Improving performance through cost allocation

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FACULTY WORKING
PAPER NO. 1355

Improving Performance Through Cost Allocation

Susan I. Cohen
Martin Loeb

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College of Commerce and Business Administration «oiS

Bureau of Economic and Business Research URt lGN

University of Illinois, Urbana-Champaign

BEBR

FACULTY WORKING PAPER NO. 1355
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
May 1987

Improving Performance Through Cost Allocation

Susan I. Cohen, Associate Professor
Department of Business Administration

Martin Loeb
University of Maryland

Abstract

Considered is an intrafirm resource allocation model with a single
principal and n agents. Each agent represents a distribution division
and the principal represents an owner who is responsible for production
of the "output" that is eventually "sold" by the agents. It is assumed
that each agent (division manager) knows the local profit function for
the division and has disutility for effort. The principal seeks to
maximize firm-wide profits net of output costs and compensation to the
agents. In this setting, which incorporates divergence of preferences
and asymmetric information, it is shown that the principal and the n
agents can strictly improve their welfare by moving from a set of com-
pensation functions that do not include any allocation of costs to
compensation functions that are based on cost allocation.

I. Introduction

In a review of the cost allocation literature, Biddle and Steinberg
[1984] conclude that "... the process of explaining and ultimately im-
proving cost allocations has just begun." (p. 35). Our goal in this
paper is to extend this process by presenting an intrafirm resource
allocation model in which the firm is better off allocating costs than
not allocating costs.

Zimmerman [1979] addresses the positive question of why firms
allocate costs for internal reporting. He presents two examples that
suggest possible uses for cost allocation. The first looks at the use
of fixed cost allocations to reduce the managers' consumption of per-
quisites. In this example, the preferences of the manager and the
superior diverge with respect to expenditures valued by the manager as
perquisites. Furthermore, there is asymmetry of information since the
superior does not know the level of expenditures and/or the (marginal)
value of these expenditures. Zimmerman claims that the cost allocation
acts as a lump-sum tax, reducing the manager's wealth and hence the
consumption of perquisites. Biddle and Steinberg [1984, p. 6], as
well-as Zimmerman [p. 509], note that using the cost allocation as a
lump-sum tax may cause the manager to seek alternative employment,
since the manager's utility has been reduced. This illustrates that a
necessary condition for a cost allocation to be useful to the superior
is that it does not decrease the welfare of the managers. In addition,
Baiman [1981, p. 109] remarks that the lump-sum tax is not really a
cost allocation in the true sense because it does not have any rela-
tionship to either the existence or size of a joint cost.

-2-

Zimraerman's second example looks at the use of cost allocations in
a decentralized firm to motivate the subunit managers to efficiently
use a centrally provided input. With this example, asymmetric infor-
mation arises because a subunit 's manager has more knowledge of its
technology and market conditions. Zimmerman here ignores the problem
of overconsumption of perquisites and does not model divergence of
preferences in any other way. This second example suggests that cost
allocations are useful because they approximate hard-to-observe costs
of service degradation, delay and future expansion.

If cost allocations are to have a role in coordinating a decen-
tralized firm, then clearly such allocations should dominate no cost
allocation. Our paper demonstrates this dominance by presenting a
model that formalizes and refines Zimmerman's examples. Our model
incorporates both asymmetric information and divergence of preferences.
We show conditions under which the use of a cost allocation mechanism
will lead to a strict improvement in the welfare of both the principal
(top management) and the agent(s) (managers). In our model, cost
allocations are useful because they signal scarcity of the centrally
provided input to the subunit managers and encourage managers to
increase their effort levels, thereby reducing a moral hazard problem.

We prove two main results. First, we show that there exist full-
cost allocations that will leave the firm better off than with no
allocation of costs. This first result is merely an existence theorem,
and does not provide a mechanism for the firm to reach this superior
position. Second, we present a cost allocation mechanism under which
the dominant strategy of each subunit manager results in higher net

-3-

profits for the firm than under no-cost allocation. In other words, we
show that the firm can use cost allocations to reach a position of
superior profits. The mechanism that we provide is not based on a full
or "tidy" allocation of costs; however, it may be thought of as
approximate full-cost allocation mechanism.

It should be pointed out that full-cost allocations are not neces-
sarily optimal. Baiman and Noel [1983] have found a setting where
cost allocations are optimal. Their model is of a principal and a
single agent and looks at the allocation of a fixed cost over time.
In contrast, our model is of a principal and many agents and looks at
the allocation of total costs in a single period. Our results are
consistent with the results in Deraski [1981], who showed that cost
allocation can have a use as a means of coordinating agents.

In the next section we describe our model of the integrated firm.
Section III contains the definition of "no-cost allocation" and the
properties of the no cost allocation reward functions. In Section III
we define a solution concept and state and prove our existence result.
In Section IV we present and analyze our dominant strategy cost allo-
cation mechanism. A brief summary is contained in Section V.

II. The Model

The focus on this paper is on presenting some (sufficient) con-
ditions under which cost allocation dominates no cost allocation, not
on presenting an optimal organizational structure. Consider an inte-
grated firm that centrally produces a product that is transferred to
and either sold by several distribution divisions, or transformed into
a final product that is then sold. Assume that the center must meet

-4-

the product demand of each division. The situation we are modelling
is one in which the divisions make binding contracts with third par-
ties; the contract must be fulfilled by the firm. Divisional profits
gross of the costs of the product transferred from the center depend
on the division manager's level of effort and the quantity of the
product transferred. All the division managers and the center know
the cost function faced by the center, but only division managers know
their own profit and utility functions ex ante, and can observe their
own levels of effort. The center observes only the realized profits of
each division ex post. Finally, every division observes the amount of
product transferred and the realized profits of every other division ex
post; that is, each division has access to all of the information that
the center has ex post. Thus, our model differs from the standard
principal-agent models, where the principal (the center) is assumed to
know the utility function(s) of the agent(s). In addition, we assume
that the principal does not have a prior distribution over the profit
and utility functions; that is, over the agents' types.

This last assumption is the one that differentiates our model
dramatically from the standard principal-agent models in the economics
and accounting literatures. However, this assumption is a common one
in the public goods literature; for example, see Groves and Ledyard
[1977a], The firm we are modelling is one in which the division man-
agers have private information about their divisions that the center
cannot acquire. This information has so many dimensions that it is
not possible for the center to form a prior distribution; that is, the
division managers' "types" cannot be parametrized in a way that would
allow the formation of a probability distribution over these types.

-5-

We are thus modelling decision making under uncertainty rather than
decision making under risk. Given the paradigm that we have chosen,
maximization of expected value is no longer an appropriate decision-
making criterion.

We assume that each division manager seeks to maximize his or her
own utility, which is a function of the reward (or incentive compen-
sation) paid by the center and the manager's effort level. In general,
the reward function may depend on whatever the principal can observe
ex post: the jointly observable transferred product, divisional real-
ized profits, and the firm's realized costs, but not on managerial
effort. As in the standard principal-agent models, the center (as the
residual claimant) seeks reward functions that result in maximum net
profits, where net profits equal the sura of divisional gross profits
less the center's cost of producing the transferred product and less
the total compensation paid to all the division managers.

