# Productivity, Inputs Misallocation and the Financial Crisis

Davide Luparello Penn State University

(First version: $8^{th}$ November, 2022
This version: )

This paper reevaluates the conventional approach to quantifying within-
industry resource misallocation, typically measured by the dispersion of an
input’s marginal product. My findings suggest that this statistic incorporates
inherent productivity heterogeneity and idiosyncratic productivity shocks,
irrespective of the input under scrutiny. Using balance sheet data from
American and European manufacturing firms, I show that total factor
productivity (TFP) volatility accounts for 7% of the variance in the marginal
product of capital, 9% for labor, and 10% for material inputs. Consequently,
this index, taken at face value, fails to identify policy-induced
misallocation for any production input. To overcome this limitation, I propose
a comparative analysis strategy driven by an identified policy variation. This
approach allows the researcher to assess induced misallocation in relative
terms whilst controlling for differences in TFP volatility. I show that the
2008 financial crisis had an uneven impact on the within-industry dispersion
of the marginal product of capital across European nations, reflecting their
differing financial sector maturity and suggesting the existence of financial
misallocative frictions. The crisis did not affect the dispersion of the
marginal product for other inputs.

Keywords: misallocation, input-output models, production, cost, capital, total
factor productivity, economic growth, aggregate productivity

JEL Codes: E22 D24 O47

##  1 Introduction

This paper reevaluates the conventional approach to quantifying within-
industry resource misallocation, typically measured by the dispersion of an
input’s marginal product ([31] and [28]). My findings suggest that this
statistic incorporates inherent total factor productivity (TFP) heterogeneity
and idiosyncratic productivity shocks, irrespective of the input under
consideration. This facet implies that, as long as it embodies economic
fundamentals and uncertainty, it cannot represent, when taken at face value, a
target for policy-makers aiming to mitigate broadly defined policy-induced
input misallocation. To address this limitation, the paper proposes a cross-
country or cross-industry comparative analysis driven by a defined policy
variation. This approach allows the researcher to assess induced misallocation
in relative terms whilst controlling for differences in TFP volatility. I show
that the 2008 financial crisis had an uneven impact on the within-industry
dispersion of the marginal product of capital across European nations,
reflecting their differing financial sector maturity and suggesting the
existence of financial misallocative frictions. In agreement with the standard
narrative about the causes and effects of the financial crisis, I found that
it did not affect the dispersion of the marginal product for other inputs.

Economics literature consistently acknowledges the overwhelming and persistent
heterogeneity in productivity and marginal products of inputs across firms,
even within narrowly defined industries. [31] and [28] argue that the
dispersion of the marginal product of inputs across firms signifies policy
frictions that obstruct the optimal allocation of resources, consequently
resulting in aggregate productivity losses. The latter authors, for instance,
famously posited that decreasing the variability of the marginal (revenue)
product of capital and labor in India and China to the level observed in the
US could increase aggregate manufacturing TFP by 40%-60% and 30%-50%,
respectively. This measure has since enjoyed widespread use in literature due
to its simplicity in quantifying aggregate misallocation 111For example, [36]
employed it to study financial misallocation.. Nevertheless, its utility in
identifying its determinants is less evident ([27]). Several previous works
have also expressed skepticism about its potential spuriousness 222Several
papers have noted that the measure proposed by [28] relies heavily on assumed
structures of technology and demand ([27, 22]). Further, [26] contends that
under general conditions, the dispersion of an input’s marginal product may
reflect economic features such as idiosyncratic demand and cost conditions,
which can hardly be deemed ”misallocative”. Identified determinants include
idiosyncratic regulations and institutions ([4, 5]), employment protection
measures ([33]), financial frictions ([25]), sub-optimal endogenous firms’
selection ([37]), noise introduced by measurement error ([24, 8]),
heterogeneity in production units and demand elasticities ([32, 7, 9]), factor
adjustment costs and productivity information frictions ([2, 16, 26, 17]).. A
notable contribution to this discourse comes from [2]. They argue that for a
dynamic input like capital, a firm’s uncertainty implies that a predetermined,
costly allocation, which is optimal in expectation, may not remain optimal
once ex-post productivity shocks occur333[11] shows that productivity
uncertainty significantly influences firms’ allocation choices and affects the
business cycle.. This realization suggests that the observed dispersion of the
marginal product of capital across firms emerges as a byproduct of information
asymmetries and dynamic adjustments. Indeed, the authors demonstrate that the
predictions from a partial equilibrium dynamic investment model, featuring
capital adjustment costs and productivity shocks, align with 80%-90% of the
dispersion of the revenue marginal product of capital observed in the data.
Building on their intuition, this paper demonstrates that the relationship
between productivity uncertainty and heterogeneity and inputs’ marginal
products holds for all production inputs. Specifically, I establish that firm-
level variations in TFP growth impact the marginal product of all production
inputs. In turn, this interaction suggests that TFP volatility contributes to
the variation of the marginal products of each input at the cross-sectional
industry level.

An important question consequently emerges: Can this measure retain its
usefulness in a comparative perspective even if, taken at face value, it
incorporates economic fundamentals and uncertainty? More specifically, if we
can unambiguously identify policy variations and control for TFP volatility,
can the cross-sectional dispersion of the marginal product of an input
maintain its value in assessing the relative misallocative impact of the
policy under scrutiny? To investigate this question, I examine the
differential impact of the 2008 financial crisis on the cross-sectional
dispersion of TFP growth and marginal products of inputs across various
European countries, attributing these differences to heterogeneous
efficiencies in their respective financial sectors. This analysis echoes the
work of [25], who formulated an open economy model of firm heterogeneity,
financial frictions, and capital adjustment costs. Their model demonstrated
that financially inefficient Southern European countries experienced
heightened dispersion of the marginal product of capital and diminished TFP at
the start of the 1990s. This situation occurred as falling real interest
rates, resultant from the euro convergence process, redirected capital inflows
towards less productive firms. Building on their insights, I document an
extension of their results: Southern European countries also faced a greater
impact from the 2008 financial crisis, which resulted in an increased TFP
volatility and a heightened dispersion of the marginal product of capital in
the years following the economic crisis.

In this study, I apply a flexible, non-parametric gross output production
function model based on [23]. I estimate this model using recent balance sheet
data from firms in the manufacturing sector across the US and various European
countries. The model accounts for TFP uncertainty, which firms encounter at
different stages of their input allocation decision-making processes. Within
the model, I consider material inputs as the only flexible input, meaning
their allocation solves a static first-order condition under imperfect
information. I take an agnostic stance on the dynamic implications of labor
and capital.444This stance is not without consideration of historical evidence
suggesting that labor tends to be sticky and subject to adjustment frictions,
particularly when considering European countries in the post-2000 years. Since
the launch of the Euro area, many reforms have been adopted to improve the
flexibility of the labor market and ease employment protection legislation.
However, the process has been slow, incomplete, and not supported by
widespread consensus. Several protests sparked, the most notable being anti-
labor reform protests in France in 2016 for the El Khomri Law and strikes in
Italy in 2014 for the Jobs Act.. Thus, the model allows for a variety of
dynamic and static distortions affecting capital and labor allocations.
Through the lens of this model, I theoretically trace the channels via which a
counterfactual, firm-level variation in TFP growth influences the marginal
products of inputs. Estimating the model yields firm-time-specific estimates
for the inputs’ marginal products, TFP growth, and its channels. I then
estimate the impact of TFP growth and its channels on the inputs’ marginal
products using country-specific linear regression models with sector-time-
specific fixed effects. Subsequently, I quantify and compare the proportion of
cross-sectional dispersion of the inputs’ marginal product explained by TFP
volatility and its channels at the industry level. Lastly, using an event
study approach, I leverage the temporal dimension of my estimates to assess
the financial crisis’s effect on the dispersion of the marginal products of
inputs in financially less advanced (Southern) versus more advanced (Northern)
European countries.

I find a strong connection between changes in TFP growth and the marginal
product of each input at the micro level. On aggregate, TFP volatility alone
accounts for an overall 7% of the dispersion of the marginal product of
capital, 9% for labor, and 10% for materials at the industry level. This
explanatory power increases with the decomposition of TFP growth variation
into its channels. While demonstrating that the cross-sectional industry-level
dispersion of the marginal product of any input captures a significant amount
of fundamental heterogeneity taken as-is, this measure still informs us about
policy-induced misallocation from a comparative perspective. For example, when
analyzing the Financial Crisis in Europe, I find the South experienced a 65%
greater increase in TFP volatility than the North. However, it also saw the
dispersion of the marginal product of capital rise by 40% more in the South,
even after accounting for TFP volatility. This finding suggests that the
crisis influenced the dispersion of the input’s marginal product through
mechanisms beyond just an increase in productivity uncertainty, with financial
sector friction heterogeneity likely playing a significant role.

The organization of this paper is as follows. In Section 2, I present the
theoretical setup. Section 3 provides details about the data used. The
estimation strategy takes center stage in Section 4. In Section 5, I showcase
the production function estimates and discuss the relationship between the
inputs’ marginal products and TFP growth at both micro and industry levels.
Section 6 delves into the application of these findings to the 2008 financial
crisis. Finally, Section 7 concludes.

##  2 The Production Function Model

Let $Y_{jt}$ the gross output and $K_{jt}$, $L_{jt}$ and $M_{jt}$ the capital,
labor, and material inputs allocations of firm $j$ at time $t$. Lowercase
variables indicate natural logarithms. The log production function is non-
parametrically specified as

$y_{jt}=f(k_{jt},l_{jt},m_{jt})+\nu_{jt}$ (1)

Where $\nu_{jt}$ is the exogenous log TFP term, specified as the composition
of a persistent and an unexpected component ([30])

$\nu_{jt}=\omega_{jt}+\varepsilon_{jt}$ (2)

From equation (2), $\omega_{jt}$ is a persistent productivity factor that firm
$j$ perfectly observes at the beginning of period $t$ but is unknown at time
$t-1$. On the other hand, $\varepsilon_{jt}$ represents a residual, short-term
idiosyncratic output fluctuation observed only as period ends555Another
observationally equivalent interpretation for $\varepsilon_{jt}$ is the ex-
post measurement error in the output. However, I keep the interpretation in
[23] as a productivity forecast error..

Let $\mathcal{I}_{jt}$ the information set that firm $j$ holds at time $t$
before the realization of $\varepsilon_{jt}$. Moreover, call
$\tilde{\mathcal{I}}_{jt}$ the information set that firm $j$ holds at the end
of period $t$. The following assumptions characterize the persistent
productivity process and the realization timing of the productivity shocks.

######  Assumption 1 (Persistent Productivity Markov Process)

$\omega_{jt}$ follows the Markovian process

$\omega_{jt}=m(\omega_{jt-1})+\eta_{jt}$ (3)

$m(.)$ is a continuous function and $\eta_{jt}$ is a mean-independent forecast
error such that

$\displaystyle\omega_{jt-1}\in\tilde{\mathcal{I}}_{jt-1},\quad\eta_{jt}\notin\tilde{\mathcal{I}}_{jt-1},\quad\eta_{jt}\perp\omega_{jt-1}\quad\forall\quad
j,t$ (4)

and

$\displaystyle E(\eta_{jt}\mid\tilde{\mathcal{I}}_{jt-1})=E(\eta_{jt})=0$ (5)
$\displaystyle
E(e^{\eta_{jt}}\mid\tilde{\mathcal{I}}_{jt-1})=E(e^{\eta_{jt}})=\mathcal{M}$

Where $\mathcal{M}$ is a scalar constant.

######  Assumption 2 (Productivity Shocks Realization Timing)

$\eta_{jt}$ realizes at the beginning of period $t$. Then, the persistent
productivity component, $\omega_{jt}$, is known at the beginning of period
$t$. The unexpected component, $\varepsilon_{jt}$, is an idiosyncratic, mean-
independent, short-term output fluctuation and realizes towards the end of
period $t$.
Then, formally

$\displaystyle\omega_{jt}\in\mathcal{I}_{jt},\quad\varepsilon_{jt}\notin\mathcal{I}_{jt},\quad\varepsilon_{jt}\in\tilde{\mathcal{I}}_{jt},\quad\omega_{jt}\perp\varepsilon_{jt}\quad\forall\quad
j,t$ (6)

And

$\displaystyle E(\varepsilon_{jt}\mid\mathcal{I}_{jt})=E(\varepsilon_{jt})=0$
(7) $\displaystyle
E(e^{\varepsilon_{jt}}\mid\mathcal{I}_{jt})=E(e^{\varepsilon_{jt}})=\mathcal{E}$

Where $\mathcal{E}$ is a scalar constant.

Furthermore, the following two assumptions characterize the inputs allocations
and the pricing behavior.

######  Assumption 3 (Inputs Allocations)

Firm $j$ allocates the period $t$ capital and labor inputs, $K_{jt}$ and
$L_{jt}$, just before period $t$ starts. Hence, these inputs are
"predetermined". On the other hand, the firm allocates intermediate materials,
$M_{jt}$, after knowing $\omega_{jt}$ but before knowing $\varepsilon_{jt}$
solving a static first-order condition. Then, intermediate materials are a
"flexible" production input.

######  Assumption 4 (Pricing Behavior)

666Assumption 4 is closely tied to the assumption on output and input markets
in [19] (page 13). Unlike the authors, I assume firms are also price takers in
the output market. Moreover, while they assume that $nominal$ inputs prices
and the final output price demand shifter follow an unspecified Markovian
process, in this paper the Markovian assumption regards $relative$ prices.

Firms are price takers in the output and input markets. The firm faces the
final output price $P_{jt}$, and $w_{jt}$, $r_{jt}$ and $\rho_{jt}$ as the
wage rate, the rental rate of capital, and the unit cost of materials,
respectively. The input prices are paid simultaneously with each input
allocation.
The inputs’ relative prices evolve according to an exogenous Markov process
not further specified, which accounts for time-inhomogeneity and may depend on
the firm’s productivity via the input allocation’s relevant information set.
Then, the following characterizations follow without loss of generality

$\displaystyle\frac{r_{jt}}{P_{jt}}=\tilde{r}_{jt}\left(\tilde{\mathcal{I}}_{jt-1},\cdots\right)$
(8)
$\displaystyle\frac{w_{jt}}{P_{jt}}=\tilde{w}_{jt}\left(\tilde{\mathcal{I}}_{jt-1},\cdots\right)$
$\displaystyle\frac{\rho_{jt}}{P_{jt}}=\tilde{\rho}_{jt}\left(\mathcal{I}_{jt},\cdots\right)$

The following timeline summarizes and contextualizes the assumptions above

  1. 1.

Firm $j$ ends period $t-1$ knowing the period’s TFP $\nu_{jt-1}$ and the final
output $Y_{jt-1}$. That is, the firm holds information set
$\tilde{\mathcal{I}}_{jt-1}$.

  2. 2.

The firm now chooses $L_{jt}$ and $K_{jt}$, period $t$ predetermined inputs,
for real wage rate $\frac{w_{jt}}{P_{jt}}$ and real rental rate of capital
$\frac{r_{jt}}{P_{jt}}$ as a function of $\tilde{\mathcal{I}}_{jt-1}$.

  3. 3.

Firm $j$ enters period $t$. $\eta_{jt}$ realizes. The firm now knows
$\omega_{jt}$ and holds information set $\mathcal{I}_{jt}$.

  4. 4.

The firm chooses the period $t$ intermediate materials allocation, $m_{jt}$,
as a function of $\mathcal{I}_{jt}$ given the real material inputs price
$\frac{\rho_{jt}}{P_{jt}}$, solving an intermediate inputs value-added
conditional maximization problem

$\operatorname*{Max}_{M_{jt}}\left[{E(F(k_{jt},l_{jt},m_{jt})e^{\nu_{jt}}\mid\mathcal{I}_{jt})-\frac{\rho_{jt}}{P_{jt}}M_{jt}}\right]$
(9)

  5. 5.

Finally, right before period $t$ ends, the firm can now observe
$\varepsilon_{jt}$ and, then, the period’s TFP $\nu_{jt}$ and final output
$Y_{jt}$. The firm now holds the information set $\tilde{\mathcal{I}}_{jt}$.

Throughout this paper, I will consistently refer to $\eta_{jt}$ as the ’ex-
ante productivity shock’ and $\varepsilon_{jt}$ as the ’ex-post productivity
shock’, relative to the material inputs allocation. Given the timing
constraints and uncertainty, TFP in this model is not necessarily Hicks-
neutral. A more detailed discussion on this topic can be found in Appendix A.
Finally, the model exhibits flexibility due to the unspecified decision
problem for the predetermined inputs. This flexibility allows the model to
accommodate a wide array of dynamic and static distortions affecting capital
and labor allocations. Examples are resale losses due to transaction costs,
the market for lemon phenomenon, the physical cost of resale and refit costs
for capital, hiring, firing, and training losses for workers ([10]), working
capital and borrowing constraints (see [13] for a discussion on their
relevance for shocks propagation on an international perspective), government
regulations, transportation costs, subsidies, and taxes ([28]).

###  2.1 Productivity Growth and the Inputs’ Marginal Product

TFP growth. Let the output elasticity of input $X$ be a non-parametric
function of the inputs allocation777Notice that ${\frac{\partial
y_{jt}}{\partial x_{jt}}}={\frac{\partial f(k_{jt},l_{jt},m_{jt})}{\partial
x_{jt}}}$ since ${\frac{\partial\nu_{jt}}{\partial x_{jt}}}=0$ by the
exogeneity assumptions on $\nu_{jt}$.

$elas^{X}(k_{jt},l_{jt},m_{jt})=\frac{\partial
f(k_{jt},l_{jt},m_{jt})}{\partial x_{jt}}\quad\forall X\in\\{K,L,M\\}$ (10)

Denote $MP_{jt}^{X}$ as the marginal product of input $X$ for firm $j$ at time
$t$. Using the chain rule, it follows that $MP_{jt}^{X}$ is a function of the
inverse input $X$ share of output and the output elasticity of input $X$.

$\displaystyle MP^{X}_{jt}=\frac{\partial Y_{jt}}{\partial
X_{jt}}=\frac{\partial Y_{jt}}{\partial y_{jt}}\frac{\partial x_{jt}}{\partial
X_{jt}}\frac{\partial y_{jt}}{\partial
x_{jt}}=\frac{Y_{jt}}{X_{jt}}elas^{X}(k_{jt},l_{jt},m_{jt})$ (11)

Taking the natural logarithm, denote for each input $mp^{X}_{jt}$ as

${mp}^{X}_{jt}=\log({MP}^{X}_{jt})$ (12)

Totally differentiating ${mp}^{X}_{jt}$ with respect to TFP growth
$\Delta\nu_{jt}$, where $\Delta$ denotes a difference between period $t$ and
$t-1$, leads to888Notice indeed that  $\frac{\partial{mp}^{X}_{jt}}{\partial
y_{jt}}=-\frac{\partial{mp}^{X}_{jt}}{\partial
x_{jt}}=\frac{\partial{mp}^{X}_{jt}}{\partial\log
elas^{X}(k_{jt},l_{jt},m_{jt})}=1$ (13)

$\frac{d{mp}^{X}_{jt}}{d\Delta\nu_{jt}}=\frac{d{y_{jt}}}{d\Delta\nu_{jt}}-\frac{d{x_{jt}}}{d\Delta\nu_{jt}}+\frac{d{\log
elas^{X}(k_{jt},l_{jt},m_{jt})}}{d\Delta\nu_{jt}}$ (14)

Equation (14) tells that three factors determine the elasticity of an input’s
marginal product with respect to a hypothetical change in the firm’s TFP
growth. The first and second terms are the output and input allocation
elasticity to TFP growth. Finally, the third term accounts for the effect of
the change in TFP growth on the input’s output elasticity.

TFP growth channels. One can further decompose TFP growth $\Delta\nu_{jt}$ as

$\displaystyle\Delta\nu_{jt}$
$\displaystyle=(\omega_{jt}-\omega_{jt-1})+(\varepsilon_{jt}-\varepsilon_{jt-1})$
(15)
$\displaystyle=m(\omega_{jt-1})+\eta_{jt}-\omega_{jt-1}+\varepsilon_{jt}-\varepsilon_{jt-1}$
$\displaystyle=g(\omega_{jt-1})+\eta_{jt}+\Delta\varepsilon_{jt}$

In the last line, I aggregate the terms related to the previous period
persistent productivity term, $\omega_{jt-1}$, in the function $g$. Assume for
simplicity that there has been no ex-post productivity shock in the previous
period (i.e., $\varepsilon_{jt-1}=0$, its unconditional mean). Then, equation
(15) simplifies into

$\Delta\nu_{jt}=g(\omega_{jt-1})+\eta_{jt}+\varepsilon_{jt}$ (16)

Using the chain and the inverse-function rules999Indeed
$\displaystyle\frac{dmp_{jt}^{X}}{d\Delta\nu_{jt}}=\left(\frac{d\Delta\nu_{jt}}{dmp_{jt}^{X}}\right)^{-1}=\left(\frac{dg(\omega_{jt-1})}{dmp^{X}_{jt}}+\frac{d\eta_{jt}}{dmp^{X}_{jt}}+\frac{d\varepsilon_{jt}}{dmp^{X}_{jt}}\right)^{-1}$
$\displaystyle=$ $\displaystyle\left(\left(\frac{\textsection
mp^{X}_{jt}}{\textsection\omega_{jt-1}}\left(\frac{\partial
g(\omega_{jt-1})}{\partial\omega_{jt-1}}\right)^{-1}\right)^{-1}+\left(\frac{\textsection
mp^{X}_{jt}}{\textsection\eta_{jt}}\right)^{-1}+\left(\frac{\textsection
mp^{X}_{jt}}{\textsection\varepsilon_{jt}}\right)^{-1}\right)^{-1}$ The symbol
$\textsection$ refers to a partial total derivative. For more details, I refer
the reader to [35], page 192. In this setup, $\frac{\textsection
mp^{X}}{\textsection\theta}$ has the interpretation of a total derivative
keeping the other exogenous variables fixed. For example  $\frac{\textsection
mp^{X}_{jt}}{\textsection\omega_{jt-1}}=\left.\frac{dmp^{X}_{jt}}{d\omega_{jt-1}}\right|_{d\eta_{jt=0},d\varepsilon_{jt}=0}$
However, since $\omega_{jt-1}\perp\eta_{jt}\perp\varepsilon_{jt}$ by
Assumption 2, I can ease the notation by replacing the partial total
derivatives with simple total derivatives
$\frac{dmp_{jt}^{X}}{d\Delta\nu_{jt}}=\left(\left(\frac{dmp^{X}_{jt}}{d\omega_{jt-1}}\left(\frac{\partial
g(\omega_{jt-1})}{\partial\omega_{jt-1}}\right)^{-1}\right)^{-1}+\left(\frac{dmp^{X}_{jt}}{d\eta_{jt}}\right)^{-1}+\left(\frac{dmp^{X}_{jt}}{d\varepsilon_{jt}}\right)^{-1}\right)^{-1}$
, I decompose the effect of a counterfactual static variation of the firm’s
TFP growth on an inputs’ marginal products into its channels

$\frac{dmp_{jt}^{X}}{d{\Delta\nu_{jt}}}=c\left(\frac{dmp_{jt}^{X}}{d\omega_{jt-1}},\frac{dmp_{jt}^{X}}{d\eta_{jt}},\frac{dmp_{jt}^{X}}{d\varepsilon_{jt}}\right)$
(17)

Expression (17) reveals three channels that determine the impact of a
counterfactual variation in TFP growth on an individual input’s marginal
product. The first channel ties into a variation in past TFP, known to the
firm at the time of every input allocation. The second and third channels
hinge on counterfactual variations in the ex-ante and ex-post productivity
shocks. The firm possesses knowledge of the second channel only during
material allocation, while the third channel always represents a post-
allocation variation.

Formally, totally differentiating $mp^{X}_{jt}$ with respect to the TFP growth
channel $\theta_{jt}$ leads to

$\frac{dmp_{jt}^{X}}{d\theta_{jt}}=\frac{dy_{jt}}{d\theta_{jt}}-\frac{d{x_{jt}}}{d\theta_{jt}}+\frac{d{\log
elas^{X}(k_{jt},l_{jt},m_{jt})}}{d\theta_{jt}}$ (18)

In Appendix B, I analytically derive the effect of a counterfactual change of
each TFP growth source on each input’s marginal product. Given the timing and
allocation assumptions, these channels affect the inputs’ marginal products
differently.

##  3 Data

###  3.1 US data

Yearly firm-level panel data for the US are sourced from Compustat. The sample
contains all publicly traded firms operating in the manufacturing sector
(NAICS 31-33) between 1960-2018. The data contain a unique firm identifier,
year of operation, the industrial sector at the 6-digit NAICS level, location,
and standard balance sheet variables. Since the data do not report quantities,
the output measure used is deflated net sales and, for the labor and capital
inputs, the number of employees and the deflated total net value of property,
plant, and equipment101010The Chilean and Colombian data used in [23] do not
report quantities either. The authors then resort to a similar approximation..
I directly source these measures from the data. However, the dataset does not
contain values for the cost of material inputs and the cost of labor, which I
construct following [29]. First, I compute the labor cost by multiplying the
number of employees by the average industry wage, retrieved from [3]. Second,
I construct the cost of materials as the cost of goods sold plus
administrative and selling expenses minus wage expenditure and capital
depreciation. Then, I use this value deflated as a measure for material inputs
allocation. I source industry-specific deflators from [3]. Appendix C.1
contains additional information on data cleaning. Table 17 in Appendix H shows
summary statistics of the variables of interest.

As many authors already pointed out (see [6] and [18]), the Compustat sample
is biased towards bigger and older companies since it is restricted to only
publicly traded firms. I compare the results using US data to those obtained
from European balance-sheet panel data, described below, to reduce the
distortions that may arise.

###  3.2 European Data

Annual firm-level harmonized balance sheet data for the European countries are
sourced from BvD-Orbis. [25] describe the dataset and its main advantages in
detail. I restrict the data to firms operating in the manufacturing sector
(NACE 10-33). The bias towards bigger firms that affects Compustat data is
absent in this dataset since balance sheet reporting is mandatory for small
private firms in European countries, and Orbis collects administrative data
from local Chambers of Commerce. I collect data from Belgium (2007-2019),
Germany (2004-2018), Spain (2000-2017), France (2000-2017), Hungary
(2007-2019), Italy (2000-2017), Poland (2004-2018), Portugal (2007-2019),
Romania (2004-2018), Slovenia (2007-2019), Sweden (2009-2019). Like Compustat,
the dataset contains a unique firm identifier, year of operation, industrial
sector up to 4-digit NACE level, and standard accounting data. The output
measure I use is deflated revenue. Moreover, I use the number of employees,
deflated fixed assets, and deflated cost of materials to measure labor,
capital, and material inputs. These quantities, as well as the cost of labor,
are present in the data. Industry-specific deflators are sourced from EU KLEMS
and, when unavailable, from OECD STAN. Appendix C.2 has additional dataset
cleaning and construction details. Table 18-37 in Appendix H shows summary
statistics of the variables of interest.

Finally, to ease the computational burden, when the number of individual firms
for each country is too big (i.e., exceeds 5,000), I randomly draw 5,000
individual firms from the country’s pool of individual firms. This approach
has been used for France, Italy, Spain, Germany, Poland, Romania, Portugal,
Slovenia, and Sweden.

##  4 Estimation

The model primitives align with the estimation strategy proposed by [23]. For
each country’s aggregated manufacturing sector, I estimate the production
function separately, employing their innovative, non-parametric method.
Detailed insights into this approach to identification and estimation appear
in Appendix D, with a brief overview provided below.

The first step unfolds by rearranging the first-order condition (FOC) from the
firm’s conditional value-added maximization problem (9). This process yields a
non-linear equation, which is estimable using least squares methods, and
allows to recover the output elasticity of material inputs and the ex-post
productivity shock at the firm-time level.

By integrating the material input elasticity, we can identify the portion of
the production function related to material inputs. Subtracting this, along
with the ex-post productivity shock, from the firm’s output allows to identify
the sum of the two remaining unobservables: the part of the production
function unrelated to material inputs, and the firm’s persistent TFP
component.

In the second stage, I approximate the Markovian process of the persistent
component of TFP and the remaining part of the production function using
polynomials. Then, I estimate the parameters of the polynomials using the
Generalized Method of Moments (GMM), leveraging the orthogonality between the
allocation of labor and capital and past persistent productivity on one side,
and the ex-ante productivity shock on the other.

The production function estimates then permit the recovery of the quantities
of interest, such as input elasticities and marginal products. I compute
standard errors by applying non-parametric bootstrap resampling 150 times at
the firm level (and 600 times for the US).

##  5 Results

###  5.1 Production Function Estimation

In Figure 1, I show the distribution of the estimated US firm-specific
average111111Average over the period the firm is active in the data. output
elasticities for each input. As one would expect, all distributions are
normally shaped and mostly lie in the unit interval: 98% of the observations
of capital elasticities, 89% of material inputs elasticities, and 83% of labor
elasticities are in between 0 and 1121212This result is in line with [23]. The
authors use Chilean and Colombian data and estimate the elasticities
distributions separately for the food products (NAICS 311), textiles (NAICS
321), apparel (NAICS 322), wood products (NAICS 331), and fabricated metal
products (NAICS 381) industries. They found that, for any given industry, at
most 2% of the labor and intermediate-input elasticities are outside the unit
interval. In contrast, for capital, the elasticities are more concentrated on
zero on average, but even in the worst case, less than 9.4% have values
outside the (0,1) range.. In the Appendix, section G.1, I report the same
distributions for the European countries in the dataset. For them, the average
firm-specific elasticities are even more concentrated in the unit
interval131313Notice that the spike in the material inputs elasticity
distribution that some of them display at 0 is a byproduct of the share
regression in equation (59), which forces the elasticity of material inputs to
be strictly positive..

Figure 1: USA - Estimated Output Elasticities

Distribution of the estimated firm-specific output elasticities of inputs for
the United States manufacturing sector, averaged over time.

Table 1 presents the estimates for each country’s average output elasticities
for each input, their sum, and the ratio of the average capital and labor
elasticities. Across all production inputs, capital typically exhibits the
lowest elasticity, with the highest being Germany’s 24%. Conversely, labor
generally possesses higher elasticity than intermediate inputs, with the US as
an exception, where the latter’s elasticity triples the average elasticity of
labor141414[23] also observed higher average output elasticity for material
inputs in their studies using Colombian and Chilean manufacturing data.. The
sum of elasticities aligns with either constant or diminishing returns to
scale. Furthermore, the ratio of average capital to labor elasticities,
indicative of the capital intensity relative to labor in the production
technology, exhibits significant heterogeneity across countries, ranging from
a low of 0.06 for Sweden to a high of 0.99 for the US.

Table 1: Average Output Elasticities

|  Capital |  Labor |  Intermediates |  Sum |  Capital/Labor
---|---|---|---|---|---
USA |  0.211 |  0.213 |  0.666 |  1.091 |  0.990
|  (0.611) |  (0.043) |  (0.008) |  (0.565) |  (1.648)
Belgium |  0.101 |  0.444 |  0.353 |  0.898 |  0.228
|  (0.011) |  (0.024) |  (0.037) |  (0.044) |  (0.042)
Germany |  (0.238 |  0.349 |  0.272 |  0.860 |  0.683
|  (0.024) |  (0.052) |  (0.029) |  (0.031) |  (0.100)
Spain |  0.147 |  0.426 |  0.419 |  0.992 |  0.345
|  (0.015) |  (0.027) |  (0.005) |  (0.015) |  (0.050)
France |  0.181 |  0.565 |  0.196 |  0.942 |  0.321
|  (0.033) |  (0.073) |  (0.035) |  (0.029) |  (0.110)
Hungary |  0.197 |  0.541 |  0.276 |  1.013 |  0.364
|  (0.025) |  (0.055) |  (0.030) |  (0.028) |  (0.106)
Italy |  0.132 |  0.435 |  0.356 |  0.923 |  0.304
|  (0.007) |  (0.010) |  (0.004) |  (0.008) |  (0.020)
Poland |  0.201 |  0.368 |  0.336 |  0.905 |  0.546
|  (0.022) |  (0.021) |  (0.051) |  (0.019) |  (0.057)
Portugal |  0.104 |  0.561 |  0.348 |  1.013 |  0.186
|  (0.489) |  (0.105) |  (0.016) |  (0.394) |  (1.109)
Romania |  0.182 |  0.519 |  0.348 |  1.049 |  0.352
|  (0.020) |  (0.036) |  (0.007) |  (0.021) |  (0.055)
Slovenia |  0.113 |  0.568 |  0.336 |  1.017 |  0.199
|  (0.048) |  (0.040) |  (0.034) |  (0.035) |  (0.080)
Sweden |  0.046 |  0.778 |  0.332 |  1.155 |  0.059
|  (0.009) |  (0.024) |  (0.012) |  (0.019) |  (0.011)

  * •

The table reports the averages across firms and years of the estimated input
elasticities, their sum, and the ratio between the averages of capital and
labor elasticities. Standard errors (in parenthesis) are computed using non-
parametric bootstrap drawing 150 times with replacement (for the USA, I
performed 600 draws with replacement).

Regarding the estimation results for TFP, I assumed the systematic component
of productivity follows a Markov process, which I approximated by a 3-rd order
degree polynomial.

$\omega_{jt}=\delta_{0}+\delta_{1}\omega_{jt-1}+\delta_{2}\omega_{jt-1}^{2}+\delta_{3}\omega_{jt-1}^{3}+\eta_{jt}$
(19)

Table 2 Section I presents the country-specific estimates for the parameter
vector $\delta$ from equation (19). For most cases, only the estimates for the
parameters $\delta_{0}$ and $\delta_{1}$ display sufficient precision to
reject the null hypothesis of equality to 0 at standard confidence levels.
Moreover, the estimates for $\delta_{1}$ range between 0.76 for Romania and
0.96 for France, leading to the conclusion that an AR(1) process (with drift)
of high persistence closely approximates the systematic productivity component
dynamics in each country. Section II of Table 2 lists the maximum likelihood
parameter estimates derived from fitting a Generalized Extreme Value
distribution, denoted as $GEV\sim(\xi,\sigma,\mu)$, to each country’s pooled
firm-time specific estimates of TFP levels ($e^{\hat{\nu}_{jt}}$). The
parameter $\xi$ shapes the distribution, while the parameters $\sigma$ and
$\mu$ determine the scale and location of the distribution. For every country,
the estimated shape parameter is positive, indicating that Frechet
distributions fit the data accurately151515The assumption of a Frechet
distribution for TFP is prevalent in the economic literature, especially in
International Trade following the foundational work of [20]. In this context,
the distribution is a result of estimation.. Expectedly, Germany’s TFP
distribution exhibits the highest mean (4.03), closely followed by Belgium
(1.49)161616Given the estimates $\hat{\xi}$, $\hat{\sigma}$, and $\hat{\mu}$,
and since $0<\hat{\xi}<1$ for each country, the expected value of the
estimated GEV distribution is computed as
$E(X)=\hat{\mu}+\hat{\sigma}(\Gamma(1-\hat{\xi})-1)/\hat{\xi}$ where
$\Gamma(.)$ is the Gamma function..

Table 2: Productivity Estimates

| I: Systematic Productivity Process | II: GEV TFP distribution
---|---|---
| $\delta_{0}$ | $\delta_{1}$ | $\delta_{2}$ | $\delta_{3}$ | $\xi$ | $\sigma$ | $\mu$
USA | $-0.366$ | $0.822$ | $-0.024$ | $-0.001$ | $0.043$ | $0.017$ | $0.032$
| $(0.143)$ | $(0.099)$ | $(0.024)$ | $(0.002)$ | $(0.001)$ | $(0.000)$ | $(0.000)$
Belgium | $0.016$ | $0.959$ | $-0.004$ | $-0.018$ | $0.015$ | $0.498$ | $1.196$
| $(0.006)$ | $(0.020)$ | $(0.019)$ | $(0.004)$ | $(0.003)$ | $(0.002)$ | $(0.003)$
Germany | $0.096$ | $0.923$ | $0.017$ | $-0.009$ | $0.098$ | $1.521$ | $2.987$
| $(0.038)$ | $(0.024)$ | $(0.034)$ | $(0.016)$ | $(0.003)$ | $(0.007)$ | $(0.010)$
Spain | $-0.017$ | $0.907$ | $0.036$ | $-0.020$ | $0.007$ | $0.277$ | $0.692$
| $(0.003)$ | $(0.019)$ | $(0.038)$ | $(0.009)$ | $(0.001)$ | $(0.001)$ | $(0.001)$
France | $0.009$ | $0.960$ | $0.017$ | $-0.009$ | $0.101$ | $0.360$ | $0.714$
| $(0.006)$ | $(0.008)$ | $(0.020)$ | $(0.009)$ | $(0.003)$ | $(0.001)$ | $(0.002)$
Hungary | $-0.078$ | $0.898$ | $-0.015$ | $-0.010$ | $0.212$ | $0.287$ | $0.503$
| $(0.015)$ | $(0.011)$ | $(0.014)$ | $(0.004)$ | $(0.005)$ | $(0.001)$ | $(0.002)$
Italy | $-0.002$ | $0.947$ | $0.115$ | $-0.004$ | $0.008$ | $0.372$ | $0.823$
| $(0.002)$ | $(0.009)$ | $(0.021)$ | $(0.007)$ | $(0.001)$ | $(0.001)$ | $(0.002)$
Poland | $-0.041$ | $0.908$ | $0.036$ | $-0.011$ | $0.092$ | $0.303$ | $0.578$
| $(0.010)$ | $(0.037)$ | $(0.033)$ | $(0.007)$ | $(0.003)$ | $(0.001)$ | $(0.002)$
Portugal | $-0.047$ | $0.922$ | $0.073$ | $0.002$ | $0.034$ | $0.260$ | $0.610$
| $(0.047)$ | $(0.044)$ | $(0.017)$ | $(0.002)$ | $(0.002)$ | $(0.001)$ | $(0.001)$
Romania | $-0.084$ | $0.761$ | $-0.009$ | $-0.041$ | $0.212$ | $0.384$ | $0.648$
| $(0.009)$ | $(0.007)$ | $(0.015)$ | $(0.009)$ | $(0.004)$ | $(0.002)$ | $(0.002)$
Slovenia | $-0.052$ | $0.900$ | $0.111$ | $0.017$ | $0.054$ | $0.270$ | $0.660$
| $(0.027)$ | $(0.039)$ | $(0.074)$ | $(0.027)$ | $(0.002)$ | $(0.001)$ | $(0.002)$
Sweden | $-0.033$ | $0.959$ | $0.165$ | $0.040$ | $0.001$ | $0.283$ | $0.616$
| $(0.012)$ | $(0.070)$ | $(0.130)$ | $(0.058)$ | $(0.002)$ | $(0.002)$ | $(0.002)$

  * •

In Section I of this table, I report the productivity process parameters in
equation (19) estimated country by country as described in Appendix D.
Standard errors (in parenthesis) are computed using non-parametric bootstrap
drawing 150 times with replacement (for the USA, I performed 600 draws with
replacement).
In Section II of this table, I report the maximum likelihood estimates from
fitting a Generalized Extreme Value distribution, $GEV\sim(\xi,\sigma,\mu)$,
for the pooled estimates of TFP in level, $e^{\nu_{jt}}$. $\xi$ is the shape,
$\sigma$ is the scale, and $\mu$ is the location parameter of the
distribution. I report maximum likelihood standard errors (in parenthesis).

The model regards the allocation of capital and labor inputs as predetermined
relative to materials, but it remains agnostic about their allocation problem.
Despite robust empirical evidence supporting the notion that capital is a
quasi-fixed input subject to depreciation, lumpy investments, and adjustment
costs (refer to [15]), there is no consensus regarding the nature of the labor
input. In Appendix E, a model-consistent test I developed allows for assessing
the null hypothesis that labor is flexible. In this context, flexible means
the input solves a static value-added maximization problem under imperfect
information. Empirical evidence lends strong support to the rejection of this
null hypothesis.

###  5.2 Productivity Growth and Inputs Marginal Products

This section delves into the relationship between TFP growth and the marginal
products of all inputs. Initially, the exploration centers on the micro-level
correlation between the marginal product of inputs and a firm’s TFP growth.
Subsequently, attention shifts to estimating the proportion of cross-sectional
industry-level dispersion of the marginal product explained by the implied TFP
volatility for all inputs.

Micro-level correlation. Figure 2 reveals a notable correlation at the firm
level between the (log) marginal products of inputs and (log) TFP growth in
the pooled data. A simple linear regression yields coefficients of $0.51$ for
capital, $0.24$ for labor, and $0.33$ for materials, all bearing strong
statistical significance. Furthermore, the data suggest a noticeable
heterogeneity in the correlation’s absolute size across different countries.
For instance, Section (b) of Figure 2 indicates a comparatively lower
correlation between labor’s marginal product and TFP growth for the US.

Figure 2: Productivity Growth and Inputs Marginal Products

In this figure, I plot the firm-time specific (log) TFP growth
($\Delta\nu_{jt}$) against the (log) marginal product ($mp_{jt}$) of capital
(a), labor (b), and material inputs (c) for all countries in the dataset. The
regression coefficients are respectively $0.51$, $0.24$, and $0.33$. The
number of observations are, respectively, 449,621, 439,387, and 451,272.

To accommodate industry-time-specific omitted variables, I introduce the
corresponding fixed effects into a country-specific linear regression model as
follows:

$mp_{jt}^{X}=\beta_{dev}\Delta\nu_{jt}+\iota_{st}+\zeta_{jt}$ (20)

Here, $\Delta\nu_{jt}$ again represents the shift in the firm-specific log TFP
($\nu_{jt}$) from period $t-1$ to period $t$, whereas $\zeta_{jt}$ serves as a
residual measurement error term.171717This residual error term can account for
measurement errors made by the econometrician or latent measurement errors in
the data balance sheet variables which have been carried on in the analysis
and incorporated in the TFP growth and marginal products estimates.
$\iota_{st}$ is a sector-time fixed effect, where the sector is identified as
the three-digit level NACE for European countries or NAICS for the US. A
closer examination of specification (20) allows us to interpret the regression
coefficient $\beta_{dev}$ causally, in comparative static terms, as the
(country-average) marginal effect of a hypothetical variation in the firm’s
TFP growth on the input’s marginal product. Each variable in this
interpretation is measured in terms of log deviations from its industry-time
mean. Formally

$\frac{\partial(mp_{jt}^{X}-E_{st}(mp_{jt}^{X}))}{\partial(\Delta\nu_{jt}-E_{st}(\Delta\nu_{jt}))}\quad\forall
X\in\\{K,L,M\\}$ (21)

where $E_{st}(.)$ denotes industry-time-specific means.

Table 3 displays the results of executing regression (20) for each country.
Consistent with the trends observed in Figure 2, all production inputs’
marginal products react substantially to changes in TFP growth, when measured
in terms of log deviations. The US exhibits the smallest effects, although all
the estimated coefficients still have high significance. I compute
elasticities of 0.51 for the marginal product of capital, 0.47 for labor, and
0.36 for materials. Interestingly, the incorporation of fixed effects does not
alter the overall elasticity of the marginal product of capital with respect
to TFP growth. However, it does lead to larger estimated elasticities for the
marginal products of labor and materials. Given the joint estimation of the
inputs’ marginal products and TFP, the discrepancy is primarily due to higher
within-group industry-time variation than between-group variation rather than
a bias caused by omitted industry-time fixed variables. As anticipated, the US
exhibits the smallest effects —0.4 for capital (compared to a peak of 0.78 for
Belgium), 0.23 for labor, and 0.21 for materials— with Spain showing the
highest effects for both labor and materials, at 0.64 and 0.48 respectively.

Table 3: The Effect of TFP Growth on the Marginal Product of Inputs

| Capital | Labor | Material Inputs
---|---|---|---
Country | $\hat{\beta}_{dev}$ | $R^{2}$ |  Adj $R^{2}$ |  N | $\hat{\beta}_{dev}$ | $R^{2}$ |  Adj $R^{2}$ |  N | $\hat{\beta}_{dev}$ | $R^{2}$ |  Adj $R^{2}$ |  N
USA | $0.398^{***}$ | 0.211 | 0.196 |  68,289 | $0.225^{***}$ | 0.208 | 0.190 | 58,294 | $0.214^{***}$ | 0.074 | 0.056 |  69,210
| $(0.028)$ |  |  |  | $(0.029)$ |  |  |  | $(0.026)$ |  |  |
Belgium | $0.776^{***}$ | 0.118 | 0.082 |  28,632 | $0.546^{***}$ | 0.265 | 0.235 | 28,631 | $0.439^{***}$ | 0.198 | 0.166 |  28,755
| $(0.083)$ |  |  |  | $(0.053)$ |  |  |  | $(0.047)$ |  |  |
France | $0.587^{***}$ | 0.223 | 0.199 |  53,833 | $0.585^{***}$ | 0.370 | 0.351 | 53,833 | $0.387^{***}$ | 0.371 | 0.351 |  53,835
| $(0.025)$ |  |  |  | $(0.019)$ |  |  |  | $(0.017)$ |  |  |
Germany | $0.495^{***}$ | 0.146 | 0.100 |  24,394 | $0.380^{***}$ | 0.272 | 0.232 | 24,346 | $0.262^{***}$ | 0.170 | 0.125 |  24,459
| $(0.052)$ |  |  |  | $(0.043)$ |  |  |  | $(0.044)$ |  |  |
Hungary | $0.548^{***}$ | 0.115 | 0.079 |  29,188 | $0.512^{***}$ | 0.303 | 0.274 | 29,168 | $0.362^{***}$ | 0.205 | 0.173 |  29,188
| $(0.031)$ |  |  |  | $(0.025)$ |  |  |  | $(0.022)$ |  |  |
Italy | $0.521^{***}$ | 0.118 | 0.085 |  50,533 | $0.367^{***}$ | 0.172 | 0.142 | 50,465 | $0.303^{***}$ | 0.200 | 0.171 |  50,643
| $(0.028)$ |  |  |  | $(0.022)$ |  |  |  | $(0.018)$ |  |  |
Poland | $0.397^{***}$ | 0.113 | 0.080 |  34,611 | $0.430^{***}$ | 0.243 | 0.215 | 34,615 | $0.272^{***}$ | 0.259 | 0.232 |  34,625
| $(0.035)$ |  |  |  | $(0.025)$ |  |  |  | $(0.021)$ |  |  |
Portugal | $0.601^{***}$ | 0.173 | 0.149 |  35,028 | $0.500^{***}$ | 0.277 | 0.257 | 35,034 | $0.438^{***}$ | 0.260 | 0.240 |  35,163
| $(0.033)$ |  |  |  | $(0.017)$ |  |  |  | $(0.015)$ |  |  |
Romania | $0.466^{***}$ | 0.120 | 0.085 |  31,508 | $0.515^{***}$ | 0.272 | 0.243 | 31,426 | $0.451^{***}$ | 0.417 | 0.394 |  31,587
| $(0.016)$ |  |  |  | $(0.010)$ |  |  |  | $(0.008)$ |  |  |
Slovenia | $0.591^{***}$ | 0.102 | 0.068 |  26,444 | $0.489^{***}$ | 0.223 | 0.193 | 26,418 | $0.330^{***}$ | 0.262 | 0.233 |  26,446
| $(0.037)$ |  |  |  | $(0.021)$ |  |  |  | $(0.014)$ |  |  |
Spain | $0.644^{***}$ | 0.128 | 0.101 |  53,934 | $0.623^{***}$ | 0.252 | 0.229 | 53,923 | $0.476^{***}$ | 0.250 | 0.226 |  54,101
| $(0.028)$ |  |  |  | $(0.021)$ |  |  |  | $(0.017)$ |  |  |
Sweden | $0.327^{***}$ | 0.115 | 0.071 |  13,228 | $0.419^{***}$ | 0.198 | 0.158 | 13,235 | $0.287^{***}$ | 0.190 | 0.149 |  13,261
| $(0.050)$ |  |  |  | $(0.027)$ |  |  |  | $(0.024)$ |  |  |
All | $0.514^{***}$ | 0.238 | 0.207 |  449,622 | $0.466^{***}$ | 0.514 | 0.493 | 439,388 | $0.359^{***}$ | 0.279 | 0.249 |  451,273
| $(0.009)$ |  |  |  | $(0.007)$ |  |  |  | $(0.006)$ |  |  |

  * •

The table reports the coefficient of the linear regression in (20).

  * •

The sector is identified at the 3-digit level of disaggregation (NACE for
European countries, NAICS for the US).

  * •

I include country-year-industry fixed effects for the regression where all
countries are pooled.

  * •

Standard errors are clustered at the individual firm level.

  * •

∗p$<$0.1.

  * •

∗∗p$<$0.05.

  * •

∗∗∗p$<$0.01.

Industry-level dispersion. The design in (20) allows aggregation at the cross-
sectional industry level. Assuming that the exogeneity assumption for the
measurement error in equation (20) also holds at the industry level, we can
estimate the following equivalent specification of equation (20) sector by
sector for a given country:

$(mp_{jt}^{X}-\overline{mp}_{st}^{X})=\beta_{s,dev}(\Delta\nu_{jt}-\overline{\Delta\nu}_{st})+\underbrace{(\zeta_{jt}-\overline{\zeta}_{st})}_{\begin{subarray}{c}\tilde{\zeta}_{jt}\end{subarray}}$
(22)

where $\overline{mp}_{st}^{X}$, $\overline{\Delta\nu}_{st}$, and
$\overline{\zeta}_{st}$ are industry-time specific averages. The
$\beta_{s,dev}$ regression coefficient thus serves as an estimator for the
sector-specific average of the marginal effect in (21). Given the exogeneity
assumption on the error term, a sum of squares transformation of equation (22)
produces the projection:

$Var_{st}(mp^{X}_{jt})=\beta_{s,var}\underbrace{Var_{st}(\Delta\nu_{jt})}_{\begin{subarray}{c}Vol_{st}(\nu_{jt})\end{subarray}}+\varphi_{st}$
(23)

Here, $Vol_{st}(\nu_{jt})$ represents the volatility of log TFP at time $t$
and sector $s$, and $\beta_{s,var}=\beta_{s,dev}^{2}$ by construction.

By employing the uncentered $R^{2}$ statistic used in linear regression
analysis, we can compute the average proportion of cross-sectional dispersion
in the marginal product of inputs explained by TFP volatility at the industry-
time level. I denote this quantity as $S^{2}$, interpreting it as the share of
dispersion in the marginal products of inputs captured by the projection in
equation (23).

$S^{2}=1-\frac{\sum^{\lvert s\rvert}_{s=1}\sum^{\lvert
t\rvert}_{t=1}(Var_{st}(mp^{X}_{jt})-\hat{\beta}_{s,var}Vol_{st}(\nu_{jt}))^{2}}{\sum^{\lvert
s\rvert}_{s=1}\sum^{\lvert t\rvert}_{t=1}Var_{st}(mp^{X}_{jt})^{2}}$ (24)

Notably, since I compute the coefficient parameter $\hat{\beta}_{s,var}$ by
construction rather than by direct OLS estimation of equation (23), the
resulting $S^{2}$ statistic is a conservative measure and can be negative.
Negative $S^{2}$ values are considered uninformative. Table 4 shows the
country-specific $S^{2}$ results. On average, the industry-time-specific TFP
volatility explains 7% of the dispersion in the marginal product of capital,
9% for labor, and 10% for materials. These results exhibit notable
heterogeneity across countries. As expected, the explanatory power is low for
the US, averaging 2.1% for capital, 2.6% for labor, and 1% for materials.
Conversely, the percentages for Portugal rise significantly: 11% for capital,
18% for labor, and 28% for materials.

Table 4: $S^{2}$ \- Total TFP

Country | Capital | Labor | Materials
---|---|---|---
USA | $2.11\%$ | $2.59\%$ | $1.01\%$
Belgium | $4.61\%$ | $13.18\%$ | $16.13\%$
France | $7.41\%$ | $11.59\%$ | $7.99\%$
Germany | $8.05\%$ | $7.30\%$ | $13.17\%$
Hungary | $9.16\%$ | $13.07\%$ | $4.16\%$
Italy | $6.86\%$ | $13.33\%$ | $22.97\%$
Poland | $6.24\%$ | $11.24\%$ | $11.77\%$
Portugal | $10.79\%$ | $17.79\%$ | $25.30\%$
Romania | $8.60\%$ | $17.70\%$ | $27.86\%$
Slovenia | $6.40\%$ | $12.09\%$ | $5.44\%$
Spain | $5.51\%$ | $10.39\%$ | $9.50\%$
Sweden | $5.77\%$ | $10.06\%$ | $7.35\%$
All | $6.88\%$ | $8.93\%$ | $10.06\%$

  * •

The $S^{2}$ statistic for the pooled dataset is computed as follows, where $c$
denotes the country

$S^{2}=1-\frac{\sum^{\lvert c\rvert}_{c=1}\sum^{\lvert
s\rvert}_{s=1}\sum^{\lvert
t\rvert}_{t=1}(Var_{cst}(mp^{X}_{jt})-\hat{\beta}_{c,s,var}Vol_{cst}(\nu_{jt}))^{2}}{\sum^{\lvert
c\rvert}_{c=1}\sum^{\lvert s\rvert}_{s=1}\sum^{\lvert
t\rvert}_{t=1}Var_{cst}(mp^{X}_{jt})^{2}}$

###  5.3 Decomposing TFP Growth: The Productivity Channels

Building upon the TFP growth decomposition presented in Section 2.1, the
analysis now pivots to ascertain how a marginal variation in each channel,
holding the others constant, determines a corresponding fluctuation in the
inputs’ marginal products at the firm level. Further, the discussion will
explore the aggregate implications of these variations.

Micro-level correlation. I substitute the TFP growth term in specification
(20), denoted as $\Delta\nu_{jt}$, with its channels from equation (15). The
transformed equation (25) reads:

$mp_{jt}^{X}=\beta_{\omega}\omega_{jt-1}+\beta_{\eta}\eta_{jt}+\beta_{\varepsilon}\Delta\varepsilon_{jt}+\iota_{st}+\zeta_{jt}$
(25)

Estimating this equation (25) for capital, labor, and materials in each
country separately generates the results displayed in Table (5), Table (6),
and Table (7) respectively.

The marginal product of capital responds strongly to variations in the ex-post
productivity shock, $\eta_{jt}$. The estimated overall elasticity stands at
1.17, ranging from a low of 0.6 for the US to a high of 1.83 for Romania.
Comparatively, changes in past productivity ($\omega_{jt-1}$) and in the
growth of ex-post productivity shock ($\Delta\varepsilon_{jt}$) induce a
milder reaction with overall positive elasticities of 0.72 and 0.48,
respectively. Notably, the effect of past productivity exhibits considerable
heterogeneity across countries, with Romania leading the pack with the highest
estimated elasticity (1.58). In contrast, the impact of the ex-post
productivity shock is more evenly distributed across countries.

Likewise, for capital, the marginal product of labor increases for all TFP
growth channels. The overall influence of the ex-ante productivity channel
mirrors that of past productivity, albeit with the ex-post productivity shock
channel inducing a comparatively milder overall effect. Hence, I estimate the
overall elasticities of the ex-ante productivity, past productivity, and ex-
post productivity channels at 0.91, 0.85, and 0.45, respectively.
Interestingly, these estimates exhibit less cross-country variation than those
for capital.

Lastly, for materials, the marginal product increases in response to the ex-
post productivity shock across all countries except for the US and Romania.
However, it decreases in response to changes in past productivity and ex-ante
productivity shock. On an overall basis, the elasticity to the ex-post
productivity shock and ex-ante productivity shock stands at 0.45 and 0.13,
respectively. This latter result likely stems from the US data. Conversely,
the overall elasticity to the past productivity channel is -0.21.

Table 5: The Effect of Productivity Channels on the Marginal Product of Inputs
- Capital

| CAPITAL
---|---
| USA | BEL | FRA | GER | HUN | ITA | POL | POR | ROM | SLO | ESP | SWE | All
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13)
$\omega_{jt-1}$ | $0.224^{***}$ | $0.723^{***}$ | $0.915^{***}$ | $0.689^{***}$ | $0.892^{***}$ | $0.943^{***}$ | $1.269^{***}$ | $1.136^{***}$ | $1.576^{***}$ | $1.388^{***}$ | $1.184^{***}$ | $1.211^{***}$ | $0.718^{***}$
| $(0.018)$ | $(0.074)$ | $(0.068)$ | $(0.045)$ | $(0.076)$ | $(0.059)$ | $(0.051)$ | $(0.096)$ | $(0.061)$ | $(0.220)$ | $(0.059)$ | $(0.128)$ | $(0.020)$
|  |  |  |  |  |  |  |  |  |  |  |  |
$\eta_{jt}$ | $0.602^{***}$ | $1.048^{***}$ | $0.906^{***}$ | $0.974^{***}$ | $1.031^{***}$ | $1.190^{***}$ | $1.370^{***}$ | $1.405^{***}$ | $1.836^{***}$ | $1.613^{***}$ | $1.269^{***}$ | $1.351^{***}$ | $1.170^{***}$
| $(0.066)$ | $(0.142)$ | $(0.109)$ | $(0.061)$ | $(0.097)$ | $(0.068)$ | $(0.059)$ | $(0.105)$ | $(0.060)$ | $(0.193)$ | $(0.062)$ | $(0.197)$ | $(0.027)$
|  |  |  |  |  |  |  |  |  |  |  |  |
$\Delta\varepsilon_{jt}$ | $0.444^{***}$ | $0.613^{***}$ | $0.426^{***}$ | $0.466^{***}$ | $0.454^{***}$ | $0.472^{***}$ | $0.339^{***}$ | $0.546^{***}$ | $0.442^{***}$ | $0.509^{***}$ | $0.573^{***}$ | $0.231^{***}$ | $0.478^{***}$
| $(0.029)$ | $(0.087)$ | $(0.041)$ | $(0.058)$ | $(0.035)$ | $(0.029)$ | $(0.056)$ | $(0.033)$ | $(0.016)$ | $(0.048)$ | $(0.030)$ | $(0.057)$ | $(0.010)$
|  |  |  |  |  |  |  |  |  |  |  |  |
N | 68,289 | 28,632 | 53,833 | 24,394 | 29,188 | 50,533 | 34,611 | 35,028 | 31,508 | 26,444 | 53,934 | 13,228 | 449,622
R2 | 0.238 | 0.172 | 0.349 | 0.245 | 0.254 | 0.183 | 0.336 | 0.230 | 0.269 | 0.204 | 0.218 | 0.152 | 0.303
Adj R2 | 0.223 | 0.138 | 0.329 | 0.204 | 0.224 | 0.153 | 0.311 | 0.209 | 0.241 | 0.173 | 0.194 | 0.109 | 0.274

  * •

The table reports the coefficient of the linear regression in (25) for
Capital.

  * •

The sector is identified at the 3-digit level of disaggregation (NACE for
European countries, NAICS for the US).

  * •

For the regression where all countries are pooled, I include country-year-
industry fixed effects.

  * •

Standard errors are clustered at the individual firm level.

  * •

∗p$<$0.1.

  * •

∗∗p$<$0.05.

  * •

∗∗∗p$<$0.01.

Table 6: The Effect of Productivity Channels on the Marginal Product of Inputs
- Labor

| LABOR
---|---
| USA | BEL | FRA | GER | HUN | ITA | POL | POR | ROM | SLO | ESP | SWE | All
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13)
$\omega_{jt-1}$ | $0.871^{***}$ | $0.804^{***}$ | $0.672^{***}$ | $1.036^{***}$ | $0.734^{***}$ | $0.925^{***}$ | $0.939^{***}$ | $0.905^{***}$ | $0.731^{***}$ | $0.831^{***}$ | $0.872^{***}$ | $0.954^{***}$ | $0.845^{***}$
| $(0.056)$ | $(0.060)$ | $(0.039)$ | $(0.051)$ | $(0.056)$ | $(0.031)$ | $(0.041)$ | $(0.066)$ | $(0.037)$ | $(0.099)$ | $(0.034)$ | $(0.056)$ | $(0.020)$
|  |  |  |  |  |  |  |  |  |  |  |  |
$\eta_{jt}$ | $0.274^{***}$ | $0.759^{***}$ | $0.763^{***}$ | $0.990^{***}$ | $0.851^{***}$ | $0.882^{***}$ | $1.037^{***}$ | $0.984^{***}$ | $1.216^{***}$ | $1.106^{***}$ | $1.158^{***}$ | $1.219^{***}$ | $0.909^{***}$
| $(0.082)$ | $(0.137)$ | $(0.061)$ | $(0.082)$ | $(0.070)$ | $(0.039)$ | $(0.041)$ | $(0.064)$ | $(0.033)$ | $(0.081)$ | $(0.040)$ | $(0.092)$ | $(0.022)$
|  |  |  |  |  |  |  |  |  |  |  |  |
$\Delta\varepsilon_{jt}$ | $0.379^{***}$ | $0.390^{***}$ | $0.486^{***}$ | $0.378^{***}$ | $0.448^{***}$ | $0.366^{***}$ | $0.419^{***}$ | $0.481^{***}$ | $0.502^{***}$ | $0.452^{***}$ | $0.561^{***}$ | $0.347^{***}$ | $0.447^{***}$
| $(0.023)$ | $(0.063)$ | $(0.035)$ | $(0.048)$ | $(0.029)$ | $(0.023)$ | $(0.043)$ | $(0.018)$ | $(0.010)$ | $(0.024)$ | $(0.022)$ | $(0.029)$ | $(0.008)$
|  |  |  |  |  |  |  |  |  |  |  |  |
N | 58,294 | 28,631 | 53,833 | 24,346 | 29,168 | 50,465 | 34,615 | 35,034 | 31,426 | 26,418 | 53,923 | 13,235 | 439,388
R2 | 0.327 | 0.524 | 0.538 | 0.646 | 0.486 | 0.387 | 0.434 | 0.430 | 0.364 | 0.427 | 0.424 | 0.420 | 0.619
Adj R2 | 0.312 | 0.504 | 0.524 | 0.627 | 0.466 | 0.364 | 0.413 | 0.414 | 0.339 | 0.405 | 0.407 | 0.391 | 0.602

  * •

The table reports the coefficient of the linear regression in (25) for Labor.

  * •

The sector is identified at the 3-digit level of disaggregation (NACE for
European countries, NAICS for the US).

  * •

For the regression where all countries are pooled, I include country-year-
industry fixed effects.

  * •

Standard errors are clustered at the individual firm level.

  * •

∗p$<$0.1.

  * •

∗∗p$<$0.05.

  * •

∗∗∗p$<$0.01.

Table 7: The Effect of Productivity Channels on the Marginal Product of Inputs
- Materials

| MATERIALS
---|---
| USA | BEL | FRA | GER | HUN | ITA | POL | POR | ROM | SLO | ESP | SWE | All
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13)
$\omega_{jt-1}$ | $0.761^{***}$ | $-0.374^{***}$ | $-0.314^{***}$ | $-0.240^{***}$ | $-0.335^{***}$ | $-0.169^{***}$ | $-0.032$ | $-0.205^{***}$ | $0.029$ | $-0.186^{*}$ | $-0.289^{***}$ | $-0.189^{***}$ | $0.132^{***}$
| $(0.024)$ | $(0.032)$ | $(0.027)$ | $(0.032)$ | $(0.036)$ | $(0.021)$ | $(0.025)$ | $(0.033)$ | $(0.023)$ | $(0.096)$ | $(0.022)$ | $(0.045)$ | $(0.024)$
|  |  |  |  |  |  |  |  |  |  |  |  |
$\eta_{jt}$ | $0.176^{***}$ | $-0.299^{***}$ | $-0.309^{***}$ | $-0.286^{***}$ | $-0.315^{***}$ | $-0.288^{***}$ | $-0.171^{***}$ | $-0.196^{***}$ | $0.045^{**}$ | $-0.138^{**}$ | $-0.204^{***}$ | $-0.086$ | $-0.207^{***}$
| $(0.040)$ | $(0.078)$ | $(0.041)$ | $(0.053)$ | $(0.046)$ | $(0.025)$ | $(0.029)$ | $(0.036)$ | $(0.022)$ | $(0.068)$ | $(0.023)$ | $(0.071)$ | $(0.020)$
|  |  |  |  |  |  |  |  |  |  |  |  |
$\Delta\varepsilon_{jt}$ | $0.406^{***}$ | $0.532^{***}$ | $0.510^{***}$ | $0.381^{***}$ | $0.460^{***}$ | $0.392^{***}$ | $0.406^{***}$ | $0.505^{***}$ | $0.490^{***}$ | $0.447^{***}$ | $0.572^{***}$ | $0.336^{***}$ | $0.445^{***}$
| $(0.021)$ | $(0.051)$ | $(0.020)$ | $(0.047)$ | $(0.025)$ | $(0.020)$ | $(0.025)$ | $(0.016)$ | $(0.009)$ | $(0.021)$ | $(0.018)$ | $(0.027)$ | $(0.007)$
|  |  |  |  |  |  |  |  |  |  |  |  |
N | 69,210 | 28,755 | 53,835 | 24,459 | 29,188 | 50,643 | 34,625 | 35,163 | 31,587 | 26,446 | 54,101 | 13,261 | 451,273
R2 | 0.541 | 0.427 | 0.495 | 0.288 | 0.344 | 0.256 | 0.299 | 0.316 | 0.437 | 0.309 | 0.338 | 0.214 | 0.324
Adj R2 | 0.532 | 0.403 | 0.479 | 0.249 | 0.318 | 0.229 | 0.273 | 0.297 | 0.415 | 0.282 | 0.317 | 0.174 | 0.296

  * •

The table reports the coefficient of the linear regression in (25) for
Materials.

  * •

The sector is identified at the 3-digit level of disaggregation (NACE for
European countries, NAICS for the US).

  * •

For the regression where all countries are pooled, I include country-year-
industry fixed effects.

  * •

Standard errors are clustered at the individual firm level.

  * •

∗p$<$0.1.

  * •

∗∗p$<$0.05.

  * •

∗∗∗p$<$0.01.

Industry-level dispersion. Adopting a similar approach to the previous
section, we can devise an equivalent specification of equation (25) with
sector-specific coefficients, under the assumption that the exogeneity of the
error term also holds at the sector level:

$\displaystyle(mp_{jt}^{X}-\overline{mp}_{st}^{X})=$
$\displaystyle\beta_{s,\omega}(\omega_{jt-1}-\overline{\omega}_{st-1})+\beta_{s,\eta}(\eta_{jt}-\overline{\eta}_{st})$
(26)
$\displaystyle+\beta_{s,\varepsilon}(\Delta\varepsilon_{jt}-\overline{\Delta\varepsilon}_{st})+\underbrace{(\zeta_{jt}-\overline{\zeta}_{st})}_{\begin{subarray}{c}\tilde{\zeta}_{jt}\end{subarray}}$

In this equation, the terms $\overline{mp}_{st}^{X}$,
$\overline{\omega}_{st-1}$, $\overline{\eta}_{st}$,
$\overline{\Delta\varepsilon}_{st}$, and $\overline{\zeta}_{st}$ represent
industry-time specific means. A sum of squares transformation, along with the
exogeneity assumption of the error term and the orthogonality assumptions on
the TFP components (see Assumptions 1 and 2), results in the following
projection:

$Var_{st}(mp^{X}_{jt})=\beta_{s,\omega
var}Var_{st-1}(\omega_{jt-1})+\beta_{s,\eta
var}Var_{st}(\eta_{jt})+\beta_{s,\varepsilon
var}\underbrace{Var_{st}(\Delta\varepsilon_{jt})}_{\begin{subarray}{c}Vol_{st}(\varepsilon_{jt})\end{subarray}}+\varphi_{st}$
(27)

Where, by construction, $\beta_{s,\omega var}=\hat{\beta}_{s,\omega}^{2}$,
$\beta_{s,\eta var}=\hat{\beta}_{s,\eta}^{2}$, $\beta_{s,\varepsilon
var}=\hat{\beta}_{s,\varepsilon}^{2}$.

Using the $S^{2}$ statistic computed for each component on the right-hand side
of equation (27), I determine the average proportion of the cross-sectional,
sectoral dispersion of the inputs’ marginal products that is attributable to
the variability of each TFP channel, $\omega_{jt-1}$, $\eta_{jt}$, and
$\Delta\varepsilon_{jt}$, all else being equal.

$\displaystyle S^{2}_{\omega-1}=1-\frac{\sum^{\lvert
s\rvert}_{s=1}\sum^{\lvert
t\rvert}_{t=1}(Var_{st}(mp^{X}_{jt})-\beta_{s,\omega
var}Var_{st-1}(\omega_{jt-1}))^{2}}{\sum^{\lvert s\rvert}_{s=1}\sum^{\lvert
t\rvert}_{t=1}Var_{st}(mp^{X}_{jt})^{2}}$ (28) $\displaystyle
S^{2}_{\eta}=1-\frac{\sum^{\lvert s\rvert}_{s=1}\sum^{\lvert
t\rvert}_{t=1}(Var_{st}(mp^{X}_{jt})-\beta_{s,\eta
var}Var_{st}(\eta_{jt}))^{2}}{\sum^{\lvert s\rvert}_{s=1}\sum^{\lvert
t\rvert}_{t=1}Var_{st}(mp^{X}_{jt})^{2}}$ (29) $\displaystyle
S^{2}_{\Delta\varepsilon}=1-\frac{\sum^{\lvert s\rvert}_{s=1}\sum^{\lvert
t\rvert}_{t=1}(Var_{st}(mp^{X}_{jt})-\beta_{s,\varepsilon
var}Var_{st}(\Delta\varepsilon_{jt}))^{2}}{\sum^{\lvert
s\rvert}_{s=1}\sum^{\lvert t\rvert}_{t=1}Var_{st}(mp^{X}_{jt})^{2}}$ (30)

One must remember that the usual remarks regarding the $S^{2}$ statistic still
apply. Moreover, the sum of $S^{2}_{\omega-1}$, $S^{2}_{\eta}$, and
$S^{2}_{\Delta\varepsilon}$ does not necessarily approximate the $S^{2}$
statistic given in Table 4 for the composite TFP growth term181818For example,
considering the $\Delta\nu_{jt}$ decomposition in equation (16),
$Var(g(\omega_{jt-1}))\neq Var(\omega_{jt-1})$ if g(.) is not linear or affine
with unitary linear coefficient..

Table 8 illustrates the results. The decomposition of TFP growth into its
components augments the overall explanatory power by harnessing this
additional, valuable source of variation. Past productivity variability
significantly accounts for the dispersion of the marginal product of inputs.
For labor, this explanatory power ranges from 12% for Portugal to 47% for
Belgium. Yet, the degree of explanation exhibits more heterogeneity across
countries for capital and materials. In relation to capital, the explained
share of variation in the marginal product oscillates between 5% for Spain and
40% for Hungary. For materials, the range extends from 2% for Spain to a
substantial 77% for the US.

Conversely, the ex-ante and ex-post productivity shock channels demonstrate a
less pronounced explanatory power. The ex-ante channel explains between 2%
(US) and 12% (Hungary and Romania) of the dispersion in the marginal product
of capital, between 4% (US) and 11% (Romania) for labor, and between 0.4% (US)
and 35% (Slovenia) for materials. Comparable estimates emerge for the ex-post
productivity shock channel. It explains between 2% (US) and 10% (Germany) of
the dispersion in the marginal product of capital, between 1% (US) and 18%
(Romania) for labor, and between 2% (US) and 31% (Romania) for materials.

Observe the notably limited explanatory power of the ex-ante and ex-post
productivity channels for each input’s marginal product dispersion,
specifically in the case of Germany. Portugal also stands out due to the
minimal capacity of past productivity to explain its capital’s marginal
product dispersion. The emergence of these large, negative results could stem
from two causes: the cross-sectional variances of these productivity
components are, on average, significantly higher than the variances of the
marginal products, or the marginal product of the input is extremely elastic
to changes in the productivity components, resulting in exceptionally high
$\beta_{s,var}$. In relation to the first point, the adopted assumption holds
that the productivity components are orthogonal at the firm-time level.
Consequently, the projection specification in (27) precludes any correlated
variation between them. Nevertheless, this correlation could still manifest in
the aggregate sector-level data. A more detailed exploration of this issue is
available in Appendix F, where I re-compute the $S^{2}$ statistics accounting
for cross-sectional sectoral co-variations between TFP growth components. The
integration of these correlated variations tends to enhance the explanatory
power of the productivity channels overall, boosting the lowest estimated
explained shares while leaving the remainder unaltered. As an outcome, each
productivity channel becomes explanatory for each input’s marginal product
dispersion191919Germany remains an exception, despite the considerable surge
in the negative estimated explained shares..

Table 8: $S^{2}$ \- Decomposed TFP | Capital | Labor | Materials
---|---|---|---
Country | $\omega_{-1}$ | $\eta$ | $\Delta\varepsilon$ | $\omega_{-1}$ | $\eta$ | $\Delta\varepsilon$ | $\omega_{-1}$ | $\eta$ | $\Delta\varepsilon$
USA | $9.83\%$ | $1.65\%$ | $2.09\%$ | $26.22\%$ | $4.12\%$ | $1.17\%$ | $77.12\%$ | $0.35\%$ | $1.66\%$
Belgium | $22.40\%$ | $6.37\%$ | $8.03\%$ | $47.08\%$ | $6.95\%$ | $10.90\%$ | $15.93\%$ | $13.48\%$ | $26.50\%$
France | $38.38\%$ | $8.52\%$ | $4.12\%$ | $40.63\%$ | $9.24\%$ | $9.18\%$ | $8.61\%$ | $19.25\%$ | $19.47\%$
Germany | $14.89\%$ | $-451.05\%$ | $-209.84\%$ | $45.92\%$ | $-35.59\%$ | $-0.62\%$ | $37.76\%$ | $-41.98\%$ | $-160.14\%$
Hungary | $39.78\%$ | $12.20\%$ | $10.47\%$ | $32.41\%$ | $13.45\%$ | $10.88\%$ | $32.39\%$ | $11.25\%$ | $6.83\%$
Italy | $17.10\%$ | $6.01\%$ | $6.14\%$ | $31.49\%$ | $7.53\%$ | $9.04\%$ | $10.16\%$ | $8.35\%$ | $28.05\%$
Poland | $22.18\%$ | $6.85\%$ | $-7.58\%$ | $25.23\%$ | $6.88\%$ | $8.45\%$ | $3.54\%$ | $2.03\%$ | $13.90\%$
Portugal | $-1217.34\%$ | $6.49\%$ | $9.75\%$ | $11.59\%$ | $-4.49\%$ | $15.31\%$ | $3.12\%$ | $3.18\%$ | $24.58\%$
Romania | $21.30\%$ | $12.41\%$ | $7.53\%$ | $15.89\%$ | $10.58\%$ | $18.11\%$ | $3.65\%$ | $2.92\%$ | $31.29\%$
Slovenia | $24.94\%$ | $5.70\%$ | $2.48\%$ | $25.02\%$ | $7.27\%$ | $5.66\%$ | $9.21\%$ | $35.28\%$ | $24.87\%$
Spain | $4.68\%$ | $-11.39\%$ | $3.93\%$ | $16.38\%$ | $4.52\%$ | $6.86\%$ | $2.30\%$ | $-7.56\%$ | $17.44\%$
Sweden | $10.38\%$ | $4.42\%$ | $4.40\%$ | $20.11\%$ | $7.62\%$ | $6.76\%$ | $3.72\%$ | $1.73\%$ | $7.91\%$

##  6 The 2008 Financial Crisis

The literature thoroughly examines the crisis, which stands out as the most
acute economic downturn since the Great Depression of 1929. The crisis took
root in the financial sector, driven by an interplay of factors such as
widespread subprime mortgage sales targeting low-income homebuyers, excessive
risk-taking by U.S. financial institutions engaging in both commercial and
proprietary trade (enabled by the 1999 Glass-Steagall Act repeal), and the
implosion of the housing financial bubble. The collapse of key banks,
exemplified by Lehman Brothers’ bankruptcy declaration in September 2008,
eroded confidence in bank solvency and resulted in a contraction of credit
availability. Consequently, the capital reallocation channel from households
to firms stalled, transforming a financial-sector crisis into a global
economic shock that decelerated the world economy. In 2009, GDP growth
plummeted to a stunning -2.5% for the US and -4.5% for the European Monetary
Union (EMU) (refer to Figure 3).

Figure 3:

The figure reports the GDP growth in the European Monetary Union (EMU) and the
USA from 2000 to 2019. Notice the fall, starting in 2007 for the USA and the
EMU, for the Financial Crisis, and the EMU’s "double dip" starting in 2011 due
to the Sovereign Debt Crisis.
Source: The World Bank.

Yet, the financial crisis wielded disparate effects across European countries.
Countries’ financial sectors function with varying degrees of efficiency. A
relatively more efficient financial sector enables better capital allocation
from households to firms and between firms. This efficiency also bolsters
resilience and adaptation to localized financial shocks, halting their spread
to real economic activity through mechanisms such as credit crunches. Observe
the time trends of productivity and inputs allocations for France and Spain in
Figures 5 and 5. France, with a relatively efficient financial sector, shows
industry-specific standard deviations for the (log) TFP level and growth and
the (log) marginal products of capital, labor, and material inputs maintaining
a more stable pattern over time. The most noticeable alteration in the series
before and after 2008 is a gradual rise in TFP dispersion to a level 30%
higher than in 2001. In contrast, Spain, with a relatively less efficient
financial sector, reveals more stark contrasts. Before 2008, the dispersions
of the inputs marginal products and TFP were stable, with a temporary apex in
2004 reaching nearly 20% more than their 2001 level. Post-2008, these
indicators began a persistent climb, with the dispersion of the marginal
product of labor and capital hitting a peak level almost 20% higher than in
2001, 50% higher for the dispersion of the marginal product of intermediates,
and 70% higher for the dispersion of TFP. TFP volatility dropped by 20% in
2008 compared to 2001, while in 2015 it was 20% higher than its 2001 level.
Figures 9-13 in Appendix G showcase the patterns for the US, Germany, Italy,
Poland, and Romania. Interestingly, while the US and Germany exhibit a stable
pattern over time with no significant variation between pre- and post-2008
periods, Italy, Poland, and Romania appear relatively more unstable with some
breaks, on average, after 2008.

Figure 4: Evolution of MPs and TFP Dispersion - FRA

Figure 5: Evolution of MPs and TFP Dispersion - ESP

The figures report the evolution of the dispersion of the (log) marginal
product of capital, labor, and material inputs and the (log) TFP for the
French and Spanish manufacturing sectors. The solid red line identifies the
Financial Crisis period. Standard deviations are computed for each NACE
3-digit level industry and then aggregated in a weighted average, using for
weights the share in revenue produced at the industry level compared to the
full country manufacturing sample.

Following the categorization in the literature ([25]), I classify Italy and
Spain as "South" countries, characterized by their financially underdeveloped
status and less efficient financial sectors. France and Germany, on the other
hand, comprise the "North" group, noted for their financial development and
more efficient financial sectors. To gauge the financial crisis’s impact on
productivity volatility and the dispersion of input’s marginal products in the
more crisis-prone South, I use an event study approach applying the [12]
Difference-in-Differences estimator, with the North serving as a control
group. This estimator permits the inclusion of multiple periods and
differentiates between short and long-term effects.

I first evaluate the financial crisis’s impact on the industry (log)
volatility of TFP, $Var_{st}(\nu_{jt}-\nu_{jt-1})$. Figure 6 showcases the
results, revealing a considerable increase in TFP volatility in the
South—about 65% on average across the nine years following the crisis.
Confirming the satisfaction of the parallel trend assumption in the pre-crisis
years, the figure depicts an immediate and lasting effect of the crisis on TFP
volatility, with a significant ATT estimate of 75% in 2009 and comparable,
statistically significant estimates in subsequent years. By breaking down TFP
growth into its components, it becomes apparent that the composite effect
largely stems from the variance of past systematic productivity,
$Var_{st-1}(\omega_{jt-1})$, and the volatility of the ex-post productivity
shock, $Var_{st}(\varepsilon_{jt}-\varepsilon_{jt-1})$ (Refer to Figure 14 in
Appendix G for more details).

Figure 6: Log TFP Volatility

The figure reports the [12] Difference-in-Difference dynamic estimates with
never treated control for the effect of the financial crisis on the industry-
specific (log) TFP Volatility. The treatment group comprises the South, i.e.
Italy (2001 - 2017) and Spain (2001 - 2017), while the control group comprises
those in the North, i.e. France (2001 - 2017) and Germany (2005 - 2017). The
industry is defined at the NACE 3-digit level. The method used is the doubly-
robust [34] estimator. Standard errors are bootstrapped using Wild bootstrap
with 999 repetitions and clustered at the panel level. $95\%$ confidence
intervals are shown. The total number of observations is 5,058.

Finally, I estimate the 2008 financial crisis’s impact on the industry-
specific (log) dispersion of the marginal products of inputs in the South.
Table 9 displays these results. Of note, the financial crisis did not
significantly affect the dispersion of the marginal products of labor and
materials in the South202020The pre-treatment ATT also reveals statistical
evidence supporting the rejection of the parallel trends assumption before the
crisis for both inputs.. A different scenario plays out for capital. As shown
in column (1), the financial crisis, on average, spurred a robust and
statistically significant increase in the industry-specific variance of the
marginal product of capital by 33%. Further, the ATT pre-treatment shows no
evidence suggesting the rejection of the parallel trends assumption for the
pre-treatment years. By observing the magnitude and significance of the year-
specific ATT, one can discern that the crisis’s effect was not only persistent
but also intensifying over time. These findings hold firm and remain virtually
unaltered when controlling for industry-specific (log) TFP volatility (columns
(2), (3), (5), (6), (8), and (9)) and industry-specific market concentration
(columns (3), (6), and (9)), measured by the (log) Herfindahl-Hirschman Index
(HHI)212121I account for the (log) industry-specific HHI to control for any
changes in the dispersion of the industry-specific marginal product of capital
due to changes in industry-specific market concentration. Figure 15 in the
Appendix G presents the [12] DiD estimates of the financial crisis’s impact on
the (log) HHI, considering the South as the treatment and the North as the
control group. The results highlight that the crisis’s effect has been
gradual, with a relative increase in market concentration in the South peaking
in 2012, but temporary..

For robustness, I reproduce the above analysis, but this time incorporating
the "East" group (comprising Poland and Romania) into the treatment group and
the US into the control group. Appendix H contains the results of this
exercise in Table 38. For labor and material inputs, the conclusions remain
unaltered. As for capital, the average ATT and year-specific ATTs exhibit a
slightly greater magnitude, with the latter also presenting heightened
statistical significance in the immediate post-crisis years.

Table 9: North-South DID Results

|  Var($mp^{K}$) |  Var($mp^{L}$) |  Var($mp^{M}$)
---|---|---|---
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
ATT | $0.334^{***}$ | $0.335^{***}$ | $0.312^{***}$ | $0.0557$ | $0.0212$ | $-0.0348$ | $0.0586$ | $0.0816$ | $-0.0247$
| $(0.0921)$ | $(0.0896)$ | $(0.0884)$ | $(0.0881)$ | $(0.0876)$ | $(0.0780)$ | $(0.102)$ | $(0.104)$ | $(0.0983)$
ATT Pre-treatment | $0.00475$ | $-0.0125$ | $-0.00292$ | $0.0796^{***}$ | $0.0572^{*}$ | $0.0569^{**}$ | $0.0746^{**}$ | $0.0811^{**}$ | $0.0913^{***}$
| $(0.0190)$ | $(0.0255)$ | $(0.0236)$ | $(0.0217)$ | $(0.0228)$ | $(0.0216)$ | $(0.0273)$ | $(0.0258)$ | $(0.0276)$
ATT 2008 | $0.195^{*}$ | $0.209^{*}$ | $0.162$ | $-0.0542$ | $-0.0264$ | $-0.00929$ | $-0.0976$ | $-0.0750$ | $-0.0964$
| $(0.0976)$ | $(0.0974)$ | $(0.0830)$ | $(0.0929)$ | $(0.0758)$ | $(0.0753)$ | $(0.104)$ | $(0.109)$ | $(0.105)$
ATT 2009 | $0.279^{**}$ | $0.268^{*}$ | $0.248^{*}$ | $0.153$ | $0.104$ | $0.0608$ | $0.101$ | $0.127$ | $0.111$
| $(0.105)$ | $(0.109)$ | $(0.0982)$ | $(0.104)$ | $(0.0962)$ | $(0.0867)$ | $(0.112)$ | $(0.108)$ | $(0.109)$
ATT 2010 | $0.223^{*}$ | $0.221^{*}$ | $0.259^{**}$ | $0.0765$ | $0.0600$ | $0.108$ | $0.115$ | $0.151$ | $0.108$
| $(0.0965)$ | $(0.0932)$ | $(0.0947)$ | $(0.100)$ | $(0.111)$ | $(0.100)$ | $(0.120)$ | $(0.122)$ | $(0.110)$
ATT 2011 | $0.194$ | $0.215^{*}$ | $0.233^{*}$ | $-0.0625$ | $-0.112$ | $-0.114$ | $-0.0234$ | $0.00638$ | $-0.136$
| $(0.0992)$ | $(0.0999)$ | $(0.0987)$ | $(0.105)$ | $(0.105)$ | $(0.0907)$ | $(0.117)$ | $(0.124)$ | $(0.128)$
ATT 2012 | $0.163$ | $0.171$ | $0.226^{*}$ | $-0.00449$ | $-0.0537$ | $-0.0707$ | $-0.0308$ | $0.0217$ | $-0.0568$
| $(0.112)$ | $(0.119)$ | $(0.106)$ | $(0.105)$ | $(0.100)$ | $(0.0881)$ | $(0.149)$ | $(0.132)$ | $(0.120)$
ATT 2013 | $0.328^{**}$ | $0.323^{**}$ | $0.334^{***}$ | $0.157$ | $0.114$ | $0.0570$ | $0.195$ | $0.196$ | $0.0684$
| $(0.105)$ | $(0.109)$ | $(0.0958)$ | $(0.132)$ | $(0.129)$ | $(0.108)$ | $(0.139)$ | $(0.133)$ | $(0.133)$
ATT 2014 | $0.432^{***}$ | $0.442^{***}$ | $0.429^{***}$ | $0.208$ | $0.169$ | $0.0291$ | $0.178$ | $0.181$ | $0.0213$
| $(0.109)$ | $(0.113)$ | $(0.101)$ | $(0.125)$ | $(0.119)$ | $(0.105)$ | $(0.160)$ | $(0.155)$ | $(0.155)$
ATT 2015 | $0.460^{***}$ | $0.457^{***}$ | $0.431^{***}$ | $0.0315$ | $-0.0160$ | $-0.136$ | $0.0638$ | $0.0727$ | $-0.0715$
| $(0.131)$ | $(0.131)$ | $(0.112)$ | $(0.131)$ | $(0.122)$ | $(0.108)$ | $(0.146)$ | $(0.143)$ | $(0.141)$
ATT 2016 | $0.530^{***}$ | $0.527^{***}$ | $0.407^{***}$ | $0.0344$ | $-0.00881$ | $-0.0888$ | $0.0604$ | $0.0853$ | $-0.0564$
| $(0.154)$ | $(0.146)$ | $(0.115)$ | $(0.125)$ | $(0.116)$ | $(0.0956)$ | $(0.149)$ | $(0.150)$ | $(0.136)$
ATT 2017 | $0.534^{***}$ | $0.518^{**}$ | $0.389^{**}$ | $0.0169$ | $-0.0182$ | $-0.183$ | $0.0256$ | $0.0489$ | $-0.138$
| $(0.157)$ | $(0.162)$ | $(0.127)$ | $(0.152)$ | $(0.158)$ | $(0.130)$ | $(0.157)$ | $(0.163)$ | $(0.140)$
$N$ | 5,149 | 5,058 | 5,058 | 5,149 | 5,058 | 5,058 | 5,149 | 5,058 | 5,058
Controls: |
Vol(TFP) | NO | YES | YES | NO | YES | YES | NO | YES | YES
HHI | NO | NO | YES | NO | NO | YES | NO | NO | YES

  * •

The figure reports the [12] Difference-in-Difference dynamic estimates with
never treated control for the effect of the financial crisis on the industry-
specific variance of the inputs marginal products.

  * •

The HHI is computed, for each year and industry, over the full sample for each
country.

  * •

All variables are in logs.

  * •

The treatment group comprises the South, i.e., Italy (2001 - 2017) and Spain
(2001 - 2017), while the control group comprises those in the North, i.e.,
France (2001 - 2017) and Germany (2005 - 2017). The industry is defined at the
NACE 3-digit level.

  * •

The method used is the doubly-robust [34] estimator. Standard errors are
bootstrapped using Wild bootstrap with 999 repetitions and clustered at the
panel level.

##  7 Conclusion

This paper revisits the conventional measure of within-industry resource
misallocation —namely, the dispersion of an input’s marginal product— and
reveals that this statistic reflects productivity uncertainty and
heterogeneity across all inputs. My analysis determines that firm-level
changes in TFP growth trigger substantial responses in the marginal product of
each input. Aggregated at the industry level, cross-sectional TFP volatility
independently explains approximately 7% of the marginal product of capital
dispersion, 9% for labor, and 10% for materials. These findings suggest that,
since this measure inherently captures economic fundamentals and firms’
imperfect information, its use as a standalone indicator of policy-induced
input misallocation (and hence as a target for policy intervention) is
questionable. However, my research also illustrates that, with identified
policy variations, we can effectively adapt this measure for cross-country or
cross-industry evaluations, thereby providing insight into relative
misallocation. For instance, my research establishes that the 2008 Financial
Crisis amplified the dispersion of the marginal product of capital in Southern
Europe by 40% more than in Northern Europe, even after accounting for TFP
volatility. This divergence reflects the respective financial sector maturity
in these regions, suggesting the presence of financial misallocative
frictions. Consistent with established narratives about the crisis’s causes
and impacts, my findings indicate no discernible effect on the dispersion of
the marginal product for other inputs.

In conclusion, I highlight several limitations of this study that could steer
future research. First, the crisis-induced misallocation may be driven not
only by financial frictions but also by idiosyncratic institutions and
regulations. Second, my theoretical and applied framework does not consider
heterogeneous firm-specific market power, a potent influence on the dispersion
of inputs’ marginal product and a potential source of misallocation that
policy intervention could address. Consequently, further research is essential
to construct a unified framework encompassing a variety of policy-induced and
non-policy-induced determinants of the dispersion of an input’s marginal
product. Such a comprehensive understanding is crucial to enhancing the
informative value of this measure for guiding policymaking.

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##  Appendix A Partially Hicks-neutral Productivity

Notice that the timing allocation and the productivity uncertainty the firm
faces allow TFP to be not necessarily Hicks-neutral, depending on the source
of variation. For the sake of explanation, the focus is restricted to labor
and materials. Only for the following, assume that labor is a flexible input,
marginal costs are constant, and the production function is Cobb-Douglas.

$F(L_{jt},M_{jt})=L_{jt}^{\alpha}M_{jt}^{\beta}$ (31)

Equating the expected marginal product of each input to its real marginal cost
delivers

$MP^{Me}_{jt}=\beta\frac{Y_{jt}}{M_{jt}}e^{m(\omega_{jt-1})+\eta_{jt}}\mathcal{E}=\frac{\rho}{P}$
(32)
$MP^{Le}_{jt}=\alpha\frac{Y_{jt}}{L_{jt}}e^{m(\omega_{jt-1})}\mathcal{M}\mathcal{E}=\frac{w}{P}$
(33)

Then, once the productivity shocks realize, the final period’s marginal
products are

$MP^{M}_{jt}=\frac{\rho}{P}\frac{e^{\varepsilon}_{jt}}{\mathcal{E}}$ (34)
$MP^{L}_{jt}=\frac{w}{P}\frac{e^{\varepsilon}_{jt}}{\mathcal{E}}\frac{e^{\eta_{jt}}}{\mathcal{M}}$
(35)

Consider first a variation in $\varepsilon_{jt}$. $\varepsilon_{jt}$ is a
post-allocation output fluctuation for each input considered. As such, it
draws a wedge between marginal costs and final period marginal products,
having an elasticity of one with the latter. Then, the final period relative
marginal product between labor and materials is unchanged, meaning that this
productivity variation is Hicks-neutral.

Consider a TFP variation coming from $\eta_{jt}$. $\eta_{jt}$ is unknown when
the firm allocates labor but known when it allocates materials. For labor,
like $\varepsilon_{jt}$, $\eta_{jt}$ is an ex-post allocation fluctuation that
draws a wedge between the final period marginal product and marginal cost and
changes its marginal product with an elasticity of one. However, since
$\eta_{jt}$ is known when the firm allocates materials, a variation of it has
no effect on its final period marginal product. A change in $\eta_{jt}$
entails a change in the allocation of materials to equate expected marginal
product and marginal cost. Since the marginal cost is unaffected by
assumption, the expected marginal product and the final period marginal
product are unaffected. Then a variation in $\eta_{jt}$ changes the final
period relative marginal product between labor and materials, and, as such, it
is not Hicks-neutral.

##  Appendix B Analytical Derivation of the Productivity Channels

To ease the notation, I eliminate the subscripts. Each variable is assumed to
be $j$ and $t$-specific unless it has a $-1$ subscript. In that case, it is
$j$ and $t-1$-specific.

First, notice that totally differentiating the final output $y$ with respect
to the productivity channel $\theta$ leads to the following

$\frac{dy}{d\theta}=\frac{df(k,l,m)}{d\theta}+\frac{dm(\omega_{-1})}{d\theta}+\frac{d\eta}{d\theta}+\frac{d\varepsilon}{d\theta}$
(36)

Notice that

$\frac{df(k,l,m)}{d\theta}=\frac{\partial f(k,l,m)}{\partial
k}\frac{dk}{d\theta}+\frac{\partial f(k,l,m)}{\partial
l}\frac{dl}{d\theta}+\frac{\partial f(k,l,m)}{\partial m}\frac{dm}{d\theta}$
(37)

And then

$\frac{dy}{d\theta}=elas^{K}\frac{dk}{d\theta}+elas^{L}\frac{dl}{d\theta}+elas^{M}\frac{dm}{d\theta}+\frac{dm(\omega_{-1})}{d\theta}+\frac{d\eta}{d\theta}+\frac{d\varepsilon}{d\theta}$
(38)

Second, it follows from developing the total differential of the log output
elasticity of input $X$ with respect to channel $\theta$

$\displaystyle\frac{d{\log(elas^{X})}}{d\theta}=\frac{1}{elas^{X}}\left(\frac{delas^{X}}{d\theta}\right)$
(39)
$\displaystyle=\frac{1}{elas^{X}}\left(\frac{\partial^{2}f(k,l,m)}{\partial
x\partial k}\frac{dk}{d\theta}+\frac{\partial^{2}f(k,l,m)}{\partial x\partial
l}\frac{dl}{d\theta}+\frac{\partial^{2}f(k,l,m)}{\partial x\partial
m}\frac{dm}{d\theta}\right)$
$\displaystyle=\frac{1}{elas^{X}}\left(\frac{\partial elas^{K}}{\partial
x}\frac{dk}{d\theta}+\frac{\partial elas^{L}}{\partial
x}\frac{dl}{d\theta}+\frac{\partial elas^{M}}{\partial
x}\frac{dm}{d\theta}\right)$

Or, equivalently

$\displaystyle\frac{d{\log(elas^{X})}}{d\theta}=\frac{\partial\log(elas^{X})}{\partial
k}\frac{dk}{d\theta}+\frac{\partial\log(elas^{X})}{\partial
l}\frac{dl}{d\theta}+\frac{\partial\log(elas^{X})}{\partial
m}\frac{dm}{d\theta}$ (40)

###  B.1 Material Inputs

From the FOC of the conditional value-added maximization problem (9), it
follows that the final period marginal product of materials has a closed form

$MP^{M}=\frac{\rho}{P}\frac{e^{\varepsilon}}{\mathcal{E}}$ (41)

Then, taking the total derivative of its log transformation
yields222222Remember that $\mathcal{E}$ is assumed to be constant.

$dmp^{M}=d\log\left(\frac{\rho}{P}\right)+d\varepsilon$ (42)

Finally, notice that combining the equation above with equation (18) delivers
an expression for the total derivative of materials allocation

$dm=dy+d\log(elas^{M})-d\log\left(\frac{\rho}{P}\right)-d\varepsilon_{jt}$
(43)

Combining the above with equation (40) delivers

$\frac{dm}{d\theta}=\frac{dy}{d\theta}+\frac{\partial\log(elas^{M})}{\partial
k}\frac{dk}{d\theta}+\frac{\partial\log(elas^{M})}{\partial
l}\frac{dl}{d\theta}+\frac{\partial\log(elas^{M})}{\partial
m}\frac{dm}{d\theta}-\frac{d\log\left(\frac{\rho}{P}\right)}{d\theta}-\frac{d\varepsilon}{d\theta}$
(44)

Substituting in equation (38) yields

$\displaystyle\frac{dm}{d\theta}$ (45)
$\displaystyle=elas^{K}\frac{dk}{d\theta}+elas^{L}\frac{dl}{d\theta}+elas^{M}\frac{dm}{d\theta}+\frac{dm(\omega_{-1})}{d\theta}+\frac{d\eta}{d\theta}+\frac{d\varepsilon}{d\theta}$
$\displaystyle+\frac{\partial\log(elas^{M})}{\partial
k}\frac{dk}{d\theta}+\frac{\partial\log(elas^{M})}{\partial
l}\frac{dl}{d\theta}+\frac{\partial\log(elas^{M})}{\partial
m}\frac{dm}{d\theta}-\frac{d\log\left(\frac{\rho}{P}\right)}{d\theta}-\frac{d\varepsilon}{d\theta}$

Then, finally

$\displaystyle\frac{dm}{d\theta}=\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}$ (46)
$\displaystyle\left(\left(elas^{K}+\frac{\partial\log(elas^{M})}{\partial
k}\right)\frac{dk}{d\theta}+\left(elas^{L}+\frac{\partial\log(elas^{M})}{\partial
l}\right)\frac{dl}{d\theta}+\frac{dm(\omega_{-1})}{d\theta}+\frac{d\eta}{d\theta}-\frac{d\log\left(\frac{\rho}{P}\right)}{d\theta}\right)$

####  B.1.1 Past productivity channel

$\displaystyle\frac{dmp^{M}}{d\omega_{-1}}=d\log\left(\frac{\rho}{P}\right)\Big{/}d\omega_{-1}$

By the allocation and pricing assumptions, by equation (42) and by the
independence of ex-post productivity shocks from past productivity levels, the
past productivity channel affects the marginal product of materials only
through a variation in the materials relative price $\frac{\rho}{P}$.

####  B.1.2 Ex-ante productivity shock channel

$\displaystyle\frac{dmp^{M}}{d\eta}=d\log\left(\frac{\rho}{P}\right)\Big{/}d\eta$

By similar steps and with the same interpretation as for the past productivity
channel.

####  B.1.3 Ex-post productivity shock channel

$\displaystyle\frac{dmp^{M}}{d\varepsilon}=1$

The result above follows by the pricing assumption that the materials’
relative price is independent of ex-post productivity shocks. The ex-post
productivity shock channel is then a mere shifter of the marginal product
since its realization generates only post-allocation output fluctuations.

###  B.2 Capital

One can develop the total derivative of the marginal product of capital with
respect to TFP growth channel $\theta$ in the following way

$\displaystyle\frac{dmp^{K}}{d\theta}$ (47)
$\displaystyle=\frac{dy}{d\theta}-\frac{d{k}}{d\theta}+\frac{d{\log
elas^{K}}}{d\theta}$
$\displaystyle=elas^{K}\frac{dk}{d\theta}+elas^{L}\frac{dl}{d\theta}+elas^{M}\frac{dm}{d\theta}+\frac{dm(\omega_{-1})}{d\theta}+\frac{d\eta}{d\theta}+\frac{d\varepsilon}{d\theta}$
$\displaystyle-\frac{d{k}}{d\theta}+\frac{\partial\log(elas^{K})}{\partial
k}\frac{dk}{d\theta}+\frac{\partial\log(elas^{K})}{\partial
l}\frac{dl}{d\theta}+\frac{\partial\log(elas^{K})}{\partial
m}\frac{dm}{d\theta}$
$\displaystyle=\left(elas^{K}+\frac{\partial\log(elas^{K})}{\partial
k}-1\right)\frac{dk}{d\theta}+\left(elas^{L}+\frac{\partial\log(elas^{K})}{\partial
l}\right)\frac{dl}{d\theta}$
$\displaystyle+\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}$
$\displaystyle\left(\left(elas^{K}+\frac{\partial\log(elas^{M})}{\partial
k}\right)\frac{dk}{d\theta}+\left(elas^{L}+\frac{\partial\log(elas^{M})}{\partial
l}\right)\frac{dl}{d\theta}+\frac{dm(\omega_{-1})}{d\theta}+\frac{d\eta}{d\theta}-\frac{d\log\left(\frac{\rho}{P}\right)}{d\theta}\right)$
$\displaystyle+\frac{dm(\omega_{-1})}{d\theta}+\frac{d\eta}{d\theta}+\frac{d\varepsilon}{d\theta}$
$\displaystyle=\biggl{(}\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\left(elas^{K}+\frac{\partial\log(elas^{M})}{\partial k}\right)$
$\displaystyle+\left(elas^{K}+\frac{\partial\log(elas^{K})}{\partial
k}-1\right)\biggr{)}\frac{dk}{d\theta}$
$\displaystyle+\biggl{(}\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\left(elas^{L}+\frac{\partial\log(elas^{M})}{\partial l}\right)$
$\displaystyle+\left(elas^{L}+\frac{\partial\log(elas^{K})}{\partial
l}\right)\biggr{)}\frac{dl}{d\theta}$
$\displaystyle-\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\frac{d\log\left(\frac{\rho}{P}\right)}{d\theta}$
$\displaystyle+\left(1+\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\right)\frac{dm(\omega_{-1})}{d\theta}$
$\displaystyle+\left(1+\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\right)\frac{d\eta}{d\theta}$
$\displaystyle+\frac{d\varepsilon}{d\theta}$

The first equality follows from substituting in equation (18), and the second
from equations (38) and (40). Substituting $\frac{dm}{d\theta}$ using equation
(46) and rearranging delivers the third equality. The fourth equality is just
a linear rearrangement.

####  B.2.1 Past productivity channel

$\displaystyle\frac{dmp^{K}}{d\omega_{-1}}$ (48)
$\displaystyle=\biggl{(}\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\left(elas^{K}+\frac{\partial\log(elas^{M})}{\partial k}\right)$
$\displaystyle+\left(elas^{K}+\frac{\partial\log(elas^{K})}{\partial
k}-1\right)\biggr{)}\frac{dk}{d\omega_{-1}}$
$\displaystyle+\biggl{(}\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\left(elas^{L}+\frac{\partial\log(elas^{M})}{\partial l}\right)$
$\displaystyle+\left(elas^{L}+\frac{\partial\log(elas^{K})}{\partial
l}\right)\biggr{)}\frac{dl}{d\omega_{-1}}$
$\displaystyle-\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\frac{d\log\left(\frac{\rho}{P}\right)}{d\omega_{-1}}$
$\displaystyle+\left(1+\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\right)\frac{dm(\omega_{-1})}{d\omega_{-1}}$

A counterfactual change in $known$, past systematic productivity does not
affect ex-ante and ex-post productivity shocks
$\left(\frac{d\eta}{d\omega_{-1}}=\frac{d\varepsilon}{d\omega_{-1}}=0\right)$
by Assumption 2. It has a direct and indirect effect on the marginal product
of capital. The indirect effect works via a variation in the allocation of the
predetermined inputs, that is, capital itself and labor, and via a change in
the relative price of materials, which changes the firm’s expectations about
the future allocation of material inputs, and then, the marginal product of
capital because of inputs interconnectedness.

####  B.2.2 Ex-ante productivity shock channel

$\displaystyle\frac{dmp^{K}}{d\eta}$ (49)
$\displaystyle=-\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\frac{d\log\left(\frac{\rho}{P}\right)}{d\eta}$
$\displaystyle+\left(1+\left(elas^{M}+\frac{\partial\log(elas^{K})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\right)$

An ex-ante productivity shock do not change the allocation of capital and
labor since they are pre-allocated by timing assumptions
$\left(\frac{dk}{d\eta}=\frac{dl}{d\eta}=0\right)$. Moreover, by Assumption 1,
it is independent of past productivity and ex-post productivity shocks
$\left(\frac{dm(\omega_{-1})}{d\eta}=\frac{d\varepsilon}{d\eta}=0\right)$.
Then, an ex-ante productivity shock affect the marginal product of capital via
a direct effect and an indirect effect through a change in the relative price
of material, which entails a change in the materials allocation and, then, a
change in the marginal product of capital by input interconnectedness.

####  B.2.3 Ex-post productivity shock channel

$\displaystyle\frac{dmp^{K}}{d\varepsilon}=1$ (50)

An ex-post productivity shock do not change the allocation of any input since
they are allocated before its realization by timing assumptions
$\left(\frac{dk}{d\varepsilon}=\frac{dl}{d\varepsilon}=\frac{dm}{d\varepsilon}=0\right)$.
Moreover, by Assumption 1, it is independent of past productivity and ex-post
productivity shocks
$\left(\frac{dm(\omega_{-1})}{d\eta}=\frac{d\varepsilon}{d\eta}=0\right)$.
Finally, the relative price of material inputs that the firm faces is
independent of the ex-post productivity shock by Assumption 4
$\left(\frac{d\log\left(\frac{\rho}{P}\right)}{d\eta}=0\right)$. the ex-post
productivity shock is only a shifter of the marginal product since its
realization generates only post-allocation output fluctuations.

###  B.3 Labor

The derivation and interpretation of the channels’ equations mirror the ones
developed for capital in Section B.2.

$\displaystyle\frac{dmp^{L}}{d\theta}$ (51)
$\displaystyle=\frac{dy}{d\theta}-\frac{d{l}}{d\theta}+\frac{d{\log
elas^{L}}}{d\theta}$
$\displaystyle=elas^{K}\frac{dk}{d\theta}+elas^{L}\frac{dl}{d\theta}+elas^{M}\frac{dm}{d\theta}+\frac{dm(\omega_{-1})}{d\theta}+\frac{d\eta}{d\theta}+\frac{d\varepsilon}{d\theta}$
$\displaystyle-\frac{d{l}}{d\theta}+\frac{\partial\log(elas^{L})}{\partial
k}\frac{dk}{d\theta}+\frac{\partial\log(elas^{L})}{\partial
l}\frac{dl}{d\theta}+\frac{\partial\log(elas^{L})}{\partial
m}\frac{dm}{d\theta}$
$\displaystyle=\left(elas^{K}+\frac{\partial\log(elas^{L})}{\partial
k}\right)\frac{dk}{d\theta}+\left(elas^{L}+\frac{\partial\log(elas^{L})}{\partial
l}-1\right)\frac{dl}{d\theta}$
$\displaystyle+\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}$
$\displaystyle\left(\left(elas^{K}+\frac{\partial\log(elas^{M})}{\partial
k}\right)\frac{dk}{d\theta}+\left(elas^{L}+\frac{\partial\log(elas^{M})}{\partial
l}\right)\frac{dl}{d\theta}+\frac{dm(\omega_{-1})}{d\theta}+\frac{d\eta}{d\theta}-\frac{d\log\left(\frac{\rho}{P}\right)}{d\theta}\right)$
$\displaystyle+\frac{dm(\omega_{-1})}{d\theta}+\frac{d\eta}{d\theta}+\frac{d\varepsilon}{d\theta}$
$\displaystyle=\biggl{(}\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\left(elas^{K}+\frac{\partial\log(elas^{M})}{\partial k}\right)$
$\displaystyle+\left(elas^{K}+\frac{\partial\log(elas^{L})}{\partial
k}\right)\biggr{)}\frac{dk}{d\theta}$
$\displaystyle+\biggl{(}\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\left(elas^{L}+\frac{\partial\log(elas^{M})}{\partial l}\right)$
$\displaystyle+\left(elas^{L}+\frac{\partial\log(elas^{L})}{\partial
l}-1\right)\biggr{)}\frac{dl}{d\theta}$
$\displaystyle-\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\frac{d\log\left(\frac{\rho}{P}\right)}{d\theta}$
$\displaystyle+\left(1+\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\right)\frac{dm(\omega_{-1})}{d\theta}$
$\displaystyle+\left(1+\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\right)\frac{d\eta}{d\theta}$
$\displaystyle+\frac{d\varepsilon}{d\theta}$

####  B.3.1 Past productivity channel

$\displaystyle\frac{dmp^{L}}{d\omega_{-1}}$ (52)
$\displaystyle=\biggl{(}\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\left(elas^{K}+\frac{\partial\log(elas^{M})}{\partial k}\right)$
$\displaystyle+\left(elas^{K}+\frac{\partial\log(elas^{L})}{\partial
k}\right)\biggr{)}\frac{dk}{d\omega_{-1}}$
$\displaystyle+\biggl{(}\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\left(elas^{L}+\frac{\partial\log(elas^{M})}{\partial l}\right)$
$\displaystyle+\left(elas^{L}+\frac{\partial\log(elas^{L})}{\partial
l}-1\right)\biggr{)}\frac{dl}{d\omega_{-1}}$
$\displaystyle-\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\frac{d\log\left(\frac{\rho}{P}\right)}{d\omega_{-1}}$
$\displaystyle+\left(1+\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\right)\frac{dm(\omega_{-1})}{d\omega_{-1}}$

####  B.3.2 Ex-ante productivity shock channel

$\displaystyle\frac{dmp^{L}}{d\eta}$ (53)
$\displaystyle=-\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\frac{d\log\left(\frac{\rho}{P}\right)}{d\eta}$
$\displaystyle+\left(1+\left(elas^{M}+\frac{\partial\log(elas^{L})}{\partial
m}\right)\left(1-elas^{M}-\frac{\partial\log(elas^{M})}{\partial
m}\right)^{-1}\right)$

####  B.3.3 Ex-post productivity shock channel

$\displaystyle\frac{dmp^{L}}{d\varepsilon}=1$ (54)

##  Appendix C Data cleaning

###  C.1 US Data

The Compustat raw dataset spans the years 1960-2022. Following [18], I deleted
all the firms that are not US incorporated. Then, I clean the dataset
according to the industry reporting. I keep only the observations that report
the industry at six-digit NAICS levels. Furthermore, I keep only firms
operating in the manufacturing sector, with NAICS codes between 310000 and
340000. I remove firms that report negative or zero sales (sale), employment
(emp), cost of goods sold (cogs) and sales and administrative expenses (xsga).
Finally, I remove all observations whose year is after the firm’s deletion
date recorded in Compustat (that is, I keep only "active" firms). Due to the
availability of the deflators, the sample is restricted to the years
1960-2018.

Once constructed cost of employees and cost of materials variables according
to [29], I replace with missing values all negative values of the cost of
materials variable. Now, the last major data cleaning problem left is to fill
the remaining missing values of the main variables for the years the firms
have been active, which threaten the dynamic structure of the production
function model. To solve this issue, I resort to focusing on the "productivity
sample" ([1]) based on the material cost variable (which is the variable with
the highest number of missing values). First, I keep only the firms with at
least two observations of the variable cost of materials. Then, I keep only
firms with at least $50\%$ of non-missing observations for this variable.
These firms compose the "productivity sample". Finally, of these firms, the
missing values for the variables output, labor, capital, cost of materials,
and cost of employees are linearly interpolated. Interpolation is limited to
the case of a maximum of three missing data points in a row. Notice that in
the cases there are not enough observations to perform the interpolation
(i.e., there are more than three missing points in a raw for a firm), the
interpolated variable will be missing for all the years for that firm.

###  C.2 European Data

European raw balance sheet data are imported from BvD-Orbis online platform
and BvD-Orbis historical data. The cleaning procedure is in line with [1],
using the firm’s unconsolidated balance sheet data. The output variable I use
is the firm’s (deflated) operating revenue. Moreover, the input variables are
measured using the number of employees (NEMP) for labor, (deflated) fixed
assets (FASS) for capital, and (deflated) material costs (MCOST) for material
inputs. The cost of employees (CEMP) is also retrieved from the BvD-Orbis.
Once the variables are downloaded, the merge between the historical and
current Orbis datasets calls for a harmonization of the status of the firm, if
"active" or "inactive". To this end, the firm’s status throughout the dataset
is updated with its most recent observation. Once the set of active firms has
been defined, as done with the US data, I focus on the "productivity sample"
([1]) based on the number of employees variable. The construction of the
"productivity sample" and the filling of the missing values of the variables
of interest for the years of activity of the firms is performed as described
above for Compustat data. Finally, variables with no information about the
industrial sector at the NACE 4-digit level and with negative or zero values
for the number of employees, revenue, fixed assets, and cost of materials
variables are eliminated. Again, according to the availability of the
deflators, the sample is restricted to the years reported.

##  Appendix D [23] \- Identification and Estimation

Taking the FOC for $M_{jt}$ in equation (9) yields

$\frac{\partial
E(F(k_{jt},l_{jt},m_{jt})e^{v_{jt}}\mid\mathcal{I}_{jt})}{\partial
M_{jt}}-\frac{\rho_{jt}}{P_{jt}}=0$ (55)

Since the firm is a price taker in the output and inputs markets and given the
distributional assumptions on $\omega_{jt}$ and $\varepsilon_{jt}$ (see
Assumption 2), it follows that the FOC reads

$\frac{\partial F(k_{jt},l_{jt},m_{jt})}{\partial
M_{jt}}e^{\omega_{jt}}\mathcal{E}-\frac{\rho_{jt}}{P_{jt}}=0$ (56)

Taking logs, and substituting
$\omega_{jt}=y_{jt}-f(k_{jt},l_{jt},m_{jt})-\varepsilon_{jt}$ delivers

$\log(\rho_{jt})=\log(P_{jt}Y_{jt})-f(k_{jt},l_{jt},m_{jt})-\varepsilon_{jt}+\log\mathcal{E}+\log\frac{\partial
F(k_{jt},l_{jt},m_{jt})}{\partial M_{jt}}$ (57)

Let $s_{jt}=ln((\rho_{jt}M_{jt})/(P_{jt}Y_{jt}))$, the (log) intermediate-
input share of output of firm $j$ at time $t$. Rearranging232323Indeed, notice
that  $\displaystyle\log\frac{\partial F(k_{jt},l_{jt},m_{jt})}{\partial
M_{jt}}-f(k_{jt},l_{jt},m_{jt})$ (58) $\displaystyle=$
$\displaystyle\log\frac{\partial F(k_{jt},l_{jt},m_{jt})}{\partial
M_{jt}}-\log F(k_{jt},l_{jt},m_{jt})$ $\displaystyle=$
$\displaystyle\log\left(\frac{\partial
F(k_{jt},l_{jt},m_{jt})}{F(k_{jt},l_{jt},m_{jt})}/\frac{\partial
M_{jt}}{M_{jt}}\right)-\log M_{jt}$ $\displaystyle=$
$\displaystyle\log\left(\frac{\partial f(k_{jt},l_{jt},m_{jt})}{\partial
m_{jt}}\right)-m_{jt}$ Where the last equality follows from $\frac{\partial
F(k_{jt},l_{jt},m_{jt})}{F(k_{jt},l_{jt},m_{jt})}/\frac{\partial
M_{jt}}{M_{jt}}=\frac{\partial f(k_{jt},l_{jt},m_{jt})}{\partial m_{jt}}$,
being the output elasticity of material inputs. equation (57) yields an
expression of the intermediate log share as a function of the log intermediate
inputs elasticity, a scalar constant, and the ex-post productivity forecast
error

$s_{jt}=\log\left(\frac{\partial f(k_{jt},l_{jt},m_{jt})}{\partial
m_{jt}}\right)+\log\mathcal{E}-\varepsilon_{jt}$ (59)

The intermediate inputs elasticity $\frac{\partial
f(k_{jt},l_{jt},m_{jt})}{\partial m_{jt}}$ can be approximated using a second
degree, complete polynomial in the capital, labor, and material costs. Then,
substituting it into equation (59) yields

$s_{jt}=\log\left(\begin{array}[]{l}\gamma_{0}+\gamma_{k}k_{jt}+\gamma_{l}l_{jt}+\gamma_{m}m_{jt}+\gamma_{kk}k_{jt}^{2}+\gamma_{ll}l_{jt}^{2}\\\
+\gamma_{mm}m_{jt}^{2}+\gamma_{kl}k_{jt}l_{jt}+\gamma_{km}k_{jt}m_{jt}+\gamma_{lm}l_{jt}m_{jt}\end{array}\right)+\log\mathcal{E}-\varepsilon_{jt}$
(60)

The $\gamma$ parameters in equation (60) can be consistently estimated via
non-linear least square up to the scalar constant $\mathcal{E}$. Call
$\hat{\gamma}^{\prime}$ this estimate242424Indeed, notice that, from equation
(60), regressing $s_{jt}$ just on
$\log\left(\begin{array}[]{l}\gamma_{0}+\gamma_{k}k_{jt}+\gamma_{l}l_{jt}+\gamma_{m}m_{jt}+\gamma_{kk}k_{jt}^{2}+\gamma_{ll}l_{jt}^{2}\\\
+\gamma_{mm}m_{jt}^{2}+\gamma_{kl}k_{jt}l_{jt}+\gamma_{km}k_{jt}m_{jt}+\gamma_{lm}l_{jt}m_{jt}\end{array}\right)$
delivers the vector $\hat{\gamma}^{\prime}$ which is an estimate for
$\gamma\mathcal{E}$..

However, since the (negative) regression residuals are estimates for the
structural ex-post productivity forecast error $\varepsilon_{jt}$, by
Assumption 2, one can use the estimated residuals to estimate
$\hat{\mathcal{E}}=(1/(JT))\sum_{(j,t)}e^{\hat{\varepsilon}_{jt}}$. Then,
having an estimate for $\mathcal{E}$, one can recover an
estimate252525$\hat{\gamma}=\hat{\gamma}^{\prime}/\hat{\mathcal{E}}$.
$\hat{\gamma}$ for the parameters vector free of the constant $\mathcal{E}$
and, then, an estimate for the intermediate input elasticity,
$\widehat{\frac{\partial f(k_{jt},l_{jt},m_{jt})}{\partial m_{jt}}}$.

Now, notice that one can invoke the Fundamental Theorem of Calculus262626For a
more detailed discussion about the use of integration for identification, see
Section IV in [23]. to derive the production function from integrating the
output elasticity of material inputs

$\int\frac{\partial f\left(k_{jt},l_{jt},m_{jt}\right)}{\partial
m_{jt}}dm_{jt}=f\left(k_{jt},l_{jt},m_{jt}\right)+\mathcal{C}\left(k_{jt},l_{jt}\right)$
(61)

The integral has a closed form solution since a complete, second-order
polynomial approximates $\frac{\partial
f\left(k_{jt},l_{jt},m_{jt}\right)}{\partial m_{jt}}$. Indeed

$\int\frac{\partial f\left(k_{jt},l_{jt},m_{jt}\right)}{\partial
m_{jt}}dm_{jt}=\left(\begin{array}[]{l}\gamma_{0}+\gamma_{k}k_{jt}+\gamma_{l}l_{jt}+\frac{\gamma_{m}}{2}m_{jt}\\\
+\gamma_{kk}k_{jt}^{2}+\gamma_{ll}l_{jt}^{2}+\frac{\gamma_{mm}}{3}m_{jt}^{2}\\\
+\gamma_{kl}k_{jt}l_{jt}+\frac{\gamma_{km}}{2}k_{jt}m_{jt}+\frac{\gamma_{m}}{2}l_{jt}m_{jt}\end{array}\right)m_{jt}$
(62)

Substituting the $\gamma$ parameters vector with its estimate $\hat{\gamma}$
is enough to retrieve the sample analog for the integral,
$\widehat{\int\frac{\partial f\left(k_{jt},l_{jt},m_{jt}\right)}{\partial
m_{jt}}dm_{jt}}$.

Finally, consider constructing the following random variable
$\mathcal{Y}_{jt}$

$\displaystyle\mathcal{Y}_{jt}$ $\displaystyle\equiv
y_{jt}-\varepsilon_{jt}-\int\frac{\partial
f\left(k_{jt},l_{jt},m_{jt}\right)}{\partial m_{jt}}dm_{jt}$ (63)
$\displaystyle=y_{jt}-\varepsilon_{jt}-f\left(k_{jt},l_{jt},m_{jt}\right)-\mathcal{C}\left(k_{jt},l_{jt}\right)$
$\displaystyle=\omega_{jt}-\mathcal{C}\left(k_{jt},l_{jt}\right)$

Where the second line follows by equation (61) and the third line by equations
(1) and (2). Notice that the variable $y_{jt}$ is observed. At the same time,
I derived estimates in the previous steps for the ex-post productivity
forecast error $\varepsilon_{jt}$ and $\int\frac{\partial
f\left(k_{jt},l_{jt},m_{jt}\right)}{\partial m_{jt}}dm_{jt}$. Then, I can
estimate $\mathcal{Y}_{jt}$. Call it $\hat{\mathcal{Y}}_{jt}$.

However, from the last line of equation (63), $\mathcal{Y}_{jt}$ is also a
function of $\omega_{jt}$, the persistent component of the TFP, and
$\mathcal{C}(.)$, the residual function from integration in equation (61).
Both need to be estimated. One can approximate the $\mathcal{C}(.)$ function
and the Markovian process of the persistent component of the TFP using second
and third-order degree complete polynomials. Formally272727The
$\mathcal{C}(.)$ function is normalized to contain no constant since this
latter cannot be separately identified later from the mean productivity,
$E[\omega_{jt}]$.

$\mathcal{C}\left(k_{jt},l_{jt}\right)=\alpha_{k}k_{jt}+\alpha_{l}l_{jt}+\alpha_{kk}k_{jt}^{2}+\alpha_{ll}l_{jt}^{2}+\alpha_{kl}k_{jt}l_{jt}$
(64)

While the Markovian process for $\omega_{jt}$ is given by

$\omega_{jt}=m(\omega_{jt-1})+\eta_{jt}=\sum_{0\leq a\leq
3}\delta_{a}\omega_{jt-1}^{a}+\eta_{jt}$ (65)

Substituting equations (64) and (65) in (63), and replacing for
$\omega_{jt-1}$ using the third line of equation (63), yields

$\displaystyle\hat{\mathcal{Y}}_{jt}=-\alpha_{k}k_{jt}+\alpha_{l}l_{jt}+\alpha_{kk}k_{jt}^{2}+\alpha_{ll}l_{jt}^{2}+\alpha_{kl}k_{jt}l_{jt}$
(66) $\displaystyle+\sum_{0\leq a\leq
3}\delta_{a}\left(\hat{\mathcal{Y}}_{jt-1}+\alpha_{k}k_{jt-1}+\alpha_{l}l_{jt-1}+\alpha_{kk}k_{jt-1}^{2}+\alpha_{ll}l_{jt-1}^{2}+\alpha_{kl}k_{jt-1}l_{jt-1}\right)^{a}+\eta_{jt}$

Since $k_{jt}$ and $l_{jt}$ are predetermined at period $t-1$ and given
Assumption 2 regarding the information set available to firm $j$ at the end of
period $t-1$, the parameters vectors $\alpha$ and $\delta$ can be estimated
from (LABEL:eq:17) using an exactly identified GMM system with the following
unconditional moment conditions.

$\displaystyle
E\left[\eta_{jt}k_{jt}^{\tau_{k}}l_{jt}^{\tau_{l}}\right]=0~{}\forall\tau_{k},~{}\tau_{l}~{}such~{}that~{}0<\tau_{k}+\tau_{l}\leq
2$ (67) $\displaystyle
E\left[\eta_{jt}\hat{\mathcal{Y}}_{jt-1}^{a}\right]=0~{}\forall
a~{}such~{}that~{}0\leq a\leq 3$

The first set of conditions contains five moments identifying the five
components of the $\alpha$ parameter vector. The last set of moments contains
four conditions identifying the four components of the $\delta$ parameter
vector.

Having the estimates $\hat{\gamma}$ and $\hat{\alpha}$ for the $\gamma$ and
$\alpha$ parameters vectors allows computing the firm-time specific output
elasticities for material inputs, capital, and labor. Then, respectively,

$\displaystyle\widehat{elas}^{M}(k_{jt},l_{jt},m_{jt})=\widehat{\frac{\partial
f(k_{jt},l_{jt},m_{jt})}{\partial m_{jt}}}=$
$\displaystyle\hat{\gamma}_{0}+\hat{\gamma}_{k}k_{jt}+\hat{\gamma}_{l}l_{jt}+\hat{\gamma}_{m}m_{jt}+\hat{\gamma}_{kk}k_{jt}^{2}+\hat{\gamma}_{ll}l_{jt}^{2}+$
(68)
$\displaystyle\hat{\gamma}_{mm}m_{jt}^{2}+\hat{\gamma}_{kl}k_{jt}l_{jt}+\hat{\gamma}_{km}k_{jt}m_{jt}+\hat{\gamma}_{lm}l_{jt}m_{jt}$
$\displaystyle\widehat{elas}^{K}(k_{jt},l_{jt},m_{jt})=\widehat{\frac{\partial
f(k_{jt},l_{jt},m_{jt})}{\partial k_{jt}}}=$ $\displaystyle
m_{jt}(\hat{\gamma}_{k}+2\hat{\gamma}_{kk}k_{jt}+\hat{\gamma}_{lk}l_{jt}+\frac{\hat{\gamma}_{mk}}{2}m_{jt})$
(69)
$\displaystyle-\hat{\alpha}_{k}-2\hat{\alpha}_{kk}k_{jt}-\hat{\alpha}_{lk}l_{jt}$
$\displaystyle\widehat{elas}^{L}(k_{jt},l_{jt},m_{jt})=\widehat{\frac{\partial
f(k_{jt},l_{jt},m_{jt})}{\partial l_{jt}}}=$ $\displaystyle
m_{jt}(\hat{\gamma}_{l}+2\hat{\gamma}_{ll}l_{jt}+\hat{\gamma}_{lk}k_{jt}+\frac{\hat{\gamma}_{ml}}{2}m_{jt})$
(70)
$\displaystyle-\hat{\alpha}_{l}-2\hat{\alpha}_{ll}l_{jt}-\hat{\alpha}_{lk}k_{jt}$

Standard errors for the estimated parameters and functionals can be computed
via non-parametric bootstrap since they are sieve M-estimators ([14], [23]).

###  D.1 Additional Functional Estimates

Having estimated the input elasticities, one can compute the firm-time-
specific inputs’ marginal products using equation (11).

$\displaystyle\widehat{MP}^{M}_{jt}=\frac{Y_{jt}}{M_{jt}}\widehat{elas}^{M}(k_{jt},l_{jt},m_{jt})$
(71)
$\displaystyle\widehat{MP}^{K}_{jt}=\frac{Y_{jt}}{K_{jt}}\widehat{elas}^{K}(k_{jt},l_{jt},m_{jt})$
$\displaystyle\widehat{MP}^{L}_{jt}=\frac{Y_{jt}}{L_{jt}}\widehat{elas}^{L}(k_{jt},l_{jt},m_{jt})$

And, taking the natural logarithm, denote for each input $X$

$\widehat{mp}^{X}_{jt}=\log(\widehat{MP}^{X}_{jt})$ (72)

Moreover, notice that, from equation (65),

$\frac{\partial\hat{m}(\omega_{j,-1})}{\partial\omega_{j,-1}}=\frac{\partial\sum_{0\leq
a\leq
3}\hat{\delta}_{a}\omega_{jt-1}^{a}}{\partial\omega_{j,-1}}=\hat{\delta}_{1}+2\hat{\delta}_{2}\omega_{j,-1}+3\hat{\delta}_{3}\omega_{j,-1}^{2}$
(73)

Furthermore, from equation (15)

$\frac{\partial\hat{g}(\omega_{j,-1})}{\partial\omega_{j,-1}}=\frac{\partial\left(\sum_{0\leq
a\leq
3}\hat{\delta}_{a}\omega_{jt-1}^{a}-\omega_{j,-1}\right)}{\partial\omega_{j,-1}}=\hat{\delta}_{1}-1+2\hat{\delta}_{2}\omega_{j,-1}+3\hat{\delta}_{3}\omega_{j,-1}^{2}$
(74)

And finally, from equation (68)

$\displaystyle\frac{\partial\widehat{elas}^{M}(k_{j},l_{j},m_{j})}{\partial
k_{j}}=\hat{\gamma}_{k}+2\hat{\gamma}_{kk}k_{j}+\hat{\gamma}_{kl}l_{j}+\hat{\gamma}_{km}m_{j}$
(75)
$\displaystyle\frac{\partial\widehat{elas}^{M}(k_{j},l_{j},m_{j})}{\partial
l_{j}}=\hat{\gamma}_{l}+2\hat{\gamma}_{ll}l_{j}+\hat{\gamma}_{kl}k_{j}+\hat{\gamma}_{lm}m_{j}$
$\displaystyle\frac{\partial\widehat{elas}^{M}(k_{j},l_{j},m_{j})}{\partial
m_{j}}=\hat{\gamma}_{m}+2\hat{\gamma}_{mm}m_{j}+\hat{\gamma}_{km}k_{j}+\hat{\gamma}_{lm}l_{j}$

From equation (69),

$\displaystyle\frac{\partial\widehat{elas}^{K}(k_{j},l_{j},m_{j})}{\partial
k_{j}}=2m_{j}\hat{\gamma}_{kk}-2\hat{\alpha}_{kk}$ (76)
$\displaystyle\frac{\partial\widehat{elas}^{K}(k_{j},l_{j},m_{j})}{\partial
l_{j}}=m_{j}\hat{\gamma}_{lk}-\hat{\alpha}_{lk}$
$\displaystyle\frac{\partial\widehat{elas}^{K}(k_{j},l_{j},m_{j})}{\partial
m_{j}}=\hat{\gamma}_{k}+2\hat{\gamma}_{kk}k_{j}+\hat{\gamma}_{lk}l_{j}+\hat{\gamma}_{mk}m_{j}$

From equation (70),

$\displaystyle\frac{\partial\widehat{elas}^{L}(k_{j},l_{j},m_{j})}{\partial
k_{j}}=m_{j}\hat{\gamma}_{lk}-\hat{\alpha}_{lk}$ (77)
$\displaystyle\frac{\partial\widehat{elas}^{L}(k_{j},l_{j},m_{j})}{\partial
l_{j}}=2m_{j}\hat{\gamma}_{ll}-2\hat{\alpha}_{ll}$
$\displaystyle\frac{\partial\widehat{elas}^{L}(k_{j},l_{j},m_{j})}{\partial
m_{j}}=\hat{\gamma}_{l}+2\hat{\gamma}_{ll}l_{j}+\hat{\gamma}_{lk}k_{j}+\hat{\gamma}_{ml}m_{j}$

###  D.2 Estimation Algorithm and Computational Issues

The first step of the estimation procedure is minimizing the residual sum of
squares in equation (60). This step is the most computationally demanding
because:

  1. 1.

The minimization problem is nonlinear. The minimum found is local and possibly
dependent on the initial condition.

  2. 2.

The log specification might allow for complex solutions.

  3. 3.

Even solving point 2. above, if the evaluation of the function at the
candidate minimum is complex, the minimization algorithm stops.

I will return to point 1. later. Regarding point 2., to avoid complex
solutions to the minimization problem, I use the nonlin Matlab routine
employing the st-region-reflective algorithm with appropriately defined (real)
lower and upper bounds for the solution. It forces the minimization routine to
limit its scope only to real solutions as long as the first function
evaluation is real, given the initial values. Otherwise, if a complex
candidate solution is found, the algorithm stops. The classic enberg-marquardt
algorithm returns a complex solution even if real bounds are specified.
Concerning point 3., The routine stops if the first evaluation of the
minimizing function is complex, even using the method above. Then, I set the
initial parameter values such that the logarithm’s argument at the procedure’s
beginning is most likely positive. I can do this by setting the initial values
for $\gamma^{\prime}_{kk}$, $\gamma^{\prime}_{ll}$, and $\gamma^{\prime}_{mm}$
positive and big enough. These parameters are key in avoiding this issue
because they multiply the only variables that are always strictly positive
($k^{2}_{jt}$, $l^{2}_{jt}$, and $m^{2}_{jt}$).

Once estimated the share regression in equation (60) and having computed
$\hat{\gamma}$$\in\textbf{R}^{10}$, the second step of the estimation
procedure involves estimating the parameters vectors
$\alpha$$\in\textbf{R}^{5}$ in equation (64) and $\delta$$\in\textbf{R}^{4}$
in equation (61). I use an inner loop-outer loop routine as suggested by [23].

  * •

I consider an initial guess for $\alpha_{0}$.

  * •

Given $\alpha_{0}$, I construct
$\tilde{\mathcal{C}}_{2}\left(k_{jt},l_{jt}\right)=\sum_{0<\tau_{k}+\tau_{l}\leq
2}\alpha_{0,\tau_{k},\tau_{l}}k_{jt}^{\tau_{k}}l_{jt}^{\tau_{l}}$.

  * •

Construct
$\omega_{jt}(\bm{\alpha_{0}})=\hat{\mathcal{Y}}_{jt}+\tilde{\mathcal{C}}_{2}\left(k_{jt},l_{jt}\right)$.
$\hat{\mathcal{Y}}_{jt}$ is observable, given the estimates for $\hat{\gamma}$
and equation (64).

  * •

Regress $\omega_{jt}(\bm{\alpha_{0}})$ on a third order sieve in
$\omega_{jt-1}(\bm{\alpha_{0}})$ (I use nonlin again).

  * •

Get an estimate for $\hat{\delta}$($\alpha_{0}$).

  * •

Given $\alpha_{0}$, $\hat{\delta}$($\alpha_{0}$) and
$\omega_{jt}(\bm{\alpha_{0}})$, I get $\hat{\eta}$ from (LABEL:eq:17), that is

$\hat{\eta}=\hat{\mathcal{Y}}_{jt}+\sum_{0<\tau_{k}+\tau_{l}\leq
2}\alpha_{\tau_{k},\tau_{l}}k_{jt}^{\tau_{k}}l_{jt}^{\tau_{l}}-\sum_{0\leq
a\leq 3}\delta_{a}(\bm{\alpha_{0}})\omega_{jt-1}^{a}(\bm{\alpha_{0}})$

  * •

Use the unconditional moment conditions
$E\left[\eta_{jt}k_{jt}^{\tau_{k}}l_{jt}^{\tau_{l}}\right]=0$ $\forall$
$\tau_{k}$, $\tau_{l}$ such that $0<\tau_{k}+\tau_{l}\leq 2$ to estimate
$\alpha$ via an exactly identified GMM system (I use the Matlab routine
nsearch).

  * •

Update the guess $\alpha_{0}$ with the newly estimated $\alpha$.

  * •

Iterate until convergence.

A final consideration regards the sample size. In particular, given the
initial values, the relatively large size of the countries’ samples negatively
affects the time of convergence in both estimation steps and the quality of
convergence to a local minimum in the first step. Taking initial guesses for