metadata: authors: 0: author: Fattal Jaef,Roberto N.
count: China
docna: 0: docna: Entry and exit, multi-product firms, and
            allocative distortions
docty: Policy Research Working Paper
subtopic: Macro-Fiscal Policy,Consumption,Fiscal & Monetary Policy,Taxation & Subsidies,Economic Adjustment and Lending,Construction Industry,Common Carriers Industry,Food & Beverage Industry,General Manufacturing,Pulp & Paper Industry,Plastics & Rubber Industry,Labor Markets,Public Sector Economics,International Trade and Trade Rules,Business Cycles and Stabilization Policies,Public Finance Decentralization and
            Poverty Reduction
teratopic: Law and Development,Private Sector Development,Industry,Macroeconomics and Economic Growth
abstracts: cdata!: This paper proposes a multi-product
            model of firm dynamics to understand the implications of
            allocative distortions for the decisions of firms to enter,
            exit, and supply products to the market. These margins of
            adjustment have been largely neglected in the literature yet
            have direct contributions to welfare and productivity. The
            paper finds that when the analysis accounts for these
            channels, the traditional focus on long-run gains in Total
            Factor Productivity from reversing misallocation strongly
            underestimates the welfare gains that accrue when accounting
            for transitional dynamics. Calibrating the distortions to
            China in 1998, the analysis finds a welfare gain of 32
            percent and a steady-state gain of 10 percent.
geo_reg_key: 517191,517198,119225
repnme: repnme: Entry and exit, multi-product firms, and
            allocative distortions
display_title: Entry and exit, multi-product firms,
            and allocative distortions
listing_relative_url: /research/2017/04/27328490/entry-exit-multi-product-firms-allocative-distortions
topicv3: Manufacturing Sectors m1326521 1281,Trade m1326924 698,Macroeconomic and Structural Policies
            m1327380 439,Taxation m1327615 1121,Labor Demand m1583073 4422
publishtoextweb_dt: 2017-04-04T00:00:00Z
historic_topic: Law and Development,Private Sector Development,Industry,Macroeconomics and Economic Growth
no_of_pages: 39
author_keys: 0: author_key: 1253388
model_extractions: WPS8023 Policy Research Working Paper 8023 Entry and Exit, Multi-Product Firms, and Allocative Distortions _Roberto N. Fattal Jaef_ Development Research Group Macroeconomics and Growth Team April 2017
Policy Research Working Paper 8023

# **Abstract**

This paper proposes a multi-product model of firm dynamics to understand the implications of allocative distortions for the decisions of firms to enter, exit, and supply products to the market. These margins of adjustment have been largely neglected in the literature yet have direct contributions to welfare and productivity. The paper finds that when the analysis accounts for these channels, the traditional focus on long-run gains in Total Factor Productivity from reversing misallocation strongly underestimates the welfare gains that accrue when accounting for transitional dynamics. Calibrating the distortions to China in 1998, the analysis finds a welfare gain of 32 percent and a steady-state gain of 10 percent.

This paper is a product of the Macroeconomics and Growth Team, Development Research Group. 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://econ.worldbank.org. The author may be contacted at rfattaljaef@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
# **Entry and Exit, Multi-Product Firms, and Allocative Distortions**<sup>_∗_</sup>

Roberto N. Fattal Jaef<sup>_†_</sup>

## _JEL CODES: O41, O43, D61_

## _Keywords: misallocation, entry and exit, multi-product, welfare_

> _∗_ We thank Andy Atkeson, Francisco Buera, Ariel Burstein, Hugo Hopenhayn, Lee Ohanian, Richard Rogerson, Luis Serven, and Venky Venkateswaran for helpful comments and suggestions. We also thank seminar participants at Stanford University, The Society of Economic Dynamics, and The World Bank’s Research Group. A previous version of this paper circulated as Entry, Exit, and Misallocation Frictions. The views presented here do not represent the views of the World Bank Group or any of it member countries. All errors are omission are the sole responsibility of the author.

> _†_ The World Bank, Development Research Group. Macroeconomics and Growth Division. 1818 H st NW MC 3-353, Washington, DC 20433; rfattaljaef@gmail.com
# **1 Introduction**

Resource misallocation across firms is pervasive in developing countries<sup>1</sup> . The evidence shows that an excessive amount of production factors are allocated to unproductive firms at the expense of the productive ones, a pattern that deviates from the optimal prescription of equalizing marginal returns across producers<sup>2</sup> . Interpreting the evidence as an outcome of underlying allocative distortions, various studies have shown that a reversal of these distortions would bring about substantial gains in Total Factor Productivity ( _TFP_ ), hence postulating resource misallocation as a promising candidate for reconciling the large differences in economic development that exist across countries. A notable omission in the majority of these studies, however, is the entry and exit of firms, and the firms’ adjustments to the number of products supplied to the market. These decisions are not only empirically relevant margins of adjustment at the firm level<sup>3</sup> but also have direct effects on welfare and _TFP_ . The goal of this paper is to bring these margins of adjustment to the forefront of the analysis of misallocation, investigating the extent to which allocative distortions affect the product-portfolio composition of firms and their decisions to enter _TFP_ . and exit production, and quantifying their implications for welfare and We find that when the response in the number of varieties is taken into account, the focus on long-run measures of aggregate productivity growth fails to capture a significant share of the welfare gain that accrues when considering transitional dynamics. For instance, a liberalization of distortions calibrated to match the pat1Salient examples in the literature establishing this fact are Hsieh and Klenow (2009) and the many subsequent replications of their methodology to other countries; and Bartelsman et al. (2013) with its respective set of replications.

> 2Our qualification of the prescription to reallocate resources across firms as optimal is accurate only under a restrictive but typically imposed set of assumptions, namely competitive factor markets and costless mobility of resources across firms. Asker et al. (2014) warn, however, about the potential efficiency in the observation of dispersion in the marginal return to capital in a context where there are adjustment costs to the mobility of capital across firms. Throughout the rest of the paper we shall continue with the initial view of interpreting dispersion in marginal returns as an outcome of distortions.

3Roberts and Tybout (1996), for instance, studied the micro patterns of job turnover in the developing economies of Chile, Colombia, Mexico and Morocco. They found entry and exit to account for about one-half of overall job turnover in these economies. Davis and Haltiwanger (1990) found this share to be about one-third in the United States. In terms of multi-variety production in developing countries, Goldberg et al. (2010) document the prevalence of multi-product firms and product turnover in India 2
provement in long-run _TFP_ , while the welfare gains accounting for the short-run . dynamics amount to 32%<sup>4</sup> While the motivation in the literature for focusing on the long-run implications of misallocation frictions stemmed primarily from the interest in thinking about these frictions as drivers of cross-country differences in _TFP_ , this motivation did not create any concern about mis-measuring welfare because the static nature of the underlying models implied that there was no distinction between the short and the long-run. Once entry, exit, and multi-variety production are taken into account, however, the number of varieties and the distribution of firms across productivity levels constitute the dynamic factors that give meaning to the distinction between short and long run. What our findings dictate is that the adjustment that occurs to these dynamic variables along the transition path is essential for fully capturing the welfare gains from reforms that reverse allocative distortions.

Our model brings the Bernard et al. (2011) model of multi-variety production into the Luttmer (2007) model of firm dynamics. The economy consists of a unit mass of differentiated products, each of which is a CES composite of a continuum of differentiated varieties. Upon entry, firms are endowed with a firm-wide productivity term and a time invariant realization of product attributes for the production of a variety of each of the products in the unit measure. Firm-wide productivity evolves stochastically over time, leading to firm growth over the life cycle conditional on survival. Product attributes are drawn independently and identically, across goods and firms, from a time-invariant probability distribution function. The firm’s portfolio of products is limited by a fixed cost of provision. Together with fixed costs of operation, these costs determine whether firms stay or exit operations and the number of products they supply to the market. Misallocation across intermediate producers is created, as in Restuccia and Rogerson (2008), through a distribution of idiosyncratic revenue taxes that are positively correlated with the distribution of firm-wide productivity of the firms. The correlated nature of distortions is intended to make the model consistent with the empirical observation of unproductive firms being too large and productive firms being too small in developing countries relative to the United States<sup>5</sup> . Since idiosyncratic distortions in the data are measured at the firm

> 4These numbers follow from calibrating distortions in the model to Hsieh and Klenow (2007)’s estimate of the regression coefficient between the log of _TFPR_ and the log of _TFPQ_ across firms in China in 1998, which amounts to 0.5, and subtracting the same estimate for the United States, which amounts to 0.1.