We capture the essential elements of the situation described above
by looking at a model with a single principal and n agents. Although
we model asymmetric information, our model may be viewed as deter-
ministic, since the only "uncertainty" concerns the private informa-
tion held by the divisions, and not a random state of the world that
must be realized at some point in the process. That is, each division
manager knows what his or her division's output would be for every
combination of effort and resource.

Our interest lies in coordination in the integrated firm, in which
the center is the residual claimant. Since the center must meet all
of the demands of the divisions, coordination has a slightly different
interpretation here. We assume that the divisions of the firm are

-6-

such that their only connection is via the center. That is, the only
externalities that exist are those created by the fact that the cost
function faced by the center is not separable, so that the marginal
cost generated by the ith division depends upon what the jth division
demands. Note that the situation we are modelling is similar to the
situation modelled by Groves and Loeb [1979]. Their model deals with
a similar coordination issue, is also deterministic, and allows for
asymmetric information. However, their model does not take into
account managerial disutility of effort nor does it examine the maxi-
mization of profits net of incentive payments made to divisional man-
agers. Therefore, the critical element of divergence of preferences
is missing from their model.

The ith agent (division manager) knows the ith division's profit

function II.(x.,e.), where x. > 0 is the quantity transferred to the ith
ill l —

division and e. e [0,e.] is the effort level of division manager i.

Th

e profit function n,(x.,e.) represents the division's profits gross

ill °

of costs of x., and is hereafter referred to as the division's gross
profit function. The center does not know the gross profit functions
and can only observe realized gross profits ex post. It is assumed
that a division cannot earn positive gross profits without selling any
output, if x. is a final good, or without any of the input when x. is
interpreted as an intermediate good. For positive levels of x. we
assume that gross profit is additively separable in transferred good
and effort. We can rewrite the gross profit function as:

0 if x. = 0

n.(x.,e.) = { l (1)

111 f (x ) + h (e.) if x. > 0 and 0 < e < e
ii li l — i — i

-7-

The interpretation of these assumptions is straightforward. Effort
is a shift parameter that changes the local profit function of the
division. For example, we could view effort as the negative of per-
quisite consumption — as effort increases, perquisite consumption de-
creases. Therefore, local fixed costs (i.e., costs that are independ-
ent of the amount of product transferred) decrease as effort increases.
This interpretation is consistent with the standard principal-agent
models found in the literature. In those models, it is necessary to
assume that the agents' utility functions are additively separable in
output and effort (that is, the cross-part ials are zero) to get the
desired results. The assumption that local profit is additively
separable in x. and e. helps to perform the same function in our model;
that is, this assumption helps to alleviate the problems created by
positive cross-partial derivatives. In addition, the assumption that
x. must be positive for positive local profit is equivalent to
assuming that the division cannot generate profits just by the provi-
sion of effort (and this without the "input" x.). The local profit in
our model is analogous to the output produced by the agent in those
models. In the usual model, the output produced by the agent is
assumed to be a function of the action (effort) taken by the agent,
where all other arguments of the output generating function (except
for the random state of the world) are ignored. In our model the ith
agent's output depends upon effort and the transferred product; thus
we have posited a somewhat different situation than appears in the
usual agency models.

-8-

2
We assume that f. is continuously twice dif f erentiable for posi-
tive x., h. is continuously twice dif f erentiable , h.(0) = 0, and h.
11 l l

is increasing in e. at a decreasing rate (i.e., h.(e.) > 0 and

h. (e.) < 0). Additionally, we assume that for x. > 0, profits are
11 l —

strictly concave in x. and that for all e. , 0 < e. < e. , there exists an
' l l—i—i

x such that n.(x ,e ) > n (x ,e ) for all x > 0. Hence, for all
l liiiii l

x. > 0, f!'(x.) < 0, and at x. , f!(x.) = 0. Let P. be the set of
l 11 ' i'ii l

all possible divisional profit functions which meet the restrictions
detailed above. We assume that the corporate center does not have any
prior beliefs about the set P.. For example, suppose that the divi-
sions were salespeople in individual territories selling a product
produced centrally. Then IT. can be viewed as revenues minus the local
costs associated with selling the product in the ith territory. In
contrast to the standard principal-agent formulation, we assume that
the principal does not have a prior distribution over P.. However, we
assume that the center knows that the demand and local curve costs
faced by the divisions are well-behaved enough so that local profit is

a -jointly concave function of x. and e. , and increasing in e. , and has

11 l

an interior maximum in x. for each e.. For expository purposes only,
we will refer to x. as "output."

Each division manager is assumed to have a utility function
separable in monetary reward, r. , and own effort e.:

U.(r.,e.) = r. - g.(e.) (2)

-9-

where g. is the disutility of effort. It is assumed that g.(e.) is an
increasing, strictly convex, continuously twice dif ferentiable func-
tion of e. and that g.(0) = 0. Also, it is assumed that as effort
approaches the maximum level, disutility of effort approaches infinity:

i.e., lim e.(e.) ♦ •. as e + e . We let G. denote the set of all
11 11 l

disutility of effort functions that meet the above conditions. In the

standard principal-agent framework, the principal knows the functional

form of the disutility of effort function g. for each agent. In the

world where the agent knows the profit function II. with certainty and

where the H. are parametrized by a single variable, the principal can

observe x. , infer agent i's "type" (i.e., infer the function II.) and

use a penalty contract to force the agent to provide the optimal

amount of effort. We explicitly assume that the g. functions are not

known by the principal and that the principal has no distribution over

the set G. . Thus, there are many pairs of II. and g. which would lead
l J l &i

to the same x. (given an R.). The center, therefore, cannot infer
l l

division i's type and its unobservable effort.

The principal (corporate center) receives the gross profits from

all the divisions and must pay for the production of the transferred

product and the incentive compensation of the n division managers.

The principal learns the realized gross profits of the ith division,

n.(x.,e.), ex post (a number not a function). The center must meet
ill

the demands of all the divisions, x, , x„ , ..., x . The cost of pro-

12 n

duction is assumed to depend only on the total amount produced,

x = x. + X. + . . . + x , and is assumed to be known by the center and
12 n

all the divisions. In addition, we shall use x - x. = Z x. to denote

1 i« J

-10-

the total amount demanded by all the divisions except division i. At

this stage of the analysis, we only restrict the cost function C(x) to

3
be strictly increasing. The most interesting cases occur when vari-
able costs are not linear or when the cost function has discrete jumps.

For these cases, the incremental production costs caused by the demand

4
of one division depends upon the demands of all other divisions. We

do not exclude piecewise linear cost functions or cost functions that
have discrete steps (although they must have a positive variable cost
component). Also, note that a reason why the firm would want to
centralize production in the first place is that there are economies
of scale — for example, large fixed production costs.

Although the center does not know the profit functions of the
divisions or the managers' disutility of effort functions, the center
knows the sets of admissable profit and disutility of effort functions.
We restrict attention to the case where real moral hazard problems
arise. Consider the extreme case where the ith division manager is
rewarded the full gross profits of division i. An interesting moral
hazard problem would exist if, in this case, the manager would select
an interior effort level; i.e., 0 < e. < e.. Restricting the admis-
sable set of profit and disutility of effort functions for manager i
to be in the subset, call it M . , of P. X G. such that h.'(0) > g.'(0),
is a sufficient condition for the existence of an interior solution.
We are assuming that the center knows that for sufficiently small
levels of effort the marginal gross profitability of effort is greater
than the marginal disutility of effort for the division manager.

