# A System Approach to Structural Identification of Production Functions with
Multi-Dimensional Productivity††thanks: Correspondence: Emir Malikov, Lee
Business School, University of Nevada, Las Vegas, Las Vegas, NV 89154-6005.
Email<EMAIL_ADDRESS>
Emir Malikov University of Nevada, Las Vegas Shunan Zhao Oakland University
Jingfang Zhang University of Kentucky

(September 1, 2022)

###### Abstract

There is growing empirical evidence that firm heterogeneity is technologically
non-neutral. This paper extends Gandhi et al.’s (2020) proxy variable
framework for structurally identifying production functions to a more general
case when latent firm productivity is multi-dimensional, with both factor-
neutral and (biased) factor-augmenting components. Unlike alternative
methodologies, our model can be identified under weaker data requirements,
notably, without relying on the typically unavailable cross-sectional
variation in input prices for instrumentation. When markets are perfectly
competitive, we achieve point identification by leveraging the information
contained in static optimality conditions, effectively adopting a system-of-
equations approach. We also show how one can partially identify the non-
neutral production technology in the traditional proxy variable framework when
firms have market power.

##  1 Introduction

Production function and productivity (growth) are fundamental economic
concepts the importance of which requires no justification among economists.
Their identification however remains a challenge. At the micro level,
identifying production functions—and firm productivity, by extension—from
observational data is not a trivial matter due to the endogeneity issue
arising from the fact that firm “productivity” capturing such factors like
tacit knowledge and managerial quality is unobserved yet must be controlled
for because it is correlated with input usage. The literature focused on
addressing these methodological issues is vast and uses myriad different
approaches, but most consider the case of scalar productivity. Until very
recently (most notably Doraszelski and Jaumandreu, , 2018), few studies have
considered the identification of production functions when latent firm
productivity is multi-dimensional in that the productivity change may not
affect marginal products of all inputs in the same proportion (i.e.,
neutrally) despite the strong evidence thereof in the data (e.g., Raval, ,
2020). In this paper we develop a structural framework for the proxy variable
estimation of production functions with multi-dimensional productivity,
including factor-augmenting and -neutral components, that does not rely on the
typically unavailable cross-sectional variation in prices for instrumentation.

Among many methods to tackling endogeneity in the production function context,
the proxy variable approach by Olley and Pakes,  (1996) and Levinsohn and
Petrin,  (2003) has become one of the most prevalent estimators in applied
productivity research in economics as evidenced, e.g., by over 15,000 Google
Scholar citations (as of the time of writing) amassed by these two papers
alone. A proxy variable methodology for the structural identification of
production functions arguably owes its popularity to not only good empirical
performance but also the relative simplicity of implementation. The
conventional proxy variable estimators as well as their many refinements and
extensions (e.g., Wooldridge, , 2009; De Loecker, , 2013; Ackerberg et al., ,
2015; Kim et al., , 2019; Gandhi et al., , 2020; Flynn et al., , 2019; Malikov
and Lien, , 2021; Malikov and Zhao, , 2021) assume that latent firm
productivity is scalar and factor-neutral. Then, under some structural timing
assumptions, this latent productivity can be expressed as a function of
(observable) either the physical investment or intermediate inputs along with
other state variables by inverting the corresponding input demand/investment
function. This identification strategy relies crucially on the scalar
unobservable assumption, whereby there exists a single latent productivity
term, which is necessary to ensure the invertibility of demand functions to
construct a proxy.

However, the usual assumption of a scalar Hicks-neutral productivity in the
production function to capture technological/productivity changes remains
inconsistent with many economic theories as well as may be too restrictive in
many empirical applications. For example, as Doraszelski and Jaumandreu,
(2018) point out, the traditional theories of both exogenous and endogenous
economic growth rest on the assumption that technological change is non-
neutral and, in particular, labor-saving. Large cross-firm heterogeneity in
variable input ratios documented in the data is at odds with the factor-
neutral productivity too, and the “biased” technological change has been
widely used to explain changes in the labor share of income (see Zhang, ,
2019; Doraszelski and Jaumandreu, , 2018, 2019; Oberfield and Raval, ,
2021).111A non-Hicksian technological change is also an important feature of
aggregate production functions in some recent macroeconomic studies (e.g., see
Baqaee and Farhi, , 2019, 2020).  Therefore, the impetus to accommodate non-
neutral productivity while estimating production functions is strong and has
garnered much attention among economists.

We extend Gandhi et al., ’s (2020) system-based proxy variable framework for
structurally identifying production functions to a more general case when
latent firm heterogeneity is multi-dimensional, consisting of factor-neutral
and factor-augmenting productivities. Following Doraszelski and Jaumandreu,
(2018), to model non-neutral technology we augment the standard proxy variable
setup by introducing labor-augmenting (Harrod-neutral) productivity in
addition to the Hicks-neutral productivity. Our focus on the labor bias of
productivity change is motivated by both its being a key element in growth
theory and its inherently unique distinction from other traditional inputs
like (physical and human) capital and intermediates, which are all producible.
As such, the marginal productivity of inputs other than labor (i.e., capital
and materials) is assumed to change equiproportionately at the rate determined
by the Hicks-neutral productivity, whereas the (relative) productivity of
labor is also affected by biased technology shift. Under separability of
variable and dynamic inputs, factor-neutral productivity scales the variable
input demands but does not change their ratio, which ensures that the
information about Harrod-neutral productivity can be teased out and identified
(separately from the Hick-neutral component) from the observed variation in
firms’ variable input ratios.

To disentangle the two components of firm productivity (neutral and labor-
biased) and to identify the production function, we trade the fully
nonparametric formulation in Gandhi et al.,  (2020) in favor of a parametric
specification of the production technology. In doing so, we are able to
explicitly utilize a known functional form of the static first-order
conditions for freely varying inputs which enables us to derive a proxy for
the labor-augmenting productivity in a closed form and use it to effectively
concentrate this non-neutral unobservable out. Namely, we assume that the
firm’s production function takes a flexible log-quadratic translog
specification. We then develop a three-step system-based estimation procedure
that makes explicit use of static first-order conditions for flexible inputs
and the Markovian properties of both productivity components for structural
identification.

Our model is closely related to Doraszelski and Jaumandreu,  (2018, 2019) and
Zhang,  (2019) who also use the information contained in the mix of flexible
inputs to separately identify Hicks- and Harrod-neutral productivities.222Our
paper is also related to Demirer,  (2020) who also considers the problem of
identifying production functions with non-neutral multi-dimensional
productivity. However, unlike ours, his methodology does not take a structural
“proxy variable” route but a more atheoretical “control function” approach to
handling endogeneity-inducing latent firm productivities. The key and
important distinction between their and our methodologies is that we develop
an alternative identification scheme that does not require external
instruments (from outside the production function) such as lagged firm-level
variation in (input) prices used in these studies. While it may be suitable
for their specific empirical applications, the validity and practicality of
using lagged cross-sectional variation in prices for identification is not
universal. Not only are such price data typically unavailable or prone to
measurement errors in micro-level production datasets (Levinsohn and Petrin, ,
2003), but their use as valid instruments may also be problematic on
theoretical grounds (see Griliches and Mairesse, , 1998; Ackerberg et al., ,
2007, 2015; Malikov and Lien, , 2021). Aside from the concerns about plausible
exogeneity of heterogeneous input prices, their strength as instruments also
implicitly relies on the strong conditions for the price evolution (see Flynn
et al., , 2019). In this paper, we therefore contribute to the literature by
developing a methodology that identifies the production function and multi-
dimensional productivity even if prices are homogeneous. To achieve
identification without external firm-level instruments, we leverage the
information contained in static optimality conditions, in effect, adopting a
system-of-equations approach.

Our paper is mainly concerned with the identification of production functions
for firms operating in perfectly competitive markets. Although the latter
assumption continues to be maintained—implicitly or explicitly—by most
productivity studies in the literature, which is partly dictated by the lack
of firm-level price data, we also discuss an extension in which we relax this
assumption by allowing for monopolistic competition in the output market. In
contrast to many other studies that deviate from the perfect competition
assumption, our setup does not rely on additional price information or a
parameterization of the demand or places restrictions on the production
technology in a pursuit of point identification. Instead, we show how one can
partially identify the production function with non-neutral productivity when
firms have market power in the traditional proxy variable framework.

We demonstrate the ability of our estimator to successfully identify multi-
dimensional firm productivity through a small set of Monte Carlo simulations
and, then, provide an empirical illustration by applying it to the firm-level
data from China’s leather manufacturing industry. We find that the labor-
augmenting productivity behaves quite differently from its factor-neutral
counterpart: it shows a larger dispersion and minor growth across years. On
the other hand, we also find that foreign direct investment (FDI), which is
arguably one of the most important productivity boosters available to firms in
developing countries, has both economically and statistically significant
effect on labor-saving productivity, whereas its effect size on Hicksian
productivity is effectively zero. This suggests that the productivity-
enhancing effect of FDI on domestic firms’ productivity has a bias towards
labor. At the same time, our estimates provide evidence that productivity
change in China’s leather industry is, overall, factor-neutral.

The rest of the paper is organized as follows. Section 2 describes the model
of production with multi-dimensional firm heterogeneity. We describe our
identification strategy in Section 3, and the detailed estimation procedure is
provided in Section 4. We examine the finite-sample performance of our
methodology using simulations and provide an empirical illustration in Section
5. The extension to imperfect competition is discussed in Section 6, and
Section 7 concludes.

##  2 A Model of Firm Production

This section describes a model of production decisions by a firm in the
presence of multi-dimensional productivity. Our model builds upon the
conceptual paradigm considered by Doraszelski and Jaumandreu,  (2018) although
we abstract away from their many application-specific nuances with the goal of
formulating a generic, versatile framework suitable for application to typical
datasets.

Consider the production process of a firm $i$ ($i=1,\dots,n$) in the time
period $t$ ($t=1,\dots,T$) in which physical capital $K_{it}$, labor $L_{it}$
and an intermediate input such as materials $M_{it}$ are being transformed
into the output $Y_{it}$ via production function $F(\cdot)$ given firm
productivity. We differentiate between factor-neutral and factor-biased
productivities. Namely, let the firm’s production technology take the
following form:

$Y_{it}=F(K_{it},\exp\left\\{\varphi_{it}\right\\}L_{it},M_{it})\exp\left\\{\omega_{it}\right\\}\exp\left\\{\eta_{it}\right\\},$
(2.1)

where, in addition to the usual log-additive Hicks-neutral productivity
$\omega_{it}$, we also allow for Harrod-neutral productivity $\varphi_{it}$
that affects firm output indirectly by “augmenting” the labor input. Both can
be persistent. The error $\eta_{it}$ is an ex-post transitory productivity
shock, which is sometimes alternatively interpreted as a classical measurement
error in the log-output.

In what follows, we characterize structural assumptions about the firm’s
technology, productivity, economic environment and its dynamic decision-making
process which facilitate a structural identification of the model.

###### Assumption 1

Among the firm’s inputs: (i) physical capital $K_{it}$ is a dynamic input
subject to adjustment frictions; (ii) labor $L_{it}$ and intermediate inputs
$M_{it}$ are freely varying inputs with no dynamic implications.

Since physical capital is subject to adjustment costs (e.g., time-to-install),
the firm optimizes $K_{it}$ dynamically at time $t-1$ rendering it a
predetermined input quasi-fixed at time $t$. Thus, $K_{it}$ is a state
variable with dynamic implications that follows the law of motion:

$K_{it}=I_{it-1}+(1-\delta)K_{it-1},$ (2.2)

where $I_{it}$ and $\delta$ are the gross investment and the depreciation
rate, respectively. Labor and materials are freely varying and are therefore
determined by the firm statically at time $t$, given the already optimized
choice of $K_{it}$. This is a fairly standard treatment of inputs in the
literature (e.g., see Olley and Pakes, , 1996; Levinsohn and Petrin, , 2003;
Wooldridge, , 2009). The setup can also be extended to allow for more inputs.
The only requirement is that there be at least one freely varying input in
addition to labor, which is necessary for identification of labor-augmenting
productivity (more on this point below).

###### Assumption 2

The production relationship between inputs, Hicks- and Harrod-neutral
productivities, and the output takes the form of (2.1). The production
function $F(\cdot)$ is (i) continuous and satisfies the standard neoclassical
assumptions, including differentiability, positive monotonicity and concavity
in inputs, and (ii) strongly separable in the partition
$(K_{it},(\exp\left\\{\varphi_{it}\right\\}L_{it},M_{it}))$ as follows:

$F(K_{it},\exp\left\\{\varphi_{it}\right\\}L_{it},M_{it})=G\left(K_{it},H\left(\exp\left\\{\varphi_{it}\right\\}L_{it},M_{it}\right)\right),$
(2.3)

where $H(\cdot)$ is homogeneous of arbitrary degree, and (iii) of the known
parametric functional form.

The assumption is that the dynamic inputs are separable from the remaining
production-function arguments. Separability is an identifying restriction that
ensures information about Harrod-neutral productivity can be teased out and
identified from the variation in the firm’s material-to-labor ratio which does
not depend on the Hicks-neutral productivity.333It is because of this reliance
on a statically optimized input ratio that we require at least two freely
varying inputs. This restriction on production technology imposes that the
marginal rate of technical substitution between the two variable inputs
(materials and labor) does not depend on dynamic inputs. Whether explicit or
not, the same assumption is made in the majority of productivity studies using
proxy variable estimators, in line with popular practice, when assuming that
the technology is Cobb-Douglas or Constant Elasticity of Substitution (CES).
The latter is also the case for Doraszelski and Jaumandreu,  (2018) and Zhang,
(2019).

In line with the literature, we model persistent firm productivities as first-
order Markov processes which we endogenize à la Doraszelski and Jaumandreu,
(2013), De Loecker,  (2013) and Malikov and Zhao,  (2021) by incorporating
productivity-enhancing and/or “learning” activities of the firm. To keep our
model as general as possible, we denote all such activities via generic
variables $X_{it}$ and $Z_{it}$ which, depending on the empirical application
of interest, may measure the firm’s R&D expenditures, FDI exposure, export
status/intensity, etc. Letting $\Xi_{it}$ be the information set available to
the $i$th firm for making the period $t$ decisions, the dynamics of
unobservables are summarized as follows.

###### Assumption 3

(i) Both components of persistent firm productivity $\omega_{it}$ and
$\varphi_{it}$ evolve according to their respective controlled first-order
Markov processes: $\mathcal{P}_{\omega}(\omega_{it}|\Xi_{it-1})=$
$\mathcal{P}_{\omega}(\omega_{it}|\omega_{it-1},X_{it-1})$ and
$\mathcal{P}_{\varphi}(\varphi_{it}|\Xi_{it-1})=$
$\mathcal{P}_{\varphi}(\varphi_{it}|\varphi_{it-1},Z_{it-1})$, where some or
all elements in $X_{it-1}$ and $Z_{it-1}$ may be common. (ii) The transitory
productivity shock $\eta_{it}$ is an i.i.d. white noise process:
$\mathcal{P}_{\eta}(\eta_{it}|\Xi_{it})=\mathcal{P}_{\eta}(\eta_{it})$.

The Markov assumptions imply the following regressions for $\omega_{it}$ and
$\varphi_{it}$:

$\displaystyle\omega_{it}$
$\displaystyle=r_{\omega}\left(\omega_{it-1},X_{it-1}\right)+\zeta_{\omega,it},$
(2.4) $\displaystyle\varphi_{it}$
$\displaystyle=r_{\varphi}\left(\varphi_{it-1},Z_{it-1}\right)+\zeta_{\varphi,it},$
(2.5)

where $r_{\omega}(\cdot)$ and $r_{\varphi}(\cdot)$ are the conditional-mean
functions of $\omega_{it}$ and $\varphi_{it}$, respectively; and
$\zeta_{it}^{\omega}$ and $\zeta_{\varphi,it}$ are mean-zero unanticipated
random innovations:
$\mathbb{E}\left[\zeta_{\omega,it}|\Xi_{it-1}\right]=\mathbb{E}\left[\zeta_{\omega,it}|\omega_{it-1},X_{it-1}\right]=\mathbb{E}\left[\zeta_{\omega,it}\right]=0$
and
$\mathbb{E}\left[\zeta_{\varphi,it}|\Xi_{it-1}\right]=\mathbb{E}\left[\zeta_{\varphi,it}|\varphi_{it-1},Z_{it-1}\right]=\mathbb{E}\left[\zeta_{\varphi,it}\right]=0$.
Also note that Assumption 3(i) places no restriction on the correlation
between two productivity components $\omega_{it}$ and $\varphi_{it}$. The two
may correlate via productivity-modifying “controls” and the innovations that
may reasonably be expected to be positively correlated.

The evolution processes in (2.4)–(2.5) implicitly assume that productivity-
enhancing activities and learning affect future firm productivity with a
delay, which is why the dependence of productivities on their controls
$X_{it}$ and $Z_{it}$ is lagged implying that the improvements in firm
productivity take a period to materialize. In
$\mathbb{E}\left[\zeta_{\omega,it}|\Xi_{it-1}\right]=\mathbb{E}\left[\zeta_{\varphi,it}|\Xi_{it-1}\right]=0$,
we effectively assume that firms do not adjust their productivity-modifying
activities in light of expected future innovations in their productivity,
which rules out their ability to systematically predict future shocks. Since
random innovations $\zeta_{\omega,it}$ and $\zeta_{\varphi,it}$ represent
uncertainty in productivity evolution as well as uncertainty in the success of
productivity-modifying activities, the firm relies on its knowledge of
contemporaneous productivities $\omega_{it-1}$ and $\varphi_{it-1}$ when
choosing the optimal level of $X_{it-1}$ and $Z_{it-1}$ at time $t-1$ while
being unable to anticipate next period’s productivity innovations. Those
innovations ($\zeta_{\omega,it}$ and $\zeta_{\varphi,it}$) are realized after
both $X_{it-1}$ and $Z_{it-1}$ have already been chosen. Analogous timing
assumptions are commonly made in the production-function models with
controlled productivity processes (Van Biesebroeck, , 2005; Doraszelski and
Jaumandreu, , 2013; De Loecker, , 2013; Malikov et al., , 2020, 2021): they
render the firm’s past productivity-modifying activities mean-orthogonal to
random innovations at time $t$, thereby helping the identification of the
learning effects.