> 5This property of the firm size distribution in developing countries has been extensively documented in the literature. See, for instance, the works of Hsieh and Klenow (2009), Neumeyer and

3
or establishment level, rather than at the product level, we assume the same degree of taxation across all products within a firm.

The justification for the divergence in welfare metrics lies in the increase in the number of firms that takes place in the long run equilibrium with distortions. In a context where entering firms’ idiosyncratic productivity grows over time conditional on survival, a profile of distortions that taxes the most productive firms and subsidizes the least productive ones constitutes a redistribution of profits from the future to the present. Combined with positive discounting of future profit streams, this redistribution is less detrimental for the expected profits of entrants than it is of average profits among incumbents. This relative advantage of entrants’ expected profitability requires an increase in entry, and hence in the number of firms, to restore the equilibrium. When allocative distortions are removed, then, a process of firm decumulation begins along which resources are reallocated away from entry and fixed operation costs and into the production of goods. This productive reallocation gives a temporary boost to consumption that is not captured in measures of long run consumption growth. The transition path is also characterized by a reallocation of products from low to high productivity firms, a feature that reinforces the temporary overshoot in consumption that is allowed for by the withdrawal of distortions.

Despite becoming an imperfect measure of welfare, the long-run gains in aggregate productivity were at the core of the motivation for exploring cross-country differences in allocative efficiency. Hence, in the second part of the quantitative analysis we seek to establish a tighter connection with the literature by offering a quantitative evaluation of the contributions of entry, exit, and multi-variety production for the purpose of accounting for cross-country differences in aggregate productivity. In order to transparently identify the contributions of the new margins, we compare the long run changes in _TFP_ in three alternative long run allocations where entry, exit, and multi-variety production are at play sequentially. First, we consider a liberalization that reallocates resources across firms according to the distribution of firms across productivity in the distorted allocation. This source of gain, which we label as static, is the one that Hsieh and Klenow (2009) and the many replications of it in other countries capture. Next, we relax the assumption of constant distribution of products and solve for the efficient product-portfolio allocation, while still holdSandleris (2009), Casacuberta and Gandelman (2009), Camacho and Conover (2010) 4
ing constant the number of firms and the exit productivity thresholds. This allocation captures the static gains plus the gains from further reallocating more products away from unproductive firms and towards productive ones. Lastly, we solve for the overall gains accounting for the efficient entry and exit decisions.

We find that while the product-reallocation channel of efficiency gain reinforces the productivity improvements from static reallocation, entry and exit decisions tend to offset it, making the differential change in _TFP_ relative to the static benchmark be in general ambiguous. The offsetting effect of the number of firms predominates at low to middle degrees of distortions, where the misallocation from high to low productivity firms is still weak in terms of harming allocative efficiency but is strong in terms of the flattening of the time-series profile of profits, which drove the increase in entry. As the log-linear relationship between distortions and productivity gets more sloped, the relative strengths revert. While ambiguous in general, the majority of the existing estimates of the elasticity between distortions ( _TFPR_ ) and productivity ( _TFPQ_ ) in developing countries fall in the range where the mitigating and . magnifying forces almost cancel out<sup>6</sup> This suggests that while the consideration of margins of adjustment that affect the number of varieties may not have a radical effect over the role attributed to resource misallocation in accounting for cross-country gaps in _TFP_ , it does substantially affect the overall gains to be reaped from efforts to implement reforms that alleviate countries from allocative distortions.

The rest of the paper is organized as follows. Section 2 presents the model, and characterizes a stationary equilibrium. In Section 3, we calibrate parameter values and perform the quantitative experiments. We begin with the welfare analysis and the shift focus to the long-run implications. Concluding remarks are in section 4.

# **2 The Economic Environment**

The model economy embeds Bernard et al. (2011)’s theory of multi-product firms into Luttmer (2007)’s closed economy model of firm dynamics, and introduces misallocation frictions in the form of idiosyncratic taxes operating at the level of the 6Hsieh and Klenow (2007) report slopes of 0.5, 0.4, and 0.1 for China, India, and the United States, respectively. Chen and Irarrazabal (2015) estimate the elasticity to be between 0.6 and 0.5 in Chile during the period of 1980-1996. Cirera et al. (2017) report values ranging between 0.4 and 0.6 in Ghana, Ethiopia, Kenya, and Cote D’ Ivoire. When subtracting the 0.1 estimate for the US, we get that all of these estimates fall between 0.3 and 0.5 which, as we show later in the paper, is around the region where the two forces neutralize each other.

5
aggregate revenue of the firm. Motivated by the empirical pattern of idiosyncratic distortions in the data, these frictions are assumed to be positively and log-linearly related to firm-wide productivity.

# **2.1 Household’s Problem**

There is a representative household with preferences of the form ∑<sup>∞</sup> _t_ =0<sup>_βt_[</sup><sup>_log_(</sup><sup>_Ct_)],</sup> where _Ct_ is the single final good produced in the economy. Lifetime utility maximization is done subject to a standard inter-temporal budget constraint of the form: where _Qt_ denote inter-temporal prices, _M_ 0 is the initial endowment of wealth (claims to the profits and losses from the initial distribution of firms), and _Tt_ represents the lump-sum tax/transfer that balances the deficit or surplus from the collection of idiosyncratic taxes and subsidies. Notice that by rebating this revenue back to (or taking it away from) the household, we are ensuring that all the welfare implications of misallocation frictions manifest their effect on solely through aggregate productivity, rather than from wasteful consumption of goods from the government. Lastly, notice that the inter-temporal accumulation of wealth pins down the economy’s interest rate, given by _Rt_ =<sup>_<u>Q</u>_</sup> _Q_<sup>_<u>t</u>_</sup><sup><u>+</u></sup> _t_<sup><u>1</u>.</sup>

# **2.2 Technologies**

There is a single final good in the economy that is produced according to the following CES composite of a continuum of measure one of intermediate inputs: We assume that there is a representative producer of the final good that operates under perfect competition.

Each of the intermediate goods is, in turn, another CES composite of a continuum of horizontally differentiated varieties of each product supplied in monopolistically competitive markets by an endogenously determined measure of heteroge6
Varieties are differentiated by the firm-wide productivity level of the producer supplying ( _ω_ ) it and the product-specific productivity attribute ( _λ_ ) . We explain later the origin of these idiosyncratic characteristics of the firms, as well as the decision involved in the mass of and the process determining producers product-portfolio. For now, it suffices to note that Ω ( _ω_ , _λ_ ) denotes the measure of producers with this productivity mix.

Profit maximization yields the following demand functions for a variety _e_<sup>_ω_+</sup><sup>_λ_</sup> of a product _k_ : where the price index of a good _k_ , _Pk_ , and the price of a given variety of the good _k_ , _pk_ ( _ω_ , _λ_ ), are in units of final good.

Price indices, in turn, are given by the following expressions: where we are adopting the final good as the numeraire, so _P_ = 1.

Product varieties are in markets and are supplied monopolistically competitive produced according to a linear production function with labor as the only input: Productivity is determined by firm-wide and product-specific terms, _e_<sup>_ω_</sup> and _e_<sup>_λ_</sup> . respectively<sup>7</sup> Combined with fixed costs of supplying goods, the latter determines the portfolio of products supplied by the firm, while the former generates heterogeneity across firms in factor demands, product provision, and profitability.

