Nonparametric Identification of Production Function, Total Factor Productivity, and Markup from Revenue Data

Nonparametric Identification of Production Function, Total Factor Productivity, and Markup from Revenue Data

Introduction

The estimation of production function and markup is a core tool used in empirical analyses of market outcomes.\footnote{griliches_mairesse_1999 and \citet*{ACKERBERG20074171} provide excellent surveys on production function estimations.} The residual of an estimated production function, total factor productivity (TFP), is widely used to measure firm-level technological efficiency (see bartelsman2000understanding and syverson2011determines for recent surveys) and its contribution to aggregate efficiency (e.g., olley1996dynamics). Researchers often estimate the elasticity of production functions to analyze technological changes (e.g., van2003productivity; doraszelski2018measuring) and price markups over marginal costs (e.g., hall1988relation; de2012markups). The estimation of firm-level markup via production function has been widely applied in various topics and complements markup estimation via demand function (e.g., \citealp*{blp1995ecta}) in economic analysis of firm's market power.

Commonly used methods of production function and markup estimation assume that a firm's output quantity can be observed as data. However, typical firm-level datasets contain only revenue, not output quantity. Therefore, in practice, many applications use revenue deflated by an industry-level price deflator as output.\footnote{A few studies use firm-level datasets that include output quantity (e.g., \citealp*{foster2008reallocation}; \citealp*{de2016prices}; lu2015trade; nishioka2019measuring). However, those quantity datasets are available only for a limited number of countries, industries, and years, and they are not easily accessible to all researchers.} For production function estimation, this practice may be justified under perfect competition where an output price is exogenous and identical across firms. However, ever since ma44ecma's pioneering study, several researchers have voiced cautions and suggested that the practice may not be justified under imperfect competition; they show that using revenue as output can significantly bias the identification of production functions (e.g., \Citealp{klette1996jae}; \citealp*{de2011product}) and TFP (e.g., \citealp*{foster2008reallocation}; \citealp*{katayama2009firm}; \citealp*{de2011product}). Furthermore, as shown in \citealp*{bond2020some}, using revenue in place of output quantity may lead to serious biases in estimation of firm's markups. Despite such criticism, the practice of using revenue in place of output quantity persists in many applications given a lack of output quantity data.

In the existing literature, it is not known whether identifying production functions and markups from firm-level revenue data is possible without imposing parametric assumptions. This paper contributes to the literature of production function and markup estimation by establishing nonparametric identification of production function, TFP, and markup from revenue data. The proof is constructive and the required assumption is similar to the standard assumption in the production function literature except that we impose additional assumptions on firm's demand function.

Following ma44ecma, klette1996jae and de2011product, we explicitly model a demand function that an individual firm faces as a function of its output and observable characteristics that are excluded from the production function.\footnote{\citet*{de2020rise} study an alternative approach using an exogenous variable to remove output price variation from revenue data. } While each of these earlier studies examines a demand function with a constant and identical demand elasticity\textemdash something that implies identical markups across firms\textemdash we consider a general nonparametric demand function that generates rich heterogeneity in various firm-level outcomes, including markups; for this reason, we can address the bias from markup heterogeneity across firms that the literature has criticized. In other respects, our method requires the standard assumptions and can be implemented using typical data found in empirical applications.

We develop a three-step identification approach that combines the control function approach developed by olley1996dynamics, levinsohn2003estimating, and \citet*{ackerberg2015identification} and the first-order condition approach recently developed by \citet*{gandhi2020identification}.\footnote{These approaches assumed quantity data or perfect competition. gandhi2020identification also examined an imperfect competition with a constant elastic demand as in klette1996jae and de2011product where markups must be constant and identical across firms. } Following levinsohn2003estimating and ackerberg2015identification, the inverse function of a material demand function serves as a control function for TFP. In the first step, we identify revenue as a function of inputs and observable demand shifters by using the control function; this first step corresponds to that of ackerberg2015identification. Our novel second step identifies the control function for TFP by applying the nonparametric identification of transformation models (e.g., horowitz1996semiparametric) examined by \citet*{ekeland2004identification} and \citet*{chiappori2015nonparametric}. By identifying the control function, TFP is identified (up to normalization) from the dynamics of inputs, without output data. In the third step, we identify a production function, markup, and a demand function, using the first-order condition for the material and the control function identified in the second step.

Our method identifies various objects from the revenue data. In our main setting, markup and output elasticities are identified up to scale; an output price, an output quantity, a gross production function, and TFP are identified up to scale and location. Identification is cross-sectional so that the identified objects can vary over time. With an additional assumption of local constant returns to scale, we identify the levels of markup and output elasticities; we may also identify an output price, an output quantity, a production function, and TFP up to location.\footnote{\citet*{flynn2019measuring} used global constant returns to scale to identify a production function. In subsection (ref), we clarify local and global constant returns to scale.} Finally, if we are willing to assume monopolistic competition (without imposing free entry), we further identify a demand system and a utility function of a representative consumer\textemdash specifically, matsuyama2017beyond's homothetic demand system with a single aggregator (HSA)\textemdash that can be used for a counter-factual analysis and a welfare analysis.\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. Section (ref) summarizes previous studies on how using revenue as output could bias the identification of production function, TFP, and markup; readers familiar with the literature can skip this section and proceed to Section (ref). Subsection (ref) explains our setting, and subsection (ref) demonstrates our three-step approach by offering a parametric example. Subsection (ref) presents our nonparametric identification results, and subsection (ref) discusses additional assumptions for fixing scale and location normalization. Subsection (ref) examines the identification of a demand system and a representative consumer's utility function. Both subsection (ref) and the Appendix present identification results in alternative settings, including endogenous labor input, endogenous firm-level observable demand shifters, unobservable demand shifters, and i.i.d. productivity shocks. Section (ref) provides concluding remarks.

Biases from Using Revenue as Output Quantity

This section summarizes possible biases in the identification of production function, TFP, and markup when revenue is used as an output quantity. We denote the logarithms of the price, output, and revenue of firm $i$ at time $t$ as $p_{it}$, $y_{it}$, and $r_{it}:=p_{it}+y_{it}$, respectively. Suppose that these variables are related via the inverse demand function $p_{it}=\psi_{it}(y_{it})$ and the revenue function $r_{it}=\varphi_{it}(y_{it}):=y_{it}+\psi_{it}(y_{it})$. Let $y_{it}=f_{t}(m_{it},k_{it},l_{it})+\omega_{it}$ be firm $i$'s production function where $\omega_{it}$ is TFP and $x_{it}:=(m_{it},k_{it},l_{it})$ is a vector of the logarithms of material, capital, and labor, respectively. To highlight the sources of biases from using revenue as output, assume that TFP is identical across firms within time $t$, with $\omega_{it}=\omega_{t}$ for all $i$. This simplification eliminates an additional and well-known source of bias, correlations between inputs and TFP.

