Homothetic Single-Aggregator in Demand Analysis

• Homothetic Single-Aggregator (HSA) is a framework that represents heterogeneous homothetic preferences with a single scalar aggregator to simplify demand and production analysis.
• Its dual aggregation geometry uses weighted geometric means of individual expenditure indices, allowing nonparametric recovery of demand structures and variable markups.
• HSA nests the CES specification as a special case, enabling counterfactual welfare analysis and empirical validation under monopolistic competition using firm-level revenue data.

Searching arXiv for the cited HSA-related papers and closely related work. Homothetic single-aggregator (HSA) denotes a homothetic demand and aggregation structure in which either a vector of varieties enters utility through a single scalar aggregator or, equivalently in dual form, a population of homothetic consumers behaves as a single representative consumer whose log-expenditure function is a weighted average of individual log-expenditure functions. In monopolistic-competition models, this restriction is economically substantive because it preserves homotheticity while allowing demand systems that are more general than CES; in recent identification results, it also makes the representative consumer’s utility nonparametrically identifiable from firm-level revenue data without observing output quantities (Chow et al., 2 Mar 2026, Sandomirskiy et al., 2024).

1. Formal definition

In the demand-system formulation used in production-and-demand identification, the representative consumer has homothetic utility

$U(q_1,\ldots,q_n)=W(A(q_1,\ldots,q_n)),$

where $A(\cdot)$ is a “single-aggregator” mapping the vector of individual varieties $q_i$ to a scalar composite, and $W(\cdot)$ is increasing. Homotheticity means $W(\cdot)$ is increasing and the indirect utility is a function of total expenditure divided by $A(\cdot)$. A leading example is the generalized CES aggregator

$$A(q_1,\ldots,q_n)=\left(\int_0^1 q_i^\rho\, di\right)^{1/\rho},$$

with $\rho \neq 0$; as $\rho \to 0$, this converges to the log-additive “translog” case, while constant $\rho$ yields standard CES (Chow et al., 2 Mar 2026).

In the dual aggregation framework, homothetic preferences are represented through expenditure functions. For a homothetic consumer with utility $A(\cdot)$0, the unit-cost expenditure takes the form

$A(\cdot)$1

where $A(\cdot)$2 is the unit-utility price index. Writing $A(\cdot)$3, the log-expenditure transform is

$A(\cdot)$4

This representation is concave in prices and affine in $A(\cdot)$5, and it is the basic object through which HSA becomes a weighted-average operation in $A(\cdot)$6-space (Sandomirskiy et al., 2024).

2. Dual aggregation geometry

For a population of $A(\cdot)$7 homothetic consumers indexed by $A(\cdot)$8, with individual price indices $A(\cdot)$9, incomes $q_i$0, total income $q_i$1, and expenditure weights $q_i$2, the aggregate log-expenditure aggregator is

$q_i$3

Exponentiating yields

$q_i$4

Thus the aggregate expenditure index is the geometric mean of individual expenditure indices, weighted by income shares (Sandomirskiy et al., 2024).

This immediately implies representative-consumer aggregation. By Shephard’s lemma and homogeneity, each individual’s Marshallian demand is

$q_i$5

and aggregate demand is

$q_i$6

A single consumer with expenditure function $q_i$7 and budget $q_i$8 therefore chooses exactly the aggregate bundle: $q_i$9 The same duality admits a direct-utility representation through an Eisenberg-Gale-type weighted Nash product,

$W(\cdot)$0

which dualizes back to $W(\cdot)$1 (Sandomirskiy et al., 2024).

3. Nonparametric identification from revenue data

In the monopolistic-competition setting of Chow, Kasahara, and Sugita, firm $W(\cdot)$2 at time $W(\cdot)$3 produces $W(\cdot)$4, faces inverse demand

$W(\cdot)$5

and is observed through revenue and inputs $W(\cdot)$6, with a transitory demand shock $W(\cdot)$7 and persistent productivity $W(\cdot)$8. The paper establishes nonparametric identification of production functions, TFP, price markups, firms’ output prices and quantities, and consumer demand from firm-level revenue data, without observing output quantities, under monopolistic competition with a fully nonparametric demand system. Under the additional restriction that demand satisfies HSA, it further nonparametrically identifies the representative consumer’s utility function from firm-level revenue data. The paper states that this overturns the widely held view—formalized by Bond, Hashemi, Kaplan, and Zoch (2021)—that output elasticities and markups are not nonparametrically identifiable from revenue data without quantity information (Chow et al., 2 Mar 2026).

The identification argument proceeds in stages. First, a reduced-form revenue function

$W(\cdot)$9

is recovered together with $W(\cdot)$0 using IV quantile regression. Under limited persistence of $W(\cdot)$1,

$W(\cdot)$2

and an IVQR procedure, such as smoothed GMM QR with lagged inputs as instruments, nonparametrically identifies $W(\cdot)$3; inversion then yields $W(\cdot)$4. Second, the material first-order condition implies

$W(\cdot)$5

strictly increasing in $W(\cdot)$6, so one inverts to obtain $W(\cdot)$7. Under an AR(1), or more general, law of motion for productivity, this yields a transformation model

$W(\cdot)$8

from which $W(\cdot)$9 and $A(\cdot)$0 are recovered up to scale and location. Third, differentiating

$A(\cdot)$1

with respect to each input and using the material FOC identifies the markup

$A(\cdot)$2

and the partial derivatives of the production function. Integrating the gradient of $A(\cdot)$3 along any path in input space under connected support recovers the full production function up to the standard location and scale normalization; the same procedure recovers the price aggregator and inverse demand (Chow et al., 2 Mar 2026).

4. Relation to CES and empirical testing

HSA strictly contains CES in the framework under discussion. CES corresponds to the special case in which the structural budget-share function is linear in $A(\cdot)$4: $A(\cdot)$5 where $A(\cdot)$6 and $A(\cdot)$7 is an additive shock. Under CES,

$A(\cdot)$8

so the revenue function is affine in $A(\cdot)$9. This characterization makes CES a restriction on the HSA class rather than its synonym (Chow et al., 2 Mar 2026).

The empirical testing strategy exploits this nesting. In the fourth step of the estimation procedure, a parametric CoPaTh-HSA form, described as constant pass-through and broader than CES, is imposed: $$A(q_1,\ldots,q_n)=\left(\int_0^1 q_i^\rho\, di\right)^{1/\rho},$$0 which nests generalized CES as $$A(q_1,\ldots,q_n)=\left(\int_0^1 q_i^\rho\, di\right)^{1/\rho},$$1. Using the estimated $$A(q_1,\ldots,q_n)=\left(\int_0^1 q_i^\rho\, di\right)^{1/\rho},$$2, $$A(q_1,\ldots,q_n)=\left(\int_0^1 q_i^\rho\, di\right)^{1/\rho},$$3, and markup $$A(q_1,\ldots,q_n)=\left(\int_0^1 q_i^\rho\, di\right)^{1/\rho},$$4 from the preceding steps, the parameters $$A(q_1,\ldots,q_n)=\left(\int_0^1 q_i^\rho\, di\right)^{1/\rho},$$5 are fit by NLS together with market-share constraints. A test of CES is therefore the null hypothesis

$$A(q_1,\ldots,q_n)=\left(\int_0^1 q_i^\rho\, di\right)^{1/\rho},$$6

In the Chilean manufacturing application, $$A(q_1,\ldots,q_n)=\left(\int_0^1 q_i^\rho\, di\right)^{1/\rho},$$7, so CES is rejected in favor of HSA (Chow et al., 2 Mar 2026).

A common misconception is that HSA is merely a relabeling of CES. The identification and testing results reject that equivalence: CES is a special case inside a broader homothetic single-aggregator class that admits variable markups, incomplete pass-through, and non-monotonic markup-size relationships (Chow et al., 2 Mar 2026).

5. Counterfactual welfare analysis

Under HSA, the representative consumer’s utility is sufficiently structured to support welfare counterfactuals without parametric assumptions on preferences. The monopolistic-competition equilibrium (MCE) is characterized firm by firm by

$$A(q_1,\ldots,q_n)=\left(\int_0^1 q_i^\rho\, di\right)^{1/\rho},$$8

together with the adding-up of shares, which solves for $$A(q_1,\ldots,q_n)=\left(\int_0^1 q_i^\rho\, di\right)^{1/\rho},$$9. The counterfactual marginal-cost-pricing equilibrium (MCPE) instead imposes

$\rho \neq 0$0

again with shares summing to one, which solves for $\rho \neq 0$1 (Chow et al., 2 Mar 2026).

Consumer welfare can then be written as

$\rho \neq 0$2

Compensating variation is defined by finding $\rho \neq 0$3 such that

$\rho \neq 0$4

and then

$\rho \neq 0$5

Under fixed aggregate revenue and Assumption A-short, firms’ profit change is

$\rho \neq 0$6

so net welfare gain is

$\rho \neq 0$7

These formulas operationalize welfare analysis directly from the recovered production, demand, and utility objects (Chow et al., 2 Mar 2026).

Empirically, the same framework is used with Chilean manufacturing data. The paper reports that market power reduces welfare by approximately $\rho \neq 0$8–$\rho \neq 0$9 of industry revenue in the three largest manufacturing industries in 1996. It also reports Monte Carlo evidence that the semiparametric estimator performs well, whereas treating revenue as output induces substantial bias (Chow et al., 2 Mar 2026).

6. Invariant domains and canonical examples

A central geometric property of HSA is invariance. A domain $\rho \to 0$0 of homothetic preferences is HSA-invariant if, whenever each $\rho \to 0$1, the single-aggregator preference $\rho \to 0$2 also belongs to $\rho \to 0$3. In the dual space, this is equivalent to convexity of the associated set of logarithmic expenditure functions: $\rho \to 0$4 The smallest closed HSA-invariant domain containing a given $\rho \to 0$5 is the closed convex hull of these $\rho \to 0$6 functions. Indecomposability is the complementary notion: a preference is indecomposable if it cannot arise as a non-trivial HSA of other members of the domain, equivalently if its $\rho \to 0$7 is an extreme point of the convex set $\rho \to 0$8 (Sandomirskiy et al., 2024).

Several standard preference classes illustrate the construction. For Cobb-Douglas preferences,

$\rho \to 0$9

the expenditure index satisfies

$\rho$0

and HSA yields

$\rho$1

so the Cobb-Douglas class is closed under HSA. For Leontief preferences,

$\rho$2

and HSA produces

$\rho$3

a homothetic form with log-mixtures of linear price indices. For linear preferences, HSA generates a “min-mixture,” and for CES preferences it mixes the $\rho$4 terms linearly inside a log before exponentiating (Sandomirskiy et al., 2024).

These examples clarify why HSA is not restricted to a single closed functional family. Some domains, such as Cobb-Douglas, are invariant under aggregation; others remain homothetic after aggregation but leave the original parametric class. This suggests that HSA is best understood as an aggregation principle in dual space rather than as one fixed direct-utility formula (Sandomirskiy et al., 2024).

7. Scope, limitations, and extensions

The principal strengths attributed to HSA in the revenue-identification framework are that it permits fully nonparametric production-demand identification from revenue only, jointly recovers the production function, markups, and representative-consumer utility, nests CES while allowing variable markups and incomplete pass-through, and enables counterfactual welfare analysis without restrictive demand functional forms (Chow et al., 2 Mar 2026). In the aggregation-theoretic framework, HSA also underlies robust welfare analysis, discrete choice and additive random-utility connections, pseudo-market mechanisms such as Fisher-CEEI, and preference identification through convex-decomposition arguments (Sandomirskiy et al., 2024).

The limitations are equally explicit. The production-and-demand application assumes monopolistic competition; strategic oligopoly would require a different identification strategy. It requires Hicks-neutral TFP as a scalar $\rho$5; factor-augmenting shocks need richer structure. It relies on homotheticity, so non-homothetic preferences lie outside HSA, and identification remains up to scale and location absent normalizations or external price or quantity indices (Chow et al., 2 Mar 2026).

The extensions proposed in the same line of work remain within this logic. They include allowing factor-augmenting technological change through a CES-like aggregator, modeling oligopoly with a game-theoretic supply side and inverse-demand system, permitting non-homothetic utility by stacking several aggregators such as nested HSA, endogenizing firm entry, exit, and dynamic investment, and introducing firm-specific input prices through discrete instruments (Chow et al., 2 Mar 2026). In the aggregation geometry literature, an additional implication is that when the convex hull of $\rho$6 is a simplex, each aggregate $\rho$7 has a unique convex decomposition, which allows recovery of the weights $\rho$8 from a single-price observation of aggregate demand (Sandomirskiy et al., 2024).

Taken together, these results position HSA as both a structural demand restriction and a geometric aggregation device. In one role it supports nonparametric identification of production, demand, markups, and welfare from revenue data under monopolistic competition; in the other it characterizes when heterogeneous homothetic consumers can be represented exactly by a single homothetic aggregator in logarithmic expenditure space (Chow et al., 2 Mar 2026, Sandomirskiy et al., 2024).