-11-

In the next section, we define the no-cost allocation benchmark,
state the center's problem, and give the properties of the no-cost
allocation reward functions.

III. No-Cost Allocation Problem

In order to define the no-cost allocation reward functions, it is
useful for us to give the exact chronology of the process we are
modelling. The center decides on a contract (based on what it can
observe) and offers this contract to the division manager. The divi-
sion manager must then decide whether or not to accept the contract;
that is, whether or not to accept employment. If the division manager
decides to accept employment, he or she then chooses a level of trans-
ferred product, which is then provided by the center. Profit is then
realized and transferred to the center. The center rewards the divi-
sion manager and keeps the residual.

We can now define what we mean by the no-cost allocation benchmark.
Clearly, no-cost allocation must mean that we are excluding C(x) as an
argument of the reward function. In addition, we exclude reward func-
tions that are nonincreasing in the transferred product. Our ration-
ale is that any reward function that is decreasing in the transferred
product implicitly defines a cost allocation (but not necessarily a
full-cost allocation). In contrast, with a full-cost allocation
reward is decreasing in transferred product and the sum of the costs
allocated to all of the divisions is exactly the firm's total cost.

Let R. (IL , . . . , JI ,x.,...,x ) be the no-cost allocation reward function
11 n 1 n

for division i. Then, the division manager will choose (x.,e.) to solve
the following:

-12-

MAX R. (II. ,. .. ,H ,x.,...,x ) - g.(e.)
11 n 1 n 11
x. ,e.

s.t. R. ( IL ,. . • ,11 .x,,..,^ ) - g.(e.) > W.
ll nl n li — i

Here W. is the manager's reservation wage. We assume that the divi-
sion manager can only observe the realized profits of the other divi-
sions ex post, and does not have a probability distribution over the
possible realizations. Therefore, the manager would not be able to
decide whether or not to accept the contract ex ante if the contract
depended upon the realized profits of the other divisions. Given the
informational assumptions that we have made, the center will restrict

attention to reward functions that depend only on (n.,x.) and are non-

11

decreasing in x.. Note that x. is still an argument of the reward
° l l °

function. The previous discussion only eliminates reward functions

that are nonincreasing in x. ; that is, reward functions that implic-
itly charge for x..

In order to determine the properties of the no-cost allocation

reward function (in addition to the aforementioned), we must state the

center's problem. The center would like to find reward functions that
solve:

MAX

< R. >.n.
l 1=1

LI Ll

I [ni(xi,e1) - R1(H1(x1,e1),x1)] - C( J x )

i=l j=l

s.t. (xi,ei) e argmax {R ( Il^x^e.) tx±) - g^e ) }

and R.(n.(x. ,e.) ,x.) - g.(e.) > W.
l ii'i'i °l l — l

for all i = 1, . . . ,n

-13-

However, the center does not know each division's (n.,g.) pair,
nor does it have a prior distribution over the set of possible profit
and disutility of effort functions, M. , for all divisions i. Thus,
there may not exist optimal reward functions in the usual sense. In
the standard principal-agent setting with unknown types, the principal
has a distribution over the agents' possible types, and selects reward
functions that maximize expected net profits, subjected to expected
utility being greater than or equal to the managers' respective reser-
vation wages (for the first best), and the maximization of expected
utility by the managers (for second best), where expectations are
taken with respect to the unknown types. The R. selected in such a
manner are ex ante optimal but not necessarily ex post optimal.

For our model we must abandon the usual notion of first or second
best optimal reward functions, as well as expected utility maximiza-
tion. Given the lack of information at the center when the reward
functions must be designed, the center has a much more difficult
problem than in the standard principal-agent models.

When expected utility maximization is no longer a feasible strategy,
other criteria must be used. We assume that the center chooses reward
functions that are ex post "rational." Specifically, the center wants
it to be the case that whenever a division manager accepts the contract
offered by the center: (1) ex post the firm would never be better off
without that division in the firm, and (2) ex post the division is not
operating on the downward sloping part of its local profit function.
Since x, is costly to the center, the center would always be better
off (ex post) with an x. chosen so that its marginal contribution is

-14-

at least zero. Clearly, some division managers may decide not to

accept the contract with the above defined kind of reward function;

and there may be situations in which the center would be better off

with a different kind of contract that does not have that property

given the actual types of managers. However, since the center does

not know the manager's types, nor have a probability distribution over

them, the center uses a "minimum regret" type strategy.

Formally, we define an admissible reward function R. to be such

1

* *
that for all (n., g.) e M. , if (x.,e.) is chosen by division manager i

(using the reward function R.), then

JL JL J? JL JL

n.(x.,e.) - R.(n.(x.,e„),x.) > 0; (3)

ill liiii —

that is, the net contribution of division i is nonnegative; and

3n« * *
^(x.,e.)>0. (4)

The incentive compensation is therefore a real-valued function of
II. and x.. Given the above assumptions, we show in the Appendix that
when there is no allocation of costs, the center will choose reward
functions for the divisions that are concave, increasing functions of
local profits alone, and such that R.(n.) < II. for all n. > 0. We
denote the set of such real valued functions by R*.

For the case where costs are not allocated, net profits are:

l n.(x*,e*) - C(x*) - I R.(n.(x*,e*)). (5)

iil * * * i-1 x x x x

-15-

In the next section, we show that for all admissable reward func-
tions that the center could select, the center and the division man-
agers do strictly better by shifting to a full allocation of costs and
adjusting the reward functions to guarantee that each division manager
suffers no loss in utility.

III. Dominance of Full Cost Allocations

Suppose that the division managers that have decided to accept the
no-cost allocation contracts have done so, and that the firm is now
comprised of those managers only. The question then arises: are
there other reward functions for the managers such that both the firm
as the residual claimant and the managers can all be made better off?
That is, given that a manager has decided to accept the no-cost allo-
cation contract and accept employment with the firm, can the firm move
to a Pareto superior outcome via a revised set of contracts? In this
section we show the existence of contracts that make both the firm and
the managers better off. Clearly, there are some managers who may
have decided not to accept the no-cost allocation contracts, but who
would now accept employment with the revised contracts. Our purpose
is to show that once the firm has been formed with the no-cost alloca-
tion contracts, a Pareto improvement can be made by using contracts
that depend on a full allocation of costs. We do not show that these
contracts are optimal; to do so we would have to consider those managers
that did not accept the no-cost allocation contracts.

It should be noted that our results on the dominance of full-cost
allocations hold for any reward functions that are (i) concave; (ii)
increasing; (iii) functions of local profits only; and (iv) strictly

-16-

less than their arguments. Thus any firm whose technology looks like
the technology of the firm we have modelled and who is currently using
a reward function of with properties (i)-(iv) can move to a Pareto
superior outcome by fully allocating costs.

With the no-cost allocation contracts, it was not necessary to
define an equilibrium concept, since the rewards of the individual
division managers were independent of the actions of the other divi-
sion managers. When we move to the full-cost allocation, a division
manager's reward will depend on the transferred product demanded by
the other division managers (but not on the unobservable effort of the
other division managers); we therefore need to define the kind of
equilibrium concept we are using and how we are using it. Since our

model is deterministic we will use the Nash equilibrium as our solu-

8
tion concept.