From the first-order condition for profit maximization, $P_{it}\left(1+\psi_{it}'(y_{it})\right)=MC_{it}$, the elasticity of revenue with respect to output is equal to the inverse of markup:

Under perfect competition where $P_{it}=MC_{it}$, the variation in revenue across firms coincides with that of output. However, they are generally different when markups vary across firms.

Suppose that, using revenue as output, a researcher identifies a true relationship between revenue and inputs, $\tilde{\varphi}_{it}(x_{it}):=\varphi_{it}(f_{t}(x_{it})+\omega_{t})$ to use $\tilde{\varphi}_{it}(x_{it})$ as a proxy for $f_{t}(x_{it})$. Prior studies show that the use of revenue as output could cause biases in three forms. First, ma44ecma and klette1996jae establish that, from ((ref)), the elasticity of $\tilde{\varphi}_{it}(x_{it})$ relates to the true elasticity of $f_{t}(x_{it})$ via markup:

Thus, output elasticities would be underestimated by the extent of markup.

Second, katayama2009firm and de2011product demonstrated a bias in TFP estimates. Let $d\omega_{t}$ be a TFP change. Suppose that a TFP change for firm $i$ is estimated as a change in revenue with inputs being fixed, $d\tilde{\omega}_{it}=\left.d\tilde{\varphi}_{it}(x_{it})\right|_{dx_{it}=0}.$ From ((ref)), we see that this TFP estimate relates to the true TFP change via markup:

Therefore, TFP would be underestimated by the extent of markup.

Finally, bond2020some show that markup estimates using the method of hall1988relation and de2012markups are generally biased when revenue elasticity is used in place of output elasticity. Suppose a firm is a price-taker of flexible input $v$. hall1988relation and de2012markups developed the following equation relating to markup and output elasticity with respect to $v$ as:

where $\alpha_{it}^{v}$ is the ratio of expenditure on input $v$ to revenue. If a researcher uses $\partial\tilde{\varphi}_{it}(x_{it})/\partial m_{it}$ instead of $\partial f_{t}(x_{it})/\partial m_{it}$ in markup equation ((ref)), then from ((ref)), the estimated markup is 1:

In such a case, the markup would be underestimated.\footnote{Result ((ref)) by bond2020some relies on the assumption that a researcher can correctly identify $\tilde{\varphi}_{it}(x_{it})$. In practice, misspecification of $\tilde{\varphi}_{it}(x_{it})$ could derive markup estimates ((ref)) that contain some information on true markups. For instance, de2012markups (2012, Section VI) show that when $f$ is Cobb\textendash Douglas, it is possible to identify the effect of firm-level variables (e.g., export) on markups.}

klette1996jae and de2011product developed methods by which to identify production functions from revenue data, by assuming a constant elastic demand function with an identical elasticity.\footnote{katayama2009firm also developed a method by which to identify production functions from revenue data. Their method allows for markup heterogeneity but requires the ability to estimate firm's marginal costs from total costs. } However, with this specific demand function, markups must be constant and identical across firms. Studies estimating markups from quantity data report substantial heterogeneity in markups across firms (e.g., \citealp*{de2016prices}; lu2015trade; nishioka2019measuring). To address the biases arising from firm-level markup heterogeneity, we extend the approach of klette1996jae and de2011product by incorporating a general nonparametric demand function that allows for variable and heterogeneous markups.

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 $y_{it}$ relates to inputs $x_{it}=(m_{it},k_{it},l_{it})'$ via the production function:

where the firm's TFP $\omega_{it}$ follows an exogenous first-order stationary Markov process given by

where we assume that neither $h(\cdot)$ nor the marginal distribution of $\eta_{it}$ change over time.\footnote{$h(\cdot)$ can include a firm's observable exogenous characteristics.}

The demand function for a firm's product is strictly decreasing in its price, and its inverse demand function is given by

where $p_{it}$ is the logarithm of output price and $z_{it}\in\mathcal{Z}$ is an observable firm characteristic that affects firm's demand (e.g., export status in de2012markups). $z_{it}$ can be either a continuous or discrete vector; in the main text below, $z_{it}$ is assumed to be continuous and exogenous\textemdash that is, $z_{it}\perp\eta_{it}$. In subsection (ref) and the Appendix, we present the identification results when $z_{it}$ is discrete and/or may correlate with $\eta_{t}$.

The inverse demand function ((ref)) generalizes the constant elastic demand function examined by ma44ecma, klette1996jae and de2011product. Although $\psi_{t}$ is nonparametric, ((ref)) implicitly makes two assumptions. First, $\psi_{t}(\cdot)$ is a common function for all firms once the observed characteristics $z_{it}$ are controlled for. This implies that unobserved demand shifters must be common for all firms\textemdash that is, $\psi_{t}$ can be written as $\psi_{t}(y_{it},z_{it},A_{t})$ where $A_{t}$ is a vector of unobserved variables and can include an aggregate price/quantity index. In subsection (ref), we discuss the case where $\psi_{t}(\cdot)$ includes a firm-level unobservable demand shifter such as quality. Second, $\psi_{t}(\cdot)$ represents a demand curve that an individual firm takes as given. This is satisfied in the case of monopolistic competition (without free entry) where each firm takes $A_{t}$ as given.

Let $\bar{r}_{it}$ and $\mathcal{\bar{R}}$ be the logarithm of (true) revenue and its support, respectively. Revenue $r_{it}$ in the data is observed with a measurement error $\varepsilon_{it}$, $r_{it}=\bar{r}_{it}+\varepsilon_{it}$. Then, from ((ref)), the observed revenue relates to output and input as follows:

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

We assume that $l_{it}$ and $k_{it}$ are predetermined at the end of the last period $t-1$, while $m_{it}$ is flexibly chosen after observing $\omega_{it}$.\footnote{In subsection (ref), we present identification when $l_{it}$ also correlates with $\omega_{it}$.} Specifically, $m_{it}=\mathbb{M}_{t}\left(\omega_{it},k_{it},l_{it},z_{it}\right)$ is chosen at time $t$ by:

where $p_{t}^{m}$ denotes the logarithm of the material input price at time $t$, which is common to all firms. A firm is assumed to be a price-taker for material input.

Equation ((ref)) highlights two identification issues raised by ma44ecma. First, $m_{it}$ correlates with the unobservable $\omega_{it}$. Second, $r_{it}$ relates to $x_{it}=(m_{it},k_{it},l_{it})$ via two unknown nonlinear functions $\varphi_{t}(\cdot,z_{it})$ and $f_{t}(\cdot)$, and two unobservables $\omega_{it}$ and $\varepsilon_{it}$.\footnote{In subsection (ref) and the Appendix, we present identification when a firm receives an i.i.d. shock $e_{it}$ to output and then, the firm's revenue includes a non-additive error, $r_{it}=\varphi_{t}(f_{t}(x_{it})+\omega_{it}+e_{it},z_{it})$.} To address these issues via a control function and a transformation model, we first make the following assumptions.

