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,

1 Address: Instituto Superior de Contabilidade e Administração de Lisboa (ISCAL/IPL), Av. Miguel

Bombarda 20, 1069-035 Lisbon, Portugal. E-mail: omgomes@iscal.ipl.pt

Dynamic budget setting

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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 decision-

making 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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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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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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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.

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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

Dynamic budget setting

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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

Dynamic budget setting

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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

Dynamic budget setting

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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 0.842 0.107 0.278 =0.21 0.871 0.095 0.232

=0.12 0.867 0.118 0.243 A=1.5 0.904 0.081 0.185 B=12 0.869 0.112 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.

Dynamic budget setting

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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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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 0.766 0.134 0.2 0.684 0.116 0.3 0.601 0.099 0.4 0.518 0.082 0.5 0.435 0.065 0.6 0.350 0.050 0.7 0.265 0.035 0.8 0.179 0.021 0.9 0.091 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.

Figure 1 – Budget shares.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

1-bj*-bi*

bi*

bj*

Dynamic budget setting

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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 decision-

making 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:

Dynamic budget setting

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(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 steady-

state values, we obtain,

(a4)

(a5)

Dynamic budget setting

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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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