###### Assumption 4

Risk-neutral firms maximize the discounted stream of life-time profits in
perfectly competitive output and factor markets with homogeneous prices.

Following the bulk of the literature, we assume perfectly competitive markets
implying that firms are price-takers which, in theory, rules out any
operationable firm-level variation in prices. In what follows, we therefore
omit prices from the list of relevant determinants entering the firm’s
decision equations: they are implicitly represented by the firm-common time
index. As noted earlier, we aim to develop an estimator that does not require
firm-level price information typically unavailable in most firm- or plant-
level production datasets. Having said that, we discuss ways to relax this
assumption and allow for monopolistic power in the output market in Section 6.

With this structural setup, the firm’s dynamic optimization problem is
described by

$\displaystyle\mathbb{V}_{t}\big{(}K_{it},\omega_{it},\varphi_{it}\big{)}=\max_{I_{it},X_{it},Z_{it}}\Big{\\{}$
$\displaystyle\
\Pi_{t}(K_{it},\omega_{it},\varphi_{it})-\text{C}^{I}_{t}(I_{it},K_{it})-\text{C}^{X}_{t}(X_{it},X_{it-1})-\text{C}^{Z}_{t}(Z_{it},Z_{it-1})\
+$ $\displaystyle\
\beta\mathbb{E}\Big{[}\mathbb{V}_{t+1}\big{(}K_{it+1},\omega_{it+1},\varphi_{it+1}\big{)}\Big{|}\Xi_{it},I_{it},X_{it},Z_{it}\Big{]}\,\Big{\\}},$
(2.6)

where $\beta$ is a time discount factor;
$(K_{it},\omega_{it},\varphi_{it})^{\prime}\in\Xi_{it}$ are the state
variables; $\Pi_{t}(\cdot)$ is the value function corresponding to the static
profit-maximization problem in (3.3), i.e., a “short-run” restricted profit
function; and $\text{C}^{\kappa}_{t}(\cdot)$ is the cost function for capital
($\kappa=I$) and productivity-enhancing activities ($\kappa=\\{X,Z\\}$). In
the above optimization problem, albeit also with dynamic implications, the
levels of productivity-enhancing activities $X_{it+1}$ and $Z_{it+1}$ are
chosen by the firm contemporaneously at time $t+1$ unlike the level of
$K_{it+1}$ which is a delayed decision made at time $t$ (via $I_{it}$). This
allows for persistence in productivity-enhancing activities but does not force
them to be subject to adjustment frictions that would also render them delayed
and, hence, predetermined.444For clarity, if the optimal decision concerning
production in period $t$ is affected by its history, then that decision is
said to be “dynamic.” If, due to adjustment frictions, a decision concerning
production in period $t$ is effectively made at $t-1$, then we say it is
“predetermined.” In this nomenclature, $K_{it}$ is dynamic and predetermined,
whereas $X_{it}$ and $Z_{it}$ are dynamic but chosen at time $t$. This
distinction is important because it does not rule out a contemporaneous
correlation between firm productivities $(\omega_{it},\varphi_{it})^{\prime}$
and $(X_{it},Z_{it})^{\prime}$. Then, solving (2) for $I_{it}$, $X_{it}$ and
$Z_{it}$ yields their respective optimal policy functions in terms of the
firm’s state variables.

Technically, our methodology can be formulated without explicit formalization
of the firm’s dynamic decisions by only describing its static optimization. We
opt to spell out the dynamics, however, to structurally contextualize the
predeterminedness of fixed inputs and productivity-modifying activities with
respect to the innovations in productivity which, otherwise, would have had to
be assumed prima facie.

##  3 Identification

The estimation of the production function in (2.1) is not trivial because of
the latent nature of firm productivity. In our case, the problem is further
complicated by the fact that unobserved productivity is two-dimensional.
Omitting $\omega_{it}$ and $\varphi_{it}$ from the production-function
regression is ill-advised because it would lead to an endogeneity problem
given that firm productivities are correlated with inputs. We tackle this
problem by adopting a control function approach à la Olley and Pakes,  (1996)
and Levinsohn and Petrin,  (2003). Specifically, we consider the
identification and estimation of the production function (2.1) by building on
Gandhi et al., ’s (2020) methodology, which we generalize to accommodate
multi-dimensional firm productivity.

Due to the presence of multiple unobservables in (2.1) and the fundamentally
different manner in which they enhance inputs, to achieve the (separable)
identification of $\omega_{it}$ and $\varphi_{it}$ we make use of a
parametric-form assumption for the production function. This makes our
methodology distinct from Gandhi et al.,  (2020) whose approach is fully
nonparametric. Forgoing a nonparametric formulation of the production function
$F(\cdot)$ in favor of a parametric specification is the price of letting firm
productivity not be restricted to a single factor-neutral dimension. We adopt
a log-quadratic translog specification for $F(\cdot)$.555E.g., see De Loecker
and Warzynski,  (2012) and De Loecker et al.,  (2016) for recent applications
of the translog production functions in the structural proxy estimation.
Namely,

$\displaystyle f(\cdot)=$ $\displaystyle\
\beta_{K}k_{it}+\tfrac{1}{2}\beta_{KK}k_{it}^{2}+\beta_{M}m_{it}+\tfrac{1}{2}\beta_{MM}m_{it}^{2}+\beta_{KM}k_{it}m_{it}\
+$ $\displaystyle\
\beta_{L}[\varphi_{it}+l_{it}]+\tfrac{1}{2}\beta_{LL}[\varphi_{it}+l_{it}]^{2}+\beta_{KL}k_{it}[\varphi_{it}+l_{it}]+\beta_{ML}m_{it}[\varphi_{it}+l_{it}],$
(3.1)

where the lower-case variables/functions denote the logs of the respective
variables/functions.

Under Assumption 2(ii), $\beta_{KM}=\beta_{KL}=0$ and the function needs be
normalized to be homogeneous of arbitrary degree in freely varying inputs.
Following Doraszelski and Jaumandreu,  (2019), we set the degree of
homogeneity to $\beta_{L}+\beta_{M}$. With this, logging both sides of (2.1),
we get the following “restricted” translog form:

$\displaystyle y_{it}$
$\displaystyle=\underbrace{\beta_{K}k_{it}+\tfrac{1}{2}\beta_{KK}k_{it}^{2}+\beta_{M}m_{it}+\beta_{L}[\varphi_{it}+l_{it}]-\tfrac{1}{2}\beta_{0}[m_{it}-\varphi_{it}-l_{it}]^{2}}_{\overline{y}_{it}}+\omega_{it}+\eta_{it},$
(3.2)

where $\beta_{0}\equiv-\beta_{MM}=-\beta_{LL}=\beta_{ML}$.

We opt for the translog specification chiefly out of convenience given its
linearity in parameters. Other functional forms could also be used; e.g., the
nested CES specification preferred by Doraszelski and Jaumandreu,  (2018) and
Zhang,  (2019). We describe how to implement our methodology under this
alternative parameterization in Appendix A.

###  3.1 A System Approach to Identification

Since freely varying inputs are non-dynamic, the risk-neutral firm’s optimal
choice of $L_{it}$ and $M_{it}$ can be modeled statically as the concentrated
expected profit-maximization problem subject to the already predetermined
optimal choice of the quasi-fixed input $K_{it}$ and both components of
persistent firm productivity $\omega_{it}$ and $\varphi_{it}$:

$\displaystyle\max_{L_{it},M_{it}}\
P_{t}^{Y}F(K_{it},\exp\left\\{\varphi_{it}\right\\}L_{it},M_{it})\exp\left\\{\omega_{it}\right\\}\theta-
P_{t}^{L}L_{it}-P_{t}^{M}M_{it},$ (3.3)

where $P_{t}^{Y}$, $P_{t}^{L}$ and $P_{t}^{M}$ are respectively the output,
labor and material prices that, given the perfect competition assumption, are
common to all firms; and $\theta\equiv\mathbb{E}[\exp\\{\eta_{it}\\}|\
\Xi_{it}]$. The value function corresponding to (3.3) yields $\Pi_{t}(\cdot)$
entering the firm’s Bellman equation (2). The corresponding first-order
conditions yield the firm’s conditional demand for $L_{it}$ and $M_{it}$.

Making use of the functional form in (3.2), the static optimality conditions
are

$\displaystyle
P_{t}^{Y}\frac{\exp\left\\{\overline{y}_{it}\right\\}}{L_{it}}\big{(}\beta_{L}+\beta_{0}[m_{it}-\varphi_{it}-l_{it}]\big{)}\exp\\{\omega_{it}\\}\theta$
$\displaystyle=P_{t}^{L},$ (3.4) $\displaystyle
P_{t}^{Y}\frac{\exp\left\\{\overline{y}_{it}\right\\}}{M_{it}}\big{(}\beta_{M}-\beta_{0}[m_{it}-\varphi_{it}-l_{it}]\big{)}\exp\\{\omega_{it}\\}\theta$
$\displaystyle=P_{t}^{M}.$ (3.5)

Taking the ratio of (3.4) and (3.5), we obtain the equation for the firm’s
optimal labor-to-material ratio:

$\frac{L_{it}}{M_{it}}=\frac{P_{t}^{M}}{P_{t}^{L}}\times\frac{\beta_{L}+\beta_{0}[m_{it}-\varphi_{it}-l_{it}]}{\beta_{M}-\beta_{0}[m_{it}-\varphi_{it}-l_{it}]},$
(3.6)

which, expectedly, does not depend on factor-neutral productivity
$\omega_{it}$ because the latter enhances both inputs equally thereby leaving
their ratio unaffected. The input ratio however is affected by the labor-
augmenting productivity, since $\varphi_{it}$ changes the relative marginal
products of labor and materials.

We can solve (3.6) for Harrod-neutral productivity $\varphi_{it}$ to arrive at

$\varphi_{it}=m_{it}-l_{it}+\frac{\beta_{L}}{\beta_{0}}-\left(\frac{\beta_{L}+\beta_{M}}{\beta_{0}}\right)S^{L}_{it},$
(3.7)

where $S^{L}_{it}\equiv
P_{t}^{L}L_{it}/\big{(}P_{t}^{L}L_{it}+P_{t}^{M}M_{it}\big{)}$ is the labor
share of the firm’s variable input cost. This expression is an operationable
proxy for unobservable $\varphi_{it}$.

First step.—We first identify the sum of $\beta_{L}$ and $\beta_{M}$
coefficients as well as nuisance parameter $\theta$ and random productivity
shocks $\\{\eta_{it}\\}$. To do so, we transform static first-order conditions
in (3.4) and (3.5) by taking their logs and subtracting (3.2) from each of
them to obtain the corresponding share equations in logs, i.e.,

$\displaystyle\ln V_{it}^{L}$
$\displaystyle=\ln\left(\theta\beta_{0}\left[\frac{\beta_{L}}{\beta_{0}}+m_{it}-\varphi_{it}-l_{it}\right]\right)-\eta_{it},$
(3.8) $\displaystyle\ln V_{it}^{M}$
$\displaystyle=\ln\left(\theta\beta_{0}\left[\frac{\beta_{M}}{\beta_{0}}-m_{it}+\varphi_{it}+l_{it}\right]\right)-\eta_{it},$
(3.9)

where $V_{it}^{L}\equiv P_{t}^{L}L_{it}/\big{(}P_{t}^{Y}Y_{it}\big{)}$ and
$V_{it}^{M}\equiv P_{t}^{M}M_{it}/\big{(}P_{t}^{Y}Y_{it}\big{)}$ are the
nominal shares of labor and material costs in total revenue, respectively.

To operationalize these equations into the estimating regression equations, we
need to tackle the unobservable $\varphi_{it}$ appearing on the right-hand
size of (3.8)–(3.9). Failure to control for it would lead to endogeneity due
to the correlation with Harrod-neutral productivity and freely varying inputs.
We control for $\varphi_{it}$ using the material-to-labor ratio proxy
function. That is, substituting for $\varphi_{it}$ in either one of the two
log-share equations using the expression in (3.7), we obtain the following
variable-input-cost-to-revenue equation in logs:

$\displaystyle\ln R_{it}$
$\displaystyle=\ln\big{(}\theta\left[\beta_{L}+\beta_{M}\right]\big{)}-\eta_{it},$
(3.10)

where
$R_{it}\equiv\big{(}P^{L}_{t}L_{it}+P^{M}_{t}M_{it}\big{)}/\big{(}P^{Y}_{t}Y_{it}\big{)}$.
The cost-to-revenue ratio $R_{it}$ is observable in the data, and the
construction thereof does not require firm-level price data: the information
on total flexible input expenditures and total revenue suffices.

The variable-input-cost-to-revenue equation in (3.10) is useful in that it
enables us to identify $\beta_{L}+\beta_{M}$, both elements of which enter the
production function of interest in (3.2), using the observable information
about expenditures on flexible inputs and revenue. Specifically, we first
identify a “scaled” sum of these two translog coefficients
$\theta\times[\beta_{L}+\beta_{M}]$ using the moment condition
$\mathbb{E}[\eta_{it}|\Xi_{it}]=0$, from which we have that

$\ln\big{(}\theta\left[\beta_{L}+\beta_{M}\right]\big{)}=\mathbb{E}[\ln
R_{it}].$ (3.11)

To identify $\beta_{L}+\beta_{M}$ net of constant $\theta$, note that $\theta$
can be identified via
$\theta\equiv\mathbb{E}\left[\exp\left\\{\eta_{it}\right\\}\right]=\mathbb{E}\left[\exp\left\\{\mathbb{E}[\ln
R_{it}]-\ln R_{it}\right\\}\right]$, which allows us to isolate
$\beta_{L}+\beta_{M}$ as follows:

$\beta_{L}+\beta_{M}=\frac{\exp\\{\mathbb{E}[\ln
R_{it}]\\}}{\mathbb{E}\left[\exp\left\\{\mathbb{E}[\ln R_{it}]-\ln
R_{it}\right\\}\right]}.$ (3.12)

Let the identified $\beta_{L}+\beta_{M}$ be denoted as
$\delta_{LM}=\beta_{L}+\beta_{M}$.

Second step.—Next, we show how to separate $\beta_{L}$ and $\beta_{M}$ and
identify $\beta_{0}$. We utilize the Markov assumption about labor-augmenting
productivity. More concretely, with $\delta_{LM}$ already identified in the
first step, the proxy function for $\varphi_{it}$ in (3.7) contains only two
unknown parameters: $\beta_{0}$ and $\beta_{L}$. Substituting this partly
identified $\varphi_{it}(\beta_{0},\beta_{L})$ expression for $\varphi_{it}$
in the Markov productivity evolution process in (2.5) and treating
$\delta_{LM}$ as an observable, we obtain

$\displaystyle
m_{it}-l_{it}+\frac{\beta_{L}}{\beta_{0}}-\frac{\delta_{LM}}{\beta_{0}}S^{L}_{it}=r_{\varphi}\left(\left[m_{it-1}-l_{it-1}+\frac{\beta_{L}}{\beta_{0}}-\frac{\delta_{LM}}{\beta_{0}}S^{L}_{it-1}\right],Z_{it-1}\right)+\zeta_{\varphi,it},$
(3.13)

which identifies $(\beta_{0},\beta_{L})^{\prime}$ as well as the mean
productivity function $r_{\varphi}(\cdot)$ on the basis of

$\mathbb{E}\left[\zeta_{\varphi,it}|\
1,m_{it-1}-l_{it-1},S^{L}_{it-1},Z_{it-1}\right]=0.$ (3.14)

Note that the nonlinear equation in (3.13) technically contains an endogenous
regressor $S^{L}_{it}$ which is not mean-orthogonal to the innovation
$\zeta_{\varphi,it}$ because the former includes information on the choice of
both $L_{it}$ and $M_{it}$ which are decided by the firm after
$\zeta_{\varphi,it}$ is realized (i.e., after $\varphi_{it}$ is updated). This
however does not impede identification of (3.13) because $S^{L}_{it}$ is not a
“free” regressor but enters the equation subject to a parameter restriction
whereby the coefficient thereon is the same as that on weakly exogenous
$S^{L}_{it-1}$. No external instrumentation for $S^{L}_{it}$ is therefore
needed.

To make our identification arguments more transparent, let unknown function
$r_{\varphi}(\cdot)$ be linear and, since it can only be identified up to a
constant, normalize $r_{\varphi}(0)=0$.666Since productivity/efficiency
measurements are relative, this is merely a restriction which implies a
“normalized” zero-mean $\varphi$ and which does not affect the relative rank
of firms based on efficiency of their labor. Qualitatively, this normalization
is akin to the typical no-intercept restriction for production functions
because an additive constant cannot be separated from the Hicksian
productivity unless the latter is also normalized to have a zero mean. More
concretely,
$r_{\varphi}=\rho_{1}\big{[}m_{it-1}-l_{it-1}+\frac{\beta_{L}}{\beta_{0}}-\frac{\delta_{LM}}{\beta_{0}}S^{L}_{it-1}\big{]}+\rho_{2}^{\prime}Z_{it-1}$.
Denoting the vector of exogenous instruments
$Q_{it-1}=(1,m_{it-1}-l_{it-1},\allowbreak
S^{L}_{it-1},Z_{it-1}^{\prime})^{\prime}$, consider now the identification of
equidimensional parameter vector
$\alpha=(\beta_{0},\beta_{L},\rho_{1},\rho_{2}^{\prime})^{\prime}$ in the
following nonlinear GMM problem:

$\displaystyle\alpha_{0}=\arg\min_{\alpha}\mathbb{E}[Q_{it-1}\zeta_{\varphi,it}(\alpha)]^{\prime}W\mathbb{E}[Q_{it-1}\zeta_{\varphi,it}(\alpha)],$
(3.15)

where $W$ is a symmetric positive-definite moment-weighting matrix. To see
that (3.15) identifies all parameters in $\alpha$, consider an information
matrix

$\displaystyle\Psi(\alpha)$
$\displaystyle=\mathbb{E}\left[Q_{it-1}\frac{\partial\zeta_{\varphi,it}(\alpha)}{\partial\alpha^{\prime}}\right]$
$\displaystyle=\mathbb{E}\left[\begin{pmatrix}1\\\ m_{it-1}-l_{it-1}\\\
S^{L}_{it-1}\\\
Z_{it-1}\end{pmatrix}\begin{pmatrix}-\frac{\beta_{L}}{\beta_{0}^{2}}(1-\rho_{1})+\frac{\delta_{LM}}{\beta_{0}^{2}}\left[S^{L}_{it}-\rho_{1}S^{L}_{it-1}\right]\\\
\frac{1-\rho_{1}}{\beta_{0}}\\\
m_{it-1}-l_{it-1}+\frac{\beta_{L}}{\beta_{0}}-\frac{\delta_{LM}}{\beta_{0}}S^{L}_{it-1}\\\
Z_{it-1}\end{pmatrix}^{\prime}\right]$ (3.16)

and note that it is full-rank (we unpack this expression in Appendix B). Thus,
the information matrix for the GMM problem in (3.15) when evaluated at the
true parameter values $\Psi(\alpha_{0})$ has a full column rank. All
parameters in $\alpha$ are therefore locally identified (see Rothenberg, ,
1971).