Assumptions (ref) (a) and (b) are standard assumptions about smooth production and demand functions. Assumption (ref) (b) $\partial\varphi_{t}(y,z)/\partial y>0$ is equivalent to that the elasticity of demand with respect to price, $-\left(\partial\psi_{t}(y,z)/\partial y\right)^{-1}$, is greater than 1; this necessarily holds under profit maximization. Therefore, Assumption (ref) (b) is innocuous as long as we analyze the outcomes of 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.

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

In the following, we identify $\varphi_{t}^{-1}\left(\cdot\right)$, $f_{t}(\cdot)$, and $\mathbb{M}_{t}^{-1}(\cdot)$ from the distribution of variables in the data. Let $v_{t}:=(k_{t},l_{t},z_{t},x_{t-1},z_{t-1})'\in\mathcal{V}:=\mathcal{K}\times\mathcal{L}\times\mathcal{Z}\times\mathcal{X}\times\mathcal{Z}$. Data includes a random sample of firms $\{r_{it},v_{it}\}_{i=1}^{N}$ from the population. For instance, the variable $x_{it}$ of firm $i$ is considered as a realization of the random variable $x_{t}$. Given a sufficiently large $N$, an econometrician can recover their joint distributions.

Assumption (ref) (a) is required for the identification of $\mathbb{M}_{t}^{-1}(\cdot)$. Assumptions (ref) (b) and (c) are additionally required for the identification of $\varphi_{t}^{-1}\left(\cdot\right)$ and $f_{t}(\cdot)$. Typical production datasets include those variables in Assumption (ref).

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

Therefore, identification requires location and scale normalization. We 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 $(m_{t1}^{*},k_{t}^{*},l_{t}^{*},z_{t}^{*})$ and $(m_{t0}^{*},k_{t}^{*},l_{t}^{*},z_{t}^{*})$ on the support $\mathcal{X}\times\mathcal{Z}$ where $m_{t0}^{*}<m_{t1}^{*}$, we denote

Note that $\partial\mathbb{M}_{t}^{-1}/\partial m_{t}>0$ 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 $b_{t}=\left(c_{3t}-c_{2t}\right)/\left(\mathbb{M}_{t}^{*-1}(m_{t1}^{*},k_{t}^{*},l_{t}^{*},z_{t}^{*})-\mathbb{M}_{t}^{*-1}(m_{t0}^{*},k_{t}^{*},l_{t}^{*},z_{t}^{*})\right)$, $a_{1t}=c_{1t}-b_{1t}f_{t}^{*}(m_{t0}^{*},k_{t}^{*},l_{t}^{*})$ and $a_{2t}=c_{2t}-b_{1t}\mathbb{M}_{t}^{*-1}(m_{t0}^{*},k_{t}^{*},l_{t}^{*},z_{t}^{*})$. 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.

Identification in a Parametric Example

Before presenting the nonparametric identification results, we demonstrate our identification approach by applying it to a simple parametric example. Consider a monopolistically competitive market where each firm $i$ faces the following constant elastic inverse demand function:

where $\alpha_{t}(z_{it})$ and $0<\rho(z_{it})\le1$ are unknown parameters.\footnote{The demand function ((ref)) can be derived from a constant elasticity of substitution (CES) utility function; $a_{t}(z_{t})$ implicitly includes aggregate expenditure and an aggregate price index.} The markup equals $1/\rho(z_{it})$ and depends on the exogenous scalar $z_{it}\in\mathcal{Z}:=\{1,0\}$ such that $z_{it}\perp\eta_{it}$. Firm $i$ has a Cobb\textendash Douglas production function and $\omega_{it}$ follows a first-order autoregressive (AR(1)) process:

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

The first-order condition for ((ref)),

determines the control function for $\omega_{it}$ as

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

For notational brevity, assume that the support $\mathcal{X}\times\mathcal{Z}$ includes two points $(m_{t1}^{*},k_{t}^{*},l_{t}^{*},z_{t}^{*})=(0,0,0,0)$ and $(m_{t0}^{*},k_{t}^{*},l_{t}^{*},z_{t}^{*})=(1,0,0,0)$. 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 $\theta_{0}=0$, $\beta_{t}(0)=0$, and $\beta_{m}(0)=1$.

Our identification approach follows three steps.

\paragraph{Step 1: Identification of Measurement Errors}

The first step removes the measurement error $\varepsilon_{it}$ in the spirit of ackerberg2015identification. Substituting ((ref)) into ((ref)) and using $\theta_{0}=0$, we obtain two expressions of $r_{it}$ as follows:

where $\phi(z_{it}):=\alpha_{t}(z_{it})+\rho(z_{it})\beta_{t}(z_{it})$. Applying the conditional moment restriction $E[\varepsilon_{it}|m_{t},z_{t}]=0$ for the second expression ((ref)), we identify $\phi(z_{it})$, $\bar{r}_{it}$ and $\varepsilon_{it}$ by \[ \phi(z_{t})=E[r_{it}-m_{it}|m_{t},z_{t}],\bar{r}_{it}=\phi(z_{it})\text{ and }\varepsilon_{it}=r_{it}-m_{it}-\phi(z_{it}). \]

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

where

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

From ((ref)) and ((ref)), we identify the parameters of the control function (under the normalization ((ref))) as:

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

The final step identifies the parameters of the demand and production functions. Comparing the two expressions of $r_{it}$ in ((ref)) and ((ref)), we obtain the following relationships:

Given that $(\beta_{t}(z_{t}),\beta_{m}(z_{t}),\beta_{k},\beta_{l})$ are identified in step 2, the first line in ((ref)) contains four equations (two equations for two values of $z_{it}\in\{0,1\}$) and five parameters $(\alpha_{t}(0),\alpha_{t}(1),\rho(0),\rho(1),\theta_{m})$. Therefore, to identify these parameters, we need a further restriction.

Following gandhi2020identification, we use as an additional restriction the first-order condition for material ((ref)). The first-order condition ((ref)) implies that the revenue share of material expenditure on the right hand side of ((ref)) is a function of $z_{it}$. Using $\varepsilon_{it}$, we obtain the revenue share of material expenditure $\exp(p_{t}^{m}+m_{it})/\exp(r_{it}-\varepsilon_{it})$ and identify it as a function of $z_{it}$ by taking its expectation conditional on $z_{it}$: \[ s(z_{t}):=E\left[\left.\frac{\exp(p_{t}^{m}+m_{it})}{\exp(\bar{r}_{it})}\right|z_{t}\right]. \] Then, we obtain an additional restriction on the parameters:

From ((ref)) and ((ref)), we identify the parameters of the demand and production functions as follows

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

The above identification argument is illustrative, but it relies on the linearity of $\mathbb{M}_{t}^{-1}(m_{it},k_{it},l_{it},z_{it})$ in $m_{it}$, which holds only under restrictive parametric assumptions. Extending the argument, the following subsection establishes nonparametric identification.

