American Journal of Political Science An Integrated Theory of Budgetary Politics and Some Empirical Tests: The US National Budget, 1791-2010 --Manuscript Draft-- Manuscript Number: AJPS-36433R1 Full Title: An Integrated Theory of Budgetary Politics and Some Empirical Tests: The US National Budget, 1791-2010 Article Type: Article Keywords: expenditures, budgets, exponential growth, path dependency, incrementalism Corresponding Author: Bryan Jones University of Texas at Austin Austin, TX UNITED STATES Corresponding Author Secondary Information: Corresponding Author's Institution: University of Texas at Austin Corresponding Author's Secondary Institution: First Author: Bryan Jones First Author Secondary Information: Order of Authors: Bryan Jones Laszlo Zalanyi, PhD Peter Erdi, PhD Order of Authors Secondary Information: Abstract: We develop a more general theory budgetary politics and examine its implications on a new dataset on US government expenditures from 1791 to 2010. We draw on three major approaches to budgeting: the decision-making theories, primarily incrementalism, and serial processing; the policy process models, basically extensions of punctuated equilibrium; and path dependency. We show that the incrementalist budget model is recursive, and its solution is exponential growth. We assess path dependency by assessing the extent to which the growth curve has a constant exponent and intercept, except when critical junctures, associated with wars or economic collapse, occur. A second type of disruption occurs in the churning that occurs during the equilibrium periods, assessed by examining the non-Gaussian character of the deviations from the growth curve for the equilibrium periods. Empirical tests indicate support for the theory, but with inconsistent findings, particularly adjustments that occur in the absence of critical junctures. Response to Reviewers: We appreciate the opportunity to revise our paper for the American Journal of Political Science. Below we highlight what we have done in response to the editor’s and the reviewers’ comments. Threading through all the reviews, and strongly highlighted by the editor, was the need to make clear the purpose of the manuscript. Reviewer 1’s critique centers almost exclusively on these points, Reviewer 2 similarly notes that “My suggestions for improvements of the paper regard first and foremost the motivation and structure of the paper.” We read Reviewer 3 as not comfortable with using ‘go-to’ arguments without incorporating them into a broader historical narrative. The earlier draft concentrated on moving our understanding of budgeting from an a- historical focus to one incorporating longer run dynamics, using some innovative methods to do so. It relied more on empirical generalizations from strong new analytical techniques and linking to well-established pieces of budgetary theory. Powered by Editorial Manager® and Preprint Manager® from Aries Systems Corporation
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American Journal of Political Science
An Integrated Theory of Budgetary Politics and Some Empirical Tests: The USNational Budget, 1791-2010
--Manuscript Draft--
Manuscript Number: AJPS-36433R1
Full Title: An Integrated Theory of Budgetary Politics and Some Empirical Tests: The USNational Budget, 1791-2010
Corresponding Author: Bryan JonesUniversity of Texas at AustinAustin, TX UNITED STATES
Corresponding Author SecondaryInformation:
Corresponding Author's Institution: University of Texas at Austin
Corresponding Author's SecondaryInstitution:
First Author: Bryan Jones
First Author Secondary Information:
Order of Authors: Bryan Jones
Laszlo Zalanyi, PhD
Peter Erdi, PhD
Order of Authors Secondary Information:
Abstract: We develop a more general theory budgetary politics and examine its implications on anew dataset on US government expenditures from 1791 to 2010. We draw on threemajor approaches to budgeting: the decision-making theories, primarilyincrementalism, and serial processing; the policy process models, basically extensionsof punctuated equilibrium; and path dependency. We show that the incrementalistbudget model is recursive, and its solution is exponential growth. We assess pathdependency by assessing the extent to which the growth curve has a constantexponent and intercept, except when critical junctures, associated with wars oreconomic collapse, occur. A second type of disruption occurs in the churning thatoccurs during the equilibrium periods, assessed by examining the non-Gaussiancharacter of the deviations from the growth curve for the equilibrium periods. Empiricaltests indicate support for the theory, but with inconsistent findings, particularlyadjustments that occur in the absence of critical junctures.
Response to Reviewers: We appreciate the opportunity to revise our paper for the American Journal of PoliticalScience. Below we highlight what we have done in response to the editor’s and thereviewers’ comments.