Although, in theory, the four instruments in $Q_{it-1}$ are enough to exactly
identify the second-step parameters of interest $\alpha$, there is a potential
to improve the finite-sample performance if additional valid instruments are
included in the estimation. Obvious candidates are the firm’s dynamic quasi-
fixed inputs $k_{it},k_{it-1},\dots$ which are weakly exogenous with respect
to the time $t$ shocks, including $\zeta_{\varphi,it}$, and relevant for the
choice of $(S^{L}_{it},S^{L}_{it-1},Z_{it-1}^{\prime})^{\prime}$ through both
the static and dynamic optimization decisions. These additional instruments
are to act as exclusion restrictions to strengthen the moment condition and
help in identification. For instance, Kim et al.,  (2019) propose a similar
simple strategy to robustify the Ackerberg et al.,  (2015) estimation
procedure.

Following identification of $\beta_{L}$, $\beta_{M}$ is identified from
$\beta_{M}=\delta_{LM}-\beta_{L}$ as a by-product. We also achieve the
identification of Harrod-neutral productivity via
$\varphi_{it}=m_{it}-l_{it}+\beta_{L}/\beta_{0}-(\beta_{L}+\beta_{M})/\beta_{0}\times
S^{L}_{it}$.

Third step.—With $(\beta_{0},\beta_{L},\beta_{M})^{\prime}$ identified, we
have effectively pinpointed the production function in the dimension of its
endogenous freely-varying inputs $L_{it}$ and $M_{it}$ thereby addressing the
Gandhi et al.,  (2020) critique, whereby the endogenous static inputs are
lacking valid internal instruments when directly included as regressors in the
proxied production function estimation. This is evident by rewriting (3.2)
with the substitution for $\omega_{it}$ using its Markov process from (2.4) as
follows:

$y_{it}^{*}=\beta_{K}k_{it}+\tfrac{1}{2}\beta_{KK}k_{it}^{2}+r_{\omega}\left(\omega_{it-1},X_{it-1}\right)+\zeta_{\omega,it}+\eta_{it},$
(3.17)

where $y_{it}^{*}\equiv
y_{it}-\beta_{M}m_{it}-\beta_{L}[\varphi_{it}+l_{it}]+\tfrac{1}{2}\beta_{0}[m_{it}-\varphi_{it}-l_{it}]^{2}=y_{it}-\beta_{M}m_{it}-\beta_{L}[m_{it}+\beta_{L}/\beta_{0}-(\beta_{L}+\beta_{M})/\beta_{0}\times
S^{L}_{it}]+\tfrac{1}{2\beta_{0}}[(\beta_{L}+\beta_{M})S^{L}_{it}-\beta_{L}]^{2}$
is already identified and, hence, equation (3.17) now contains no endogenous
regressors that need instrumentation. However, we still need to deal with
unobservability of $\omega_{it-1}$.

To identify the rest of the production function, i.e.,
$(\beta_{K},\beta_{KK})^{\prime}$, we proxy for latent Hicks-neutral
$\omega_{it-1}$ in (3.17) by inverting the conditional material demand
function implied by the static first-order condition in (3.5):

$\displaystyle\omega_{it}=$ $\displaystyle\
\overbrace{\ln\left[\frac{P_{t}^{M}}{P_{t}^{Y}}\right]-\ln\theta-\ln\big{(}\beta_{M}-\beta_{0}[m_{it}-\varphi_{it}-l_{it}]\big{)}+(1-\beta_{M})m_{it}-\beta_{L}[\varphi_{it}+l_{it}]+\tfrac{1}{2}\beta_{0}[m_{it}-\varphi_{it}-l_{it}]^{2}}^{m^{*}_{it}}$
$\displaystyle\ -\beta_{K}k_{it}-\tfrac{1}{2}\beta_{KK}k_{it}^{2},$ (3.18)

where, given the already identified
$(\beta_{0},\beta_{L},\beta_{M},\theta)^{\prime}$ and $\varphi_{it}$ in the
first two steps, $m^{*}_{it}$ is observable. Substituting for $\omega_{it-1}$
using this proxy, from (3.17) we derive

$y_{it}^{*}=\beta_{K}k_{it}+\tfrac{1}{2}\beta_{KK}k_{it}^{2}+r_{\omega}\left(\big{[}m^{*}_{it-1}-\beta_{K}k_{it-1}-\tfrac{1}{2}\beta_{KK}k_{it-1}^{2}\big{]},X_{it-1}\right)+\zeta_{\omega,it}+\eta_{it},$
(3.19)

where, under our structural assumptions, all regressors are weakly exogenous
because the productivity innovation $\zeta_{\omega,it}$ is realized at time
$t$ after the firm had already optimized its dynamic input $K_{it}$ (at time
$t-1$) for the period $t$ production and, obviously, after the lagged flexible
inputs contained inside $m_{it-1}^{*}$ were chosen. That is, all right-hand-
side variables in (3.19) can self-instrument. This identifies
$(\beta_{K},\beta_{KK})^{\prime}$ as well as the mean productivity function
$r_{\omega}(\cdot)$ from

$\mathbb{E}\left[\zeta_{\omega,it}+\eta_{it}|\
k_{it},k_{it-1},m_{it-1}^{*}(m_{it-1},l_{it-1}),X_{it-1}\right]=0.$ (3.20)

In fact, given the exogeneity of regressors,
$\gamma=(\beta_{K},\beta_{KK},r_{\omega}(\cdot))^{\prime}$ can be identified
as a solution to the following nonlinear sieve M-problem:

$\displaystyle\gamma_{0}=\arg\min_{\gamma}\mathbb{E}\left[\varrho_{it}(k_{it},k_{it-1},m_{it-1}^{*},X_{it-1};\gamma)^{2}\right],$
(3.21)

where
$\varrho_{it}(k_{it},k_{it-1},m_{it-1}^{*},X_{it-1};\gamma)\equiv\zeta_{\omega,it}+\eta_{it}$
is the residual function from (3.19).

###### Remark 1

One can alternatively operationalize the third step using the inverted
conditional labor demand implied by (3.4) to construct a proxy function for
$\omega_{it}$ using observable $l_{it}^{*}$:

$\displaystyle\omega_{it}=$ $\displaystyle\
\overbrace{\ln\left[\frac{P_{t}^{L}}{P_{t}^{Y}}\right]-\ln\theta-\ln\big{(}\beta_{L}+\beta_{0}[m_{it}-\varphi_{it}-l_{it}]\big{)}+(1-\beta_{L})l_{it}-\beta_{M}m_{it}-\beta_{L}\varphi_{it}+\tfrac{1}{2}\beta_{0}[m_{it}-\varphi_{it}-l_{it}]^{2}}^{l^{*}_{it}}$
$\displaystyle\ -\beta_{K}k_{it}-\tfrac{1}{2}\beta_{KK}k_{it}^{2},$ (3.22)

or using a convex combination of the two inverted static input demands. Under
the model assumptions, all these proxies are numerically equivalent.

With $(\beta_{K},\beta_{KK})^{\prime}$ identified, we can also recover Hicks-
neutral productivity either via the proxy in (3.1) or from the production
function (3.2) as

$\displaystyle\omega_{it}=y_{it}-\beta_{K}k_{it}-\tfrac{1}{2}\beta_{KK}k_{it}^{2}-\beta_{M}m_{it}-\beta_{L}[\varphi_{it}+l_{it}]+\tfrac{1}{2}\beta_{0}[m_{it}-\varphi_{it}-l_{it}]^{2}-\eta_{it},$
(3.23)

with the translog parameters, Harrod-neutral productivity $\varphi_{it}$ and
the transitory shock ${\eta}_{it}$ successfully identified in the three steps.

On a final note, our methodology is robust to the Ackerberg et al.,  (2015)
critique that focuses on the inability of structural proxy estimators to
separably identify the additive production function and Hicksian productivity
proxy. Such an issue normally arises in the wake of perfect functional
dependence between freely varying inputs appearing both inside the unknown
production function and productivity proxy. Our third-step equation (3.19)
does not suffer from such a problem because it contains no endogenous variable
input on the right-hand side, the corresponding parameters of which have
already been identified from the variable-input-cost-to-revenue equation and
Harrodian productivity process in the first two steps.

###  3.2 Unidentification of the Standard Proxy Approach

We now show that, were one to pursue the standard proxy variable approach, the
production function (3.2) would be unidentified in the absence of external
instruments (outside the production function) for freely varying inputs.

Normally, to estimate the production function via proxy variable technique,
one makes use of the Markovian nature of unobservables and proxies for them by
inverting the firm’s optimality conditions. Specifically, along the lines of
Doraszelski and Jaumandreu, ’s (2018) empirical methodology, the estimating
model consists of two stochastic equations: (i) the production function in
(3.2) combined with the law of motion for Hicksian productivity in (2.4):

$\displaystyle y_{it}=$ $\displaystyle\
\beta_{K}k_{it}+\tfrac{1}{2}\beta_{KK}k_{it}^{2}+\beta_{M}m_{it}+\beta_{L}[\varphi_{it}+l_{it}]-\tfrac{1}{2}\beta_{0}[m_{it}-\varphi_{it}-l_{it}]^{2}\
+$ $\displaystyle\
r_{\omega}\left(\omega_{it-1},X_{it-1}\right)+\zeta_{\omega,it}+\eta_{it}$
(3.24)

and (ii) the the law of motion for Harrodian productivity in (2.5):

$\displaystyle\varphi_{it}=r_{\varphi}\left(\varphi_{it-1},Z_{it-1}\right)+\zeta_{\varphi,it},$
(3.25)

in both of which the unobservables $\varphi_{it}$ and $\omega_{it}$ are
“controlled” for using their deterministic proxy expressions in (3.7) and
(3.1), respectively.

The two-equation model in (3.2)–(3.25) suffers from endogeneity originating in
the correlation between freely varying inputs $m_{it}$ and $l_{it}$ (and
functions thereof) and contemporaneous random productivity innovations. The
remaining covariates are predetermined and can self-instrument. To identify
the model, one is to use a system of moment restrictions on the two errors
$(\zeta_{\omega,it}+\eta_{it},\zeta_{\varphi,it})^{\prime}$ à la

$\displaystyle\mathbb{E}\left[A_{it}\begin{pmatrix}\zeta_{\omega,it}+\eta_{it}\\\
\zeta_{\varphi,it}\end{pmatrix}\right]=0,$ (3.26)

where $A_{it}$ is a block-diagonal matrix of exogenous instruments and their
functions.

Following the bulk of literature, one may be tempted to make use of higher-
order777The first-order lags are already utilized in the estimation inside
Markov processes and thus self-instrument. lagged inputs
$(k_{it-2},l_{it-2},m_{it-2},\dots)^{\prime}$ and productivity-modifying
controls $(X_{it-2}^{\prime},Z_{it-2}^{\prime},\dots)^{\prime}$ to instrument
for $m_{it}$ and $l_{it}$ given that such lags meet the weak exogeneity
requirement for valid instruments under the structural assumptions. However,
the identification can only be achieved if these additional lags provide
additional relevant (exogenous) variation for $M_{it}$ after conditioning on
the already included self-instrumenting variables. Intuitively, these
additional lags must be relevant to meet the rank condition for identification
of the model.

It happens that despite the apparent abundance of internal instruments, model
(3.2)–(3.25) remains unidentified because all such relevant instruments for
$m_{it}$ and $l_{it}$ that were initially excluded from production function
(3.2) are now used to proxy for the unobserved $\varphi_{it-1}$ and
$\omega_{it-1}$. This is the key argument in the Gandhi et al.,  (2020)
critique of the proxy estimators. More formally, consider the endogenous
$m_{it}$ input which, according to the conditional material demand implied by
the first-order condition in (3.5), is given by the following implicit
function:

$\displaystyle 0=$ $\displaystyle\
\ln\left[P_{t}^{Y}/P_{t}^{M}\right]+\ln\theta+\ln\big{(}\beta_{M}-\beta_{0}\big{[}m_{it}-l_{it}-r_{\varphi}\left(\varphi_{it-1},Z_{it-1}\right)-\zeta_{\varphi,it}\big{]}\big{)}-(1-\beta_{M})m_{it}\
+$ $\displaystyle\
\beta_{L}\big{[}l_{it}+r_{\varphi}\left(\varphi_{it-1},Z_{it-1}\right)+\zeta_{\varphi,it}\big{]}-\tfrac{1}{2}\beta_{0}\big{[}m_{it}-l_{it}-r_{\varphi}\left(\varphi_{it-1},Z_{it-1}\right)-\zeta_{\varphi,it}\big{]}^{2}\
+$ $\displaystyle\
\beta_{K}k_{it}+\tfrac{1}{2}\beta_{KK}k_{it}^{2}+r_{\omega}\left(\omega_{it-1},X_{it-1}\right)+\zeta_{\omega,it},$
(3.27)

where we have substituted for $\varphi_{it}$ and $\omega_{it}$ using their
respective laws of motion. An analogous expression exists for $l_{it}$ which,
when combined with (3.2), expectedly shows that both static inputs are a
function of
$(K_{it},\varphi_{it-1},\omega_{it-1},X_{it-1}^{\prime},Z_{it-1}^{\prime},P_{t}^{Y},P_{t}^{M},P_{t}^{L},\zeta_{\varphi,it},\zeta_{\omega,it})^{\prime}$.
Comparing these determinants of $m_{it}$ and $l_{it}$ with the variables
entering the two equations in (3.2)–(3.25), it is evident that the only
sources of variation in $m_{it}$ and $l_{it}$, which have not already been
included as a self-instrumenting variable, are the prices
$(P_{t}^{Y},P_{t}^{M},P_{t}^{L})^{\prime}$ and the unobservable innovations
$\zeta_{\varphi,it}$ and $\zeta_{\omega,it}$.

Assuming the price-taking behavior of competitive firms, aggregate prices
$(P_{t}^{Y},P_{t}^{M},P_{t}^{L})^{\prime}$ provide very little—however
theoretically valid—identifying variation in practice (even in long panels) as
studied by Gandhi et al.,  (2020). This means that, conditional on the already
included predetermined variables (or their proxies), there is practically no
other relevant exogenous variables from within the production model that may
be used to instrument for the endogenous $m_{it}$ and $l_{it}$ because, for
them to be relevant in predicting $m_{it}$ and $l_{it}$, they would have to
correlate with $\zeta_{\varphi,it}$ and $\zeta_{\omega,it}$, which is the only
source of “free” variation left in static inputs. The correlation with these
productivity innovations would however violate the exogeneity requirement
thereby invaliding the instruments. Therefore, both flexible inputs lack
excluded relevant internal instruments, and the model in (3.2)–(3.25) is
unidentified.

Note that, although not explicitly discussed, this unidentification problem is
overcome by Doraszelski and Jaumandreu,  (2018) along the lines of their
earlier work in Doraszelski and Jaumandreu,  (2013) by incorporating external
instruments such as demand shifters and, importantly, lagged firm-level
variation in input prices. However, while suitable for their empirical
application, the validity and practicality of using lagged firm-level prices
for identification is not universal. Not only are the price data often
unavailable or prone to measurement errors (Levinsohn and Petrin, , 2003), but
the use of prices may also be problematic on theoretical grounds (see
Griliches and Mairesse, , 1998; Ackerberg et al., , 2007, 2015; Flynn et al.,
, 2019).

Specifically, the validity of input prices as exogenous instruments is
normally justified by invoking the assumption of perfectly competitive
markets. However, if firms were indeed price-takers, in theory, one should not
observe the firm-level variation in prices and, without such a variation,
prices cannot be used as operational instruments. Even with the aggregate
prices varying exogenously across space, such a variation may be insufficient
for identification as discussed earlier. If a researcher does observe the
variation in prices across all individual firms, the latter variation may be
reflecting differences in the firms’ market power and/or the quality of
inputs/outputs. For instance, if the firm-level variation in input prices
reflects differential quality in inputs, then random updates in prices that
render lagged prices usable instruments888That is, not perfectly dependent
with contemporaneous prices that enter estimating equations directly. are
likely related to productivity innovations because a more productive firm is
to use more productive, higher-quality inputs (Flynn et al., , 2019). Thus, be
it due to the market power or quality differentials, the variation in prices
will then be endogenous to firms’ decisions and hence cannot help the
identification (also see Gandhi et al., , 2020). Furthermore, putting the
issue of exogeneity aside, Flynn et al.,  (2019) raise concerns about the
strong conditions on the price evolution processes that must be satisfied for
the lagged prices to have any strength as instruments. We therefore pursue an
alternative identification strategy which does not require firm-level
variation in prices.