Nonparametric Identification

Step 1: Identification of Measurement Error

The first step removes the measurement error $\varepsilon_{it}$. Substituting the control function $\omega_{it}=\mathbb{M}_{t}^{-1}(m_{it},k_{it},l_{it},z_{it})$, the revenue function ((ref)) can be written as:

where $\phi_{t}(x_{t},z_{t}):=\varphi_{t}\left(f(x_{t})+\mathbb{M}_{t}^{-1}\left(x_{t},z_{t}\right),z_{t}\right)$. From Assumption (ref), $\phi_{t}(\cdot)$ is continuously differentiable. From $E\left[\varepsilon_{it}|x_{t},z_{t}\right]=0$, we can identify $\phi_{t}(\cdot)$, $\bar{r}_{it}$, and $\varepsilon_{it}$ as:

Hereafter, $\phi_{t}(\cdot)$, $\bar{r}_{it}$, and $\varepsilon_{it}$ are assumed to be known.\footnote{As will be shown, $\omega_{it}$ is identified in step 2 independently of step 1. Therefore, one can think of an alternative approach that first identifies $\omega_{it}$ and then regresses $r_{it}$ on $(x_{it},z_{it},\omega_{it})$ to obtain $E\left[r_{it}|x_{it},z_{it},\omega_{it}\right]$ instead of $E\left[r_{it}|x_{it},z_{it}\right]$. However, it is not possible to identify $E\left[r_{it}|x_{it},z_{it},\omega_{it}\right]$ because $\omega_{it}=\mathbb{M}_{t}^{-1}(x_{it},z_{it})$ is a deterministic function of $(x_{it},z_{it}$). Once $(x_{it},z_{it})$ are conditioned, there is no remaining source of variation in $\omega_{it}$.}

Step 2: Identification of Control Function and TFP

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

where $\bar{h}_{t}\left(x_{t-1},z_{t-1}\right):=h\left(\mathbb{M}_{t-1}^{-1}(m_{t-1},k_{t-1},l_{t-1},z_{t-1})\right)$. As $\partial\mathbb{M}_{t}^{-1}/\partial m_{it}>0$, given the values of $(k_{it},l_{it},z_{it})$, 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\textendash A3, A5, and A6 in chiappori2015nonparametric.\footnote{Assumption (ref) (c) corresponds to Assumption A4 of chiappori2015nonparametric.}

We can relax Assumption (ref)(b) by allowing $z_{t}$ and $l_{t}$ to correlate with $\eta_{t}$, which we discuss this in subsection (ref). Assumption (ref)(d) holds without loss of generality because we can choose any two points on the support of $\omega_{t}$ without changing the essence of our argument. Assumption (ref)(f) can be interpreted as a generalized rank condition, thus implying that a given exogenous variable $q_{t-1}$ has a causal impact on $(m_{t},k_{t},l_{t},z_{t})$. Suppose $g_{\eta}\left(\eta\right)>0$ for all $\eta\in\mathbb{R}$. Then, as will be shown below (in ((ref))), Assumption (ref)(f) holds if and only if

for some $(\tilde{x}_{t-1},\tilde{z}_{t-1})$ and some $q_{t-1}\in\{k_{t-1},l_{t-1},m_{t-1},z_{t-1}\}$. This condition is equivalent to (1) $\omega_{t-1}$ has a causal impact on $\omega_{t}$ ($h'(\omega_{t-1})\neq0$) and (2) $q_{t-1}$ has a causal impact on $m_{t-1}$, ($\partial\mathbb{M}_{t-1}/\partial q_{t-1}\neq0$). These conditions must be satisfied for at least one exogenous variable $q_{t-1}$ and some point $(\tilde{x}_{t-1},\tilde{z}_{t-1})$.

Proposition (ref) shows that the control function is identified from the distribution of $(m_{it},v_{it})$.

Step 3: Identification of Production Function and Markup

The final step identifies production function, markup and other remaining objects. From $\bar{r}=\phi_{t}(x_{t},z_{t})=\varphi_{t}(f_{t}(x_{t})+\mathbb{M}_{t}^{-1}\left(x_{t},z_{t}\right),z_{t})$ and the monotonicity of $\varphi_{t}$, differentiating $\varphi_{t}^{-1}(\phi(x_{t},z_{t}),z_{t})=f_{t}(x_{t})+\mathbb{M}_{t}^{-1}\left(x_{t},z_{t}\right)$ with respect to $q_{t}\in\{m_{t},k_{t},l_{t}\}$ and $z_{t}$ gives:

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

Rearranging the first-order condition, we obtain the Hall-De Loecker-Warzynski markup equation:

We establish the following proposition.

The output price for individual firms is identified as

Comparison to Existing Identification Approaches

Our approach follows the spirits of existing identification approaches, but it does differ from them in terms of implementations. First, step 2 distinguishes our approach from the standard control function approach (e.g., ackerberg2015identification). In step 2, we identify the control function from the dynamics of the inputs, and without using any output measure. To clarify why this approach is necessary, consider an alternative approach that uses an output measure. Specifically, in the second step, we substitute $\omega_{it}=\varphi_{t}^{-1}(\bar{r}_{it},z_{it})-f_{t}(x_{it})$ into ((ref)) and obtain the alternative transformation model: \[ \varphi_{t}^{-1}(\bar{r}_{it},z_{it})=f_{t}(x_{it})+\tilde{h}_{t}(\bar{r}_{it-1},x_{it-1},z_{t-1})+\eta_{it} \] where $\tilde{h}_{t}(\bar{r}_{t-1},x_{t-1},z_{t-1}):=h(\varphi_{t-1}^{-1}(\bar{r}_{t-1},z_{t-1})-f_{t}(x_{t-1}))$. Since this model also belongs to the class of transformation models examined by chiappori2015nonparametric, one might think that we could have identified $\varphi_{t}(\cdot)$ and $f_{t}(\cdot)$ from the conditional distribution function $G_{\bar{r}_{t}|w_{t}}(\bar{r}_{t}|w_{t})$ of $\bar{r}_{t}$ given $w_{t}:=(x_{t},z_{t},\bar{r}_{t-1},x_{t-1},z_{t-1})$. This is not possible, however, because once $(x_{t},z_{t})$ is conditioned on, $\bar{r}_{t}=\phi_{t}(x_{t},z_{t})$ loses all variations. Therefore, the derivatives of $G_{\bar{r}_{t}|w_{t}}$ with respect to past variables $(\bar{r}_{t-1},x_{t-1},z_{t-1})$ are always 0, which violates the condition corresponding to Assumption (ref) (f).

