metadata: authors: 0: author: David,Joel Michael
1: author: Venkateswaran,Venky
2: author: Cusolito,Ana Paula
3: author: Didier Brandao,Tatiana
count: World
docna: 0: docna: Capital Allocation in Developing Countries
docty: Policy Research Working Paper
subtopic: Pulp & Paper Industry,Plastics & Rubber Industry,Construction Industry,Business Cycles and Stabilization Policies,Food & Beverage Industry,Common Carriers Industry,Textiles, Apparel & Leather Industry,General Manufacturing,Public Sector Administrative and Civil
            Service Reform,De Facto Governments,Democratic Government,Public Sector Administrative & Civil
            Service Reform,Administrative & Civil Service Reform,International Trade and Trade Rules,Economic Conditions and Volatility,Trade and Services
teratopic: Industry,Governance,Social Protections and Labor,Macroeconomics and Economic Growth,International Economics and Trade
abstracts: cdata!: This paper investigates the sources of
            capital misallocation across a group of 11 developing and
            developed countries. The main findings are (i) technological
            frictions, namely, adjustment costs and uncertainty, account
            for only a modest share of observed misallocation, leaving
            ample scope for other factors; (ii) heterogeneity in
            firm-level technologies potentially explains between
            one-quarter and one-half; but (iii) dispersion in markups is
            much smaller; and (iv) after accounting for these factors,
            on average, at least 50 percent of misallocation within each
            of these countries remains unexplained, suggesting a large
            role for additional, potentially distortionary factors.
            These factors are largely attributable to a component that
            is correlated with firm size/productivity and one that is
            essentially permanent to the firm. The paper reports a broad
            set of moments describing firm-level investment dynamics and
            detailed parameter estimates on a country-by-country basis,
            with an eye toward future work in this area.
repnme: repnme: Capital Allocation in Developing Countries
display_title: Capital Allocation in Developing Countries
listing_relative_url: /research/2019/10/31449489/capital-allocation-developing-countries
topicv3: Macroeconomic Vulnerability and Debt
            m1326057 1104,Manufacturing Sectors m1326521 1281,Trade m1326924 698,Trade in Services m1326928 780,Employment and Wage Data m1327013 2055
publishtoextweb_dt: 2019-10-01T00:00:00Z
historic_topic: Industry,Governance,Social Protections and Labor,Macroeconomics and Economic Growth,International Economics and Trade
no_of_pages: 31
author_keys: 0: author_key: 1672469
1: author_key: 1003705
2: author_key: 582096
model_extractions: Policy Research Working Paper 9031

# Capital Allocation in Developing Countries

_Joel M. David Ana Paula Cusolito Venky Venkateswaran Tatiana Didier_ Finance, Competitiveness and Innovation Global Practice October 2019
Policy Research Working Paper 9031

# **Abstract**

This paper investigates the sources of capital misallocation across a group of 11 developing and developed countries. The main findings are (i) technological frictions, namely, adjustment costs and uncertainty, account for only a modest share of observed misallocation, leaving ample scope for other factors; (ii) heterogeneity in firm-level technologies potentially explains between one-quarter and one-half; but (iii) dispersion in markups is much smaller; and (iv) after accounting for these factors, on average, at least 50 percent of misallocation within each of these countries remains unexplained, suggesting a large role for additional, potentially distortionary factors. These factors are largely attributable to a component that is correlated with firm size/ productivity and one that is essentially permanent to the firm. The paper reports a broad set of moments describing firm-level investment dynamics and detailed parameter estimates on a country-by-country basis, with an eye toward future work in this area.

This paper is a product of the Finance, Competitiveness and Innovation Global Practice. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at acusolito@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Capital Allocation in Developing Countries<sup>_∗_</sup>

Joel M. David<sup>_†_</sup> FRB Chicago Ana Paula Cusolito<sup>_‡_</sup> World Bank Venky Venkateswaran<sup>§</sup> NYU Stern/FRB Minneapolis Tatiana Didier<sup>_¶_</sup> World Bank

> _∗_ The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Banks of Chicago or Minneapolis or the Federal Reserve System.

> _†_ joel.david@chi.frb.org.

> _‡_ acusolito@worldbank.org.

> §vvenkate@stern.nyu.edu.

> _¶_ tdidier@worldbank.org.
# **1 Introduction**

A large literature in macroeconomics and development explores the ‘misallocation’ of resources across firms, i.e., dispersion in static marginal products of inputs, and the implications for crosscountry differences in aggregate productivity and per capita income/output.<sup>1</sup> Because marginal products are not directly observed, the extent of such misallocation is typically inferred from dispersion in the _average_ product of these inputs (revenue or value-added/input ratios) – under standard assumptions, the dispersions in the two are identical (for example, with common production technologies and demand elasticities). Recently, attention has turned towards examining the role of particular factors in generating the observed misallocation. Examples include technological frictions such as adjustment costs or imperfect information, unobserved heterogeneity at the firm level, for example, in production technologies or markups, as well as firm-specific ‘distortions’ stemming from economic policies or other institutional features.<sup>2</sup> From a policy standpoint, disentangling the role of these forces is paramount – any prescription for economic reforms aimed at reducing misallocation must be based on a careful understanding of the underlying determinants and nature – e.g., efficient or inefficient – of the observed average product dispersion.

In a recent contribution, David and Venkateswaran (2017) develop an empirical methodology to disentangle a number of sources of _measured_ capital misallocation – dispersion in the average (revenue) product of capital ( _arpk_ ) – using observable data on firm revenues (or value-added) and inputs. This paper applies their method to a number of countries, with a particular focus on developing ones. Specifically, we use the Orbis database in conjunction with a number of country-specific firm-level censuses/surveys to compile data for 11 countries – nine developing and two developed (Japan and the US) – and implement that methodology on this set of countries. In addition to furthering our understanding of the sources of capital misallocation, our analysis also aims to provide a broader perspective on firm investment dynamics in these countries, which may help to guide future research.

The essence of the method in David and Venkateswaran (2017) is that by targeting an appropriately chosen set of moments, one can jointly estimate the contributions of capital adjustment costs, firm-level uncertainty and a broad class of other firm-specific factors to observed _arpk_ dispersion. This class includes factors that are correlated with firm fundamentals, e.g., productivity or demand, as well as those orthogonal to fundamentals. The latter can be further

> 1Seminal contributions include Hsieh and Klenow (2009) and Restuccia and Rogerson (2008). Restuccia and Rogerson (2017) and Hopenhayn (2014) provide excellent recent overviews of this line of work. 2See, e.g., work by Asker et al. (2014) on adjustment costs, Buera, Kaboski, and Shin (2011), Moll (2014), Gopinath et al. (2017) and Midrigan and Xu (2014) on financial frictions, David et al. (2016) on uncertainty and Peters (2016) on markup dispersion.  Following  Hsieh and Klenow (2009) and Restuccia and Rogerson (2008),    a large number of studies focus on the role of distortions along the lines that we model here.

2
broken down into transitory and permanent components. Thus, the specification allows for a rich correlation structure for these factors, both over time as well as with firm characteristics. Further, it is also possible to provide bounds for the role of two specific factors, namely, firmlevel heterogeneity in markups and production technologies.<sup>3</sup> In the body of the paper, we describe these steps in more detail and report key data moments as well as detailed parameter estimates on a country-by-country basis, along with the contributions of each of these forces to observed _arpk_ dispersion.