Threading through all the reviews, and strongly highlighted by the editor, was the needto make clear the purpose of the manuscript. Reviewer 1’s critique centers almostexclusively on these points, Reviewer 2 similarly notes that “My suggestions forimprovements of the paper regard first and foremost the motivation and structure of thepaper.” We read Reviewer 3 as not comfortable with using ‘go-to’ arguments withoutincorporating them into a broader historical narrative.
The earlier draft concentrated on moving our understanding of budgeting from an a-historical focus to one incorporating longer run dynamics, using some innovativemethods to do so. It relied more on empirical generalizations from strong newanalytical techniques and linking to well-established pieces of budgetary theory.
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What we missed in this exercise is how close we were to being able to formulate andtest a general theory of the politics of budgeting that extended existing understandingstemporally (Reviewer 2 indicates the weaknesses of current budgetary studies is thatthey do not “reveal anything about the more precise budget dynamics over time” (thereviewer is referring to a particular paper, but the critique is general). Our contributionin this draft in essence incorporates all three major approaches to the study ofbudgeting: decision-making approaches (in which individual budget actors are thefocus, with the major approach being incrementalism), policy process approaches (inwhich the system of actors are the focus, with punctuated equilibrium being the majortheoretical perspective), and historical institutionalism (with its focus on pathdependency and critical junctures, which has not seriously been used in budgetstudies, but clearly should be). This is what we have done in this version; in essence,we have “gone long”. This has the advantage of offering a top-down approach to whatwe have done, highlighting how closely the overall data series fits the theory of‘disrupted exponential incrementalism”, as we have termed it (suggestions for a betterterm welcome). It also allows for highlighting explicitly where the data do not fit theseries, and there are several places where it does not. This is particularly true for thePost WWII period, as we note in the conclusion.
This approach responds to the first two comments by the editor, which emphasize thatthe paper was unpersuasive in its motivation, referring to comments by both Reviewer1 and Reviewer 2. It also addresses the third comment by the editor, “I would like tosee more than making an empirical generalization”. And it addresses, we think,Reviewer 2’s observation that It would be helpful in further developing this researchagenda if expressions such as ‘the potential of churning within periods of steadyexponential growth’ ‘stable but nevertheless noisy periods’ and permanent ratchets’could be integrated into a more systematic conceptualization of budget changes”.
Specifics
We have addressed the editor’s third comment, based on observations by Reviewer 2)that some technical material could be put into an on-line appendix. That on-lineappendix also includes details on the construction of the dataset.
We have addressed the editor’s fourth comment, and Reviewer 2’s suggestion that weprovide an example of the differences in real dollars the shift from linear to exponentialmakes, as well as the change from an exponent of say .03 to .04 (a real shift from ouranalysis) on page 14. We have tried to address Reviewer 3’s request for a morehistorical type example on p.11.
We’ve added some brief explanatory text below the figures to link the technicaldiagrams directly to the development of the argument in the paper. This is becomingmore common in political science papers, and we think it works especially well in thispaper.
Reviewer 3 is most explicit in tying budget history to the budget trends we detect, andraises some important issues. All of his/her points are reasonable and intriguing, butin the end we ran out of room to address these matters in sufficient detail. However wehave discussed what we see as the most important historical puzzle: the failure ofnegative feedback processes to restore exponential equilibrium in some times forsome policies. In the last paragraph we note that the final period in particular does notfit the exponential equilibrium very well, and speculate a little about why.
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An Integrated Theory of Budgetary Politics and Some Empirical Tests: The US National Budget, 1791-2010
Manuscript ( Not to include ANY author-identifying information)
Abstract
We develop a more general theory budgetary politics and examine its implications
on a new dataset on US government expenditures from 1791 to 2010. We draw on three
major approaches to budgeting: the decision-making theories, primarily incrementalism,
and serial processing; the policy process models, basically extensions of punctuated
equilibrium; and path dependency. We show that the incrementalist budget model is
recursive, and its solution is exponential growth. We assess path dependency by assessing
the extent to which the growth curve has a constant exponent and intercept, except when
critical junctures, associated with wars or economic collapse, occur. A second type of
disruption occurs in the churning that occurs during the equilibrium periods, assessed by
examining the non-Gaussian character of the deviations from the growth curve for the
equilibrium periods. Empirical tests indicate support for the theory, but with inconsistent
findings, particularly adjustments that occur in the absence of critical junctures.
