Identification and Estimation of Production Function and Consumer Demand Function under Monopolistic Competition from Revenue Data

Identification and Estimation of Production Function and Consumer Demand Function under Monopolistic Competition from Revenue Data

Introduction

\epigraph{ The derivatives of the function $\phi$ [with respect to inputs] will be random for \dots\ differences in the prices paid or received by various firms if we drop, as we realistically must, the assumption of perfect competition.}{--- Marschak and Andrews (1944, p. 145)}

The estimation of production functions and markups is a central tool in empirical analyses of market outcomes, underpinning research on firm-level productivity (bartelsman2000understanding; syverson2011determines), aggregate productivity and misallocation (olley1996dynamics; hsieh2009misallocation), technological change (van2003productivity; doraszelski2018measuring), and the evolution of market power (hall1988relation; de2012markups; de2020rise).\footnote{griliches_mairesse_1999 and \citet*{ACKERBERG20074171} provide excellent surveys of production function estimation methods.} A common assumption underlying many estimation methods is that firms\textquoteright output quantities are observable. In practice, however, most firm-level datasets contain only revenue, and researchers typically deflate revenue using an industry-level price deflator.\footnote{A few studies employ datasets with firm-level quantity information (e.g., \citealp*{foster2008reallocation};\citealp*{doraszelski2013, doraszelski2018measuring};\citealp*{de2016prices}; lu2015trade; nishioka2019measuring), but such data are typically limited to specific countries, industries, and time periods and remain inaccessible to most researchers.}

Following ma44ecma's pioneering critique,\footnote{Marschak and Andrews were acutely aware that economic data typically come in the form of monetary values (sales, value added) rather than physical counts (tons, bushels). In the introduction to ma44ecma, they explicitly flag the danger of conflating these concepts. In Footnote 3, attached to the definition of net output ($x_0$), they write: “[S]ome changes in notation will be needed later to distinguish between physical output; (gross) revenue; net revenue (`net output'): see \S\S4 and 9.”} a large body of subsequent research has shown that replacing output quantity with revenue can severely bias estimates of production function parameters (e.g., klette1996jae; \citealp*{de2011product}), TFP (e.g., \citealp*{foster2008reallocation}; \citealp*{katayama2009firm}), and markups (\citealp*{bond2021some}). \citet*{bond2021some} formalize this difficulty, arguing that "in the usual setting in which the researcher observes only revenue, and does not have separate information on the price and quantity of output, the output elasticity for a flexible input is not identified non-parametrically from estimation of the revenue production function" (p. 2). Under a nonparametric demand function, the identification challenge has two dimensions: revenue becomes a nonparametric function of inputs, unobserved TFP, and an unobserved demand shock; and flexible inputs may be correlated with both TFP and the demand shock.\footnote{As noted by Klette and Griliches (1996), Marschak and Andrews (1944) were the first to recognize these two identification challenges and to criticize the practice of replacing output quantities with revenue in production function estimation.} Despite these concerns, researchers continue to use revenue in place of quantity due to the scarcity of firm-level quantity data.\footnote{Researchers also rely on revenue when products differ in quality, since physical output alone may not reflect true production, though such practices often lack theoretical foundations.}

In this paper, we establish that, under monopolistic competition, nonparametric identification of the production function, total factor productivity, consumer demand, and counterfactual welfare effects is in fact possible using firm-level revenue data, without observing output quantity. We consider the same demand-side setting as bond2021some, in which an individual firm faces a nonparametric demand function that depends on its output, observable characteristics, and an unobserved demand shock; for identification, we allow this shock to be transitory.\footnote{In the appendix, we also consider identification under persistent demand shocks (e.g., AR(1)) by utilizing lagged firm characteristics (such as R&D) as instruments or by explicitly modeling persistent unobserved quality heterogeneity.} Our identification proof is constructive: it yields closed-form mappings from observables to the objects of interest. The proof combines standard assumptions maintained in the proxy variable literature levinsohn2003estimating, such as strict monotonicity of input demand, with the first-order condition approach of \citet*{doraszelski2013,doraszelski2018measuring} and gandhi2020identification, while imposing additional nonparametric restrictions on firms' demand functions.

To address the two identification challenges described above, we develop a three-step approach that combines the control function method of olley1996dynamics, levinsohn2003estimating, and \citet*{ackerberg2015identification} with the first-order condition approach of \citet*{doraszelski2013, doraszelski2018measuring} and \citet*{gandhi2020identification} in a novel way. The key insight is to treat the control function---the inverse of a material demand function, which serves as a proxy for TFP---not merely as an auxiliary device, but as an object of nonparametric identification in its own right. In the first step, we identify the unobserved transitory demand shock that nonlinearly affects revenue by combining the control function as in ackerberg2015identification with the instrumental variable quantile regression of chernozhukov2005iv. Intuitively, this step separates demand-side variation from productivity variation in revenue, exploiting the fact that lagged information shifts inputs and TFP but not the transitory demand shock. In the second step, we identify the control function for TFP by applying the nonparametric identification of transformation models (e.g., horowitz1996semiparametric) examined by \citet*{ekeland2004identification} and \citet*{chiappori2015nonparametric}. This step recovers TFP (up to normalization) from the dynamics of inputs and the demand shock, without requiring output quantity data, because the control function maps observable inputs and the identified demand shock into unobserved productivity. In the third step, we identify the production function, markups, and the demand function using the first-order condition for materials and the control function identified in the second step.

Our method identifies several key objects from revenue data. In our main setting, markups and output elasticities are identified up to scale, while the output price, output quantity, TFP, gross production function, and consumer demand function are identified up to scale and location---normalizations that are standard in nonparametric settings. Proposition (ref) in Appendix (ref) formally characterizes this equivalence class, showing that any existing identification result for output elasticities or markup levels from revenue data necessarily imposes location and/or scale normalizations, whether explicitly or implicitly. Identification is cross-sectional, allowing these objects to vary over time. With the additional assumption of local constant returns to scale, we identify the levels of markups and output elasticities, and identify the output price, output quantity, TFP, production function, and consumer demand function up to location.\footnote{\citet*{flynn2019measuring} impose global constant returns to scale to identify a production function. In Subsection (ref), we clarify the distinction between local and global constant returns to scale.}

By assuming the homothetic single-aggregator (HSA) demand system of matsuyama2017beyond, we further identify the firms\textquoteright demand system and the representative consumer\textquoteright s utility function nonparametrically, without imposing parametric functional form restrictions.\footnote{In the Appendix, we establish identification of two additional families of homothetic demand systems proposed by matsuyama2017beyond, the homothetic demand systems with direct implicit additivity (HDIA) and those with indirect implicit additivity (HIIA), and discuss how to conduct counterfactual analyses using them.} The HSA class---which nests CES while permitting variable markups, incomplete pass-through, and non-monotonic relationships between firm size and markups (see matsuyama2023non; matsuyama2025homothetic for comprehensive reviews)---has become a leading framework for monopolistic competition with heterogeneous firms. To our knowledge, the nonparametric identification of the consumer\textquoteright s utility function within the HSA class from firm-level data is a new result; it enables counterfactual welfare analysis under flexible demand without the specification errors that arise from imposing CES.

Our result bridges two literatures that have largely operated in isolation: the production function literature, which recovers supply-side objects from revenue data, and the demand estimation literature, which recovers consumer-side objects from market-level data. In the existing production function literature, empirical measures of productivity or markups constructed from revenue data neither permit welfare evaluation nor counterfactual analysis, because the underlying consumer demand system and utility function remain unidentified. Our result enables counterfactual welfare analysis even when only revenue data are available. In particular, we show how to compute a counterfactual marginal-cost-pricing equilibrium and compare it with the observed monopolistic-competition equilibrium, thereby enabling a structural evaluation of firms\textquoteright market power and its welfare consequences through counterfactual changes in prices, quantities, consumer utility, and firm profits.

While the nonparametric identification results establish informational sufficiency in principle, practical estimation with moderate sample sizes requires additional structure. We develop a semiparametric estimator that assumes a Cobb-Douglas production function but leaves the demand system unrestricted. The estimation proceeds in three steps. First, we nonparametrically estimate the transitory demand shock using the smooth GMM IV quantile regression of \citet*{firpo2022gmm}, which ensures quantile monotonicity. Second, we estimate the control function via the profile likelihood estimator of \citet*{linton2008estimation}. Third, we recover the production function, markups, and TFP. This three-step procedure provides a standalone estimate of the production function without parametric demand assumptions. In a fourth step, for counterfactual welfare analysis and testing the CES restriction, we estimate the CoPaTh-HSA demand system of matsuyama2020constant.

Simulation results show that our estimator performs well in recovering structural parameters, markups, and TFP. Applying the estimator to Chilean plant-level data from the three largest manufacturing industries (SIC 31, 32, and 38), we find evidence of misspecification under the CES demand system in favor of the HSA demand system. Our counterfactual welfare analysis reveals that market power results in welfare losses of approximately 3%--6% of industry revenue in the three largest Chilean manufacturing industries in 1996. To put this in context, these losses exceed standard Harberger-triangle calculations and are broadly consistent with the welfare costs of markups estimated by edmond2023costly and the productivity losses from misallocation analyzed by baqaee2020productivity, though our estimates are derived from a different structural framework.

Our analysis contributes to a growing literature that addresses the revenue-versus-quantity problem in production function estimation. klette1996jae, de2011product, and gandhi2020identification impose a CES demand structure under which log revenue is linear in log inputs and log TFP; in contrast, we identify the demand system and the representative consumer\textquoteright s utility function nonparametrically, without relying on parametric assumptions. \citet*{flynn2019measuring} achieve identification under global constant returns to scale; we require only local constant returns to scale for certain normalizations. Recent work by demirer2025factor studies factor misallocation using revenue data, doraszelski2021reexamining reexamine the internal consistency of the de2012markups method under market power, deridder2025hitchhiker provides practical guidance on production function estimation, and kirov2025measuring develops alternative approaches to measuring markups. Our framework complements these contributions by providing nonparametric identification of both the production and demand sides from revenue data. On the demand side, our identification of the HSA utility function contributes to the growing literature on non-CES demand systems under monopolistic competition (matsuyama2017beyond; matsuyama2023non; matsuyama2025homothetic), providing a new empirical foundation for counterfactual welfare analysis within this class.\footnote{One frequently sees within the literature an assumption of market structure for the identification of demand and supply side objects. For example, \citet*{blp1995ecta} identify firm-level marginal costs by specifying oligopolistic competition; meanwhile, \citet*{ekeland2004identification} and \citet*{heckman2010nonparametric} identify various demand and supply side objects of a hedonic model by exploiting the properties of perfect competition.}

The remainder of this paper is organized as follows. Subsection (ref) introduces the model and setting, while Subsection (ref) illustrates our three-step identification strategy through a parametric example. Subsection (ref) establishes the main nonparametric identification results, and Subsection (ref) discusses alternative settings. Subsection (ref) introduces additional assumptions to fix scale and location normalizations, and Subsection (ref) discusses the identification of the demand system and counterfactual analysis. Section (ref) presents our semiparametric estimator. Section (ref) reports simulation results. Section (ref) provides an empirical application. Section (ref) concludes. The Appendix contains proofs and simulation details, and presents identification results for several extensions, including endogenous labor input, endogenous or discrete firm characteristics, persistent demand shocks, unobserved quality heterogeneity, and two additional families of homothetic demand systems proposed by matsuyama2017beyond.