While the estimates vary by country, there are a number of broad patterns that hold across all the countries in our sample: (i) adjustment costs and informational frictions – although important drivers of investment dynamics – generate only modest _arpk_ dispersion; (ii) correlated, or size-dependent, factors – which implicitly ‘tax’ more productive firms, play a more significant role, especially in developing countries;<sup>4</sup> (iii) a significant portion of the observed _arpk_ dispersion can be attributed to permanent factors, i.e., firm-level fixed effects; (iv) heterogeneity in production technologies (i.e., firm-level differences in output elasticities of capital and other inputs in the production function) can potentially account for between about one-quarter and one-half of dispersion in _arpk_ . This is an important result, as it suggests that a non-negligible portion of observed dispersion may not entail a ‘misallocation’ at all; and (v) markup dispersion is generally modest.<sup>5</sup> Taken together, these latter two factors – markup and technology heterogeneity – can explain as much as 50% of observed _arpk_ dispersion. But this leaves a substantial unexplained component, which seems to stem from other – potentially distortionary – factors. The rest of the paper is organized as follows. Section 2 describes our data sources and reports key moments used in the estimation. Sections 3 and 4 contain our main results. Section 5 concludes.

# **2 The Data**

We use data compiled from a number of different sources. First, we use the Bureau van Dijk Orbis database. Orbis contains firm-level data (including both public and private firms) for a number of countries (although the extent of the coverage varies widely).<sup>6</sup> We focus mainly

> 3The data to compute markup dispersion are only available  in four of the 11 countries, highlighting some of the data hurdles future work along these lines may have to overcome.

> 4‘Correlated’ distortions of this kind have been emphasized in, e.g., Restuccia and Rogerson (2008), Guner et al. (2008), Bartelsman et al. (2013), Buera, Moll, and Shin (2013), Buera and Fattal-Jaef (2016), Hsieh and Klenow (2014) and Bento and Restuccia (2016).

> 5Note that this refers to dispersion in markups within an industry, not the average level of markups or the dispersion across industries.

> 6For a comprehensive description of the Orbis database, see Kalemli-Ozcan et al. (2015). Gopinath et al. (2017) also use the Orbis data in a misallocation context, albeit for a developed country, Spain.

3
on developing countries (and Japan) and use countries that have sufficient data on revenues, capital and a measure of labor input (the minimum requirements for the analysis).<sup>7</sup> We use only firms in the manufacturing sector and, given our focus on within-industry moments, use only country-year-industry cells with at least 10 firms. The data cover the period 2002-2010. There are seven countries in our Orbis sample: Argentina (ARG), Brazil (BRA), Japan (JPN), Malaysia (MYS), Taiwan, China (TWN), Thailand (THA) and Turkey (TUR).

We supplement this sample with data on four additional countries. First, we use data on US publicly traded firms from Compustat. Second, we use data on Chinese manufacturing firms from the Annual Surveys of Industrial Production conducted by the China National Bureau of Statistics. Next, we use data in Colombia (at the establishment level) from the Colombian Annual Manufacturers Survey and lastly, data in Mexico (also at the establishment level) from the Mexican Annual Industrial Survey.<sup>8</sup> Throughout the paper, we report results first for the seven Orbis countries (in alphabetical order) and then for the four additional countries (again in alphabetical order).

Our use of multiple data sources has the advantage of yielding a wider set of countries on which we can perform our analysis – nine developing and two developed. A disadvantage is that direct cross-country comparisons become more difficult, given differences in coverage, sample sizes, the set of firms included, etc.<sup>9</sup> However, we will show that despite these difference, our study leads to a number of robust cross-country patterns with respect to the sources of observed misallocation.

**Investment Moments.** The key input into our estimation procedure is a set of carefully chosen moments from data on firm-level capital and revenues, both of which are directly available in our data. Since we are interested in capital allocation across firms within an industry, we extract the firm-level idiosyncratic components of each data series by regressing the measured values on industry-by-year fixed effects and working with the residuals. Industries are defined at the four-digit SIC code level.

We report a number of moments for each of the countries in Table 1, grouped into four categories. The first group describes the stochastic process of firm-level productivity.<sup>10</sup> In our theoretical framework, which uses a standard Cobb-Douglas production function and constant

> 7The measure of labor varies depending on reporting within each country.  For Malaysia and Taiwan, China,  we have measures of the wage bill. For the remaining countries, we have number of employees. 8For a more detailed description of these data sets, see, e.g., David and Venkateswaran (2017) and the references therein.

> 9It turns out that these issues crop up even using a single cross-country database. For example, as previously noted, the coverage differs widely even within the Orbis database alone.

> 10As is well known, in standard models of misallocation/firm dynamics, measures of productivity can also be interpreted as idiosyncratic demand factors.

4
elasticity of substitution demand, firm-level productivity can be inferred from revenues and capital. Formally, _ait_ = _yit − αkit_ where _ait_ denotes (in logs) the productivity of firm _i_ at time _t_ , _yit_ (log) revenues, _kit_ (log) capital and _α_ is a composite parameter that depends on other demand and production-side parameters (see equation (1) in Appendix A for details). For simplicity, we use a common value of _α_ across countries and set it equal to 0.62.<sup>11</sup> Firm productivity is assumed to follow an AR(1) process and so is parameterized by its persistence 2 ( _ρ_ ) and the variance of the innovations in the process ( _σ_ ), which are reported in Table 1. _μ_ In the second panel of the table, we report two commonly-used moments of firm-level in2 vestment, the serial correlation ( _ρ_ ( _i, i −_ 1 ) ) and variance ( _σ_ ( _i_ ) ). We measure net investment as the period-over-period percentage change in the firm’s capital stock, i.e., _iit_ = _kit_ +1 _− kit_ (reported capital stocks in the data are year-end, which we map to _kit_ +1). The third panel of the table reports these two moments for investment growth rates (denoted ∆ _i_ ) along with the correlation of investment growth with lagged productivity growth (denoted ∆ _a −_ 1). Finally, the last panel of Table 1 reports moments related to the average (revenue) product of capital, i.e., (in logs) _arpkit_ = _yit − kit_ . The table reports the correlation of the _arpk_ 2 with productivity, denoted _ρ_ ( _arpk, a_ ) and its cross-sectional dispersion, _σ_ ( _arpk_ ) , a standard indicator of misallocation, which represents the key object of interest.

Table 1 uncovers a number of noteworthy patterns. First, across all countries, firm-level productivity is very persistent and volatile, especially when compared to the variability of investment.<sup>12</sup> The serial correlation of investment is relatively modest. These patterns also hold for investment growth rates, which also show low variability and a low serial correlation (indeed, the latter are significantly negative). Turning to the last panel, the average product of capital tends to be significantly correlated with productivity – in other words, high (low) productivity firms tend to have less (more) capital than would seem to be dictated by their fundamentals. Finally, the dispersion in the average product of capital is everywhere positive and significant, suggesting a substantial degree of ‘misallocation’, even in the two developed countries. In the next section, we use the moments in Table 1 to shed light on the underlying drivers of this dispersion in the _arpk_ . In the following section, we add additional data on other inputs into production (i.e., labor and intermediate inputs, where available) to further explore the extent to which _σ_ 2 ( _arpk_ ) reflects a true misallocation – i.e., dispersion in _marginal_ products – or whether unobserved heterogeneity across firms confounds the mapping from the former to

> 11In their baseline analysis, David and Venkateswaran (2017) use a slightly higher value in their study of  China, based on evidence suggesting a relatively higher share of capital in production. Their estimates regarding the sources of misallocation are very close to the ones in this paper, suggesting that our results are somewhat robust to allowing for country-specific curvature parameters.

