4.2.1 Persistence in Local Elasticities

Fixed effects regression. First, we regress RTS estimates from our baseline GNR method on firm size, firm age, time dummies, and firm fixed effects. If differences in RTS are largely permanent, firm fixed effects should absorb most of the variation. This is exactly what we find: of the total RTS variance of \(0.052^{2}\), firm fixed effects (with a variance of \(0.045^{2}\)) account for 75% of the variation when controlling for firm age and size. We find similarly strong persistence in U.S. manufacturing: RTS has a variance of \(0.058^{2}\) and fixed effects account for 65% of the total variation after controlling for other firm observables.

Autocovariance structure. Second, following the earnings dynamics literature (e.g., Abowd and Card (1989); Karahan and Ozkan (2013)), we use the autocovariance structure of RTS estimates to decompose firm-level RTS into a firm-specific fixed effect, a persistent AR(1) component, and a fully transitory component (see Appendix D for details). Consistent with the fixed effects results, only 10.5% of the total RTS variance is attributable to purely transitory shocks. Firm fixed effects and the highly persistent component (with an estimated persistence parameter of 0.94) account for 38.9% and 50.6% of the total variation, respectively.

4.2.2 Permanent Technological Differences: Clustering Analysis

These persistence results are consistent with two interpretations: firms may operate genuinely different production technologies, or they may share a common non-homothetic technology but maintain persistently different input bundles—for instance, due to financial constraints or other frictions. To distinguish between these interpretations, we turn to cluster-specific production function estimation. As detailed in Section 2.3, we implement an iterative clustering procedure that groups firms with similar technologies. Since cluster assignments are constant over each firm’s life cycle, this approach directly asks whether today’s large, high-RTS firms operated a more scalable technology even when they were small. The resulting RTS gap between the top and bottom deciles is 7.6 p.p.—larger than the baseline gap of 5.7 p.p.—reflecting the additional flexibility that cluster-specific production functions afford relative to the common-technology benchmark (Figure 4).

The clustering framework allows us to separate (i) within-cluster RTS variation driven by non-homotheticities along a common technology from (ii) between-cluster RTS variation, reflecting genuine technology differences. To isolate the technology component, we replace firm-level RTS with the corresponding cluster-average RTS. The P90-P10 RTS gap declines only modestly, from 7.6 p.p. to 6.3 p.p., implying that 83% of the observed dispersion reflects differences in production technologies rather than non-homotheticities along a common production function. Crucially, the resulting clusters are not simply partitions of the input space: firms in different clusters overlap substantially in their input usage, confirming that the between-cluster RTS differences reflect genuinely distinct technologies.

Figure 4 – RTS and Size: Differences in Technology vs. Local Elasticities

Our results are robust to alternative ways of grouping firms. Appendix B reports estimates using two simpler clustering schemes based on input shares and firm size. First, given the importance of input shares in identifying factor elasticities, we cluster firms within industries based on input shares and revenue percentiles over their lifecycle. Second, to capture persistent technology differences across firms with different growth trajectories, we group firms by their maximum revenue percentile attained over their life cycle. Across these alternative classifications, the RTS–size gradient and the implied RTS gaps remain similar to our baseline estimates, with most RTS dispersion reflecting persistent differences across clusters rather than variation along a common technology.

Taken together with the persistence evidence from the fixed-effects regressions and the autocovariance analysis, these results indicate that most RTS heterogeneity reflects ex-ante technological differences across firms rather than transitory variation in local elasticities. These findings are consistent with the endogenous entrepreneurship model with heterogeneous RTS in Section 5.

Accounting for Firm Size: Technology versus TFP Differences. A natural question is how much of the within-industry firm-size distribution reflects differences in technology (RTS) versus differences in productivity (TFP). Answering it requires addressing two conceptual challenges.

First, we need genuine between-firm technology heterogeneity, not merely local elasticity variation from firms operating at different points along a common non-homothetic production function. Our iterative-clustering approach directly addresses this issue by estimating separate production functions for groups of firms with similar technologies. To construct the counterfactual, we define a common technology within each industry as the coefficient-wise average of the cluster-level production functions.¹⁷ Under this common technology, firms share the same production function but differ in TFP; allowing cluster-specific production functions, by contrast, captures technology heterogeneity across firms.

A second complication is that TFP levels are not directly comparable across technologies. We therefore normalize TFP to have mean zero within each cluster, so that firm-level TFP captures productivity relative to the cluster average. Differences in mean TFP across clusters—the production function intercept—are attributed to technology heterogeneity.