> 7Notice that we are treating the product attribute as part of the productivity of the firm. As in Bernard et al. (2011) identical results are obtained if the product attributes enter revenues and profitability through the household’s demand functions.

7
In terms of the distortions that generate misallocation in the model, these take the form of idiosyncratic taxes and subsidies to the firm’s revenue. We adopt the assumption that there is heterogeneity in the degree of taxation across firms but we assume this rate to be the same across products within a firm. This is the assumption that we consider suits best the nature of the data from which we calibrate the functional form of the distribution of distortions. Since firm-level data are typically reported at the firm or establishment level, reported statistics about distributions of _TFPR_ and _TFPQ_ can only be informative about firm-wide or establishment-wide, rather than product specific, distortions. Thus, the combination of misallocation frictions with the multi-product feature of the model creates another layer of misallocation across firms, namely the distribution of products across producers. However it obliges us to abstract from a channel of misallocation within firms.

While we specify a particular functional form governing the degree of relationship between idiosyncratic distortions and physical productivity once we get to the quantitative analysis, at this stage we require that whatever functional form we impose, it does not create reversals in the positive relationship that exists between productivity and profitability in the undistorted equilibrium. That is, while it is certainly the case that distortions that correlate positively with productivity flatten the positive relationship between productivity and profitability, we rule out cases where a relatively unproductive firm is turned more profitable than a relatively more productive one. We find empirical support for this assumption in that all of the existing estimates in the literature of the regression coefficient between the log of _TFPR_ and the log of _TFPQ_ are far below one.

Letting _τω_ denote the idiosyncratic distortion corresponding to a firm with firmwide productivity _e_<sup>_ω_</sup> , the price and the labor demand associated with the provision of variety _e_<sup>_λ_</sup> of a product _k_ solves the following static profit maximization problem: which yields the following expressions for labor demand, output, revenues, and profits 8
Notice that we have suppressed the _k_ subscript denoting the type of good in the price index of the firms’ profit functions in anticipation of the fact that, as a result of the assumptions about the independence of the distribution of product attributes and the commonality of the fixed cost of provision across products and firms, goods are identical to each other. This will become more evident below.

# **2.3 Product-Portfolio Determination**

Firms earn positive profits from the provision of each good. To establish a limit to the menu of products actually supplied to the market, we assume that such activity entails a fixed labor-denominated cost of marketing equal to _f p_ . Furthermore, we assume these fixed costs to be identical across goods and across firms, and constant over time. The total profits earned from the provision of a given variety of a product, then, are equal to: Given the assumption of no reversals in the ranking of profits and productivity induced by the introduction of distortions, the total profits of the firms are strictly increasing in firm productivity and product attributes. Therefore, the fixed cost of provision implies that there exists a cutoff level such that varieties with attributes above it are supplied, and varieties with attributes below it are dropped. The cutoff attribute is the one at which the profits of a given product attribute are equal to zero. Solving for it in the expression above yields: where Π, defined below, collects all the aggregate variables and parameters that determine the factor of proportionality of firms’ outcomes with respect to idiosyncratic 9
variables: Firms with higher firm-wide productivity confront a lower threshold for the attribute that guarantees positive profitability from the provision of a given product and, hence, are able to supply a wider range of products to the market. Idiosyncratic distortions interfere with this decision. Firms that are taxed at a positive rate would find it harder to supply products to the market, while firms that are subsidized will find it easier. This mechanism of transmission of misallocation frictions to aggregate productivity is new to the literature and is one of the forces that we are interested in characterizing quantitatively.

# **2.4 Distribution and Evolution of Idiosyncratic Productivity**

Firms are endowed, upon entry, with a realization of firm-wide productivity and a continuum of attributes for each of the in the unit mea- product product types sure. We assume that entrants start off at a firm-wide productivity level consistent with the ratio of the average size of entrants to incumbents in the US data, and then fan out over time according to a stochastic process. Attributes for each product, on the other hand, are drawn independently and identically from a known distribution _F_ � _e_<sup>_λ_�</sup> . Unlike firm-wide productivity, we assume that the product attribute is constant throughout the life-cycle of the firm.

Firm dynamics are driven by the stochastic process for firm-wide productivity. I parameterize such process taking a discrete-time random walk approximation to a Brownian motion with drift _μ_ and variance _σ_<sup>2</sup> . Following Stokey (2008), I assume that given current log-productivity _ω_ , next period’s log-productivity could give an upward jump of size _h_ , with probability _α_ , or a downward jump also of size _h_ , with 1 _− α_ probability ( ) . The discrete time approximation of the drift and variance of the process is The appeal of the discrete time random walk approximation of the Brownian motion is that it easily maps into the model of Luttmer (2007), who shows that the resulting stationary distribution from such a stochastic process displays a right-tail 10
of the cumulative distribution function that is of the Pareto type, which constitutes an accurate characterization of the right tail of the firm size distribution in the United States. In our model with multi-product firms the mapping from firm-wide productivity to size is mediated by the product-attribute distribution but, given the latter, it is still true that there is a tight link between the parameters of the binomial process and the tail of the size distribution.

# **2.5 Aggregation within Firms**

Let us first characterize the aggregation of product-specific outcomes within firms. This is greatly simplified by the _iid_ assumption of the product-attribute distribution. Together with the law of large numbers, they imply that averages across the unit-continuum of products are equal to the corresponding average of each product. Aggregate productive employment, revenue, and profits within the firm are thus given by: The expressions show the transparent way in which firm-wide productivity interacts with the product-scope decision of the firm in turning into observable variables at the firm level, such as employment and sales. Conditional on a product portfolio, determined by the cutoff-attribute of the marginal product supplied to the market _e_<sup>_λ_ ̄(</sup><sup>_ω_)</sup> , more productive firms employ more workers, generate more revenue, and are more profitable. This is the standard relationship between productivity and size implied by single product models. With multi-product firms, the elasticity of firm-level variables with respect to firm-wide productivity gets magnified by the endogeneity of the product scope of the firms. More productive firms also get to produce a wider range of products, which in turns expands the demand for labor, the sales, and the profits. By the same token, this same force operates to magnify the distortive effect of allocative distortions. Not only will firms with higher taxes produce and earn less 11
from each product, but they shall also provide fewer varieties to the market. This is a novel layer of misallocation that we are able to take into account by means of our multi-product structure.

# **2.6 Entry and Exit**

Until now, we have dispensed from carrying a time subscript in the description of the model, as we were characterizing static decisions for which a time dimension was irrelevant. To decide about entry and exit, however, firms are looking ahead into future profitability so, unless we restrict to stationary allocations, we must bring a time subscript into the model.

The value of an operating producer with productivity _e_<sup>_ω_</sup> is characterized by the following equation: where _πt_ ( _ω_ ) is defined in equation 2.12, with the addition of a time subscript to the aggregate variables absorbed in Π _t_ . Firms discount the future at the market interest factor, _Rt_ , augmented by an exogenous death probability _δ_ , orthogonal to firm characteristics. The main goal of this shock is to account for the exit of firms from across the entire size distribution, a feature that we see in the data but would be absent in the model if exit were to be purely endogenous.

The value of the firm, then, is given by with _xt_ ( _ω_ ) encoding the exit decision of the firm, being equal to 1 if it stays in operation, and 0 otherwise.

In terms of entry, prospective entrants compare the value of a labor-denominated entry cost _fe_ with the value under the entrant’s productivity _e_<sup>_ωe_</sup> . Free entry from an of ensures that costs and valuation are infinite pool producers entry expected equalized in equilibrium: Notice that we are assuming a one period time to build before the entrant starts 12
operations.

