
organization it is convenient to think of price as a policy
variable and computer capacity as determined by the
amount demanded at that price. Users try to maximize
their utility. The following variables are needed to express the behavior of a user in mathematical terms.
= price of commodity one
P 2 = price of commodity two
PI
q il
=
amount of commodity one used by user i
q i2
=
amount of commodity two used by user i
Ii
=
budget assigned to user i by the organization
Ui
=
an index of the satisfaction derived by user i
where
APPENDIX
Long range planning with only
internal financing: Modell
Resource allocation decision of users-In the planning
horizon under consideration, the organization has
available to it a fixed dollar amount, K, available for
expenditure. Detailed decisions about resource allocation are decentralized by assigning budgets to users.·
A user is any decision-making unit in the organization that is free to make its own resource allocation
decisions subject to the budget constraints determined
by the central organization. Assume all users receive
funds only from the central organization, and that one
**This statement assumes an elastic demand for computer
services by users who are, at the margin spending hard money
allocations on computers. If the demand, schedule of these users
is inelastic, allocating soft money to other users will reduce the
cash outlays of the organization in the short-run.
tIf soft money budgets have been used to encourage demand
because of excess capacity, and the amount of excess capacity is
reduced, then reducing the amounts of soft money allocated may
be useful.
503
(1)
Equation (1) implies that user satisfaction is determined by the amounts of each commodity he consumes.
With respect to the utility function U i , the following
assumptions are made
8U i
>0
all i, j
82U i
<0
8qil
all i, j
8qii
=
1,2.
(2)
1, 2.
(3)
The ith user faces the following problem:
Maximize
(4)
subject to
and
j
=
1, 2.
(5)
Using a Lagrange multiplier technique the necessary
From the collection of the Computer History Museum (www.computerhistory.org)
504
Fall Joint Computer Conference, 1968
con.ditions for a maximum for the user can be 'determined by differentiating (6) partially with respect
to qil, qi2, and Xi.
cPi(qil, qi2, Xi) = U i(qil, qi2)
+ Xi(I i -
P1qil - P 2qi2)
(6)
The simultaneous solution of equations (7) and (8),
which are necessary for an optimum, require that
aUi / aU i
~
-;-;= PPI2,and therefore that
uq il
uq i2
The partial derivatives are:
(7)
(15)
acPi
aqi2
aU i
aqi2
-- = -
~i
P2 = 0
(8)
Substituting this in (14) and rearranging, gives
(9)
Let q *ii be the values that satisfy this maximization
problem. In general, the optimum values will depend on
the values of PI, P 2 and Ii. Thus
all i, j = 1, 2
(10)
Equation (10) is the demand schedule of the ith user
for commodity j. From equation (4), q* i1 can be expressed as
(11)
Let QI and Q2 be the total demand by all users for these
two commodities. Then, if there are n users,
n
Qi =
(16)
Noting that P 2 is always positive (since it is a price), it
follows from condition (2) and (3) that the utility of the
ith user will increase at a decreasing rate as his income
increases ..
Now consider how the ith user's utility changes as a
function of a change in P 2. Differentiating equation
(13) partially with respect to P 2 gives
(17)
n
2: q*ij = 2: dij(P1, P
i-l
1 aU i
2,
Ii)
i=1
From equation (11),
The optimum allocation at the central organization
level depends on how each user's utility changes as a
function of his budget-allocation and the prices of the
inputs he uses. Consider the budget allocations first.
Using equation (11), the utility of the ith user, if he is at
an optimum position, can be expressed as:
Substituting equations (15) and (18) into (17), and
rearranging, gives
(13)
(19)
From the collection of the Computer History Museum (www.computerhistory.org)
Use of Hard and Soft Money Budgets
505
It follows from conditions (2) and (3) that the utility of
the ith user will decrease as P 2increases.
Central organization decisions-The central organization's decisions are how much money to allocate to each
user, and what price to set for computer services. Assume the organization's objective is to maximize a
linear combination of user utilities. (More complicated
functions of user's utility might also be consistent with
the assumption of no goal conflict between users and
the organization; but these complications will not be
considered in this paper.) The organization's objective
function can be written as
fa
Do
=
2: 1J.i u,*
(20)
i=1
i = 1,2, ... n
The organization's budget constraint is that the amount
it can allocate to users is the sum of its fixed resources,
K,plus the profits of the computing center. The
revenues of the computing center are P 2Q2. Its costs are
(21)
Thus the budget constraint of the organization is
K
+ P2Q2 -
fa
O2 =
2: Ii
(22)
i=1
The necessary conditions for a maximum at the
organization level, can be obtained from the Lagrangian
expression
fa
2: IL' u,*
8 =
=
AO
[1 -
aq* i2 (P2 - ~)],
ali
aQ2
1
= 1, 2, •.. n.