> 12As a benchmark, without adjustment costs or other distortions, capital is given by 2 _kit_ = ~~1~~ 1 _~~−~~ aαit_ , so the variability of investment should be<sup><u>1</u></sup> 1 _−_ = _~~α~~_<sup>1</sup> = 6 _._ 092 _._ 38<sup>2</sup> times as volatile as productivity growth.

5
|||Produ|ctivity|Table 1: M<br>Invest|oments o<br>ment|f Investment D<br>Inve|ynamics<br>stment Growth||Avg. Pro|d. of K|
|---|---|---|---|---|---|---|---|---|---|---|
||Num. Obs.|_ρ_|_μ_<br>_σ_<br>2|_ρ_(_i, i−_1)|_σ_<br>2 (_i_)|_ρ_(∆_i,_∆_a−_1)|_ρ_(∆_i,_∆_i−_1)|_σ_<br>2 (∆_i_)|_ρ_(_arpk, a_)|_σ_<br>2 (_arpk_)|
|ARG|993|0_._89|0_._05|0_._19|0_._04|0_._31|_−_0_._36|0_._06|0_._58|0_._54|
|BRA|1389|0_._90|0_._08|0_._13|0_._05|0_._35|_−_0_._39|0_._09|0_._60|0_._65|
|CHN|797047|0_._91|0_._14|0_._04|0_._08|0_._25|_−_0_._36|0_._14|0_._68|0_._92|
|COL|44909|0_._95|0_._09|0_._13|0_._04|0_._28|_−_0_._35|0_._07|0_._61|0_._98|
|MEX|3208|0_._93|0_._07|0_._17|0_._01|0_._17|_−_0_._39|0_._02|0_._69|0_._79|
|MYS|548|0_._95|0_._06|0_._31|0_._02|0_._18|_−_0_._29|0_._03|0_._86|0_._73|
|TWN|2076|0_._96|0_._04|0_._34|0_._03|0_._21|_−_0_._36|0_._04|0_._66|0_._57|
|THA|4025|0_._95|0_._07|0_._26|0_._06|0_._27|_−_0_._32|0_._08|0_._57|0_._88|
|TUR|830|0_._89|0_._08|0_._11|0_._05|0_._37|_−_0_._38|0_._09|0_._57|0_._56|
|JPN|60720|0_._98|0_._03|0_._13|0_._02|0_._17|_−_0_._40|0_._03|0_._48|0_._43|
|USA|34260|0_._93|0_._08|0_._25|0_._04|0_._13|_−_0_._30|0_._06|0_._55|0_._45|

_Notes_ : This table reports moments in firm-level investment dynamics across eleven countries. The data sources are described in Section 2 of the main text. The first panel reports the persistence and variance of the innovations in the firm-level productivity process, _ρ_ and _σ_<sup>2</sup> , respectively. The second panel _μ_ reports the serial correlation and volatility of investment rates, _ρ_ ( _i, i−_ 1) and _σ_<sup>2</sup> ( _i_ ) . The third panel reports the correlation of investment growth rates with lagged innovations in productivity, _ρ_ (∆ _i,_ ∆ _a−_ 1) , and the serial correlation and volatility of investment growth, _ρ_ (∆ _i,_ ∆ _i−_ 1) and _σ_<sup>2</sup> (∆ _i_ ) . The last panel reports the correlation of the average revenue product of capital with productivity, _ρ_ ( _arpk, a_ ) and the variance of the average revenue product of capital, _σ_<sup>2</sup> ( _arpk_ ) . All series are regressed on industry-by-year fixed effects to extract the firm-level variation within an industry, where industries are defined at the four-digit SIC code level.
# **3 The Sources of Misallocation**

What are the sources of dispersion in the _arpk_ ? David and Venkateswaran (2017) use a rich model of investment dynamics and show that combining the information contained in the moments reported in Table 1 can shed light on this question. In particular, it is possible to _simultaneously_ measure the contributions of (i) adjustment frictions (ii) information frictions and (iii) a broad class of other firm-specific factors influencing investment decisions.

**The model.** The environment is an extension of the canonical Hsieh and Klenow (2009) framework to include dynamic considerations in firms’ investment decisions. We defer a full description to Appendix A and only discuss the key elements here. First, adjustment frictions are modeled using a standard quadratic adjustment cost function, along the lines of, e.g., Cooper and Haltiwanger (2006), Bloom (2009) and Asker, Collard-Wexler, and De Loecker (2014).

Next, informational considerations are introduced along the lines of David, Hopenhayn, and Venkateswaran (2016) by assuming that the firm, at the time of making its period- _t_ investment choice, has a noisy signal of its future productivity, _ait_ +1. In a Gaussian setting, a sufficient statistic for this uncertainty is the posterior variance of the firm, denoted V . Given the AR(1) assumption on productivity, V takes on values between zero and _σμ_ 2: if V = 0 , the firm is perfectly informed about _ait_ +1 at the end of period _t_ . If V = _σ_ 2 the firm has no information _μ_ about the following period’s innovation (but obviously knows its current productivity).<sup>13</sup> Finally, all other factors influencing investment choices are assumed to take the form of a wedge (an implicit proportional ‘tax’ on the cost of capital) in the firm’s optimality condition. The model yields the following log-linearized law of motion for capital: where _ξ_ is a composite parameter that summarizes the severity of adjustment costs and _β_ the discount rate, the expectation operator E _it_ reflects the possibility that the firm has imperfect information about future productivity and _τit_ is the effective wedge that acts on the firm’s investment decisions. We assume that _τit_ has 3 components: 13 <u>V</u> In what follows, we report values fo ~~r~~ _σ_ , which denotes the posterior uncertainty as a percent of the prior<sup>2</sup> _μ_ (values of V are easily calculated by multiplying by _σ_<sup>2</sup> as reported in Table 1). This statistic takes on values between _μ_ zero and one and is increasing in the extent of uncertainty.

7
The first component is correlated with firm productivity, _ait_ , with the degree of correlation governed by _γ_ . If _γ <_ 0 , the distortion discourages (encourages) investment by firms with higher (lower) productivity, the empirically relevant case. The opposite is true if _γ >_ 0 . The remaining components of _τit_ are both uncorrelated with _ait_ , but differ in their time-series properties. The term _χi_ is permanent while _εit_ is iid over time. Thus, the severity of factors 2 other than adjustment/information frictions is summarized by three parameters: _γ_ , _σχ_ , which 2 is the cross-sectional variation in the fixed component, and _σε_ , which is the volatility of the iid shocks to the wedge.