# **2.7 Equilibrium and Macroeconomic Variables**

The remaining step in the characterization of the equilibrium is to aggregate the outcomes of the firm up to the level of the macroeconomy. Key for the aggregation is the distribution of firms across firm-wide productivity, which we denote with _Mt_ ( _ω_ ). This distribution evolves according to the following law of motion: The expression establishes that a fraction (1 _− δ_ ) _α_ of firms with productivity less than or equal to _ω_<sup>_′_</sup> _− h_ survives the exogenous exit shock and transitions to a productivity level that is less than or equal to _ω_<sup>_′_</sup> . A fraction (1 _− δ_ ) (1 _− α_ ) of the mass of firms with productivity between _ω_<sup>_′_</sup> and _ω_<sup>_′_</sup> + _h_ survives the exit shock and jumps downward to have productivity less than or equal to _ω_<sup>_′_</sup> . There is also an inflow of new firms to this group which is given by the mass of entrants, conditional on productivity being equal to the productivity level assumed for entry, _e_<sup>_ωe_</sup> . Endogenous exit will be driven by the mass of firms that transition downwards from the productivity cutoff , (1 _− δ_ ) _αMt_ ( _<u>ω</u>_ + _h_ ).

A _competitive equilibrium_ in this economy is: 1) a sequence of aggregate consumption decisions from the household _{Ct}_<sup>∞</sup> _t_ =0<sup>,2)sequencesofprices,labordemands,</sup> value functions, product cutoffs, and exit cutoffs for the producers of varieties, ∞ _pk_ , _t_ ( _ω_ , _λ_ ), _lt_ ( _ω_ , _λ_ ) , _Vt_ ( _ω_ ), _eλt_ ( _ω_ ), _<u>ωt</u>_ � � _t_ =0<sup>; 4) a sequence of final good quantities and</sup> ∞ demand functions for intermediate variety � _Yt_ , _q_<sup>_d_</sup> _k_ , _t_<sup>(</sup><sup>_ω_,</sup><sup>_λ_)</sup> � _t_ =0<sup>; 5) a sequence of mea-</sup> sures of firms _{Mt_ ( _ω_ ) _}_<sup>∞</sup> _t_ =0<sup>anditslawofmotion(equation2.13),6)asequenceof</sup> entrants _{Me_ , _t}_<sup>∞</sup> _t_ =0<sup>, 7) a sequence of prices and transfers</sup><sup>_{wt_,</sup><sup>_Rt_,</sup><sup>_Pt_,</sup><sup>_Qt_,</sup><sup>_Tt}_; 8) a dis-</sup> tortion profile _G_ ( _τ|ω_ ), a distribution of productivity at entry _G_ ( _ω_ ), a distribution of product attributes _F_ ( _λ_ ), and a stochastic process for firm-wide productivity; and 9) an initial wealth of the household _M_ 0 such that: a) given 7, 5, and 9, 1 solves household’s optimization problem, b) given 7, 4 and 8, 2 solves the incumbents’ dynamic optimization problem, c) given _pk_ , _t_ ( _ω_ ), 4 solves the final good sector’s profit maximization problem, d) _Me_ , _t_ is such that the free entry condition is satisfied in every

13
period, and e) markets clear in every period: where _Lp_ , _t_ , _L f c_ , _t_ , and _L f p_ , _t_ are the aggregate demands for labor in production, fixed costs of operation, and fixed cost of provision of products, defined by: The term Λ _t_ stands for the statistic of the joint distribution of productivity and product attributes that characterizes across outcomes in the econ- aggregates firm-level omy, and is given by A _stationary competitive equilibrium_ is one in which the distribution of productivity has become stationary, and where aggregate variables and prices have become constant.

# **2.7.1 Aggregate Variables**

Prior to the quantitative analysis, it is instructive to derive an analytical characterization of aggregate output and aggregate productivity in the model<sup>8</sup> . To derive it, we start substituting the aggregate component of firms’ variable profits defined in equation 2.9 into equation 2.14 for aggregate labor demand in production. Then, solving for aggregate output we get: 8Once again, to simplify notation, we avoid carrying time subscripts in the notation with the understanding that the aggregate variables just defined are static in the stationary equilibrium and timevarying in the analysis of transitions.

14
As it is familiar in models of monopolistic competition with CES demand systems, the price of a variety of product _κ_ produced by a firm with productivity _e_<sup>_ω_</sup> , product attribute _e_<sup>_λ_</sup> , and idiosyncratic distortion _τω_ is given by Firms with higher productivity and higher product attribute offer lower prices. Revenue taxes and subsidies are passed on to the consumer, to the extent allowed by the elasticity of substitution.

Plugging the individual prices into the price index for good _k_ , we get Notice that the expression justifies the earlier claim that price indices were symmetric across products. This is guaranteed by the assumption of an independent and identically distribution of product attributes and the assumption of a common fixed cost of provision across products and firms, which together imply that _F_ ( _λ_ ) and _eλ_ ( _ω_ ) are identical for all _k_ .

Exploiting this symmetry and recalling that the price of the final good is chosen to be the numeraire, it follows that the price index of each product _k_ is also equal to unity, _P_ = 1. Also, substituting into the price index of the final good in equation 2.3, we can solve for the equilibrium wage rate as: where we have subsumed the aggregation of productivities and product attributes Λ<sup>_ω_</sup> . that determine marginal costs into the term We can go back to equation 2.15 and get the following expression for _GDP_ and _TFP_ in the model: 15
Changes in the entry and exit of firms affect _TFP_ through the consequent change in the number of firms and through the demand for labor that goes into entry and _L_ . fixed operation costs, which determines _p_ Changes in the firms’ portfolio of products also affect output through changes in _Lp_ , because of the labor-intensive nature of the fixed costs of supplying goods. Furthermore, it affects _TFP_ by shaping the distribution of products across firms of different productivities, controlled by _eλ_ ( _ω_ ).

# **3 Quantitative Analysis**

We turn now to the quantitative evaluation of the model. As emphasized throughout the paper, we are interested in measuring the welfare gains from liberalizations that eliminate misallocation frictions. We seek to provide a quantitative answer accompanied with an understanding of how the various forces at work in the model contribute to shaping the gains.

## **3.1 Calibration**

We must choose parameter values for the elasticity of substitutions _σ_ and _ρ_ , the subjective discount factor of the household _β_ , the size of the labor force _L_ , and the set of parameters governing the process of firm dynamics, entry and exit: entry and fixed operation costs _fe_ and _fc_ , the size and probability of the jump in the binomial process _h_ and _α_ , and the exogenous exit rate _δ_ . Furthermore, we must specify values for the fixed cost of supplying varieties to the market, _f p_ , as well as the shape parameter of the distribution of product attributes _η_ . We calibrate these parameters working with the undistorted stationary allocation of the model, taking the United States as empirical target. Table 1 summarizes the parameter values.

The strategy of the calibration is as follows. For the elasticity of substitution, we set _ρ_ = _σ_ = 3, which lies in the middle of estimates of substitutability found in the trade and industrial organization literature, and is the value chosen by Hsieh and Klenow (2009) in their measurement of misallocation in the United States, China, and India<sup>9</sup> . Since our assumptions imply no heterogeneity of prices across products,

> 9See Broda and Weinstein (2006) for a range of estimates of the elasticity of substitution for US

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Table 1: Parameter Values and Calibration Targets

|Parameter|Value|Target|
|---|---|---|
|_ρ_=_σ_|3|Hsieh and Klenow (2009), Broda and Weinstein (2006)|
|_β_|1<br>1.05|Interest Rate of 5%|
|_δ_|0.02|Employment-Based Exit Rate of Large Firms of 2%|
|_α_|0.467|Slope Log of Right Tail of Empl. Based Size Distribution= -0.2|
|_h_|0.25<br>_η_|Std Dev. of Employment Growth of Large Firms|
|_ωe_|_e_<sup>_ωe_ </sup>=1|Size of Entrants = 6% of Median Incumbent (Luttmer 2010)|
|_fc_<br>_fe_|0.1|Exit Rate of 5%|
|_fp_|210.95|Av. Fraction of products per frm=0.0024|
|_η_|2|Distribution of Output-Share byProduct in Multi Product Firms|

distortions will not create changes in relative prices across products, so only the elasticity of substitution across varieties within a product category matters for aggregate variables. Thus we set the elasticities of substitution to be identical to each other. We choose the discount factor so that _β_<sup><u>1</u></sup><sup>_−_1equalsarealinterestrateof5%,and</sup> normalize the size of the labor force to be equal to 1.