We then construct counterfactual revenues for each firm-year observation by successively imposing common TFP, common technology, or both. A Shapley–Owen decomposition exercise shows that technology heterogeneity accounts for the majority of firm-size dispersion within industries: it explains 68%, 73%, and 71% of the p99–p50, p95–p50, and p90–p50 revenue gaps, respectively. The remaining dispersion is, by construction, explained by TFP differences. These patterns reinforce the paper’s central finding: the size advantage of large firms primarily reflects the technology they operate rather than higher productivity within a common technology.

4.3.1 Markups and Market Power

Our RTS estimates are based on revenue elasticities of the three inputs. A key concern in the literature (e.g., Bond et al. (2021)) is that identifying revenue-based or physical production functions typically requires either price and quantity data or strong parametric assumptions about demand and technology. In particular, using revenue data alone may lead to unknown biases in estimates of markups and output elasticities. However, recent evidence (e.g., De Ridder et al. (2022)) suggests that such biases are modest in practice and that relative variation in markups and elasticities is well identified even with revenue-based data. Consistent with this view, we show that relative variation in RTS—our main object of interest—is robust across multiple estimation methods. Specifically, we present three sets of theoretical and empirical arguments supporting our interpretation that the observed positive relationship between RTS and firm size primarily reflects technological differences, rather than variation in markups (e.g., De Loecker et al. (2020)), or monopsony markdowns (e.g., De Loecker et al. (2016) and Burstein et al. (2024)).

Role of markups. First, if larger firms charge higher markups or markdowns—as implied by models with oligopolistic competition (e.g., Atkeson and Burstein (2008)) or monopolistic competition under log-concave demand systems (e.g., Edmond et al. (2023))—then physical RTS would increase even more strongly with firm size than revenue-based RTS. This follows because the physical output elasticity equals the revenue elasticity multiplied by the firm’s markup. We directly estimate firm-level markups following De Loecker and Warzynski (2012) and find that markups increase with firm revenue in our data (Figure A.3)—consistent with De Loecker et al. (2020)— indicating that the size gradient is larger for physical RTS than for revenue-based RTS.

Controlling for market power. Second, while our baseline method permits markdowns in capital and labor markets, it does not account for markups in output markets or markdowns on intermediate inputs. We then extend the GNR approach to explicitly control for both types of market power using firms’ output market shares as proxies for unobserved price elasticities (following De Loecker et al. (2020) and De Loecker et al. (2016)). Relaxing the perfect competition assumption, we allow firms to face downward-sloping demand and adjust the FOC for intermediate inputs accordingly.¹⁸ If markups (or markdowns) are a significant determinant of input expenditure shares, we should find that our estimates of the intermediate input elasticity are sensitive to the inclusion of these controls. Controlling for market share barely changes the size gradient of the intermediate input elasticity (Figure A.12), the main driver of RTS differences along the firm-size distribution.

Demirer method. Third, we apply Demirer (2025)’s method, which explicitly allows for heterogenous markups across firms. Yet, it requires several stronger assumptions on production and input choices. Specifically, we impose the following restrictions on the general firm problem (Section 2.1): (i) Firms \(j\) in industry \(\mathfrak{i}\) share a common, weakly homothetic separable production function \(F^{\mathfrak{i}}(K_{jt},h^{\mathfrak{i}}(L_{jt},\omega ^{M}_{jt}M_{jt}))\) where \(h^{\mathfrak{i}}\) is homogeneous in both inputs and \(\omega ^{M}_{jt}\) is intermediate-augmenting productivity. (ii) Both \(M_{jt}\) and \(L_{jt}\) are flexible inputs, optimally chosen, and firms are price takers in both input markets. (iii) Firms have price setting power in output markets via a demand function \(P_{jt}=P^{\mathfrak{i}}_{t}(Y_{jt})\). See Demirer (2025) for additional technical assumptions.¹⁹

Under these different assumptions, the estimated RTS-size gradient remains robust and, if anything, becomes even steeper: RTS increases by about 10 percentage points from the bottom half to the top 5% of the firm-size distribution (Figure 5). These results are consistent with the view that physical RTS increases more strongly with firm size than revenue-based RTS, as expected if markups increase with firm revenue—a relationship we confirm again using the Demirer methodology (Figure A.3).

Factor-augmenting productivity shocks. Another potential concern is that larger firms may have higher intermediate input elasticities simply because they use intermediate inputs more efficiently, reflecting factor-specific productivity shocks. Since our results remain robust under Demirer (2025)’s method, which accommodates factor-augmenting productivity shocks, this concern is likewise alleviated.