These five distinct forces all contribute to dispersion in _arpk_ : adjustment costs, uncertainty and the three components of the wedge, parameterized by _ξ_ , V , _γ_ , _σ_ 2 and _χ σ_ 2. The main _ε_ contribution of David and Venkateswaran (2017) is an empirical strategy to estimate these five parameters from firm-level data on revenues and capital. Specifically, they use a simulated method-of-moments estimator targeting the following five moments from Table 1: the variability and serial correlation of investment growth, the correlation of investment growth with lagged changes in _a_ , the correlation of _arpk_ with _a_ and the cross-sectional dispersion in _arpk_ . We follow their approach and report the resulting parameter estimates in Table 2.<sup>14</sup> The implications for _arpk_ dispersion and aggregate productivity are presented in Table 3 and discussed in detail there.

Table 2: Parameter Estimates

||||O<br>|ther Factors<br>|<br>|
|---|---|---|---|---|---|
||Adjustment Costs|Uncertainty|Correlated|Permanent|Transitory|
||_ξ_|_μ_<br>V/_σ_<br>2|_γ_|_σ_<br>2<br>_χ_|_σ_<br>2<br>_ε_|
|ARG|0_._19|0_._67|_−_0_._79|0_._36|0_._00|
|BRA|0_._12|0_._71|_−_0_._67|0_._42|0_._00|
|CHN|0_._16|0_._63|_−_0_._63|0_._51|0_._00|
|COL|0_._54|0_._61|_−_0_._55|0_._60|0_._01|
|MEX|0_._13|0_._58|_−_0_._82|0_._42|0_._00|
|MYS|0_._83|0_._49|_−_0_._94|0_._18|0_._00|
|TWN|0_._20|0_._58|_−_0_._65|0_._32|0_._00|
|THA|0_._29|0_._61|_−_0_._58|0_._59|0_._00|
|TUR|0_._15|0_._72|_−_0_._61|0_._37|0_._00|
|JPN|2_._05|0_._49|_−_0_._35|0_._32|0_._06|
|USA|1_._38|0_._42|_−_0_._33|0_._29|0_._03|

> 14The estimation also uses standard values for other parameters governing production and demand. We report the values for these parameters in Table 5 in Appendix A.

8
**Parameter estimates.** There are a number of takeaways from Table 3. First, we find evidence of economically meaningful adjustment costs in all countries. Our estimates in the US are within the range, albeit toward the lower end, of previous values found in the literature.<sup>15</sup> If anything, they appear to be higher in the two developed countries in our sample, Japan and the US, as compared to the developing ones. Second, the estimates suggest that firms make investment decisions under considerable uncertainty. The information friction tends to be more severe in the developing countries. For example, the posterior uncertainty as a share of the prior ranges from lows of 42% in the US and 49% in Japan to as high as 72% in Turkey. Finally, the last three columns show the estimates for factors other than adjustment/information frictions. Turning first to the correlated component, the negative values of _γ_ suggest that, across all countries, these factors act to disincentivize investment by more productive firms. The estimates of _γ_ are more negative in the developing countries, suggesting the magnitude of these factors is larger in those countries. The estimates of the fixed component of the distortion are quite substantial across the board, while the transitory component appears to be negligible in most of the countries.

What features of the data lead us to these estimates? David and Venkateswaran (2017) develop a formal identification argument; here, we restrict ourselves to the key intuition. First, as we saw in Table 1, the volatility of investment is relatively low. By itself, this would suggest a significant role for adjustment frictions. However, quadratic adjustment costs tend to induce strong serial correlation in investment, since they create incentives to smooth capital accumulation over time. But, as we saw, the autocorrelation of investment is relatively low, which translates to relatively modest estimates for adjustment costs.<sup>16</sup> The low volatility of investment along with the high correlation between the _arpk_ and productivity together point to large negative values for _γ_ , which acts to dampen the responsiveness of investment to shocks without unduly raising the serial correlation. The non-trivial correlation of current investment decisions with lagged productivity changes – in other words, that firms do not seem to respond immediately to shocks – suggests learning and therefore, the degree of uncertainty. Finally, given all of these factors, matching the observed degree of cross-firm dispersion in _arpk_ requires substantial variation in the fixed-effect, _χi_ .

> 15Our estimate is smaller than that in Asker et al. (2014) and closer to (and slightly higher than) those              in Cooper and Haltiwanger (2006) and Bloom (2009) (although it should be noted that each of these papers      uses different data and considers different additional elements in their specification of these costs, making direct comparisons difficult).

> 16An obvious concern with this argument is the possibility for non-convex (e.g., fixed) costs of adjustment. David and Venkateswaran (2017) extend their baseline model to allow for such costs but find that they are quite small in the Chinese and US samples.

9
**Implications.** With these parameter estimates in hand, we compute their implications for measured misallocation and the resulting effects on measures of aggregate total factor productivity (TFP). We report these calculations in Table 3. Each horizontal panel contains the results for a single country. The top row of each panel displays the contribution of each of the factors to _arpk_ dispersion. First, we calculate these values under the assumption that only the factor of interest is operational, i.e., in the absence of the others, so that the contribution of each one is measured relative to a frictionless benchmark. Second, we recalculate the contributions of the non-technological factors, i.e., the components of the distortion, holding the technological frictions, i.e., adjustment/information, fixed at their estimated values. This second calculation takes into account interaction effects between the factors, whereas the first does not. The columns labeled ‘In Isolation’ display the contributions of distortions on their own; the columns labeled ’With Interactions’ their contributions (or the decrease in _arpk_ dispersion that would arise from eliminating them) in the presence of adjustment/information frictions. The second row of each panel expresses these contributions as a percentage of the total _arpk_ dispersion measured in the data. The last row translates these values into implied losses in aggregate TFP, in other words, by how much would aggregate productivity improve if one could eliminate the various factors (the mapping from _arpk_ dispersion to TFP losses is detailed in expression (2) in Appendix A).

The results show that adjustment costs and information frictions play a relatively modest role in generating the observed dispersion in _arpk_ . Together, these two forces account for 2 between about 5% (Taiwan, China) and about 20% (the US) of total _σ_ ( _arpk_ ) . The resulting

# TFP

losses are also modest – at their highest, they lead to losses of about 4% (China). In contrast, other firm-specific factors account for a large portion of the observed dispersion, everywhere upwards of about 80%. These effects are essentially entirely driven by the correlated and fixed components of the distortion, though there is some variation across countries in their relative importance. In other words, the main drivers of observed _arpk_ dispersion manifest themselves in firm-level data by (i) disproportionately reducing (increasing) the investment of high (low) productivity firms relative to what their fundamentals would dictate and (ii) as an extremely persistent firm-level effect.