Second, ackerberg2015identification identify a structural value-added function, $y_{it}=v_{t}(k_{it},l_{it})+\omega_{it}$, which under perfect competition derives from a Leontief production function $y_{it}=\min\left\{ v_{t}(k_{it},l_{it})+\omega_{it},a+m_{it}\right\} $. However, the structural value-added function is difficult to employ under imperfect competition because $y_{it}<v_{t}(k_{it},l_{it})+\omega_{it}$ can occur. Note that the maximum output capacity $y_{it}^{*}:=v_{t}(k_{it},l_{it})+\omega_{it}$ is determined before a firm chooses $m_{it}$ and $y_{it}$. Therefore, if $y_{it}^{*}$ is large\textemdash due, for example, to a large shock on $\omega_{it}$\textemdash then the profit maximizing output $y_{it}$ can be lower than $y_{it}^{*}$.\footnote{As ackerberg2015identification explains, under perfect competition, if $y_{it}<y_{it}^{*}$, then the optimal output is 0 since the output becomes linear in material. Since firms in a dataset have positive outputs, $y_{it}=y_{it}^{*}$ holds for firms observed in a dataset. However, under imperfect competition, it is possible to have $y_{it}<y_{it}^{*}$ and the optimal output is strictly positive. } Intuitively speaking, when increases in TFP double, a firm can preclude a price drop by increasing its output by less than double.

Third, our approach uses the first-order condition for material in a way different from that seen in gandhi2020identification, whose step identifies the material elasticity $\partial f_{t}(x_{t})/\partial m_{t}$ from the first-order condition ((ref)): \[ \ln\text{\ensuremath{\frac{\exp(p_{t}^{m}+m_{it})}{\exp\left(r_{it}\right)}}}=\ln\frac{\partial f_{t}(x_{it})}{\partial m_{it}}-\ln\frac{\partial\varphi_{t}^{-1}(r_{it}-\varepsilon_{it},z_{it})}{\partial r_{t}}-\varepsilon_{it} \] under the assumption of perfect competition where $\ln\partial\varphi_{t}^{-1}(r_{it}-\varepsilon_{it},z_{it})/\partial r_{it}=0$ for all $i$. Under imperfect competition, when the markup depends on revenue $r_{it}-\varepsilon_{it}$, $\partial f_{t}(x_{t})/\partial m_{t}$ cannot be identified solely from the first-order condition.

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\textemdash 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 \[ \frac{\partial\varphi_{t+1}^{-1}(r,z)/\partial r}{\partial\varphi_{t}^{-1}(r,z)/\partial r}=\frac{b_{t+1}}{b_{t}}\frac{\partial\varphi_{t+1}^{*-1}(r,z)/\partial r}{\partial\varphi_{t}^{*-1}(r,z)/\partial r}. \] 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: \[ \frac{\partial f_{t+1}(x)/\partial q}{\partial f_{t}(x)/\partial q}=\frac{b_{t+1}}{b_{t}}\frac{\partial f_{t+1}^{*}(x)/\partial q}{\partial f_{t}^{*}(x)/\partial q}\text{ and }\frac{\omega_{it+1}-E\left[\omega_{it+1}\right]}{\omega_{it}-E\left[\omega_{it}\right]}=\frac{b_{t+1}}{b_{t}}\left(\frac{\omega_{it+1}^{*}-E\left[\omega_{it+1}^{*}\right]}{\omega_{it}^{*}-E\left[\omega_{it}^{*}\right]}\right) \] for $q\in\{m,k,l\}$.

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.

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 it is weaker than the assumptions used in some other studies on markups. Markup is sometimes estimated as the ratio of revenue $\exp(r_{it})$ to total costs $TC_{it}$ under the assumption that a cost function is linear in output $TC_{it}=MC_{it}y_{it}$ with constant marginal costs $MC_{it}$. The linear cost function requires the following assumptions that are stronger than Assumption (ref): (1) constant returns to scale globally holds for all $x\in\mathcal{B}$; (2) all three inputs are flexible and (3) a firm is a price taker of all three inputs. Under Assumption (ref), marginal costs may increase in output, especially in the short run, when dynamic inputs such as capital require adjustment costs.

With Assumption (ref), the scale normalization parameter $b_{t}$ can be identified for all periods as follows. Let $f_{t}(x)$ be the identified production function and $f_{t}^{*}(x)$ be the true one where $f_{t}(x_{t})=a_{t}+b_{t}f_{t}^{*}(x_{t})$ from ((ref)). For $x\in\mathcal{B}$, we have \[ b_{t}=b_{t}\left(\frac{\partial f_{t}^{*}(x)}{\partial m}+\frac{\partial f_{t}^{*}(x)}{\partial k}+\frac{\partial f_{t}^{*}(x)}{\partial l}\right)=\frac{\partial f_{t}(x)}{\partial m}+\frac{\partial f_{t}(x)}{\partial k}+\frac{\partial f_{t}(x)}{\partial l}. \] 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\textemdash 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; this condition is usually 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.

Identification of Demand System and Utility Function

Given that we have identified each firm's output price and quantity, it is possible to identify with additional assumptions a system of demand functions and a homothetic utility function of a representative consumer. The identified demand system and the identified utility function can be used to undertake counterfactual analysis and welfare analysis.

We consider an HSA system matsuyama2017beyond, which can be expressed as a system of direct demand functions or of inverse demand functions. The two systems are self-dual in the sense that either can be derived from the other. We consider a system of inverse demand functions. Let $P_{it}:=\exp(p_{it})$ and $Y_{it}:=\exp(y_{it})$ be the levels of price and quantity of firm $i$'s output at time $t$, respectively. Let $N_{t}$ be the set of firms in the industry and $\Phi_{t}:=\sum_{i\in N_{t}}P_{it}Y_{it}$ be the industry expenditure. The inverse demand function for product $i$ is given by \[ P_{it}=\frac{\Phi_{t}}{Y_{it}}S_{t}\left(\frac{Y_{it}}{A_{t}\left(\mathbf{Y}_{t},\mathbf{z}_{t}\right)},z_{it}\right). \] where $S_{t}(\cdot,z_{it})$ provides the budget share of product $i$, $\mathbf{Y}_{t}:=(Y_{1t},...,Y_{Nt})\in\mathscr{\bar{Y}}:=\exp\left(\mathcal{Y}\right)^{N}$ is a vector of consumption, $\mathbf{z}_{t}:=(z_{1t},...,z_{Nt})$ is a vector of observable demand shifters and $A_{t}(\mathbf{Y}_{t},\mathbf{z}_{t})$ is the aggregate quantity index summarizing interactions across products.\footnote{If the utility function is CES $U_{t}(\mathbf{Y}_{t},\mathbf{z}_{t})=\left[\sum_{i=1}^{N}Y_{it}^{\rho(z_{it})}\right]^{1/\rho(z_{it})}$, then the inverse demand function is given by $P_{it}=\frac{\Phi_{t}}{Y_{it}}\left(\frac{Y_{it}}{U_{t}(\mathbf{Y}_{t},\mathbf{z}_{t})}\right)^{\rho(z_{it})}$. In this case, the quantity index is the same as the utility function, but they are generally different.} Since $S_{t}(\cdot)$ is nonparametric, the HSA system can nest various demand functions used in the literature such as the constant elastic demand from the CES utility, the symmetric translog demand (feenstra2003homothetic; feenstra2017globalization), or the constant response demand (Mrázová and Neary, mrazova2017not,MRAZOVA2019102561).\footnote{A HSA version of the constant response demand (Mrázová and Neary, mrazova2017not,MRAZOVA2019102561) can be formulated as for example, $P_{it}=\frac{\beta\Phi_{t}}{Y_{it}}\left[\left(\frac{Y_{it}}{A_{t}\left(\mathbf{Y}_{t},\mathbf{z}_{t}\right)}\right)^{\alpha}+\gamma z_{it}\right]^{\delta}$ where firm $i$'s markup is given by $\mu_{it}=\frac{1}{\alpha\beta}+\frac{\gamma z_{it}}{\alpha\beta\left(Y_{it}\right)^{\alpha}}$. See matsuyama2017beyond regarding how the HSA nests the translog demand.}