A Nash equilibrium is a best-replay equilibrium; that is, if you
manage to reach it, given that the other players use their Nash stra-
tegies, your best response is to play the Nash strategy. In the tra-
dition of much of the economics literature, we do not describe how the

9
Nash strategy is reached. We do show that if the firm gets to the

Nash equilibrium (with the full-cost allocation contracts), then that
solution strictly dominates the no-cost allocation solution. Given
the notion of Nash as a best replay equilibrium, it is not inconsis-
tent with our previous analysis to have the reward functions depend on
the realized transferred products of all of the divisions. Since the
divisions can observe every other division's realized profits and

-17-

transferred product ex post, we have not violated the spirit of the

Nash equilibrium concept.

To summarize, we show existence of reward functions such that

at equilibrium the firm and the managers are strictly better off.

Let R.(y) be an arbitrary reward function in R*. We show that if
1

y is gross divisional profits, the principal (center) and the n agents
(divisions) can all do strictly better by ad-justing y downward by a
full allocation of costs, and adding a constant to the reward function.
We first show that a division's profits net of the full-cost alloca-
tion are strictly greater when the argument of R. includes the cost
allocation than when the argument of R. does not include the cost
allocation.

Suppose that costs are fully allocated so that the argument of the

reward function is now realized net divisional profits; i.e.,

n
y = II.(x.,e.) - C.(x.;x-x.) where C.(x.;x-x.) _> 0 and £ C.(x.;x-x.) = C(x)

i = l
We require that for each i the C. function be increasing, lower semi-

• . . 10

continuous, and quasi-convex in x. , and nondecreasing in x-x..

Note that if the cost function C is lower semi-continuous and quasi-
convex, and C. = C/n, then these criteria are met. Also, if there is

a "dual" allocation C = (l/n)FC + (x /x)VC, where FC and VC are fixed

i i

and variable costs, respectively, and where VC is either linear or

quadratic in x, then the criteria on the C. are again met. Finally,

if C is convex, C. = (x./x)C, and the center knows that the firm will
l l

be operating in that portion of the cost function where marginal costs
are greater than or equal to average cost, C. will again meet the
requirements. Thus, for common (full) allocation schemes, and

-18-

reasonable cost functions, the C. functions will be increasing, lower

semi-continuous, and quasi-convex in x. , and nondecreasing in x-x..

1 1

Moreover, we only seek, to demonstrate the existence of conditions

12
whereby the firm is strictly better off fully allocating costs.

* *
As in the previous section, we let (x.,e.) denote the output and

effort levels selected by the ith division's manager when costs are not

allocated. When costs are allocated, let the n-tuple

((x. ,e,), .... (x ,e )) of output and effort define a Nash equilibrium.
linn

Then (x.,e.) maximizes:
11

Ri(Hi(xi,e.)-Ci(xi;x-x.)) - g^e.). (6)

In addition, for an accepted contract, division managers must receive

positive compensation, and thus x. > 0. Hence, (x.,e ) maximizes

l 11

R.(f.(x.)+h.(e.)-C.(x.;x-x\)) - g.(e.). (7)

1111111 l 11

For tractability, it is assumed that (x.,e.) satisfies the first-order
necessary conditions for an interior maximum.

We now show that for each division, net profits are higher with
cost allocations than with no allocation. That is, we prove that:

JL JU JU JU

f,(x.) + h.(e.) - C.(x :x-x) > f.(x.) + h.(e ) - C. ( x . ; x*-x. ) . (8)
ii 11 lii 11 li 11 l

PROPOSITION 1: For every R. e R* , for i = 1, 2, ..., n, the net
l

profit (net of allocated costs) in equilibrium for division i is
higher with an increasing, lower semi-continuous, quasi-convex full
cost allocation than with no cost allocation.

-19-

Proof :

By definition, (x.,e.) satisfies the following first-order conditions:

R!(f.(x*)+h.(eX))f!(x*) = 0 (9)

1111111

and

R!(f.(x*)+h.(e*))h!(e*) - g!(e*) = 0. (10)

l l l l l l l *i i

Since R. > 0, (9) implies that f.(x.) = 0. Note that the choice of x.

is independent of e.. Also by definition, (x.,e.) satisfies the fol-

l li

lowing first order conditions

R!(f.(x.)+h.(e.)-C.(x.;x-x.))[f!(x.)-c!(x.;x-x\)] =0 (11)

11111 li l li li l

and

R!(f.(x.)+h.(e.)-C.(x.;x-x.))h:(e.) - g[(e.) = 0, (12)

1111111 ill li

where C. is defined as the first derivative of C from the right.
i l &

(Since C. is lower semi-continuous, this derivative exists). Since
RJ > 0, (11) yields

f!(x.) - c!(x.;x-x.) = 0. (13)

11111

Thus, x. is also selected independently from e.. Since f.(x ) = 0,
l ill

C. > 0 for all x., and f. is strictly concave, (13) implies that
i 11 J ' r

x. > x.. From concavity of R. , we have R. nonincreasing in its argu-
11 l i

*

ment. This with x. > x. gives us:

-20-

R!(f.(x.)+h.(e.)-C.(x.;x-x.))
ill ii 11 I —

R'.(f.(x.)+h.(e.))
111 11 —

R'.(f.(x*)+h.(e.)) for all e. > 0. (14)

lilii l

When e. = e. , we have:
l l

R!(f .(x.)+h.(e.)-C.(x.;x-x.)) > R* ( f . (x*)+h. (e. ) ) . (15)

ill ii ii l—iii ii

Since h! > 0, (15) implies Chat:

0 = R^(f.(x.)+h.(e.)-C.(xi;x-x.))hJ(e.) - g^(e.) >

R*(fi(x*)+h.(e.))h:(e.) - g^(e.). (16)

As R. (f . (x. )+h. (e. ))h. (e. ) - g„(e.) is decreasing in e. (from concavity),
111111111 l

*
(10) and (16) together imply that e. < e..

By strict concavity (in x.) of f., quasi-convexity of C. and (13)

we have x. as a unique maximizer of f. - C . Therefore,
i ii

f.(x.) - C.(x.;x-x.) > f.(x.) - C.(x.;x-x\). (17)

ii ii l ii ii i

*
As x. > x. for all j and C. is nondecreasing in x-x. , (17) then gives

us :

a * /\ Jc AAA

f,(x.) - C.(x.;x-x.) > f,(x.) - C.(x.;x-x.). (18)

ii ii l ii ii i

*

Finally, using e. > e. and h. increasing, we get:

f.Uj) + h (e ) - C^xjx-x^ > f.(x*) + h.(e*) - C. (x*;x*-x*) . (19)

-21-

We have therefore shown that net divisional profits are higher with an
ex ante cost allocation.

• The next step in the demonstration that there exist reward func-
tions based on full-cost allocations that dominate reward functions
based on no allocation, is to show that divisional profits net of cost

allocation and managerial disutility of effort are higher at (x.,e.)

* *
than at (x.,e.), i =1, 2, ..., n.