Manuscript word count: 8736.
1
An Integrated Theory of Budgetary Politics and Some Empirical Tests 1
Three-quarters of a century ago, V.O. Key (1940) commented that no budget theory
existed. Key was discussing a normative theory of budget allocation, but he recognized the
limits of any normative theory unsupported by empirical study. Today we have several
budgetary theories, and an increasing number of studies of budgetary allocations across
several countries. Yet it is fair to say that we continue to lack a comprehensive empirically-
based budgetary theory. If we are to achieve that more comprehensive theory, we will
need to be able both to unify the strong theories we have at present, and to extend them
through much longer time periods than scholars have been able to do to date. In this paper,
we develop a more general theory of budgetary allocations, termed disrupted exponential
incrementalism. Then we examine the implications of the theory on a new dataset on US
government expenditures from 1791 to 2010, a synthesis of data from two separate
sources.
To develop a more unified budgetary theory, we draw on three major approaches to
budgeting: the decision-making theories, primarily incrementalism of Wildavsky and his
colleagues, and the serial processing model of Padgett; the policy process models,
extensions of punctuated equilibrium; and path dependency. While the former two
approaches have focused explicitly on budgets, the latter has not, although quite a few
mentions and informal analyses exist in the budget literature.
Decision-Making Theories
1 We appreciate the assistance of (redacted) in the development of the ideas and the analysis of the
data in this paper.
2
Decision-making theories focus on how budget actors decide allocations. Actors
themselves are grouped by institutional role, so the decision-making theories focus both on
institutional interactions and the cognitive capacities of the actors involved. Budget
decisions are complex, and environmental constraints too limited and conflicting to impose
deterministic solutions. Consequently, the decision-making capacities of budget actors are
often critical to the choices made. Because problems are multifaceted and the time
available to devote to the task limited, decision heuristics often strongly affect the patterns
of choices.
Budget decisions, however, are not made by single decision-makers, but rather in a
complex setting of multiple actors across different institutions and agencies (Padgett
1981). In the United States, budgeting requires complex cooperation between the
executive and the legislative branches. Formal rules and procedures govern these
interactions in complex patterns that do not apply to all programs equally. Mandatory
programs—those whose rules of determining payments are set by statute—require
changes in law as well as budgets to change budgetary outcomes. Discretionary programs
can be changed in a budget bill, but even then budget makers can face complex constraints.
If agencies have signed multiyear contracts, those contracts must be factored into budget
changes, which can be particularly problematic in the case of budget cuts. Agency requests
for budget allocations are affected by signals from the bureaucratic hierarchy within which
it is embedded; the Office of Management and Budget; the demands of congressional
oversight and appropriations committees; and the actual allocations received in the
previous year (Padgett 1981; Carpenter 1995).
3
In the early 1960s, Aaron Wildavsky (1964) conducted a systematic study of
budgeting within federal agencies, focusing on the strategies the participants used in the
process. These strategies were for the most part fairly simple, and reduced to adjustments
based on the existing budgetary base. Incrementalist models postulate that reasonably
simple heuristic decision rules govern budgeting, and that these rules empirically can be
reduced to the following maxim: “Grant to an agency some fixed mean percentage of that
agency’s base, plus or minus some stochastic adjustment to take account of special
circumstances” (Davis, Dempster, and Wildavsky 1966: 535).
In the model, there are two types of actors, requesters and appropriators. An
agency’s current year budget request is some percentage of its last year’s appropriation,
plus some adjustment factor. Appropriators grant some percentage of its request, plus or
minus an adjustment factor.
Rn = βBn-1 + ξn , and Bn = γRn + ζn
Where Rn is the request in year n, Bn is the agency’s budgetary allocation in year n, and ξn
and ζn are the random adjustment factors – serially independent, normally distributed.2
As consequence, this year’s appropriation is a percentage of last year’s
appropriation, plus or minus an adjustment factor:
Bn = γ(βBn-1 + ξn ) + ζn
Bn = δBn-1 + ηn ; where δ = γβ and ηn = (γ ξn + ζn) [Equation 1]
2 The above equations are stochastic difference equations. However, to keep our line of argument as
simple as possible, we follow the DDW formulation and avoid the use of the complicated formalism of
stochastic equations.