It is also instructive to compare the characteristics of the two developed countries to the developing ones. First, as shown in Table 1, the total amount of _σ_ 2 ( _arpk_ ) is smaller in the developed countries, suggesting _a priori_ a lesser overall degree of misallocation. Thus, the absolute magnitude of each of the factors tends to be smaller in the developed countries. Table 3 reveals that as a _share_ of total dispersion, adjustment/information frictions tend to have a larger role in these countries, particularly in the US. Another key difference between the two sets of countries is in the magnitude of correlated factors: as shown in Table 3, the estimate of 10
Table 3: The Sources of Misallocation

|||||In Isolation|Other|Factors<br>W|ith Interactio|ns|
|---|---|---|---|---|---|---|---|---|
||Adjustment Costs|Uncertainty|Correlated|Permanent|Transitory|Correlated|Permanent|Transitory|
|**ARG**|||||||||
|Effect on_σ_<sup>2</sup>(_arpk_)|0_._00|0_._03|0_._16|0_._36|0_._00|0_._14|0_._36|0_._00|
|% of total_σ_<sup>2</sup>(_arpk_)|(1%)|(6%)|(29%)|(67%)|(0%)|(25%)|(67%)|(0%)|
|Effect on TFP|0_._00|0_._01|0_._07|0_._16|0_._00|0_._06|0_._16|0_._00|
|**BRA**|||||||||
|Effect on_σ_<sup>2</sup>(_arpk_)|0_._00|0_._05|0_._19|0_._42|0_._00|0_._17|0_._42|0_._00|
|% of total_σ_<sup>2</sup>(_arpk_)|(1%)|(8%)|(29%)|(64%)|(0%)|(26%)|(64%)|(0%)|
|Effect on TFP|0_._00|0_._02|0_._08|0_._18|0_._00|0_._07|0_._18|0_._00|
|**CHN**|||||||||
|Effect on_σ_<sup>2</sup>(_arpk_)|0_._01|0_._09|0_._33|0_._51|0_._00|0_._31|0_._51|0_._00|
|% of total_σ_<sup>2</sup>(_arpk_)|(1%)|(9%)|(36%)|(55%)|(0%)|(33%)|(55%)|(0%)|
|Effect on TFP|0_._00|0_._04|0_._14|0_._22|0_._00|0_._13|0_._22|0_._00|
|**COL**|||||||||
|Effect on_σ_<sup>2</sup>(_arpk_)|0_._02|0_._05|0_._30|0_._60|0_._01|0_._30|0_._60|0_._00|
|% of total_σ_<sup>2</sup>(_arpk_)|(3%)|(6%)|(31%)|(61%)|(1%)|(31%)|(61%)|(0%)|
|Effect on TFP|0_._01|0_._02|0_._13|0_._26|0_._00|0_._13|0_._26|0_._00|
|**MEX**|||||||||
|Effect on_σ_<sup>2</sup>(_arpk_)|0_._00|0_._04|0_._36|0_._42|0_._00|0_._33|0_._43|0_._00|
|% of total_σ_<sup>2</sup>(_arpk_)|(1%)|(5%)|(45%)|(53%)|(0%)|(41%)|(54%)|(0%)|
|Effect on TFP|0_._00|0_._02|0_._16|0_._18|0_._00|0_._14|0_._19|0_._00|
|**MYS**|||||||||
|Effect on_σ_<sup>2</sup>(_arpk_)|0_._02|0_._03|0_._54|0_._18|0_._00|0_._50|0_._18|0_._00|
|% of total_σ_<sup>2</sup>(_arpk_)|(3%)|(4%)|(73%)|(25%)|(0%)|(68%)|(25%)|(0%)|
|Effect on TFP|0_._01|0_._01|0_._23|0_._08|0_._00|0_._22|0_._08|0_._00|

Continued on next page
Table 3: The Sources of Misallocation – Continued

|||||In Isolation|Other|Factors<br>W|ith Interactio|ns|
|---|---|---|---|---|---|---|---|---|
||Adjustment Costs|Uncertainty|Correlated|Permanent|Transitory|Correlated|Permanent|Transitory|
|**TWN**|||||||||
|Effect on_σ_<sup>2</sup>(_arpk_)|0_._00|0_._02|0_._23|0_._32|0_._00|0_._22|0_._32|0_._00|
|% of total_σ_<sup>2</sup>(_arpk_)|(1%)|(4%)|(40%)|(56%)|(0%)|(39%)|(56%)|(0%)|
|Effect on TFP|0_._00|0_._01|0_._10|0_._14|0_._00|0_._10|0_._14|0_._00|
|**THA**|||||||||
|Effect on_σ_<sup>2</sup>(_arpk_)|0_._01|0_._05|0_._24|0_._59|0_._00|0_._24|0_._59|0_._00|
|% of total_σ_<sup>2</sup>(_arpk_)|(1%)|(5%)|(28%)|(67%)|(0%)|(27%)|(67%)|(0%)|
|Effect on TFP|0_._00|0_._02|0_._11|0_._26|0_._00|0_._10|0_._26|0_._00|
|**TUR**|||||||||
|Effect on_σ_<sup>2</sup>(_arpk_)|0_._01|0_._05|0_._14|0_._37|0_._00|0_._12|0_._37|0_._00|
|% of total_σ_<sup>2</sup>(_arpk_)|(1%)|(10%)|(25%)|(67%)|(0%)|(22%)|(67%)|(0%)|
|Effect on TFP|0_._00|0_._02|0_._06|0_._16|0_._00|0_._05|0_._16|0_._00|
|**JPN**|||||||||
|Effect on_σ_<sup>2</sup>(_arpk_)|0_._02|0_._01|0_._07|0_._32|0_._06|0_._08|0_._32|0_._00|
|% of total_σ_<sup>2</sup>(_arpk_)|(5%)|(3%)|(16%)|(73%)|(14%)|(18%)|(73%)|(0%)|
|Effect on TFP|0_._01|0_._01|0_._03|0_._14|0_._03|0_._03|0_._14|0_._00|
|**USA**|||||||||
|Effect on_σ_<sup>2</sup>(_arpk_)|0_._05|0_._03|0_._06|0_._29|0_._03|0_._08|0_._29|0_._00|
|% of total_σ_<sup>2</sup>(_arpk_)|(11%)|(7%)|(14%)|(65%)|(6%)|(17%)|(65%)|(0%)|
|Effect on TFP|0_._02|0_._01|0_._03|0_._13|0_._01|0_._03|0_._13|0_._00|

_Notes_ : The contribution of adjustment costs is calculated under the assumption that they are the only source of dispersion in _arpk_ , i.e. with all other factors set to zero. The contribution of uncertainty is computed under the same assumption. The contributions from other factors are calculated in two ways. The first (columns ‘In Isolation’) assumes that all other forces are set to zero while the second (columns ‘With Interactions’) holds adjustment costs/uncertainty at their estimated values. For each country, the first row reports the effect of each force on _σ_<sup>2</sup> ( _arpk_ ) ; the second row expresses these effects as a percentage of the observed _σ_<sup>2</sup> ( _arpk_ ) ; the third row calculates the implied TFP losses.
_γ_ is smallest (in absolute value) in the developed countries, indicating that productivity or sizedependent distortions are relatively less important there. For example, these factors account 2 for 14% and 16% of _σ_ ( _arpk_ ) in Japan and the US, respectively. In contrast, the figures tend to be substantially higher in the developing countries, where correlated factors explain from 25% (Turkey) to 73% (Malaysia) of total _arpk_ dispersion. The implied TFP losses in these latter countries are thus substantial, ranging from 6% to as high as 23%.