To achieve identification without external firm-level instruments, we build
upon Gandhi et al., ’s (2020) ideas by effectively augmenting a system in
(3.2)–(3.25) with an additional (simultaneous) equation for the optimal
variable-input-cost-to-revenue ratio in (3.10).

##  4 Estimation Procedure

We now describe how to empirically implement our identification strategy
outlined in Section 3.1. We estimate the unknown production-function
parameters and the two components of persistent firm productivity
$\varphi_{it}$ and $\omega_{it}$ via a three-stage procedure. If the
functional form of productivity conditional-mean functions $r_{\omega}(\cdot)$
and $r_{\varphi}(\cdot)$ in the evolution processes (2.4)–(2.5) is known, the
estimation becomes fully parametric which, among other things, can streamline
asymptotic inference. For instance, a go-to choice in applied productivity
research involving the proxy-variable estimation of production functions is to
assume that firm productivity is a linear AR(1) process (e.g., see Zhang, ,
2017, 2019; Kim et al., , 2019; Ackerberg et al., , 2020; Grieco et al., ,
2020; Mo et al., , 2021). To maximize impact among practitioners, in what
follows, we also assume that both $r_{\omega}(\cdot)$ and $r_{\varphi}(\cdot)$
are linear (parametric) functions.999We can also justify this linearity as a
sieve approximation using linear polynomials. However, in this case the
estimation will no longer be “parametric” but semiparametric. See Appendix C.
We discuss a semiparametric alternative to our estimator in which the unknown
functions are approximated using sieves in Appendix C.

In the first step, we consistently estimate $\beta_{L}+\beta_{M}$ via a sample
analogue of (3.12):

$\widehat{\delta}_{LM}\equiv\widehat{\left(\beta_{L}+\beta_{M}\right)}=\frac{\exp\\{\frac{1}{nT}\sum_{it}\ln
R_{it}\\}}{\frac{1}{nT}\sum_{it}\exp\left\\{\left(\frac{1}{nT}\sum_{it}\ln
R_{it}\right)-\ln R_{it}\right\\}},$ (4.1)

where the denominator is
$\widehat{\theta}=\frac{1}{nT}\sum_{it}\exp\left\\{\left(\frac{1}{nT}\sum_{it}\ln
R_{it}\right)-\ln R_{it}\right\\}$.

We then proceed to the second-step estimation of the Harrodian productivity
process in (3.13). We parameterize the unknown function
$r_{\varphi}(\varphi_{it-1},Z_{it-1})$ using a linear function,101010As noted
earlier, we normalize $r_{\varphi}(0)=0$ since $\varphi_{it}$ can be
identified up to a constant only. In practice, this implies that
$r_{\varphi}(\cdot)$ is parameterized using a linear function with no
intercept. with the unobservable $\varphi_{it-1}$ substituted for by the proxy
function from (3.7) and $\delta_{LM}$ replaced with its estimate from the
first step. The second-step equation is then estimated via nonlinear GMM along
the lines of (3.15):

$\displaystyle\widehat{\alpha}=\arg\min_{\alpha}\Big{[}\tfrac{1}{nT}\sum_{it}\mathbb{Q}_{it-1}\epsilon_{it}(D_{it};\alpha)\Big{]}^{\prime}W\Big{[}\tfrac{1}{nT}\sum_{it}\mathbb{Q}_{it-1}\epsilon_{it}(D_{it};\alpha)\Big{]},$
(4.2)

where
$\alpha=(\beta_{0},\beta_{L},\rho_{\varphi,1},\rho_{\varphi,2})^{\prime}$,
$D_{it}=(m_{it}-l_{it},m_{it-1}-l_{it-1},S^{L}_{it},S^{L}_{it-1},Z_{it-1}^{\prime})^{\prime}$,
and the corresponding residual function is

$\displaystyle\epsilon_{it}(D_{it};\alpha)=$ $\displaystyle\
m_{it}-l_{it}+\frac{\beta_{L}}{\beta_{0}}-\frac{\widehat{\delta}_{LM}}{\beta_{0}}S^{L}_{it}\
-\rho_{\varphi,1}\left[m_{it-1}-l_{it-1}+\frac{\beta_{L}}{\beta_{0}}-\frac{\widehat{\delta}_{LM}}{\beta_{0}}S^{L}_{it-1}\right]-\rho_{\varphi,2}Z_{it-1}.$
(4.3)

Following our earlier arguments, here we expand the instrument vector to
include capital as additional instruments:
$\mathbb{Q}_{it-1}=(1,m_{it-1}-l_{it-1},S^{L}_{it-1},Z_{it-1}^{\prime},k_{it},k_{it-1},\dots)^{\prime}$.
With $\big{(}\widehat{\beta}_{0},\widehat{\beta}_{L}\big{)}^{\prime}$ in hand,
we construct $\widehat{\beta}_{M}=\widehat{\delta}_{LM}-\widehat{\beta_{L}}$
as well as the estimator of Harrod-neutral productivity via
$\widehat{\varphi}_{it}=m_{it}-l_{it}+\widehat{\beta}_{L}/\widehat{\beta}_{0}-\widehat{\delta}_{LM}/\widehat{\beta}_{0}\times
S^{L}_{it}$.

To estimate the third-step equation in (3.19), we first construct estimators
of $y_{it}^{*}$ and $m_{it}^{*}$ using the results from steps one and two:
$\widehat{y}_{it}^{*}=y_{it}-\widehat{\beta}_{M}m_{it}-\widehat{\beta}_{L}[\widehat{\varphi}_{it}+l_{it}]+\tfrac{1}{2}\widehat{\beta}_{0}[m_{it}-\widehat{\varphi}_{it}-l_{it}]^{2}$
and
$\widehat{m}^{*}_{it}=\ln\left[P_{t}^{M}/P_{t}^{Y}\right]-\ln\widehat{\theta}-\ln\big{(}\widehat{\beta}_{M}-\widehat{\beta}_{0}[m_{it}-\widehat{\varphi}_{it}-l_{it}]\big{)}+(1-\widehat{\beta}_{M})m_{it}-\widehat{\beta}_{L}[\widehat{\varphi}_{it}+l_{it}]+\tfrac{1}{2}\widehat{\beta}_{0}[m_{it}-\widehat{\varphi}_{it}-l_{it}]^{2}$.
Then, using a linear parameterization for $r_{\omega}(\cdot)$ with
$\omega_{it-1}$ replaced by its proxy, we estimate
$\gamma=(\beta_{K},\beta_{KK},\rho_{\omega,0},\rho_{\omega,1},\rho_{\omega_{2}})^{\prime}$
via nonlinear sieve least squares in line with (3.21):

$\displaystyle\widehat{\gamma}=\arg\min_{\gamma}\sum_{it}\Bigg{[}\widehat{y}_{it}^{*}-\beta_{K}k_{it}-\tfrac{1}{2}\beta_{KK}k_{it}^{2}-\rho_{\omega,0}-\rho_{\omega,1}\big{[}\widehat{m}^{*}_{it-1}-\beta_{K}k_{it-1}-\tfrac{1}{2}\beta_{KK}k_{it-1}^{2}\big{]}-\rho_{\omega,2}X_{it-1}\Bigg{]}^{2}.$
(4.4)

Using the obtained $(\widehat{\beta}_{K},\widehat{\beta}_{KK})^{\prime}$
estimates, we then construct Hicks-neutral productivity via
$\widehat{\omega}_{it}=y_{it}-\widehat{\beta}_{K}k_{it}-\tfrac{1}{2}\widehat{\beta}_{KK}k_{it}^{2}-\widehat{\beta}_{M}m_{it}-\widehat{\beta}_{L}[\widehat{\varphi}_{it}+l_{it}]+\tfrac{1}{2}\widehat{\beta}_{0}[m_{it}-\widehat{\varphi}_{it}-l_{it}]^{2}-\widehat{\eta}_{it}$
using
$\widehat{\eta}_{it}=\ln\big{(}\widehat{\theta}\widehat{\delta}_{LM}\big{)}-\ln
R_{it}$ from step one.

The outlined three-step estimator is consistent and asymptotically normal.
This is easy to establish by recasting all three steps in a multiple-equation
system GMM framework which, conveniently, also permits the derivation of an
asymptotic variance-covariance matrix that accounts for a multi-step nature of
the estimator (see Newey, , 1984). Essentially, we can rewrite our sequential
estimator as a simultaneous multiple-equation system of moment restrictions
where the instrument sets vary across equations.

Specifically, referring to all unknown parameters collectively as
$\Lambda=\big{(}\beta_{0},\beta_{L},\beta_{M},\beta_{K},\beta_{KK},\theta,\rho_{\varphi,1},\rho_{\varphi,2},$
$\rho_{\omega,0},\rho_{\omega,1},\rho_{\omega,2}\big{)}^{\prime}$, the fully
expanded third-step error after the substitutions for $y^{*}_{it}$,
$\varphi_{it}$ and $m^{*}_{it}$ is

$\displaystyle r_{it}(\Lambda)=$ $\displaystyle
y_{it}-{\beta}_{M}m_{it}-{\beta}_{L}\left[m_{it}+\frac{\beta_{L}}{\beta_{0}}-\frac{\beta_{L}+\beta_{M}}{\beta_{0}}S^{L}_{it}\right]+\frac{1}{2\beta_{0}}\left[-{\beta_{L}}+(\beta_{L}+\beta_{M})S^{L}_{it}\right]^{2}-\beta_{K}k_{it}-\tfrac{1}{2}\beta_{KK}k_{it}^{2}-\rho_{\omega,0}-$
$\displaystyle\rho_{\omega,1}\Bigg{\\{}\ln\left[\frac{P_{t}^{M}}{P_{t}^{Y}}\right]-\ln\left[\theta(\beta_{L}+\beta_{M})(1-S^{L}_{it})\right]+(1-{\beta}_{M})m_{it}-{\beta}_{L}\left[m_{it}+\frac{\beta_{L}}{\beta_{0}}-\frac{\beta_{L}+\beta_{M}}{\beta_{0}}S^{L}_{it}\right]+\frac{1}{2\beta_{0}}\left[-{\beta_{L}}+(\beta_{L}+\beta_{M})S^{L}_{it}\right]^{2}-$
$\displaystyle\beta_{K}k_{it-1}-\tfrac{1}{2}\beta_{KK}k_{it-1}^{2}\Bigg{\\}}-\rho_{\omega,2}X_{it-1},$

and we can rewrite the three estimation steps in the form of their equivalent
multiple-equation moment restrictions:

$\mathbb{E}\left[\boldsymbol{f}(\Lambda)\right]\equiv\mathbb{E}\begin{bmatrix}\ln
R_{it}-\ln\big{(}\theta\left[\beta_{L}+\beta_{M}\right]\big{)}\\\
\exp\left\\{\ln\big{(}\theta\left[\beta_{L}+\beta_{M}\right]\big{)}-\ln
R_{it}\right\\}-\theta\\\
\left(m_{it}-l_{it}+\frac{\beta_{L}}{\beta_{0}}-\frac{\beta_{L}+\beta_{M}}{\beta_{0}}S^{L}_{it}\
-\rho_{\varphi,1}\left[m_{it-1}-l_{it-1}+\frac{\beta_{L}}{\beta_{0}}-\frac{\beta_{L}+\beta_{M}}{\beta_{0}}S^{L}_{it-1}\right]-\rho_{\varphi,2}Z_{it-1}\right)\mathbb{Q}_{it-1}\\\
r_{it}(\Lambda)\dfrac{\partial
r_{it}(\Lambda)}{\partial(\beta_{K},\beta_{KK},\rho_{\omega,0},\rho_{\omega,1},\rho_{\omega_{2}})^{\prime}}\end{bmatrix}=\mathbf{0},$
(4.5)

consisting of three blocks, where the first two moments correspond to the
sample estimator of $\delta_{LM}=\beta_{L}+\beta_{M}$ and $\theta$ (first
block), the middle $\dim(\mathbb{Q})$ moments correspond to the GMM estimation
of $\alpha=(\beta_{0},\beta_{L},\rho_{\varphi,1},\rho_{\varphi,2})^{\prime}$
in (4.2) (second block) and and the remaining 5 orthogonality conditions
correspond to the nonlinear least-squares estimation of
$\gamma=(\beta_{K},\beta_{KK},\rho_{\omega,0},\rho_{\omega,1},\rho_{\omega_{2}})^{\prime}$
in (4.4) (third block).

The benefit of interpreting our sequential multi-step estimator as solving a
GMM problem corresponding to a system of nonlinear moment equations in (4.5)
simultaneously is that the standard large-$n$ limit results for a class of
such GMM estimators apply here. Furthermore, using the moment equivalents in
(4.5) also facilitates asymptotic inference based on the optimal covariance
matrix of GMM estimators
$\mathbb{V}\big{[}\widehat{{\Lambda}}\big{]}=\big{[}\mathbb{E}\frac{\partial\boldsymbol{f}({\Lambda})}{\partial{\Lambda}^{\prime}}\big{]}^{-1}\mathbb{E}[\boldsymbol{f}({\Lambda})\boldsymbol{f}({\Lambda})^{\prime}]\big{[}\mathbb{E}\frac{\partial\boldsymbol{f}({\Lambda})}{\partial{\Lambda}}\big{]}^{-1}$
which, if desired, can also be robustified using usual off-the-shelf methods.
Having said that, should one find evaluating the analytical covariance matrix
tedious or in the case when asymptotic inference is difficult to justify,
bootstrap provides an alternative avenue for hypothesis testing. We discuss
how to approximate the sampling distribution of our estimator via wild
residual block bootstrap111111Other bootstrap procedures can also be used.
that takes into account a sequential nature of our methodology in Appendix D.

##  5 Finite-Sample Performance

In this section, we examine the finite-sample performance of our methodology.
We first demonstrate its ability to successfully identify the multi-
dimensional firm productivity and the production-function parameters in a
small Monte Carlo study. Then, we apply our estimator to the firm-level data
to provide an empirical illustration in practice.

###  5.1 Simulations

We conduct simulations to evaluate the performance of our proposed estimator
in finite samples. Our data generating process (DGP) draws from those used by
Grieco et al.,  (2016), Gandhi et al.,  (2020) and Malikov and Zhao,  (2021).
More specifically, we consider a balanced panel of $n=\\{100,200,400,1600\\}$
firms operating during $T=10$ time periods.121212We have also experimented
with 5 and 50 periods. The results are qualitatively unchanged. Each panel is
simulated 1,000 times. We let that the true production technology take a
(restricted) translog form with a two-dimensional firm productivity given in
(3.2), where we set $\beta_{K}=0.2$, $\beta_{KK}=-0.01$, $\beta_{M}=0.5$,
$\beta_{L}=0.25$ and $\beta_{0}=-0.05$. Given the DGPs for the production
variables below, this choice of parameter values facilitates that the
monotonicity and curvature properties of the production function are satisfied
in the generated data; firms exhibit decreasing returns to scale.

The persistent firm productivity components are generated as follows. To
simplify matters, we abstract away from productivity modifiers ($X$ and $Z$)
and model the Hicks- and Harrod-neutral productivities as exogenous linear
AR(1) processes:

$\displaystyle\omega_{it}$
$\displaystyle=\rho_{\omega,0}+\rho_{\omega,1}\omega_{it-1}+\zeta_{\omega,it},$
(5.1) $\displaystyle\varphi_{it}$
$\displaystyle=\rho_{\varphi,1}\varphi_{it-1}+\zeta_{\varphi,it},$ (5.2)

where we set $\rho_{\omega,0}=0.2$, $\rho_{\omega,1}=0.6$ and
$\rho_{\varphi,1}=0.9$. The innovations are drawn independently as
$\zeta_{\omega,it}\sim\ i.i.d.\ \mathbb{N}(0,\sigma_{{\omega}}^{2})$ and
$\zeta_{\varphi,it}\sim\ i.i.d.\ \mathbb{N}(0,\sigma_{{\varphi}}^{2})$, with
$\sigma_{{\omega}}=\sigma_{{\varphi}}=0.04$. The initial levels of Hicks- and
Harrod-neutral productivities $\omega_{i1}$ and $\varphi_{i1}$ are drawn from
$\mathbb{U}\left(-1,1\right)$ identically and independently distributed over
$i$. The random transitory productivity shocks $\\{\eta_{it}\\}$ entering the
production function are drawn from $\eta_{it}\sim\ i.i.d.\
\mathbb{N}\big{(}0,\sigma_{\eta}^{2}\big{)}$ with $\sigma_{\eta}=0.07$.

We assume the following about evolution of the firm’s state variables.
Physical capital, a dynamic predetermined input, is set to evolve according to
$K_{it}=I_{it-1}+\left(1-\delta_{i}\right)K_{it-1}$, where the firm-specific
depreciation rates $\delta_{i}\in\left\\{0.05,0.075,0.10,0.125,0.15\right\\}$
are distributed uniformly across $i$, and the investment function takes the
following form:

$I_{it-1}=K_{it-1}^{\iota_{1}}\left[\exp\left\\{\omega_{it-1}\right\\}\right]^{\iota_{2}}\left[\exp\left\\{\varphi_{it-1}\right\\}\right]^{\iota_{3}},$
(5.3)

where $\iota_{1}=0.8$ and $\iota_{2}=\iota_{3}=0.1$. The initial level of
capital is generated as $K_{i1}\sim\ i.i.d.\ \mathbb{U}\left(10,200\right)$.

The optimal labor and materials series (the freely varying inputs) are
generated by numerically solving the firm’s static first-order conditions in
(3.4)–(3.5) after having already generated the series of
$\left(K_{it},\omega_{it},\varphi_{it}\right)^{\prime}$ for each firm and time
period. When doing so, we normalize $P_{t}^{L}=P_{t}^{M}=\theta\ \forall\ t$
and also intentionally assume away any temporal variation in output prices:
$P_{t}^{Y}=1$ for all $t$. In such a scenario, changes in the firm’s labor-to-
materials ratio are driven by the improvements in labor-augmenting
productivity only.

