An Application of Economic Modeling Methods to Management
Accounting: Dynamic Budget Setting
Orlando Gomes1
Instituto Superior de Contabilidade e Administração de Lisboa (ISCAL/IPL)
&
Business Research Unit (UNIDE/ISCTE-IUL)
- July, 2013 –
Abstract
Economics and managerial sciences heavily resort to analytical models in order to
interpret the reality. These models are intended to be stylized representations of
observable phenomena; they should be as simple and flexible as possible, but at the
same time sufficiently comprehensive and rigorous to offer relevant insights on the
issues under debate. Most of the models developed in the mentioned disciplines share
some transversal features: they consider a representative agent who has decisions to
make; the decision-maker is typically a rational agent, that desires to plan the future
and that adopts an optimal behavior. The benchmark model that is built over these
fundamental guidelines may be adapted to many areas of knowledge. In this
communication, such generic analytical structure is characterized and an example of
its adaptation to the field of management accounting is offered; in particular, we
sketch a model relating budget setting within a given organization.
Areas: A7) Teaching and research in Accounting; A5) Management Accounting.
European Accounting Association Code: M1
1. Introduction
Economic thought has progressed a lot in the last few decades. Sophisticated
statistical tools allow today for a better understanding of empirical evidence,
Address: Instituto Superior de Contabilidade e Administração de Lisboa (ISCAL/IPL), Av. Miguel
Bombarda 20, 1069-035 Lisbon, Portugal. E-mail: [email protected]
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Dynamic budget setting
powerful logical arguments have been built to provide solid explanations for many
observable phenomena and models were constructed to help us think about the
world that surround us.
Concerning theory, Economics relies at the present on a series of benchmark
models, in turn of which many important issues have been and continue to be
subject to careful analysis and discussion. Although these models are of various
natures, and not always a framework on one field can just be adapted to approach
other kinds of problems, there is a common trait on much of the built theoretical
apparatus.
The most prominent features of economic models are as follows:
(1) there is an implicit notion of rationality; Economics have to do with decisionmaking and this suggests that the individuals have the ability to choose. Choices
respect to the capacity to weight benefits and costs and a rational decision is the
one for which the selected option is the alternative involving higher expected net
benefits;
(2) rational agents take optimal decisions. This observation is vital to understand
that most of the Economic problems one might conceive have to do with
maximization or minimization of some objective function;
(3) individuals plan for the future. If agents rationally optimize their behavior, it
will not be reasonable to think that they take static decisions. They establish plans
for the future and optimize intertemporally, meaning that, in principle, all
economic problems have a dynamic nature. Furthermore, not only problems are
intertemporal, they are also forward-looking; the past just determines the current
state and it is the expected future behavior that effectively matters.
According to the previous arguments, any model constructed to explain economic
phenomena is necessarily a dynamic model, where rational agents search for the
solution that best serves their pre-specified goals. Therefore, it should not be a
surprise that most of the techniques developed to explain economic phenomena
have acquired the form of dynamic optimization tools. Some influential works that
served as the basis for today’s economic research are structured in turn of the idea
of dynamic optimization; these works include Stokey and Lucas (1989), Woodford
(2003), Barro and Sala-i-Martin (2004) or Ljungqvist and Sargent (2004).
Dynamic optimization is pervasive in economic thought. We can find it in economic
growth theory (e.g., Romer (1986), Lucas (1988)), in the study of business cycles
(e.g., Kydland and Prescott (1982), Long and Plosser (1983)), in the search and
matching approaches to unemployment (e.g., Pissarides (1979), Mortensen
(1982)) and in monetary policy analysis (e.g., Clarida, Gali and Gertler (1999),
Svensson and Woodford (2003)), just to cite a few examples.
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Dynamic budget setting
Techniques on economic modeling, as characterized above, are straightforward to
adapt and apply to many other fields of study. In this specific case, we will argue
that the setting that puts together rational thinking, optimization and dynamic
analysis is the ideal setup to approach problems in the field of management
accounting. A specific example will be developed; this relates to departmental
budget setting in an intertemporal context.