4.3.2 Cobb-Douglas with RTS Heterogeneity.

Another potential concern is that our results may be sensitive to the flexible functional form assumed in the GNR method. To address this, we reestimate the production function by imposing homogeneous relative factor elasticities while still allowing for RTS differences, i.e., \(F_{jt}(K_{jt},L_{jt},M_{jt})=\left (K^{\varepsilon _{K}}_{jt}L^{\varepsilon _{L}}_{jt}M^{\varepsilon _{M}}_{jt}\right)^{\eta _{jt}}\). Consistent with our baseline findings, the Cobb-Douglas series in Figure 5 shows that RTS increases with firm size by about 10 p.p., with a steeper rise in the bottom half of the revenue distribution and a more moderate increase among the largest firms relative to our baseline.

4.3.3 Intangible capital.

We also analyze the importance of including intangibles in our measure of firms’ capital stock and reestimate their production functions using Canadian data. In theory, including intangibles affects measured productivity, the output elasticity of capital, and therefore RTS. In particular, if larger firms invest disproportionally more in intangible capital, omitting intangibles would understate their capital elasticity (and thus RTS) and overstate their TFP. Consistent with this intuition, we find that the positive relationship between firm size and RTS becomes even stronger when intangible capital is included. As shown in Figure 5, the P95–P50 RTS gap increases from \(0.08\) in the baseline to \(0.20\) when intangible capital is incorporated.

4.3.4 Ranking firms by employment or value added.

Appendix Figure A.5 presents the results when ranking firms, within industry, by employment or value added rather than by revenue. While the patterns for RTS are similar, the output elasticities display distinct variations: firms with high employment or high value added exhibit higher labor elasticities, whereas the intermediate input elasticity shows only a small increase among the largest firms. This pattern arises mechanically from the ranking criterion: firms with high employment or value added are, by construction, more labor-intensive. Therefore, we prefer to rank firms by revenue—a factor-neutral approach—in our primary analysis.

4.4.1 Firm Dynamics

Heterogeneity in RTS has significant implications not only for the firm size distribution but also for firms’ growth trajectories over the life cycle. Firms with higher RTS are expected to grow faster to reach their larger optimal sizes, compared to firms with similar TFP but lower RTS. To analyze these life-cycle patterns, we construct a balanced panel of Canadian firms from our baseline estimates and group firms by their initial production function characteristics. Specifically, we focus on firms born between 2002 and 2005 that are observed for 12 consecutive years. We group firms based on their initial RTS and initial TFP, demeaned at the industry level. We then track their average log revenue, again demeaned within industries, over their life cycle (Figure 7).

(a) RTS (b) TFP

Figure: Figure 7 – Life Cycle of Firms Starting with Different RTS and TFP

Firms with higher initial RTS and TFP start with higher revenues relative to their industry peers. More importantly, firms with higher initial RTS (Panel A) exhibit significantly faster growth: firms in the top 10% of the initial RTS distribution grow about 30 log points over 10 years, whereas firms in the bottom 90% grow by only about 20 log points. This evidence supports our interpretation that high-RTS firms operate more scalable technologies, enabling substantially greater life-cycle growth. We also rank firms by their average growth rates over 12 years and find that the top 1% fastest-growing firms (“gazelles”) exhibit an average RTS of 0.98 compared to 0.95 among those below the 90th percentile. Similarly, Guntin and Kochen (2025) recently show that a firm dynamics model with ex-ante heterogeneity in production functions is required to explain empirical life cycle trajectories of the largest firms.

In contrast, Panel B shows that firms entering with high TFP, while initially larger, do not grow faster than other firms in their industry. Indeed, higher initial TFP is associated with slightly lower subsequent growth, which can be explained by TFP being a mean-reverting process. These finding suggest that highly productive firms might have low RTS, which constrains their growth (Hurst and Pugsley (2011)).

While these results focus on surviving firms, we also examine the effects of RTS and TFP heterogeneity on firm exit. We estimate probit regressions of firm exit on TFP percentile and RTS (Table A.15). Across specifications, higher RTS is associated with a lower probability of firm exit. The effect of TFP on firm exit is smaller, and varies in sign across specifications. We conclude that, from an ex ante perspective, RTS heterogeneity better predicts differences in firm growth and survival over the life cycle than TFP heterogeneity.

Finally, we investigate whether firms with varying RTS respond differently to aggregate shocks. We use two types of shocks: changes in industry-level TFP, and the 2007-2008 global financial crisis. We estimate regressions of firm revenue growth on RTS, the aggregate shock, and their interaction (Table A.16). The results show that firms with higher RTS respond more strongly to aggregate shocks, consistent with greater scalability (see also Smirnyagin (2023), Clymo and Rozsypal (2025), and Argente et al. (2024)).

4.4.2 Role of RTS Heterogeneity in Wealth and Wage Inequality

We conclude this section by examining how firm-level RTS relates to the wealth of firm owners and the wages of workers. First, we analyze how production function parameters vary with the equity wealth of business owners. A key advantage of our dataset is that we can link firms to their ultimate individual owners using administrative records from the Shareholder Information in Corporate Tax Files. We calculate each individual’s equity wealth by aggregating the value of the firms they own, weighted by ownership shares. For each owner, we then compute an equity-value-weighted average RTS and TFP percentile across the firms they own. Figure 8a shows that wealthier individuals tend to own firms with higher RTS. That is, owners at the top of the wealth distribution are more likely to own firms with more scalable production technologies. In addition, TFP is also increasing in owner wealth, but in a concave manner, particularly at the top of the distribution.

It is well established that large firms tend to pay higher wages than smaller firms for similar workers (Brown and Medoff (1989)). Given our results, a natural question is whether this firm size-wage premium is driven by higher TFP or greater scalability (RTS) among large firms. To disentangle these factors, we rank firms by their average wage and compute the average RTS and TFP across wage deciles. Figure 8b shows that higher-paying firms tend to have significantly higher RTS, while the relationship between wages and TFP is weaker and less systematic. These findings suggest that RTS heterogeneity is an important driver of the firm size-wage premium.

5.2.1 Model Setup

Time is discrete and there is a unit continuum of agents who derive log utility from consumption. They discount the future at rate \(\tilde{{\beta}}\) and face a constant death probability \(p\in [0,1)\). Thus, their effective discount factor is \(\beta =(1-p)\cdot \tilde{{\beta}}\), and they maximize \(\mathbb{E}\left [\sum _{t\geq 0}\beta ^{t}\ln (c_{t})\right]\). Agents choose between employment and entrepreneurship, \(o\in \left \{W,E\right \}\). A worker earns labor income \(w\cdot h\), 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, hire labor \(\ell\) and capital \(k\) at rental rates \(w\) and \(R\) to produce output \(z\cdot f(k,\ell)^{\eta}\), where \(f(\cdot)\) is a constant-returns-to-scale production function. The pair \((z,\eta)\) denotes entrepreneurial productivity \(z\) and project scalability \(\eta\), both treated as firm characteristics and assumed to follow a joint first-order Markov process. Consistent with our empirical findings, we model RTS as a persistent firm characteristic and abstract from non-homotheticities, which play a secondary role.

Asset markets are incomplete. Agents can invest their wealth \(a\geq 0\) in an annuity that pays an interest rate \(r\). Upon death, agents are replaced by an equal number of newborns who start with zero wealth. We parameterize financial frictions by \(\lambda \in [0,1]\), assuming that a fraction \(\lambda\) of total input expenditures must be financed with the entrepreneur’s own wealth. When this constraint binds, it generates a wedge between marginal products and input prices. As a result, static profit maximization yields a net profit of

\[\begin{align*} \pi (a,z,\eta) & =\max _{k\geq 0,\ell \geq 0}z\cdot f(k,\ell)^{\eta}-w\cdot \ell -R\cdot k\\ & \text{s.t.}w\cdot \ell +R\cdot k\leq \frac{a}{\lambda}, \end{align*}\]

implying input choices \(k(a,z,\eta),\ell (a,z,\eta)\) and output \(y(a,z,\eta)\).²¹ Thus, the agent’s dynamic problem can be written in recursive form as

\[\begin{align*} V(a,h,z,\eta) & =\max _{a'\geq 0,c\geq 0,o\in \left \{W,E\right \}}{u(c)}+\beta \cdot \mathbb{E}[V(a',h',z',\eta ')]\\ & \text{s.t}.\quad c+a'=\mathbb{{I}}_{o=W}\cdot w\cdot h+\mathbb{{I}}_{o=E}\cdot \pi (a,z,\eta)+(1+r)\cdot a. \end{align*}\]

We assume a competitive financial intermediary that invests in physical capital, subject to depreciation rate \(\delta\), and issues the annuities.

Equilibrium. We relegate the standard definition of equilibrium to Appendix E.2.

5.2.2 Calibration

We calibrate both the \((\eta,z)\)- and the \(z\)-economy to match the same set of observable moments of the firm size distribution and entrepreneurship dynamics. This isolates the role of RTS heterogeneity while holding fixed all other aspects of the environment. We follow a standard calibration strategy. First, we briefly discuss fixed common parameters. We set the death probability to \(\frac{1}{80}\), corresponding to an expected life expectancy of 80 years.²² We use a Cobb-Douglas production function \((f)\) with capital share \(\alpha =0.4\) and depreciation rate \(\delta =0.05\). Labor efficiency units \(h\) follow a log-normal AR(1) process with autocorrelation of \(0.9\) and cross-sectional standard deviation of \(1.3\), with mean normalized to \(\mu _{h}=-\frac{\sigma ^{2}_{h}}{2}\). We estimate this process directly from the data on individual post-tax earnings. We calibrate both economies at \(\lambda =0.3\), implying that 30% of input expenditures must be financed with the owner’s wealth, and then vary \(\lambda\) in counterfactuals.²³

\((z)\)-Economy: We jointly calibrate five parameters \((\beta,\eta,\sigma _{z},\rho _{z},\xi _{z})\) to match six empirical moments as summarized in the middle column of Table II. The effective discount factor \(\beta\) primarily influences the aggregate capital-output ratio. The (common) RTS parameter \(\eta\) is closely tied to the fraction of the population engaged in entrepreneurship, as it governs the share of income accruing to entrepreneurs. Productivity \(z\) follows a log-normal AR(1) process with normalized mean \(\mu _{z}=-\frac{\sigma ^{2}_{z}}{2}\). Its persistence \((\rho _{z})\) shapes entry into and exit from entrepreneurship, while its cross-sectional dispersion, \(\sigma _{z}\), is central for matching the firm size distribution. To better capture the right tail, we model the top 1% of the \(z\)-distribution with a Pareto tail, where \(\xi _{z}\) governs tail thickness. Overall, the model matches the targeted moments closely.

\((\eta,z)\)-Economy: We extend the calibration of the \(z\)-economy by additionally disciplining heterogeneity in \(\eta\) to match the observed dispersion of RTS along the revenue distribution (rightmost column of Table II). Specifically, we model \(\eta\) as a truncated normal AR(1) process on the interval \((0,1)\) with parameters \((\mu _{\eta},\sigma _{\eta},\rho _{\eta})\). We set \(\rho _{\eta}=0.98\), matching the high persistence of RTS documented in the empirical analysis. The mean \(\mu _{\eta}\) helps determine the fraction of entrepreneurs, while the cross-sectional dispersion \(\sigma _{\eta}\) is closely linked to the difference in average RTS between the top 5% and the bottom 50% of firms ranked by revenue.²⁴ We allow \(z\) and \(\eta\) to be correlated, but rather than imposing the empirically estimated joint distribution of \((\eta,z)\) directly, we calibrate the TFP process residually. This serves two purposes. First, it ensures that the \((\eta,z)\)-economy matches the same firm size distribution as the \(z\)-economy. Second, when firms operate production functions with different RTS, relative TFP levels are not directly comparable across firms, so the empirical joint distribution of \((\eta,z)\) cannot be taken at face value.²⁵ Formally, we specify log TFP as \(\ln z=\sqrt{(1-\rho ^{2}_{z,\eta}}\tilde{{z}}+\rho _{z,\eta}\cdot \frac{\sigma _{z}}{\sigma _{\eta}}\cdot (\eta -\mu _{\eta})\), where \(\tilde{{z}}\) follows an independent normal AR(1) process with parameters \((\sigma _{z},\rho _{z},\mu _{z}=-\frac{\sigma ^{2}_{z}}{2})\). Under this parameterization, \(\sigma _{z}\) denotes the cross-sectional standard deviation of log TFP. Intuitively, both \(\rho _{z,\eta}\) and \(\sigma _{z}\) influence moments of the firm size distribution: when empirical RTS dispersion is small, the model requires greater residual TFP dispersion \(\sigma _{z}\) to match observed revenue concentration, while larger RTS dispersion implies a more negative correlation parameter \(\rho _{z,\eta}\).

In total, we calibrate six parameters to match seven empirical moments. This model version also matches the targeted moments closely and does not require a Pareto tail in \(z\) to replicate the right tail of the firm-size distribution; observed heterogeneity in RTS, combined with log-normal \(z\), is sufficient.

5.2.3 Quantitative Findings

The two model economies are observationally equivalent in terms of the fraction of entrepreneurs, the persistence of entrepreneurship, the firm-size distribution, and the capital-output ratio. We evaluate the efficiency losses associated with the same financial friction in both economies. Figure 10 compares output losses as the financial friction parameter \(\lambda\) increases from the unconstrained case (\(\lambda =0\)) to \(\lambda =1\) across stationary equilibria. For \(\lambda =0.3\), corresponding to entrepreneurs financing 30 cents of each dollar of input expenditure with own wealth, the \((\eta,z)\)-economy—disciplined by empirical heterogeneity in both TFP and RTS—exhibits an output loss of 18.3 log points relative to the frictionless benchmark. By contrast, the conventional \(z\)-economy with homogeneous RTS incurs a significantly smaller loss of 7.4 log points. Thus, allowing for empirically measured RTS heterogeneity—while holding fixed the same observables and calibration targets—amplifies output losses from financial frictions by 147%.

Figure 10 – Output losses from financial friction in dynamic model

To understand this finding and connect it to the analytical mechanism in Section 5.1, we 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-accumulation of capital. Panel A of Table III shows that static misallocation of production factors across firms contributes 10.6 log points in the \((\eta,z)\)-economy—more than half of the total GDP loss and twice as much as in the conventional \(z\)-economy. This is precisely the channel highlighted in the analytical discussion: consistent with Proposition 1, a given input wedge distorts factor demand more for high-\(\eta\) firms. The dynamic setting further magnifies this effect through selection and accumulation: Consider two hypothetical superstar entrepreneurs that are initially poor and face the same financial friction. The one distinguished by high productivity (high \(z\)) can operate profitably even at small scale and therefore outgrows the constraint relatively quickly. In contrast, the one distinguished by high scalability (high \(\eta\)) is less profitable at small scale and consequently struggles to expand when constrained, leading to larger and more persistent misallocation.

The majority of the remaining output loss is due to the under-accumulation of capital. Misallocation of talent across occupations also contributes slightly more in the \((\eta,z)\)-economy, but remains relatively small in both economies. The \(\lambda\)-friction primarily misallocates production factors across firms rather than distorting occupational choice in this environment. We chose a simple and transparent calibration strategy with a small number of parameters, deliberately abstracting from additional elements such as fixed costs of entry and exit that could magnify the importance of the occupational choice channel.

In the benchmark scenario, we increase \(\lambda\) from \(0\) to \(0.3\) in both economies. Panel B of Table III shows that the results are even stronger when we instead equate alternative observable moments, such as the aggregate debt-to-capital ratio or the dispersion of log marginal input products. For these exercises, we continue to raise \(\lambda\) from \(0\) to \(0.3\) in the \(z\)-economy, which generates an aggregate debt-to-capital ratio of \(0.708\) and a cross-sectional standard deviation of log marginal products of \(0.144\). We then adjust \(\lambda\) in the \((\eta,z)\)-economy—raising it from 0 to 0.797 to replicate the debt ratio, or to 0.454 to match the marginal product dispersion. In these cases, the \((\eta,z)\)-economy generates output losses that are \(192-264\%\) larger than those in the \(z\)-economy.

The macro-development literature (Buera et al. (2011); Midrigan and Xu (2014); Moll (2014)) shows that misallocation losses from financial frictions are modest when firms differ only in TFP under a common homothetic technology, but grow larger once accounting for technology choice along the fixed-cost/marginal-cost margin, which locally generates an RTS-firm size gradient. Our framework abstracts from technology choice, and as such the entry margin contributes little to misallocation (as reflected by the small contribution of entrepreneurial talent misallocation in Table III). Instead, our results highlight that static misallocation is substantially amplified when empirically measured differences in RTS among existing firms are taken into account, even in the absence of technology choice.

5.2.4 Intermediate Inputs and Pre-determined Capital

Our baseline model deliberately adopts a streamlined entrepreneurial framework to isolate the quantitative role of RTS heterogeneity relative to the standard TFP-only calibration. We now consider extensions that introduce empirically relevant features of production such as intermediate inputs and alternative timings of input choices. These modifications are not intended to microfound RTS heterogeneity, but to verify that the baseline mechanism—stronger distortions from input wedges for high-RTS firms—remains robust in environments closer to those used in the empirical analysis. While they affect the overall level of misallocation—consistent with related literature—these extensions do not overturn our central result: conditioning on the same observables, heterogeneity in RTS substantially amplifies misallocation relative to TFP heterogeneity alone.

We first modify the production function to allow for intermediate inputs: \(y=z\cdot k^{\alpha _{K}}\cdot \ell ^{\alpha _{L}}\cdot m^{\eta -\alpha _{K}-\alpha _{L}},\) where \(\alpha _{K}\) and \(\alpha _{L}\) are fixed across firms, and \(\eta\) governs firm-level RTS. In this formulation, differences in RTS arise entirely from heterogeneity in intermediate input elasticities. Consistent with the data, firms with higher \(\eta\) operate at larger scale and use intermediate inputs more intensively.

We consider three alternative model setups that differ in the formulation of the financial constraint and in the timing of input choices. Throughout, the calibration strategy closely follows the baseline model. A detailed description of these extensions, along with their calibration and results, is provided in Appendix E.3.

Symmetric constraint on all inputs. First, we maintain a financial constraint that applies symmetrically across inputs: \(w\cdot \ell +R\cdot k+m\leq \frac{a}{\lambda}\). Relative to the baseline, misallocation losses increase in both economies, reflecting that the presence of intermediate inputs magnifies distortions (see, e.g., Baqaee and Farhi (2019)). Importantly, RTS heterogeneity continues to generate much larger misallocation from financial frictions: total output losses equal 46.3 log points in the \((\eta,z)\)-economy, compared to 9.3 log points in the \(z\)-economy—an amplification of 398%, relative to 112% in the baseline (row 2 of Table A.8 in Appendix E.3).

Flexible intermediates and a constraint on capital and labor only. Next, consistent with our empirical approach, we treat intermediate inputs as fully flexible and impose the financial constraint only on capital and labor: \(w\cdot \ell +R\cdot k\leq \frac{a}{\lambda}\). In this case, firms with higher \(\eta\) are effectively less exposed to the financial friction, because constrained expenditures account for a smaller share of total costs, weighted by factor elasticities (\(\frac{\alpha _{L}+\alpha _{K}}{\eta}\)). Relative to the baseline, two offsetting effects are at work: intermediates continue to amplify distortions, while the financial constraint becomes less restrictive. Despite this, RTS heterogeneity continues to magnify misallocation, with losses more than tripling (+214%) relative to the homogeneous-RTS case (row 3 of Table A.8).

Pre-determined capital. Finally, we modify the timing of input choices by assuming that capital is chosen one period in advance, so that period-\(t\) capital is predetermined at \(t-1\), before current shocks are realized. Under this assumption, the model satisfies the identifying conditions underlying the GNR estimation approach. Using model-simulated data, we verify that applying the GNR procedure with sufficiently many clusters recovers the true parameter values.²⁶ Relative to the baseline, the overall level of misallocation generated by the financial friction \(\lambda\) is lower, consistent with existing evidence that when input choices are risky, financial constraints are secondary to risk wedges (David et al. (2022); Boar et al. (2022)). Even so, RTS heterogeneity continues to amplify misallocation: output losses from \(\lambda\) are 81% higher in the \((\eta,z)\)-economy than in the \(z\)-economy (row 4 of Table A.8).

Summary. Introducing intermediate inputs and alternative timings of input choices affects the overall level of misallocation in familiar ways. Nonetheless, across all extensions, the central insight remains unchanged: conditioning on the same observables, heterogeneity in RTS substantially amplifies misallocation from financial frictions relative to models with TFP differences alone.

10.1.1 Variable Construction

Revenue. We use the revenue measure that is computed by Statistics Canada for constructing the National Account. This measure is derived by summing up relevant terms from the T2 Corporate Income Tax Return Form.

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 capital stock. We make use of information on the first book value of tangible capital observed in the dataset, annual tangible capital investment, and amortization. Specifically, the capital stock \(K\) of firm \(i\) in year \(t\) is computed as \(K_{i,t}=K_{i,t-1}+Invest_{i,t}-Amort_{i,t},t>t^{0}_{i}\), where \(t^{0}_{i}\) is the first year we observe the book value of the tangible capital of firm \(i\). The initial year capital stock \(K\) is calculated as the book value of tangible capital net of accumulated tangible capital amortization. Tangible investment includes investments in building and land, computers, and machines and equipment. While the production function estimation uses only the constructed capital stock \(K_{i,t}\) as an input, whenever we report capital’s share in revenue (e.g., Table I and Figure 1) we convert the stock into a capital cost measure, \(RK_{i,t}\), using a a user cost of \(R=15\%\). This choice affects only the reporting of input shares and has no bearing on the estimation itself. In addition, we construct a 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.

Intermediates: We measure intermediate inputs as the total expenses not related to capital and labor. Specifically, the measure is computed as the sum of operating expenses and costs of goods sold net of capital amortization. The operating expenses and costs 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 from the 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 firm C) and constructs a tracked share of ownership of firms by each ultimate individual shareholder. We merge the ownership information with the firm panel dataset and calculate total individual equity wealth as the ownership share weighted sum of the value of all holding firms. Firm value is calculated as total assets net of total liabilities.

Linked employer-employee information: We obtain linked employer-employee and earnings information from the T4 Statement of Remuneration 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 multiple T4 records in a year if she works for more than one firm. For multiple job holders, we keep the job that offers the highest earnings of the year and call it the main job. In addition, we drop workers with annual earnings from the main job that are lower than 5,000 CAD.

10.1.2 Sample Selection

Several steps are taken to construct the estimation sample. First, we drop firms with missing industry information. Second, we exclude the initial year in which a firm’s book value of tangible capital is observed, along with all prior observations, as we cannot use the PIM to construct the capital stock for these observations. Third, we drop firm-year observations with missing and nonpositive revenue, labor, capital, and intermediate input values. We further drop the observations whose one-year lagged revenue or inputs are missing or non-positive, as our identification strategy requires using lagged labor input as the instrument. Fourth, we drop the observations with extreme factor shares, that is, the ones with a ratio of wage-bill-to-revenue below the 1st percentile or above the 99th percentile, with a ratio of wage bill-to-value-added below the 1st percentile or above the 99th percentile, with a ratio of intermediate-input-to-revenue above 0.95 or below 0.05, and with a ratio of capital-stock-to-revenue above the 99.9th percentile. This sample selection procedure leaves us with around 4.3 million firm-year observations. We convert all monetary variables to 2002 Canadian dollars. Summary statistics are reported in Table A.2.

12.3.1 Adding intermediates, ex-post capital choice as in baseline

First, we maintain the baseline timing of input choices: capital is chosen after observing the realization of shocks, as are labor and intermediate inputs. We estimate the parameters in Table A.7 using the exact same strategy as in our baseline model versions. Rows 2 and 3 in Table A.8 show the resulting misallocation costs from raising the financial friction \(\lambda\) from 0 to 0.3, in both the calibration with and without RTS heterogeneity. Row 2 contains the results from the model that imposes the financial constraint on intermediates as well, while row 3 treats intermediates as fully flexible in line with the assumptions of our empirical approach (only capital and labor are subject to the constraint).

12.3.2 Adding intermediates, and pre-determined capital

Finally, we also change the timing of input choices, in addition to adding intermediate inputs in production: period \(t\) capital is chosen in period \(t-1\), so prior to the realization of period \(t\) shocks. This specification is rich enough such that the identification assumptions of GNR hold. The computation becomes more involved, as an agent’s inter-temporal choice includes (i) net savings \(a'\), (ii) capital \(k'\) (part of net savings), (iii) and occupation \(o'\). In our numerical solution, we exploit that when using resources after production \(x\) (“cash-on-hand”) as endogenous state variable, there is no need to keep track of the two assets separately; instead, the problem becomes a portfolio choice problem conditional on occupational choice. The agent’s dynamic problem is:

\[\begin{align*} V(x,h,z,\eta) & =\max _{c\geq 0,a'\geq 0,k'\geq 0,o'\in \left \{W,E\right \}}{u(c)}+\beta \cdot \mathbb{E}[V(x'_{o'}(a',k',h',z',\eta '),h',z',\eta ')]\\ \text{s.t}. & \quad c+a'=x,\\ & x_{W}(a,k,h,z,\eta)=w\cdot h+(1+r)\cdot a-R\cdot k,\\ & x_{E}(a,k,h,z,\eta)=\pi (a,k,z,\eta)+(1+r)\cdot a-R\cdot k, \end{align*}\]

where \(x_{W}\) (\(x_{E})\) denotes cash-on-hand of workers (entrepreneurs),³⁰

This formulation is the natural extension of our general setup to pre-determined capital. The financial constraint applies to capital and labor, for every shock realization. The interpretation is, as before, that a fraction \(\lambda\) of the expenditures on capital \(R\cdot k_{t}\) and labor \(w\cdot \ell _{t}\) required for production in period \(t\) need to be financed with the owner’s period \(t\) net wealth, \(a_{t}.\) The difference to before is that capital \(k_{t}\) is chosen before the realization of period \(t\) shocks, while the labor choice \(\ell _{t}\) is made (as before) after observing current shocks.

Table A.9 displays the calibration, following again the same strategy as in the previously discussed model versions. Row 4 of Table A.8 shows the resulting static misallocation costs from raising the financial friction \(\lambda\) from 0 to 0.3. The overall level of static misallocation associated with \(\lambda\) is now much smaller, because capital is still chosen ex-ante, and so even with \(\lambda =0\), marginal products of capital are not equalized. Instead, eliminating the \(\lambda\) friction only removes dispersion in marginal labor input products (intermediates are fully flexible, and hence marginal intermediate input products are fully equalized, regardless of the value of \(\lambda\)). We re-iterate that our main point is not the overall level of misallocation associated with financial frictions, but rather the additional amount of misallocation (+81% in this model version) generated by allowing for realistic RTS heterogeneity in line with our empirical results.