PROPOSITION 2: For each 1-1, 2, ..., n,

f.(x.) + h1(ejL) - C.(x.;x-x.) - g^e.) >

f.(x*) + h.(e*) - C.(x*;x*-x*) - g.(e*). (20)

Proof:

Rewriting f.(x.) + h.(e.) as H.(x.,e.), using Proposition 1 and
& l l 11 ill

the fact that y > R.(y) for y > 0, we have:

[n (x ,e )-C (x.;x-x )] - [n. (x. ,e . )-C (x. ;x*-x. ) ] >

R.([H.(x. ,V)-C.(x.fx-x.)] - [n.(x.,e.R(x.;x*-x.)]). (21)

111111 i 11111 i

From concavity of R. for y > 0 (and R. > 0) , we get:

R.([n.(x. ,e.)-C.(x. ;x-x.)] - [n. (x. ,e. )-C. (x. ; x-x. ) J ) >
l ill 11 l ill 11 l —

R.(n.(x. ,e.)-Ci(x.;x-x.)) - R. ( H. ( x. ,e. )-C. ( x . ; x*-x . ) ) . (22)

Therefore,

-22-

~ ~ * * * *

[ll.(x. ,e.)-C.(x.;x-x.)] - [n. (x. ,e . )-C. (x. ;x*-x. ) ] >
ill 11 1 ill 11 l

R.(n.(x. ,e.)-C.(x.;x-x.)) - R. ( II. (x . ; x*-x. )-C. (x. ; x*-x. ) ) . (23)
1111 11 l ill l 11 l

By definition of (x.,e ),

R.(n.(x. ,e.)-C.(x.;x-x.)) - g.(e.) > R. ( H. (x. ,e . )-C . (x . ;x-x\ ) ) - g.(e.). (24

1111 11 l li — iiii 11 l 11

Since x. > x. for all i, C. nondecreasing in x-x , and R. > 0, we have:

R.(n.(x. ,e.)-C.(x. ,x-x.)) - g.(e.)
1111 11 i ii —

R.(n.(x. ,e.)-C.(x.;x*-x.)) - g.(e.). (25)

1111 ii l ii

Rewriting (25) and using (23) gives us:

[Il.(x. 9e.)-C.(x.;x-xJ] - [ H. ( x. ,e . )-C. (x . ; x*-x. ) ] >
lii ii i ill ii i

g.(e.) - g.(e*) > 0. (26)

*
Finally, rearranging terras in (26), and noting that x. and x. are both

positive so that we can substitute f.(x.) + h.(e.) for II.(x.,e.), we
K 1111 ill

have our desired result that:

f.(x.) + h.(e.) - C.(x.;x-x.) - g^e.) >

f^x*) + h.(e*) - C.(x*;x*-x*) - g^e*). (27)

D

We can now use Propositions 1 and 2 to demonstrate our main result.
That is, if the center currently uses reward functions R , R , ..., R
based only on gross divisional profits, there exist reward functions

-23-

R, , R_, .... R based on divisional profits net of a full allocation
1' 2 n

of costs such that the center is strictly better off and each division
manager is strictly better off. It will therefore be in the best
interests of the center and division managers to renegotiate the con-
tracts to use a compensation scheme based on a full-cost allocation.

THEOREM: There exists R, , R„ , .... R , which depend on divisional
1 2 n

profits minus a full-cost allocation, that lead to an outcome that

Pareto dominates the outcome realized with R, , R„ , .... R , when the

1 ' 2' n '

R. are concave, increasing functions that are less than their argu-
ments.

Proof :

Let e. be the difference between gross profits net of allocated
costs and disutility of effort at (x.,e.) and (x.,e.)« That is

e. = [n.(x. ,e1)-Ci(x.;x-xi)-gi(e.)]

- [^(x ,e )-C (x ;x-x )-g (e )]. (28)

By Proposition 2, e. > 0.

*
Since e. >^ e. and g. is increasing, we have that:

[n.(x. ,e.)-C.(x.;x-x.)] - [ n. (x* ,e*)-C . ( x. ; x*-x*) ] > e.. (29)
iiiii l 11111 i l

We can now define R as:

i

R.(n.(x. ,e.)-C.(x. ;x-x.)) = K. + R. ( JT. ( x . ,e . )-C . (x. ; x-x . ) ) (30)
1111 ii l l iiii li i

-24-

where

e.
K. = MlUx^e*)) - g.(e*) - ^(n-Cx^.e.) - C.(x\;x-x.) + g.(e.) + f: (31)

Clearly, (x.,e\), i-1, 2, ..., n form a Nash equilibrium when division
managers are rewarded according to (30). At equilibrium, the division
manager's utility is:

R1(a1(xJ,e1)) - g1(e1) + Y~ , (32)

so that the division managers' utilities have all increased by a
strictly positive amount. At the (x\,e\) equilibrium, the center's
profits are:

I [n.(x.,e.)-R,(n.(x ,e )-C (x ; x-x ))] - C(x) =
i = l

e.
I [ni(xi,ei)-R.(ni(x*,e*))-g.(e.)+gi(e*) - 3- -C.Cx. ;x-x±)} . (33)

i=l

Without the cost allocation, the center's profits are:

-* .^M (34)

I [ni(xi,ei)-Ci(xi;x*-xi)-Ri(ni(x.,e.))]

Therefore, the benefit of the cost allocation can be calculated as the
difference between the r.h.s. of (33), and (34):

-25-

n

T {[II. (x. ,e.)-C.(x.;x-x.)-g.(e.)]
*■, "■ 1 I 1 11 l 11

i = l

- [n.(x*,e*)-C.(x*;x*-x*)-g.(e*)]} - J ^- = V ^ > 0. (35)
11111 ill .,2 .,2

1=1 1=1

0

Thus, the center and all the divisions are strictly better off

with a full allocation of costs and the reward functions R. , i = l, 2,

l

..., n. Although the center does not know the values of K. a priori,

the proof given above demonstrates that there exist reward functions

that are Pareto superior to reward functions based on no allocation of

costs. We have not shown that the R. are optimal reward functions,

l

only that they dominate reward functions that are optimal in the class
R*.

IV. A DOMINANT STRATEGY COST ALLOCATION MECHANISM

In this section, we give an explicit formulation of compensation
functions R.,R„,...,R that are based upon a cost allocation (although
not on a full cost allocation) and that result in an outcome that

Pareto dominates the outcome achieved with the R 's. Furthermore,

l

with the cost allocation that we present, each division's manager
can explicitly calculate his or her best input and effort levels
without knowledge of the current behavior of the other divisions'
managers.

We define new compensation functions based on an allocation of
costs by:

-26-

R.[tt.,x.] = R.[tt.(x. ,e.)-C.(x.;x*-x*)] + R. [ tt. (x* ,e*) ]
111 111111 1 1111

- R.U„(x*,e*) - C.(x*;x*-x*)]. (35)

1111 11 l

Let K. and (x.,e.) be defined as follows:
l 11

- * * it it - it it

K. = R.[ir.(x. ,e.)] - R. [ tt. (x. ,e . )-C. (x. ; x*-x. ) ] (36)

l 1111 1111 11 l

and

(x.,e.) e argmax{R. [tt.(x. ,e.)-C.(x. ;x*-x*)] - g.(e.) + K. }. (37)

11 111111 l 11 l

The following theorem shows that for R. 's in R* , the associated
to l

R.'s will result in higher total profits for the firm, while causing
no loss in utility for the divisions' managers. Note that with the
R. compensation functions, the manager of division i is "charged" a
portion of total input costs that would have been incurred if the
other divisions continued to demand their no cost allocation level of
inputs. The K. term, which can be explicitly computed by the Center,
is included to ensure that the manager of division i suffers no loss
in utility in the switch from no-cost allocation to cost allocation.