The communication is structured as follows. Section 2 characterizes the specificity
of management accounting problems. Section 3 presents an analytical model that
exemplifies the adaptation of dynamic optimization tools to the field of
management accounting; a budget setting problem is modeled. In section 4, the
model’s implications are discussed and a brief extension is developed. Section 5
concludes.
2. Analytical modeling in management accounting
Following Demski (2006), we might state that management accounting deals with
formalized measurement and reporting inside a firm. In the own words of this
author (page 365),
‘In broad terms we study such things as (1) organizational arrangements,
including divisionalized structures, alliances and allocation of decision
rights; (2) decision methods and frames; (3) evaluation and
compensation, including costing systems; (4) governance structures; and
(5) the comparative advantage of the accounting system with its
elaborate, nested controls and professional management. Moreover, we
do this in a variety of settings, real and imagined, using a variety of
methods.’
From the above sentence, we can infer that management accounting is related with
the treatment of data and to the organization of information in order to assist
decision-makers within a firm to better optimize procedures and to choose the
best strategic options for the company. Moreover, this should be done by devising
a conceptual structure and a series of models that are supposed to help equipping
professionals in this field with the tools to think about the problems in their sphere
of action.
In the mentioned paper, Joel Demski claims that analytical modeling is of primary
importance to obtain additional insights on management accounting challenges.
This author presents a general structure of analysis, which has many points of
contact with what we have characterized in the introduction as being the typical
modeling setup of economic problems. Given a set of independent variables, some
controllable and some uncontrollable (these may be designated state variables),
the agent will want to maximize the value of a dependent variable; this implies
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Dynamic budget setting
finding optimal values for the independent variables (or optimal trajectories over
time, in the cases when we have a dynamic problem).
In a management accounting context, the mentioned author provides an example:
the dependent variable may be a vector of marginal cost estimates, the control
variable eventually takes the form of a tentative production schedule and the state
variables might be a series of shocks that will hit the decision process. Independent
variables are combined through some kind of function that delivers an output,
which corresponds to the value obtained for the dependent variable; this function
could be, in the advanced example, some kind of ABC procedure.
As described, the problem might be sophisticated in multiple directions. Because
organizations are complex and involve many agents and decisions, there are many
types of control variables, which may be controlled in many different levels of the
organization. Likewise, state variables vary with the problem at hand; variables
that exert influence over a given process will eventually have no meaning for other
problems faced within the organization. Finally, the mechanism through which
variables are connected to each other certainly changes with the specific problem
being dealt with. Associated with this mechanism, we may add to the problem
many other relevant features that turn the modeling structure more elaborated,
sophisticated and complex, namely, strategic behavior, information constraints,
bounded rationality, among other.
As we associate new elements, we gain in comprehensiveness but we certainly lose
in simplicity and, consequently, in the ability to understand phenomena in a
straightforward way. Thus, a careful balance between a framework that is simple
but at the same time comprehensive is required although, we have to recognize, it
is a difficult exercise.
In Brekelmans (2000, page 6), management accounting is defined as a set of
systems which
‘provide information to assist managers in their planning and control
activities and are designed to help decision making within the company.
Management accounting systems are especially useful in hierarchical and
complex company’s where a single manager cannot process all the
relevant information needed to manage the company, or when it is too
costly to obtain and analyze all relevant information. The information
provided by management accounting systems does not have to satisfy any
laws or rules which usually is the case for financial information prepared
for external constituencies, such as investors, creditors and suppliers. Its
only purpose is to be beneficial to the decision making and control by the
managers of the company.’
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Dynamic budget setting
This definition is a good starting point when addressing how management
accounting can be thought at an analytical level. It aims at contributing to decision
making, it gains a decisive role in complex business environments and there is
freedom in the way problems might be approached, because the goal is to assist
managers in planning future activities and not to inform, in a compulsory way, the
stakeholders of the organization about its past results and perspectives about
future activity.