4
We refer to Equation 1 as the basic incrementalist equation. The above equations
model process incrementalism, which in turn implies outcome incrementalism. The
converse is also true: if we do not observe outcome incrementalism, process
incrementalism cannot be the full story.
DDW tested the basic incrementalist model repeatedly on budget requests and
congressional appropriations for 53 non-defense agencies for 1947-63, using a linear
regression framework. They found excellent fits, but the coefficients for the equations
were not constant. In a second paper Davis, Dempster, and Wildavsky (1974) attempted to
integrate external influences into the linear model, with little success. These studies were
repeated by many other scholars in different settings with similar results (see Padgett
1980: 354 for a summary).
Several scholars critiqued the DDW regression-centered approach as leading to
overestimates of incrementalism (Wanat 1974, Padgett 1980). Padgett (1980) argued for a
stochastic process approach to studying budget behavior, and showed that incrementalism
implied a Gaussian distribution of budget changes for single, homogenous programs, and a
Student’s t distribution for aggregations of programs with heterogeneous parameters.
Padgett performed tests on data from a limited period; Jones, Baumgartner, and True
performed more complete tests on US budget authority after 1947. Their stochastic studies
of Office of Management and Budget subfunctions for the full period indicated that the
incremental model as a general explanation of program-level budget change could not be
sustained (Jones, Baumgartner, and True 1996; True, Jones, and Baumgartner, 1999).
Subsequently, numerous studies in various political settings have confirmed decisively that
5
budget change distributions are not distributed as the incremental theory predicts (Jones,
et al. 2009).
Padgett’s serial processing model (1980, 1981) offered a strong improvement on
the classic incremental models by showing a path by which the external political and policy
forces could be transferred to internal budget dynamics. His model incorporated
sequential incremental adjustments and an external constraint, the fiscal climate. By
serially calculating a comparison between the budget choice for a program and the overall
fiscal constraint, Padgett derived probability distributions consistent with the model.
Policy Process Theories
As Breunig and Koski (2012: 50) note, incrementalism was developed “in a context
in which budgeting decisions are removed from political and policy considerations.”
Indeed, the primary distinction between decision-making and policy process models is that
the later explicitly incorporates these forces. Policy process models incorporate policy and
political considerations, and as a consequence view the political system holistically,
conceiving of inputs (information) flowing into the system, and the system responding to
these flows. But the response is not proportional to the information. Rather resistance, or
friction, in the system blocks action until the political system responds by shifting quickly,
resulting in a pattern of budgetary responses that are not smooth, but rather highly
punctuated. Most of the time program budgets are highly incremental, changing only
marginally, but occasionally they change very rapidly (a good summary is Ryu 2011). The
implications of this approach have been extensively tested using stochastic process
methods (Breunig and Jones 2011). Looking at program-level data, researchers found that
the distribution of budget changes is highly leptokurtic, exactly the indicator of this kind of
6
budget changes (True, Jones, and Baumgartner 1999; Jones, Sulkin and Larsen 2003; Jones
and Baumgartner 2005; Jones et.al. 2009). The general findings of the tests of the policy
process models show that most programs experience only marginal adjustments most of
the time. The large-scale budget changes happen only in rare circumstances. Incremental
budget adjustments are embedded in a broader system of policymaking, which can involve
non-incremental punctuations (Howlett and Migone 2011).
The broader empirical tests of the policy process models, by showing how increases
in institutional friction as a policy moves from proposal to enactment to budgetary
allocation (Jones, Sulkin and Larsen 2003; Jones and Baumgartner 2005), rule out simple
decision rules, including both process incrementalism and serial processing, as
explanations of patterns of budgetary allocations. These rules may explain the choices of
single actors, but cannot be generalized to budgetary systems.