THEOREM: The compensation functions R. ,R_,...,R lead to an outcome
that Pareto dominates the outcome achieved with the compensation func-
tions R, ,R ,...,R , when the R fs are concave, increasing functions of
1 2 n i

divisional profits alone and are less than their respective arguments.

-27-

Proof : For the outcome achieved with the R.'s to be Pareto superior

the outcome achieved with the R.'s it is sufficient that:

1

n

2 {ir.(xi,ei)-Ri[TT1(x1,e.)-C.(xi;x*-x*)]-Ki} - C(x) >
i=l

n

I (TT.Cx^.ep-R.U.Cx^e*)]} - C(x*), (38)

i=l

and

R.[ir.(x.,e.)-C.(x.;x*-x*)] + K. - g.(e.)
iiiiii i i ii —

R [it (x* e*)] - g.(e*) for all i=l,2,...,n. (39)

From the definition of K. , the left-hand-side of (39) equals the

l

right-hand-side of (39) when (x.,e.) = (x*,e*). Since (x.,e.) is

l ' l ii l l

chosen to solve:

MAX R. [tt.(x. ,e.)-C.(x. ;x*-x*)] + K. - g.(e.),
(x.,e.) iiiiii i ill

l l

the inequality in (39) is immediate.

Recall that ir.(x.,e.) = f.(x.) + h.(e ). The first-order condi-
lii ii ii

tions for the above problem are:

R'.[f.(x.)+h.(e.)-C.(x.;x*-x*)][f!(x.)-c!(x".;x*-x*)] = 0 (40)

1111111 l 1111 i

and

R![f.(x.)+h.(e\)-C.(x.;x*-x*)]h'(e ) - g!(e.) = 0. (41)

ill ii ii iii ii

Since R* > 0, (40) yields:

-28-

f!(x.) - c!(x.;x*-x*) = 0. (42)

i i l i l

Thus x. is selected independently from e.. Since f.(x*) = 0 and
l l 11

f.(») - C.(») is quasi-concave, (42) implies that x. > x.. From con-
11 11

cavity of R ( •) , we have R non-increasing in its argument; this along
i i

with x* > x. gives us:
l l

R![f .(x.)+h.(e.)-C.(x.;x*-x*)] > r! [ f . (x . )+h. (e . ) ] >
l l 11111 l — l l l l l —

R'.[f.(x*)+h.(e.)] for all e. > 0. (43)

11111 l

When e. = e. , we have:
l l

R![f.(x.)+h.(e.)-C.(x.;x*-x*)] > r! [ f . (x*)+h. (e. ) ] . (44)

ill 11 11 l—iii 11

Since h' > 0, (44) and (41) imply that

0 = Rj[fi(xi)+hi(ei)-Ci(xi;x*-x*)]h^(ei) - g!(e.) >

R^[fjL(x*)+hi(ei)]h](e1) - gUe.). (45)

From the definition of (x*,e*) we know that:

R^[f.(x*)+h.(e*)]h|(e*) - g^(e*) = 0. (46)

From R. concave, h. concave and g. convex, the right-hand-side of (46)

is decreasing in e.. Thus, (45) and (46) together imply that e. < e..
& l l—i

By quasi-concavity in x. of f.(«) - C. ( •) , and (38) we have x as

ill l

the unique maxiraizer of f.(0 - C.(«). Therefore,

f.(x.) - C^x.jx^x*) > f.(x*) - C.(x*;x*-xJ). (47)

-29-

JL ""

Since e. < e. , and h.(») is increasing, we have
1 — 1 ' l °

f .(x.) + h.(e.) - C.(x.;x*-x*)
11 11 11 l

> f.(x*) + h.(e*) - C.(x*;x*-x*). (48)

11 11 11 l

Rewriting (48) results in:

TT.(x.,e.) - C.(x.;x*-x*) > Tr.(x*,e*) - C.(x*;x*-x*) (49)
ill 11 i ill ii i

Using R.(y) < y for all y > 0 and (49), we get:

R.[{TTi(x.,e.)-C.(x.;x*-x*)}-{TT.(x*,e:)-C.(x*;x*-x*)}]

< {1r.(xi,e~i)-C.(x.;x*-x*)}-{1Ti(x*,e*)-C;L(x*;x*-x*)}] (50)

Concavity of R.(«) and R. > 0 together imply that:

R.[TTi(x.,e.)-Ci(x.;x*-x*)]-R.[7T.(x*,e*)-C;L(xJ;x*-x*)]

< R.[{TTi(xi,e"i)-C.(xi;x*-x*)}-{TT.(x*,e*)-Ci(x*;x*-x*)}].

(51)

Combining (50) and (51) gives:

R.[Tr.(x.,e.)-C.(x.;x*-x*)]-R.[Tr.(x*,e*)-C.(x*;x*-x*)]
l l I' i i i' i 1111 ii l

< {TT.(xi,ei)-C.(x.;x*-x*)}-{TTi(x*,e*)-C.(x*;x*-x*)}.

(52)

-30-

Rearranging terms in (52) yields:

TT.(x.,e )-C (x ;x*-x*)-R [tt.(x ,e.)-C (x ;x*-x*)]
liiii iiiiiii i

+ R.[Tr.(x*,e*)-C (x*;x*-x*)] > tt. (x*,e*)-C. (x*; x*-x*) .
1111 ii i ill li i

(53)

Subtracting R. [ it. (x* ,e*) ] from both sides of (53) and using the
definition of K. gives us:

tt.(x. ,e.)-C.(x.;x*-x*)-R.[Tr.(x. ,e". )-C. (x. ; x*-x*) ] - K.
ill ii l 1111 ii l l

> 7r.(x*,e*)-C.(x*;x*-x*)-R.[TT.(x*,e*)]. (54)

ill ii l 1111

n
Since (54) holds for all i = l,2,...,n and since C( •) = £ C.(0, we

i=l *
get:

n

I {tt.(x. ,e.)-R.[TT.(x.,e.)-C.(x.;x*-x*)] - K. }
._. 111111111 l l

n

- I C.(x.;x*-x*) >
.,ii i

1=1

n

I {^(x^ep-R^Cx* e*)]} - C(x*). (55)

i = l

As x* > x for all i = l,2,...,n and C. is non-decreasing in x-x. , we

have

n n

C(x) = I C.(x.;x-x.) < Y C.(x.;x*-x*). (56)

..ii i — . L, i l i

i = l i = l

-31-

Using (55) and (56) together we get:

n _

I {Tr.(x.,e.)-R.[7T.(x.,e.)-C.(x.;x*-x*)] -K.}-C(x)

*■« i i i liii li i i

1=1

> I (T:.(x*,e*)-R.[Tr.(x*,e*)]} - C(x*), (57)

. L, l li l l l l

which is just (9), what we set out to prove.

n

Note that the cost allocated to division i is based on the amount
of input that division i actually demands and the total amount of input
that the other divisions would have demanded without allocated costs.
This is necessary in order for the division managers to be able to
explicitly compute their input demands and effort levels. If we view
the input demands of the divisions when costs are not allocated as
"historical" in some sense, then division i is being allocated costs
based on its current input demand and the historical demand of the
other divisions.