For identification of a demand system, we make assumptions regarding the market structure.

The assumption of monopolistic competition follows klette1996jae and de2011product, with the inverse demand function becoming a symmetric function of the firm's own output, as in ((ref)).

The demand elasticity equals $(\mu-1)/\mu$ when $\mu$ is markup. If the markup is identified up to scale, then the demand elasticity is not uniquely identified. Therefore, we need to fix the scale normalization to identify the demand function.

Assumption (ref) is satisfied when Proposition (ref) holds.

An HSA demand system can be constructed as follows. Suppose $\varphi_{t}^{-1}(\bar{r}_{t},z_{t})$ is identified from Proposition (ref); taking its inverse function obtains the revenue function $\varphi_{t}(y_{t},z_{t})$. Fixing a realized data point of $\mathbf{Y}_{t}^{0}:=(Y_{1t}^{0},...,Y_{Nt}^{0})\in\mathscr{\mathscr{\bar{Y}}}$ and $\mathbf{z}_{t}^{0}:=(z_{1t}^{0},...,z_{Nt}^{0})\in\mathcal{Z}^{N}$, we let $\Phi_{t}:=\sum_{i\in N_{t}}\exp\left(\varphi_{t}\left(\ln Y_{it}^{0},z_{it}^{0}\right)\right)$ be the consumer's budget, which is taken as given. For given $(\mathbf{Y}_{t},\mathbf{z}_{t})\in\mathscr{\bar{Y}}\times\mathcal{Z}^{N}$, we define a vector of market shares $S_{t}(\mathbf{Y}_{t},\mathbf{z}_{t}):=\left(S_{t}(Y_{1t},z_{1t}),....,S_{t}(Y_{Nt},z_{Nt})\right)$ such that \[ S_{t}(Y_{it},z_{it}):=\frac{\exp\left(\varphi_{t}\left(\ln Y_{it},z_{it}\right)\right)}{\Phi_{t}}. \] The quantity index $A_{t}(\mathbf{Y}_{t},\mathbf{z}_{t})$ is identified as follows. First, since $\sum_{i\in N_{t}}S_{t}\left(Y_{it}^{0},z_{it}^{0}\right)=1$, by construction, $A_{t}(\mathbf{Y}_{t}^{0},\mathbf{z}_{t}^{0})=1$ holds for the data point $(\mathbf{Y}_{t}^{0},\mathbf{z}_{t}^{0})$. For other values $(\mathbf{Y}_{t},\mathbf{z}_{t})\in\mathscr{\bar{Y}}\times\mathcal{Z}^{N}$, we can obtain $A_{t}(\mathbf{Y}_{t},\mathbf{z}_{t})$ by solving \[ \sum_{i\in N_{t}}S_{t}\left(\frac{Y_{it}}{A_{t}(\mathbf{Y}{}_{t},\mathbf{z}{}_{t})},z{}_{it}\right)=1. \] Since $S_{t}(\cdot,z_{it})$ is continuous and strictly increasing, $A_{t}(\mathbf{Y}_{t},\mathbf{z}_{t})$ is uniquely determined. Then, we obtain the inverse demand function for all $(\mathbf{Y}_{t},\mathbf{z}_{t})\in\mathscr{\bar{Y}}\times\mathcal{Z}^{N}$:

Applying the result of matsuyama2017beyond (2017, Proposition 1 and Remark 3), the following proposition establishes that the HSA demand system ((ref)) constructed above can be derived from a unique consumer preference, and that it is possible to identify an associated utility function. Appendix (ref) supplies the proof.

Identification in Alternative Settings

Endogenous Labor Input

Identification is possible when $l_{t}$ correlates with $\eta_{t}$. In the spirits of ackerberg2015identification and the dynamic generalized method of moment approach (e.g., arellano1991some,arellano1995another,blundell1998initial,blundell2000gmm), we provide identification using lagged labor $l_{t-1}$ as an instrument for $l_{t}$. Specifically, we follow the approach of ackerberg2015identification, which assumes (1) $l_{t}$ correlates with $l_{t-1}$ and (2) the firm's profit maximization problem regarding $m_{t}$ conditional on $l_{t}$ is expressed by ((ref)), which allows the material demand to be written as $m_{it}=\mathbb{M}_{t}(\omega_{it},k_{it},l_{it},z_{it})$. This approach has the advantage of being consistent with various data generating processes regarding the choice of $l_{t}$.\footnote{See ackerberg2015identification for examples of such data-generating processes. For example, $l_{t}$ can be chosen at time $t$ with adjustment costs; a firm can face an auto-correlated firm-specific wage; or $l_{t}$ can be chosen at time $t-1$ or at an intermediate time between $t$ and $t-1$. }

Identifying $\mathbb{M}_{t}^{-1}(x_{t},z_{t})$ using $l_{t-1}$ as an instrument for $l_{t}$ is nontrivial because the model ((ref)) includes $l_{t-1}$ in $\bar{h}_{t}\left(x_{t-1},z_{t-1}\right)$. It is not possible to use the variation of $l_{t-1}$ simultaneously for two purposes (i.e., identifying $\bar{h}_{t}\left(x_{t-1},z_{t-1}\right)$ and instrumenting $l_{t}$). Therefore, we proceed to identification in two steps. We first identify $\bar{h}_{t}\left(x_{t-1},z_{t-1}\right)$ (up to location) and then use $l_{t-1}$ to identify $\mathbb{M}_{t}^{-1}(m_{t},k_{t},l_{t},z_{t})$.