The mentioned author highlights two important areas where a careful and detailed
study is likely to produce meaningful results in the research field that we are
addressing:
1) Production and capacity planning. This theme has to do with the identification
of the exact amount of costs and revenues that each stage of production generates.
An historical perspective may be relevant, however we should be mainly
concerned with expectations about the future, i.e., with how one can estimate costs
in which the activity will incur and revenues that are likely to be generated in the
productive process. Forecasting these values is not easy, given the complexity of
many activities, that involve multi-product settings, where complementarity and
substitutability relations between different products exist; furthermore, costs are
many times hard to measure, for instance the opportunity costs of the sub-optimal
use of the available resources.
2) Budget setting. This is another area that typically one associates with the
domain of management accounting. Budgets are conceived in order to plan the
employment of resources. A budget involves a financial amount that, at the
beginning of a given period, is attributed to some task, project or department in
order to attain a specific objective that is designed with anticipation. Budget
setting may serve other goals besides planning, e.g., control, motivation,
communication or evaluation. From a strict planning point of view, conceiving a
budget in a complex organization is a form of coordination; it has to do with the
best possible allocation of resources one might accomplish in order to minimize
the uncertainty that is inherent when addressing future expected outcomes.
In the specific case of Brekelmans (2000), the issue of setting budget goals is
modeled within a framework where a top manager has to set budget targets for the
subdivisions of the company. In this environment, subordinate managers have the
incentive to cooperate with the manager at the highest level, but such a scenario
might lead to information asymmetries and agency costs. Budgets of subdivisions
must be coordinated in order to avoid a result where small deviations of each
individual budget might compromise the feasibility of the firm’s budget as a whole.
5
Dynamic budget setting
3. A budget setting analytical model
In order to illustrate how the dynamic methods by now widely used in Economics
can be adapted to the structure of management accounting problems, we focus on
budget setting. The model to propose will involve the features mentioned in the
introduction, namely an intertemporal perspective, an optimization mechanism
and a fully rational behavior by the part of the involved agents.
The structure of analysis will be simple, since it only has an illustrative purpose;
over this benchmark model, one can then apply some sophistication in order to get
a deeper understanding of the problem at hand. The framework is essentially
inspired in a search and matching type of model, where we match the will of an
organization’s department in obtaining the resources required to develop the
activities that should potentially be assigned to it and the possibility of the firm as
a whole in transferring the funds that such activities or projects demand.
Start by assuming that the company has a maximum budget that can be allocated
to the activities of one of its departments, which is invariant in time; let this be
represented by B. Next, let bt(0,1) be the share of such potential budget that is
effectively assigned to the department at a given time period t. We assume that
time is discrete and that the decisions within the firm are taken at the current
period t=0, given an undefined future horizon; thus t = 0, 1, 2, …
In this framework, both the top manager (i.e., the organization as a whole) and the
managers responsible for the department have a choice to make. The company
decides about the value of the budget to attribute to the department (with a ceiling
B); the department will have to evaluate how much effort is worth doing in
convincing top management to attribute financial resources to the activities that
take place in that department. Under this perspective, we need to define a new
variable, xt, which refers to the costs that the department incurs in order to get
more funds from the organization. We can think of this variable as the outcome of
the project proposals that the department presents to the firm; the larger the value
of xt, the more resources the department spends in convincing the firm of its
capabilities to generate value and, hopefully, the larger will be the amount of funds
directed to the activity of the department.
The way in which the department can obtain additional funds will be modeled
through a matching function, which will have as inputs the project proposals
variable xt and the funds not yet allocated to the department, i.e., (1- bt)B. Both of
these inputs contribute for a larger output, with this output corresponding to the
funds effectively attributed to the department at a given period.
The main characteristic of the matching function, that we present as f(xt,(1- bt)B),
relates to the complementarity between its arguments: a large effort from the
department in convincing the firm to attribute it additional funds will be useless
6
Dynamic budget setting
whenever the firm has no more funds available; in the same way, no fund transfer
will occur when the firm has yet resources to give to the department, but the
department does not make an effort to earn the transference of these resources.
