Expenditure Density: Theory & Applications

• Expenditure density is a multi-context measure that allocates expenditure mass across preference distributions, municipal scales, or decile-based data.
• It underpins analytical frameworks in consumer theory, public procurement, and econophysics by linking aggregation, scale effects, and statistical reconstructions.
• This concept aids in deriving welfare bounds, adjusting per-capita benchmarks, and achieving closed-form aggregations in nonhomothetic models.

Searching arXiv for the cited topic and papers. arXiv search query: "Expenditure Density arXiv homothetic expenditure function procurement expenditure density" “Expenditure density” is not a single standardized term across the arXiv literature. It appears in at least three distinct technical senses: as a probability measure over logarithmic expenditure functions in consumer aggregation theory; as a size-dependent or size-adjusted measure of public spending intensity in scaling analyses of procurement; and as a fitted distributional object, often a survival function or a probability density reconstructed from grouped expenditure data, in econophysics and statistical-distribution studies. These usages differ in ontology and method, but each formalizes how expenditure mass is distributed across agents, municipalities, transaction sizes, or expenditure levels (Sandomirskiy et al., 2024).

1. Terminological scope and major meanings

In the most formal microeconomic usage, the paper “The geometry of consumer preference aggregation” identifies the object closest to an expenditure density as a Borel probability measure $\mu$ over individual preferences, or over their parameterizations, which induces continuous aggregation in the space of logarithmic expenditure functions. For a finite population, the role of a density is played by income shares $\beta_k = b_k/B$, and aggregation is

$\ln E^{agg} = \sum_k \beta_k \ln E_k.$

For a continuum, the corresponding expression is

$$\ln E(\mathbf{p}) = \int \ln E_\theta(\mathbf{p})\, d\mu(\theta).$$

In this framework, the measure $\mu$ is the mathematically precise object that governs how log-expenditures are averaged across the population (Sandomirskiy et al., 2024).

In urban-scaling and public-finance work, “expenditure density” refers instead to expenditure intensity relative to population size. “Scaling Behavior of Public Procurement Activity” models municipal procurement expenditure as

$Y = a P^\beta,$

with per-capita expenditure density

$y_{pc} \equiv \frac{Y}{P} = a P^{\beta-1}.$

When $\beta < 1$, per-capita expenditure density decreases with population, which the paper interprets as economies of scale (Curado et al., 2020).

In econophysics, the phrase is used in a more distributional sense. “A statistical physics analysis of expenditure in the UK” defines expenditure density in the standard probabilistic sense as the probability density function

$f(x) = \frac{dF(x)}{dx} = -\frac{dS(x)}{dx},$

derived from cumulative or survival functions fitted to decile expenditure data (Oltean et al., 2014). By contrast, “An Econophysical dynamical approach of expenditure and income distribution in the UK” uses “expenditure density” for a fitted cumulative probability curve over ordered intertemporal expenditure differences, specifically a polynomial survival function on decile-ranked cumulative changes rather than a normalized density in the strict sense (Oltean et al., 2014).

These usages are conceptually related but non-equivalent. A plausible implication is that “expenditure density” functions as a family resemblance term: it always denotes a structure that allocates expenditure mass across some domain, but the domain may be preference space, municipal populations, transaction values, or household expenditure levels.

2. Expenditure density in aggregation theory

The aggregation framework in “The geometry of consumer preference aggregation” assumes each consumer has a convex, monotone, continuous, homothetic preference represented by a non-decreasing, continuous, concave, $1$-homogeneous utility $\beta_k = b_k/B$0. The unit-expenditure function is

$\beta_k = b_k/B$1

and the indirect utility satisfies

$\beta_k = b_k/B$2

Marshallian demand is

$\beta_k = b_k/B$3

for almost all $\beta_k = b_k/B$4, and expenditure shares are

$\beta_k = b_k/B$5

These identities make the logarithmic expenditure function the natural object for aggregation (Sandomirskiy et al., 2024).