\paragraph{Identification of $\bar{h}_{t}\left(x_{t-1},z_{t-1}\right)$.}

Assumptions (ref) (i) and (ii) simply modify Assumption (ref) such that $l_{t}$ may correlate with $\eta_{t}$. Assumption (ref) (iii) is innocuous because it is satisfied if the firm's survival probability at time $t$ conditional on $(x_{t-1},z_{t-1})$ is not 0.

The conditional distribution of $m_{t}$ given $v_{t}$ satisfies

\[ G_{m_{t}|v_{t}}(m_{t}|v_{t})=G_{\eta_{t}|l_{t}}\left(\mathbb{M}_{t}^{-1}(m_{t},k_{t},l_{t},z_{t})-\bar{h}_{t}\left(x_{t-1},z_{t-1}\right)|l_{t}\right). \] Taking the derivatives of both sides with respect to $q_{t}\in\{m_{t},k_{t},z_{t}\}$ and $q_{t-1}\in\{k_{t-1},l_{t-1},m_{t-1},z_{t-1}\}$ and their ratios, we identify $\partial\mathbb{M}_{t}^{-1}(m,k_{t},l_{t},z_{t})/\partial q_{t}$ and $\partial\bar{h}(x_{t-1},z_{t-1})/\partial q_{t}$ as follows:

where $\left(\tilde{x}_{t-1},\tilde{z}_{t-1}\right)\in\mathcal{A}_{q_{t-1}}$ and $\left(\tilde{x}_{t},\tilde{z}_{t}\right)\in\mathcal{A}_{m_{t}}(x_{t-1},z_{t-1})$. Note that ((ref)) is the same as in ((ref)). Thus, following the same steps as those in the proof for Proposition (ref), we identify $\partial\mathbb{M}_{t}^{-1}(m,k_{t},l_{t},z_{t})/\partial q_{t}$ up to scale, and then $\partial\bar{h}\left(x_{t-1},z_{t-1}\right)/\partial q_{t-1}$ up to scale from ((ref)).

Define $d_{l}\left(l_{t}\right):=\mathbb{M}_{t}^{-1}(m_{t0}^{*},k_{t}^{*},l_{t},z_{t}^{*})$ for $(m_{t0}^{*},k_{t}^{*},z_{t}^{*})$ in ((ref)) and $d:=\bar{h}_{t}\left(x_{t-1}^{*},z_{t-1}^{*}\right)$ for some point $(x_{t-1}^{*},z_{t-1}^{*})\in\mathcal{X}\times\mathcal{Z}$. Integrating the identified elasticities in ((ref)) and ((ref)), we obtain

where function $d_{l}(l_{t})$ and constant $d$ are unknown objects to be identified; $\Lambda_{lt}\left(x_{t},z_{t}\right)$ and $\Lambda_{ht}\left(x_{t-1},z_{t-1}\right)$ are identified and thus treated as known functions.\footnote{Specifically, $\Lambda_{lt}\left(x_{t},z_{t}\right)$ and $\Lambda_{ht}\left(x_{t-1},z_{t-1}\right)$ are given by

}

\paragraph{Identification of $\mathbb{M}_{t}^{-1}(m_{t},k_{t},l_{t},z_{t})$.}

Defining $H_{it}:=\Lambda_{lt}\left(x_{it},z_{it}\right)-\Lambda_{ht}\left(x_{it-1},z_{it-1}\right)$ as a known variable, we rewrite model ((ref)) as \[ H_{it}=d-d_{l}(l_{it})+\eta_{it}. \] From $l_{t-1}\bot\eta_{t}$, we obtain the following moment condition for nonparametric instrument variable (IV) identification:

For instance, if $f_{t}$ is Cobb-Douglas as in ((ref)), then $d_{l}(l_{t})=-\theta_{l}\left(l_{t}-l_{t}^{*}\right)$ from ((ref)), and the moment condition ((ref)) becomes that for linear IV regression: \[ E\left[H_{it}-d-\theta_{l}(l_{it}-l_{it}^{*})|l_{it-1}\right]=0. \] A standard procedure of linear IV regression identifies $(d,\theta_{l})$ if $l_{it}$ sufficiently correlates with $l_{it-1}$.

Following the literature on nonparametric IV (e.g, newey2003instrumental), we assume that $l_{t-1}$ satisfies the following completeness condition.

With Assumption (ref), the moment condition ((ref)) uniquely identifies $\{d,d_{l}(l_{t})\}$.\footnote{The proof is as follows. Suppose $\{\tilde{d},\tilde{d}_{l}(l_{it})\}$ also satisfies the moment condition ((ref)). Then, it holds that $E\left[\tilde{d}-d+\tilde{d}_{l}(l_{it})-d_{l}(l_{it})|l_{it-1}\right]=0$ a.s. The completeness condition implies $\tilde{d}-d+\tilde{d}_{l}(l_{it})-d_{l}(l_{it})=0$ a.s. Since $\tilde{d}_{l}(l_{t}^{*})=d_{t}(l_{t}^{*})=0$ from Assumption (ref), $\tilde{d}=d$ holds so that $\tilde{d}_{0}(l_{it})=d_{0}(l_{it})$.} Since $E[\varepsilon_{t}|x_{t},z_{t}]=0$, step 1 continues to identify $\phi_{t}(\cdot)$. Therefore, once $\mathbb{M}_{t}^{-1}(m_{t},k_{t},l_{t},z_{t})$ is identified, step 3 identifies all the same objects as before.

Endogenous Firm Characteristics

Firm characteristics $z_{t}$ may correlate with $\eta_{t}$. For simplicity, we again assume that $l_{t}$ is exogenous. We show that even in the absence of any IV for $z_{t}$, we can identify the markup and the production function. If valid IVs for $z_{t}$ are available, all the same objects can be identified as before.

We modify Assumption (ref) so that $z_{t}$ may correlate with $\eta_{t}$.

\paragraph{Identification without Instrument Variables.}

The conditional distribution of $m_{t}$ given $v_{t}$ satisfies

\[ G_{m_{t}|v_{t}}(m|v_{t})=G_{\eta_{t}|z_{t}}\left(\mathbb{M}_{t}^{-1}(m,k_{t},l_{t},z_{t})-\bar{h}_{t}\left(x_{t-1},z_{t-1}\right)|z_{t}\right). \] Taking the derivatives of both sides with respect to $m$, $q_{t}\in\{m_{t},k_{t},l_{t}\}$ and $q_{t-1}\in\{k_{t-1},l_{t-1},m_{t-1},z_{t-1}\}$, we obtain ((ref)) and ((ref)). Following the same steps as in subsection (ref), we identify $\partial\mathbb{M}_{t}^{-1}(m,k_{t},l_{t},z_{t})/\partial q_{t}$ and $\partial\bar{h}\left(x_{t-1},z_{t-1}\right)/\partial q_{t-1}$ up to scale.