Table 1: Simulation Results for the Proposed Three-Step Estimator

$n=100$ $n=200$ $n=400$ $n=1600$ Mean RMSE MAE Mean RMSE MAE Mean RMSE MAE
Mean RMSE MAE $\beta_{K}$ 0.1900 0.5316 0.4115 0.1705 0.3619 0.2821 0.2077
0.2549 0.2039 0.2003 0.1277 0.1003 $\beta_{KK}(\times 10)$ –0.0856 0.8633
0.6704 –0.0518 0.5927 0.4635 –0.1123 0.4167 0.3335 –0.1006 0.2096 0.1644
$\beta_{L}$ 0.2502 0.0038 0.0030 0.2502 0.0024 0.0019 0.2500 0.0018 0.0014
0.2500 0.0009 0.0007 $\beta_{M}$ 0.4997 0.0038 0.0030 0.4998 0.0024 0.0019
0.4999 0.0018 0.0014 0.4999 0.0009 0.0007 $\beta_{0}(\times 10)$ –0.5024
0.0165 0.0124 –0.5008 0.0104 0.0081 –0.5003 0.0070 0.0056 –0.5004 0.0036
0.0029 $\rho_{\varphi,1}$ 0.8990 0.0135 0.0107 0.8995 0.0094 0.0074 0.8998
0.0062 0.0050 0.8998 0.0031 0.0024 $\rho_{\omega,1}$ 0.5981 0.0381 0.0305
0.5988 0.0268 0.0211 0.5996 0.0198 0.0156 0.5998 0.0095 0.0076
$\rho_{\omega,0}$ 0.2082 0.6385 0.4987 0.2338 0.4380 0.3418 0.1891 0.3083
0.2475 0.1993 0.1546 0.1217 Notes: The true parameter values are
$\beta_{K}=0.20$, $\beta_{KK}\times 10=-0.10$, $\beta_{L}=0.25$,
$\beta_{M}=0.50$, $\beta_{0}\times 10=-0.50$, $\rho_{\varphi,1}=0.90$,
$\rho_{\omega,1}=0.60$ and $\rho_{\omega,0}=0.20$. Throughout, $T=10$.

We estimate our model via the three-step algorithm outlined in Section
4.131313Following our discussion, we include $k_{it}$ as an additional
instrument in the second-step estimation. For each simulation repetition, we
obtain point estimates of the production-function and productivity-process
parameters and then report the mean, the root mean squared error (RMSE) and
the mean absolute deviation (MAD) of these point estimates computed over 1,000
simulations. Table 1 reports these simulation results. The results are
encouraging and show that, with a modestly large sample size, our methodology
recovers the true parameters fairly well, thereby lending support to the
validity of our identification strategy. Of all parameters, those obtained in
the third step are the most imprecisely estimated. This is unsurprising given
that their (nonlinear) estimation relies on the generated regressors estimated
in not one but two previous steps all of which are measured with sampling
error. But overall, as expected of consistent estimators, the estimation
becomes more stable as $n$ grows.

###  5.2 Empirical Illustration

We showcase our methodology by applying it to study the multi-dimensional
productivity heterogeneity among Chinese manufacturing firms. We let the
Hicks- and Harrod-neutral productivities share the same scalar productivity
shifter, i.e., $X_{it-1}=Z_{it-1}$ in (2.4) and (2.5). Given the well-
documented importance of inbound FDI—as a vehicle of international
technological diffusion—for productivity advances among domestic firms in the
recipient countries (see Malikov and Zhao, , 2021, for more discussion and
references to the related literature), we use a measure of the foreign equity
share as a productivity-modifying “control,” with an objective to examine the
potentially differential factor-neutral and labor-saving effects of FDI on
firm productivity. Through their foreign investors, domestic firms gain access
to intangible productive “knowledge" assets from abroad such as new
technologies, proprietary know-hows, more efficient and innovative marketing
and management practices, established relational networks, reputation, etc.,
which can help boost their productivity. Whether such knowledge/technology
transfers are neutral or biased remains, however, unexamined.

Data.—Our data come from the Chinese Industrial Enterprises Database survey
conducted by China’s National Bureau of Statistics (NBS). We focus on the
“leather, fur, feather and the related products” industry (SIC 2-digit code
19) because China is the largest leather producing country in the world,
representing more than a quarter of the annual global production, and is one
of the world’s largest leather exporters and importers. Also, a relatively
large share of firms (15.2%) in this industry are foreign-invested.

The production variables are standard. The firm’s capital stock ($K_{it}$) is
the net fixed assets deflated by the price index of investment into fixed
assets. Labor ($L_{it}$) is measured as the total wage bill plus benefits
deflated by the GDP deflator. Materials ($M_{it}$) are the total intermediate
inputs, including raw materials and other production-related inputs, deflated
by the purchasing price index for industrial inputs. The output ($Y_{it}$) is
defined as the gross industrial output value deflated by the producer price
index. The price indices are obtained from NBS and the World Bank. The four
variables are measured in thousands of real RMB. Our sample period runs from
1998 to 2007, and the operational sample is an unbalanced panel of 11,167
firms with a total of 31,287 observations.141414We exclude observations with
missing values for production variables as well as a small number of likely
erroneous observations with the foreign equity share values outside the unit
interval.

The summary statistics of data are reported in Table E.1 in Appendix E. For
all variables, the mean values are much larger than their medians, consistent
with their distributions being right-skewed, and their inter-quartile ranges
are wide. The large heterogeneity of variable distributions suggests that a
more flexible model of the production process, such as ours, is needed to
better characterize the production relationship between inputs and output.
Also note that all variable statistics are larger for foreign-invested firms
($Z>0$) compared with their wholly-domestically-owned counterparts ($Z=0$).
This difference provides a further motivation to incorporate the information
about firms’ exposure to foreign investment into the analysis, which we
accomplish by conditioning productivity evolution processes on the foreign
equity share.

Results.—We report the estimated input elasticities, returns to scale (RTS)
and the productivity-process parameters in Table 2. Note that, albeit
parametrically, we model the production function using a log-quadratic form,
which is why the estimated elasticities and RTS are observation-specific. As
discussed in Section 4, we assume that both $r_{\omega}(\cdot)$ and
$r_{\varphi}(\cdot)$ are linear given the wide popularity of an AR(1)
assumption for productivity processes among practitioners. Therefore, the
estimated productivity parameters are fixed and global. Lastly, following the
discussion of our methodology, our main results are obtained using the
inverted conditional material demand to proxy for Hicksian productivity. While
in theory the estimator is invariant to the choice of a variable input for the
role of $\omega$ proxy, whether the latter is the case in practice or not,
effectively, provides an indirect test of our model assumptions. The
counterpart of Table 2 containing the results obtained using the labor proxy
as well as the average of labor and material proxies are provided in Appendix
F. The estimates differ little and are qualitatively unchanged.

Table 2: Estimates of the Production Function and Productivity Parameters
Panel A: Elasticities and Returns to Scale
---
| Mean | 1st Qu. | Median | 3rd Qu.
Capital elasticity | 0.0361 | 0.0213 | 0.0368 | 0.0507
| (0.0080) | (0.0094) | (0.0081) | (0.0099)
Labor elasticity | 0.0950 | 0.0458 | 0.0864 | 0.1287
| (0.0003) | (0.0001) | (0.0002) | (0.0004)
Material elasticity | 0.7377 | 0.7041 | 0.7463 | 0.7869
| (0.0020) | (0.0019) | (0.0021) | (0.0022)
RTS | 0.8688 | 0.8540 | 0.8695 | 0.8834
| (0.0081) | (0.0095) | (0.0082) | (0.0100)
Panel B: Productivity Parameters
Labor-Augmenting |  | Factor-Neutral
$\rho_{\varphi,0}$ | 0.0000 |  | $\rho_{\omega,0}$ | 0.6525
| — |  |  | (0.1026)
$\rho_{\varphi,1}$ | 0.5723 |  | $\rho_{\omega,1}$ | 0.7562
| (0.0244) |  |  | (0.0307)
$\rho_{\varphi,2}$ | 0.0726 |  | $\rho_{\omega,2}$ | 0.0020
| (0.0183) |  |  | (0.0084)
Notes: The productivity processes are parameterized as follows:
$\varphi_{it}=\rho_{\varphi,1}\varphi_{it-1}+\rho_{\varphi,2}Z_{it-1}+\zeta_{\varphi,it}$
with $\rho_{\varphi,0}$ normalized to 0, and
$\omega_{it}=\rho_{\omega,0}+\rho_{\omega,1}\omega_{it-1}+\rho_{\omega,2}Z_{it-1}+\zeta_{\omega,it}$.
Bootstrap standard errors are in parentheses.

Per the results in Panel A of Table 2, the manufacturers of leather, fur,
feather and related products in China show a large material elasticity and
relatively small elasticities of capital and labor. This oversized importance
of intermediate inputs (materials) compared with the capital and labor inputs
in the production process of Chinese manufacturing firms is confirmed by other
studies that used the same dataset (see, e.g., Brandt et al., , 2017; Zhao et
al., , 2020; Malikov and Zhao, , 2021; Malikov et al., , 2021). The implied
estimates of RTS have the mean value of 0.8688 (median is 0.8695) with the
rather narrow inter-quartile range of 0.029. Statistically, all firms exhibit
decreasing returns to scale, i.e., diseconomies of scale. This inference is
based on the RTS point estimate being statistically less than 1 at the 5%
significance level.

Panel B of Table 2 reports the estimated productivity parameters for factor-
netural and labor-augmenting components of firm productivity. As discussed in
Section 3.1, because function $r_{\varphi}(\cdot)$ is identified only up to a
constant, the intercept coefficient for $\varphi_{it}$ is normalized to 0.
Comparing the autoregressive coefficients $\rho_{\varphi,1}$ and
$\rho_{\omega,1}$, we find that Harrod-neutral productivity is not as
persistent over time as is Hicks-neutral productivity. Interestingly, we find
that the foreign equity share—a productivity modifier of interest—has both
economically and statistically significant marginal effect on labor-saving
productivity, whereas the effect size on factor-neutral productivity is
insignificant and effectively zero. From this we can conclude that, at least
in China’s leather industry, the productivity-boosting effect of FDI on
domestic firms’ productivity has a bias towards labor. Thus, better/new
technologies and more efficient business practices that firms “import” and
learn from abroad through their foreign investors, as commonly argued in the
FDI literature, appear to be primarily labor-saving as opposed to boosting
marginal productivity of all factors. This is a novel empirical finding. More
concretely, our point estimate of $\rho_{\varphi,2}$ implies that a 10
percentage point increase in the firm’s foreign equity share boosts its
expected future labor-biased productivity by about 0.73%. While this effect
size might at first appear to be too modest, it is imperative to remember that
$\rho_{\varphi,2}$ only captures a short-run impact of FDI on labor-augmenting
productivity (that is,
$\partial\mathbb{E}[\varphi_{it+1}|\Xi_{it}]\big{/}\partial Z_{it}$) and does
not account for dynamic effects over time. Obviously, owing to the persistence
of productivity, the cumulative implications of receiving more FDI are
expected to be bigger in the long run. In fact, under temporal stationarity of
$\varphi_{it}$ we have that the long-run effect of a 10 percentage point
increase in the firm’s foreign equity share on its labor-augmenting
productivity is estimated at $0.73/(1-0.5723)\approx 1.7$%.

(a)

(b)

Figure 1: Distributions of Hicks- and Harrod-Neutral Productivities: (a) in
levels, (b) in growth rates

Figure 1 compares the empirical distributions of the two estimated
productivities. We report box-plots of the estimated $\varphi_{it}$ and
$\omega_{it}$ by year in Figure 1(a) and of their annual changes in Figure
1(b). For the ease of comparison, the medians of both productivity terms are
normalized to zero in the year 1998. We see that the distributions of Harrod-
and Hicks-neutral productivities behave quite differently. First, according to
Figure 1(a), the labor-augmented productivity has a larger cross-sectional
variation, as exhibited by wider inter-quartile ranges and longer whiskers.
Second, in each year, the Hicks-neutral productivity is distributed almost
symmetrically across individual firms, whereas the labor-augmented
productivity is heavily skewed to the right. Third, based on the medians in
each year, the Hicks-neutral productivity steadily shifts up over time, but we
cannot say the same about the labor-augmented productivity. Instead, it is
mainly the the upper/right whiskers of the labor-augmented productivity that
generally shift up over time, suggesting the presence of persistently more
labor-efficient firms that keep becoming more productive.

In Figure 1(b), we plot the box-plots of annual productivity changes in logs,
i.e., $\varphi_{it}-\varphi_{it-1}$ and $\omega_{it}-\omega_{it-1}$. Since
both the $\varphi_{it}$ and $\omega_{it}$ are attached to the log inputs and
output, their changes can be approximately interpreted as the corresponding
within-firm productivity growth rates. We see that, compared with factor-
neutral productivity, the labor-augmenting productivity also exhibits a larger
cross-sectional variation in its growth. The median growth rate of Hicks-
neutral productivity is non-negative across all years, whereas that of labor-
augmenting productivity oscillates around zero, with essentially a nil
cumulative effect.

To see the cumulative and total impact of the growth in these two productivity
components on the industry output, we calculate the aggregate industry output-
weighted Harrod-neutral and Hicks-neutral productivities and plot their trends
in Figure 2(a). These two aggregate productivities are depicted using solid-
circle and dashed-triangle lines, respectively, and for comparability, both of
them are normalized to zero in the year 1998. We find that, during our sample
period, the industry-level factor-neutral productivity was steadily rising and
increased by around 20%. However, the aggregate labor-augmenting productivity
peaked in 2002 right after China’s accession to the WTO and decreased since
then, although with a rebound in the last year to a level close to that of the
beginning of the sample period. This trend is generally consistent with the
labor-to-material ratio box-plot in Figure 2(b). Were the labor-augmenting
productivity to increase significantly in our sample period, we would have
expected the labor-to-material ratio to decrease over time too. Such labor-
saving technological advances have been documented by Doraszelski and
Jaumandreu,  (2018) and Zhang,  (2019) for different countries/industries. In
our case, however, the post-2002 downward trend of labor-augmented
productivity is in line with the upward shift in the labor-to-material ratios
observed in the data.151515The graph for the output-weighted average labor-to-
material ratios looks similar. The widening whiskers over the years in Figure
2(b) are also consistent with the increasing variation of labor-augmenting
productivity over time documented in Figure 1(a).

(a)

(b)

Figure 2: Productivity and Flexible Input Ratio Trends

Now, note that, because the firm’s output is log-linear in $\omega_{it}$, the
magnitude of Hicks-neutral productivity growth is directly corresponding to
the output growth. However, this is not the case for the non-neutral
productivity $\varphi_{it}$ that affects output via labor. The marginal effect
of $\varphi_{it}$ on the firm’s (log) output depends on the labor elasticity.
Therefore, to make a fair comparison between Hicks- and Harrod-neutral
productivities in terms of their effects on output growth, we follow
Doraszelski and Jaumandreu,  (2018) in computing the product of the labor
elasticity $\partial f_{it}/\partial l_{it}$ and $\varphi_{it}$ and refer to
it as the labor-augmenting productivity in output terms. In Figure 2(a), we
plot the industry output-weighted average for it using a dot-dashed-cross
line. The line is almost flat, implying that the labor-augmenting productivity
had no material effect on the industry output growth during our sample period.
Thus, our estimates provide evidence that the overall productivity growth in
China’s leather industry in 1998–2007 was factor-neutral.

##  6 Extension to Imperfect Competition

As argued by Battisti et al.,  (2022), empirical studies have so far produced
very limited evidence on the magnitude of non-neutral technological change
with market imperfections.161616In their study, Battisti et al.,  (2022)
present a new approach and estimate the skill-biased technical change from the
production side while allowing for labor-market inefficiencies using country-
level data. Our methodology and its underlying identification scheme presented
above are developed under the assumption that firms operate in a perfectly
competitive output market. Although this assumption continues to be
maintained—implicitly or explicitly—by most productivity studies, which is
partly dictated by the typical unavailability of firm-level price
data,171717This is because, unless the quantity information about
output/inputs is observed in the data (which is rare in practice), researchers
commonly assume perfect competition with homogeneous prices to justify
deflating nominal revenues and expenditures using price indices to obtain real
values. there has been notable effort recently aimed at extending proxy
variable production-function estimators to accommodate market power. Besides
usually requiring the data on exogenous demand shifters, most such methods
also tend to rely on (observable) exogenous heterogeneous input prices and/or
isoelastic demand specifications for identification (e.g., see De Loecker, ,
2011; De Loecker and Warzynski, , 2012; De Loecker et al., , 2016; Doraszelski
and Jaumandreu, , 2013, 2019). Others resort to restrictions on the production
technology such as the constant returns to scale (see Flynn et al., , 2019;
Raval, , 2020) or abandon the structural “proxy variable” paradigm in favor of
a more “atheoretical” control function approach to handling endogeneity-
inducing unobservable firm productivity (Demirer, , 2020).

Motivated by this emerging literature, we show how to relax the perfect
competition assumption and allow for monopolistic power in the output market.
We do so while retaining the assumption of competitive homogeneous factor
prices, given our earlier discussion concerning the use of firm-level
variation in input prices for identification. In Appendix G, we first show
point non-identification of the production function in (3.2) when firms
exhibit market power even if one observes exogenous demand shifters in the
data. We then discuss how with some additional but quite reasonable
assumptions about unobservables one can still set identify the production
function. We derive this partial identification result without requiring that
the information on demand shifters be available and rely on the same set of
observables that are used in our methodology for the case of perfectly
competitive markets.