(25)
af IS
. the marglna
. I cost to t h e organlza
. t'Ion 0 f
Note that -Q
a
2
an additional unit of computer output. The expression
(Po - :~o)
is the marginal contribution earned by the
organization from the computer center as a result fo a
unit increase in the quantity demanded at a constant
aq· * (
af ) .
.
price. The term a~:
P 2 - aQ2 IS the computmg
center's "marginal profit" (revenue minus costs) for
each marginal dollar allocated to user i's budget. The
.
h
' ht
expression [ 1 - aqi2*(
ali P 2 - af)]
aQ2 occumngont
eng
hand side of equation (25) is thenet amount by which
the organization's remaining resources are reduced for
each marginal dollar allocated to user i. This can be
called the net budget drain of a dollar to user i. The left
hand side of equation (25) is the increase in utility to
the organization from each marginal dollar allocated to
user i. Since the term AO is common to all n such equations, equation (25) says that an optimal allocation of
budgets to users is one in which the ratio of the marginal
utility derived by the organization from a dollar allocated to user i, to the corresponding net budget drain,
is the same for all users.
If computer services are priced at marginal cost,
af ,then the net budget d"
P 2 = -Q
raIn IS uru'ty. Suppose
i=1
a
2
aq * the marginal propensity to use computer
that ~,
ali
Differentiating 8 partially with respect to Ii, and
setting equal to zero gives
services, is positive for all users. If the price of computer
services is greater than their marginal cost, that is, if
P2
a8
aI ..
i = 1,2, ... n
Using equation (16), a sufficient condition for
is that
P.i
.
(!..) au,
._-
P 2 aq*i2
a8
aI, =
0
> afQ
a 2' then the net budget drain is less than unity.
In this case the computing center, at the margin, is
"profitable" for the organization. * In these circumstances, the organization, ip. maximizing its utility
would make larger budget allocations to users with a
high propensity to spend money on computers than
would be the case if marginal cost equals price. Simi• A computing center could be profitable at the margin, but still
a2f
.
show a deficit if -:- < O. Computer center costs very likely have
aQ2
this characteristic.
From the collection of the Computer History Museum (www.computerhistory.org)
506
Fall Joint Computer Conference, 1968
larly, if the computing center is unprofitable at the maraf
h
..
gin, so the P2 < aQ2' t e orgaruzatIOn would allocate
relatively smaller budget amounts to users with a high
marginal propensity to use computer services.
Next, consider the problem of optimum price setting.
Differentiating (23) partially with respect to P 2 gives
ceive no outside financing, or so little, that the optimal
budget allocation of the organization requires that they
receive additional funds. This category of users can be
called internally financed users. There are additional
(n-m) users who receive from outside more funds than
they would receive from the organization. These are
externally financed users.
Rewriting equation (26) to distinguish between these
two categories gives
(26)
From equation (19) aU i
aP 2
=
q*i2
(~)
P2
aU i
aq*~
•
First consider an extreme situation. From equation
Therefore
(19), if for any user q \2
=
q*i2 [JLi
-
=
0, then
:~:
=
all internally financed users are in this situation. Suppose also that for i > m, JLi = 0; that is the organization receives no direct satisfaction from computer usage
of externally financed users. Under these conditions) a
(~) a~i ]
(27)
P 2 aq i2
aO
.
suffi Clent
cond"ItIOn for -P
i
O' th t
a 2 = IS a
SUbstituting the right hand side of equation (25) for the
expression in square brackets in equation (27), equation
(26) can be written as
00
n
~
{
[a---ar.*
q
-q*i2 Ao 1 -
'/.=1
i2 (
P2
-
'I.
af )]}
~
q
$2
From the above, it follows that a necessary condition
for a universIty "lend" maximum is that
~)
aq* i2
i
i=l
=
aq*i2
ali
(28)
af (
Q2 + aQ 2 P 2
af)
-
O. Suppose
aQ 2
af
Condition (28) will be satisfied if P 2 = aQ2 ' that is if
computer services are priced at their marginal cost of
production.
Effect of outside financing for some users: Model II
Suppose now that there are m users who either re-
(30)
The sum of the first two terms on the left is the marginal revenue of the computing center. Thus equation
(30) implies that the organization should set prices so
that marginal cost equals marginal revenue. This is the
profit maximizing price.