For a finite population with budgets $\beta_k = b_k/B$6, expenditure functions $\beta_k = b_k/B$7, total budget $\beta_k = b_k/B$8, and income shares $\beta_k = b_k/B$9, aggregate preference is characterized by

$\ln E^{agg} = \sum_k \beta_k \ln E_k.$0

together with the gradient identity

$\ln E^{agg} = \sum_k \beta_k \ln E_k.$1

Hence aggregate Marshallian demand equals the sum of individual Marshallian demands. The paper also gives an Eisenberg–Gale dual characterization:

$\ln E^{agg} = \sum_k \beta_k \ln E_k.$2

In this finite case, the “density” is atomic: $\ln E^{agg} = \sum_k \beta_k \ln E_k.$3 (Sandomirskiy et al., 2024).

For continuous populations, the aggregation formula becomes

$\ln E^{agg} = \sum_k \beta_k \ln E_k.$4

where $\ln E^{agg} = \sum_k \beta_k \ln E_k.$5 is a Borel probability measure on types or preferences. This measure pushes forward onto the space of logarithmic expenditure functions and determines both aggregate expenditure and aggregate demand via $\ln E^{agg} = \sum_k \beta_k \ln E_k.$6. The paper further shows that for closed invariant domains every preference admits a Choquet representation over indecomposable preferences,

$\ln E^{agg} = \sum_k \beta_k \ln E_k.$7

with $\ln E^{agg} = \sum_k \beta_k \ln E_k.$8 supported on extreme points. In this setting, the canonical expenditure density is the Choquet measure supported on indecomposables (Sandomirskiy et al., 2024).

The same paper characterizes aggregation-invariant domains by convexity of the set of logarithmic expenditure functions: a domain $\ln E^{agg} = \sum_k \beta_k \ln E_k.$9 is invariant if and only if $$\ln E(\mathbf{p}) = \int \ln E_\theta(\mathbf{p})\, d\mu(\theta).$$0 is convex. Examples include the Cobb–Douglas domain, domains defined by monotonicity of expenditure shares, and the full homothetic domain. This is central because expenditure density is not merely descriptive here; it is the primitive object that generates aggregate preferences, welfare bounds, and identification results (Sandomirskiy et al., 2024).

3. Distributional and decile-based expenditure densities

In statistical-distribution studies, expenditure density is constructed from grouped household data, usually deciles. “A statistical physics analysis of expenditure in the UK” starts from household expenditure $$\ln E(\mathbf{p}) = \int \ln E_\theta(\mathbf{p})\, d\mu(\theta).$$1, defines the cumulative distribution $$\ln E(\mathbf{p}) = \int \ln E_\theta(\mathbf{p})\, d\mu(\theta).$$2 and survival function $$\ln E(\mathbf{p}) = \int \ln E_\theta(\mathbf{p})\, d\mu(\theta).$$3, and identifies the expenditure density as

$$\ln E(\mathbf{p}) = \int \ln E_\theta(\mathbf{p})\, d\mu(\theta).$$4

Using UK Office for National Statistics decile data, the paper constructs empirical survival points from either mean expenditure by decile or lower limit on expenditure. For mean expenditure, the empirical points are

$$\ln E(\mathbf{p}) = \int \ln E_\theta(\mathbf{p})\, d\mu(\theta).$$5

while for lower-limit expenditure they are

$$\ln E(\mathbf{p}) = \int \ln E_\theta(\mathbf{p})\, d\mu(\theta).$$6

These are then fitted by parametric models (Oltean et al., 2014).

The same paper uses two main models. The Fermi–Dirac specification, displayed on logarithmic axes, is written as

$$\ln E(\mathbf{p}) = \int \ln E_\theta(\mathbf{p})\, d\mu(\theta).$$7

equivalently

$$\ln E(\mathbf{p}) = \int \ln E_\theta(\mathbf{p})\, d\mu(\theta).$$8

This yields a logistic-type density

$$\ln E(\mathbf{p}) = \int \ln E_\theta(\mathbf{p})\, d\mu(\theta).$$9

The paper reports high goodness of fit, with $\mu$0 around $\mu$1–$\mu$2 across mean disposable, mean gross, lower-limit disposable, lower-limit gross, and a clothing-and-footwear subset (Oltean et al., 2014).

The alternative model is a first-degree polynomial survival fit on raw axes,

$\mu$3

For the UK data, these linear fits are weaker than the Fermi–Dirac fits but remain robust, with $\mu$4 generally around $\mu$5–$\mu$6 depending on the dataset and construction. The paper emphasizes the novelty of the “lower limit on expenditure” construction, which anchors the survival function at $\mu$7 for $\mu$8 and tests robustness to alternative decile assignments (Oltean et al., 2014).