Both the decision-making and the policy process models were developed to explain
changes in budget allocations to programs. Yet a more general theory must also address
more aggregate budgetary allocations across longer periods of time. The dynamics of long-
range budget changes may not be consistent with programmatic dynamics. So we turn to
concepts more normally found in historical institutionalism: path dependency and critical
junctures.
Path Dependency
The notion of path dependency is encompassing to the point of vagueness, as Page
(2006) has lucidly shown. He distinguishes four distinct meanings of the term, one of
which, self-reinforcement, is generally what is mean by budgetary path dependency. In self-
reinforcement, choices put in place mechanisms which themselves operate to sustain the
7
choice (Pierson 2004; Mahoney 2000; Baumgartner and Jones 2009; Howlett and Rayner
2006). Institutions, including those established by enabling statutes for specific policies,
budgetary routines and procedures, and informal norms all operate to reinforce budgetary
dynamics (Wildavsky 1964; Myers 2011; Dufour 2008; Begg 2007). This observation is the
key to measuring the long-term effects of budgetary path dependency. Budgetary
incrementalism is a type of self-reinforcing path dependency. If all agencies are behaving
incrementally, then the full budget of a political system will obviously also be incremental;
indeed, even if the agencies include disjoint behavior, the full budget will be incremental
within limits, as we discuss below.
Path dependency alone cannot account for major disjoint change, so historically-
centered studies of path dependency generally incorporate the concept of critical junctures
Jones, Bryan D., Tracy Sulkin, and Heather Larsen. 2003. Policy Punctuations and
American Political Institutions. American Political Science Review 97: 151-70.
Jones, Bryan D., et al. 2010. A General Empirical Law of Public Budgets. 2009. American
Journal of Political Science 53: 855-73.
Key, V.O. Jr. 1940. The Lack of a Budgetary Theory. American Political Science Review 34:
1137-
Lewis-Beck, Michael and Tom Rice. 1985. Government Growth in the United States.
Journal of Politics
Mahoney, James. 2000. Path Dependency in Historical Sociology. Theory and Society 29:
507-48.
Myers, Roy. 2011. Path Dependence in the Federal Budget Process. Seattle WA: Paper
presented at the American Political Science Association Meetings, September 3.
Niskannen, William. 1971. Bureaucracy and Representative Government. Chicago: Aldine-
Atherton.
Page, Scott. 2006. Path Dependence. Quarterly Journal of Political Science 1: 87-115.
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Padgett, John F. 1980. Bounded Rationality in Budgetary Research. American Political
Science Review 74: 354-72.
Padgett, John F. 1981. Hierarchy and Ecological Control in Federal Budgetary Decision
Making. American Journal of Sociology 87: 75-128.
Peacock, Alan T., and Jack Wiseman. 1961[1994]. The Growth of Public Expenditures in the
United Kingdom. 2nd Ed., reprinted. Aldershot, England: Gregg Revivals.
Pierson, Paul. 2000. Increasing Returns, Path Dependency, and the Study of Politics.
American Political Science Review 94: 251-67.
Pierson, Paul. 2004. Politics in Time. Princeton: Princeton University Press.
Sparrow, Bartholomew. 1996. From the Outside In: World War II and the Development of
the American State. Princeton: Princeton University Press..
True, James L., Bryan D. Jones, and Frank R. Baumgartner. 1999. Punctuated Equilibrium
Theory. In Paul Sabbatier, ed. Theories of the Policy Process. Boulder, Colo:
Westview.
Wanat, John (1974). "Bases of Budgetary Incrementalism." American Political Science
Review 68: 1221 -28.
Whitman, D.A. 1995. The Myth of Democratic Failure. Chicago: University of Chicago Press.
Wildavsky, Aaron. 1964. The Politics of the Budgetary Process. Boston: Little, Brown.
1
Appendix to Accompany
An Integrated Theory of Budgetary Politics and Some Empirical Tests: The US National Budget, 1791-2010
This appendix presents some details about the construction and validation of
the data series used in the analyses, and it addresses several more technical
methodological issues of modeling and testing that involve more extensive technical
elements. It also includes some extended analyses that reinforce the general
conclusions in the paper.