If we compute the total of all the allocated costs, and compare
that to actual total cost, we see that with this mechanism more than
total costs are allocated to the divisions. Specifically, since
x. > x. and C.(») is nondecreasing in x-x., it is straightforward
to demonstrate that

n _ n

C(x) = T C.(x.;x-x.) < V C.(x.;x*-x*)
.,ii l — . , 11 l
1 = 1 1 = 1

n

< I C.(x7;x*-x:) = C(x*). (58)

i=l X 1 X

-32-

V. Summary

Suppose a principal (the center) compensates agents (division
managers) according to reward functions that are based on divisional
profits gross of any allocation of costs; and suppose that these
reward functions are concave, increasing, and less than their respec-
tive arguments. We have demonstrated that for all such reward func-
tions that the center could select, it is al-ways in the interests of
the center and the divisions to renegotiate a new set of compensation
functions based on a full allocation of costs. Thus, we demonstrated
the existence of conditions under which full cost allocations dominate
no allocations.

Our analysis may be viewed as a formal demonstration of Zimmerman's
[1979] assertion that cost allocations can dominate no cost allocation.
Our formalization explicitly considered the welfare of division
managers as well as the welfare of the principal, and explicitly
modelled both divergence of preferences and asymmetric information. We
allowed the cost function to have several jumps so that an increase in
output may impose additional (capacity) "fixed" costs as well as
variable costs. Hence, the implications of allocating total costs
differ from those of allocating only variable costs. Our results are
robust in the sense that most common full cost allocation methods can
lead to Pareto improvements in welfare, in a wide range of environments,

In proving the existence of full cost allocations that dominate no
cost allocation, we relied on the concept of a Nash equilibrium. How-
ever, the issue of the actual computation of the Nash equilibrium
input and effort levels by the division managers was not resolved.

-33-

Cost allocation mechanisms that resolve this computation problem were
given. With these mechanisms, the ith division manager's compensation
is a function of the ith division's current input demand and the
historical demand of the other n-1 divisions. These mechanisms result
in a dominant strategy equilibrium, thereby solving the computation
problems, but the allocation of costs is only approximate. This sug-
gests that in order to resolve the computational problems associated
with the full cost allocation model, one must be willing to sacrifice
the notion of a tidy allocation of costs in the current period based
only on the actions (demands) in the current period.

The intuition underlying our results is as f ollows : at the no-
cost allocation equilibrium the manager could trade a small amount of
additional effort for a small amount of transferred product and leave
local profits unchanged, costs lower, and firm profits gross of reward
greater. However, since effort is costly to the manager and the trans-
ferred product is not, the manager has no incentive to do so. From
the firm's point of view, both the transferred product and effort are
costly (the latter because the firm must pay the manager for effort in
order to meet the manager's reservation wage). When the division is
charged for the transferred product, the manager's relative prices for
effort and transferred product change, and the manager increases
effort and reduces consumption of the transferred product, giving the
firm larger profits gross of rewards but net of costs. For this cost
allocation scheme to Pareto dominate the no-cost allocation scheme,
the decrease in costs must be greater than the increased reward paid
to the manager to compensate for the increase in effort. Since R. is

-34-

concave, the firm saves more in costs of producing the transferred
product than the manager incurs as a result of increased effort.
Thus, a transfer from the center to the manager exists that makes both
parties better off.

Finally, we again point out that we have not shown that cost allo-
cations are an optimal method of coordinating decentralized units of
an organization. Certainly for cost allocations to be optimal it is
necessary that they dominate no-cost allocation. We have seen that
this necessary condition is met.

-35-

Appendix
We prove that in the no cost allocation case, the center chooses
reward functions that are concave, increasing functions of local prof-
its alone, and such that R.(lL) < II. for all It. > 0. We demonstrate

111 1

this by proving four separate assertions.

ASSERTION 1: Let R.(n.,x.) be the reward function for division i.

2
Without loss of generality, we can restrict R.'s domain to R such

that R. is increasing in its first argument.

Proof:

2
By contradiction. Assume there exists (a,b) e R such that R. is non-
increasing in its first argument at (a,b).

CASE 1

There exists no (H.,g ) e M such that R (a,b) = R (II. (x ,e ) ,x ) for

li l l liiii

some (x.,e.). Then (a,b) is not part of the relevant domain of R. and
may be excluded without loss of generality.

CASE 2

There exists (n.,g.) z M. such that for some (x.,e.), R.(a.b) =
11 l ill

R. ( II. (x. ,e. ) ,x. ) . We claim that (x.,e.) will never be chosen by divi-
sion manager i. Using the fact that g. is strictly increasing in e. ,
II. is strictly increasing in e. , and R.(a,b) is non-increasing in its
first argument at (a,b), we have that there exists some e. < e. such
that:

-36-

R.(a,b) - g^e.) = R1(ni(x.,e.),x.) - g.(e.)

< R.(iri(x.,ei),7.) - g^e.).

Therefore, since there exists (x.,e.) that makes division manager i

11 °

strictly better off than at (x.,e.), the latter will never be chosen
by the division manager. We may therefore, without loss of generality,
exclude (a,b) from the relevant domain of R..

D

For an accepted contract, the following self-selection constraint
must be met:

* *
(x.,e.) e Argmax R. (n. (x. ,e. ) ,x. ) - g.(e.)
11 11111 11

(x.,e.) e T. (Al)

where T. = {(x. ,e. ) |0 < x. and 0 < e. < 7. }.

l li1 — l —l—i

The existence of a solution to the problem in (Al) leads to our second
assertion.

ASSERTION 2: Let R.(n,,x.) be the reward function for division
l 11

manager i. Then R is concave in its argument.

Proof:

A necessary condition for a solution to (Al) to exist for a particular

(II ,g ) e M is quasi-concavity of the raaximand in (x ,e ). A neces-
i i l i l

sary condition for quasi-concavity of the maxiraand in (Al) for all

(JI.,g.) e M. is concavity of R.. Therefore, a necessary condition for

14
a solution to (Al) to exist for all (n.,g ) e M is concavity of R. .

ill i

-37-

The third assertion depends directly on our definitions of no-cost
allocation, and "reasonable" reward functions.

ASSERTION 3: Let R.(lT.,x.) be the reward function for division i.
1 11

Then for the no cost allocation benchmark, R. "depends only on II..

l i

Proof:

The self-selection constraint can be rewritten, at (x.,e.), as

3R. an. 3R.

1 * + T± = 0.

an. ax. ax

ii i

and

3R. an.

TiT IT ~ gi = °'
i i

9Ri

By definition of no-cost allocation, we know that — — is non-negative,

3x .

l

3R. . .

i * a

Assume that -r — is strictly positive. Then (x.,e.) will be chosen
3x. J v i» x

x

such that

3n. 3R. 3R.
i / i

3x. 3x. an. *
i ii

3R. 3R. an.

From assertion 1, -rr- > 0; from assumption, — — > 0. Thus, — — < 0

Oil . dx . 3x .

* * 1 1 i

at (x.,e.), for all II.. From our definition of admissable reward
ii l

9lIi , * *N

functions, we excluded all R. where - — < 0 at (x.,e.), for all n. .