In terms of product attributes, we assume these are distributed Pareto, with shape parameter _η_ and lower bound _e_<sup>_λmin_</sup> = 1. Besides its analytical tractability, this parameterization is consistent with the behavior of multi-product firms in the US, in the distribution of across within particular output-shares products multi-product firms. Bernard et al. (2010) report the output-share by rank of the product, for firms of various degrees of multiple-good production (4, 6, 8, and 10 products). They run a regression of log of product rank, and the log of the output share of the product. the of this to be to 0.5. In the we They find coefficient regression equal appendix, show that this target maps into a parameter value of _η_ = 2 for the Pareto distribution of product-attributes.

Our to calibrate the cost of of also relies on strategy fixed provision goods � _f p_ � data from Bernard et al. (2010)<sup>10</sup> . They show that out of a potential of 1,440 products, the average number of products supplied by a multi-product firm is 3.5. Given that our product space lives in the continuum of measure one, this gives us a target of 3.5/1440 to be matched by the product range of the average multi-product firm in the model. Formally, _f p_ is chosen so that: imports at a 4-digit disaggregation level. 10Data taken from Table 1 in Bernard et al. (2010).

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which amounts to devoting 17% of the labor force to the task of confronting fixed costs involved in supplying products to the market.

Despite the empirical merits and the analytical tractability of the Pareto distribution, the lower limit in its support carries the implication that some sufficiently productive multi-product firms in the model will find it profitable to supply all of the products in the continuum. This is problematic for two reasons. First, according to Bernard et al. (2010), there is no evidence of such firms in the data for the manufacturing sector in the United States. Second, it introduces a discontinuity in the elasticity of firm size with respect to firm-wide productivity. To see this, consider the expression for firm-wide employment in the undistorted economy that results from aggregating across products within a firm with firm-wide productivity _e_<sup>_ω_</sup> under a Pareto distribution of product attributes and an arbitrary lower bound _e_<sup>_λmin_</sup> : Since _e_<sup>_λ_ ̄(</sup><sup>_ω_)</sup> is decreasing in firm-wide productivity, it can be shown that there exists a sufficiently large but finite value of _e_<sup>_ω_</sup> such that the cutoff-attribute is lower than the lower bound of the Pareto distribution. In such case, once the full product-space has been covered, the firm behaves as if it was a single product firm, with an elasticity with respect to firm-wide productivity that switches from _η_ to 1.

In spite of these complications, we find that a negligible fraction of firms (0.02%) in the undistorted stationary equilibrium hits the lower bound and supplies all the products to the market under our final calibration. Thus, the gains in tractability that are allowed for by the adoption of a Pareto distribution, in particular the transparent way in which the product-portfolio decision of the firms magnifies the elasticity of size with respect to productivity, more than compensate the costs, so we keep it as the choice for parameterizing the distribution of product attributes.

Having characterized more sharply the aggregate employment level within a firm, we can proceed with the description of the strategy to calibrate the parameters 18
governing the stochastic process of firm-wide productivity. Taking logs of firm-wide employment in the region where the product range is less than one, we get: where Ψ = <u>�(</u> _fρp_<sup>(</sup><sup>_η_</sup> _−_<sup>_−_</sup> 1<sup>1</sup> <u>)</u><sup>)</sup> <u>�</u> _η−η_ 1<sup>collectsparametersrelatedtothefixedcostofprovisionof</sup> products, the elasticity of substitution across varieties within a product, and the tail of the Pareto distribution of product attributes.

The moment in the data that we target to pin down the step size _h_ is the standard deviation of the distribution of employment growth rates for large firms, which in the US economy is equal to 0.25. It is easy to show that, in the stationary equilibrium : of our model, such variance is approximately equal to<sup>11</sup> Thus, for a given value of _η_ , the standard deviation of employment growth rates in the cross-section of large firms in the United States implies that _h_ = 0.25/ _η_ . We take into account that there exists a small fraction of large firms supplying all the products, for which _η_ no longer plays a role. For these, the variance of employment growth rates is driven entirely by the step size in the process for productivity, so we set _h_ = 0.25 for such group.

In terms of the probability of technological upgrading, _α_ , we set it to match properties of the right tail of the U.S. employment based size distribution. Luttmer (2010) highlights the linearity of the right tail of the US establishment and firm size distribution across employment, as well as the stationarity of the distribution over time, . using various sources of US micro-data<sup>12</sup> For a given value of the productivity step size _h_ and a given value of the exogenous exit probability _δ_ , the slope of the right tail of the firm size distribution in the model is determined by _α_ . To ensure that we are capturing the linear portion of the size distribution, we focus on the slope implied

> 11 ˆ 2 2 2 The exact value for the variance is _Var_ � _L_ � = ( _ηh_ ) _−_ [ _ηh_ ] (2 _p −_ 1) . The approximation is exact in the case of a stochastic process with zero drift, namely _p_ = 0.5. We show below that our calibrated value of _p_ is 0.467, which allows us to qualify the approximated value of the variance as a close approximation. The sole advantage of it is that we can independently identify the values of the parameters of the stochastic process that match their counterparts in the data.

> 12County Business Patterns Database, statistics from the Small Business Administration and the Business Dynamics Statistics from the census.

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by the ratio of the change in the logarithm of the fraction of employment accounted for by firms with 1,000 or more and 5,000 or more employees, relative to the logdifference of the employment levels, which is equal to -0.2 in the data. Formally, we set _p_ so as to match where _G_ (.) here denotes the cumulative distribution function of the employment weighted firm size distribution in the United States. Targeting this slope translates into a 72% share of employment accounted for by the top 10% largest firms, which is consistent with the data for the U.S.’ manufacturing sector.

Regarding the probability of the exogenous exit shock, this parameter controls the exit rate among the largest firms. Thus, we calibrate it to replicate the exit rate among large firms in the employment-weighted size distribution which, according to the Small Business Administration (SBA)<sup>13</sup> , is equal to 0.02.

With respect to the realization of firm-wide productivity for entering firms, we assume that all entrants get a productivity draw of _e_<sup>_ωe_</sup> = 1 and then fan-out over the space of idiosyncratic productivity according to its stochastic process. The target driving this choice is the average size of an entrant relative to the average incumbent in the U.S.’ size distribution, which Luttmer (2007) reports to be equal to 6%. The entry and fixed production costs are set at _fe_ = 1 and _fc_ = 0.1 respectively. Together with the parameters of the stochastic process for firm-wide productivity, particularly the standard deviation of the shocks to idiosyncratic productivity, the choice of _fc_ and _fe_ imply an exit rate of 5% .

The last portion of the calibration refers to the distribution of distortions. A pervasive feature in developing countries is that idiosyncratic distortions exhibit a of correlation with the distribution of across strong degree physical productivities firms. This is a key property of the data, since it is the one that creates a non-trivial response in the dynamic decisions of the firms, such as the decision to enter and . exit the economy<sup>14</sup> To honor such relationship, we propose the following log-linear functional form for the relationship between idiosyncratic distortions and the id13This is the exit rate that applies to firms larger than or equal to 500 workers in the year 2002. 14It can be shown that uncorrelated distortions, although damaging for allocative efficiency in terms of products and labor, have neutral effects on entry and exit. We resume this point once we explore the quantitative performance of the model.