Construction of the Data Series1
As indicated in our paper, there is no single expenditure series for the US
Federal Government since the founding of the Republic. Two separate data series
are available for US Federal Expenditures. These are : Historical Statistics of the
United States: Millennial Edition database TABLE Ea636–643 (compiled by the
Treasury Department); and Office of Management and Budget, Historical Statistics,
Table 3.1(compiled by the Office of Management and Budget). The Treasury Series
runs from 1791 to 1970, and the OMB series covers 1940 to the present. There exist
some differences in the two series in the period of overlap, particularly regarding
the Domestic and Defense categorizations. These differences were particularly
severe during WWII and the Korean War. Figure A.1 displays the divergences.
1 We appreciate the efforts of Frank Baumgartner and John Lovett of the University of North Carolina, who were instrumental in the development of the series.
Manuscript ( Not to include ANY author-identifying information)
2
Figure A1: Differences between the Treasury and OMB Series, Adjusted for Inflation, for the Overlap Period, Defense and Domestic Expenditure
From this data, we constructed two synthetic series by merging data from
the US Treasury with data from OMB. We report two different synthetic series,
because there are considerable differences because OMB adjusts its series for any
changes in categorization rules back to 1940. The series labeled Treasury Synthetic
uses Treasury data from 1791 through 1970, OMB afterward. OMB Synthetic, uses
Treasury numbers until 1940, OMB afterward.
The US used a July 1 – June 30 Fiscal Year from 1789 to 1842; a January 1 –
December 31 Fiscal Year from 1843 to 1976, and an October 1 – September 30
3
Fiscal year since 1977. We adjusted by doubling the reported expenditure in 1943,
and simply neglected the transition quarter reported for 1977 by OMB.
The Consumer Price Index (CPI) is used to adjust for inflation because it is
the only measure available for the full series. Data source was Historical Statistics of
the United States: Millennial Edition database TABLE Cc1-2. We supplemented from
OMB's Historical Statistics, and recalculated to base year 2000 = 100. For 1791-
1970, The Statistical Abstract base year 1982-84 is reported. Adjustments were
made by calculating how it was proportionally different from the 1982-1984 base
(1982-84 =172.2), and calculated a new CPI by multiplying the proportional
difference (0.5872) by the 1982-1984 number. We compared this base with the old
Bureau of Labor Statistics number (base 1967) versus the 1982-1984 base, and got
similar results from 1913-1987.
We conducted analyses with each of the two synthetic series. Separate
analyses on each series indicated some minor differences. While we detected
nothing that would change our general conclusions, we present analyses based on
the OMB Synthetic, as we have more confidence in OMB’s system.
The data are available at Policyagendas.org.
Methodological Issues
In this section, we summarize some modeling and testing issues in more
detail.
Issues in Developing the Models:
1. Incrementalism implies exponential growth:
4
Taking the original incremetalist model (Equation 1) where δ>1 represents
exponential growth and taking the logarithm and expressing the budget with the
starting budget B0: Bn = B0*δn + Σi δi * ηi . The expected value of Bn is B n = B0* δn Þ
lnB n = lnB0 + n*ln(δ). This describes exponential growth with an average slope ln
δ.
2: Some issues with linear estimates of incrementalism.
We have shown that the incremental models were estimated using linear
regression equations, and that the proper approach is geometric growth. The linear
estimating approach actually incorporates assumptions that are incorrect, at least if
followed over a more extended period of time.
One issue is the specification of the error term. The incrementalist models
build in a random error component conceived to be the sum of a series of special
one-time adjustments that behave according to the central limit theorem, and is
assumed to be additive—that is, it is just added to the linear equation. If the budget
is growing by a constant percent, such an error term does not grow in proportion to
the budget. As a consequence, the incremental models imply that the error will
shrink and finally disappear with time—something that obviously can’t happen
given the substantive interpretation. In the original linear model (Equation 1
above), Bn = δBn-1 + ηn implies that Bn/Bn-1 = δ + ηn/Bn-1 . Then ηn/Bn-1 0 as Bn-1
infinity. This is obviously not true.
3. A Note on Exponential Growth
Exponential growth models are an essential component of the study of the
growth of populations. This suggests that students of government size and growth
5
examine some of the basic findings in this field. The ‘pure form’ of exponential
incrementalism presented in this paper would require several assumptions:
assumptions: the system is not destabilized by exogenous events (path dependent,
closed-system incrementalism), and budget growth is not limited by the ‘carrying
capacity’.