Basically, this means that f(xt,0)= f(0,(1-bt)B)=0. Moreover, we can consider that
constant returns to scale exist in this matching relation, and therefore we might
define a simple Cobb-Douglas specification for it; specifically, we take
, with A>0 and (0,1). Constant returns to scale
simply mean that the matching result is independent of the scale of the budget
amounts involved in the analysis; obviously, decreasing or increasing returns are
viable assumptions, cases in which matching would fall or would increase,
respectively, with the size of the amounts involved in the relation.
Besides increasing with matching, we assume that the budget of the department
will fall over time at a constant rate (0,1). This systematic cut in the
department’s budget is the way the company has to force the department to
innovate and present new projects; if they are not presented, the budget will
progressively shrink and fall to zero. In order to survive, the department has to
present new projects and hope for a matching with the firm’s will to attribute
funds, in order for this to release the budget resources the department needs to
survive and develop profitable activities.
With the previous arguments in mind, we can present the dynamics of the
department’s budget under the form of the following difference equation,
, b0 given
(1)
According to equation (1), the department’s budget increases with matching and
falls, if nothing else occurs, as the organization attributes a lower budget to the
department at period t+1 than it had transferred to it at period t. The equation is
presented in a recursive form and, hence, the value of the budget at t=0 has to be
established exogenously from the mechanics of the model, i.e., it has to be given.
Next, we have to approach the profitability of the department. The only meaningful
costs that it faces are the ones relating the project proposals, i.e. xt; its revenues
will be an increasing function of the corresponding budget, f(btB), with f’>0. The
larger the budget, the more revenues the department is able to access by pursuing
its activity; we may consider that the mentioned activity is subject to diminishing,
constant or increasing marginal returns relatively to the budget resources; the
following functional form is adopted: f(btB)= (btB), with ,>0.
Having defined revenues and costs, the profit generated by the activities of the
department at each period t will be given by
. However, the
instantaneous profit function will not be the concern of the firm; this intends to
7
Dynamic budget setting
maximize profits under an intertemporal perspective and, thus, will consider in its
optimization problem the flow of profits from the current period until some future
horizon. Taking an infinite horizon (i.e., an activity that has no expected moment to
cease), the objective function for the profits of the department is:
(2)
with (0,1) the discount factor (although the problem is solved by taking a long
horizon, close in time periods have a larger weight in the decisions to be taken
than periods far away in time, i.e., the value of  is necessarily lower than 1).
The problem faced by the department (which has an identical goal to the one of the
firm, that is to obtain the highest possible profit level), is then to maximize (2)
subject to constraint (1).
In this problem, we identify two endogenous variables, which are xt and bt. The two
variables have different natures; the first is a control variable that is dependent
solely on the behavior of the department and the second is, in this context, a state
variable: the budget that the company attributes to the department is not a
decision of the top management or of the managers of the department in isolation;
it is the result of the interaction between the two, given a pre-specified mechanism
through which the budget is transferred for the department’s activity.
In appendix, we show how this problem might be approached resorting to optimal
control techniques. In the next section, we take a numerical example to illustrate
the long-term steady-state outcome.
4. Numerical example and extensions
The appendix solves the dynamic budget setting model and reaches a steady-state
solution that we now characterize through a numerical example. Consider the
following values of parameters: β=0.96; =0.25; α=0.75; =0.75; =0.1; A=1; B=10.
For the above array of parameters, the corresponding equilibrium values, that will
prevail after the adjustment towards the steady-state is concluded, are, in this case,
the following:
and
. The value of
indicates that in this
circumstance, in the equilibrium, the department will optimally access 84.7% of
the budget the firm has available to allocate to this department. We can also
measure, in this circumstance, the profits of the department, which correspond to
. A meaningful result is the ratio between resources
spent to obtain funds and the profits the department is able to capture; this is
. This value means that for each monetary unit of profit generated
8
Dynamic budget setting
by this department, it has spent 0.283 monetary units in resources to acquire
funds to be able to generate such profits.
Changing values of parameters, this equilibrium is disturbed. In table 1, we
maintain the already established parameter values except for the change suggested
in the table. For these individual changes, we present the equilibrium values of the
various variables,
Parameter
change
β=0.94
=0.21
=0.12
A=1.5
B=12
0.842
0.871
0.867
0.904
0.869
0.107
0.095
0.118
0.081
0.112
0.278
0.232
0.243
0.185
0.240
Table 1 – Steady-state for different parameter values
The results in the table are intuitive:
1) A smaller discount factor signifies stronger impatience relatively to future
outcomes, what makes the department achieve a lower amount of funds in
equilibrium. The resources spent to get them will also fall, both in absolute
level and as a percentage of the profits generated in the department;
2) If the firm withdraws a relatively lower amount to the department’s budget
at each period independently of the new projects the department presents
(lower ), the equilibrium budget will rise and, in this example, the
resources used to increase the budget will fall;
3) Concerning the productivity of the department, translated by parameter ,
its increase makes the department’s budget increase although, in this case,
with the cost of being needed additional resources to achieve such outcome;
4) A stronger efficiency on matching the department’s intentions and the
firm’s funding also increases and lowers ;
5) Finally, an increase in the potential budget as a favorable impact on the
department’s budget, although with the negative impact of higher
associated project proposal costs.
We can think of a more sophisticated framework, namely a framework that
considers more than one department. In what follows, we characterize the setup
with two departments. Generalizing this to more than two units within the
company would be straightforward.
9
Dynamic budget setting
With two departments, i, j, one can write two constraints, one for each of the
departments, concerning the time evolution of the assigned budget. These are
similar to (1); from the point of view of the first department,
,
given
(3)
The problem is identical to the previous one, except for the fact that, from the
perspective of the firm, the available budget must be allocated to both
departments. This changes the matching function, but everything else in the
problem remains identical.
There are two ways in which one can address this more sophisticated problem.
First, department decisions may be simultaneous; second, one may approach them
as being sequential, with department i solving the problem after department j. This
second version is simpler to approach because the two problems become
independent; the choice of i takes the previous choice of j as given, and thus its
problem is the same as before, but now having as reference not the whole budget B
but a percentage of this, i.e.,
. We will recover our example to illustrate
this simpler case below; relatively to the first case, we just present some intuitive
remarks.
Under simultaneous decisions, we can no longer solve each department’s problems
separately. In this circumstance, we have to put ourselves in the perspective of the
firm, that wishes to maximize overall profits, i.e.,
(4)
The maximization of (4) is subject to two constraints, which are (3) and a similar
constraint concerning the budget of department j. Now, the problem not only has
two constraints, but it has also the double of the endogenous variables: there are
two budget variables and two project proposal variables, one of each for each
department. This problem is harder to solve, and we do not pursue an analytical
study here. One result is, nevertheless, obvious. If parameters relating the activity
of both departments (namely, , α, ,  and A) are identical between them, then the
results will be symmetric: both departments will access an identical budget,
independently of the overall budget assigned by the firm. The impact of qualitative
differences in parameters is intuitive and it would be easy to confirm if one
approached the concrete dynamics of the setup. Namely, a better matching or a
higher productivity in one department, relatively to the other, would lead to a
budget allocation that would benefit more the first of the assumed departments.
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Dynamic budget setting
The second possibility, sequential decisions, is the case in which the share of the
budget attributed to department i will depend on the budget already assigned to
the other department. Assume the same parameter values as before and that the
firm has already allocated to department j a budget share
. Under our
model’s specification and the assumed parameter values for the activity of
department i, we compute the following equilibrium budget share for this
department:
. In this case, a percentage equal to 0.099 of the firm’s
budget remains unassigned in the equilibrium.
Other examples may be taken. Table 2 presents a wide variety of possibilities, for
different values of the budget of department j.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.766
0.684
0.601
0.518
0.435
0.350
0.265
0.179
0.091
0.134
0.116
0.099
0.082
0.065
0.050
0.035
0.021
0.009
Table 2 – Department i equilibrium budget share, given department j budget share.