Thus at one extreme, with all users internally financed, the rule for an optimum is that the price should
equal marginal costs. At the other extreme, with all
users externally financed, the rule for an optimum is
that the price should be set so that marginal cost equals
marginal revenue, which of course implies a higher
price. If some users are in each category, the optimum
price would be somewhere between these two levels.
Hard money verus soft money ; No external
financing : Model I I I
Hard money can be used for any purpose. Soft money
is usable only for computer expenditures. Let hli be the
amount of hard money allocated to the ith user, and let
ali be the amount of soft money allocated to him.
The effective constraint facing an individual user will
depend on the relative sizes of P 2 • q\2 and Bli. If
P 2 q\2 > ali, then the deficit in the user's soft money
From the collection of the Computer History Museum (www.computerhistory.org)
Use of Hard and Soft Money Budgets
budget must be made up with hard money, and there is
effectively only one budget constraint, equation (4),
with Ii = Ji + IIi. This case has already been considered. If P 2q\2 = ,Ii, then there are effectively two
budget constraints, as follows:
507
Noting from equation (32) that aq
*i2 = - qP*i2 ,and
aP2
2
from equation (31) that aq *il = 0
aP2
'
(31)
and
(38)
(32)
This case will be considered below. Finally if P 2q *i 2 <
,Ii, then computer services are essentially a free good to
the ith user and the only effective constraint is equation
(31}.
In cases where equations (31) and (32) are the effective constraints, the user's final decision can be thought
of as the result of maximizing the following Lagrangian
expression
-
Plqil)
+
K
+ P 2Q2
n
L:
A,(81 i - P 2qi2)
n
n
i=1
i=1
L: hL + L:
,I,
n
= P1Ql
hhI i
The necessary conditions for a maximum are
and
L:
81,
=
P 2Q2. Substituting and
i:=1
i=1
acf>
- C2 =
But if equations (31) and (32) hold for every user, then
cf>(qil, qi2, Ah, A,) = U i (qil, qi2)
+ Ah(hl i
Next consider the organization's problem of optimally
allocating hard and soft money to the n users. By
analogy to equation - (22) the organization's overall
budget constraint can be written
simplifying gives the following budget constraint
aU i
(39)
-aqil = -aqil - -AhP l = 0 ,
In this case the central organization will seek to
maximize
and
n
(34)
8 =
L: ~iU i + AO (K -
f(Q·2) - P 1Ql)
(40)
i=1
and equations (3J) and (32).
Next consider the effect on a user's level of utility as
a result of a change in the values of ,Ii, hL, and P 2.
Differentiating partially with respect to hI" gives
(35)
From equation (31), the second term on the right will
aq*·
1
be zero. From equation (32), _ _
",2 = - . Thus
a,Ii
P2
It follows from equation (31) and the definition of Q2
h aQ2
aQI
1
..
t at ahI = 0 and that ahI, = PI . SubstItutIng these
i
relations, and equation (37), equation (41) can be
.
aU i
1
wrItten as ~i - a
* (-p1 ) - AO = O. Thus
q il
. (36)
aU i
,..s-aq*il
11..
Similarly, consider a change in the hard money budget.
(37)
AO =
(42)
PI'
Similarly, differentiating (40) with respect to ,Ii,
gives
From the collection of the Computer History Museum (www.computerhistory.org)
508
Fall Joint Computer Conference, 1968
(43)
(45)
(44)
Since capacity is fixed, and the price must be free to
adjust demand to the available supply, the only decision
variables under the control of the university are the
hard and soft money budgets of each user. Thp, necessary conditions for a maximum are
Rearranging equation (43) gives
aU i
aq*i2
JLi--
AO =
af
aQ2
Equations (42) and (44) are both necessary conditions for an optimum when hard and soft money
budgets are used, provided that soft money is not so
plentiful that equation (32) is violated. Taken together,
these two equations are identical to equation (15). The
latter is a necessary condition for an optimum when
only hard money budgets are considered. Thus the optimum allocation of hard and soft money is one that
yields the same expenditure pattern for every user as if
only hard money had been used.
Hard money verus soft money: Computer capacity
fixed : Model IV
Up to this point the assumption has been that the
amount of computer capacity available could be varied
to meet the demand. It has also been assumed that the
organization has been free to choose an optimal price
for computer services. These assumptions are appropriate if the time horizon is long enough so that
capacity can be varied. In this section, these assumptions are reversed. Assume that capacity is fixed,
and that the price of computer services varies to allocate the available supply among users. These l:J.,ssumptions are appropriate to a short time horizon in which
capacity changes are not feasible. However since user
budgets are assumed to be variable, the time horizon
must be somewhat longer than the budget period.