##  7 Concluding Remarks

The literature on proxy variable identification of production functions is
dominated by models that accommodate a single source of firm heterogeneity: a
scalar Hicks-neutral productivity. In this paper, we contribute to the
relatively thin but emerging strand of the literature that seeks to generalize
these proxy methods to allow for non-neutral production technology by
considering the identification of a translog production function when latent
firm productivity is multi-dimensional, with biased labor-augmenting and
factor-neutral components. In contrast to the available alternatives, our
model can be identified under weaker data requirements, notably, without
relying on the firm-level input price information as instruments. This
tremendously increases the practical value of our methodology by making it
applicable to most production datasets. When markets are perfectly
competitive, we achieve point identification by leveraging the information
contained in static optimality conditions, in effect, adopting a system-of-
equations approach. We also show how one can set identify the production
function with non-neutral productivity in the traditional proxy variable
framework when firms have market power.

## Appendix

##  Appendix A Identification under the CES Specification

In Section 3, we present our identification strategy for a restricted translog
production function. To show that the same strategy can also be applied to
other production function specifications, here we detail its application to a
CES functional form, another widely-used specification for production
technology. Specifically, let the firm’s production technology $F(\cdot)$ with
labor-saving productivity take the following form:

$F(\cdot)=\left\\{\beta_{K}K_{it}^{-\frac{1-\sigma}{\sigma}}+\left[\exp\\{\varphi_{it}\\}L_{it}\right]^{-\frac{1-\sigma}{\sigma}}+\beta_{M}M_{it}^{-\frac{1-\sigma}{\sigma}}\right\\}^{-\frac{\nu\sigma}{1-\sigma}},$
(A.1)

where $\nu$ and $\sigma$ are the elasticities of scale and substitution,
respectively; and $\beta_{K}$ and $\beta_{M}$ are the distribution parameters,
with that corresponding to labor implicitly normalized to unity since it
cannot be identified separately from $\varphi_{it}$.

Among others, the CES form in (A.1) has been used by Doraszelski and
Jaumandreu,  (2013). Besides the usual monotonicity and curvature assumptions,
it is easy to confirm that it also satisfies separability per our Assumption
2(ii). Namely, $F(\cdot)$ above is a nested CES that is separable in capital
as follows:
$F(K_{it},\exp\\{\varphi\\}L_{it},M_{it})=G\left(K_{it},H\left(\exp\\{\varphi_{it}\\}L_{it},M_{it}\right)\right)$,
where

$\displaystyle G\left(\cdot\right)$
$\displaystyle=\left\\{\beta_{K}K_{it}^{-\frac{1-\sigma}{\sigma}}+H\left(\exp\\{\varphi_{it}\\}L_{it},M_{it}\right)^{-\frac{1-\sigma}{\sigma}}\right\\}^{-\frac{\nu\sigma}{1-\sigma}},$
$\displaystyle H\left(\cdot\right)$
$\displaystyle=\left\\{\left[\exp\\{\varphi_{it}\\}L_{it}\right]^{-\frac{1-\sigma}{\sigma}}+\beta_{M}M_{it}^{-\frac{1-\sigma}{\sigma}}\right\\}^{-\frac{\sigma}{1-\sigma}},$

and $H\left(\cdot\right)$ is normalized to be linearly homogeneous (i.e., a
unitary scale elasticity within the labor-and-materials pair of inputs), and
the elasticity of substitution between capital and the labor-and-materials
aggregator is the same as that between labor and materials within the
aggregator.

Making use of the “known” functional form of $F(\cdot)$, from the static
first-order conditions for flexible inputs $L_{it}$ and $M_{it}$ we obtain the
following expression for the Harrod-neutral productivity $\varphi_{it}$:

$\varphi_{it}=\frac{1}{1-\sigma}\left(m_{it}-l_{it}\right)-\frac{\sigma}{1-\sigma}\ln(\beta_{M})+\frac{\sigma}{1-\sigma}\left(\ln
P^{M}_{t}-\ln P^{L}_{t}\right).$ (A.2)

The above equation is the counterpart of (3.7) when the production function
takes a CES functional form, and it provides a proxy for the latent
$\varphi_{it}$ expressed as a function of the production function parameters
and observed data.

Next, we combine (A.2) with the law of motion for labor-augmenting
productivity, which is specified in (2.5), into an estimating equation for
parameters of the production function in (A.1). More concretely, substituting
$\varphi_{it}$ and $\varphi_{it-1}$ in (2.5) with (A.2), we have

$\displaystyle\frac{1}{1-\sigma}\left(m_{it}-l_{it}\right)-\frac{\sigma}{1-\sigma}\ln(\beta_{M})+\frac{\sigma}{1-\sigma}\left(\ln
P^{M}_{t}-\ln P^{L}_{t}\right)$ $\displaystyle=\quad$ $\displaystyle
r_{\varphi}\left(\left[\frac{1}{1-\sigma}\left(m_{it-1}-l_{it-1}\right)-\frac{\sigma}{1-\sigma}\ln(\beta_{M})+\frac{\sigma}{1-\sigma}\left(\ln
P^{M}_{t-1}-\ln P^{L}_{t-1}\right)\right],Z_{it-1}\right)+\zeta_{\varphi,it}.$
(A.3)

The above equation is the counterpart of (3.13) in the translog case. The two
unknown parameters $(\sigma,\beta_{M})^{\prime}$ and the mean function
$r_{\varphi}(\cdot)$ can be estimated via nonlinear least squares, given the
exogeneity of regressors, viz., $\mathbb{E}\left[\zeta_{\varphi,it}\ |\ 1,\
m_{it-1}-l_{it-1},\ \ln P^{M}_{t}-\ln P^{L}_{t},\ \ln P^{M}_{t-1}-\ln
P^{L}_{t-1},\ Z_{it-1}\right]=0$. Also note that, because of the particular
functional form of the CES specification, we are able to recover all
parameters pertaining to variable inputs in a single step, whereas in the case
of translog, we do so in two steps.

With $\sigma$ and $\beta_{M}$ identified from (A), we can also estimate
$\varphi_{it}$ following (A.2). To see the identification of the remaining
parameters in the production function, i.e., $\beta_{K}$ and $\nu$, take the
logarithm of the production function and substitute for $F(\cdot)$ using
(A.1):

$\displaystyle y_{it}=$
$\displaystyle-\nu\frac{\sigma}{1-\sigma}\ln\left\\{\beta_{K}\overbrace{K_{it}^{-\frac{1-\sigma}{\sigma}}}^{K_{it}^{*}}+\overbrace{\left[\exp\\{\varphi_{it}\\}L_{it}\right]^{-\frac{1-\sigma}{\sigma}}+\beta_{M}M_{it}^{-\frac{1-\sigma}{\sigma}}}^{H_{it}^{*}}\right\\}+\omega_{it}+\eta_{it},$
$\displaystyle=$
$\displaystyle-\nu\frac{\sigma}{1-\sigma}\ln\bigg{\\{}\beta_{K}K_{it}^{*}+H_{it}^{*}\bigg{\\}}+r_{\omega}\left(\omega_{it-1},X_{it-1}\right)+\zeta_{\omega,it}+\eta_{it},$
(A.4)

where we have replaced $\omega_{it}$ with its law of motion in the second
equality. The new variables $K_{it}^{*}\equiv
K_{it}^{-\frac{1-\sigma}{\sigma}}$ and
$H_{it}^{*}\equiv\left[\exp\\{\varphi_{it}\\}L_{it}\right]^{-\frac{1-\sigma}{\sigma}}+\beta_{M}M_{it}^{-\frac{1-\sigma}{\sigma}}$
are effectively data because they are defined using observable inputs and the
already identified parameters along with labor-augmenting productivity. Also,
remember that $\frac{\sigma}{1-\sigma}$ is known as well.

To address the unobservability of $\omega_{it-1}$, we proxy for it by
inverting the conditional material demand function implied by the
corresponding static first-order condition for $M_{it}$:181818One can also
operationalize this step using the inverted conditional labor demand instead.
Both proxies are equivalent.

$\displaystyle\omega_{it}$ $\displaystyle=\overbrace{\ln P^{M}_{t}-\ln
P^{Y}_{t}+\frac{1}{\sigma}m_{it}-\ln\left(\beta_{M}\right)}^{m^{*}_{it}}+\left(1+\nu\frac{\sigma}{1-\sigma}\right)\ln\bigg{\\{}\beta_{K}K_{it}^{*}+H_{it}^{*}\bigg{\\}}-\ln\left(\theta\nu\right)$
(A.5)

where $m^{*}_{it}$ is already identified and, hence, observable. The above
proxy for the latent Hicks-neutral productivity is the counterpart of (3.1).
Replacing $\omega_{it-1}$ in (A) with the lag of (A.5), we have the second-
step estimating equation that identifies the remaining unknown parameters
$(\nu,\beta_{K})^{\prime}$:

$\displaystyle y_{it}=$
$\displaystyle-\nu\frac{\sigma}{1-\sigma}\ln\left\\{\beta_{K}K_{it}^{*}+H_{it}^{*}\right\\}+r_{\omega}\left(\left[m^{*}_{it-1}+\left(1+\nu\frac{\sigma}{1-\sigma}\right)\ln\bigg{\\{}\beta_{K}K_{it-1}^{*}+H_{it-1}^{*}\bigg{\\}}-\ln\left(\theta\nu\right)\right],X_{it-1}\right)+\zeta_{\omega,it}+\eta_{it}.$
(A.6)

Per the structural assumptions, $K_{it}^{*}$, $K_{it-1}^{*}$, $H_{it-1}^{*}$,
$m_{it-1}^{*}$ and $X_{it-1}$ are all mean-orthogonal to the composite
innovation $\zeta_{\omega,it}+\eta_{it}$ in (A.6) and can self-instrument. The
same cannot be said about $H_{it}^{*}$ that also appears in the equation,
since it includes information on the choice of both $L_{it}$ and $M_{it}$
which are decided by the firm after $\zeta_{\omega,it}$ is realized (i.e.,
after $\omega_{it}$ is updated). However, analogous to the case with the
second-step estimation under the translog specification in (3.13) in Section
3.1, the endogeneity of $H_{it}^{*}$ does not impede identification of (A.6)
because $H_{it}^{*}$ is not a “free” regressor but enters the equation subject
to a parameter restriction whereby its distribution parameter is normalized to
1 and requires no estimation. No external instruments for $H_{it}^{*}$ are
therefore needed. As such, we can identify $(\beta_{K},\nu)^{\prime}$ as well
as the mean productivity function $r_{\omega}(\cdot)$ from (A.6) based on the
following moments:

$\mathbb{E}\left[\zeta_{\omega,it}+\eta_{it}|\
1,k_{it}^{*},k_{it-1}^{*},m_{it-1}^{*},H_{it-1}^{*},X_{it-1}\right]=0.$ (A.7)

##  Appendix B Expanded $\Psi(\alpha)$ from (3.1)

Suppressing the firm index $i$, we have that

$\displaystyle\Psi(\alpha)=\begin{pmatrix}\left\\{-\frac{\beta_{L}}{\beta_{0}^{2}}(1-\rho_{1})+\frac{\delta_{LM}}{\beta_{0}^{2}}\left[\mathbb{E}[S^{L}_{t}]-\rho_{1}\mathbb{E}[S^{L}_{t-1}]\right]\right\\}&\frac{1-\rho_{1}}{\beta_{0}}&\left\\{\mathbb{E}[\breve{m}_{t-1}]+\frac{\beta_{L}}{\beta_{0}}-\frac{\delta_{LM}}{\beta_{0}}\mathbb{E}[S^{L}_{t-1}]\right\\}&\mathbb{E}[Z_{t-1}^{\prime}]\\\
\left\\{-\frac{\beta_{L}}{\beta_{0}^{2}}(1-\rho_{1})\mathbb{E}[\breve{m}_{t-1}]+\frac{\delta_{LM}}{\beta_{0}^{2}}\left[\mathbb{E}[\breve{m}_{t-1}S^{L}_{t}]-\rho_{1}\mathbb{E}[\breve{m}_{t-1}S^{L}_{t-1}]\right]\right\\}&\frac{1-\rho_{1}}{\beta_{0}}\mathbb{E}[\breve{m}_{t-1}]&\left\\{\mathbb{E}[\breve{m}_{t-1}^{2}]+\frac{\beta_{L}}{\beta_{0}}\mathbb{E}[\breve{m}_{t-1}]-\frac{\delta_{LM}}{\beta_{0}}\mathbb{E}[\breve{m}_{t-1}S^{L}_{t-1}]\right\\}&\mathbb{E}[\breve{m}_{t-1}Z_{t-1}^{\prime}]\\\
\left\\{-\frac{\beta_{L}}{\beta_{0}^{2}}(1-\rho_{1})\mathbb{E}[S^{L}_{t-1}]+\frac{\delta_{LM}}{\beta_{0}^{2}}\left[\mathbb{E}[S^{L}_{t-1}S^{L}_{t}]-\rho_{1}\mathbb{E}[(S^{L}_{t-1})^{2}]\right]\right\\}&\frac{1-\rho_{1}}{\beta_{0}}\mathbb{E}[S^{L}_{t-1}]&\left\\{\mathbb{E}[\breve{m}_{t-1}S^{L}_{t-1}]+\frac{\beta_{L}}{\beta_{0}}\mathbb{E}[S^{L}_{t-1}]-\frac{\delta_{LM}}{\beta_{0}}\mathbb{E}[(S^{L}_{t-1})^{2}]\right\\}&\mathbb{E}[S^{L}_{t-1}Z_{t-1}^{\prime}]\\\
\left\\{-\frac{\beta_{L}}{\beta_{0}^{2}}(1-\rho_{1})\mathbb{E}[Z_{t-1}]+\frac{\delta_{LM}}{\beta_{0}^{2}}\left[\mathbb{E}[S^{L}_{t}Z_{t-1}]-\rho_{1}\mathbb{E}[S^{L}_{t-1}Z_{t-1}]\right]\right\\}&\frac{1-\rho_{1}}{\beta_{0}}\mathbb{E}[Z_{t-1}]&\left\\{\mathbb{E}[\breve{m}_{t-1}Z_{t-1}]+\frac{\beta_{L}}{\beta_{0}}\mathbb{E}[Z_{t-1}]-\frac{\delta_{LM}}{\beta_{0}}\mathbb{E}[S^{L}_{t-1}Z_{t-1}]\right\\}&\mathbb{E}[Z_{t-1}Z_{t-1}^{\prime}]\end{pmatrix}$

where $\breve{m}_{t-1}=m_{t-1}-l_{t-1}$ is the logged material-to-labor ratio.

##  Appendix C Semiparametric Sieve Estimation

In what follows, we describe how to empirically implement our methodology
semiparametrically, with the unknown functions $r_{\varphi}(\cdot)$ and
$r_{\omega}(\cdot)$ approximated using linear sieves. Sieves globally
approximate unknown nonparametric (i.e., infinite-parameter) functions using a
sequence of less complex parameter spaces that are characterized by a finite
number of “parameters” which effectively reduces the estimation problem to a
parametric estimation when implemented in practice, with the caveat being that
the complexity of such an approximation increases with the sample size to
ensure consistency. For more on sieves, see an excellent review by Chen,
(2007).

The first-step estimation remains the same as described in Section 4. In the
second step, however, we now approximate the unknown function
$r_{\varphi}(\varphi_{it-1},Z_{it-1})$ using a linear series of
$[\dim(Z)+1]$-variate polynomial basis functions
$\\{\mathscr{P}_{r,n}(\varphi_{it-1},Z_{it-1})\\}_{r=1}^{R_{n}}$ without the
intercept term, with the degree of approximation complexity $R_{n}\to\infty$
slowly with $n$. As before, the unobservable $\varphi_{it-1}$ is replaced with
a proxy function from (3.7), and $\delta_{LM}=\beta_{L}+\beta_{M}$ is replaced
with its estimate from the first step. Modifying the GMM problem in (4.2) to
accommodate a series approximation of $r_{\varphi}(\cdot)$, the second-step
equation is now estimated for a given $R_{n}$ via semiparametric nonlinear
sieve GMM. Thus, letting
$\alpha=(\beta_{0},\beta_{L},\rho_{1},\dots,\rho_{R_{n}})^{\prime}$, we have
that

$\displaystyle\widehat{\alpha}=\arg\min_{\alpha}\Big{[}\tfrac{1}{nT}\sum_{it}\mathbb{Q}_{it-1}\epsilon_{it}(D_{it};\alpha)\Big{]}^{\prime}W\Big{[}\tfrac{1}{nT}\sum_{it}\mathbb{Q}_{it-1}\epsilon_{it}(D_{it};\alpha)\Big{]},$
(C.1)

where the approximated residual function is

$\displaystyle\epsilon_{it}(D_{it};\alpha)\approx
m_{it}-l_{it}+\frac{\beta_{L}}{\beta_{0}}-\frac{\widehat{\delta}_{LM}}{\beta_{0}}S^{L}_{it}-\sum_{r=1}^{R_{n}}\rho_{r}\mathscr{P}_{r,n}\left(\left[m_{it-1}-l_{it-1}+\frac{\beta_{L}}{\beta_{0}}-\frac{\widehat{\delta}_{LM}}{\beta_{0}}S^{L}_{it-1}\right],Z_{it-1}\right),$
(C.2)

and the instrument vector $\mathbb{Q}_{it-1}$ now needs to include not only
the linear terms $(m_{it}-l_{it},Z_{it-1},S_{it-1}^{L})^{\prime}$ but also the
additional higher-order terms from a polynomial expansion of
$(m_{it}-l_{it},Z_{it-1},S_{it-1}^{L})^{\prime}$.

Then, just like in a fully parametric case, with the
$\big{(}\widehat{\beta}_{0},\widehat{\beta}_{L}\big{)}^{\prime}$ estimates in
hand, we can recover
$\widehat{\beta}_{M}=\widehat{\delta}_{LM}-\widehat{\beta}_{L}$ as well as
estimate Harrod-neutral productivity as
$\widehat{\varphi}_{it}=m_{it}-l_{it}+\widehat{\beta}_{L}/\widehat{\beta}_{0}-\widehat{\delta}_{LM}/\widehat{\beta}_{0}\times
S^{L}_{it}$.