# **4 Heterogeneity in Markups and Technologies**

In this section, we dig deeper into the firm-specific factors that seem to generate much of the observed _arpk_ dispersion. In particular, we investigate two potential sources – unobserved heterogeneity in markups and production technologies. In our baseline setup, all firms within an industry (1) had homogeneous production technologies and (2) were monopolistically competitive facing CES demand curves and therefore, set identical markups. As a result, any firm-level heterogeneity in technologies and/or markups would show up in our estimates as other firmspecific factors. In what follows, we relax both of these assumptions. To focus on the role of these two forces, the analysis abstracts from adjustment/information frictions in firms’ input decisions. This is largely in the interest of simplicity, but it is also supported by the relatively modest role played by these dynamic considerations in our baseline estimates.

**Markup dispersion.** To explore the potential for markup dispersion, we follow the methodology in De Loecker and Warzynski (2012). We first extend our production function to include raw materials. Under two key assumptions: (i) the materials elasticity in production is common across firms (within an industry) and (ii) the materials choice is a static, otherwise undistorted (besides the markup) decision, we can directly measure these firm-specific markups using the average revenue product of materials, _arpm_ (the inverse of the share of revenues paid to intermediates). This result implies that the cross-sectional dispersion in _arpm_ can be mapped one-for-one into dispersion in markups (for detailed derivations, see Appendix B).

Data on intermediate inputs are only available for four of the countries in our sample – China, Colombia, Mexico and the US.<sup>17</sup> In the first column of Table 4, we report the (withinindustry) dispersion in the _arpm_ for these four countries. In the first column of the second panel of the table, we report the dispersion in _arpk_ that arises from markups (this is simply equal 2 to the markup dispersion itself, given by _σ_ ( _arpm_ )) and the share of observed _arpk_ dispersion 17In the US, we construct the measure of intermediate expenditures as sales less operating income before depreciation less wage bill. Following, e.g., Keller and Yeaple (2009), the wage bill is imputed as the number of employees multiplied by the average industry wage calculated using data from the NBER-CES Manufacturing Industry Database (available at http://www.nber.org/nberces/).

13
markups can explain. The results show that markup dispersion can account for between 4% 2 (China) and 18% (Colombia) of the observed _σ_ ( _arpk_ ) . Although the shares are generally significant (except in China), these findings seem to suggest that markup heterogeneity is unlikely to be the primary force behind the unexplained _arpk_ dispersion.

Table 4: Heterogeneous Markups and Technologies

|||M|oments||Effe<br>(% of|ct on_σ_<br>2 (_arpk_)<br>total_σ_<br>2 (_arpk_))|
|---|---|---|---|---|---|---|
||_σ_<br>2 (_arpm_)|_σ_<br>2 (_arpk_)|_σ_<br>2 (_arpn_)|cov(_arpk, arpn_)|Markups|Capital Elasticities|
|ARG|-|0.55|0.31|-0.01|-|0.32<br>(58%)|
|BRA|-|0.74|0.52|0.09|-|0.40<br>(54%)|
|CHN*|0.05|1.37|0.76|0.41|0.05<br>(4%)|0.48<br>(37%)|
|COL*|0.22|1.49|0.87|0.40|0.22<br>(18%)|0.59<br>(48%)|
|MEX*|0.13|1.18|0.50|0.31|0.13<br>(12%)|0.35<br>(33%)|
|MYS|-|0.93|0.49|0.35|-|0.25<br>(27%)|
|TWN|-|0.62|0.30|0.18|-|0.20<br>(32%)|
|THA|-|0.95|0.76|0.22|-|0.48<br>(51%)|
|TUR|-|0.84|0.47|0.09|-|0.41<br>(49%)|
|JPN|-|0.56|0.29|0.08|-|0.24<br>(44%)|
|USA*|0.06|0.52|0.35|0.23|0.06<br>(14%)|0.16<br>(38%)|

_Notes_ : This table reports estimates of (within-industry) dispersion in firm-level markups and production technologies. The first panel displays the data moments used for these calculations – dispersion in the average revenue products of materials ( _arpm_ ), capital ( _arpk_ ) and labor ( _arpn_ ), and the covariance of the _arpk_ and _arpn_ . For the countries denoted with an asterisk, the _arpk_ and _arpn_ moments pertain to markup-adjusted average revenue products, calculated by subtracting the firm-specific markup (measured by _arpmit_ ) from _arpkit_ and _arpnit_ , as detailed in equations (4) and (5). The second panel of the table displays the contributions of the two forms of heterogeneity to the dispersion in _arpk_ . Values in parentheses express these contributions as percentages of total observed dispersion.

**Technology heterogeneity.** Next, we analyze the contributions of dispersion in production technologies. Specifically, we generalize the production function to allow for firm-specific output elasticities of capital and labor. The key insight that informs our analysis is the observation

14
that, all else equal, with firm-specific elasticities, a high output elasticity of capital would be associated with a low output elasticity of labor. As a result, to the extent this form of heterogeneity is present, it would induce the average revenue products of capital and labor to ˆ move in opposite directions – specifically, firms with a high _αit_ will, _ceteris paribus_ , tend to have a low _arpk_ and a high _arpn_ and _vice versa_ . We can exploit this implication to derive an upper bound on the potential for this type of heterogeneity based on the observed covariance of _arpk_ and _arpn_ (we derive the bound in Appendix B).<sup>18</sup> Table 4 reports the results from computing this bound in each country. In the last three columns of the first panel, we show the values of the moments that are used in the bound calculation: _σ_ 2 ( _arpk_ ) , _σ_ 2 ( _arpn_ ) and cov ( _arpk, arpn_ ) . The column labeled ‘Capital Elasticities’ shows the implied dispersion in _arpk_ arising from heterogeneity in firm-level technologies. With the caveat that these represent upper bounds, they point to a substantial role for this form of heterogeneity: depending on the country, it can explain between about one-quarter and onehalf of total _arpk_ dispersion. A promising, if challenging, area for future research would be to further explore this channel by estimating firm-level production functions. This would likely require data with a significant panel dimension.

# **5 Conclusion**

This paper has explored the determinants of capital allocations – in particular, the sources of dispersion in the average revenue product of capital, a standard indicator of capital misallocation – across a number of developing and developed countries. Figure 1 summarizes the main findings. Our results suggest that much of the observed dispersion stems not from technological and informational frictions but, rather, from other firm-specific factors, both ones systematically correlated with firm productivity/size and ones that are relatively persistent. The former is particularly important in the developing countries in our sample. We also find that unobserved heterogeneity in demand and production technologies can potentially account for a significant portion of the observed _arpk_ dispersion.

The obvious next step in this research agenda is to explore specific candidates for the sizable firm-specific factors our estimation uncovers, along the lines of our analysis in Section 4. Our results will hopefully prove useful both in identifying suitable candidates as well as in guiding the empirical strategy to measure their impact. For example, policies and/or frictions which induce transitory wedges that are unrelated to firm size seem relatively less promising. On the

> 18Markups, on the other hand, are a source of positive comovement between the average revenue products of all inputs – see Appendix B for details. For the countries where we have materials data, we can use it to adjust  the observed _cov_ ( _arpk, arpn_ ) to control for the effect of markups. The numbers reported in Table 4 for China, Colombia, Mexico and USA are after this adjustment (again, see Appendix B for details).