A related but conceptually distinct construction appears in “An Econophysical dynamical approach of expenditure and income distribution in the UK.” There, the object of analysis is not the level distribution in one year but the distribution of changes in expenditure across deciles between two years. For decile differences $\mu$9, the ordered differences $Y = a P^\beta,$0 are cumulatively summed:

$Y = a P^\beta,$1

The y-axis is a decile-based cumulative population probability. For mean expenditure, the points are

$Y = a P^\beta,$2

and for lower-limit expenditure they are

$Y = a P^\beta,$3

The fitted object is

$Y = a P^\beta,$4

usually specialized to

$Y = a P^\beta,$5

The paper explicitly notes that this is a fitted cumulative probability curve, best interpreted as a survival function, not a normalized probability density function (Oltean et al., 2014).

Concrete reported examples include mean gross expenditure difference for 2010/2009 with $Y = a P^\beta,$6, $Y = a P^\beta,$7, $Y = a P^\beta,$8, and lower limit on disposable expenditure difference for 2011/2010 with $Y = a P^\beta,$9, $y_{pc} \equiv \frac{Y}{P} = a P^{\beta-1}.$0, $y_{pc} \equiv \frac{Y}{P} = a P^{\beta-1}.$1. Most values for the coefficient of determination are above $y_{pc} \equiv \frac{Y}{P} = a P^{\beta-1}.$2, while higher-degree polynomials and longer intervals are said to increase goodness of fit (Oltean et al., 2014).

4. Scale-dependent expenditure density in public spending and transparency data

“Scaling Behavior of Public Procurement Activity” uses expenditure density in a population-scaling sense. Public procurement expenditure $y_{pc} \equiv \frac{Y}{P} = a P^{\beta-1}.$3 is the monetary value spent by municipal governments through contracts to purchase Works, Goods, and Services. The empirical relation with population $y_{pc} \equiv \frac{Y}{P} = a P^{\beta-1}.$4 is modeled as

$y_{pc} \equiv \frac{Y}{P} = a P^{\beta-1}.$5

The paper defines per-capita expenditure density as

$y_{pc} \equiv \frac{Y}{P} = a P^{\beta-1}.$6

If $y_{pc} \equiv \frac{Y}{P} = a P^{\beta-1}.$7, then $y_{pc} \equiv \frac{Y}{P} = a P^{\beta-1}.$8 declines with $y_{pc} \equiv \frac{Y}{P} = a P^{\beta-1}.$9, which the paper interprets as economies of scale in public spending (Curado et al., 2020).

The study analyzes 278 mainland Portuguese municipalities over 2011–2018 using a curated municipal dataset of 310,819 contracts totaling €16.9 billion, derived from an initial raw dataset of 930,513 contracts. Total municipal procurement expenditure scales sub-linearly with population throughout 2011–2018. Works and Goods are sub-linear; Services are near-linear for larger municipalities but sub-linear when all municipalities are included. For Works, the paper reports a transition in $\beta < 1$0 from approximately $\beta < 1$1 before 2014/16 to approximately $\beta < 1$2 thereafter. For Services, $\beta < 1$3 for larger municipalities; for Goods, the scaling is clearly sub-linear across the full sample (Curado et al., 2020).

The same paper argues that per-capita comparisons are biased whenever $\beta < 1$4. It therefore introduces the Scale-Adjusted Indicator,

$\beta < 1$5

A positive SAI indicates spending above the amount predicted by population; a negative SAI indicates spending below the predicted level. The empirical distribution of SAIs is approximately Normal across years and types, with one exception, and the paper states that SAIs are uncorrelated with population size and show no heteroscedasticity (Curado et al., 2020).

The thresholded transparency study “Nonlinear Rank Scaling and Hidden Structure in NHS Expenditure Transparency Data” uses a different spending-intensity notion. There, expenditure density refers to how NHS spending is distributed and concentrated across suppliers, expense types and areas, and transaction sizes, with particular emphasis on the distortions induced by the £25,000 transparency threshold. The paper models rank-ordered totals using piecewise power laws of the form

$\beta < 1$6

or piecewise

$\beta < 1$7

This operationalizes density as concentration in top ranks versus dispersion in the tail (Mohammed et al., 24 Jun 2025).