The discussion of user level maximization in the previous section is applicable here as well, since the individual user can still control the amount of computer
service he uses. The only difference is that an increase
in demand by one user now leads to a higher price (and
therefore decreased demands from other users) whereas
before it would have led to increased capacity. It is
assumed that each user accounts for a small enough
part of the total usage so that he ignores the effects of
his own behavior on the level of prices.
For the central organization, the Lagrangian expression to be maximized
1
=
1,2, "', n
(46)
and
ao
a8I i· =
JLi
aU i
-a
I.
9~
+
8P 2
Ao (Q2 -a
I. - 1)
= 0,
8i
1
= 1,2,
"',n
(47)
aP2
To find a more meaningful expression for 8hI i and
aP2 , consider the aggregate demand for computer
asIi
services when there are both hard and soft money budget
allocations. By analogy to equation (10), let
.
f h ·th
be the demand schedule for computer servlCes
0 tel
user. The aggregate demand schedule is then
n
Q2 =
L: d* i2 (P
l,
P 2 , hL, 8L)
i=l
(49)
Using the formula for implicit differentiation, the
relevant partial derivatives are
8Q2
aP 2
ahI i
ahIi
-aQ2
aP 2
-
8q*i2
8hL
-
8Q2
8P 2
From the collection of the Computer History Museum (www.computerhistory.org)
(50)
Use of Hard and Soft Money Budgets
and
Oq*i2
OBl i
OQ2
OP 2
OQ2
OBli
--=
OQ2
OP 2
OP2
08Ii
(51)
08
i
1
1
i
since, by definition, these users spend all increments of
hard money on noncomputer uses, and all increments of
soft money on computers. For these soft-money users,
the necessary condition (46) can be expressed as
OUi
/J.i OkI.i =
1.
(53)
Ao
In equation (47), the expression
The expression on the right in equation (54) is
1
m'
where m is the aggregate price elasticity of demand for
computer services. Thus equations (47) can be written
as
OUi
1
08Ii
m
-+ 1.
P.i- =
computer services is price elastic. That is, a one percent
decline in the price of computer services will lead to a
more than one percent increase in the quantity demanded. This means that 11 < -1. Furthermore, since
o ~ ai ~ 1, the following inequality will hold true
for all users.
For soft money users, OqI* i2 = O,and OqI* i2
Ok
509
(55)
Ao
~.
-+
1 ~ ~, + 1 ~
11
11
BIBLIOGRAPHY
users equations (46) and (47) are satisfied if
1 M GREENBERGER
2
1
= 1,2, ... , n
(57)
When computer capacity was assumed to be variable,
a necessary condition for an optimum budget allocation was that the ratios on the left of equations (53)
and (55) be identical and equal to unity for all users,
which in turn implied that the optimum budget allocation was one that made all users hard money users.
Under the present assumptions this conclusion does
not necessarily hold. With capacity fixed, prices elastic,
and all users having strictly concave utility functions,
it may be desirable to give some users more soft money,
and less hard money than they would have received if
capacity were variable.
The logic for using soft money under these circumstances can be explained in more intuitive terms as
follows. If capacity is fixed, an additional dollar of soft
money allocated to a soft-money user will be spent entirely on computer services. This will represent less of a
drain on the organization's resources, than a dollar of
hard money allocated to such a user (all of which
would be spent on non-computer goods) or a dollar (of
hard or soft money) allocated to a hard money user
(part of which will be spent on non-computer goods).
The dollar of soft money allocated to a soft-money
user will however cause some hard money to be spent
outside the organization. By trying to spend his money
on computer services, the soft-money user will drive up
the price of these services, making them less attractive
to hard money users, who will reduce their demand for
computer services, and increase their demand for other"
services.
Oq*'2
oq* i2
.
For hard money users, --~ = - - . Denotmg by
0 1,
okI,
~i, the fraction of each additional dollar of income
devoted to computer services by the ith hard money
oq* ,2
~i
user, then 081, = P for hard money users. For these
8
1,
The priority problem and computer time-sharing
Man Sci 12 8881966
2 H KANTER A MOORE
N SINGER
The allocation oj computer time by university computer centers
(56)
J Bus 41 375 1968
3 SSMIDT
Flexible pricing of computer services
In practice it seems likely that the demand for
Man Sci 14000 1968 Also see papers by Marchand Nielsen and
Singer in this volume
From the collection of the Computer History Museum (www.computerhistory.org)
From the collection of the Computer History Museum (www.computerhistory.org)