15
<!-- Start of picture text --> 100 100<br>90 90<br>80 80<br>70 70<br>60 60<br>50 50<br>40 40<br>30 30<br>20 20<br>10 10<br>0 0<br>ARG BRA CHN COL MEX MYS TWN THA TUR JPN USA<br>Adj. Costs Information Markups Technologies Unexplained<br>Percent of Total Dispersion<br><!-- End of picture text --> Figure 1: The Sources of Misallocation empirical front, our analysis could help researchers investigate particular forces while controlling for others in a tractable, albeit reduced-form, way. A recent example is David, Schmid, and Zeke (2018), who propose a theory based on heterogeneity in firm-level risk premia that delivers a fixed wedge in firms’ capital choices of exactly the type our results ascribe a large role to. To quantify this mechanism, they adopt an approach that is robust to the presence of other distortions, modeled along the lines we have outlined here.

Our methodological insights should also serve to guide future data collection efforts. They show how additional data, e.g., on other inputs into production, can be very useful in disentangling the drivers of average product dispersion. The availability of richer and more detailed firm-level data would help us go beyond the relatively simple calculations in this note to more sophisticated techniques.

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# **Appendix**

## **A Theoretical Framework**

This section lays out the theoretical environment underlying our exercise. It reproduces the main features of the setup outlined in Section 2 of David and Venkateswaran (2017). For further details and full derivations we refer the reader to that paper.

**The environment.** The economy is populated by a continuum of firms, indexed by _i_ , which produce a set of intermediate goods using a Cobb-Douglas technology:

A representative final good firm operating in a competitive market frictionlessly bundles these intermediate goods using a standard CES aggregator where _θ ∈_ (1 _, ∞_ ) is the elasticity of substitution between intermediate goods and _A_<sup>ˆ</sup> _it_ represents a firm-specific component, which can capture idiosyncratic firm productivity or demand factors. This yields a standard demand function for intermediate good _i_ : where _Pit_ denotes the relative price of good _i_ in terms of the final good. Revenues for firm _i_ at time _t_ are

**Input choices.** Firms hire labor on a period-by-period basis under full information at a competitive wage _Wt_ . At the end of each period, firms choose investment in new capital, which becomes available for production in the following period. Investment is subject to quadratic adjustment costs, given by

19
where _ξ_<sup>ˆ</sup> parameterizes the severity of the adjustment cost and _δ_ is the rate of depreciation.

To capture additional factors that may influence firm investment decisions (in addition to productivity/demand or the level of previously installed capital), we introduce a class of idiosyncratic ‘wedges’ that appear in the firm’s optimization problem as proportional taxes on the flow cost of capital. These wedges can arise, for example, from distortionary government policies, additional un-modeled market frictions (e.g., financial frictions) or unobserved heterogeneity in markups/production technologies. We denote these wedges by _itT_<sup>_K_</sup> .

We assume the economy is in a stationary equilibrium in which all aggregate variables remain constant (and henceforth suppress the time subscript on aggregates). After optimizing over the static labor decision, the firm’s problem in a stationary equilibrium can be written in recursive form as _·_ where E _it_ [ ] denotes the firm’s expectations conditional on _I it_ , the information set of the firm at the time of making its period _t_ investment choice. We describe this set explicitly below. The term 1 _− β_ (1 _− δ_ ) is the user cost per unit of capital. The constant _G_ is given by<sup>_<u>α</u>_</sup> <u>2 1 1 1</u> _G_ ductivity/demand and _≡_ (1 _− α_ 2 ) _~~α~~ W_ ~~2~~ 1 _−α_ 2 _Y θ α_ 1 _−α_ 2 ,<sup>_<u>α</u>_</sup> _A_ <u>1</u> _it ≡ A_<sup>ˆ1</sup> _it_<sup>_−α_2</sup> is a simple transformation of idiosyncratic pro- _≡_ 1 _− α_ 2<sup>is the curvature of operating profits (revenues net of wages).</sup> From here on, we simply refer to _Ait_ as firm-level productivity.

**Equilibrium.** A _stationary equilibrium_ in this economy is defined as (i) a set of value and policy functions for the firm, _V_ ( _Kit, I it_ ) _, Nit_ ( _Kit, I it_ ) and _Kit_ +1 ( _Kit, I it_ ) _,_ (ii) a wage _W_ and (iii) a joint distribution over ( _Kit, I it_ ) such that (a) taking as given wages and the law of motion for _I it_ , the value and policy functions solve the firm’s optimization problem, (b) the labor market clears and (c) the joint distribution remains constant through time.

**Characterization.** We use perturbation methods to characterize the equilibrium. In particular, we log-linearize the firm’s optimality conditions and laws of motion around _Ait_ = _A_<sup> ̄</sup> (the unconditional average level of productivity) and _Tit_<sup>_K_</sup> = 1 (i.e., no distortions). David and Venkateswaran (2017) show that the setup leads to the following log-linearized Euler equation:

20
where we use lower-case to denote natural logs. The parameter _ξ_ is a composite parameter that captures the degree of adjustment costs and _τit_ summarizes the effect of _Tit_<sup>_K_</sup> on the firm’s investment decision.<sup>19</sup>

**Stochastic processes.** We assume that the productivity _Ait_ follows an AR(1) process in logs with normally distributed i.i.d. innovations, i.e.,

where _ρ_ is the persistence and _σ_ 2 _μ_ the variance of the innovations.

We adopt a specification for the distortion, _τit_ , that allows for a rich correlation structure, both over time as well as with firm-level productivity. Specifically, _τit_ takes the form: where the parameter _γ_ controls the extent to which _τit_ co-moves with productivity. If _γ <_ 0 , the wedge discourages (encourages) investment by firms with higher (lower) productivity – arguably, the empirically relevant case. The opposite is true if _γ >_ 0 . The uncorrelated component of _τit_ has both permanent and iid (over time and across firms) parts, denoted _χi_ and _εit_ respectively. 2 2 Thus, the severity of these factors is summarized by 3 parameters: ( _γ, σo , σχ_ ) .

**Information.** Next, we spell out _I it_ , the information set of the firm at the time of choosing _Kit_ +1. This includes the entire history of productivity realizations through period _t_ , i.e., _{ ait − s }_<sup>_∞_</sup> _s_ =0<sup>.  GiventheAR(1)assumption, thiscanbesummarizedbythemostrecentobserva-</sup> tion, namely _ait_ . The firm also observes a noisy signal of the following period’s innovation:

where _eit_ +1 is an i.i.d., mean-zero and normally distributed noise term. This is in essence an idiosyncratic ‘news shock,’ since it contains information about future productivity. Finally, firms also perfectly observe the uncorrelated transitory component of distortions, _εit_ (as well as the fixed component, _χi_ ) at the time of choosing period _t_ investment. They do not see the correlated component but are aware of its structure, i.e., they know _γ_ .

21
Thus, the firm’s information set is given by _I it_ = ( _ait, sit_ +1 _, εit, χi_ ) . Direct application of Bayes’ rule yields the conditional expectation of productivity, _ait_ +1: There is a one-to-one mapping between the posterior variance V and the noisiness of the signal, 2 2 _σ_ (given the volatility of productivity, _σ_ ). In the absence of any learning (or ‘news’), i.e., when _e μ σe_ 2 approaches infinity, V = _σμ_ 2, that is, all uncertainty regarding the innovation in productivity, _μit_ +1, remains unresolved at the time of investment. In this case, we have a standard one period time-to-build structure with E _it_ [ _ait_ +1 ] = _ρait_ . At the other extreme, when _σ_ 2 _e_ approaches zero, V = 0 and the firm becomes perfectly informed about _μit_ +1 so that E _it_ [ _ait_ +1 ] = _ait_ +1.