The study covers NHS England and 37 of 42 Integrated Care Boards over approximately two years, using 993 monthly files and 1,956,196 entries. For outgoing expenditure, there are 1,805,513 transactions totaling £302.5 billion. The paper reports extreme concentration by amount: among 17,062 suppliers, a changepoint appears near rank $\beta < 1$8, and the top 221 suppliers, representing 1.3%, receive £237,795,574,537, or 81.6% of spending. By transaction count, a different changepoint appears near rank $\beta < 1$9; the top regime generates 89.7% of transactions (Mohammed et al., 24 Jun 2025).

The £25,000 threshold removes much of the low-value mass from view. Of all records, 665,231, or 34.0%, are above threshold and 1,290,965, or 66.0%, are below. For outgoing expenditure alone, 634,536 transactions, or 35.2%, are above threshold and 1,170,977, or 64.8%, are below. Below-threshold transactions are dominated by Delegated GP services: 996,249 transactions, or 77.2% of the sub-£25k set. The paper therefore argues that the threshold truncates the long tail, artificially increases apparent concentration, and induces features such as a histogram valley at £25k (Mohammed et al., 24 Jun 2025).

5. Expenditure density in nonhomothetic CES aggregation

“Aggregation and Closed-Form Results for Nonhomothetic CES Preferences” uses the term in yet another precise way: as the probability density of household total expenditures in the cross section. Utility is defined implicitly over a continuum of goods by

$f(x) = \frac{dF(x)}{dx} = -\frac{dS(x)}{dx},$0

with elasticity of substitution $f(x) = \frac{dF(x)}{dx} = -\frac{dS(x)}{dx},$1. Under the paper’s assumptions, Hicksian demand is

$f(x) = \frac{dF(x)}{dx} = -\frac{dS(x)}{dx},$2

and the expenditure function is

$f(x) = \frac{dF(x)}{dx} = -\frac{dS(x)}{dx},$3

The paper then derives a closed-form mapping between utility and expenditure under assumptions on the joint distribution of price and taste parameters and on the Gamma distribution of $f(x) = \frac{dF(x)}{dx} = -\frac{dS(x)}{dx},$4 (Bohr et al., 2023).

In this framework, the expenditure density is explicitly denoted $f(x) = \frac{dF(x)}{dx} = -\frac{dS(x)}{dx},$5 and interpreted as the cross-sectional density of household total expenditures. The paper assumes an Amoroso distribution,

$f(x) = \frac{dF(x)}{dx} = -\frac{dS(x)}{dx},$6

with the economically relevant case $f(x) = \frac{dF(x)}{dx} = -\frac{dS(x)}{dx},$7, support $f(x) = \frac{dF(x)}{dx} = -\frac{dS(x)}{dx},$8, and normalization $f(x) = \frac{dF(x)}{dx} = -\frac{dS(x)}{dx},$9. Assumption 3 links its shape parameter to preference heterogeneity:

$1$0

The paper notes that the Amoroso family nests or limits to Gamma, Weibull, Fréchet, and exponential distributions as special cases, and Lognormal and Pareto as limiting cases (Bohr et al., 2023).

The analytical payoff is closed-form aggregation. Individual expenditure shares at household expenditure $1$1 are

$1$2

Aggregating over the expenditure density yields

$1$3

The tractability depends on the choice of $1$4 and on the link $1$5, which turns the relevant Laplace transform into a rational expression with gamma-function coefficients (Bohr et al., 2023).

This usage differs sharply from the homothetic aggregation paper. In the homothetic case, the density is over preferences or log-expenditure functions; here, it is over household total expenditures. A plausible implication is that the two papers address complementary aggregation margins: one along heterogeneity in tastes, the other along heterogeneity in total expenditure.

6. Methodological contrasts, misconceptions, and related structures

A common misconception is to treat all uses of “expenditure density” as if they referred to an ordinary probability density over expenditure levels. This is incorrect. In the aggregation geometry paper, the object is a probability measure over preferences or over logarithmic expenditure functions, not over scalar expenditure levels (Sandomirskiy et al., 2024). In the procurement-scaling paper, the term refers to per-capita expenditure intensity and scale-adjusted residuals rather than to a normalized density function (Curado et al., 2020). In the NHS transparency paper, it denotes concentration across ranks, categories, and transaction-size ranges, not a univariate expenditure pdf (Mohammed et al., 24 Jun 2025).