The smoothing parameter $R_{n}$ can be controlled indirectly by selecting the
optimal degree of polynomial expansion $d_{n}$ via generalized cross-
validation of Craven and Wahba,  (1979):

$d_{n}^{*}=\arg\min_{d_{n}}\frac{\tfrac{1}{nT}||(\text{I}_{nT}-{\Pi}_{R_{n}})\widehat{\boldsymbol{\varphi}}||^{2}}{\left[1-\tfrac{1}{nT}\text{tr}\\{{\Pi}_{R_{n}}\\}\right]^{2}}\quad\text{with}\
R_{n}=R_{n}(d_{n}),$ (C.3)

where, for a given $(\beta_{0},\beta_{L})^{\prime}$,
${\Pi}_{R_{n}}=\mathbb{P}_{R_{n}}(\mathbb{P}_{R_{n}}^{\prime}\mathbb{P}_{R_{n}})^{-1}\mathbb{P}_{R_{n}}^{\prime}$
is a projection matrix defined using the matrix of basis functions
$\mathbb{P}_{R_{n}}$ constructed by stacking up
$P_{R_{n}}(\widehat{\varphi}_{it-1},Z_{it-1})=[\mathscr{P}_{1,n}(\widehat{\varphi}_{it-1},Z_{it-1}),\dots,$
$\mathscr{P}_{R_{n},n}(\widehat{\varphi}_{it-1},Z_{it-1})]^{\prime}$ in the
ascending order of index $i$ first then index $t$. The column vector
$\widehat{\boldsymbol{\varphi}}$ is stacked up similarly but using
$\\{\widehat{\varphi}_{it}\\}$.

To estimate the third-step equation in (3.19), we construct estimators of
$y_{it}^{*}$ and $m_{it}^{*}$ using the results from steps one and two:
$\widehat{y}_{it}^{*}=y_{it}-\widehat{\beta}_{M}m_{it}-\widehat{\beta}_{L}[\widehat{\varphi}_{it}+l_{it}]+\tfrac{1}{2}\widehat{\beta}_{0}[m_{it}-\widehat{\varphi}_{it}-l_{it}]^{2}$
and
$\widehat{m}^{*}_{it}=\ln\left[P_{t}^{M}/P_{t}^{Y}\right]-\ln\widehat{\theta}-\ln\big{(}\widehat{\beta}_{M}-\widehat{\beta}_{0}[m_{it}-\widehat{\varphi}_{it}-l_{it}]\big{)}+(1-\widehat{\beta}_{M})m_{it}-\widehat{\beta}_{L}[\widehat{\varphi}_{it}+l_{it}]+\tfrac{1}{2}\widehat{\beta}_{0}[m_{it}-\widehat{\varphi}_{it}-l_{it}]^{2}$.
Then, approximating unknown $r_{\omega}(\cdot)$ using $[\dim(X)+1]$-variate
polynomial sieves of degree $d^{\prime}_{n}$, i.e.,

$\displaystyle r_{\omega}(\omega_{it-1},X_{it-1})$
$\displaystyle\approx\sum_{r^{\prime}=1}^{R^{\prime}_{n}}{\pi}_{r^{\prime}}\mathscr{P}_{r^{\prime}}\left(\big{[}\widehat{m}^{*}_{it-1}-\beta_{K}k_{it-1}-\tfrac{1}{2}\beta_{KK}k_{it-1}^{2}\big{]},X_{it-1}\right),$
(C.4)

where $\omega_{it-1}$ is replaced with its proxy and
$R^{\prime}_{n}(d^{\prime}_{n})$ increases with the sample size, we estimate
$\gamma=(\beta_{K},\beta_{KK},$ $\pi_{1},\dots,\pi_{R^{\prime}_{n}})^{\prime}$
via semiparametric nonlinear sieve least squares in line with (4.4):

$\displaystyle\widehat{\gamma}=\arg\min_{\gamma}\sum_{it}\Bigg{[}\widehat{y}_{it}^{*}-\beta_{K}k_{it}-\tfrac{1}{2}\beta_{KK}k_{it}^{2}-\sum_{r^{\prime}=1}^{R^{\prime}_{n}}{\pi}_{r^{\prime}}\mathscr{P}_{r^{\prime}}\left(\big{[}\widehat{m}^{*}_{it-1}-\beta_{K}k_{it-1}-\tfrac{1}{2}\beta_{KK}k_{it-1}^{2}\big{]},X_{it-1}\right)\Bigg{]}^{2}.$
(C.5)

Analogous to the second step above, $R^{\prime}_{n}$ can be cross-validated
indirectly by selecting the optimal degree of polynomial expansion
$d^{\prime}_{n}$ via generalized cross-validation.

Using the obtained $(\widehat{\beta}_{K},\widehat{\beta}_{KK})^{\prime}$
estimates, we then can construct the estimates of Hicks-neutral productivity
as
$\widehat{\omega}_{it}=y_{it}-\widehat{\beta}_{K}k_{it}-\tfrac{1}{2}\widehat{\beta}_{KK}k_{it}^{2}-\widehat{\beta}_{M}m_{it}-\widehat{\beta}_{L}[\widehat{\varphi}_{it}+l_{it}]+\tfrac{1}{2}\widehat{\beta}_{0}[m_{it}-\widehat{\varphi}_{it}-l_{it}]^{2}-\widehat{\eta}_{it}$
using $\widehat{\eta}_{it}$ from step one.

Inference.—Due to a multi-step nature of our methodology and the presence of
nonparametric components, computation of the asymptotic variance of the
semiparametric estimator above is not that simple. For statistical inference
in this case, we therefore use bootstrap; the algorithm is described in
Appendix D.

##  Appendix D Bootstrap Inference

We approximate the sampling distribution of our estimator via wild residual
block bootstrap that takes into account a panel structure of the data as well
as a sequential nature of our multi-step estimation procedure. Concretely, the
bootstrap algorithm is as follows.

  1. 1.

Compute the three steps of our estimation procedure using the original data.
Denote the obtained point estimates of all the parameters and functions using
a “hat.” Correspondingly, let the (negative of) first-step residuals be
$\\{\widehat{\eta}_{it}\\}$, the second-step residuals be
$\\{\widehat{\zeta}_{\varphi,it}\\}$, and the third-step residuals be
$\\{\widehat{\zeta_{\omega,it}+\eta_{it}}\\}$. Recenter these.

  2. 2.

Generate bootstrap weights $\xi_{i}^{b}$ for all cross-sectional units
$i=1,\dots,n$ from the Mammen,  (1993) two-point mass distribution:

$\displaystyle\xi_{i}^{b}=\begin{cases}\frac{1+\sqrt{5}}{2}&\text{with
prob.}\quad\frac{\sqrt{5}-1}{2\sqrt{5}}\\\ \frac{1-\sqrt{5}}{2}&\text{with
prob.}\quad\frac{\sqrt{5}+1}{2\sqrt{5}}.\end{cases}$ (D.1)

Next, for each observation $(i,t)$ with $i=1,\dots,n$ and $t=1,\dots,T$,
jointly generate a new bootstrap first-step disturbance
$\eta_{it}^{b}=\xi_{i}^{b}\times\widehat{\eta}_{it}$, a new bootstrap second-
step disturbance
$\zeta_{\varphi,it}^{b}=\xi_{i}^{b}\times\widehat{\zeta}_{\varphi,it}$, and a
new bootstrap third-step disturbance
$(\zeta_{\omega,it}+\eta_{it})^{b}=\xi_{i}^{b}\times(\widehat{\zeta_{\omega,it}+\eta_{it}})$.

  3. 3.

Generate a new bootstrap first-step outcome variable via $\ln
R_{it}^{b}=\ln\left[\widehat{\theta}(\widehat{\beta}_{M}+\widehat{\beta}_{L})\right]-\eta_{it}^{b}$
for all $i=1,\dots,n$ and $t=1,\dots,T$.

  4. 4.

Generate a new bootstrap second-step outcome variable recursively as
$(m_{it}-l_{it})^{b}=-\frac{\widehat{\beta}_{L}}{\widehat{\beta}_{0}}+\left(\frac{\widehat{\beta}_{L}+\widehat{\beta}_{M}}{\widehat{\beta}_{0}}\right)S^{L}_{it}+\widehat{r}_{\varphi,1}\left[(m_{it-1}-l_{it-1})^{b}+\frac{\widehat{\beta_{L}}}{\widehat{\beta_{0}}}-\left(\frac{\widehat{\beta}_{L}+\widehat{\beta}_{M}}{\beta_{0}}\right)S^{L}_{it-1}\right]+\widehat{r}_{\varphi,2}Z_{it-1}+\zeta_{\varphi,it}^{b}$
for all $i=1,\dots,n$ and $t=1,\dots,T$. To initialize at $t=0$, we set
$(m_{i0}-l_{i0})^{b}=(m_{i0}-l_{i0})$.

  5. 5.

Generate a new bootstrap third-step outcome variable via
$y_{it}^{*b}=\widehat{\beta}_{K}k_{it}+\frac{1}{2}\widehat{\beta}_{KK}k^{2}_{it}+\widehat{r}_{\omega,0}+\widehat{r}_{\omega,1}\Big{[}\widehat{m}^{*}_{it-1}-\widehat{\beta}_{K}k_{it}-\frac{1}{2}\widehat{\beta}_{KK}k^{2}_{it}\Big{]}+\widehat{r}_{\omega,2}X_{it-1}+(\zeta_{\omega,it}+\eta_{it})^{b}$
for all $i=1,\dots,n$ and $t=1,\dots,T$.

  6. 6.

Recompute the first step using $\\{\ln R_{it}^{b}\\}$ in place of $\\{\ln
R_{it}\\}$. Denote the obtained parameter estimates as
$\big{(}(\widehat{\beta}_{M}+\widehat{\beta}_{L})^{b},\widehat{\theta}^{b}\big{)}^{\prime}$.

  7. 7.

Recompute the second step using $\\{(m_{it}-l_{it})^{b}\\}$ in place of
$\\{(m_{it}-l_{it})\\}$ and using $\\{(m_{it-1}-l_{it-1})^{b}\\}$ in place of
$\\{(m_{it-1}-l_{it-1})\\}$. Denote the obtained parameter and function
estimates as
$\big{(}\widehat{\beta}_{M}^{b},\widehat{\beta}_{L}^{b},\widehat{\beta}^{b}_{0},\widehat{r}_{\varphi}(\cdot)\big{)}^{\prime}$.
Also obtain
$\widehat{\varphi}^{b}_{it}=m_{it}-l_{it}+\frac{\widehat{\beta}^{b}_{L}}{\widehat{\beta}^{b}_{0}}-\left(\frac{\widehat{\beta}^{b}_{L}+\widehat{\beta}^{b}_{M}}{\widehat{\beta}^{b}_{0}}\right)S^{L}_{it}$.

  8. 8.

Recompute the third step using $y_{it}^{*b}$ in place of $y_{it}^{*}$. When
re-estimating the equation, also use $m_{it}^{*b}$ in place of $m_{it}^{*}$,
where
$m_{it}^{*b}=\ln\left[\frac{P_{t}^{M}}{P_{t}^{Y}}\right]-\ln\theta^{b}-\ln\big{(}\widehat{\beta}^{b}_{M}-\widehat{\beta}^{b}_{0}[m_{it}-\widehat{\varphi}^{b}_{it}-l_{it}]\big{)}+(1-\widehat{\beta}^{b}_{M})m_{it}-\widehat{\beta}_{L}[\widehat{\varphi}^{b}_{it}+l_{it}]+\tfrac{1}{2}\widehat{\beta}^{b}_{0}[m_{it}-\widehat{\varphi}^{b}_{it}-l_{it}]^{2}$
for all $i=1,\dots,n$ and $t=1,\dots,T$. Denote the obtained parameter and
function estimates as
$\big{(}\widehat{\beta}_{K}^{b},\widehat{\beta}_{KK}^{b},\widehat{r}^{b}_{\omega}(\cdot))^{\prime}$.

  9. 9.

Repeat steps 2 through 8 of the algorithm $B$ times.

Let the estimand of interest be denoted by $\mathcal{E}$, e.g., the firm $i$’s
capital elasticity $\epsilon_{K,it}\equiv\beta_{K}+\beta_{KK}k_{it}$ at time
$t$. To perform hypothesis testing, we can use the empirical distribution of
$\\{\widehat{\mathcal{E}}^{1},\dots,\widehat{\mathcal{E}}^{B}\\}$ to obtain a
bootstrap estimator of $\mathbb{V}\big{[}\widehat{\mathcal{E}}\big{]}$
following the standard methods.

##  Appendix E Data Summary

Table E.1: Data Summary Statistics

$Z=0$ (Obs #: 27,094) $Z>0$ (Obs #: 4,193) Mean 1st Qu. Median 3rd Qu. Mean
1st Qu. Median 3rd Qu. Output ($Y$) 29,719 10,554 19,353 37,684 38,611 14,483
28,312 52,405 Capital ($K$) 3,863 926 2,215 4,877 4,933 1,296 3,003 6,534
Labor ($L$) 2,140 729 1,370 2,659 3,069 1,199 2,181 4,018 Materials ($M$)
20,448 7,159 13,251 25,941 26,833 9,665 19,220 36,612 Foreign equity share
($Z$) 0.47 0.21 0.41 0.72 Notes: The three inputs and the output are in
thousands of real RMB. The foreign equity share is a unit-free proportion.

##  Appendix F Additional Empirical Results

The two tables below report the estimates of the production function and
productivity parameters obtained when using the inverted conditional labor
demand in (1) [Table F.1] and the average of the inverted conditional material
and labor demands in (3.1) and (1) [Table F.2] to proxy for latent
$\omega_{it}$. Since this proxy is used in the third step of our estimator,
only the estimates of capital elasticity and the Hicks-neutral productivity
process are affected; the remaining parameters are the same as those reported
in Table 2.

Table F.1: Estimates of the Production Function and Productivity Parameters
using the Labor Proxy Panel A: Elasticities and Returns to Scale
---
| Mean | 1st Qu. | Median | 3rd Qu.
Capital elasticity | 0.0436 | 0.0283 | 0.0443 | 0.0587
| (0.0053) | (0.0067) | (0.0053) | (0.0080)
Labor elasticity | 0.0950 | 0.0458 | 0.0864 | 0.1287
| (0.0003) | (0.0001) | (0.0002) | (0.0004)
Material elasticity | 0.7377 | 0.7041 | 0.7463 | 0.7869
| (0.0020) | (0.0019) | (0.0021) | (0.0022)
RTS | 0.8763 | 0.8610 | 0.8771 | 0.8915
| (0.0057) | (0.0071) | (0.0058) | (0.0083)
Panel B: Productivity Parameters
Labor-Augmenting |  | Factor-Neutral
$\rho_{\varphi,0}$ | 0.0000 |  | $\rho_{\omega,0}$ | 0.8082
| — |  |  | (0.0928)
$\rho_{\varphi,1}$ | 0.5723 |  | $\rho_{\omega,1}$ | 0.6840
| (0.0244) |  |  | (0.0247)
$\rho_{\varphi,2}$ | 0.0726 |  | $\rho_{\omega,2}$ | 0.0045
| (0.0183) |  |  | (0.0084)
Notes: The productivity processes are parameterized as follows:
$\varphi_{it}=\rho_{\varphi,1}\varphi_{it-1}+\rho_{\varphi,2}Z_{it-1}+\zeta_{\varphi,it}$
with $\rho_{\varphi,0}$ normalized to 0, and
$\omega_{it}=\rho_{\omega,0}+\rho_{\omega,1}\omega_{it-1}+\rho_{\omega,2}Z_{it-1}+\zeta_{\omega,it}$.
Bootstrap standard errors are in parentheses.

Table F.2: Estimates of the Production Function and Productivity Parameters
using the Average of Material and Labor Proxies Panel A: Elasticities and
Returns to Scale
---
| Mean | 1st Qu. | Median | 3rd Qu.
Capital elasticity | 0.0392 | 0.0243 | 0.0399 | 0.0539
| (0.0065) | (0.0077) | (0.0066) | (0.0089)
Labor elasticity | 0.0950 | 0.0458 | 0.0864 | 0.1287
| (0.0003) | (0.0001) | (0.0002) | (0.0004)
Material elasticity | 0.7377 | 0.7041 | 0.7463 | 0.7869
| (0.0020) | (0.0019) | (0.0021) | (0.0022)
RTS | 0.8719 | 0.8570 | 0.8726 | 0.8866
| (0.0068) | (0.0080) | (0.0069) | (0.0091)
Panel B: Productivity Parameters
Labor-Augmenting |  | Factor-Neutral
$\rho_{\varphi,0}$ | 0.0000 |  | $\rho_{\omega,0}$ | 0.7115
| — |  |  | (0.0975)
$\rho_{\varphi,1}$ | 0.5723 |  | $\rho_{\omega,1}$ | 0.7277
| (0.0244) |  |  | (0.0279)
$\rho_{\varphi,2}$ | 0.0726 |  | $\rho_{\omega,2}$ | 0.0033
| (0.0183) |  |  | (0.0083)
Notes: The productivity processes are parameterized as follows:
$\varphi_{it}=\rho_{\varphi,1}\varphi_{it-1}+\rho_{\varphi,2}Z_{it-1}+\zeta_{\varphi,it}$
with $\rho_{\varphi,0}$ normalized to 0, and
$\omega_{it}=\rho_{\omega,0}+\rho_{\omega,1}\omega_{it-1}+\rho_{\omega,2}Z_{it-1}+\zeta_{\omega,it}$.
Bootstrap standard errors are in parentheses.

##  Appendix G Extension to Imperfect Competition

Our methodology and its underlying identification scheme in Sections 2–3 are
developed under the assumption that firms operate in a perfectly competitive
output market. In this appendix, we discuss how to relax this assumption and
allow for monopolistic power in the output market in our setup.