**Optimal investment.** With this structure, we can explicitly solve for the firm’s (log-linearized) optimal investment policy:

where – The coefficients _ψ_ 1 _ψ_ 4 depend only on production (and preference) parameters, including the adjustment cost, and are independent of assumptions about information and distortions. Note that the coefficient _ψ_ 1 is increasing and _ψ_ 2- _ψ_ 4 decreasing in the severity of adjustment costs, 1 _ξ_ . If there are no adjustment costs (i.e., _ξ_ = 0 ) , _ψ_ 1 = 0 and _ψ_ 2 = _ψ_ 3 = _ψ_ 4 = . At the 1 _− α_ other extreme, as _ξ_ tends to infinity, _ψ_ 1 approaches one and _ψ_ 2- _ψ_ 4 go to zero. Intuitively, as adjustment costs become large, the firm’s choice of capital becomes more autocorrelated and less responsive to productivity and distortions.

**Aggregation.** Aggregate output in this economy can be expressed as

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where _k_ and _n_ denote the (logs of the) aggregate capital stock and labor inputs, respectively. Aggregate TFP, denoted by _a_ , is given by where _a_<sup>_∗_</sup> is aggregate TFP if static average (revenue) products of capital (in logs, _arpkit_ = 2 _pityit − kit_ ) are equalized across firms and _σarpk_ is the cross-sectional dispersion in the _arpkit_ . Thus, aggregate TFP is monotonically decreasing in the extent of _arpk_ dispersion, summarized in this log-normal world by _σarpk_ 2 . The effect of _σ_ 2 on aggregate TFP depends on the elasticity _arpk_ of substitution, _θ_ , and the relative shares of capital and labor in production. The higher is _θ_ , that is, the closer we are to perfect substitutability, the more severe the losses from variation in valueadded/capital ratios. Similarly, fixing the degree of overall returns to scale in production, for a larger capital share, _α_ ˆ 1, a given degree of dispersion has larger effects on aggregate outcomes.

**Calibration.** As noted in the text, for simplicity, we hold ‘standard’ demand and technology parameters fixed across countries in all of our exercises. Table 5 report the values we use. We set the elasticity of substitution, _θ_ , equal to 6 and the capital and labor output elasticities ˆ to one-third and two-thirds, respectively. These values yield _α_ = 1 _−_ _<u>α</u>_ <u>1</u> _α_ 2 = 1 _−_ <u>(</u> 1 ~~(~~ _−_ 1 _−θ_<sup>~~1~~</sup> <u>)</u> _θ_<sup>1</sup> _α_ ~~)~~ 1 _α_ ˆ 2<sup>= 0</sup><sup>_._62.</sup>

The discount rate, _β_ , is set to 0.95 annually and the rate of depreciation, _δ_ , to 0.10. Table 5 summarizes these values.

Table 5: Calibration of Common Parameters

|Parameter|Description|Value|
|---|---|---|
|_θ_|Elasticity of substitution|6|
|_α_ˆ1|Capital share|0.33|
|_α_ˆ2|Labor share|0.67|
|_β_|Discount rate|0.95|
|_δ_|Depreciation rate|0.10|

# **B Heterogeneity in Markups/Technologies**

We extend the production function to include materials and to allow for (potentially timevarying) heterogeneity in capital intensities. Specifically, the output of firm _i_ at time _t_ is: 23
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where _Mit_ denotes intermediate or materials input. The price of these inputs, potentially firmspecific, is denoted _Pit_<sup>_M_</sup> . We assume that the choice of intermediates is undistorted except for the firm-specific markup (and intermediate goods price). As noted in the text, for simplicity, we abstract from adjustment/information frictions in firms’ capital and labor choices. Both decisions are made subject to a factor-specific ‘distortion’ (in addition to the markup), denoted _T_<sup>_K_</sup> and _T_<sup>_N_</sup> , respectively.

_it it_ Under this structure, the firm’s cost minimization problem is where _Rt_ is the cost of capital. The optimality conditions yield expressions for the average revenue product of the inputs: or and where _MCit_ is the Lagrange multiplier on the constraint (i.e., the marginal cost) and _τ it_<sup>_K_</sup> and _N τit_ are the logs of the capital and labor wedges. Expression (3) shows that dispersion in the _arpm_ maps one-for-one into the dispersion in (log) markups, i.e., Observed average revenue products of capital and labor are combinations of the firm-specific production elasticities, as well as markups and distortionary factors. Importantly, the expressions reveal that the capital elasticity, _α_ ˆ _it_ has opposing effects on the average products of capital and labor. Specifically, firms with a high _α_ ˆ _it_ will, _ceteris paribus_ , tend to have a low _arpk_ and a high _arpn_ . This property enables us to use the observed covariance of the average products to bound the extent of variation in _α_ ˆ _it_ .

We assume that the firm-specific distortions, _τ_<sup>_K_</sup> and _τ_<sup>_N_</sup> are uncorrelated with log _α_ ˆ _it_ . Then, 25
_it it_ given the observed second moments of ( _arpkit, arpnit_ ) , an upper bound on the variation in log _α_ ˆ _it_ can be obtained by assuming that the correlation between _τ it_<sup>_K_</sup> and _τ it_<sup>_N_</sup> to one. Intuitively, this 26
maximizes potential for variation in _α_ ˆ _it_ , which is a source of negative correlation between _arpkit_ and _arpnit_ .

To implement this calculation, first consider the case when we have data on materials, i.e. we can measure firm-specific markups. Define ‘markup-adjusted’ average products as with second moments: from which we can solve for the correlation of the distortions, Given the observed values of second moments of _arpk_ and _a rpn_ , the maximum possible variation in _α_ ˆ _it_ is one where _ρ τ it_<sup>_K_</sup> _, τ it_<sup>_N_</sup> = 1 . To calculate this upper bound numerically, we assume that _α_ ˆ _i_ is distributed as a truncated log-normal, with a lower bound of zero and upper bound of _ζ_<sup>ˆ</sup> . 2 We then simulate 100,000 firm observations and find _σ_ log _α_ ˆ that sets _ρ τ_<sup>_K_</sup> _it, τ_<sup>_N_</sup> _it_ to one.

To see the intuition more clearly, note that Using this, the inequality in (6) can be re-arranged to yield: 2
_α_ ̄ _._ + _ζ_<sup>ˆ</sup> _− α_ ̄ ( 7 ) _σa_<sup>2</sup> _rpk_<sup>+</sup><sup>_σ_</sup> _arpn_<sup>2</sup> Expression (7) reveals the main insight: the more positive the observed covariance between _arpkit, arpnit_ , the lower is the scope for heterogeneity in _α_ ˆ _it_ .

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For countries where we do not have materials data, a similar logic holds, with one modification. Because we can no longer explicitly adjust for markup variation, we treat them as part of the unobserved _τ it_<sup>_K_</sup> and _τ_<sup>_N_</sup> _it_ . Then, the observed second moments of the (unadjusted) _arpk_ and _arpn_ directly enter expression (6), rather than the moments of the markup-adjusted counterparts.

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