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iosyncratic firm-wide productivities: The appeal of the expression is that it provides a transparent mapping between _γ_ , the slope, and the coefficient of a regression between the log of _TFPR_ and the log of _TFPQ_ , which is a statistic that is normally reported in empirical papers about misallocation. In order to understand the strength of the mechanism at various degrees of distortions, we experiment with a range of values of _γ_ . In choosing this range, we restrict ourselves to values that do not revert the positive relationship between productivity and profitability. Doing so would carry the counterfactual prediction of making the most productive firms be the ones exiting the market endogenously.

We consider the following values: _γε {_ 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 _}_ . As a form of reference, according to Hsieh and Klenow (2007), _γ_ would be roughly equal to 0.1 in the United States, around 0.5 in China 1998 and about 0.4 in India in 1987.

# **3.2 Welfare Analysis**

This subsection begins the quantitative exploration measuring the welfare gains to be reaped from liberalizations that eliminate idiosyncratic distortions. We follow the tradition of thinking about permanent consumption compensations that should be given to households in order to make them indifferent between different allocations. The benchmark comparison is between keeping the household at a stationary equilibrium with distortions, relative to transitioning towards a stationary equilibrium without frictions. When we investigate long run gains only, the comparison is between the level of consumption in the distorted steady state against jumping straight towards the undistorted steady state one. The results are reported in figure 3.1. The black line reports the welfare gains accounting for both transitional and long run gains, while the gray line reports the latter only.

Given the Pareto optimality of the undistorted allocation, welfare gains are exthe of distortion The are also pectedly increasing throughout space slopes. gains sizable, requiring an increase of up to 55% in permanent consumption to compensate the household if it were to be preserved at a stationary equilibrium with distortions. For instance, at the level of distortions consistent with China in 1998, _γ_ = 0.5, the welfare gains from liberalization amount to almost 40%. For India in 1987, with 21
<!-- Start of picture text --> Welfare Gains Liberalization<br>60<br>50<br>Long Run Gain<br>Welfare Gain<br>40<br>30<br>20<br>10<br>0<br>−10<br>0 0.1 0.2 0.3 0.4 0.5 0.6<br>Distortion Slope<br>% welfare gain<br><!-- End of picture text --> Figure 3.1: Welfare Gains From Liberalizing Distortions

# _γ_ = 0.4, the gains reach 32%<sup>15</sup> .

The figure also reveals a significant divergence between the welfare gains with and without consideration of transition dynamics. The gray line, which reports the welfare gains from steady state to steady state improvements in consumption, shows that ignoring transition dynamics significantly underestimates the overall potential gains in welfare. Continuing with the examples of China and India, long run gains . are only half as large as the overall gains<sup>16</sup> The divergence between the two measures of welfare gain arises as a result of the temporary gain in consumption that the liberalization of distortions allows in the early years of the transition path. We show this in figure 3.2, which reproduces the time series of consumption along the transition path from liberalizations at the bottom, middle, and top of the range of distortions that we consider: _γ ∈{_ 0.1, 0.3, 0.6 _}_ .

15All elasticities are read off Table 4 in Hsieh and Klenow (2007) . Albeit at a weaker degree, there is also evidence of correlated distortions in the United States, which we use as the efficient benchmark in our calibration. Had we subtracted the estimated elasticity for the US from the estimates for China and India, the corresponding slopes of distortions would have been 0.3 and 0.2 respectively, leading to welfare gains of 17% and 10%/

> 16Given the abstraction from physical capital in our model, our finding that the full dynamics of _TFP_ matter for the quantification of the welfare gains is a verification of a more general theoretical result established in Basu et al. (2012), who show that, to a first order approximation and for general conditions about the production side of the economy, the present value of _TFP_ and the growth rate of the capital stock are sufficient for characterizing welfare in a country.

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<!-- Start of picture text --> Consumption Dynamics<br>70<br>gamma=0.6<br>gamma=0.3<br>60<br>gamma=0.1<br>50<br>40<br>30<br>20<br>10<br>0<br>−10<br>0 20 40 60 80 100<br>Periods<br>% change from distorted SS<br><!-- End of picture text --> Figure 3.2: Consumption Dynamics along Transition Paths We can see that consumption overshoots upon reform and persists for several periods above the level at which it converges in undistorted steady state.

In order to uncover the forces that allow for such temporary boost, figure 3.3 below shows the time series of the total number of firms in the economy, the shares of labor allocated into the production of goods, entry costs, fixed costs of operation and the fixed costs of supplying goods; and the total number of products per firm. The number of firms, the mass of entrants, and the number of products per firm are measured as ratios with respect to their distorted steady state values, while the labor shares are expressed as absolute deviations from the initial shares.

The salient feature of the figure is the process of decumulation of firms that takes place in the transition towards the undistorted stationary equilibrium. As the economy converges to an allocation with fewer firms, a result that we shall explain in greater detail below, resources are reallocated away from entry and fixed costs of operation and into the production of goods. The number of products per-firm increases sharply, in reflection of the efficient reallocation of products towards highly productive firms that takes place in response to the reform. However, since the number of firms is falling, the total number of varieties and the requirements of labor to confront fixed cost of provision also fall during the transition, which further reinforces the reallocation of labor towards productive activities and further contributes to sustaining consumption above the undistorted stationary level.

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<!-- Start of picture text --> Mass of Firms Mass of Entrants<br>1 1<br>0.8 0.8<br>0.6 0.6<br>0.4 0.4<br>0.2 0.2<br>0 0<br>0 20 40 60 80 100 0 20 40 60 80 100<br>Share of Labor in Production Share of Labor in Provision of Goods<br>0.15 0<br>0.12 −0.01<br>0.09<br>−0.02<br>0.06<br>0.03 −0.03<br>0 −0.04<br>0 20 40 60 80 100 0 20 40 60 80 100<br>Share of Labor in Entry and Fixed Costs Number Products per Firm<br>0 16<br>−0.025 13<br>−0.05 10<br>−0.075 7<br>−0.1 4<br>−0.125 1<br>0 20 40 60 80 100 0 20 40 60 80 100<br>gamma=0.6 gamma=0.3 gamma=0.1<br><!-- End of picture text --> Figure 3.3: Understanding Sources of Welfare Gain: Transition Dynamics **Note:** The number of firms, the mass of entrants, and the number of products per firm are measured as ratio with respect to their distorted steady state values. Labor shares are expressed as absolute deviations from the initial shares.being 24
As mentioned above, the increase in number of firms in the economy’s distorted stationary equilibrium is a key property of the model’s response to allocative distortions that explains the divergence in results between measures of welfare gain. As can be read from figure 3.3, the number of firms is between 30 to 70% lower in the undistorted than in the distorted stationary allocations.

The rise in the number of firms in the distorted allocation is a result of the following two forces: an increase in the measure of entrants and a decrease in the rate of exit. The rise in firm entry emerges as an outcome of the interaction between two features of the calibration that are in the data: the stochastic for grounded process productivity upon entry which, conditional on survival, gives firms an increasing pattern of productivity growth over the life-cycle; and the positive relationship between distortions and idiosyncratic productivity. From the perspective of an entrant, a profile of distortions that taxes high productivity firms and subsidizes low productivity ones creates a redistribution of profits from the future to the present. In the cross-section of incumbents, this pattern of taxation redistributes profits from high to low productivity firms. Since firms discount future profit streams at a positive interest rate, the overall decline in expected-profitability of entrants will be lower than the average decline in profitability among incumbents. In order to restore equilibrium in the labor market and the balance in the free entry condition, an increase in entry will follow. It is easy to show that had any of the above-mentioned properties of the calibration been absent, distortions would have not had any effect on the .<sup>17</sup> The decline in the exit rate, in turn, is also attributable to the positive relationship between distortions and productivity. By taxing the high productivity firms, labor demand is relaxed and factors prices fall. This allows for marginal firms that would have been left out of the market to sustain operations profitably. In addition, these firms are aided by subsidies to producers in the left tail of the productivity distribution. Even though the decline in the exit rate carries a decline in aggregate productivity per-firm, it also contributes to increasing the number of firms for, at a given number of entrants, it sustains a larger number of producers in the stationary equilibrium.