Since $E[\varepsilon_{t}|x_{t},z_{t}]=0$, Lemma (ref) continues to hold and $\phi_{t}(\cdot)$ is identified. Therefore, using ((ref)) and the first-order condition ((ref)) with the identified derivatives of $\mathbb{M}_{t}^{-1}(\cdot)$, it is possible to identify markup ((ref)) and output elasticities ((ref)) up to scale. Integrating the output elasticities, we can identify the production function, following ((ref)).

Applying Propositions (ref) and Proposition (ref), it is possible to identify the changes in markup and output elasticities overtime and the levels of markup and elasticities, respectively.

\paragraph{Identification with Instrument Variables.}

To identify $\varphi_{t}^{-1}(\cdot)$ and $\mathbb{M}_{t}^{-1}(\cdot)$, we need a set of IVs $\zeta_{t}$ for $z_{t}$. A candidate for $\zeta_{t}$ is $z_{t-1}$ if $z_{t-1}$ correlates with $z_{t}$.

Following similar steps by which to derive ((ref)), we obtain

and ((ref)), where $d_{z}(z_{t}):=\mathbb{M}_{t}^{-1}(m_{t0}^{*},k_{t}^{*},l_{t}^{*},z_{t})$ is an unknown function to be identified; $\Lambda_{zt}\left(x_{t},z_{t}\right)$ is identified and treated as a known function.\footnote{Specifically, $\Lambda_{zt}\left(x_{t},z_{t}\right)$ is given by

} Defining $H_{it}^{zh}:=\Lambda_{zt}\left(x_{it},z_{it}\right)-\Lambda_{ht}\left(x_{it-1},z_{it-1}\right)$ as a known variable, we rewrite model ((ref)) as \[ H_{it}^{zh}=d-d_{z}(z_{it})+\eta_{it}. \] From Assumption (ref), the moment condition, $E[\eta_{it}|\zeta_{it}]=E\left[H_{it}^{zh}-d+d_{z}(z_{it})|\zeta_{it}\right]=0$, identifies $\{d,d_{z}(z_{t})\}$.

Alternative Settings

The Appendix presents the identification results in three alternative settings. The identification argument remains the same but requires some additional steps.

\paragraph{Discrete Firm Characteristics.}

Observable firm characteristics $z_{t}$ may constitute a discrete variable. Appendix (ref) provides a proof.

\paragraph{Unobservable Firm-Level Demand-Shifter.}

The identification can incorporate an unobserved demand shifter $\xi_{it}$, which can be called quality. Let $y_{it}^{\dagger}:=y_{it}+\xi_{it}$ and $p_{it}^{\dagger}:=y_{it}-\xi_{it}$ be the quality-adjusted output and the quality-adjusted price, respectively. We consider the following inverse function and revenue function:

where $\omega_{it}^{\dagger}\equiv\omega_{it}+\xi_{it}$ is a composite of TFP and quality. In Appendix (ref), we show that ((ref)) derives from a representative consumer's maximization problem where $\exp(\xi_{it})$ enters the utility function in a multiplicative manner with quantity. In ((ref)), higher quality allows a firm to earn more revenue for a given output. We assume that $\tilde{\omega}_{it}$ follows a first-order Markov process $\omega_{it}^{\dagger}=h\left(\omega_{it-1}^{\dagger}\right)+\eta_{it}$.

Under the current setting, the model structure becomes identical to the main model where $(p_{it},y_{it},\omega_{it})$ are replaced with $(p_{it}^{\dagger},y_{it}^{\dagger},\omega_{it}^{\dagger})$. Therefore, applying precisely the same steps, we can identify all functions identified in Section 3 and the quality-adjusted variables $(p_{it}^{\dagger},y_{it}^{\dagger},\omega_{it}^{\dagger})$.

\paragraph{IID Productivity Shock.}

As an alternative error structure, we consider an i.i.d. production shock $e_{it}$ to output instead of a measurement error $\varepsilon_{it}$. Then, the firm's observed revenue $r_{it}$ and inputs $x_{it}$ are related as follows:

A firm chooses $m_{it}$ at time $t$ by maximizing the expected profit:

where $\mathcal{I}_{it}$ is the set of information for the firm that includes all past variables and all time$-t$ variables except $e_{it}$. The identification of the control function $\omega_{it}=\mathbb{M}_{t}^{-1}\left(m_{it},k_{it},l_{it},z_{it}\right)$ remains the same because $\mathbb{M}_{t}^{-1}(\cdot)$ continues to be a function of the same variables.

In the second step, the revenue function ((ref)) is written as:

Model ((ref)) also belongs to the class of transformation models studied by chiappori2015nonparametric. Therefore, by applying the nonparametric identification of a transformation model and using the first-order condition for the material, we can identify $\varphi_{t}(\cdot)$ and $f_{t}(\cdot)$ up to scale and location from the conditional distribution of $r_{it}$ given $(x_{it},z_{it})$ under the assumptions similar to those for Proposition (ref). As an additional complication, the first-order condition includes expectation with respect to $e_{t}$. Therefore, we first identify the distribution of $e_{t}$ to derive the first-order condition. Appendix (ref) provides a proof.

Because of the i.i.d. shock $e_{it}$, the realized value of $\partial\varphi_{t}^{-1}(r_{it},z_{it})/\partial r_{t}$ no longer equals the markup. We identify the markup from the cost minimization, following hall1988relation and de2012markups. As shown in Appendix (ref), the equation for the markup $\mu_{it}$ becomes \[ \mu_{it}=\frac{\partial f_{t}(x_{it})/\partial m_{it}}{\exp(p_{t}^{m}+m_{it})/\exp\left(r_{it}-e_{it}\right)}. \] The difference from the original Hall-De Loecker-Warzynski markup equation ((ref)) is $\exp\left(r_{it}-e_{it}\right)$ instead of $\exp\left(\bar{r}_{it}\right)=\exp\left(r_{it}-\varepsilon_{it}\right)$. While $\bar{r}_{t}=\phi_{t}(x_{t},z_{t})$ in ((ref)) is a deterministic function of $(x_{t},z_{t})$, $r_{t}-e_{t}$ is generally not. Therefore, the markups are different across firms even after being conditioned on $(x_{t},z_{t})$.

Concluding Remarks

The current study developes constructive nonparametric identification of production function and markup from revenue data. Our method simultaneously addresses two fundamental identification issues raised in the literature of production function estimation since ma44ecma\textemdash namely, correlations between inputs and TFP, and biases from markup heterogeneity when revenue is used as output. Under standard assumptions, when revenue is modeled as a function of output (rather than a mere proxy for output) and firm's observed characteristics, various economic objects of interest can be identified from revenue data. In an ongoing follow-up research, we provide an estimation procedure and plan to estimate these objects from an actual dataset.