~„ 1 oX . 11 1

3R. i

This gives us — — = 0, or R. depending only on n. .
3x . i i

□

-38-

The center will choose reward functions R. that depend only on

1

IT. and that are strictly increasing in II.. Our fourth assertion is
1 l

that the division's reward will be positive only if it produces posi-
tive profits, and that the division's reward will always be less than
its profits.

ASSERTION 4: Let R.(n.) be the reward function for division i. Then,
for all (n.,gi) e Mj.,

(i) R.(a) > 0 only if a > 0; and
(ii) R.(a) < a for all a > 0.

Proof:

* *
(i) Let (x.,e.) denote the output level and effort level chosen by

division manager i, if the contract is accepted. A necessary con-
dition for the contract (employment under the compensation plan) to be

* *

accepted is R. ( II. (x. ,e. ) ) > w. , where w. > 0 is the division manager's
K l l l ' l — l ' l b

reservation utility level. Since a division manager knows n.(x.,e.)

i ii

prior to accepting the contract, and reward functions are chosen for

all (H.,g.) e M, , the contract must have the property that

* * * *

R. ( II.(x. ,e. )) > 0. If it were the case that R. ( II. (x. ,e . ) ) > 0 and
i I I i 1111

H.(x.,e ) < 0, then the center would not want the division manager to

accept the contract. Thus, the center will restrict its attention to

* * * *

reward functions such that R. ( II. (x. ,e . ) ) > 0 if n.(x.,e.) > 0, and

1111 ill

* * * *

R. ( II. (x. ,e. )) = 0 if n.(x.,e.) < 0. Again, as reward functions are
1111 ill —

chosen for all admissible profit and disutility of effort functions,
it follows that the center can restrict attention to reward functions
such that R.(y) > 0 only if y > 0, and R.(y) = 0 if y <_ 0. Since

-39-

II,(x.,e.) > 0 only if x. > 0, it must be the case that for an accepted
iiii

* 15

contract, x. > 0. For an accepted contract, the self-selection

constraint (Al) can be written as

* *

(x.,e.) e Argmax R. ( f . (x. )+h. (e. ) ) - g.(e )
11 11111 11

(x. ,e.) e T.
11 l

(ii) Suppose that for some a > 0, R.(a) > a, and that there exists

* *

(II., g.) e M. such that JI.(x.,e.) = a. From the definition of admis-
li l ill

sable reward functions, such an R. is not admissable.

l

□

-40-
Footnotes

See Harris, Kriebel and Raviv [1982] for more on this point.

2
We denote the first and second derivatives by single and double

primes, respectively.

3
Further restrictions are placed on the cost functions in Section

III.

4
See Zimmerman [1979, p. 515] for more on this point.

This is similar in spirit, but not the same as the minimum regret
strategy defined in Luce and Raiffa [1957].

f.
Note also that the properties of the reward functions do not

exclude a simple sharing of a division's gross profits.

Clearly, the game we model is one in which the managers accept
the contract myopically; that is, they (the managers) assume that the
contract is not going to be changed in the future. Therefore, those
potential managers who have not joined the firm have no interest in
any change in the reward scheme.

g
It should be noted that the Nash equilibrium is ex post rational,

in the same sense as the center's strategy in choosing the no-cost
allocation reward functions.

9
Groves and Ledyard [1985] provide a cogent defense of the use of

Nash equilibria as the solution concept in the type of situation we
have modelled:

. . .we turn to the Nash equilibrium of the complete
information game. We do not suggest that each agent
knows all of [the environment] when they compute
[their strategies], just as in real markets no auc-
tioneer knows the excess demand function when equil-
ibrium prices are calculated. We do suggest, how-
ever, that the Complete Information Nash game-
theoretic equilibrium [strategies] may be the pos-
sible "equilibrium" of [some] iterative process,
i.e., the stationary messages, just as the demand-
equal-supply price is thought of as the "equil-
ibrium" of some unspecified market dynamic process.

[p. 30]

10

For our proof to go through, we do not need Ci to be quasi-
convex in x-^ , but the somewhat weaker condition that f^ - C^ be
strictly quasi-concave in x^ .

-41-

It should be noted that this condition is analagous to Zimmerman ' s
conditions on page 518 for his Case 1. He points out that if marginal
cost is greater than average cost, allocating overhead dominates no
overhead allocation.

12

When Ci = (xj/x)C, we are using a transfer pricing scheme where

the price is average cost. We will thus demonstrate that there exist

conditions under which average cost transfer pricing dominates no cost

allocation.

13

All of our results on dominance in full cost allocations can be

obtained for compensation functions in the larger class of increasing,

quasi-concave functions whose first derivatives are strictly less than

one. However, optimal reward functions for the no allocation problem

must meet the additional restriction that they are less than their

arguments.

14

Since R^ is concave and defined over all of its domain, it is

necessarily continuous. In addition, there exists only a finite num-
ber of points where Rj is not dif ferentiable [see Rockafellar, p. 83
and p. 246]. Therefore, without loss of generality we can let the
partial derivatives be from either the right or left, as long as we
are consistent. In that case, the first and second order necessary
conditions have their usual interpretation.

We can further restrict the set of reward functions to be ones
that would be accepted for some w^ > 0. Thus, we need only concern
ourselves with the properties of a reward function R$_(y) over positive
values; i.e., where y > 0.

-42-

Ref erences

Baiman, S. , "Discussants Comments on 'The Concept of Fairness in the
Choice of Joint Cost Allocation Methods'," Joint Cost Allocation,
ed. by S. Moriarity (Center for Economic and Management Research,
University of Oklahoma, 1981), pp. 106-109.

Baiman, S. and J. Noel, "A Role for the" Allocation of Fixed Costs
Within an Agency Model," unpublished working paper, Carnegie-
Mellon University (1983).

Biddle, G. and R. Steinberg, "Allocation of Joint and Common Costs,"
Journal of Accounting Literature (Spring, 1984), pp. 1-45.

Demski, J., "Cost Allocation Games," Joint Cost Allocations, ed. by
S. Moriarity (Center for Economic and Management Research,
University of Oklahoma, 1981), pp. 142-173.

Groves, T. and J. Ledyard, "Incentive Compatibility Ten Years Later,"

Discussion Paper No. 648, Center for Mathematical Studies in

Economics and Management Science, Northwestern University (April,
1985).

Groves, T. and M. Loeb, "Incentives in a Divisionalized Firm,"
Management Science (March, 1979), pp. 221-230.

Harris, M. , "Discussion of 'Models in Managerial Accounting'," Journal
of Accounting Research (Supplement, 1982), pp. 149-152.

Harris, M. , C. Kriebel and A. Raviv, "Asymmetric Information, Incen-
tives and Intrafirm Resource Allocation," Management Science
(June, 1982), pp. 604-620.

Holmstrora, B. , "Moral Hazard in Teams," Bell Journal of Economics
(Autumn, 1982), pp. 324-340.

Luce, R. D. and H. Raiffa, Games and Decisions, Wiley, New York (1957).

Rockafellar, R. T. , Convex Analysis, Princeton University Press,
Princeton, New Jersey (1970).

Zimmerman, J., "The Costs and Benefits of Cost Allocations," The
Accounting Review (July, 1979), pp. 504-521.

D/143

IECKMAN

INDERY INC.

JUN95

«, N MANCHESTER,1
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