A last feature of the transition that is worth is the non- dynamics highlighting

> 17That was the case, for instance, in Restuccia and Rogerson (2008) where the missing ingredient justifying the neutrality of entry in their model is the lack of firm dynamics upon entry.

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monotonicity of the convergence. Two properties of the model play a key role in generating this pattern: the non-negativity of entry and the productivity distribution of entrants relative to incumbents. The non-negativity of entry prevents a sudden adjustment in the number of firms. The fastest the economy can reduce the number of firms is by shutting down entry, increasing the exit productivity cutoff, and waiting for the negative drift in incumbents’ productivity and the exogenous exit shocks to drive firms out of the market. This explains the smooth decline of the number of firms and the protracted increase in the shares of labor allocated to production. The point of entry of new firms in the productivity distribution, in turn, is the key ingredient for understanding the eventual recovery in the number of firms. Newcomers to the market enter at a lower productivity level than the average incumbent. This means that during the periods where there is zero entry, production gets concentrated at the top of the productivity distribution, leaving a lower mass of firms around the marginal productivity cutoff. When entry resumes, it creates an inflow of firms that takes time to populate the left tail. Until this happens, entry exceeds exit, so the number of firms reverses its trajectory and converges to the new stationary level from below.

our shows that for the in Summarizing, analysis accounting dynamic response the economy’s number of varieties through changes in the number and allocation of products, and the entry and exit of firms, magnifies the welfare gains from alleviating misallocation far beyond the gains that can be captured by focusing on long-run measures of consumption growth. Despite the merit that resource misallocation has in the context of explaining cross-country differences in _TFP_ (merit that we reassess below with our richer framework), our results motivate the adoption of a broader measure of welfare when discussing benefits and costs of reforms that are aimed at alleviating allocative inefficiencies.

# **3.3 Long-Run Implications of Misallocation Frictions**

In this section we focus on understanding the long run implications of misallocation frictions in the context of our model. While the previous section emphasized the importance of transition dynamics for welfare assessments, the steady state conseof distortions were at the core of the initial motivation for about quences thinking misallocation as a potential driver of the large differences in income and productivity that exist across countries. For this reason, we here connect with the bulk of the 26
literature by providing a quantitative evaluation of the contribution of entry, exit, and multi-variety production for the ability of idiosyncratic distortions to play such role.

In order to transparently identify the contributions of the new margins, we compare the long run changes in _TFP_ in three alternative long run allocations of our model were entry, exit, and multi-variety firms are at play sequentially. More specifically, recall the expression that defines measured _TFP_ in the model: We construct a benchmark allocation where all the gains from liberalizations accrue due to the efficient reallocation of resources among a given set of producers, and among a given distribution of products across firms. This allocation, which we label _static,_ is the one that resembles the type of gains that are captured in Hsieh and Klenow (2009) and the many replications thereafter, and the one that we consider an adequate benchmark to gauge the contributions of entry, exit, and multi-variety production. To construct it, we set all taxes and subsidies to zero while keeping the distribution of cutoff attributes _e_<sup>_λ_ ̄(</sup><sup>_ω_)</sup> for the provision of a product, the number and distribution of firms _M_ ( _ω_ ), and the share of labor allocated to the production of goods<sup>_L_</sup> _L_<sup>_<u>p</u>_,fixedattheirdistortedallocation’slevel.Then,inordertoassessthe</sup> contribution of the multi-product structure of production, we construct an allocation where we allow the distribution of products across firms _e_<sup>_λ_ ̄(</sup><sup>_ω_)</sup> and the labor share in production<sup>_L_</sup> _L_<sup>_<u>p</u>_to adjust to the efficient level,while keeping constant the number</sup> of firms and the distribution of firms across firm-wide productivity at the distorted economy’s allocation. We label this allocation as _product-channel._ Lastly, we show the long-run gains in the _full_ model which, when compared to the previous two, identifies the contribution of entry and exit. Results are shown in figure 3.4.

A comparison between the solid black line, corresponding to the static benchmark, and the dashed line, corresponding to the gains with consideration of the product-reallocation channel, shows that the latter margin significantly magnifies 27
<!-- Start of picture text --> Long Rung TFP Gains<br>50<br>Full<br>45<br>Static<br>40 Product−Channel<br>35<br>30<br>25<br>20<br>15<br>10<br>5<br>0<br>−5<br>−10<br>0 0.1 0.2 0.3 0.4 0.5 0.6<br>Distortion Slope<br>% change from distorted SS<br><!-- End of picture text --> Figure 3.4: Long Run TFP Gains from Liberalizing Distortions: the Role of the Number of Products and the Number of Firms the improvements in _TFP_ that would emerge in the long run from a liberalization of misallocation frictions. Subsidies to unproductive firms and taxes to productive ones reallocate products from the latter to the former. Because the favored firms are relatively less productive, a lower amount of output per product is produced under a given aggregate allocation of labor to production, reinforcing the decline in _TFP_ in<sup>18</sup> . the distorted allocation and enhancing the gains when these are reversed The consideration of firm entry and exit, illustrated by the gray line in the graph, brings a countervailing force to the analysis which makes the overall contribution of the response in the number of varieties be in general ambiguous. As explained earlier in the context of understanding transition dynamics, distortions that correlate positively with firm-wide productivity induce an increase in entry and a decline in the exit rate that increases the total number of firms. This increase is reversed in the undistorted long-run equilibrium, property which under a _CES_ production structure constitutes a drag on aggregate productivity.

18Given that there is a small fraction of firms that cannot expand beyond the full range of products, the total number of products falls slightly in the _product-channel_ allocation when removing allocative distortions. While this force tends to weaken the _TFP_ gains, it is partially compensated by the increase in share of labor allocated to the production of goods that is allowed for by the lower demand for labor that is required to confront the fixed costs of provision. It can be shown, though, that the predominant force driving the productivity gains in this allocation is given by the efficient reallocation of products to more productive firms.

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The of the in _TFP_ in the full model as a function of non-monotonicity response the underlying degree of distortions is worth exploring. The figure shows that the countervailing effect of entry and exit prevails at liberalizations from low to middle values of the slope between distortions and idiosyncratic productivity, while it is . dominated by the labor and product reallocation channels at more severe values<sup>19</sup> To understand this feature of the results, recall that the rise in entry in the distorted allocation was critically determined by the interaction between an increasing trajectory of profitability upon entry (conditional on survival), a profile of distortions that taxed productive firms more heavily, and discounting of future profit streams. As we move along the range of distortion slopes, the redistribution of profits from higher to lower productivity values flattens the upward sloping trajectory of profits. Hence, an additional increase in distortions carries a weaker effect than when distortions were first put in place. On the other hand, the damaging effect of distortions on allocative efficiency increases together with the severity of distortions. Thus, there exists a distortion level beyond which the latter force dominates the former, so that liberalizing distortions becomes more productivity-enhancing than implied by the static benchmark<sup>20</sup> .

Going back to the question of how much cross-country differences in allocative distortions contribute to understanding cross-country differences in _TFP_ , our findings indicate that entry, exit, and multi-variety production can enhance the role attributed to resource misallocation provided we are considering countries with sufficiently high degrees of correlation between idiosyncratic distortions and idiosyncratic productivity. Based on existing studies, the majority of the estimates of the elasticity between distortions ( _TFPR_ ) and productivity ( _TFPQ_ ) in developing countries fall in the range where the magnifying forces and the countervailing ones . roughly balance out<sup>21</sup> Had it not been for the consideration of the dynamic impli-

> 19Notice that there are levels of distortion slopes where the undistorted _TFP_ is even lower than the distorted one. This feature stresses even further the need to account for transitional dynamics, given the knowledge that the undistorted allocation is Pareto optimal and, hence, transitioning to it should deliver higher welfare to the household.

20Notice that the argument relies in a sense of continuity in the evolution of the strengths of the two forces affecting _TFP_ as a function of distortions, a feature that we have only explored numerically.