A second misconception is to identify all decile-based cumulative constructions with proper distribution functions. The dynamic UK econophysics paper explicitly models a survival curve over ordered cumulative expenditure differences between years. It does not fit a normalized probability density function, and its linear polynomial can formally exceed the $1$6 range outside the observed support. The paper identifies this as a limitation of extrapolation rather than a probabilistically exact density model (Oltean et al., 2014).

A third misconception is to assume that per-capita expenditure is always an unbiased comparison metric. The procurement paper states the opposite: per-capita comparisons assume $1$7, and when $1$8 they are biased by population. SAIs are introduced precisely to remove this size effect (Curado et al., 2020).

Across these literatures, several related structures recur. Survival functions are central in the UK decile studies [(Oltean et al., 2014); (Oltean et al., 2014)]. Convex geometry and Choquet measures are central in homothetic preference aggregation (Sandomirskiy et al., 2024). Rank-size and segmented power laws are central in the NHS transparency analysis (Mohammed et al., 24 Jun 2025). Parametric cross-sectional densities such as the Amoroso family are central in nonhomothetic CES aggregation (Bohr et al., 2023). This suggests that “expenditure density” is best understood not as one formula but as a context-dependent representation of where expenditure mass is located and how it aggregates.

7. Applications and analytical significance

In consumer theory, expenditure density enables aggregation, identification, and welfare analysis. In “The geometry of consumer preference aggregation,” unknown population heterogeneity is summarized by a measure $1$9, and robust welfare bounds are obtained by optimizing welfare over all $\beta_k = b_k/B$00 that generate a given aggregate $\beta_k = b_k/B$01. The paper states that equivalent variation is convex in $\beta_k = b_k/B$02 and compensating variation is concave, implying that the representative consumer gives the most pessimistic equivalent variation and the most optimistic compensating variation. It also links welfare uncertainty for small price changes to the variance of demand under the expenditure density (Sandomirskiy et al., 2024).

In nonhomothetic CES models, the expenditure density $\beta_k = b_k/B$03 is the key ingredient that makes aggregation analytically feasible. It also enters the Euler equation implications of the model and shapes aggregate sectoral shares through cross-sectional inequality in expenditures. The paper emphasizes that changes in the distribution of $\beta_k = b_k/B$04 alter aggregate demand weights and ideal price indices even at fixed mean expenditures (Bohr et al., 2023).

In public procurement, expenditure density supports fair cross-municipality comparisons and policy diagnostics. The Portuguese procurement paper uses SAIs and clustering based on SAIs by contract type and year to derive local characterizations of municipalities. It reports four clusters, with distinctive profiles in Works, Goods, and Services and links to regional, political, debt, and EU-funds patterns. The central analytical point is that scale-aware expenditure density measures outperform naive per-capita benchmarking when $\beta_k = b_k/B$05 (Curado et al., 2020).

In transparency analysis, expenditure density exposes hidden concentration and threshold-induced opacity. The NHS study shows that top suppliers receive a dominant share of spend while most transactions lie below the reporting threshold. The paper argues that lowering or eliminating thresholds and standardizing schemas would restore the dense field of small suppliers and small payments, improving oversight and benchmark validity (Mohammed et al., 24 Jun 2025).

In econophysics, expenditure density is used to characterize expenditure distributions and distributional dynamics from grouped data. The UK static paper finds that both Fermi–Dirac and polynomial distributions are robust in describing expenditure distributions, while the dynamic paper argues that polynomial survival functions over decile expenditure changes capture macroeconomic stress, including crisis-period anomalies such as slope sign reversals and lower $\beta_k = b_k/B$06 [(Oltean et al., 2014); (Oltean et al., 2014)].

Taken together, the literature shows that expenditure density is a structurally important object wherever expenditure must be aggregated, normalized, compared across scale, or reconstructed from grouped or truncated data. The specific formalization depends on whether the underlying problem concerns preference aggregation, cross-sectional heterogeneity, public-spending scaling, or rank-based concentration.