Let the output price be no longer $P^{Y}_{it}\neq P^{Y}_{t}\ \forall\ i$. To
allow firms to have some market power, we assume they produce differentiated
products and operate in a monopolistically competitive market. Let each firm
face a downward-sloping (residual) inverse demand function of the following
generic form:191919More generally, the residual demand that a firm faces can
also depend on its rivals’ prices. While we assume this away, one may be able
to account for such substitution effects by replacing rivals’ prices with an
aggregate price index or dummies, although it may substantially increase the
number of parameters to be estimated. $P_{it}^{Y}=D(Y_{it}^{e},U_{it})$, where
$Y_{it}^{e}=Y_{it}\exp\\{-\eta_{it}\\}$ is the expected, or “planned,” output
quantity net of an unanticipated ex-post productivity shock $\eta_{it}$, and
$U_{it}$ is a vector of demand shifters known to the firm at time $t$, i.e.,
$U_{it}\in\Xi_{it}$.202020We can also allow for random price/demand shocks by,
say, augmenting the demand equation with a log-additive unanticipated i.i.d.
shock akin to $\eta_{it}$:
$P_{it}^{Y}=D(Y_{it}^{e},U_{it})\exp\\{\varepsilon_{it}\\}$ with
$\mathbb{E}[\varepsilon_{it}|\Xi_{it}]=\mathbb{E}[\varepsilon_{it}]=0$. This
would only result in an additional multiplicative constant entering the firm’s
static first-order conditions in expectation [see eqs. (G.1)–(G.2)] thereby
having no material impact on the analysis. But given our discussion concerning
the oft-problematic use of firm-level variation in input prices for
identification in Section 3.2, we retain the assumption of competitive
homogeneous factor prices.

In what follows, we fist show point non-identification of the production
function in (3.2) when firms exhibit market power even if one observes
exogenous demand shifters in the data. We then discuss how, with some
additional but quite reasonable assumptions about unobservables, one can still
set identify the production function, and we derive this result without
requiring that the information on demand shifters be available: we only rely
on the same set of observables that are used in our main methodology for the
case of perfectly competitive markets from Section 3.

### Point Non-Identification of the Production Function

For a risk-neutral firm with market power, the firm’s static optimality
conditions are now given by

$\displaystyle\left(1+\frac{1}{\delta(P^{Y}_{it},U_{it})}\right)P_{it}^{Y}\frac{\exp\\{\overline{y}_{it}\\}}{L_{it}}\big{(}\beta_{L}+\beta_{0}[m_{it}-\varphi_{it}-l_{it}]\big{)}\exp\\{\omega_{it}\\}\theta$
$\displaystyle=P_{t}^{L},$ (G.1)
$\displaystyle\left(1+\frac{1}{\delta(P^{Y}_{it},U_{it})}\right)P_{it}^{Y}\frac{\exp\\{\overline{y}_{it}\\}}{M_{it}}\big{(}\beta_{M}-\beta_{0}[m_{it}-\varphi_{it}-l_{it}]\big{)}\exp\\{\omega_{it}\\}\theta$
$\displaystyle=P_{t}^{M},$ (G.2)

where $\delta(P^{Y}_{it},U_{it})<-1$ is the price elasticity of demand. The
ratio of these optimality conditions however remains unchanged [see eq. (3.6)]
implying that the proxy for Harrod-neutral productivity $\varphi_{it}$ is also
unchanged [see eq. (3.7)]. But the new proxy for Hicks-neutral productivity
does need to explicitly account for the firm’s market power (here we continue
to use the inverted material demand):

$\displaystyle\omega_{it}=$ $\displaystyle\
\overbrace{\ln\left[\frac{P_{t}^{M}}{P_{it}^{Y}}\right]-\ln\theta-\ln\big{(}\beta_{M}-\beta_{0}[m_{it}-\varphi_{it}-l_{it}]\big{)}+(1-\beta_{M})m_{it}-\beta_{L}[\varphi_{it}+l_{it}]+\tfrac{1}{2}\beta_{0}[m_{it}-\varphi_{it}-l_{it}]^{2}}^{m^{*}_{it}}$
$\displaystyle\
-\beta_{K}k_{it}-\tfrac{1}{2}\beta_{KK}k_{it}^{2}+\ln\mu(P^{Y}_{it},U_{it}),$
(G.3)

where
$\mu(P^{Y}_{it},U_{it})=\left(1+\frac{1}{\delta(P^{Y}_{it},U_{it})}\right)^{-1}$
is a markup (a price-to-marginal-cost ratio).

Now, consider the first step of our methodology. Because the new first-order
conditions in (G.1)–(G.2) contain the demand elasticity, the variable input
share equations will also have to account for markups:

$\displaystyle\ln V_{it}^{L}$
$\displaystyle=\ln\left(\theta\beta_{0}\left[\frac{\beta_{L}}{\beta_{0}}+m_{it}-\varphi_{it}-l_{it}\right]\right)-\ln\mu(P^{Y}_{it},U_{it})-\eta_{it},$
(G.4) $\displaystyle\ln V_{it}^{M}$
$\displaystyle=\ln\left(\theta\beta_{0}\left[\frac{\beta_{M}}{\beta_{0}}-m_{it}+\varphi_{it}+l_{it}\right]\right)-\ln\mu(P^{Y}_{it},U_{it})-\eta_{it}.$
(G.5)

Controlling for $\varphi_{it}$ using the (unchanged) material-to-labor ratio
proxy function in (3.7), we thus obtain the following variable-input-cost-to-
revenue equation for a monopolistically competitive firm:

$\displaystyle\ln R_{it}$
$\displaystyle=\ln\big{(}\theta\left[\beta_{L}+\beta_{M}\right]\big{)}-\ln\mu(P^{Y}_{it},U_{it})-\eta_{it},$
(G.6)

in which all regressors are weakly exogenous with respect to an ex-post shock
$\eta_{it}$.

The two components $\ln\big{(}\theta\left[\beta_{L}+\beta_{M}\right]\big{)}$
and $\ln\mu(P^{Y}_{it},U_{it})$ in (G.6) are however not separably identified.
The unknown function $\mu(\cdot)$ can be identified up to a scale only. With
the lack of point identification of the firm’s markup
$\mu(P^{Y}_{it},U_{it})$, the production-function parameters
$(\beta_{L},\beta_{M})^{\prime}$ cannot be point identified
either.212121$\theta$ is point identified because the random shocks
$\\{\eta_{it}\\}$ are point identified via
$\eta_{it}=\ln\big{(}\frac{\theta\left[\beta_{L}+\beta_{M}\right]}{\mu(P^{Y}_{it},U_{it})}\big{)}-\ln
R_{it}$, where
$\ln\big{(}\frac{\theta\left[\beta_{L}+\beta_{M}\right]}{\mu(P^{Y}_{it},U_{it})}\big{)}=\mathbb{E}[\ln
R_{it}|P^{Y}_{it},U_{it}]$ is some unknown conditional mean function of $\ln
R_{it}$ estimatable via least squares. To see this more clearly, consider the
special case of isoelastic demand whereby $\mu(P^{Y}_{it},U_{it})=\mu\
\forall\ i,t$. In this case, we can only point identify
$(\beta_{L}+\beta_{M})/\mu$ from (G.6) and, consequently, we cannot point
identify $(\beta_{0},\beta_{L})^{\prime}$ from (3.13) either, with the similar
implications for $(\beta_{K},\beta_{KK})^{\prime}$ in the third step. Thus,
all production-function parameters can only be point identified up to a scale
of the firm’s markup $\mu$. More generally, the point non-identification of
production function when firms exercise monopolistic power (although when
productivity is uni-dimensional) is also discussed in Flynn et al.,  (2019).

Interestingly, despite the lack of point identification of both the
production-function parameters and the markup, the variable-input-cost-to-
revenue equation under imperfect competition in (G.6) may nonetheless provide
some useful information about markups if $P^{Y}_{it}$ and $U_{it}$ are
observed in the data. Namely, oftentimes markups per se are of little policy
relevance and, instead, economists focus on their relation with some other
correlates or their distribution across firms. For instance, one might be
interested in testing if exporters enjoy a greater price-setting power that do
wholly domestically oriented firms (e.g., De Loecker and Warzynski, , 2012; De
Loecker et al., , 2016). Alternatively, we may be interested in temporal
dynamics of markups (e.g., De Loecker et al., , 2020; De Loecker and Eeckhout,
, 2020), be it on average or in terms of their cross-firm dispersion. Such
analyses of markups are customarily done by regressing the estimated (log)
markups on the variables of interest such export status or time trend/dummies.
We can still accomplish the latter using “scaled” markup estimates from (G.6).
Namely,
$\ln\big{(}\tfrac{\theta\left[\beta_{L}+\beta_{M}\right]}{\mu(P^{Y}_{it},U_{it})}\big{)}$
in (G.6) is some unknown function of $(P^{Y}_{it},U_{it}^{\prime})^{\prime}$
easily estimable via nonparametric least squares by regressing $\ln R_{it}$ on
$(P^{Y}_{it},U_{it}^{\prime})^{\prime}$ with intercept. Then, so long as the
production function is correctly specified and thus $\beta_{L}+\beta_{M}$ is a
constant, we can regress the recovered scaled log-markup
$\ln\big{(}\tfrac{\mu(P^{Y}_{it},U_{it})}{\theta\left[\beta_{L}+\beta_{M}\right]}\big{)}=-\ln\big{(}\tfrac{\theta\left[\beta_{L}+\beta_{M}\right]}{\mu(P^{Y}_{it},U_{it})}\big{)}$
on whatever variables of interest, with the only (and unimportant) implication
being a bias in the intercept. Analogously, we can study the dispersion in
markups or changes therein over time by analyzing the ratios of
$\tfrac{\mu(P^{Y}_{it},U_{it})}{\theta\left[\beta_{L}+\beta_{M}\right]}$
across firms or the shifts in their distributions over time.

### Partial Identification of the Production Function

Although the production function is not point-identified when firms have
monopolistic power in the output markets, we can still achieve its partial
identification. To set identify the production-function parameters
$(\beta_{K},\beta_{KK},\beta_{M},\beta_{L},\beta_{0})^{\prime}$, we build upon
Demirer, ’s (2019) framework which we modify to admit multi-dimensional firm
productivity.

Since most production datasets contain no information on exogenous firm demand
shifters, here we can also relax the assumption that demand heterogeneity
$U_{it}$ be observable. But with the introduction of new unobservables, we
need to formalize their relation to other unobservables that are known to the
firm such as productivity. Defining the vector of observable (by an
econometrician) state variables as
$O_{it}=(K_{it},P_{t}^{L},P_{t}^{M},X_{it}^{\prime},Z_{it}^{\prime})^{\prime}$,
we augment our assumptions as follows.

######  Assumption 5 (Replaces Assumption 4)

(i) Risk-neutral firms maximize the discounted stream of life-time profits in
perfectly competitive factor markets with homogeneous prices. The output
market is monopolistically competitive, and the firm’s downward-sloping
inverse (residual) demand function is given by
$P_{it}^{Y}=D(Y_{it}^{e},U_{it})$ and $U_{it}\in\Xi_{it}$ is a vector of
unobservable (to an econometrician) demand shifters known to the firm when
making time $t$ decisions. (ii)Conditional on observable $O_{it-1}$, demand
heterogeneity $U_{it}$ is jointly independent of firm productivity
$\varphi_{it}$ and $\omega_{it}$.222222Assumption 5(ii) can be relaxed by
making the independence be conditional on $\varphi_{it-1}$ and $\omega_{it-1}$
thereby, in effect, assuming that $U_{it}$ is jointly independent of the
contemporaneous innovations in firm productivities.

In the above, $Y_{it}^{e}=Y_{it}\exp\\{-\eta_{it}\\}$ is the expected/planned
output quantity as defined earlier. Since the presence of markups does not in
any way affect the ratio of firm’s static first-order conditions, thereby
providing a deterministic proxy for labor-augmenting productivity given by
(3.7), we can utilize it to substitute latent $\varphi_{it}$ out of the
production function (3.2) to arrive at

$\displaystyle y_{it}$
$\displaystyle=\beta_{K}k_{it}+\tfrac{1}{2}\beta_{KK}k_{it}^{2}+(\beta_{L}+\beta_{M})m_{it}+\tfrac{1}{2\beta_{0}}\left[\beta_{L}^{2}-(\beta_{L}+\beta_{M})^{2}\big{(}S_{it}^{L}\big{)}^{2}\right]+\omega_{it}+\eta_{it}$
$\displaystyle=\underbrace{\tfrac{\beta_{L}^{2}}{2\beta_{0}}+\beta_{K}k_{it}+\tfrac{1}{2}\beta_{KK}k_{it}^{2}+(\beta_{L}+\beta_{M})m_{it}-\tfrac{(\beta_{L}+\beta_{M})^{2}}{2\beta_{0}}\big{(}S_{it}^{L}\big{)}^{2}}_{\overline{y}_{it}(v_{it};\beta)}+\
\omega_{it}+\eta_{it},$ (G.7)

which now contains only additive unobservables and where
$v_{it}=(k_{it},m_{it},l_{it},P_{t}^{M},P_{t}^{L})^{\prime}$ and
$\beta=(\beta_{K},\beta_{KK},$
$\beta_{L},\beta_{M},\beta_{0})^{\prime}$.232323Recall that $S^{L}_{it}$ is a
deterministic function of $P_{t}^{M},P_{t}^{L},\exp\\{m_{it}\\}$ and
$\exp\\{l_{it}\\}$. In what follows, we seek to set-identify $\beta$.

We first establish the relationship between flexible inputs and firm
productivity, which facilitates the proxy variable approach to tackling
unobservability of the latter. Consistent with our primary methodology, we use
materials to control for productivity.

###### Proposition 1

Under Assumptions 1–3 & 5 and some additional regularization of the curvature
of the production function and the firm’s downward-sloping (residual) demand
function, the firm’s conditional demands for $M_{it}$ is weakly increasing in
$\omega_{it}$ and $\varphi_{it}$, conditional on all other state variables
entering the expected static profit maximization problem
$(K_{it},P_{t}^{L},P_{t}^{M},U_{it}^{\prime})^{\prime}$.

The proposition signs partial derivatives of the material demand with respect
to both components of firm productivity and is easy but tedious to show, which
involves differentiation of first-order conditions in (G.1)–(G.2) with respect
to productivities (see Appendix H). Intuitively, the firm (i) substitutes
materials for labor conserving the latter ceteris paribus as the labor input
becomes more productive when $\varphi_{it}$ rises and (ii) uses more materials
(as well as labor) expanding the production when its overall total factor
productivity $\omega_{it}$ improves increasing the marginal products of static
inputs. For convenience, denote the material demand as
$M_{it}=\mathcal{M}(K_{it},P_{t}^{L},P_{t}^{M},U_{it},\varphi_{it},\omega_{it})=\mathcal{M}(K_{it},U_{it},\varphi_{it},\omega_{it})$,
where we suppress input prices because they provide little operationable
information due to the lack of cross-firm variation under our assumptions.

To derive moment inequalities that partially identify the firm’s production
function, we need to tighten our assumptions by formalizing the relationship
between the two components of firm productivity. As noted in Section 2,
Assumption 3(i) places no restriction on the relation between $\omega_{it}$
and $\varphi_{it}$ but, to identify the production function when firms have
market power, its now needs be regulated. We do so by letting the labor-
augmenting productivity $\varphi_{it}$ be stochastically increasing in factor-
neutral Hicksian productivity $\omega_{it}$, conditional on productivity
“controls.” Thus, we extend our Assumption 3(i) as follows.

###### Assumption 6

Conditional on productivity-modifying controls
$(X_{it-1}^{\prime},Z_{it-1}^{\prime})^{\prime}$, the distribution of
$\varphi_{it}$ is stochastically increasing in $\omega_{it}$.

More specifically, Assumption 6 means that
$\mathcal{P}_{\varphi}(\varphi_{it}|\omega_{it}^{H},X_{it-1},Z_{it-1})$ first-
order stochastically dominates
$\mathcal{P}_{\varphi}(\varphi_{it}|\omega_{it}^{L},X_{it-1},Z_{it-1})$ iff
$\omega_{it}^{H}\geq\omega_{it}^{L}$. In this, we intuitively assume that the
firms that are more productive in general (in all factors) are likely to also
be more productive in labor. Then, the set identification of the production-
function parameters $\beta$ is obtained based on the following proposition,
according to which the material proxy can be used to stochastically order the
log-additive (and only) unobservable $\omega_{it}+\eta_{it}$ entering the
production function in (G).

###### Proposition 2

For some cutoff value $\widetilde{\mathrm{m}}$ for the materials input, let
$\mathbb{S}=\\{(o,\widetilde{\mathrm{m}}):\Pr(m_{it}<\widetilde{\mathrm{m}}|O_{it-1}=o)\in(0,1)\\}$
denote the common support. Under Assumptions 1–3 & 5–6 and by Proposition 1,
for $(O_{it-1},\widetilde{\mathrm{m}})\in\mathbb{S}$ we have

$\mathbb{E}\left[y_{it}-\overline{y}(v_{it};\beta)|O_{it-1},m_{it}>\widetilde{\mathrm{m}}\right]\geq\mathbb{E}\left[y_{it}-\overline{y}_{it}(v_{it};\beta)|O_{it-1},m_{it}<\widetilde{\mathrm{m}}\right].$
(G.8)

The proof is in Appendix H. By this proposition, when comparing high-materials
(those with $m_{it}>\widetilde{\mathrm{m}}$) and low-materials (those with
$m_{it}<\widetilde{\mathrm{m}}$) firms, the firms that use more inputs are
more Hicks-productive on average. In
$y_{it}-\overline{y}(v_{it};\beta)=\omega_{it}+\eta_{it}$, it is evident that
the focus here is on factor-neutral productivity without the need to
characterize labor-augmenting productivity. This is possible owing to the
availability of a deterministic proxy of the known functional form for
$\varphi_{it}$ afforded to us by our parametric specification of the firm’s
separable production technology, which enable us to concentrate Harrod-neutral
productivity from production model in (G).

Conditional on $o$ and $\widetilde{\mathrm{m}}$, partially identified $\beta$
parameters are a nonlinear half-space, and the identified set
$\mathcal{B}^{*}$ is the intersection of these half-planes: