Scalable vs. Productive Technologies∗
Mons Chan† Guangbin Hong‡

Joachim Hubmer§ Serdar Ozkan¶

Sergio Salgadok

July 11, 2024

Abstract
Do larger firms have more productive technologies or are their technologies more scalable, or both? We use administrative data on Canadian and US
firms to estimate flexible nonparameteric production functions. Our estimation results in a joint distribution of output elasticities of capital, labor, andintermediate inputs—therefore, returns to scale (RTS)—along with total factor
productivity (TFP). We find significant heterogeneity in both RTS and TFP
across firms. Larger firms operate technologies with higher RTS, both acrossand within industries. Higher RTS for large firms are entirely driven by higher
intermediate input elasticities. Descriptively, these align with higher intermediate input revenue shares. We then incorporate RTS heterogeneity into anotherwise standard incomplete markets model with endogenous entrepreneurship that matches the observed heterogeneity in TFP and RTS. In this model,
we find that the efficiency losses of financial frictions are more than twice aslarge relative to the conventional calibration that loads all heterogeneity on
TFP and imposes a common RTS parameter.
Keywords: Production function heterogeneity, returns to scale, misallocation∗

For helpful comments, we thank seminar participants at the Barcelona GSE Summer Forum, SED, Bonn Univer-sity, Goethe University, University of Wisconsin, and Wharton. The views expressed herein are those of the authorsand do not reflect the ones of the Federal Reserve Bank of St. Louis, the Federal Reserve System or the Boardof Governors. Ozkan acknowledges financial support from the Canadian Social Sciences and Humanities ResearchCouncil. Click here for the latest version.
†

Queen’s University; mons.chan@queensu.ca
University of Toronto; g.hong@mail.utoronto.ca§
University of Pennsylvania; jhubmer@sas.upenn.edu¶
St. Louis Fed, University of Toronto; serdar.ozkan@gmail.com
k
The Wharton School-University of Pennsylvania; ssalgado@wharton.upenn.edu‡

1Introduction
The large and persistent firm heterogeneity in total factor productivity (TFP)

has been extensively documented within industries and for different countries andtime periods (see Syverson (2011) for an overview). The earlier literature focused on
this dimension of heterogeneity to explain the firm size distribution (at least within
industries). For example, the misallocation literature (pioneered by Restuccia andRogerson (2008) and Hsieh and Klenow (2009)) quantified the efficiency costs of differences in the marginal product of capital across firms with different TFP (a notable
exception is David and Venkateswaran (2019)). Moreover, models of entrepreneurshipwith heterogeneous TFP and decreasing returns to scale production functions havebeen used to explain persistent differences in rates of return and wealth inequality(e.g., Cagetti and De Nardi (2006)).
In this paper, we allow for more general heterogeneity in production technologies
among firms—focusing on differences in returns to scale (RTS)—and investigate howthese differences shape the firm size distribution. We employ a broad set of estimation
methods and use panel data on firms to document significant heterogeneity in production technologies across firms. Our benchmark method builds on Gandhi, Navarro
and Rivers (2020) (henceforth GNR), and provides us with a joint distribution of
output elasticities of labor, capital, and intermediate inputs—thereby, RTS—as well
as TFP, at the firm-year level. We then study whether larger firms have technologies
that are more productive (high TFP) or more scalable (high RTS). Finally, we show
why the answer to this question matters in an application to efficiency costs of misallocation from financial frictions, indicating the potential role of RTS heterogeneity
for an array of quantitative questions, such as optimal taxation of capital, firm hiring
decisions (as in Gavazza et al. (2018)) or firm growth and cyclicality (as in Clymo
and Rozsypal (2023)).
In our main empirical analysis, we use administrative panel data for the universeof incorporated Canadian firms—which account for more than 90% of private businesssector output—from 2001 to 2019 from the National Accounts Longitudinal Microdata
file. This dataset contains firms’ balance sheet information on revenues as well as thetotal cost of labor, capital, and intermediate inputs. The measurement of labor andintermediate inputs is consistent with Statistics Canada’s national accounts, and we1

construct the capital stock for each firm using the perpetual-inventory method (asstandard in the literature). After sample selection, our sample includes 4.3 million
firm-year observations. We also corroborate our main findings from Canada withsimilar empirical results for manufacturing firms in the US using administrative datafrom the Annual Survey of Manufactures and Economic Census.
In our benchmark approach, we estimate nonparametric production functions
building on the method of GNR.1 This technique leverages standard assumptions
on profit maximization, adjustment costs, and input choice timing to recover firmlevel measures of TFP, output elasticities, and returns to scale. The non-homothetic
production function is identified from variation in input expenditure shares, as well
as from the covariance between in input and output levels, controlling for the endogeneity of inputs to TFP. Intuitively, higher returns-to-scale firms are those withhigher expenditure shares and/or a higher covariance of input and output levels.
We apply the GNR method to each 2-digit NAICS industry, and for each year in
our sample. In addition to considerable heterogeneity in TFP (as the previous literature also documented), we find large differences in RTS among Canadian firms (evenwithin industries). The mean of our estimated RTS equals 0.97, with considerable
variation across industries.2 More novel, we document significant variation in RTS
within 2-digit NAICS industries. The average within-industry differential between
the 90th and 10th percentiles (P90-P10) equals 0.07. These differences imply that, fora given level of TFP, when increasing the input bundle by 1%, the output of the firmat the 90th percentile increases by 7% more relative to the increase in output of thefirm at the 10th percentile. Interpreted as deviations from constant returns to scale,these differences are large.3 Furthermore, these differences are mainly due to ex-ante
1

In particular, our estimation involves two steps. First, we employ the factor share or firstorder condition approach to nonparametrically identify the output elasticity of the fully flexible
intermediate input, and transitory productivity shocks. In the second step, we use these estimates
to identify the output elasticities of capital and labor along with the persistent TFP shocks byexploiting dynamic panel (or proxy) variable conditional moment restrictions based on lagged input
decisions.
2
A few papers have documented heterogeneity in RTS. For instance, Gao and Kehrig (2017)
reports RTS across different industries in the US. Demirer (2020) finds heterogeneity in output
elasticities and RTS across firms, industries and countries, often with RTS greater than one. Noneof these papers, however, look at the relation between RTS, TFP, and firm size within industries.
3E.g., with Cobb-Douglas production functions in an efficient economy, the elasticity of optimal
1
firm output to firm TFP is 1−RT
S . This elasticity is 3.3 times larger for a firm with RTS of 0.97
compared to a firm with RTS of 0.90.

2

production technology heterogeneity. Using our firm-year level estimates, we run apanel regression for RTS and find that firm fixed effects account for 86% of the overallvariation after controlling for firm size and age. Thus, while we estimate production
functions at the 2-digit industry-year level, we interpret heterogeneity between firms
as mainly stemming from permanent differences.
The dispersion in the output elasticities of capital, labor, and intermediates islarger than dispersion in RTS, meaning that firms’ technologies differ both in theirrelative factor intensities and in the overall RTS. The P90-P10 of our estimated intermediate input elasticity and of the labor elasticity are both equal to 0.35, while
the P90-P10 dispersion of the capital elasticity is lower, at 0.07. The estimated output elasticity patterns align with the corresponding revenue shares of the respective
input. In particular, the intermediate input shares in firm revenue closely mirror theestimated intermediate input elasticities, which is expected as our estimation treats
them as flexible input. For labor and capital, the correlation between observed revenue shares and estimated elasticities is less than perfect but still strongly positive.
Again, this is expected as these two inputs are potentially subject to adjustment costs
and input market power, and capital is pre-determined.
Next, we investigate the variation in production technologies across the revenue
distribution. Our main empirical finding is that RTS is increasing in firm size, especially above the median: When pooling firms across the economy we find that, on
average, the largest 5% of firms exhibit RTS that are 10 percentage points (pp) higher
compared to those in the bottom 50%. Similarly, within 2-digit NAICS industries,
the largest 5% firms within industries exhibit RTS that are 8 pp higher compared tothe bottom 50%. When we zoom into the respective output elasticities, we find thatthe overall increase in RTS along the revenue distribution is entirely accounted forby an increase in the output elasticity of intermediate inputs. In contrast, labor andcapital elasticities tend to decline along the revenue distribution, though this findingis less robust across samples and specifications.
We find that TFP increases in firm revenue until the top 10% of the revenue
distribution, after which it then flattens out. RTS instead increases in a convex
fashion at the top of the revenue distribution, indicating a more important role for
RTS differences in shaping the right tail of the firm revenue distribution. In sum, our3empirical results indicate that the largest firms tend to feature the highest RTS, andnot necessarily the highest TFP as commonly assumed.
Our results are robust along several dimensions. We find similar results whether
we estimate a production function across the entire economy or within narrow industry groups. Methodologically, our main findings are unchanged when (i) clusteringfirms according to input shares and estimating separate production functions withinclusters, (ii) including intangibles in the definition of capital, and (iii) imposing homogeneous relative factor elasticities while still allowing for RTS differences. Finally,
we also document a positive relation between plant size and RTS in the US manufacturing sector.
We also show that high-RTS firms grow faster over the life-cycle and are less likely
to exit, in addition to having a higher output level. Furthermore, high-paying firms
on average feature higher RTS. These secondary findings indicate that incorporating
realistic RTS heterogeneity is important for a wide range of applications, including
wage inequality.
Finally, we link firms to their owners. Many models of wealth inequality (buildingon Quadrini (2000); Cagetti and De Nardi (2006)) feature entrepreneurs that operate
production technologies with heterogeneous TFP but common (and decreasing) RTS.
Hence, in these models, the right tail of the wealth distribution is populated byhouseholds operating the highest TFP technologies. Testing this assumption, we
first document that wealthier households invest in firms with higher RTS. Second,firm TFP increases with owner wealth, but flattens out at the top of the wealthdistribution, where RTS increases even more steeply. These findings suggest that
wealthier households tend to invest in firms with high RTS (i.e., in more scalable
technologies), and not necessarily in high-TFP firms as commonly assumed; therefore,
they have important implications for the optimal taxation of wealth and income fromwealth (e.g., Guvenen et al. (2023); Gaillard and Wangner (2021)).
To investigate the implications of our findings, we incorporate heterogeneous RTS
into a standard incomplete markets heterogeneous agents model with endogenous entrepreneurship (e.g., Quadrini (2000); Cagetti and De Nardi (2006)). In the model,
agents have an occupational choice between supplying their stochastic efficiency unitsof labor or operating a private business. The private business technology is stochastic.

4

Inputs choices are constrained by a friction λ such that, for each dollar spent on inputs, the entrepreneur needs to finance at least a fraction λ with their private wealth.
More novel, an entrepreneur’s output depends not only on a standard stochastic idiosyncratic TFP term z, but also on a stochastic idiosyncratic RTS term η. We theninvestigate the output and productivity effects of changing the financial conditionsin this economy. That is, we evaluate the impact of raising λ from zero—the unconstrained case—to positive values all the way up to one—the fully constrained casewhere all expenditures have to be financed with the owner’s equity.
Our main exercise compares the effects of increasing the friction λ in two economies:
the conventional z-economy, where there is no variation in η and which fundamentallyloads all heterogeneity on TFP, and the (η, z)-economy, which features the observed
joint distribution of RTS and TFP that we estimate in our empirical work. We
calibrate the two economies such that they agree on key observable moments; inparticular, both economies feature the same firm size distribution.
Our main finding is that in the (η, z)-economy, financial frictions generate morethan double the output losses relative to the z-economy. Static misallocation of production factors accounts for the majority of output losses in both economies, and isabout twice as large in the (η, z)-economy compared to the z-economy. To provide
intuition, we derive an analytical result in a static endowment economy. We showthat a given marginal input product wedge translates into more misallocation if theconstrained firms have relatively higher RTS—a feature that our dynamic model generates endogenously. Dynamic effects also lead to more output losses in the economy
with RTS heterogeneity, both because of underaccumulation of capital and because
the selection intro entrepreneurship is relatively more distorted. If a potential superstar (but currently poor) entrepreneur has a business idea that is highly productive(i.e., has a high z), then her business is already profitable even at a small scale, and
it is relatively easier to outgrow the friction. Instead, if the business idea is highly
scaleable (i.e., high η), but not necessarily profitable at a small scale, it is harder
to outgrow the friction, and she might never enter entrepreneurship.4 Given theseresults, we conclude that taking into account RTS heterogeneity is crucial for a broad
set of quantitative questions related to misallocation such as wealth inequality and4Some entrepreneurs have a high z but a low η, and hence a smaller optimal scale. This can
explain why some entrepreneurs do not expect to grow their firms as in Hurst and Pugsley (2011).

5

optimal taxation of capital.5

2

Empirical Methodology

2.1

Estimating Returns to Scale

Our main empirical approach builds on the production function estimation methodology developed by GNR. We use a slightly modified GNR procedure to estimate a
flexible non-parametric revenue production function which allows us to recover firm
and time-varying measures of total factor productivity (TFP) and returns to scale(RTS). This production function relates each firm-year pair’s output Yjt to its inputs,
including capital stock Kjt , labor input Ljt , and intermediate inputs Mjt , with thefollowing assumption:
Assumption 1. The firm’s production function takes the following general form inlevels Yjt = F (Kjt , Ljt , Mjt )eνjt and in logs yjt = f (kjt , `jt , mjt ) + νjt where f isa continuous and differentiable function which is strictly concave in mjt and νjt isHicks-neutral productivity.
The traditional challenge in the production function estimation literature is how toseparate productivity shocks that affect firm’s output from input choices. Following
GNR, we use the first-order conditions of the firm as well as timing assumptions on
the nature of productivity and input choices to form moment conditions. We illustrate
the details below.
The estimation technique of GNR provides several advantages relative to othermore standard methods. First, it offers a robust estimation method of gross production functions which is important if we want to obtain precise estimates of output
elasticities and thus returns to scale. Second, the non-parametric identification strategy minimizes specification error when measuring both input elasticities and the productivity term which is crucial if we aim to understand the relation between firm-levelproductivity, returns to scale, and firm size.
Following the literature, we rely on standard timing assumptions to identify theparameters of the firm production function. Define Ijt as the information set available
5E.g., in ongoing work we study the entrepeneurial activity of New Money and Old Moneyhouseholds (à la Hubmer et al. (2024)), focusing on differences in their production technologies.

6

to firm j when it enters period t. Ijt contains all the information relevant to the firm
(i.e., firm productivity, current capital stock, and so on) when it makes its period-t
decisions. Following GNR, we define any input Xt ∈ Ijt as predetermined. Any such
input is thus a function of the previous period’s information set: Xt (Ijt−1 ). We willtreat capital as a predetermined input. Inputs which are not predetermined (andthus are set in period t) we define as variable. We define any input which is variable
and where the optimal choice of Xt is a function of lagged values of itself as dynamic.
We will depart from GNR in assuming that labor is a dynamic input. Finally, wedefine an input which is variable but not dynamic as flexible. Intermediate inputs
will be treated as flexible in our framework. This implies that both Kjt and Lj,t−1
are elements of Ijt , but Ljt and Mjt are not.
Assumption 2. Capital (Kjt ∈ Ijt ) is predetermined and a state variable. Labor
input (Ljt ∈/ Ijt ) is dynamic, such that Ljt−1 ∈ Ijt is a state variable. Intermediate
inputs (Mjt ∈/ Ijt ) are flexible, so that Mjt−1 ∈/ Ijt .
The Hicks-neutral productivity term νjt can be decomposed into a persistent component, ωjt , which is known to the firm when it makes input decisions, and a transitory
component, εjt , which is unknown to the firm when making input decisions in periodt. Notice, the changes in the productivity terms can be due to both technology shocks
and market demand shifts.
Assumption 3. The permanent productivity component, ωjt ∈ Ijt , is observed by thefirm prior to making period-t decisions and is first-order Markov, such that E[ωjt |Ijt−1 ] =E[ωjt |ωjt−1 ] = h(ωjt−1 ) for some continuous function h(.). The purely transitory productivity innovation, εjt ∈/ Ijt , is i.i.d. across firms and time and it is not observedby the firm prior to making period-t decisions, with Pε (εjt |Ijt ) = Pε (εjt ).
We normalize E[εjt ] = 0 and define ξjt = ωjt − E[ωjt |ωjt−1 ] implying E[ξjt |Ijt−1 ] =0.
Assumption 4. We assume that demand for intermediate input mjt = M (kjt , `jt , ωjt )is strictly monotone in ωjt .
Note that this conditional (on period-t labor) demand function is critical in identifying the production function while allowing labor to be a dynamic (and not predeter7

mined) input. We also make the following assumption about firm’s profit maximizing
behavior and environment:
Assumption 5. Firms maximize short-run expected profits and are price takers inboth output and intermediate input markets. Denote the common output price index
for period t as Pt and the common intermediate price index as ρt .
Following GNR, the assumptions 1 to 5 give us the following first-order condition
for the firm’s profit maximization problem in period t with respect to Mjt :Pt∂
F (Kjt , Ljt , Mjt )eωjt E = ρt ,∂Mjt

where E ≡ E[eεjt ] is a constant. Multiplying both sides by Mjt /Yjt , plugging in the
production function, and rearranging provides our first estimating equation:
sjt = ln E + ln D(kjt , `jt , mjt ) − εjt≡ ln(DE (kjt , `jt , mjt )) − εjt(1)

where sjt ≡ ln(ρt Mjt /Pt Yjt ) is the log revenue share of intermediate input expenditureand D(kjt , `jt , mjt ) ≡ ∂m∂ jt f (kjt , `jt , mjt ) is the output elasticity of materials. Sinceby assumption 3 we have E[εjt ] = 0, we can use equation 1 to identify εjt and DE .
Given εjt = ln DE (kjt , `jt , mjt ) − sjt , we can identify the constant E, which
subsequently provides the elasticity D(kjt , `jt , mjt ) = DE (kjt , `jt , mjt )/E. Once weknow D(kjt , `jt , mjt ) and εjt , we can integrate the elasticity up, to estimate the restof the production function non-parametrically.6 In particular we haveZ
D(kjt , `jt , mjt ) ≡∂f (kjt , `jt , mjt )dmjt = f (kjt , `jt , mjt ) − Ψ (kjt , `jt )
∂mjt

(2)

Define ỹjt ≡ yjt − εjt − D(kjt , `jt , mjt ) = Ψ (kjt , `jt ) + ωjt . Plugging in the structure ofωjt from assumption 3, we get our second estimating equation:ỹjt = Ψ (kjt , `jt ) + h(ỹjt−1 − Ψ (kjt−1 , `jt−1 )) + ξjt
6(3)This result requires one further technical assumption on the support of (kjt , `jt ) – see Assumption 5 in GNR.

8

where ỹjt is observable given the first-stage estimates of εjt and D(kjt , `jt , mjt ). Our
assumptions on the firm’s information set give us E[ξjt |kjt , `jt−1 , kjt−1 , ỹjt−1 , `jt−2 ] = 0,
which we use with equation 3 to identify Ψ, h, and thus ξjt .7
Our estimation procedure follows GNR in using a standard sieve-series estimator
to non-parametrically identify the input elasticities and production function. We
proceed in two steps. First, we estimate equation 1 with a complete second-degree
polynomial in kjt , `jt , and mjt using nonlinear least squares. This estimator solves
!#2

"
min
0
γ

X
j,t

ε2jt =

X

0X

sjt − lnrk r` rmγrk ,r` ,rm kjt`jt mjt(4)

rk +r` +rm ≤2j,t0
rk r` rmcE (kjt , `jt , mjt ) = P
which gives us estimates of εˆjt and D
rk +r` +rm ≤2 (γ̂rk ,r` ,rm kjt `jt mjt ).
We can then recover Eb = E[exp(ε̂jt )] and the input elasticity

Xb jt , `jt , mjt ) =D(krk r` rm(γ̂rk ,r` ,rm kjt`jt mjt )rk +r` +rm ≤20
b We then integrate the estimated flexible input elasticity to recoverwhere γ̂ ≡ γ̂ /E.

b jt , `jt , mjt ) =
D(k



X
rk +r` +rm ≤2mjtγ̂r ,r ,r k rk `r` mrm
rm + 1 k ` m jt jt jtb jt , `jt , mjt ), that is the component of
which allows us to recover ỹˆjt = yjt − ε̂jt − D(koutput unrelated to variation in intermediate inputs.
In the second step, we estimate equation 3 using GMM, by approximating Ψ(kjt , `jt )and h(ωjt−1 ) using complete (separate) second- and third-degree polynomials respectively. Since we can identify both the constant of integration and TFP only up to an
additive constant, we follow GNR in normalizing Ψ to have mean zero, which implies
that the constant of integration will show up in the firm productivity level. This gives
us the following second-stage estimating equation:!aỹjt = −X
0<τk +τ` ≤27

τk τ `ατk ,τ` kjt
`tj +

X

δa ỹjt−1 +Xτk`ατk ,τ` kjt−1`τtj−1+ ξjt , (5)0<τk +τ` ≤20≤a≤2This differs from the moments used by GNR, who assume that labor is predetermined.

9where a is the degree of the polynomial. Since E[ξjt |kjt , `jt−1 , Ijt−1 ] = 0, the only endogenous variable is `jt . Thus we can use functions of the set {kjt , kjt−1 , `jt−1 , mjt−1 , ỹjt−1 }τk τ`aas instruments. In particular, our moments are E[ξjt ỹjt−1] and E[ξjt kjt`jt−1 ] for all

0 ≤ a ≤ 2 and 0 < τk + τ` ≤ 2, leaving us exactly identified.8 This provides usˆ jt ≡ ĥ(ω̂jt−1 ).
with estimates of the production function as well as ω̂jt , ξˆjt , and ω̄
The estimated production function also provides estimates of the output elasticities
of capital and labor, which combined with the previously estimated output elasticity
for intermediate inputs, gives us a firm-level measure of returns to scale. This is the
specification we use to obtain the main empirical results in the next section.

2.2

Identification and Intuition

While GNR provides a rigorous identification argument for their strategy (and
we defer to them for the details), here we discuss the intuition behind our estimation
results. We use the non-parametric identification strategy of GNR to estimate returns
to scale for two reasons. First, this method allows us to identify the output elasticities
for a gross production function, while other common estimation methods (such as
Ackerberg et al. (2015)) are typically only able to identify value-added production
functions. As we show below, variation in the output elasticity of intermediate inputs
is a key driver of variation in RTS, necessitating identification of the gross production
function to learn about RTS.
Second, the non-parametric approach provided by GNR allows us to estimate
a non-homothetic production function whereby the output elasticities and overall
RTS vary by inputs and input shares, thereby, potentially varying across the firm
size distribution. In particular, we first recover the output elasticity of intermediate
inputs, εYMjt = εYM (kjt , `jt , mjt ), which is a function of input levels through the shapeof the production function. This elasticity is identified from the mean intermediate
expenditure share of revenue, as well as from covariation between expenditure sharesand input levels. Intuitively, if the true underlying production function is CobbDouglas, then the expenditure share will be uncorrelated with input levels and theoutput elasticity will be a constant equal to the mean expenditure share. This direct
relationship is implied by the assumption that firms are price-takers in intermediate8

As pointed out by GNR, this implies that the estimator is a sieve-M estimator, which allows
us to do inference treating the polynomials as if they were the true parametric structure.

10input markets and do not face adjustment costs when choosing the level of mjt .
We also allow for adjustment costs for capital and labor, and make no assumptions
about the optimality of either input choice, driving an (unknown) wedge between
expenditure shares and output elasticities. Instead, the output elasticities of capital εYKjt = εYK (kjt , `jt , mjt ), and labor εYLjt = εYL (kjt , `jt , mjt ) are identified jointly
from their contribution to the variation in intermediate expenditure shares (e.g.,∂D(kjt , `jt , mjt ) for labor) and their covariation with (residualized) output (e.g.,∂`jt∂ỹ ). We define the firm’s returns to scale as ηjt = η(kjt , `jt , mjt ) = εYKjt + εYLjt +∂`jt jt
εYMjt .9 Thus, in the data, a high RTS firm will be a firm which has a high intermediate

input expenditure share of revenue, or a high correlation between output and capital
and/or labor.10

3

Data and Sample Selection
Our main dataset is the Canadian Employer-Employee Dynamics Database of

Statistics Canada (CEEDD), which is a set of linkable administrative tax files covering
the universe of tax-paying Canadian firms and individuals between 2001 and 2019. We
obtain the balance sheet and income statement information on firms from the National
Accounts Longitudinal Microdata File, which covers all incorporated firms.11 We
use the total revenue and total wage bill variables constructed by Statistics Canada
based on the corresponding corporate tax return line items. These same variables
are used in calculation of the national account system. Therefore, our micro data is
consistent with aggregate measures. We construct total tangible capital by employing
the perpetual-inventory method (PIM), using information on the first book value oftangible capital observed in the dataset, annual tangible capital investment, andamortization. Intermediate inputs are calculated as the sum of operating expenses
9

While the notation in this section assumes a common production function for all firms, inpractice we allow the production function to vary across different groupings such as detailed industry
and capital-ratio clusters. See Section 4 for details. Conditional on a firm grouping, our procedure
still allows output elasticities and RTS to vary by firms’ input levels.
10Note that due to the non-homotheticity of the production function, these correlations are functions of input levels and thus vary across firms.
11
Our CEEDD dataset also covers all unincorporated firms in Canada. Unincorporated firms in
Canada are typically small size businesses owned by self-employed individuals. We do not include
these firms because they do not report capital stock. According to Baldwin and Rispoli (2010),
unincorporated firms account for 9.5% of the total GDP in the economy in 2005, with the sharedeclining since mid-1990s.

11and costs of goods sold net of capital amortization. We measure the wage bill as
the total worker compensation used to construct the National Account. All nominal
monetary values are converted to 2002 real Canadian dollars.
We link firms to their owners using administrative records from the Shareholder
Information in Corporate Tax Files. In particular, Schedule 50 of T2 form provides
information on shareholders who own at least 10% of shares, the percentage of shares
owned by each shareholder above the 10% threshold, and the type of shares owned(common or preferred). Statistics Canada also tracks chained ownership by individuals (e.g., if individual A owns some shares of firm B and firm B owns some of firm C)
and constructs a dataset of ultimate individual shareholder. We merge the ownership
information and demographic characteristics (i.e., age and gender) to the firm panel
dataset and calculate total equity wealth for each individual, measured as the shareweighted sum of the book value of all holding firms.
To construct the estimation sample, we start from firm-year observations with
non-missing values in total revenue, capital stock, wage bill, intermediate input, andindustry code. For the first few firm-year observations of capital stock the PIM
method relies heavily on the initial available book value. We then further drop theobservations with outlier factor shares. Specifically, we drop firm-year observations
with (i) wage bill-to-revenue ratio below the 1st percentile or above the 99th percentile, (ii) wage bill-to-value added ratio below the 1st percentile or above the 99th
percentile, (iii) intermediate input-to-revenue ratio greater than 0.95 or smaller than0.05, and (iv) capital-to-revenue ratio above the 99.9th percentile. This sample selection leaves us with 4.3 million firm-year observations and 620 thousand firms with anaverage of 6.9 observations per firm. Summary statistics are reported in Table I. 12
US Manufacturing Sector. As a robustness exercise, we perform similar analysis
using data from from the US Economic Census and the Annual Survey of Manufactures (ASM). Our sample considers all manufacturing plants available in the dataset
between 1978 and 2019. We measure (revenue) productivity for all plants with information on total value of shipments, real capital stock, wages of plant workers, andinformation on intermediate inputs and materials in 2019 US dollars.
12

The firm-level distributions of these variables are similar to economy-wide microdata in othercountries. E.g., Chan et al. (2024) find very similar distributions of log revenues and inputs in
administrative data covering the entire Danish private sector.

12

Table I – Summary Statistics
Revenue
Intermediates
Wage Bill
Capital Stock

Mean
13.73
13.18
12.35
11.29

Median
13.54
12.99
12.19
11.26

St.dev P10
P50
P90
P99
1.39 12.13 13.54 15.60 17.75
1.52 11.41 12.99 15.21 17.46
1.30 10.82 12.19 14.07 16.04
1.82
9.02 11.26 13.54 15.97

Note: Table I shows cross-sectional moments of the distribution of log revenues, log intermediate
inputs, log wage bill, and log capital stock. All variables in 2002 Canadian dollars. The total numberof observation is 4.3 million firm-years.

4

Empirical Results
In this section, we apply the GNR method to each of the 23 2-digit NAICS in-

dustries in our Canadian administrative data (see Table OA.3 for a list of these industries), which provide estimates of output elasticities of inputs for all the firm-year
observations in our sample along with their TFPs. Below, we first present unconditional moments of these estimated parameters from this pooled sample. We then show
how they vary over the firm size distribution as well as over the life cycle. Although
GNR analytically proves identification for their method, we present the features ofthe data that have pronounced effects on our parameter estimates as an informal
identification argument. We further relate our results to wage and wealth inequality
as well as lifecycle growth of firms.

4.1

Unconditional Heterogeneity in Production Technologies

We start by discussing cross-sectional moments of the distribution of firm technologies. In particular, we first calculate within-industry moments from the distribution of firm-level estimates and then we average across industries. Our results,
shown in Table II, reveal considerable heterogeneity in the estimated productivity,
output elasticities, and returns to scale across firms. Starting with within-industry
TFP differences, we find that the 90th -to-10th percentile gap (P90-P10) of firm-level
TFP is 0.31. This implies that a firm at the 90th percentile produces about 36.2%
more output than the firm at the 10th percentile, with the same inputs and holding
output elasticities constant. This is substantially lower than previous estimates of
productivity dispersion even for narrow 6-digit industries in Canada and the USA,13

Table II – Distribution of Production Function Estimates
Mean
TFP
RTS

St. dev

P10

P50

P90

P99

Panel A: Main Estimates—
0.16-1.71 -1.55
0.97
0.04
0.93 0.97

-1.40
1.01

-1.06
1.07

0.60
0.33
0.03

0.79
0.49
0.07

0.99
0.64
0.12

0.61
0.28
0.09

0.85
0.50
0.51

0.93
0.72
2.16

Panel B: Output Elasticities
Intermediates
Labor
Capital

0.61

0.14

0.33

0.14

0.03

0.03

0.43
0.15
0.00

Panel C: Input SharesIntermediates
Labor
Capital

0.61

0.18

0.29

0.15

0.23

0.48

0.36
0.11
0.01

Note: Table II shows cross-sectional moments of the distribution of firm-level log TFP, returns toscale (RTS), and the elasticities of output with respect to Intermediate inputs, labor and capital.
To obtain these estimates, we apply the method of Section 2 within two-digits NAICS and calculatecross-sectional moment within the same cell. Then we average across all estimated values weighting
by the number of observations in each cell. The total number of observation is 4.3 million firm-years.

which typically find P90-P10 TFP ratios closer to 2 (see for instance, De Loecker
and Syverson (2021) and Syverson (2011)). The difference stems from the use of aflexible non-parametric production function estimation, and from using the wage billas our measure of labor input rather than the number of workers or the total numberof hours (see Fox and Smeets (2011)). Using a similar method, Chan et al. (2024)
estimate TFP for the entire Danish private sector and find a P90-P10 ratio of 0.54.
We find similar results for manufacturing plants in the US.
The average return-to-scale (RTS) is estimated to be 0.97 with a P90-P10 of 0.07.
This implies that with a 1% higher input bundle, the firm at the 90th percentile
produces 7.3% (= e0.07 − 1) more output than the firm at the 10th percentile, holding
productivity constant. Interpreted as deviations from constant returns to scale, thesedifferences are large. For example, with Cobb-Douglas production functions in an1efficient economy, the elasticity of optimal firm output to firm TFP is 1−RT
. ThisS

elasticity is 3.3 times larger for a firm with RTS of 0.97 compared to a firm with

14

Table III – Within-Industry Variance of Elasticity Estimates
RTS
K-elasticity L-elasticity
Fraction of variation (variance) within industry
2-digit NAICS 23.3%
61.9%
65.9%
4-digit NAICS 22.0%
57.8%
58.6%
Standard deviation within industry
2-digit NAICS 0.052
0.031
0.152
4-digit NAICS 0.051
0.030
0.143

I-elasticity
72.7%
63.6%
0.149
0.139

Note: Table III shows the within-industry (NAICS2 and NAICS4) variations for the three output
elasticities and return-to-scale estimates. It includes both the within-industry fraction of total
variance and the within-industry standard deviation.

RTS of 0.90. Around 75% of firms have return-to-scale below 1, that is, they operatedecreasing return-to-scale technologies. Consistent with earlier literature (e.g., Basuand Fernald (1997); Ruzic and Ho (2023); Gao and Kehrig (2017)) we find large
differences in average RTS across industries (Table OA.3), ranging from 0.59 (forhealthcare) to 1.03 (for Management of Companies and Enterprises). On top ofthese large between-industry differences, we further find significant within-industry
variation. For example, average within industry P50-P10 and P90-P50 are around0.03 and 0.04, respectively. In fact, as shown in Table III, a quarter of the total RTS
variance in our data is accounted for by within-industry differences.
By construction, heterogeneity in RTS is explained by the significant heterogeneityin estimated output elasticities as shown in Panel B of II. The output elasticity withrespect to intermediate inputs has the highest average value of 0.61, followed bylabor with an average elasticity of 0.33, and capital with an average elasticity of
0.03.13 Moreover, labor and intermediate input elasticities are also more dispersed
across firms, compared to the capital elasticity. For example, the average within
industry P90-P10 for intermediate, labor, and capital inputs are 0.35, 0.35, and 0.07,
respectively. Furthermore, variance decompositions reveal that more than 60% of theoverall variation for each output elasticity is explained by within-industry differences,meaning that within-industry heterogeneity accounts for a larger fraction of the overall
13

We find an elasticity of capital that is much lower than typical estimates. This is because
our measure of capital stock only considers plants, equipment, and buildings (tangible capital) andignores any other form of capital such as intangible capital, inventories, cash and so on, which areincluded in the aggregate measure of capital.

15

Table IV – Correlation of Output Elasticity EstimatesIntermediates
Labor
CapitalBetween-Industry Variation
Intermediates Labor Capital
-0.3
-0.7
-0.3
-0.4
-0.7
-0.4
-

Within-Industry Variation
Intermediates Labor Capital
-0.9
-0.4
-0.9
0.0
-0.4
0.0
-

Note: Table IV shows the correlation coefficients of the output elasticity estimates of the three inputs.
The between-industry results show the weighted correlation of the average output elasticities of each
NAICS2 industry, the within-industry results demean the output elasticities at NAICS2 level.

firm-level variation for output elasticities compared to RTS. This is partly because theintermediate input elasticity is negatively correlated with the capital and the labor
elasticity within industries (see Table IV).
One potential concern is that the observed dispersion in TFP, output elasticities,
and RTS is driven by transitory movements of the same firm over the production function rather than fixed firm characteristics. To evaluate whether this is the case, we
run a panel regression of each of our estimates on firm size, firm age, time dummies,
and firm fixed effects. Intuitively, if the dispersion is mostly due to fixed differences
between firms, the firm fixed effects should absorb most of the variation in our estimates. We find that this is the case: of a standard deviation of RTS of 0.052, 86%
is explained by firm fixed effects after controlling for firm age and size (the standard
deviation of the FE is 0.045). We find similar results for the rest of our estimates(elasticities and TFP) and in US manufacturing.14 We conclude that the observed
heterogeneity in RTS is mostly explained by permanent between-firm differences,consistent with the model of heterogenous RTS that we describe in Section 5.

4.2

Production Technologies over the Firm-Size Distribution

We now turn to the systematic variation of our estimates over the revenue distribution.
Returns to scale by firm size. Figure 1a shows a binned scatter plot of average RTS by firm revenue for a pooled sample of all firm-year observations acrossindustries. For this figure, we use the firm-year estimates that result from estimating
14

In US data, we find a standard deviation of RTS of 0.058, of which 0.048 is accounted for byfirm fixed effects.

16Figure 1 – Returns To Scale by Firm Size(a) Pooled Sample(b) Within-Industry Variation
Average Within-Industry Returns to Scale

Average Returns to Scale

1.04
1.02
1
.98
.96
.94
0

20
40
60
80
Percentiles of the Sales Distribution

100

.08
.06
.04
.02
0
-.02
0

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

Notes: Figures 1a and 1b show the average RTS within 5%-quantiles of the firm-revenue distributions. In Panel (B),RTS are demeaned by industry averages.

production functions within 2-digit NAICS. We find that firms in the bottom two
fifths of the revenue distribution have, on average, similar (decreasing) RTS of about
0.94. As we move to higher percentiles of the revenue distribution, however, RTS
increases monotonically and strongly by firm revenue from 0.94 for firms below the40th percentile to 1.04 for those in the top 5%. These differences imply that firms
in the top 5% of the revenue distribution produce 11% more with a 1% higher input
bundle compared with those below the 40th percentile, holding TFP constant.15
The variation in Figure 1a reflects within- and between-industries heterogeneity. For example, manufacturing firms are, on average, larger and therefore overrepresented in the upper end of the revenue distribution, and manufacturing industries also have higher average RTS. Therefore, some of the overall variation is driven
by heterogeneity across industries. To isolate the role of within-industry differences,
we rank firms into 20 quantiles within their industries according to their revenues
and demean firm RTS by industry averages. Figure 1b shows average demeaned RTS
over the within-industry revenue distribution. Again, firms below the median havetechnologies with similar RTS, whereas above the median RTS increases steeply byfirm revenue: the average RTS of firms in the top 5% of the revenue distribution is15

Note that RTS is not fixed over time and firms are still subject to adjustment costs. Therefore,
the fact that some firms have increasing returns to scale does not necessarily mean that they canincrease their supply indefinitely. Furthermore, other studies commonly estimate RTS to be above 1
for some industries or firms (e.g., Gandhi et al. (2020) and Demirer (2020) find average RTS above
1 across multiple industries and countries).

17Figure 2 – Intermediate Input Elasticities by Firm Size(b) Intermediate Inputs Share.1.1
Average Intermediate Inputs Share

Average Intermediate Inputs Elasticity(a) Intermediate Inputs Elasticity

.05

0

-.05

-.1

.05

0

-.05
0

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

0

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

Notes: Figure 2 shows the average estimated factor elasticities within 5%-quantiles of the revenue distribution; Figure
2b shows the intermediate inputs revenue share. In both cases we demean the within-quantile average by NAICS2
industry averages.

8 p.p. higher relative to the average RTS of firms in the bottom half of the revenue
distribution. This variation is almost as large as the variation in the pooled sample(10 p.p.). Thus, we conclude that most of the variation in RTS by firm size is due towithin-industry differences.
Output elasticities by firm size. As we discussed in Section 2, we measure RTS
as the sum of output elasticities with respect to inputs. Therefore, the significant
increase in RTS is driven by either of these inputs or a combination of them. Our
analysis shows, however, that it is the intermediate input elasticity that entirely
accounts for the positive relation between RTS and firm revenue. As shown in Figure
2a, the intermediate input elasticity monotonically increases from -0.09 (relative tothe industry average) for the bottom 5% of firms by revenue, to approximately zerofor firms around the median, and to 0.09 for the top 5% of firms. This 9 p.p. gap
in intermediate input elasticities between the top 5% and median firms thus explains
all of the 8 p.p. gap in RTS in this range. Furthermore, Figure 2b shows thatthe corresponding variation in the intermediate input revenue share is very similar—
with larger firms spending more on intermediate inputs relative to small firms as a
portion of their revenue—as expected since our estimation treats them as flexible
input. This finding also highlights that estimating gross output production functions
is essential: the sum of capital and labor elasticities is declining across the firm revenue
distribution, indicating that the use of value added production functions can lead to18

Figure 3 – Returns To Scale and Elasticities of Intermediate Inputs in Manufacturing(a) Returns to Scale
USA

Canada

.06
.04
.02
0

Canada All

.3

-.02

USA Manuf.
Canada Manuf.

Average Intermediate Input Elasticities
.4
.5
.6
.7

Average Returns to Scale.08(b) Intermediate Input Elasticity

0

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

0

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

Notes: Figure 3a shows the average RTS within percentiles of the sales growth distribution demeaned by industry
averages for the Canadian and US manufacturing sectors. Canadian results shown within 5%-quantiles of the revenue
distribution. Figure 3b shows the average intermediate input elasticity for the US and Canadian manufacturing sectors
and for the entire Canadian private sector within percentiles of the sales distribution.

misleading results.16
Total factor productivity by firm size. Intuitively, one would expect that firms
at the top of the firm size distribution are also the most productive. To see if thisis the case, we rank firms into 100 bins according to their RTS within each industry.
Then, we measure the relative TFP of a firm as its TFP rank within these narrow bins.
Figure 4 shows the average TFP percentile across the firm revenue distribution forthe pooled sample. Similar to previous literature (see Leung et al. (2008) or Baldwin
et al. (2002)), we find that relative TFP increases by firm size until the top decileof the revenue distribution. For the top 10% of the largest firms, however, relative
TFP flattens out. This contrasts with the previously documented pattern of RTS byfirm size, which increases in a convex fashion at the top of the revenue distribution(Figure 1). Therefore, we conclude that the largest firms tend to feature the highest
RTS, and not necessarily the highest TFP as commonly assumed.
4.2.1

Robustness Checks

US-Manufacturing. Our results are not driven by particular features of the Canadian economy but are also present in other countries and industries. In particular,
16

We show capital and labor elasticities in Appendix Figure OA.2. In Canada, both are, on
average, declining in firm size. In US manufacturing, the labor elasticity is also declining, while thecapital elasticity is u-shaped in firm revenue.

19

our main results hold within the US manufacturing sector as well. First, Figure 3
displays the firm-level RTS relative to its industry average, showing that RTS is ushaped with respect to firm revenue, with a larger rise at the top, increasing about 9
p.p from the middle 50th to the top 1% of the revenue distribution. For comparison,
we also add a corresponding series for the Canadian manufacturing sector to thisfigure, which shows a similar pattern. Second, most of the increase is again due to asignificant rise in the output elasticity of intermediate inputs, that rises from 0.4 at
the bottom of the firm-size distribution to around 0.55 for the largest firms within
US manufacturing.
Cobb-Douglas specification. One additional concern is that our results are driven
by the particular method we use in our estimation. To address this concern, we first we
re-estimate the production function by imposing homogeneous relative factor elasticities while still allowing for RTS differences. The basic idea is to isolate heterogeneity
in RTS regardless of different relative output elasticities. In particular, we estimate
the production function by restricting to homogeneous relative output elasticities
while allowing for heterogeneity in RTS, that is, we assume the production function
γK γL ηjt
. Then, the returns-to-scale parameteris specified as Yjt = eνjt · MjtγM Kjt
Ljt
ηjt is estimated from the first stage of the estimation procedure described in Section
2. Similarly to the baseline results, the Cobb-Douglas series in Figure 5 show that
RTS increases with firm size by about 10 p.p., with a stronger increase in the bottom
half and a weaker increase in the top half of the revenue distribution relative to ourbaseline results.
Clustering specification. In a second robustness exercise, we re-estimate firms’
production technology within clusters of firms with similar characteristics. The motivation for this exercise is that we ideally want to estimate a separate production
function for each firm, but this is econometrically not feasible. Clustering similarfirms is feasible way of approaching this ideal. In practice, we cluster firms using
information on their average levels and growth rates of output, capital stock, labor
expenditure, and intermediate input expenditures. We standardize these firm-levelvariables and then apply the k-means clustering algorithm, with 20 clusters. We further impose that a firm belongs to only one cluster throughout its lifecycle. Finally,
we then estimate the nonparametric production function by each cluster. After esti-

20

30

Average TFP Percentile
40
50

60

Figure 4 – Firm Size and Productivity

0

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

Notes: Figure shows the average firm TFP rank within percentiles of the within-industry revenue distribution. The
TFP rank is calculated within percentiles of the RTS distribution.

mation, we sort firms by revenue within an industry, as we do in our baseline results.
The Cluster series in Figure 5 shows that while these estimates are less smooth along
the revenue distribution, the main patterns are as in the baseline results: RTS are
similar within the bottom half of firms, and then increase by close to 10 p.p. fromthe median to the top 5% of firms within an industry.
Intangible capital specification. Finally, we include a measure of intangibles inour measure of the capital stock and re-estimate firms’ production functions. In theory, including intangibles in firms’ capital stock changes measured productivity, theoutput elasticity of capital, and therefore RTS. In particular, if larger firms invest
disproportionally more in intangible capital, then excluding it from the capital stock
measure leads to underestimation of the capital elasticity (and of RTS) and overestimation of TFP for larger firms. Indeed, the Intangible Capital series in Figure 5
shows that the positive relation between firm size and RTS becomes even stronger
when including intangible capital.17
Ranking firms by employment or value added. Appendix Figure OA.3 shows
our main findings when ranking firms, within industry, by employment or value added
instead of by revenue. The patterns for RTS are very similar. However, the output
elasticities show different patterns: firms with high employment or high value added
tend to have higher labor elasticities, whereas the intermediate input elasticity shows17

The positive relation between firm size and conditional TFP percentile remains unchanged,
perhaps due to the fact firms that have the same RTS also have similar intangible intensities.

21

only a small uptick for the largest firms. This is somewhat mechanical, as we except
high employment or high value added firms to be labor-intensive by construction of
the ranking. For this reason, we prefer to rank firms by firm revenue—a factor-neutral
approach—in our main approach. It is reassuring that the pattern for RTS by firm
size is very stable across the three ranking methods.
Markups. Our RTS estimates are based on revenue elasticities of the three inputs.
One may be concerned that the positive relation between RTS and firm size could be
driven by markup variation (e.g. De Loecker et al. (2020)). Ideally, we would want
to separately estimate physical output elasticities and markup. However, doing sowould require firm-level information on output prices and physical quantities, which
we do not have in the Canadian or US datasets. Even with such output price andphysical quantity information, it is challenging to estimate the physical elasticity for
multi-product firms, in the absence of information on product-specific inputs. It is
important to note that if larger firms charge higher markups—as implied by models with oligopolistic competition (Atkeson and Burstein, 2008) or with monopolisticcompetition and log-concave demand systems (Edmond et al. (2023))—then physicalRTS would be increasing even more with firm size compared to measured revenue
RTS.18 Indeed, when we estimate firm-level markups in our data following the approach of De Loecker and Warzynski (2012), we find that markups increase with firmsize. Specifically, we adopt the value-added and translog production function specification of De Loecker and Warzynski (2012) and conduct the estimation by industry.
Figure OA.1 in the Appendix shows that, on average, markups monotonically increase with firm revenue, consistent with De Loecker et al. (2020). These theoreticaland empirical considerations reinforce our interpretation of measured RTS differences
along the firm-size distribution as representing differences in production technologies.

4.3

Identification: Output Elasticities and Input Shares

Typically, estimates of output elasticities of inputs reflect their corresponding revenue input shares. In fact, for Cobb-Douglas production functions, output elasticities
are exactly equal to (average) input shares. Our specification is more flexible than
Cobb-Douglas, and the GNR method does not solely rely on the first order condi18

It can easily be shown that the physical output elasticity is the product of revenue elasticity
and markup.

22

Figure 5 – Returns to Scale and Firm Size for Different Specifications-.05

Average Returns to Scale
0
.05
.1

.15

Cobb Douglas
Cluster
Intangible Capital

1

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

Notes: In the “Cobb-Douglas” specification, we estimate the production function by restricting to homogeneous
relative output elasticities while allowing for heterogeneity in RTS. In the “Cluster” specification, we apply the k-meansclustering algorithm (20 clusters) and estimate the nonparametric production function within clusters. Intangible
capital is constructed using PIM. In all specifications, we sort firms based on sales within industry, and RTS isdemeaned by industry averages.

tions (FOCs) of profit-maximizing firms. Nevertheless, output elasticities tend to bepositively correlated with the respective factor shares, as we discuss here.
Figure 6 shows (demeaned) output elasticities for all three inputs, when orderingfirms by their intermediate input shares (Figure 6a), by their labor shares (Figure 6b),and by their capital shares (Figure 6c).19 For all three inputs, the respective output
elasticity is positively correlated with the corresponding input share. Intermediate
input-intensive firms have higher intermediate input elasticities, labor-intensive firms
have higher labor elasticities, and capital-intensive firms have higher capital elasticities. For intermediate inputs, the positive correlation is very strong, which is expected
since we treat them as flexible input and we use the firm’s FOC to estimate the intermediate input elasticity (see equation (1)). On the other hand, for labor and capital,
our estimation does not rely on firms FOCs. Yet, we find that their input shares arealso strongly positively correlated with their corresponding output elasticities. These
results suggest that heterogeneity in output elasticities, therefore in RTS, reflects
differences in input shares.20
19

In the Appendix, we plot the relationship between input shares and firm revenue in Figure
OA.4. We find that the intermediate input share increases with firm revenue for the entire Canadianprivate sector, within manufacturing, and within manufacturing in the US. Instead, labor and capitalshares decline by firm revenue.
20
These findings suggest that high-RTS firms should have relatively low profit shares. This isindeed the case: Figure OA.5 in the Appendix shows that, on average across firms, the EBITDA-

23

Figure 6 – Output Elasticities and Factor Shares of Revenue

Average Output Elasticity
-.2
0
.2

Average Output Elasticity
-.2
0
.2

.4

(b) Labor share.4(a) Intermediate input share

Labor

CapitalIntermediate Inputs

-.4

-.4

Intermediate Inputs
-.4

-.3
-.2
-.1
0
.1
.2
Within-Industry Intermediate Input Share

.3

-.3

-.2

Labor

Capital

-.1
0
.1
.2
Within-Industry Labor Share

.3

-.1

Average Output Elasticity
-.05
0

.05

(c) Capital share

Intermediate Inputs
-.05

Labor

Capital0.05.1Within-Industry Capital Share.15

Notes: Figure 6 shows the relation between the input revenue shares defined as the ratio between the total cost ofintermediate inputs, the total wage bill, and the total value of capital stock, divided by firm revenue, and the estimatedoutput elasticity. Firms are ordered by the respective factor shares on the horizontal axis. The vertical axis shows
averages of estimated output elasticities, demeaned within 2-digit NAICS industry.

4.4

Firm Lifecycle Profile

Intuitively, heterogeneity in returns to scale has important implications not only
for the firm size distribution, but for the evolution of each firm over the life-cycle. In
particular, one would expect that firms with high RTS tend to grow faster and reach ahigher optimal size relative to firms with the similar TFP but lower RTS. To examine
these life-cycle patterns, we construct a balanced panel of firms and group thembased on their production function characteristics when they entered the economy.
Specifically, we construct a sample of firms which i) are born between 2002 and 2005,
revenue ratio correlates negatively with RTS.

24

Table V – Probit Regressions of Firm Exits(1)
T F P P ercentile -0.001***(0.000)RT S-0.600***(0.020)N
4.1M
ConstantY
Industry FE
YFirst-difference
Pseudo R2
0.010

(2)
0.005***(0.000)-4.527***(0.088)
3,.4M
Y
Y
Y
0.018

Note: Robust standard error reported. We first-difference both regressors in Column (2).

and ii) are observed for at least 12 consecutive years in the sample. We group firms
based on their initial RTS demeaned at the industry level and initial TFP. Figure 7
shows the average log revenue, also demeaned at the industry level, against firm age
for different firm groups. We find that firms with high initial RTS and TFP both startwith higher revenue relative to other firms of the same age within the same industry.
More interestingly, firms with higher initial RTS grow significantly faster than thosewith low initial RTS. Firms in the top 10% of the initial RTS distribution grow about
30 log points over the next 10 years, whereas those in the bottom 90% grow, on
average, only about 20 log points. This finding corroborates our interpretation thatfirms with high measured RTS operate technologies that are more scalable, allowing
them to grow significantly more over their lifecycle.
In contrast, Figure 8b shows that firms that enter with high TFP, while initiallylarger, do not grow faster than other firms in the same industry.21 In fact, higher
initial TFP is associated with slightly smaller subsequent firm growth rates. Thispattern is consistent with TFP as a mean-reverting process and might explain whysome firms do not expect to growth, as documented by Hurst and Pugsley (2011). Our
results indicate that these firms, although highly productive, might be characterized
by low RTS, and hence small optimal size.
The previous results focus on the life-cycle patterns of surviving firms. We also
study how RTS and TFP heterogeneity affect firm exit. Specifically, we estimate a21

By construction, the three TFP groups in Figure 8b all have identical average RTS.

25

Figure 7 – Life cycle of firms starting with different RTS and TFP(b) TFP-.2

3

Average Log Revenue
-.2
0
.2

RTS Groups
Top 10%P50 to P90
Below P50

TFP Groups
Top 10%P50 to P90
Below P50-.4

Average Log Revenue
-.1
0
.1

.2

.4

(a) Returns to Scale

6

9
12
3
6
9
12
Firm Age
Firm Age
Notes: Figures 8a and 8b compares the life-cycle profile of revenue between firms with different initial RT S and T F P .
It is constructed using a balance panel of firms which 1) are born between 2002 and 2005, and 2) survive for at least
12 years. We demean firms’ initial RT S at the NAICS2 industry level and construct the T F P as described before.
We bin firms into three groups based on their demeaned initial RT S in Figure 8a and three groups based on initial
T F P in Figure 8b. Firm log revenue is also demeaned at the NAICS2 industry level.

probit regression of firm exit on TFP percentile and RTS. The results are reported
in Table V. In Column (1), we use the levels of standardized production parameters,
and in Column (2) we use the first-differenced parameters, controlling for NAICS2
industry fixed effects in both columns. We see that in both specifications, a higher
RTS is associated with a lower probability of firm exit. The effect of TFP on firm
exit is significantly smaller, with opposite signs in levels and first-differences. We conclude that from an ex ante perspective, RTS rather than TFP heterogeneity predicts
differences in firm growth over the life cycle.

4.5

Implications for Wealth and Wage Inequality

We conclude this section by looking at the relation between wealth, wage, andreturns to scale. First, we look at how the production function parameters vary byowner’s equity wealth. The share of self-employed and business owners in Canadaequals 11.7% of individual tax filers. The average number of firms owned by a businessowner is 1.96, with a standard deviation of 2.34. We calculate the equity wealth ofeach individual by aggregating the value of the firms they own, weighted by the ownership shares. Then, for each individual, we calculate their RTS and TFP percentile
by taking a equity-value weighted average of the firms they own. Figure 8 show theresults. Similarly to our results by firm size, we find that high-wealth individuals tend26

Figure 8 – Returns to Scale and Productivity by owner’s wealth(a) Returns to Scale(b) TFP
65

Average TFP Percentile

Average Returns to Scale

1.02

1

.98

.96

.94

60
55
50
45
40

0

20
40
60
80
Percentiles of the Wealth Distribution

100

0

20
40
60
80
Percentiles of the Wealth Distribution

100

Notes: Figure 8 shows the average RTS and TFP by percentiles of owners’ equity wealth distribution.

to own firms that on average have higher RTS. In other words, wealthier firm owners’
production technologies are more scalable. In addition, conditional on within-sector
RTS, the TFP rank is also increasing in owners’ wealth. Given that firms owned byrich households tend to have high RTS and high TFP, it is hard to disentangle thestrength of each for wealth accumulation. We use the model of the next section toshed some light on this issue.
It is well known that large firms tend to pay higher wages to similar workers
than smaller firms (Bloom et al., 2018; Brown and Medoff, 1989). Given our results,
however, it is unclear if this relation derives from large firms having higher TFP or
higher RTS. To distinguish between these two explanations, we rank firms in our
sample by their average wage and compute average RTS and TFP within firm wage
deciles. We find that firms that pay on average more to their workers tend to havehigher RTS—and therefore tend to be larger—than firms with low average wage, as
shown in Panel A of Figure 9. In contrast, there is much less variation in TFP acrossthe firm wage distribution as shown in panel B of Figure 9. We conclude that the
wage-firm size relation is mostly driven by heterogeneity in RTS.

27

Figure 9 – Returns to Scale and Productivity and Firm Average Wage(a) Returns to Scale(b) TFP
65

Average TFP Percentile

Average Returns to Scale

.99

.98

.97

.96

60
55
50
45
40

1

2

3

4
5
6
7
8
Deciles of Firm Average Wage

9

10

1

2

3

4
5
6
7
8
Deciles of Firm Average Wage

910

Notes: Figure 9 shows the average RTS and TFP by deciles of firms’ average wage.

5

Misallocation with RTS Heterogeneity
So far, we have shown sizable heterogeneity in RTS across incorporated firms in

Canada and manufacturing firms in the US. More importantly, larger firms are likely
to have more scalable production technologies. In this section, we show why thisheterogeneity matters in an application to efficiency costs of misallocation, indicating
the potential role of RTS heterogeneity for an array of quantitative questions, suchas optimal taxation of capital, firm recruiting intensity (Gavazza et al., 2018), or firm
cyclicality (Clymo and Rozsypal, 2023). In particular, we embed firm heterogeneity in
RTS (η) and TFP (z) in a quantitative model of entrepreneurship. Our main application compares the aggregate effects of financial frictions in a model with heterogeneity
in both η and z—the (η, z)-economy—to the standard setting with heterogeneity in z
only—the z-economy. To build intuition, we begin by deriving an analytical result ina static endowment economy. We then quantify the mechanism in a dynamic setting.

5.1

Analytical Result in an Endowment Economy

We consider an endowment economy with aggregate factor supply normalized toone, X = 1. There is a continuum of firms i ∈ [0, 1], producing perfectly substitutable
goods. Firms i ∈ [0, χ], where χ ∈ (0, 1), are constrained, facing an input price wedgeτ ≥ 0. Their production technologies are iso-elastic: each firm is characterized by28

a pair of parameters (η, z) with decreasing RTS η ∈ (0, 1) and TFP z, and their
output is given by y = f (x; z, η) = z · xη . The residual fraction of firms i ∈ (χ, 1]
is unconstrained and has constant RTS.22 The following proposition characterizes
misallocation in terms of the share of the economy that is constrained as well as thereturns to scale of constrained firms:
Proposition 1. Consider an interior equilibrium where the output share of constrained firms is below one. Then, up to a second order approximation around thefirst best (τ = 0), the percent output loss associated with τ is given by∆ ln Y (τ ) =

τ2
2
|{z}

Z χ
·
|0

Z χ
wi · di{z }size of friction output share of constrained firms

·
0

|

ηi
wi
·
· di
wj dj 1 − ηi
0
{z
}

Rχ

RT S
avg. 1−RT
constrained firmsSy?

where wi ≡ Y i? denotes the relative output of firm i in the first best.
The proposition states that misallocation is proportional to the size of the friction, the output share of constrained firms, as well as increasing and convex in the(weighted-average) RTS of constrained firms. Therefore, for a given friction, misallocation is more severe when constrained firms exhibit higher RTS. In addition,
due to the convexity of misallocation in RTS, higher dispersion of RTS also translates into more misallocation. Intuitively, a given input price wedge translates intoa larger quantity adjustment when RTS are high—when marginal products decline
more slowly—as constrained firms reduce their inputs more. Therefore, misallocationis larger. Firm TFP, on the other hand, matters only indirectly via the output shareof constrained firms. We illustrate this in Figure 10, which depicts the marginal input
product of firms that would be “large” in the first-best, and which matter the mostfor misallocation: the solid blue line represents the conventional setting where large
firms have high TFP (z̄), the dashed red line represents an economy where theselarge firms have high RTS (η̄). For a given wedge τ , misallocation—the area under
the curve—is larger in the η̄-economy.

22

Alternatively, unconstrained firms exhibit decreasing RTS but there is free entry in that sector,
such that there are constant RTS at the sectoral level.

29

Figure 10 – Efficiency costs in endowment economy

5.2

Quantitative Dynamic Model

We now consider a dynamic economy in the tradition of Quadrini (2000); Cagetti
and De Nardi (2006), where the set of constrained firms emerges endogenously. We
use this model to quantify misallocation in an economy where firms differ in both
RTS and TFP, as in our data, and compare it with the misallocation in an economy
where firms differ only in TFP.
5.2.1

Model setup

Time is discrete and there is a continuum of agents of mass one, who derive log
utility from consumption. They discount the future at rate β̃ and face a constantdeath probability p ∈ [0, 1). Thus, their effective discount factor is β = (1 − p) · β̃and they maximizeXβ t ln(c).

t≥0Agents face an occupational choice between employment as a worker and entrepreneurship, o ∈ {W, E}. A worker’s labor income equals w · h, where w denotes the wage
rate and h efficiency units of labor supply, which follow a first-order Markov process.
Entrepreneurs are price-takers in input and output markets, using labor ` and capital k at rental rates w, R to produce output z · f (k, `)η , where f (·) is a constant
RTS production function. The pair (z, η) denotes entrepreneurial productivity z andscalability of their project η, and follows a joint first-order Markov process. Assetmarkets are incomplete, and agents can invest their wealth a ≥ 0 into an annuity,
paying an interest rate r. Upon death, individuals are replaced by the same number ofnewborn households who start with zero wealth. We parameterize financial frictions
30

by λ ∈ [0, 1], and assume that a fraction λ of total input expenditures needs to befinanced by the entrepreneur’s own wealth. Thus, static profit maximization entails
a net profit ofπ(a, z, η) = max z · f (k, `)η − w · ` − R · kk≥0,`≥0s.t. w · ` + R · k ≤a,λimplying input choices k(a, z, η), `(a, z, η) and output y(a, z, η).23 Thus, the dynamic
agent problem can be written in recursive form as
V (a, h, z, η) = 0

maxa ≥0,c≥0,o∈{W,E}u(c) + β · E[V (a0 , h0 , z 0 , η 0 )]s.t. c + a0 = Io=W · w · h + Io=E · π(a, z, η) + (1 + r) · a.
We assume that there is a competitive financial intermediary, investing in physicalcapital with depreciation rate δ, and issuing the annuities.
5.2.2

Equilibrium

We consider the stationary equilibrium of this model, which is described by a setof prices (r, R, w) such that:
1. Agents optimize, giving rise to decision rules a0 (θ), c(θ), o(θ), k(θ), `(θ), y(θ),
where θ = (a, z, h, η) summarizes the individual’s state, as well as an ergodic
distribution G(θ).
2. The financial intermediary maximizes profits, implying R = r + δ − p · (1 + r).

23

We assume that the friction affects all inputs symmetrically to focus on overall firm size distortions, without introducing additional distortions on relative input use (as would be the case, e.g.,with a collateral constraint on k only). Relatedly, we also do not include intermediate inputs in
our production function here. Including them would tend to magnify the level of efficiency costs offinancial frictions, but not change the comparison between the (η, z)- and the z-economy.

31

3. Given G(θ), all markets clear:ZL≡Zh · dG(θ) =o=WZ`(θ) · dG(θ) (labor market)
Zo=E

1
a · dG(θ) =k(θ) · dG(θ) (capital market)
K≡
1−p
Z
Zo=E
Y ≡ c(θ) · dG(θ) + δ · K =y(θ) · dG(θ) (goods market)
o=E

5.2.3

Calibration

The main idea is to calibrate both the (η, z)- and the z-economy to the same
set of observable moments of the firm size distribution and entrepreneurship dynamics. Before going there, we briefly discuss fixed common parameters. We set the1
death probability to 80, corresponding to an expected life expectancy of 80 years.24

We use a Cobb-Douglas production function (f ) with capital share α = 0.4 anddepreciation rate δ = 0.05. We model the process for labor efficiency units h as alog-normal AR(1) with autocorrelation of 0.9, cross-sectional standard deviation of
σ2

1.3, and normalized mean µh = − 2h , which we estimate using Canadian data on
individual post-tax earnings. We calibrate both economies at λ = 0.3, meaning that
30% of input expenditures need to be financed with the owner’s wealth, and thenvary λ in counterfactuals.25(z)-Economy: We jointly calibrate a set of five parameters (β, η, σz , ρz , ξz ) to matcha set of six empirical moments as summarized in the right column of Table VI. We now
provide intuition on how the parameters are identified: the effective discount factor β
affects the aggregate capital-output ratio. The (common) RTS parameter η is closely
related to the population fraction of entrepreneurs, as it impacts their income share.
2

We model the z-process as log-normal AR(1) with normalized mean µz = − σ2z . Its
auto-correlation (ρz ) affects the transition into (and out of) entrepreneurship. Finally,
the firm size distribution is closely related to the cross-sectional dispersion of z(σz ).
We also stretch out the top 1% of the z- distribution with a Pareto tail, where ξz
24

The death rate affects in particular wealth accumulation at the bottom of the wealth distribution, as newborns enter with zero wealth. The bottom 50% wealth share equals 3.4% in the(η, z)-model and 2.2% in the z-model, in the ballpark of the value for Canada of 4.9%.
25
Defining the debt d of entrepreneurs as d = max {0, k − a}, the aggregate debt-to-capital ratio
equals 81% in the (η, z)-model and 71% in the z-model, in the range of the value for Canada of 70%.

32

Table VI – Dynamic model: Targeted moments and calibrated parameters
Data
A. Targeted moments
Fraction entrepreneurs
Transition rate W→E
Top 10% revenue share
Top 1% revenue share
Top 0.1% revenue share
RTS: Top 5% vs Bottom 50% (by revenue)
Capital-output ratio
B. Internally calibrated parameters
Mean RTS
Standard deviation RTS
Standard Deviation TFP
Persistence TFPPareto tail TFPCorrelation (z, η)Discount factorModel(η, z)-economy z-economy

0.117
0.021
0.799
0.522
0.282
0.083
2.970

0.117
0.021
0.799
0.522
0.282
0.073
2.971

0.117
0.021
0.804
0.515
0.284
0*
2.970

µη
ση
σz
ρz
ξz
σz,η
β

0.794
0.049
0.571
0.952

0.683

-0.219
0.889

0.910
0.970
2.880
0.902

Notes: Steady state calibration of the (η, z)- and z-economy (both at λ = 0.3). * not targeted.

denotes the tail coefficient. This allows the model to match the right tail of the firm
size distribution as well. Overall the fit of the model to the targeted moments is
excellent.
(η, z)-Economy: In a nutshell, we repeat the calibration of the z-economy, and in
addition discipline the heterogeneity in η by matching the observed dispersion of RTS
along the revenue distribution (middle column of Table VI). Concretely, we model η
as a truncated normal AR(1) in the interval (0, 1) with parameters (µη , ση , ρη ). We
ex ante fix the auto-correlation to a high value of ρη = 0.98, which is the persistence of RTS in our empirical analysis. Its mean µη again controls the fraction ofentrepreneurs. Its cross-sectional standard deviation ση is closely linked to the difference in the average RTS of the top 1% vs. the bottom 50% of firms ordered by revenue.
We also allow z and η to be correlated by setting log TFP ln z = z̃ + ση,z · σσηz · (η − µη ),2where z̃ follows a normal AR(1) process with parameters (σz , ρz , µz = − σ2z ). Intuitively, both ση,z and σz affect moments of the firm size distribution: e.g., if theempirical dispersion in RTS was small, then a high residual TFP dispersion σz would
be required to match the high revenue concentration of firms. Conversely, if theobserved dispersion in RTS was very high, then the calibration would infer a more
33

negative value for the correlation parameter ση,z . We calibrate the TFP parametersresidually in this way, instead of directly feeding the estimated joint distribution of ηand z into the model, because when firms operate different production functions thatdiffer in η, inferred relative TFP depends on the unit choice. This is the case in ourmodel setting, as well as in some of our empirical approaches (when we cluster firms,
such that firms within an industry do not all share the same production function).26
In sum, we calibrate six parameters to match seven moments. This model version
also provides an excellent fit to the data. It is noteworthy that it does not require a
Pareto tail in z to replicate the right tail of the firm size distribution; the observed
heterogeneity in RTS combined with a log-normal z suffices.
5.2.4

Quantitative findings

The two model economies are observationally equivalent in terms of the fraction ofentrepreneurs, the persistence of entrepreneurship, the firm size distribution, and theratio of wealth (capital) to output. We now evaluate the efficiency losses associatedwith the same financial friction in both economies. Figure 11 below compares theoutput losses induced by raising the financial friction from the unconstrained case of
λ = 0 all the way up to λ = 1, across stationary equilibria of the two model versions.
We see that, e.g., if entrepreneurs need 30 cents of their own wealth to finance a dollar
of input expenditure, the (η, z)-economy with heterogeneity in both TFP and RTS—
disciplined by our empirical estimates—features an output loss of 16.3 log points,
relative to the frictionless economy. However, in the conventional z-economy that
imposes homogeneous RTS, the output loss is significantly lower, at 7.4 log points.
26

To see this, consider a simple example of two firms, j = 1, 2, that differ in their RTS (η1 > η2 )
ηand TFP levels. Assume that their production function is given by yj = zj · `j j , and that RTS (aunit-free elasticity), as well as input and output levels are known. Then the ratio of their measured
TFP is given by −η1
z1
y1
`1
1
=
·
·
η1 −η2 ,z2y2`2`| 2 {z }unit dependance

which depends on the level of the input `; therefore, it depends on the unit of account. In particular,
relative TFP of the higher-RTS firm is inversely proportional to the unit of account. Therefore,
depending on the choice of unit (e.g., hours vs. full-time equivalents), for the same data, one canfind any relationship (both in sign and magnitude) between TFP of these two firms. Therefore,
when firms operate different production functions that differ in RTS, relative TFP does not havethe usual cardinal interpretation. For a similar argument about unit dependance in the context ofhouse price elasticities, see Greaney (2019).

34

Figure 11 – Output losses from financial friction in dynamic model
1

0.95

0.9

0.85

0.8

0.75
0

0.2

0.4

0.6

0.8

1

Therefore, incorporating realistic heterogeneity in RTS, while otherwise matching thesame observables, amplifies the output losses due to the financial friction by 120%.
To understand this finding, we additively decompose the total log output loss into
three terms: (i) static misallocation of production factors holding fixed occupational
choice, (ii) misallocation of talent across occupations, and (iii) under-accumulationof capital. Panel A of Table VII shows that static misallocation of production factors
across firms contributes 9.5 log points in the (η, z)-economy, more than half of the total
GDP loss, and almost twice as much as in the conventional z-economy. This is thechannel highlighted in our analytical discussion in Section 5.1, and our quantitative
findings are in line with Proposition 1. Under-accumulation of capital contributes the
majority of the remaining output loss. Misallocation of talent across occupations is
also a little bit larger in the (η, z)-economy, but rather small in both economies: theλ-friction mainly misallocates production factors across firms, but does not distortthe decision to enter entrepreneurship vs. employment much. We deliberately used arather conservative, simple and transparent calibration strategy, avoiding, e.g., fixed
costs of entry and exit, which would tend to magnify the overall importance of thatchannel.
Our findings are robust to alternative approaches of making the financial friction
comparable in both economies. In our benchmark scenario we raise λ from 0 to 0.3 inboth economies. Panel B of Table VII shows that our findings are even stronger wheninstead equating observable moments such as the aggregate debt-to-capital ratio or35

Table VII – Dynamic model: Decomposition of output losses and robustness(η, z)-economy z-economy
A. Decomposition of output losses going from λ = 0 to λ = 0.3
Total log GDP loss
16.3
7.4
. . . due to misallocation of production factors
9.5
5.0. . . due to misallocation of talent
0.6
0.5. . . due to K accumulation
6.2
1.9
B. Total log GDP loss: robustness to alternative comparisons
Equating λ16.3
7.4
Equating aggregate debt/capital ratio
24.9
7.4
Equating dispersion in log marginal products
20.1
7.4
Notes: Panel A additively decomposes the total (steady state) log GDP loss going from λ = 0 to λ = 0.3 into: (i)misallocation of production factors (starting from the λ = 0.3 steady state, fixing K, L, and occupational status,
allowing for efficient reallocation of K, L across firms); (ii) misallocation of talent (in addition allowing for efficientchange of occupational status), and (iii) dynamic under-accumulation of capital. Panel B reports the total log GDP
loss in alternative scenarios where we raise λ from 0 to 0.3 in the z-economy, and from 0 to x in the (η, z)-economy,
where x is chosen to match the debt ratio, resp. marginal input product dispersion of the z-economy with λ = 0.3.

the dispersion in log marginal input products. For these exercises, we continue toraise λ from 0 to 0.3 in the z-economy, which generates an aggregate debt-to-capitalratio of 0.708 and a cross-sectional standard deviation of log marginal products of
0.144 in the latter case. We then raise λ from 0 to values of 0.803 (resp. 0.488) inthe (η, z)-economy, in these two scenarios, to replicate the debt ratio (resp. marginal
product dispersion). As a result, the (η, z)-economy generates output losses whichare 172 − 236% larger than the ones in the conventional z-economy.

6

Conclusion
In this paper, we have documented significant heterogeneity in firms’ scalability(their RTS), even within narrowly defined industries. RTS heterogeneity is large,
highly persistent, and systematically linked to firm size: larger firms tend to be thosethat have higher RTS. Our empirical analysis also showed that differences in the
output elasticity of intermediate inputs drive the positive relation between firm size
and RTS, whereas labor and capital elasticities are jointly decreasing in firm size.
The documented RTS heterogeneity has important implications for our understanding of the drivers of firm growth, the determinants of the firm size distribution,and the distributional impact of financial constraints and taxes. To demonstrate this,we developed a quantitative model with firm heterogeneity in TFP—as in typical36

models of entrepreneurship and firm dynamics—and in RTS. We used this model
to evaluate the efficiency and output costs of financial frictions. When large firms
are characterized by high RTS—as we documented empirically—as opposed to high
TFP—the conventional view—efficiency costs are magnified. We provide intuition
for this result in a static setting, and then quantify the mechanism in the dynamic
model. We show that the same friction generates more than twice the efficiency andoutput costs in an economy with both RTS and TFP heterogeneity, as observed inour data, compared to the conventional calibration that loads all observed firm heterogeneity on TFP dispersion. These findings suggest that allowing for realistic RTS
heterogeneity also has important implications for optimal wealth and capital income
taxation.

37

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40

Online Appendix for “Scalable vs. Productive Technologies”
Mons Chan1

A

Guangbin Hong2

Joachim Hubmer3 Serdar Ozkan4
Salgado5

SergioData Appendix
We describe how we construct the variables and the estimation sample in thissection.

A.1

Variable Construction

Revenue We use the revenue measure which is computed by Statistics Canada forconstructing the National Account. This measure is derived by summing up relevantterms from the T2 Corporate Income Tax Return Form terms.
Labor: We use the total worker compensation, which is also computed by Statistics
Canada for constructing the National Account. This measure includes wages, salaries,
and commissions paid to all the workers employed within a year.
Capital: We employ the Perpetual-Inventory Method (PIM) to construct the capitalstock. We make use of information on the first book value of tangible capital observedin the dataset, annual tangible capital investment, and amortization. Specifically,
capital stock K of firm i in year t is computed as Ki,t = Ki,t−1 +Investi,t −Amorti,t , t ≥t0i and t0i is the first year we observe the book value of tangible capital of firm i. The
initial year capital stock K is calculated as the book value of tangible capital net ofaccumulated tangible capital amortization. Tangible investment include investment ofbuilding and land, computer, and machines and equipment. In addition, We constructa capital stock measure that includes intangible capital. We also follow the PIM for
intangibles and make use of information on the book value of intangible capital,
annual intangible capital investment and amortization.

1

Queen’s University; mons.chan@queensu.ca
University of Toronto; g.hong@mail.utoronto.ca3
University of Pennsylvania; jhubmer@sas.upenn.edu
4
St. Louis Fed, University of Toronto; serdar.ozkan@gmail.com
5
The Wharton School-University of Pennsylvania; ssalgado@wharton.upenn.edu
2Intermediates: We measure intermediate inputs as the total expenses not related
to capital and labor. Specifically, it is computed as the sum of operating expenses
and costs of goods sold net of capital amortization. The operating expenses andcosts of good sold variables are also constructed by Statistics Canada to replicate the
National Account, and neither of them encompasses worker compensation.
Firm owner and wealth information: We obtain ownership information fromthe Schedule 50 Shareholder Information of T2 Corporate Tax Files. Schedule 50
provides information of the filing firms on their shareholders with at least 10% of
shares, the percentage of shares owned by each shareholder, and the type of shares
owned (common or preferred). Statistics Canada tracks chained ownership by individuals (e.g. individual A owns a share of firm B and firm B owns a share of firmC) and constructs a tracked share of ownership of firms by each ultimate individual
shareholder. We merge the ownership information to the firm panel dataset and calculate total individual equity wealth as the ownership share weighted sum of the valueof all holding firms. Firm value is calculated as total asset net of total liabilities.
Linked employer-employee information: We obtain linked employer-employee
and earnings information from the T4 Statement of Renumeration Paid form. The
T4 files provide job-level earnings information with individual and firm identifiers,
where a job is defined as a worker-firm pairing. A worker can have more than one T4
records in a year if she works for more than one firms. For multiple job holders, we
keep the job that offers the largest earnings of the year and call it the main job. In
addition, we drop workers with annual earnings from the main job lower than 5000
CAD.

A.2

Sample Selection

We convert all the monetary variables to be denominated in 2002 Canadian dollars.
Several steps are taken to construct the estimation sample. Second, we drop firms
with missing industry information. Second, we drop the first-year observation that
we observe a firm’s book value of tangible capital and the observations before, as
we cannot the the PIM to construct capital stock for these observations. Third, we
drop firm-year observations with missing and non-positive revenue, labor, capital, andintermediate input values. We further drop the observations whose one-year lagged
revenue or inputs are missing or non-positive, as our identification strategy requires
2

using lagged labor input as the instrument. Fourth, we drop the observations withextreme factor shares, that is, the ones with wage bill-to-revenue ratio below the 1st
percentile or above the 99th percentile, with wage bill-to-value added ratio below
the 1st percentile or above the 99th percentile, with intermediate input-to-revenue
ratio above 0.95 or below 0.05, and with capital stock-to-revenue ratio above the
99.9th percentile. This sample selection procedure leaves us with around 4.3 million
firm-year observations.

B

Proof of Proposition 1
W.l.o.g., set the productivity of the CRTS sector to 1. Then, the equilibrium

input price equals 1. Given τ ≥ 0, the input choice and output of constrained firm iare:
xi (τ ) =ηi · zi
1+τ

1
 1−η

i

1
1−ηi

and

yi (τ ) = zi


·

ηi
1+τ

ηi
 1−η

i

.

By market clearing, aggregate input and output of unconstrained firms both equalZ χ
1−xi (τ )di.
0

Thus, we can write the aggregate misallocation loss as
xi (0)di −(yi (0) − yi (τ )) di −∆Y (τ ) = Y − Y (τ ) =

Z χ

Z χ

Z χ

?

xi (τ )di
0

0

Z0 χ



(yi (0) − yi (τ )) − (xi (0) − xi (τ )) di
"
!#
ηi !
1

 1−η

 1−η
Z χ
i
i
1
1
=
yi? ·
1−
−η· 1−
di
1+τ
1+τ
0
{z
}
|
=

0

≡Li (τ )Perform a second-order approximation of Li (τ ) around τ = 0. Since Li (0) = L0i (0) = 0
2

y?

ηi
ηi
and L00i (0) = 1−η
, it follows that Li (τ ) ≈ τ2 1−η
. Using the definition wi ≡ Y i? , theii3proposition follows:Z χ
2∆Y (τ )
ηi1
? τ∆ ln Y (τ ) =
y
·
di
≈
·
i
∗
∗
Y
Y
2 1 − ηi
0
Z χ
ηi
τ2
·
wi ·
di
=
2
1 − ηi
0
Z χ
Z χ
w
τ2
ηi
Rχ i
wi · di ·
=
·
di.
·
2
wj · dj 1 − ηi
0
0
0

C

Additional Figures and Tables
Table OA.1 – Summary statistics for Manufacturing Firms
Revenue
Intermediates
Labor
Capital

Mean
14.15
13.56
12.91
12.03

Median
13.95
13.35
12.77
11.98

St.dev
1.58
1.68
1.49
1.99

P50-P10
1.67
1.76
1.67
2.39

P90-P50
2.31
2.46
2.12
1.87

P99-P50
4.67
4.93
4.10
5.27

Notes: Table shows moments of the distribution of revenues, intermediate inputs, labor, and capital stock in log real
Canadian dollars for the Canadian Manufacturing Sector.. The total number of observations is 436,000.

Table OA.2 – Distribution of production function parameters for manufacturing Firms
Returns to Scale

Mean
1.00

Intermediates
Labor
Capital

0.57
0.40
0.03

Median St.dev P50-P10
1.00
0.02
0.02
Output Elasticities
0.56
0.14
0.16
0.41
0.13
0.17
0.03
0.03
0.03

P90-P50
0.02

P99-P50
0.07

0.18
0.15
0.04

0.37
0.28
0.09

Notes: Table shows moments of the distribution of estimates RTS and output elasticities for the Canadian Manufacturing Sector. The total number of observations is 436,000.

4

Table OA.3 – Average production estimates by industry
Industry
Agriculture
MiningEnergyConstruction

NAICS
N
rtscale melast
11
37,6001.00
0.53
21
16,500
1.00
0.46
22
2,500
1.00
0.59
23
738,300
1.00
0.55
31
69,1001.01
0.61
Manufacturing
32
119,700
1.01
0.59
33
247,100
1.00
0.55
Wholesale Trade41
366,400
0.99
0.71
44
614,400
1.00
0.75
Retail Trade45
185,400
1.00
0.71
48
109,300
0.99
0.58
Transportation and warehousing
49
13,3001.01
0.63
Information and cultural
51
39,2001.00
0.56
Finance and insurance
52
33,6000.65
0.57
Real estate
53
69,1001.01
0.54
Professional Services
54
260,000
0.98
0.48
Management of companies and enterprises
55
27,7001.03
0.59
Administrative and support
56
186,800
1.00
0.53
Education
61
26,7000.98
0.51
Healthcare
62
111,300
0.59
0.40
Arts, entertainment and recreation
71
66,0000.98
0.51
Accommodation and food services
72
552,500
0.99
0.59
Other Services
81427,600
0.77
0.54
Notes: The numbers of observations are rounded to the nearest hundreds.

5

lelast
0.41
0.44
0.34
0.41
0.37
0.38
0.42
0.26
0.22
0.27
0.36
0.33
0.41
-0.05
0.40
0.47
0.39
0.42
0.45
0.05
0.44
0.37
0.16

kelast
0.05
0.10
0.07
0.04
0.03
0.03
0.03
0.02
0.02
0.02
0.05
0.04
0.04
0.13
0.07
0.03
0.05
0.04
0.03
0.14
0.03
0.04
0.06

Figure OA.1 – Estimated Markups and Firm Revenue

Average Markups

.06

.04

.02

0

-.02
0

20
40
60
80
Percentiles of the Sales Distribution

100

Notes: Figures OA.1 shows estimated markup across the firm-size distribution. We follow the value-added translog
production function method as in De Loecker and Warzynski (2012). We estimate the production function by industry.In all figures, we sort firms by within-industry revenue ranks and plot the average within ranks. The figure shows
the markup relative to the industry average.

6

Figure OA.2 – RTS and Output Elasticities for Canada and the US
Canada All

USA Manuf.
Canada Manuf.

Canada All

.3

-.02

Average Returns to Scale
0
.02
.04
.06

USA Manuf.
Canada Manuf.

(b) Intermediate Inputs
Average Intermediate Input Elasticities
.4
.5
.6
.7

.08

(a) Returns to Scale

0

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

0

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

Canada All

USA Manuf.
Canada Manuf.

Canada All

0.3

Average Labor Elasticity
.35
.4
.45

Average Capital Elasticity
.05
.1
.15

USA Manuf.
Canada Manuf.

(d) Capital.2.5(c) Labor

0

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

0

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

Notes: Figures OA.2 shows returns to scale and output elasticities for the US manufacturing sector, for the Canadianprivate sector, and for the Canadian manufacturing sector. In all figures, we sort firms by within-industry revenue
ranks and plot the average within ranks. Panel A shows the returns to scale relative to the industry average.

7

Figure OA.3 – Results by Employment and Value Added(b) RTS and Value Added

.08

.08

.06

.06

Average Returns to Scale

Average Returns to Scale(a) RTS and Employment

.04
.02
0
-.02

.04
.02
0
-.02

0

20
40
60
80
Percentiles of the Employment Distribution

1000(c) Elasticities and Employment
Labor
CapitalIntermediate

.05

0

.1

.05

0

5
20
40
60
80
100
Within-Industry Percentiles of the Employment Distribution

5
20
40
60
80
100
Within-Industry Percentiles of the Value Added Distribution(e) Revenue Shares and Employment
Labor
CapitalIntermediate.4.2(f) Revenue Shares and Value Added.8

Average Revenue Share

Average Revenue Share

.6

Labor
CapitalIntermediate

-.05

-.05

.8

100

(d) Elasticities and Value Added

Average Input Elasticity

Average Input Elasticity

.1

20
40
60
80
Percentiles of the Value Distribution

0

.6

.4

.2

0
5
20
40
60
80
100
Within-Industry Percentiles of the Employment Distribution

Labor
CapitalIntermediate
5
20
40
60
80
100
Within-Industry Percentiles of the Value Added Distribution

Notes: Figure OA.3 shows results sorting firms by within-industry employment ranks (left column) and within-industry
value added ranks (right column). We use the intermediate input and labor costs and the value of the capital stock
to construct the revenue shares.

8

Figure OA.4 – Input Revenue Shares Across the Firm Revenue Distribution(b) Canada Manufacturing(a) Canada Economy.8.6

Average Revenue Share

Labor
CapitalIntermediate

.4

.2

0

Labor
CapitalIntermediate

.6

.4

.2

0
5

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

5

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution(c) US Manufacturing.8

Average Revenue Share

Average Revenue Share

.8

.6

.4

.2

Labor
CapitalIntermediate

0
0

20
40
60
80
100
Within-Industry Percentiles of the Sales Distribution

Notes: Figure OA.4 shows revenue shares across the firm-size distribution for Canada and for the US Manufacturing.
In each plot, we sort firms by within-industry revenue ranks and then average the revenue share across all firms
within corresponding percentiles. We use the intermediate input and labor costs and the value of the capital stock toconstruct the revenue shares. Results for Canada presented in 5 percentiles groups.

9-.015

Average Returns to Scale
-.01
-.005
0

.005

Figure OA.5 – Profits and Returns to Scale

-.2

-.1
0
.1
.2
.3
EBITA Revenue Ratio Relative to Industry Mean

Notes: Figure OA.5 plots the relastionship between the returns to scale and the ebita-revenue ratio. EBITA iscomputed as total revenue net of intermediate inputs and labor costs. Both variables are demeaned at the industry
level.

10