Identification

Setting

We denote the logarithm of physical output, material, capital, and labor as $y_{it}$, $m_{it}$, $k_{it}$, and $l_{it}$, respectively, with their respective supports denoted as $\mathcal{Y}$, $\mathcal{M}$, $\mathcal{K}$, and $\mathcal{L}$. We collect the three inputs (material, capital, and labor) into a vector as $x_{it}:=(m_{it},k_{it},l_{it})'\in\mathcal{X}:=\mathcal{M}\times\mathcal{K}\times\mathcal{L}$.

At time $t$, output $y_{it}$ is related to inputs $x_{it}=(m_{it},k_{it},l_{it})'$ through the production function

where $z_{it}^{s}$ is a vector of exogenous characteristics with support $\mathcal{Z}_{s}$ that may affect either the functional form of $f_{t}(\cdot)$ or the level of total factor productivity (TFP) (e.g., ownership status).\footnote{Throughout the paper, the subscript $t$ on functions (e.g., $f_t$, $\psi_t$) indicates that the structural form is time-varying (common to all firms at time $t$), while the subscript $it$ denotes firm-specific realizations.} Firm-level productivity $\omega_{it}$ follows a first-order Markov process given by

where $\eta_{it}$ is an innovation to productivity that is serially uncorrelated, and $z_{it-1}^{h}$ is a vector of lagged characteristics with support $\mathcal{Z}_{h}$ that may affect the productivity process (e.g., previous import status as in kasahara2008).

The demand function for a firm's product is strictly decreasing in its price, and its inverse demand function is given by \[ p_{it}=\tilde{\psi}_{t}(y_{it},z_{it}^{d},\epsilon_{it}), \] where $z_{it}^{d}$ is an observable firm characteristic with support $\mathcal{Z}_{d}$ that affects firm's demand (e.g., firm's export status in de2012markups) while $\epsilon_{it}$ represents an unobserved demand shock.

We assume the demand shock $\epsilon_{it}$ has limited persistence to facilitate identification via lagged instruments, and interpret $\epsilon_{it}$ as the demand fluctuation remaining after conditioning on observable characteristics. Specifically, $\epsilon_{it}$ is generated by

Therefore, conditional on $z_{it}^{d}$, the underlying innovation $\zeta_{it}$ has a transitory effect on the demand shock $\epsilon_{it}$. Consequently, $\epsilon_{it}$ is serially correlated over $\upsilon$ periods but its persistence is limited: $\epsilon_{it}$ is independent of $\epsilon_{i,t-s}$ for $s \geq \upsilon + 1$. In contrast, an innovation to productivity $\eta_{it}$ has a permanent effect on future productivity in ((ref)). This difference between the demand and supply shock specifications in ((ref)) and ((ref)) captures the idea that demand shocks are temporary while supply shocks are permanent nelson1982trends. Appendix (ref) provides an alternative identification approach and shows that identification remains possible under persistent demand shocks when a supply-side instrument is available. Appendix (ref) discusses how productivity $\omega_{it}$ may capture persistent differences in product quality across firms, where the firm's output $y_{it}$ can be interpreted as a quality-adjusted measure of output quantity; from this perspective, heterogeneous demands attributable to quality differences are captured by $\omega_{it}$.

As shown in matzkin2003nonparametric, the identification of a non-additive unobservable $\epsilon_{it}$ has to be up to its monotonic transformation. Let $F_{\epsilon_{t}}$ be the c.d.f. of $\epsilon_{it}$. Without loss of generality, we transform $\epsilon_{it}$ to a uniform variable, using $u_{it}:=F_{\epsilon_{t}}(\epsilon_{it})$,

Given $t$, $u_{it}$ cross-sectionally follows an independent and identical uniform distribution.

The inverse demand function ((ref)) is non-parametrically specified and generalizes the constant-elasticity demand function examined by ma44ecma, klette1996jae, and de2011product. Equation ((ref)) implicitly imposes two key assumptions. First, $\psi_{t}(\cdot,z_{it}^{d},u_{it})$ is common across firms once we control for observed demand characteristics $z_{it}^{d}$ and a transitory scalar unobserved demand shock $u_{it}$. Second, $\psi_{t}(\cdot,z_{it}^{d},u_{it})$ represents the demand curve that each individual firm takes as given. This assumption is satisfied under monopolistic competition (without free entry), where $\psi_{t}$ can be expressed as $\psi_{t}(y_{it},z_{it}^{d},u_{it},a_{t})$, with $a_{t}$ denoting a vector of aggregate price and quantity indices which each firm treats as exogenous.

Let $r_{it}$ and $\mathcal{R}$ be the logarithm of revenue and its support, respectively. Then, from ((ref)), the observed revenue relates to output and input as follows:

where $\varphi_{t}(y_{it},z_{it}^{d},u_{it}):=\psi_{t}(y_{it},z_{it}^{d},u_{it})+y_{it}.$

We make the following timing assumption.

Assumptions (ref)(a)(b) specify the timing structure, which is similar to that in gandhi2020identification.\footnote{Stochastic independence between $(l_{it},k_{it})$ and $\eta_{it}$ is stronger than the standard mean independence assumption $E[\eta_{it}\mid\mathcal{I}_{it}]=0$, where $\mathcal{I}_{it}$ is the firm's information set at the beginning of period $t$. Most identification results below require only mean independence; full stochastic independence is used in the IVQR step (Proposition (ref)) and the control function argument (Proposition (ref)).} In Appendix (ref), we present identification results when $l_{it}$ is also endogenous. The continuity requirement in Assumption (ref)(c) can be relaxed, but the exogeneity of $\left(z_{it}^{s},z_{it}^{d},z_{it-1}^{h}\right)$ remains an important---albeit potentially strong---assumption, though it is commonly maintained in the empirical literature.\footnote{In Appendix (ref), we further discuss identification when these variables are discrete and endogenous, under the availability of suitable instruments.} Assumption (ref)(d) is also standard in most empirical applications.\footnote{ We treat deflated expenditures on materials as measures of inputs. This abstracts from unobserved heterogeneity in material prices. For instance, geographically segmented input markets may induce systematic price differences across regions.} Appendix (ref) shows that the identification strategy extends to such environments when material prices vary across firms as a function of observed characteristics.

Under Assumption (ref), the firm chooses $m_{it}=\mathbb{M}_{t}\left(\omega_{it},k_{it},l_{it},z_{it}^{s},z_{it}^{d},u_{it}\right)$ at time $t$ to maximize the profit:

where $p_{t}^{m}$ denotes the logarithm of the material price at time $t$.

Equation ((ref)) highlights two identification issues, originally raised by ma44ecma. First, $m_{it}$ correlates with two unobservables $\omega_{it}$ and $u_{it}$. Second, $r_{it}$ relates to $x_{it}=(m_{it},k_{it},l_{it})$ via two unknown nonlinear functions $\varphi_{t}(\cdot,z_{it}^{d},u_{it})$ and $f_{t}(\cdot)$. From the first-order condition for profit maximization, $P_{it}\left(1+\partial\psi_{t}(y_{it},z_{it}^{d},u_{it})/\partial y_{it}\right)=MC_{it}$, where $P_{it}$ and $MC_{it}$ denote the price and marginal cost of output, respectively, the elasticity of revenue with respect to output equals the inverse of the markup:

Thus, the revenue elasticity relates to the output elasticity via markup: \[ \frac{\partial\varphi_{t}(f_{t}(x_{it})+\omega_{it},z_{it}^{d},u_{it})}{\partial v_{it}}=\frac{MC_{it}}{P_{it}}\frac{\partial f_{t}(x_{it})}{\partial v_{it}}\text{ for }v_{it}\in\{m_{it},k_{it},l_{it}\}. \]

For identification, we make the following assumptions.

Assumptions (ref)(a)(b) are standard assumptions about smooth production and demand functions. In Assumption (ref)(b), the condition $\partial\varphi_{t}(y_{it},z_{it}^{d},u_{it})/\partial y_{it}>0$ is equivalent to that the elasticity of demand with respect to price, $-\left(\partial\psi_{t}(y_{it},z_{it}^{d},u_{it})/\partial y_{it}\right)^{-1}$, being greater than 1; this necessarily holds under profit maximization. Assumption (ref)(c) is a standard assumption in the control function approach that uses material as a control function for TFP levinsohn2003estimating,ackerberg2015identification. Assumption (ref)(d) requires the demand shock and the productivity shock are independent.

Let $w_{it}:=(k_{it},l_{it},z_{it}^{s},z_{it}^{d})$ be observable exogenous variables at $t$. The inverse function of the material demand function with respect to TFP \[ \omega_{it}=\mathbb{M}_{t}^{-1}(m_{it},w_{it},u_{it}) \] is used as a control function for $\omega_{it}$. Since $\partial\varphi_{t}(y_{it},z_{it}^{d},u_{it})/\partial y_{it}>0$, there exists the inverse function $\varphi_{t}^{-1}(\cdot,z_{it}^{d},u_{it})$ so that the revenue function $r_{it}=\varphi_{t}(f_{t}(x_{it},z_{it}^{s})+\omega_{it},z_{it}^{d},u_{it})$ can be written as:

Let $v_{it}:=(w_{it},u_{it},m_{it-1},w_{it-1},u_{it-1},z_{it-1}^{h})'\in\mathcal{V}:=\mathcal{W}\times[0,1]\times\mathcal{M}\times\mathcal{W}\times[0,1]\times\mathcal{Z}_{h}$, where $\mathcal{W}:=\mathcal{K}\times\mathcal{L}\times\mathcal{Z}_{s}\times\mathcal{Z}_{d}$. We assume that the data constitute a random sample of $N$ firms observed over multiple periods, $\{\{r_{is},m_{is},v_{is}\}_{s=t-\upsilon-2}^{t}\}_{i=1}^{N}$, drawn from the population. Given a sufficiently large $N$, the econometrician can consistently recover the corresponding population joint distributions.

Our objective is to identify $\{\varphi_{t}^{-1}(\cdot),f_{t}(\cdot),\mathbb{M}_{t}^{-1}(\cdot)\}$ from the population joint distribution of $\{r_{is},m_{is},v_{is}\}_{s=t-\upsilon-2}^{t}$. Let $\{\varphi_{t}^{*-1}(\cdot),f_{t}^{*}(\cdot),\mathbb{M}_{t}^{*-1}(\cdot)\}$ be the true model structure that satisfies ((ref)). Then, for any $(a_{1t},a_{2t},b_{t})\in\mathbb{R}^{2}\times\mathbb{R}_{++}$,

also satisfy ((ref)). Hence, the true structure $\{\varphi_{t}^{*-1}(\cdot),f_{t}^{*}(\cdot),\mathbb{M}_{t}^{*-1}(\cdot)\}$ is observationally equivalent to the structure ((ref)) and is therefore identified only up to location and scale normalization $(a_{1t},a_{2t},b_{t})$ from restriction ((ref)).

Normalization is unavoidable absent further assumptions. Because the unobserved output level $y_{it}$ enters the revenue function through the unknown nonlinear function $\varphi_t$ in (ref), it has no natural scale or location, so the structural functions are identified only up to a class of transformations. Proposition (ref) in Appendix (ref) formally characterizes this equivalence class for $(\varphi_t^{-1}, f_t, \mathbb{M}_t^{-1})$; accordingly, any existing identification result for output elasticities or markup level from revenue data either explicitly or implicitly imposes location and/or scale normalizations.

We may fix $(a_{1t},a_{2t},b_{t})$ in ((ref)) by fixing the values of $\{\varphi_{t}^{-1}(\cdot),f_{t}(\cdot),\mathbb{M}_{t}^{-1}(\cdot)\}$ at some points. Specifically, choosing two points $(m_{t0}^{*},w_{t}^{*},u_{t}^{*})$ and $(m_{t1}^{*},w_{t}^{*},u_{t}^{*})$ on the support $\mathcal{M}\times\mathcal{W}\times[0,1]$ where $m_{t0}^{*}<m_{t1}^{*}$, we denote

Note that $\partial\mathbb{M}_{t}^{-1}/\partial m_{it}>0$ implies that $c_{2t}<c_{3t}$. Then, there exists a unique one-to-one mapping between $(c_{1t},c_{2t},c_{3t})$ in ((ref)) and $(a_{1t},a_{2t},b_{t})$ in ((ref)) such that $b_{t}=\left(c_{3t}-c_{2t}\right)/\left(\mathbb{M}_{t}^{*-1}(m_{t1}^{*},w_{t}^{*},u_{t}^{*})-\mathbb{M}_{t}^{*-1}(m_{t0}^{*},w_{t}^{*},u_{t}^{*})\right)$, $a_{1t}=c_{1t}-b_{t}f_{t}^{*}(m_{t0}^{*},k_{t}^{*},l_{t}^{*},z_{t}^{s*})$ and $a_{2t}=c_{2t}-b_{t}\mathbb{M}_{t}^{*-1}(m_{t0}^{*},w_{t}^{*},u_{t}^{*})$. Thus, we can fix the value of $(a_{1t},a_{2t},b_{t})$ by choosing arbitrary values $(c_{1t},c_{2t},c_{3t})\in\mathbb{R}^{3}$ that satisfies $c_{2t}<c_{3t}$. In particular, we impose the following normalization that corresponds to (N2) in chiappori2015nonparametric.

As chiappori2015nonparametric demonstrates, this choice of normalization makes the identification proofs transparent. In Section (ref), we discuss how we can use additional restrictions and data to identify the normalization parameters $(a_{1t},a_{2t},b_{t})$.

Identification in a Parametric Example: Generalized CES Demand with Heterogeneity

Before presenting the nonparametric identification results, we demonstrate our identification approach by applying it to a simple parametric example without exogenous covariates, i.e., where $(z_{it}^{d},z_{it}^{s},z_{it}^{h})$ is empty. Consider a monopolistically competitive market where each firm $i$ faces the following constant elastic inverse demand function with heterogeneity:

where $\alpha_{t}(\cdot)$ and $\rho(\cdot)$ are unknown functions, where $0<\rho(\cdot)\le1$.\footnote{The demand function (ref) can be derived from a constant elasticity of substitution (CES) utility function, where the elasticity of substitution parameter is heterogenous across firms, depending on $u$. The term $\alpha_{t}$ implicitly captures aggregate expenditure and an aggregate price index.} We assume that $\rho'(u)<0$, which implies that the markup $1/\rho(u)$ is increasing in $u$.

Firm $i$ has a Cobb--Douglas production function with the TFP $\omega_{it}$ that follows a first-order autoregressive (AR(1)) process:

where $\{\theta_{0},\theta_{m},\theta_{k},\theta_{l},h_{1}\}$ are unknown parameters. The firm's revenue function is expressed as:

Denote the ratio of material cost to revenue as $s_{it}^{m}:=\frac{\exp(p_{t}^{m}+m_{it})}{\exp(r_{it})}$. Then, the first-order condition for ((ref)) can be written as

which, in turn, determines the control function for $\omega_{it}$ as

where $\beta_{t}(u_{it})=\left(p_{t}^{m}-\alpha_{t}(u_{it})-\rho(u_{it})\theta_{0}-\ln\rho(u_{it})\theta_{m}\right)/\rho(u_{it})$, $\beta_{m}(u_{it})=\left(1-\rho(u_{it})\theta_{m}\right)/\rho(u_{it})>0$, $\beta_{k}=-\theta_{k}$ and $\beta_{l}=-\theta_{l}$.

For notational brevity, assume that the support $\mathcal{X}$ includes two points $(m_{t0}^{*},k_{t}^{*},l_{t}^{*})=(0,0,0)$ and $(m_{t1}^{*},k_{t}^{*},l_{t}^{*})=(1,0,0)$. Following Assumption (ref), we fix the location and scale of $f_{t}(\cdot)$ and $\mathbb{M}_{t}^{-1}(\cdot)$ by imposing the following normalization:

which implies $\theta_{0}=0$, $\beta_{t}(0.5)=0$, and $\beta_{m}(0.5)=1$.

Our identification approach follows three steps.

Step 1: Identification of the Demand Shocks

The first step identifies the demand shock $u_{it}$. Substituting $\omega_{it}=\mathbb{M}_{t}^{-1}(m_{it},k_{it},l_{it},u_{it})$ and using $\theta_{0}=0$, we obtain

where the second equality uses $\beta_{k}=-\theta_{k}$ and $\beta_{l}=-\theta_{l}$ from ((ref)), so that $\rho(u_{it})(\theta_{k}+\beta_{k})=\rho(u_{it})(\theta_{l}+\beta_{l})=0$, and $\tilde{\phi}_{t}(u_{it}):=\alpha_{t}(u_{it})+\rho(u_{it})\beta_{t}(u_{it})$ with $\tilde{\phi}'_{t}(u)=-\theta_{m}\rho'(u)/\rho(u)>0$ for all $u$.

From ((ref)), we have $\Pr[r_{it}-m_{it}\leq\tilde{\phi}_{t}\left(u\right)]=u\quad\,\text{for all }u\in[0,1]$ because $\Pr[r_{it}-m_{it}\leq\tilde{\phi}_{t}\left(u\right)]=\Pr[\tilde{\phi}_{t}\left(u_{it}\right)\leq\tilde{\phi}_{t}\left(u\right)]=u$ by the monotonicity of $\tilde{\phi}_{t}(\cdot)$. Therefore, the quantile of $r_{it}-m_{it}$ identifies $u_{it}$ while the moment condition $E\left[1\left\{ r_{it}-m_{it}\le\tilde{\phi}_{t}\left(u\right)\right\} -u\right]=0$ for $u\in[0,1]$ identifies $\tilde{\phi}_{t}(\cdot)$.

Alternatively, from the first-order condition ((ref)) and the monotonicity of $\rho(\cdot)$ with $\rho'(\cdot)<0$, the demand shock $u_{it}$ is identified as the quantile of $1/s_{it}^{m}$. This equivalence arises because the quantile of $r_{it}-m_{it}$ coincides with that of $1/s_{it}^{m}=\exp(r_{it}-m_{it}-p_{t}^{m})$.

\paragraph{Step 2: Identification of Control Function and TFP}

The second step identifies the control function $\mathbb{M}_{t}^{-1}(\cdot)$. Substituting ((ref)) into the AR(1) process ((ref)) leads to

Since $\mathbb{M}_{t}^{-1}(m_{it},k_{it},l_{it},u_{it})$ is linear in $m_{it}$ from ((ref)), we can rearrange ((ref)) as:

where

$\tilde{\eta}_{it}={\eta_{it}}/{\beta_{m}(u_{it})},$ and $\delta_{m}(u_{it},u_{it-1})={h_{1}\beta_{m}(u_{it-1})}/{\beta_{m}(u_{it})}$. For a given $(u_{it},u_{it-1})$, ((ref)) is a linear model. Since $E\left[\left.\tilde{\eta}_{it}\right|v_{it}\right]=E\left[\left.\eta_{it}\right|v_{it}\right]/\beta_{m}(u_{it})=0$, where $v_{it}:=(k_{it},l_{it},x_{it-1},u_{it},u_{it-1})$, we can identify $\{\gamma(\cdot,\cdot)$, $\gamma_{k}(\cdot)$, $\gamma_{l}(\cdot)$, $\delta_{m}(\cdot,\cdot)$, $\delta_{k}(\cdot)$, $\delta_{l}(\cdot)\}$ in ((ref)) from the conditional moment restriction $E\left[\left.\tilde{\eta}_{it}\right|v_{it}\right]=0$.

Then, because $\beta_{m}(0.5)=1$, we can recover $(\theta_{k},\theta_{l},h_{1})$ from ((ref)) as \[ \theta_{k}=-\beta_{k}=\gamma_{k}(0.5),\ \theta_{l}=-\beta_{l}=\gamma_{l}(0.5),\text{ and }h_{1}=-\frac{\delta_{k}(0.5)}{\gamma_{k}(0.5)}=-\frac{\delta_{l}(0.5)}{\gamma_{l}(0.5)}. \] Also, applying the normalization ((ref)) to ((ref)), we have $\gamma(0.5,u_{t-1})=h_{1}\beta_{t-1}(u_{t-1})$. Then, $\beta_{m}(u)$ and $\beta_{t}(u)$ are identified from ((ref))-((ref)) as \[ \beta_{m}(u)=\frac{\gamma_{k}(0.5)}{\gamma_{k}(u)}=\frac{\gamma_{l}(0.5)}{\gamma_{l}(u)}\text{ and }\beta_{t}(u)=\gamma(0.5,u_{t-1})-\frac{\gamma(u,u_{t-1})\gamma_{k}(0.5)}{\gamma_{k}(u)}. \] Given the identification of $(\beta_{k},\beta_{l},\beta_{m}(\cdot),\beta_{t}(\cdot))$, we can identify $\omega_{it}$ from ((ref)).

\paragraph{Step 3: Identification of Production Function, Markup, and Demand Function}

The identification of $\rho(u)$ follows from substituting ((ref)) into $\beta_{m}(u)=\left(1-\rho(u)\theta_{m}\right)/\rho(u)$, and rearranging the terms, which yields \[ \rho(u_{it})=\frac{1-s_{it}^{m}}{\beta_{m}(u_{it})}=\frac{1-s_{it}^{m}}{\gamma_{k}(0.5)/\gamma_{k}(u_{it})}. \] Therefore, the markup $1/\rho(u_{it})$ is identified.

The first order condition ((ref)) implies that the revenue share of material expenditure is a function of $u_{it}$, which we denote by $s(u)$, such that $s_{it}^{m}=s(u_{it})$. In particular, $s(0.5)$ represents the median revenue share of material expenditure. Then, the identification of $\theta_{m}$ follows from the identification of $\rho(u)$ and the first order condition ((ref)) as

Appendix (ref) further discusses counterfactual analysis under the CES specification.

\paragraph{The Identification under Normalization}

In view of the first-order condition $\rho(u_{it})\theta_{m}=s_{it}^{m}$, it is clear from the argument above that the markup level cannot be separately identified from the material input coefficient $\theta_{m}$ without imposing the normalization restriction $\beta_{m}(0.5)=1$.

More generally, the parameters are identified under the scale and location normalization of $f_{t}(\cdot)$ and $\mathbb{M}_{t}^{-1}(\cdot)$ in ((ref)). Let $\theta_{i}$ ($i=0,m,k,l$) and $\beta_{j}(u_{t})$ ($j=t,m,k,l$) be those parameters identified above and let $\theta_{j}^{*}$ and $\beta_{i}^{*}(u_{t})$ be the true parameters. Then, there exist unknown normalization parameters $(a,b)\in\mathbb{R}\times\mathbb{R}_{+}$ such that \[ \theta_{0}=a+b\theta_{0}^{*},\ \beta_{t}=a+b\beta_{t}^{*},\ \theta_{i}=b\theta_{i}^{*},\ \beta_{j}(u_{t})=b\beta_{j}^{*}(u_{t}). \] We can fix the normalization by imposing further restrictions. For instance, if constant returns to scale $\theta_{m}^{*}+\theta_{k}^{*}+\theta_{l}^{*}=1$ holds, then the scale parameter $b$ can be identified as \[ b=b\left(\theta_{m}^{*}+\theta_{k}^{*}+\theta_{l}^{*}\right)=\theta_{m}+\theta_{k}+\theta_{l}=\frac{s(0.5)}{1-s(0.5)}-\beta_{k}-\beta_{l}. \] We discuss in subsection (ref) additional assumptions for fixing normalization.

The above identification argument is illustrative but relies on the linearity of $\mathbb{M}_{t}^{-1}(m_{it},k_{it},l_{it},u_{it})$ under restrictive parametric assumptions. The next subsection establishes nonparametric identification in a more general framework presented in Section (ref).

Nonparametric Identification

Step 1: Identification of the Demand Shocks

Substituting $\omega_{it}=\mathbb{M}_{t}^{-1}(m_{it},w_{it},u_{it})$ into the revenue function ((ref)), we can rewrite it as

We impose the following assumptions.

Assumption (ref)(a) implicitly imposes restrictions on the shape of the demand function. The Appendix (ref) shows that Assumption (ref)(a) holds if and only if $\frac{\partial\varphi_{t}}{\partial u}\frac{\partial\sigma_{t}}{\partial y}>\frac{\partial\varphi_{t}}{\partial y}\frac{\partial\sigma_{t}}{\partial u}$, where $\sigma_{t}(y,z^{d},u):=-1/\left(\frac{\partial\psi_{t}(y,z^{d},u)}{\partial y}\right)>0$ denotes the demand elasticity. Since $\frac{\partial\varphi_{t}}{\partial u}>0$ and $\frac{\partial\varphi_{t}}{\partial y}>0$, a sufficient condition for Assumption (ref)(a) is that an increase in the demand shock $\epsilon_{it}$ makes demand less elastic (i.e., increases the markup), while an increase in consumption makes demand more elastic (i.e., decreases the markup).

Under Assumption (ref)(a), given values of $(m,w)$, $\phi_{t}(m,w,\cdot)$ in ((ref)) can be interpreted as the quantile function of revenue $r$. Although $m_{it}$ is endogenous and correlated with $u_{it}$, Assumption (ref)(a)--(c) and equation ((ref)) imply that $u_{it}$ is independent of $(m_{it-\upsilon-1},w_{it-\upsilon})$ while $u_{it}$ is serially correlated with $u_{is}$ for $s=1,...,v$. Then, we have $\Pr[r_{it}\leq\phi_{t}\left(m_{it},w_{it},u\right)|m_{it-\upsilon-1},w_{it-\upsilon}]=u$ for all $u\in[0,1]$.\footnote{This follows because \[

\] where the second equality follows from the monotonicity of $\phi_{t}\left(m,w,\cdot\right)$ while the last equality holds because $u_{it}\perp \! \! \! \perp(m_{it-\upsilon-1},w_{it-\upsilon})$.}

Assumption (ref)(b), referred to as the completeness condition, implies the following uniqueness property: for any two candidate functions $\phi_{t}^{1}$ and $\phi_{t}^{2}$ and any fixed $u\in[0,1]$, $E\left[\left.1\left\{ r_{it}\le\phi_{t}^{1}\left(m_{it},w_{it},u\right)\right\} \right|m_{it-\upsilon-1},w_{it-\upsilon}\right]=E\left[\left.1\left\{ r_{it}\le\phi_{t}^{2}\left(m_{it},w_{it},u\right)\right\} \right|m_{it-\upsilon-1},w_{it-\upsilon}\right]$ a.s. implies that $\phi_{t}^{1}(\cdot,\cdot,u)=\phi_{t}^{2}(\cdot,\cdot,u)$ almost surely. Then, following chernozhukov2005iv, the moment condition

identifies $\phi_{t}(\cdot)$, and the demand shock $u_{it}$ is identified as $u_{it}=\phi_{t}^{-1}(r_{it},m_{it},w_{it})$ under Assumption (ref).

Hereafter, $\phi_{t}(\cdot)$ and $u_{it}$ are assumed to be known.

Step 2: Identification of Control Function and TFP

From ((ref)), the control function $\omega_{it}=\mathbb{M}_{t}^{-1}(m_{it},w_{it},u_{it})$ satisfies

where $\bar{h}_{t}\left(m_{it-1},w_{it-1},u_{it-1},z_{it-1}^{h}\right):=h_{t}\left(\mathbb{M}_{t-1}^{-1}(m_{it-1},w_{it-1},u_{it-1}),z_{it-1}^{h}\right)$. As $\partial\mathbb{M}_{t}^{-1}/\partial m_{it}>0$, given the values of $(w_{it},u_{it})$, the dependent variable in ((ref)) is a monotonic transformation of $m_{it}$. Therefore, the model ((ref)) belongs to a class of transformation models, the identification of which chiappori2015nonparametric analyze.

We make the following assumption, which corresponds to Assumptions A1--A3, A5, and A6 in chiappori2015nonparametric.\footnote{Assumption (ref) (c) corresponds to Assumption A4 of chiappori2015nonparametric.}

We can relax Assumption (ref)(b) by allowing $z_{it}^{h}$ and $l_{it}$ to correlate with $\eta_{it}$ as discussed in Appendix (ref). The sign restriction in Assumption (ref)(d) holds without loss of generality because we can choose any two points in place of $\{0,1\}$ on the support of $\omega_{it}$ without changing the essence of our argument.

Assumption (ref)(c) relaxes the "full support" condition (i.e., that the support is a Cartesian product of intervals) typically required in identification proofs of this type chiappori2015nonparametric. By requiring only that the support $\mathcal{M}\times \mathcal{V}$ be an open, connected set, we accommodate data structures where inputs are highly persistent (e.g., $k_{it} \approx k_{it-1}$), which often results in "diagonal" support shapes. As long as the support is connected, identification is achieved via line integration as detailed in the proof.

Assumption (ref)(f) can be interpreted as a generalized rank condition. Suppose $g_{\eta_{t}}\left(\eta_{it}\right)>0$ for all $\eta_{it}\in\mathbb{R}$. Then, as will be shown below in ((ref)), Assumption (ref)(f) holds if either

holds for some vector of lagged variables $\left(\tilde{m}_{it-1},\tilde{w}_{it-1},\tilde{u}_{it-1},\tilde{z}_{it-1}^{h}\right)$ in the projection $\text{Proj}_{v}(\mathcal{M}\times \mathcal{V})$ and some instrument $q_{it-1}\in\{m_{it-1},k_{it-1},l_{it-1},z_{it-1}^{s},z_{it-1}^{d},u_{it-1},z_{it-1}^{h}\}$. The latter condition is equivalent to (i) $\omega_{it-1}$ has a causal impact on $\omega_{it}$ ($\partial h/\partial\omega_{it-1}\neq0$) and (ii) $q_{it-1}$ has a causal impact on $\omega_{it-1}$ ($\partial\mathbb{M}_{it-1}^{-1}/\partial q_{it-1}\neq0$). These conditions must be satisfied for at least one exogenous variable $q_{it-1}$ at some point in the support.

Proposition (ref) shows that the control function is identified from the distribution of $(m_{it},v_{it})$. The identification of TFP also follows from Proposition (ref) as $\omega_{it}=\mathbb{M}_{t}^{-1}(m_{it},w_{it},u_{it})$.

Step 3: Identification of Production Function, Markup, and Demand Function

The final step identifies the production function, markup, and demand function. From $\text{\ensuremath{\phi_{t}}(\ensuremath{m_{it}},\ensuremath{w_{it}},\ensuremath{u_{it}})}=\varphi_{t}\left(f_{t}(x_{it},z_{it}^{s})+\mathbb{M}_{t}^{-1}\left(m_{it},w_{it},u_{it}\right),z_{it}^{d},u_{it}\right)$ and the monotonicity of $\varphi_{t}$, differentiating $\varphi_{t}^{-1}(\phi_{t}(m_{it},w_{it},u_{it}),z_{it}^{d},u_{it})=f_{t}(x_{it},z_{it}^{s})+\mathbb{M}_{t}^{-1}\left(m_{it},w_{it},u_{it}\right)$ with respect to $q_{it}^{s}\in\{m_{it},k_{it},l_{it},z_{it}^{s}\}$ and $q_{it}^{d}\in\{z_{it}^{d},u_{it}\}$ gives:

Note that $\partial\varphi_{t}^{-1}(r_{it},z_{it}^{d},u_{it})/\partial r_{it}=\left(\partial\varphi_{t}(y_{it},z_{it}^{d},u_{it})/\partial y_{it}\right)^{-1}$ represents the markup from ((ref)). If the markup $\partial\varphi_{t}^{-1}(r_{it},z_{it}^{d},u_{it})/\partial r_{it}$ were known, then equations ((ref)) and ((ref)) could identify $\partial f_{t}(x_{it},z_{it}^{s})/\partial q_{it}^{s}$ and $\partial\varphi_{t}^{-1}(r_{it},z_{it}^{d},u_{it})/\partial q_{it}^{d}$ given that $\mathbb{M}_{t}^{-1}(m_{it},w_{it},u_{it})$ is identified. However, since the markup is unknown, identification requires further restriction. Following \citet*{doraszelski2013,doraszelski2018measuring} and gandhi2020identification, we use the first-order condition with respect to the material as an additional restriction.

Rearranging the first-order condition, we obtain the markup equation used by de2012markups:

We establish the following proposition.

The output quantity and price for individual firms are identified as $y_{it}=\varphi_{t}^{-1}(r_{it},z_{it}^{d},u_{it})$ and $p_{it}=\psi_{t}(y_{it},z_{it}^{d},u_{it})=r_{it}-\varphi_{t}^{-1}(r_{it},z_{it}^{d},u_{it})$, respectively.

Comparison to Existing Identification Approaches

Our setup extends existing identification analyses of production functions by allowing prices to depend on output through an inverse demand function and by incorporating transitory unobserved demand shocks as a source of heterogeneous markups. While our approach builds on existing identification methods, our use of control functions and the IVQR framework differs from conventional formulations.

First, because the model includes both productivity and demand shocks, the standard control function approach cannot account for two sources of unobserved heterogeneity. We therefore assume that demand shocks are transitory while productivity shocks are persistent and use the IVQR approach to identify demand shocks in Step 1.

Second, Step 2 identifies the control function from the dynamics of input choices without relying on output measures, distinguishing our approach from the standard control function framework (e.g.,olley1996dynamics; levinsohn2003estimating; ackerberg2015identification).

Third, ackerberg2015identification identify a structural value-added function, $y_{it}=\tilde{f}_{t}(k_{it},l_{it})+\omega_{it}$, derived under perfect competition from a Leontief production function $y_{it}=\min\{\tilde{f}_{t}(k_{it},l_{it})+\omega_{it},a+m_{it}\}$. This formulation is difficult to apply under imperfect competition because $y_{it}<\tilde{f}_{t}(k_{it},l_{it})+\omega_{it}$ can occur. The maximum output capacity $y_{it}^{*}:=\tilde{f}_{t}(k_{it},l_{it})+\omega_{it}$ is determined before a firm chooses $m_{it}$ and $y_{it}$, so when $y_{it}^{*}$ is large---e.g., due to a high productivity shock---a profit-maximizing firm may produce $y_{it}<y_{it}^{*}$.\footnote{As noted by ackerberg2015identification, under perfect competition $y_{it}<y_{it}^{*}$ implies zero output, so only firms with $y_{it}=y_{it}^{*}$ are observed. Under imperfect competition, however, positive output with $y_{it}<y_{it}^{*}$ is possible.} Intuitively, when TFP doubles, a firm may avoid a large price decline by expanding output less than proportionally.

Fourth, our approach differs from gandhi2020identification in the use of the first-order condition for materials. Their method identifies the material elasticity $\partial f_{t}(x_{it},z_{it}^{s})/\partial m_{it}$ from the first-order condition ((ref)) in their first step: $\ln\frac{\partial f_{t}(x_{it},z_{it}^{s})}{\partial m_{it}}=\ln\text{\ensuremath{\frac{\exp(p_{t}^{m}+m_{it})}{\exp\left(r_{it}\right)}}}+\ln\frac{\partial\varphi_{t}^{-1}(r_{it},z_{it}^{d},u_{it})}{\partial r_{it}}$, assuming perfect competition or identical constant markups where $\partial\varphi_{t}^{-1}(r_{it},z_{it}^{d},u_{it})/\partial r_{it}$ becomes a common constant for all $i$. Under imperfect competition with variable markups, when the markup depends on revenue $r_{it}$, $\partial f_{t}(x_{it},z_{it}^{s})/\partial m_{it}$ cannot be identified solely from the first-order condition.

Extensions to Alternative Settings

We demonstrate in the Appendix that our identification strategy extends to settings where the baseline assumptions are relaxed. First, we establish identification with endogenous labor inputs in Appendix (ref) by incorporating labor adjustment costs, which allow lagged labor to serve as a valid instrument. Second, we address endogenous firm characteristics in Appendix (ref) by employing a control function approach for triangular systems. Third, Appendix (ref) confirms that our identification results---particularly the instrumental variable quantile regression in Step 1---remain valid when firm characteristics are discrete rather than continuous.

We also address the restrictiveness of assuming limited persistence in demand shocks through two extensions. First, Appendix (ref) demonstrates that identification holds under persistent shocks (e.g., AR(1)) if a lagged supply shifter, such as supply-driven R&D ($z_{it-1}^{h}$), serves as the instrument. Because $z_{it-1}^{h}$ shifts productivity but is excluded from demand, it satisfies the completeness condition despite serial correlation. Second, Appendix (ref) explicitly models persistent unobserved quality. This captures the persistent component of demand heterogeneity, allowing $\epsilon_{it}$ to represent only the remaining high-frequency, transitory fluctuations.

Fixing Normalization across Periods

Let $(\varphi_{t}^{-1}(\cdot),f_{t}(\cdot),\mathbb{M}_{t}^{-1}(\cdot))$ be a model structure for period $t$ identified by using Propositions (ref) and (ref) under the normalization in Assumption (ref). Let $(\varphi_{t}^{*-1}(\cdot),f_{t}^{*}(\cdot),\mathbb{M}_{t}^{*-1}(\cdot))$ denote the true model structure. Since the structure is identified up to scale and location normalization, there exist period-specific location and scale parameters $(a_{1t},a_{2t},b_{t})\in\mathbb{R}^{2}\times\mathbb{R}_{+}$ such as

Generally speaking, the location and scale normalization differ across periods---that is, $(a_{1t},a_{2t},b_{t})\neq(a_{1t+1},a_{2t+1},b_{t+1})$. For the identified objects to be comparable across periods, we need to fix normalization across periods by assuming that some object in the model is time-invariant. The subsection discusses these additional assumptions.\footnote{klette1996jae and de2011product identify the levels of markups and output elasticities from revenue data by using a functional form property of a demand function. They consider a constant elastic demand function leading to $\varphi_{t}(y_{it},z_{it})=\alpha y_{it}-(\alpha-1)z_{it}$ where $z_{it}$ is an aggregate demand shifter, which is an weighted average of revenue across firms, and $\alpha$ is an unknown parameter. This formulation implies $\varphi_{t}^{-1}(r_{it},z_{it})=(1/\alpha)r_{it}+(1-1/\alpha)z_{it}$ and imposes a linear restriction $\partial\varphi_{t}^{-1}(r_{it},z_{it})/\partial r_{it}+\partial\varphi_{t}^{-1}(r_{it},z_{it})/\partial z_{it}=1$, which fixes the scale parameter $b_{t}$.}

Scale Normalization

From ((ref)), the ratio of identified markups across two periods relates to the ratio of true markups as \[ \frac{\partial\varphi_{t+1}^{-1}(r_{it+1},z_{it+1}^{d},u_{it+1})/\partial r}{\partial\varphi_{t}^{-1}(r_{it},z_{it}^{d},u_{it})/\partial r}=\frac{b_{t+1}}{b_{t}}\frac{\partial\varphi_{t+1}^{*-1}(r_{it+1},z_{it+1}^{d},u_{it+1})/\partial r}{\partial\varphi_{t}^{*-1}(r_{it},z_{it}^{d},u_{it})/\partial r}. \] Therefore, the ability to identify how true markups change over two periods requires identification of the ratio of scale parameters, $b_{t+1}/b_{t}$. Similarly, the ratio of identified output elasticities across periods and that of identified TFP deviation from the mean are related to their true values via the ratio of scale parameters:

for $q\in\{m,k,l,z^{s}\}$.

To identify $b_{t+1}/b_{t}$, we consider the following assumptions.

Assumption (ref)(a) holds, for example, if the productivity shock $\omega_{it}$ follows a stationary process because stationarity requires that the distribution of $\eta_{it}$ does not change over time. Assumption (ref)(b) assumes that the elasticity of output with respect to one input does not change over time for some known interval; meanwhile, under Assumption (ref)(c), returns to scale in production technology does not change for some known interval of inputs.

The proof is given in Appendix (ref).

Local Constant Returns to Scale

We consider the following local constant returns to scale that strengthens Assumption (ref)(c).

Assumption (ref) is stronger than Assumption (ref)(c) but weaker than those used in some studies of markup estimation. In particular, markups are often estimated as the ratio of revenue $\exp(r_{it})$ to total cost $TC_{it}$ under the assumption of a linear cost function $TC_{it}=MC_{it}y_{it}$ with constant marginal cost $MC_{it}$. Such a linear cost function requires stronger conditions than Assumption (ref): (i) global constant returns to scale for all $x\in\mathcal{X}$, (ii) full flexibility of all inputs, and (iii) price-taking behavior in all input markets. By contrast, under Assumption (ref), marginal cost may increase with output, especially in the short run when dynamic inputs such as capital entail adjustment costs.

With Assumption (ref), the scale normalization parameter $b_{t}$ can be identified for all periods as follows. Let $f_{t}(x_{t},z_{t}^{s})$ be the identified production function under Assumption (ref) and $f_{t}^{*}(x_{t})$ be the true one where $f_{t}(x_{t},z_{t}^{s})=a_{t}+b_{t}f_{t}^{*}(x_{t},z_{t}^{s})$ from ((ref)). For $x\in\mathcal{B}$, we have \[ b_{t}=b_{t}\left(\frac{\partial f_{t}^{*}(x,z^{s})}{\partial m}+\frac{\partial f_{t}^{*}(x,z^{s})}{\partial k}+\frac{\partial f_{t}^{*}(x,z^{s})}{\partial l}\right)=\frac{\partial f_{t}(x,z^{s})}{\partial m}+\frac{\partial f_{t}(x,z^{s})}{\partial k}+\frac{\partial f_{t}(x,z^{s})}{\partial l}. \] Given that we have identified the scale parameter $b_{t}$ in ((ref)), we have established the following proposition.

Location Normalization

Suppose that scale normalization $b_{t}$ is already identified---for example, from Proposition (ref). Define

Then, ((ref)) is written as

From ((ref)), the growth rates (log differences) of the identified output and TFP between $t$ and $t+1$ are related to their true values as follows:

Therefore, to identify the growth rates of output and TFP, we need to identify the changes in the location parameters. To do so, we can use an industry-level producer price index $P_{t}^{*}$, which is often available as data, to identify the change in the location parameters. Suppose that $P_{t}^{*}$ is a Laspeyres index

where $\tilde{N}$ is a known set (or a random sample) of products. $p_{i0}^{*}$ and $y_{i0}^{*}$ are firm $i$'s log true price and log true output at the base period, respectively. The following argument holds for forms of a price index (other than Laspeyres) as long as the price index is a known function of prices that is homogenous of degree $1$, which is typically satisfied.

Assumption (ref)(b) is innocuous, implying that any output change between $t$ and $t+1$ when inputs are fixed at $\bar{x}$ is attributed to a TFP change.

Using the aggregate price index, we can identify the change in the location parameters and identify the growth of TFP and output.

The proof is given in Appendix (ref).

Identification of HSA Demand System, Utility Function, and Counterfactual Welfare Effects

Given that we have identified each firm's output price and quantity, it is possible to nonparametrically identify a Homothetic Single Aggregator (HSA) system of demand functions and the associated utility function of a representative consumer under an additional homotheticity shape restriction. Furthermore, we may identify the welfare effects of counterfactual experiments. matsuyama2017beyond further propose two additional families of homothetic demand systems---the homothetic demand system with direct implicit additivity (HDIA) and that with indirect implicit additivity (HIIA). In the Appendix (ref), we show that the analysis in this subsection extends to the HDIA and HIIA cases. Notably, the HDIA family nests the Kimball aggregator, which is widely used in Macroeconomics, as a special case.

Identification of the HSA demand system and consumer preference

\paragraph{The HSA demand system}

We adopt the Homothetic Single Aggregator (HSA) demand system proposed by matsuyama2017beyond.\footnote{The HSA system can be expressed as a system of direct demand functions or of inverse demand functions. These two systems are self-dual, meaning either can be derived from the other. matsuyama2023non and matsuyama2025homothetic provide comprehensive reviews of flexible extensions to the CES demand system, including the HSA framework.} Let $N_{t}$ denote the number of firms in the industry. The HSA demand system is characterized by the structural budget-share function $\mathfrak{s}_{t}^{*}(\cdot,z_{it}^{d},u_{it})$, which returns the log market revenue share of product $i$ as a function of its log relative output quantity, defined by $y_{it}-q_{t}(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})$. This relationship is expressed as:

where $\mathbf{y}_{t}:=(y_{1t},...,y_{N_{t}t})\in\mathcal{Y}^{N_{t}}$ is a vector of consumption, $\mathbf{z}_{t}^{d}:=(z_{1t}^{d},...,z_{N_{t}t}^{d})$ is a vector of observable demand shifters, $\mathbf{u}_{t}:=(u_{1t},...,u_{N_{t}t})$ is a vector of demand shocks, and $q_{t}(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})$ is the aggregate quantity index summarizing interactions across products.\footnote{If the utility function is CES, $U(\mathbf{y}_{t})=\left[\sum_{i=1}^{N_{t}}\exp\left(\rho y_{it}\right)\right]^{1/\rho}$, the inverse demand becomes $p_{it}=\rho\left(y_{it}-\ln U(\mathbf{y}_{t})\right)+\Phi_{t}-y_{it}$. Here, $\ln U(\mathbf{y}_{t})=q_{t}\left(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t}\right)$, meaning the quantity index coincides with the utility function, though they generally differ in the HSA framework.} In equilibrium, the quantity index $q_{t}(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})$ is uniquely determined by the adding-up constraint on market shares:

Because $\mathfrak{s}_{t}^{*}(\cdot)$ is nonparametric, the HSA framework nests various demand systems established in the literature, including the CES and symmetric translog demand systems (feenstra2003homothetic; feenstra2017globalization).\footnote{See matsuyama2020constant for details on how the HSA nests translog demand.}

\paragraph{Identification of the HSA demand system}

To identify the demand system, we maintain the following assumptions. Let $\Phi_{t}$ denote the log of aggregate expenditure for a representative consumer, where $\Phi_{t}=\ln\left(\sum_{i=1}^{N_{t}}\exp(r_{it})\right)$ in equilibrium.

Assumption (ref)(a) explicitly specifies the baseline aggregate state $\left(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t}\right)$ under which the reduced-form functions are defined. Assumption (ref)(b) imposes the standard framework of monopolistic competition klette1996jae,de2011product, implying that $N_{t}$ is sufficiently large such that single-firm strategic decisions have a negligible impact on aggregate quantities. Assumption (ref)(c) provides the normalization condition required to uniquely determine the structural function $\mathfrak{s}_{t}^{*}(\cdot)$, as discussed below. Assumption (ref)(d) holds when the conditions of Proposition (ref) hold.

Treating $q_{t}(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})$ and $\Phi_{t}$ as fixed, we define the reduced-form revenue function ((ref)) from the structural budget-share function:

Correspondingly, the reduced-form inverse demand function is related to $\mathfrak{s}_{t}^{*}\left(\cdot\right)$ by: \[ p_{it}=\Phi_{t}+\mathfrak{s}_{t}^{*}\left(y_{it}-q_{t}\left(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t}\right),z_{it}^{d},u_{it}\right)-y_{it}=:\psi_{t}\left(y_{it},z_{it}^{d},u_{it}\right). \] In this context, the function

represents the reduced-form budget-share function, which is known conditional on $\Phi_{t}$ and $\varphi_{t}\left(\cdot\right)$.

Conditional on the aggregate state $(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})$, the reduced-form revenue $\varphi_{t}(\cdot)$ and budget-share function $\mathfrak{s}_{t}(\cdot)$ are identified up to location normalization via Proposition (ref). When evaluating these reduced-form functions ((ref)) and ((ref)) across different values of $(y_{it},z_{it}^{d},u_{it})$, the quantity index $q_{t}(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})$ is held constant at the baseline state $(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})$. Consequently, these reduced-form functions are not invariant to changes in the aggregate state. As indicated by ((ref)), $\varphi_{t}(\cdot)$ shifts in response to changes in $\mathbf{y}_{t}$ because the quantity index $q_{t}(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})$ depends on the consumption vector $\mathbf{y}_{t}$.

Nonetheless, we may identify the structural function $\mathfrak{s}_{t}^{*}\left(\cdot\right)$ from the reduced-form function $\mathfrak{s}_{t}\left(\cdot\right)$ in ((ref)) under the equilibrium constraint ((ref)) as follows. Evaluating ((ref)) and ((ref)) at the fixed baseline state $\left(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t}\right)$, the reduced-form and structural budget share functions are related as:

Let $\Delta{q}_{t}(\tilde{\mathbf{y}}_{t},\tilde{\mathbf{z}}_{t}^{d},\tilde{\mathbf{u}}_{t}):={q}_{t}(\tilde{\mathbf{y}}_{t},\tilde{\mathbf{z}}_{t}^{d},\tilde{\mathbf{u}}_{t})-{q}_{t}(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})$ denote the change in the quantity index induced by a shift from the baseline state $(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})$ to a new state $(\tilde{\mathbf{y}}_{t},\tilde{\mathbf{z}}_{t}^{d},\tilde{\mathbf{u}}_{t})$. Evaluating ((ref)) at the argument $(y_{it},z_{it}^{d},u_{it})=(\tilde{y}_{it}-\Delta{q}_{t}(\tilde{\mathbf{y}}_{t},\tilde{\mathbf{z}}_{t}^{d},\tilde{\mathbf{u}}_{t}),\tilde{z}_{it}^{d},\tilde{u}_{it})$ while keeping ${q}_{t}(\mathbf{y}_{t},{\mathbf{z}}_{t}^{d},\mathbf{u}_{t})$ as constant, we have:

for any alternative state $(\tilde{\mathbf{y}}_{t},\tilde{{\mathbf{z}}}_{t}^{d},\tilde{\mathbf{u}}_{t})$. By substituting ((ref)) into the constraint ((ref)), we can uniquely identify $\Delta{q}_{t}(\tilde{\mathbf{y}}_{t},\tilde{\mathbf{z}}_{t}^{d},\tilde{\mathbf{u}}_{t})$ by solving:

Given the identified quantity index change $\Delta q_{t}(\cdot)$, we are able to identify the structural budget share function $\mathfrak{s}_{t}^{*}\left(\cdot\right)$ from the reduced-form function $\mathfrak{s}_{t}\left(\cdot\right)$ via ((ref)), up to the unknown value of the baseline index ${q}_{t}(\mathbf{y}_{t},{\mathbf{z}}_{t}^{d},\mathbf{u}_{t})$.

The preceding discussion demonstrates that $\mathfrak{s}_{t}^{*}(\cdot)$ cannot be separately identified from the baseline aggregate quantity index ${q}_{t}(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})$. Specifically, given the identified reduced-form function $\mathfrak{s}_{t}(\cdot)$ and the change in the index $\Delta{q}_{t}(\cdot)$, there exist multiple pairs $\{\mathfrak{s}_{t}^{*}(\cdot),{q}_{t}(\mathbf{y}_{t},\mathbf{z}_{t}^{d},\mathbf{u}_{t})\}$ that satisfy ((ref)). For this reason, we impose Assumption (ref)(c) to uniquely determine $\mathfrak{s}_{t}^{*}(\cdot)$ but the fundamental properties of the preference structure to be characterized in Proposition (ref) are invariant to this specific normalization.

Applying the result of matsuyama2017beyond (Proposition 1 and Remark 3), the following proposition establishes that the HSA demand system constructed above can be derived from a unique representative consumer preference, and that it is possible to identify an associated utility function. Appendix (ref) supplies the proof.

Counterfactual analysis

We conduct a short-run partial equilibrium counterfactual analysis by maintaining the following assumptions.

Under Assumption (ref)(a), firms respond to counterfactual interventions by adjusting their material inputs, outputs, and prices in the short run. Assumption (ref)(b) implies that the aggregate income of the representative consumer remains invariant to the counterfactual interventions. This is a reasonable approximation when analyzing a specific industry that is small relative to the aggregate economy. For large-scale interventions, however, this assumption can be relaxed to allow aggregate income to adjust endogenously in response to changes in aggregate profits induced by the intervention.

\paragraph{Monopolistic Competition Equilibrium}

Using the identified HSA demand system $\{\mathfrak{s}_{t}(\cdot),\Delta q_{t}(\cdot)\}$, we calculate a monopolistic competition equilibrium (MCE) as follows. Define the inverse production function $m_{it}=\chi_{it}\left(y_{it}\right)$ such that $y_{it}=f_{t}\left(\chi_{it}\left(y_{it}\right),k_{it},l_{it},z_{it}^{s}\right)+\omega_{it}$ for given $(k_{it},l_{it},\omega_{it},z_{it}^{s})$; namely, $\chi_{it}(y_{it}):=f_{t}^{-1}\left(y_{it}-\omega_{it},k_{it},l_{it},z_{it}^{s}\right)$.

Equilibrium outputs and the quantity index $(\mathbf{y}_{t}^{m},\Delta q_{t}^{m})$ in an MCE are obtained from the first order condition for profit maximization in ((ref)), equivalently written in view of ((ref))-((ref)), and the market share condition ((ref)):

for $i=1,...,N_{t}$, where $\frac{\partial\chi_{it}(y_{it}^{m})}{\partial y_{it}}=\left(\frac{\partial f_{t}}{\partial m}\right)^{-1}$. The above system can be extended to incorporate policies such as taxes and subsidies to investigate their effects.

\paragraph{Welfare Costs of Firm's Market Power}

In the empirical section below, we quantify the deadweight loss attributable to firm's market power by considering the transition to a counterfactual Marginal Cost Pricing Equilibrium (MCPE), in which firms set their prices equal to their marginal costs. Specifically, equilibrium outputs and quantity index $(\mathbf{y}_{t}^{c},\Delta q_{t}^{c})$ are obtained by solving:

for $i=1,...,N_{t}$.

The consumer welfare cost of firm's market power can be calculated as the utility change:

An alternative welfare measure is the Compensating Variation (CV), which is constructed in monetary terms as follows. For a given counterfactual log income $\Phi_{t}^{c}$, define the counterfactual output and quantity index $(\mathbf{y}_{t}^{c*}(\Phi_{t}^{c}),\Delta q^{c*}(\Phi_{t}^{c}))$ that solve the price-taking condition: \[ \underbrace{\exp\left(\mathfrak{s}_{t}\left(y_{it}^{c*}-\Delta q_{t}^{c*},z_{it}^{d},u_{it}\right)+\Phi_{t}^{c}-y_{it}^{c*}\right)}_{\text{Price}}=\underbrace{\frac{\exp\left(p_{t}^{m}+\chi_{it}\left(y_{it}^{c*}\right)\right)}{\exp(y_{it}^{c*})}\frac{\partial\chi_{it}(y_{it}^{c*})}{\partial y_{it}}}_{\text{Marginal Cost}} \] for $i=1,...,N_{t}$ with $\sum_{i=1}^{N_{t}}\exp\left(\mathfrak{s}_{t}\left(y_{it}^{c*}-\Delta q_{t}^{c*},z_{it}^{d},u_{it}\right)\right)=1$. We then find the counterfactual income $\Phi_{t}^{c*}$ that achieves the same utility as in the benchmark MCE:

The compensating variation is defined as $CV_{t}:=\exp\left(\Phi_{t}^{c*}\right)-\exp\left(\Phi_{t}\right)$. It measures the change in consumer welfare when moving from an MCPE to an MCE and quantifies the consumer's loss due to firms' market power. Since consumers require less income under competitive pricing to achieve the same utility, $CV_{t}<0$: the magnitude $|CV_{t}|$ is the monetary gain from eliminating market power, and the consumer's welfare improvement from transitioning to an MCPE is $-CV_{t}>0$.

To evaluate the overall welfare change from an MCE to an MCPE, the consumer gain, measured by $-CV_{t}$, can be compared with firms' profit loss. Under Assumption (ref)(b), aggregate revenue is fixed. Thus, the change in total profits is solely the negative change in total material costs:

Note that if $\Pi_{t}^{c}<0$, this counterfactual implies firms operate at a loss, representing a short-run equilibrium before firm exit occurs. The overall welfare change associated with a transition from an MCE to an MCPE is therefore given by $\Delta\Pi_{t}-CV_{t}$.

Semiparametric Estimator

The identification results in Section (ref) are nonparametric: revenue data alone suffice to recover the production function, demand system, and welfare-relevant objects without functional-form restrictions. While a fully nonparametric sieve estimator could in principle be constructed from the identified equations, it would demand sample sizes far exceeding typical manufacturing datasets. We therefore impose Cobb-Douglas production and AR(1) productivity dynamics, following the standard practice of pairing nonparametric identification with parametric specifications olley1996dynamics,levinsohn2003estimating,ackerberg2015identification,gandhi2020identification.

We develop a semiparametric estimator that is applicable for the panel data with $T\geq4$. We assume the Cobb-Douglas production function:

and TFP follows an AR(1) process:

We introduce assumptions on the production function to normalize scale and location parameters. First, ((ref)) normalizes one location parameter by $f_{t}(0,0,0)=0$. As another location normalization, we assume $E\left[\omega_{it}\right]=0$, which naturally arises from a stationary AR1 process ((ref)). Finally, we assume the constant returns to scale, $\theta_{m}+\theta_{k}+\theta_{l}=1$, which normalizes the scale parameter.

The control function becomes separable under the Cobb-Douglas assumption:

Substituting ((ref)) into the revenue function, we obtain

where $\phi_{t}$ depends only on $(m_{t},u_{it})$ and increases in $m_{t}$ and $u_{t}$. In Appendix (ref), we derive a separable control function similar to ((ref)) in a more general setting and show that ((ref)) holds when the production function is separable with respect to $m_{it}$. \\

Substituting ((ref)) into ((ref)), we obtain the transformation model as

\paragraph{Step 1: Estimation of the Quantile of Demand Shocks}

The first step estimates $\phi_{t}\left(m_{it},u_{it}\right)$ and $u_{it}$ by IV quantile regression. A traditional approach to IV quantile regression estimates $\phi_{t}\left(\cdot,u\right)$ from the moment condition ((ref)) for a fixed quantile point $u$. This approach often yields a non-monotonic and non-smooth function in $u$, which is problematic for our identification using uniquely identified $u_{it}$ and derivatives of $\phi_{t}$. To overcome this, we use the smoothed GMM quantile regression of \citet*{firpo2022gmm}. Their approach stacks moment conditions over all quantile points so we can estimate the smooth sieve function and impose $\partial\phi_{t}/\partial u_{it}>0$. For the approximation of $\phi_{t}(m_{it},u_{it})$, we employ the basis $B_{\phi}(m_{it},\tau)$ that consists of a constant term, a B-spline basis of degree 3 with 2 interior knots in $m_{it}$, a cubic polynomial in $u_{it}$, and interactions of the B-spline in $m_{it}$ with $u_{it}$ and $u_{it}^{2}$. firpo2022gmm also replace the indicator in ((ref)) with a smooth kernel CDF to ease computation.

We partition $[0,1]$ into $L=100$ equal parts and let $\mathbb{T}=\{0.01,\ldots,0.99\}$. The moment condition is

where $\mathcal{K}_{1}(\cdot)$ is a smooth kernel CDF with bandwidth $b_{n1}$ and $B_{IV}(m_{it-2})$ is the B-spline basis of degree $S_{1}=3$ with $K_{1}=2$ interior knots in $m_{it-2}$ as instruments. Following firpo2022gmm, we use the rule-of-thumb bandwidth and the kernel CDF of horowitz1998bootstrap: \[ \mathcal{K}_{1}(s):=\left[\frac{1}{2}+\frac{105}{64}\left(s-\frac{5}{3}s^{3}+\frac{7}{5}s^{5}-\frac{3}{7}s^{7}\right)\right]1\{s\in[-1,1]\}+1\{s>1\}. \] The number of moment conditions ((ref)) is the number of IVs times the number of quantile $(S_{1}+K_{1}+1)\times(L-1)$. As $L$ is usually a large number, the moment condition ((ref)) typically overidentifies $\alpha_{t}$ so that we use GMM. firpo2022gmm derive a expression of the optimal GMM weight matrix and showed it does not depend on the parameter $\alpha_{t}$ so that its estimation completes in one step. Monotonicity in $m_{it}$ and $u_{it}$ is imposed via linear constraints on the derivatives of the basis functions. The demand shocks $\hat{u}_{it}$ are then estimated by numerically inverting $\hat{\phi}_{t}(m_{it},\hat{u}_{it})=r_{it}$. The same procedure is applied to $t-1$ to estimate $\hat{u}_{it-1}$.

\paragraph{Step 2: Estimation of the control function}

The second step estimates the transformation model ((ref)). We use the Profile Likelihood (PL) estimator developed by \citet*{linton2008estimation}. From ((ref)) for $q_{it}=m_{it}$ and ((ref)), the conditional density of $m_{it}$ given $v_{it}$ is written as

To approximate $\lambda_{t}(m_{it},u_{it})$, we use the basis $B_{\lambda}(m_{it},u_{it})$ that is the Kronecker product of B-spline bases of degree 3 with 1 interior knot in $m_{it}$ and $u_{it}$. We do not assume a parametric distribution on $\eta_{it}$. Let $\partial_{m}B_{\lambda}(m_{it},u_{it})$ be the derivatives of the B-spline bases $B_{\lambda}(m_{it},u_{it})$ so that $\partial_{m}B_{\lambda}(m_{it},u_{it})^{T}c_t$ approximates $\frac{\partial\lambda_{t}(m_{it},u_{it})}{\partial m_{it}}$. Thus, the log-likelihood function is written as \[ \sum_{i=1}^{n}\left\{ \ln g_{m_{t}|v_{t}}\left(m_{it}|v_{it}\right)\right\} =\sum_{i=1}^{n}\left\{ \ln g_{\eta_{t}}\left(\eta_{it}\right)+\ln\partial_{m}B_{\lambda}(m_{it},u_{it})^{T}c_t\right\} . \] where ${g}_{\eta_{t}}(\eta)$ is the corresponding (Gaussian) kernel density with the bandwidth chosen by Silverman's Rule. We obtain estimates of $\eta_{it}$ as follows. {For given $(c_t,\rho)$, define $\lambda_{it}(c_t):=B_{\lambda}(m_{it},u_{it})^{T}c_t$, $\tilde k_{it}(\rho):=k_{it}-\rho k_{it-1}$, $\tilde l_{it}(\rho):=l_{it}-\rho l_{it-1}$, and $\tilde B_{\lambda}(m_{it-1},u_{it-1},\rho):=\rho B_{\lambda}(m_{it-1},u_{it-1})$, and ((ref)) implies that

Therefore, for each $(c_t,\rho)$, let $\eta_{it}(c_t,\rho)$ be the residual from regressing $\lambda_{it}(c_t)$ on $\tilde k_{it}(\rho)$, $\tilde l_{it}(\rho)$, and $\tilde B_{\lambda}(m_{it-1},u_{it-1},\rho)$. Then, the PL estimator $(\tilde{c}_{t},\tilde\rho)$ is defined as

}

\paragraph{Step 3: Estimation of production function, markup, TFP, and output}

With estimated $(\tilde{c}_{t},\tilde\rho)$, we estimate $(\theta_{k},\theta_{l},c_{t-1})$ by regressing $B_{\lambda}(m_{it},\hat{u}_{it})^{T}\tilde{c}_{t}$ on $\tilde k_{it}(\tilde\rho)$, $\tilde l_{it}(\tilde\rho)$, and $\tilde B_{\lambda}(m_{it-1},u_{it-1},\tilde\rho)$:

From ((ref)) and ((ref)), the material elasticity is identified as $\theta_{m}=\frac{\partial\lambda_{t}\left(m_{it},u_{it}\right)}{\partial m_{it}}\left(\frac{s_{it}^{m}}{\partial\phi_{t}\left(m_{it},u_{it}\right)/\partial m_{it}-s_{it}^{m}}\right)$, where $s_{it}^{m}=\exp(p_{t}^{m}+m_{it})/\exp\left(r_{it}\right).$ Following this equation, we estimate $\theta_{m}$ as follows:

where $\partial_{m}B_{\phi}(m_{it},\hat{u}_{it})$ is the derivatives of bases $B_{\phi}(m_{it},\tau)$ so that $\partial_{m}B_{\phi}(m_{it},\hat{u}_{it})^{T}\hat{\alpha}_{t}$ estimates $\partial\phi_{t}\left(m_{it},u_{it}\right)/\partial m_{it}$. Using the constant returns to scale, we obtain the normalized production parameters as $\hat{\theta}_{j}=\tilde{\theta}_{j}/\tilde{b}_{t}$ for $j\in\{m,k,l\}$ where $\tilde{b}_{t}=\tilde{\theta}_{m}+\tilde{\theta}_{k}+\tilde{\theta}_{l}$. Then, we estimate markups as follows:

With the mean-zero restriction on $\omega_{it}$, the location parameter $\hat{a}_{2t}=n^{-1}\sum_{i=1}^{n}\left[\tilde{\lambda}_{it}-\tilde{\theta}_{k}k_{it}-\tilde{\theta}_{l}l_{it}\right]$ is estimated. The estimated TFP, output, and price are

\paragraph{Step 4: Estimation of parametric CoPaTh-HSA demand system}

Our estimation steps of production function above do not assume any parametric demand system. Thus, in theory, one can estimate a fully nonparametric HSA demand system as described in Section (ref). However, in our empirical application, we estimate a parametric HSA demand system to obtain more stable estimates from a dataset with a moderate sample size. In particular, we consider a HSA demand system with the CoPaTh (constant pass-through) demand function with incomplete pass-through by matsuyama2020constant:

where the quantity index $q_{t}(\mathbf{y}_{t},\epsilon_{t})$ is implicitly defined by the market share constraint ((ref)) for a given output vector $\mathbf{y}_{t}=\left(y_{1t},...,y_{Nt}\right)$ and a given demand shock vector $\mathbf{\epsilon}_{t}=\left(\epsilon_{1t},...,\epsilon_{Nt}\right)$. Appendix (ref) derives the log-variable version of the CoPaTh demand ((ref)) from matsuyama2020constant's original formulation.

As explained in Section (ref), we estimate the following reduced-form revenue function instead of the structural form ((ref)):

where we impose the normalization $q_{t}(\mathbf{y}_{t},\boldsymbol{\epsilon}_{t})=0$, as stated in Assumption (ref)(c), by assuming that the observed data correspond to the baseline aggregate state. Consequently, the implied markup is \[ \frac{P_{it}}{MC_{it}}=\left(\frac{\partial\varphi_{t}(y_{it},\epsilon_{it})}{\partial y_{it}}\right)^{-1}=1+\epsilon_{it}\exp(\beta_{t}y_{it}-\gamma_{t}). \]

Taking the limit as $\beta_{t}\to0$ on the right-hand side of ((ref)) yields a linear revenue function $\varphi_{t}(y_{it},\epsilon_{it})=\alpha_{t}+\rho_{it}y_{it}$, where $\alpha_{t}=\Phi_{t}+\delta_{t}$ and $\rho_{it}=\frac{1}{1+\epsilon_{it}}$. Thus, the CoPaTh demand system in ((ref)) nests the generalized CES demand with heterogeneous markups defined in ((ref)). Under this special case, the markup is independent of $y_{it}$ and pass-through is complete. Moreover, if $\epsilon_{it}=\epsilon$ is common across all firms, then the system collapses to the standard CES demand framework with a homogeneous markup.

With the estimated outputs $\hat{y}_{it}$ and markups $\hat{\mu}_{it}$ from Step 3, the composite nonlinear least square estimator of demand parameters $(\beta_{t}$, $\gamma_{t}$, $\delta_{t})$ is defined as, \[ {(\hat{\beta}_{t},\hat{\gamma}_{t},\hat{\delta}_{t})'}\in\arg\min_{\beta_{t},\delta_{t},\gamma_{t}}\sum_{i}\left(r_{it}-\left(\Phi_{t}+\delta_{t}-\frac{1}{\beta_{t}}\ln\left(\frac{\exp(-\beta_{t}\hat{y}_{it}+\gamma_{t})+\epsilon_{it}}{1+\epsilon_{it}}\right)\right)\right)^{2} \] \[ +\sum_{i}\left(\hat{u}_{it}-\left(\text{quantile}(\epsilon_{it})\right)\right)^{2}, \] subject to

where $\text{quantile}(\epsilon_{it})$ is the empirical quantile of $\epsilon_{it}$ among all firms and $\Phi_{t}=\ln\left(\sum_{i}\exp(r_{it})\right)$ is the industry revenue. The estimation utilizes the theoretical relationship between markups and demand shocks and the market share restriction. \footnote{For model-consistency, we estimate the HSA demand system only using the firms with estimated markups $\hat{\mu}_{it}>1$.}

Simulation

This section presents the finite sample performance of our proposed estimator, comparing to that of the conventional method, when firms charge small but heterogeneous markups under the HSA demand system. We consider a simple data-generating process (DGP) in which firms set variable markups, calibrated to Chilean manufacturing data. Further details of the DGP and simulation design are provided in the Appendix (ref).

Consider $N$ firms in a market and $t\in\{1,2,...,T\}$ period. Each firm produces one variety of differentiated goods and faces the HSA-CoPaTh demand function ((ref)). The demand shock $\epsilon_{it}$ follows an MA(1) process: $\epsilon_{it}=0.5\zeta_{it-1}+\zeta_{it}$, where $\zeta_{it}\sim\text{Unif}[0,0.3]$. The production function takes the Cobb-Douglas form ((ref)) where $\omega_{it}$ follows an AR1 process $\omega_{it}=0.8\omega_{it-1}+\eta_{it},\eta_{it}\sim N\left(0,\left(0.05\right)^{2}\right)$. Capital $k_{it}$ and labor $l_{it}$ are predetermined and follow exogenous laws of motion explained in Appendix (ref).

The production function parameters are set to $\left(\theta_{m},\theta_{k},\theta_{l}\right)=\left(0.4,0.3,0.3\right)$. We choose the structural parameters of the HSA demand system, $(\Phi_{t},\delta_{t},\beta_{t})=(20,-6.5,0.21)$, and allow parameter $\gamma_{t}$ to vary across simulations to satisfy the normalization $q_{t}(\mathbf{y}_{t},\boldsymbol{\epsilon}_{t})=0$ in each simulation. Specifically, for each period, we find equilibrium outputs $\{y_{it}^{m}\}_{i=1}^{N_{t}}$ and $\gamma_{t}$ in an MCE by solving the first order conditions and the market share condition analogous to ((ref)):

where $\Xi_{it}=\ln\theta_{m}+(\theta_{k}k_{it}+\theta_{l}l_{it}+\omega_{it})/\theta_{m}$ and $p_{t}^{m}=0$. Appendix (ref) show its derivation. We simulate 100 replications of $N=600$ firms and $T=5$ periods, with the following summary statistics of the resulting markups:

In addition to our proposed estimator, we also consider the widely-used estimator proposed by ackerberg2015identification(ACF) applied with quantity data and the ACF estimator applied with revenue data. gandhi2020identification(GNR) showed the difficulty of identification in the DGP that ACF assumed where a firm-level unobserved shock is a scalar, TFP. The GNR criticism is not applicable for the current DGP with two unobserved shocks. However, to show our point is different from the GNR critique, we employ the ACF method with constant returns to scale (CRS) restriction (i.e., $\theta_{m}+\theta_{k}+\theta_{l}=1$) that \citet*{flynn2019measuring} proposed to address the GNR criticism.

Figure (ref) shows the histograms of 100 estimates of $(\theta_{m},\theta_{k},\theta_{l})$ from the ACF method with revenue data and quantity data. While using quantity data yields estimates that are tightly clustered around the true values, using revenue data substantially biases the estimation of the production function. The simulation result confirms the long criticism in the literature against the ad hoc use of revenue data as output.

Figure (ref) shows the histograms of 100 estimates for $(\theta_{m},\theta_{k},\theta_{l})$ from our proposed estimator and $(\beta_{t},\delta_{t})$ at $t=T$ on the HSA demand system. Since $\gamma_{t}$ varies across simulation, we report the histograms of estimation errors $\hat{\gamma}_{t}-\gamma_{t}$. They are tightly clustered around their true values, suggesting that our method recovers the structural parameters very well. Figure (ref) shows the scatter plot of true versus estimated TFPs and markups for the first 20 Monte Carlo simulations. The strong alignment of points along the 45-degree lines accompanying with the low RMSEs and high correlations suggest that our method precisely estimates TFPs and markups.

Empirical Application: Chilean Manufacturing Sector

The semiparametric estimator is applied to the Chilean manufacturing plant dataset, derived from the census conducted by the Chilean Instituto Nacional de Estadística, covering all plants with 10 or more employees from 1993 to 1996. The primary objective of this empirical application is to illustrate the practical feasibility of our method in a well-studied empirical environment, rather than to uncover new stylized facts. The Chilean manufacturing census has been used extensively in the literature, making it a natural benchmark for assessing how our approach performs relative to existing empirical frameworks. In addition, we use this setting to examine whether the commonly imposed CES demand specification is rejected by the data in favor of the more flexible HSA demand system, and to quantify the welfare losses associated with markups through the counterfactual experiments discussed in Section (ref).

Following the standard approach in this literature, we define labor input as the number of workers, material input as materials cost, and revenue as income plus the value of capital produced for own use, with all nominal values deflated using industry-specific deflators. Capital input is constructed as the sum of deflated values for buildings, machinery, and vehicles using the perpetual inventory method. Our analysis focuses on the three largest manufacturing industries in 1996, corresponding to 2-digit SIC codes 31 (Food, Beverage, and Tobacco), 32 (Textiles, Apparel, and Leather Products), and 38 (Metal Products, Electric and Non-electric Machinery, Transport Equipment, and Professional Equipment). We exclude plants with non-positive capital, as well as those with material cost-to-revenue ratios below zero, above one, or in the bottom and top two percentiles of the distribution, in order to remove observations that are inconsistent with the production model or likely to reflect reporting errors and extreme measurement noise.

Result

Tables (ref) and (ref) report estimates of the structural parameters for the Chilean manufacturing industries obtained using our method. Notably, the estimates of $\beta$ is statistically significantly different from zero. Since the HSA demand system nests the CES demand system as the special case $\beta=0$, this provides evidence against the CES specification in favor of the more flexible HSA demand system in Chilean manufacturing industries. Appendix (ref) shows that these findings are robust under decreasing returns to scale.

Figures (ref) and (ref) show scatter plots of observed revenue $r_{it}$ against fitted revenue from the HSA demand system in Step 4, along with quantile--quantile plots comparing the estimated demand shocks $\epsilon_{it}$ from Step 4 with the corresponding nonparametric estimates $u_{it}$ from Step 1. Observed revenue aligns closely with fitted revenue under the parametric HSA restriction, and the quantiles of $\epsilon_{it}$ generally track those of $u_{it}$ along the 45-degree line. While the fit is not perfect, these patterns provide supportive evidence for the HSA demand specification, particularly for Industries 31 and 32. Appendix (ref) shows that these patterns are preserved under decreasing returns to scale.

Counterfactual Welfare Analysis

Using the estimated HSA demand system, we quantify the consumer utility loss attributable to firm's market power by calculating the compensation variation for a counterfactual marginal cost pricing equilibrium (MCPE) as described in Section (ref).

In our counterfactual analysis, we hold fixed the structural primitives: the production function $f_{t}$, the demand shock distribution $G_{u}$, and the structural taste parameter $\gamma_{t}$. The counterfactual equilibrium under a marginal cost pricing equilibrium (MCPE) recomputes the aggregate index $\Delta q_{t}^c$ and firm-level outcomes, ensuring that the equilibrium condition ((ref)). Specifically, we implement the following procedure.

First, we calculate a monopolistic competition equilibrium (MCE), utilizing the estimated structural parameters from Tables (ref) and (ref) and the firm-level states $(k_{it},l_{it},\hat{\omega}_{it},\hat{\epsilon}_{it})$. Since the observed output values do not exactly satisfy the equilibrium conditions ((ref))---due to model misspecification and estimation error for $\gamma_{t}$---we recompute the equilibrium output vector $\{y_{it}^{m}\}_{i=1}^{n}$ and parameter $\gamma_{t}^{new}$ under the normalization $q_{t}^{m}=0$ by jointly solving the conditions in ((ref)), which are simplified as:

where $\Xi_{it}=\ln\hat{\theta}_{m}+(\hat{\theta}_{k}k_{it}+\hat{\theta}_{l}l_{it}+\hat{\omega}_{it})/\hat{\theta}_{m}$ and $p_{t}^{m}$ is normalized to zero. The resulting output vector and parameters are fully consistent with our HSA demand system, which mitigates model misspecification bias and addresses estimation errors in the counterfactual analysis.

Second, we consider a marginal cost pricing equilibrium (MCPE) for given counterfactual log income $\Phi_{t}^{c}$. We find an output vector $\mathbf{y}_{t}^{c}(\Phi_{t}^{c})$ and quantity index $\Delta{q}_{t}^{c}(\Phi_{t}^{c})$ by solving ((ref)), which are simplified as:

We assume the same income case $\Phi_{t}^{c}=\Phi_{t}$ for our benchmark, while also analyzing how potential changes in income might affect the equilibrium outcome.

To calculate the compensating variation for transitioning from an MCE to an MCPE, we solve for the counterfactual income $\Phi_{t}^{c*}$ that results in a zero utility change, as defined in ((ref)):

The compensating variation is then calculated as $CV_{t}:=\exp(\Phi_{t}^{c*})-\exp(\Phi_{t})$, which quantifies the consumer's welfare change from firm's market power.

Finally, we calculate firms' profit loss. In the case of $\Phi_{t}^{c}=\Phi_{t}$, the total profit change ((ref)) is expressed as

since $\chi_{it}\left(y_{it}\right)=(y_{it}-\hat{\theta}_{k}k_{it}-\hat{\theta}_{l}l_{it}-\hat{\omega}_{it})/\hat{\theta}_{m}$.

From Table (ref), we found empirical evidence that under our HSA demand system market power in these industries results in consumer's welfare losses of approximately 10%--15% and profit gains of approximately 4%--11%, with overall welfare losses of 3%--6% of industry revenue in the three largest Chilean manufacturing industries in 1996. To put these findings in context, our estimates far exceed the classic Harberger-triangle calculation of less than 0.1% of GDP harberger1954. They are more consistent with recent general-equilibrium analyses that find substantially larger welfare costs. baqaee2020productivity estimate that eliminating misallocation would raise output by approximately 15%, while edmond2023costly find welfare costs of variable markups of around 7.5% in consumption-equivalent terms. Our estimates, though derived from a different structural framework, are comparable in magnitude---reinforcing the view that the welfare costs of imperfect competition are economically significant. Appendix (ref) shows that these estimates are broadly robust under moderate decreasing returns to scale of 0.9.

Concluding Remarks

This paper develops constructive nonparametric identification of production functions and markups from revenue data, simultaneously addressing the two fundamental challenges in production function estimation since ma44ecma when revenue is used as output: input-TFP correlation and markup heterogeneity bias. By modeling revenue as a function of output, observed characteristics, and an unobserved demand shock, we identify various economic objects of interest under standard assumptions. Our semiparametric estimator, implementable with standard datasets, performs well in simulation. Applied to Chilean manufacturing plant data, we find evidence of CES demand misspecification and estimate that market power generates welfare losses of approximately 3%--6% of industry revenue in the three largest manufacturing industries in 1996.

While our identification results establish that firm-level revenue data suffice to recover production functions and consumer demand without physical quantity data, the proof relies on structural assumptions that merit discussion. First, we assume perfectly competitive input markets; monopsony power would make input prices firm-specific and endogenous, breaking the link between expenditure shares and marginal products. Second, we assume Hicks-neutral, scalar productivity; factor-augmenting technological change requires additional structure or data for identification. Third, we assume monopolistic competition; in oligopolistic markets, strategic interaction makes the first-order conditions underlying our markup identification insufficient without a fully specified equilibrium model. Relaxing any of these assumptions is an important direction for future work.