The meaning of carrying capacity in population dynamics is clear, but not so
clear in government budgeting. Like a biological population, no budgetary system
can grow exponentially without limits, unless the carrying capacity of the system,
basically the vibrancy of the economic base of government, is growing similarly. In
democracies, the tolerance of the public for taxes or other revenue-raising methods
also factors into the budget system’s carrying capacity.
Statistical Methods
Method 1 and Method 2 Compared:
Method 1 examines the cumulative sum of the budget values—roughly the
numerical integration of the series. The definite integral requires an additive
constant to satisfy the initial conditions of the integration. We estimated this
constant based on the period 1791-1810, and applied it to the full series. Each of
the stable periods had to be estimated separately, so this technique is exploratory
because we could not estimate a full model.
Method 2 examines rates of change instead of budgetary levels. We based
our analysis on the change of the logarithm of the budget for two reasons. One is the
need for a correction of the error term assumption in the incremetalist model. As
we noted above, if Equation 1 were the true description of budgeting process it
6
would yield vanishing relative fluctuations over time. Re-arranging Equation 1,
Bn/Bn-1 = δ + ηn /Bn-1 [Equation 4]
This suggests that with increasing budgets the fluctuations should disappear. Figure 5
shows that as we expect the fluctuations remain.
[Figure 5 about here]
The logarithm of the left side of Equation 3 is the derivative of the log-budget
(with one-year budget sampling)
ln(Bn /Bn-1) = ln(Bn ) - ln(Bn-1)
which should give a flat line graphically over time where exponential incremetalism
holds.
As the fluctuations continue to be significant through time (Figure 5), we propose
the modification of exponential incrementalism:
Bn /Bn-1 = δ + ηn
yielding multiplicative noise in the untransformed time series of the budget:
Bn = (δ+ ηn) * Bn-1. [Equation 5]
Depending on the noise term, Equation 5 describes a discrete geometric Brownian
Motion process. This year's budget is proportional to the previous year’s both in the
adjustment and in the noise term. Here δ>1 represents the exponential growth and the
standard deviation of η in general must be much smaller than 1 to get a realistic budget
function. It follows from Equation 5 that the residuals around Bn has a lognormal
distribution if ln(Bn /Bn-1 ) is normally distributed. We show that this is the case during
the stable periods.
7
The other reason for a logarithmic specification is heuristic: logarithmic change
weights equally a 200% increase and a 50% decrease, which is a more natural
comparison given that percentage changes are unbounded on the upside but constrained
at 100% on the downside.
Algorithm Used to Fit Least Squares Segments in Method 2
Using a systematic process, we fit various least-squares trend line models to
the full derivative of the log budget series. Our algorithm proceeded this way. First
all the possible two-line segments were fitted and the best pair was selected based
on least square deviations from the two line segments. We define ‘best fit” as the fit
that gives the largest decrease in error compared to the previous fit or among the
similar fits. Next all the possible three-line segments were fit was done and the best
fitting was chosen. Similarly for four. But at five segments the combinatorial need
for computational power gets too large for computation. For five segments and
more we assumed that the breakpoints of the previous fittings would remain in
subsequent fittings, and divided the remaining segments each into best-fitting pairs.
Further Examination of Critical Junctures
Figure A.2 presents a further examination of the critical junctures in the US
Budget series by graphing year-to-year changes in the ratio of the defense budget to
the total budget and the derivative of logarithm of the total budget. It can be seen
that when the whole budget changes in a dramatic way, changes in the ratio of the
defense to total expenditures does so as well. The big exception is the period of the
Great Depression, during which domestic expenditures increased while changes in
defense/total spending remained constant.
8
Figure A.2: Year-to-Year Changes in the Ratio of the Defense to the Total Budget and the Log of the Derivative of the Total Budget
9
Reference
Carter, Susan B., Scott Sigmund Gartner, Michael R. Haines, Alan L. Olmstead, Richard Sutch, Gavin Wright, eds. Historical Statistics of the United States, Millennial Edition. Cambridge University Press.