Results in table 2 are graphically presented in figure 1, for a better visualization.
1
0,9
0,8
0,7
0,6
1-bj*-bi*
0,5
bi*
0,4
bj*
0,3
0,2
0,1
0
Figure 1 – Budget shares.
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Dynamic budget setting
Fig. 1 indicates that the larger the budget of the other department is, the lower will
necessarily be the budget that department i will be able to optimality capture.
Note, as well, that the unassigned budget shrinks as the larger is the already
allocated budget to department j.
5. Conclusion
Models are simplified representations of the reality. In a famous contribution to
Economics, Robert Solow once wrote
‘All theory depends on assumptions that are not quite true. That is what
makes it theory. The art of successful theorizing is to make the inevitable
simplifying assumptions in such a way that the final results are not very
sensitive. A “crucial” assumption is one on which the conclusions do
depend sensitively, and it is important that crucial assumptions be
reasonably realistic. When the results of a theory seem to flow specifically
from a special crucial assumption, then if the assumption is dubious, the
results are suspect.’ (Solow (1956), page 65).
To build a convincing theory, one needs to simplify the reality, to take a series of
assumptions and one has to be secure that such assumptions are adequate to deal
with the required depth the problem that is being approached. Once this is done,
the constructed model becomes a relevant theoretical structure that helps us think
about the reality.
Economics have progressed a lot as new models destined to explain observable
phenomena have gained life. This is true in almost every area in which decisionmaking is present. In this text, we have applied some basic tools of economic
dynamic analysis to a problem of management accounting, namely a problem of
budget setting. The framework is useful to understand how departments inside an
organization may act in order to guarantee funds from the firm with the goal of
achieving an optimal outcome under an intertemporal perspective.
Appendix – Resolution of the Optimal Control Budget Setting Problem
To maximize profits as presented in equation (2), subject to the dynamic
constraint (1), we need to apply dynamic optimization techniques, which require
writing the current value Hamiltonian function, which takes the form:
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Dynamic budget setting
(a1)
In expression (a1), variable pt+1 corresponds to a shadow-price or co-state
variable, in this case evaluated at period t+1. First-order conditions are as follows,
(a2)
(a3)
The following transversality condition should also be considered,
(a4)
Equations (1) and (a3) constitute a two-dimensional dynamic system, with two
endogenous variables, which are the control and the state variables, i.e., xt and bt.
The co-state variable pt can be suppressed from the system by resorting to relation
(a2). However, it is not possible to analyze the system, in what regards its
transitional dynamics, given that we cannot express it under the form
),
). Nevertheless, assuming that the equilibrium is stable, i.e.,
that convergence towards a long-term steady-state will occur, we can analyze the
scenario in which the system will rest in the long-term equilibrium.
To find steady-state expressions, define values
,
and
. Replacing variables in (1), (a2) and (a3) by the respective steadystate values, we obtain,
(a4)
(a5)
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Dynamic budget setting
Equation (a4) allows to find a unique steady-state value for the steady-state level
of the budget the department will have access to in equilibrium; although, such
value cannot be computed, in its general form, given the shape of the equation. The
equilibrium value of the resources employed by the department in order to get
funds from the firm, x*, is dependent on the value of b*. Equilibrium values can,
thus, be obtained only through numerical examples. Some examples are presented
in the main text.
If more than one department is assumed, the above equilibrium relations are
slightly changed, given the influence that the other department’s (j) budget
allocation has over the budget of department i. From the point of view of
department i, the equilibrium relations come
(a6)
(a7)
Taking expression (a6) for both departments, we find a relation between the two
budget shares, which is
(a8)
Relation (a8) is satisfied for every values of
such that
. Because all
parameter values are identical, departments have access to the exact same share of
the firm’s budget. The result changes whenever different parameter values across
departments are taken. This reasoning applies to simultaneous budget setting and
not to sequential budget allocations (see main text).
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