> 21Hsieh and Klenow (2007) report slopes of 0.5 and 0.4 for China and India, respectively. Chen and Irarrazabal (2015) estimate the elasticity to be between 0.6 and 0.5 in Chile during the period of 19801996. Cirera et al. (2017) report values ranging between 0.4 and 0.6 in Ghana, Ethiopia, Kenya, and Cote D’ Ivoire. However, not every study of misallocation reports statistics about the degree of relationship between distortions ( _TFPR_ ) and productivity ( _TFPQ_ ), so there are much fewer estimates of this elasticity than there are estimates of the dispersion.

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cations of reforms, the long-run analysis would have mistakenly led us to conclude that there is not much to be lost, in terms of capturing the full extent of the costs of misallocation, from ignoring the effect of distortions over the number of varieties in the economy. The findings from this section, then, do nothing but reinforce our message of motivating the adoption of a broader measure of welfare that internalizes these dynamic gains.

# **3.3.1 The Role of Multi-Variety Production**

While the analysis above was designed to isolate the marginal effect of entry, exit, and multi-product firms, it was not well suited to establish a comparison between the results of our model with multi-product firms against those stemming from a model with single-product units. The reason is that the models are not comparable unless the calibration of the shock process for firm-wide productivity is adjusted to make sure that both models match the same targets in the data. To appreciate this, recall that the relationship between productivity and size in the multi-product model is intermediated by the product scope of the firm. This source of magnification in the mapping from productivity to employment dispersion is absent in the singleproduct model, hence it must be compensated through an increase in the calibrated value of the variance of the shock process. This leads us to set _h_ = 0.25.

We proceed by constructing the same _full_ and _static_ allocations that we did in the context of the analysis of the benchmark model, comparing the overall magnitude of the _TFP_ gains across models, and identifying the contribution of entry and exit relative to the static gains depending on whether the multi-product channel is at play or not. The results are illustrated in Figure 3.3.1.

The salient property of the figure is the remarkable difference in the contribution of the static gains in shaping the overall improvements in productivity across models. While the gains under the full model are slightly lower in the multi-product than in the single-product version, the static gains only amount to at most 15% in the former, significantly below the 50% gain that can be reached in the latter. This means that, in order to reach similar gains overall, the changes in the number of varieties are playing different roles depending on whether these are entirely driven by changes in the number of firms or if they are also driven by changes in the allocation of products across firms. Like before, entry and exit contribute negatively to the long-run changes in _TFP_ that are triggered by a dismantlement of distortions. How30
<!-- Start of picture text --> Multi−Product Model Single Product Model<br>60 60<br>55 Full 55 Full<br>Static Static<br>50 50<br>Prod. Channel<br>45 45<br>40 40<br>35 35<br>30 30<br>25 25<br>20 20<br>15 15<br>10 10<br>5 5<br>0 0<br>−5 −5<br>−10 −10<br>0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6<br>Distortion Slope Distortion Slope<br>% change from distorted SS<br><!-- End of picture text --> Figure 3.5: Long Run Gains from Liberalizing Distortions: Multi-Product vs SingleProduct Models ever, the offsetting effect of these margins is countervailed by the additional gains that accrue due to efficient product reallocation. Therefore, while abstracting from the latter would not lead to substantially different magnitudes of _TFP_ gains from reversals of allocative distortions, changes in the number of varieties would mistakenly be attributed to have a negative contribution over these gains if the number of varieties is affected by entry and exit decisions only.

The key driver of the difference in the magnitude of the static gains lies in the change in the distribution of _TFPQ_ that is induced by the firms’ incentives to adjust their portfolio of products. When firms are multi-product, _TFPQ_ at the firm level is a combination of the exogenous firm-wide productivity level and the endogenously determined number of products supplied by the firms. Allocative distortions that correlate positively with firm-wide productivity induce a reallocation of products from high to low productivity firms that compresses the distribution of _TFPQ_ in the distorted stationary equilibrium. Therefore, when computing the gains from reallocation holding fixed the distribution of products across firms, as in the _static_ allocation, we are holding back the scope for efficiency gains by reallocating labor among firms that are not too different from each other in terms of their measured physical productivity<sup>22</sup> . This is not the case in the single-product version of the

> 22The intuition that the properties of the underlying distribution of _TFPQ_ determine the detrimen-

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model, where the dispersion of firms across the space of _TFPQ_ is not affected by the existence of allocative distortions (except only through changes in the exit cutoff) and hence is purely determined by the properties of the shock process. Notice, then, the importance of having re-calibrated the variance of these shocks.

Overall, the results of this section shed light on the importance of jointly considering margins of adjustment that contribute to aggregate productivity through changes in the number of varieties. It highlights that focusing on entry and exit only would mistakenly attribute a negative contribution to the variety channel, that may become overturned once changes in varieties that operate through multi-product incumbents are taken into account. Furthermore, it emphasizes the importance of recalibrating the properties of the distribution of idiosyncratic productivity when comparing the effect of allocative distortions in models with endogenous _TFPQ_ against models where this object is exogenous.

# **4 Concluding Remarks**

Allocative distortions have acquired a pivotal role among economists in thinking about growth. The central hypothesis is that by inducing a sub-optimal allocation of resources across firms, this type of distortion could become an important driver of the large productivity gaps that we observe across countries. While this hypothesis led to promising results, it focused mostly on accounting for a static allocative channel through which misallocation can be detrimental to productivity. However, less attention was given to the goal of understanding the implications of these distortions for other firm-level decisions that, besides being empirically relevant margins of adjustment, have direct effects over welfare and productivity. In this paper we provided an integrated investigation of the effects of allocative distortions for three particular channels: the entry, exit, and product-provision decisions of firms.

We found that the consideration of these dynamic factors motivate the redirection of the interest away from long-run metrics of welfare, such as _TFP_ , into a broader definition of welfare that accounts for transitional dynamics. This is because when dynamic factors were taken into account, we found sizable gains in welfare materializing along the transition path from and equilibrium contaminated with allocative tal effects of a given distribution of distortions on aggregate productivity is formalized in Hopenhayn (2014).

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distortions towards an undistorted allocation. The overall welfare gains more than doubled the ones that one would have come-up from the consideration of the longrun effects on _TFP_ only. Given the costly process involved in the implementation of reforms that liberalize countries from distortive policies and frictions, our analysis suggests that once the effect of these distortions on entry, exit, and product-provision decisions of firms are taken into account, the payoffs from these efforts are higher than previously thought.

Despite our proposal to think about broader measures of welfare, our study reconnected with the original motivation for thinking about misallocation by providing an assessment of how entry, exit, and multi-product firms mattered for the ability _TFP_ . of misallocation to generate large long-run losses in In this respect, we found a less monotone message, in the sense that while multi-variety provided another layer of misallocation that magnified the efficiency losses from it, we also found that the correlated pattern of distortions with the underlying distribution of productivity encouraged the entry of more firms into the economy, which partly offset the long-run costs from these distortions.

The paper’s approach to modeling misallocation was very stylized. Even though the allowed me to characterizations of the re- resulting tractability provide sharp of the to such and enabled me to make sponse economy policies quantitative predictions, the large welfare gains associated with their removal calls for a deeper investigation of what the sources of misallocation really are. Several candidates have already being explored in the literature, such as financial frictions, firing costs, and heterogeneous mark-ups<sup>23</sup> , although none of these seems to be accounting for most of the misallocation that has been documented in the data.

The model incurred in additional simplifications that were shown to have a direct impact on its mechanisms. One such simplification consisted of the technology of entry, treated here as a linear function of labor. Some researchers have modeled entry as a result of an occupational choice decision where agents select themselves into working for a wage or undertaking entrepreneurial activities according to their talent. In this environment, it costs more than just one unit of labor to switch a worker out of the labor force and start a new firm. In the formulation of my model, the entry cost has the interpretation of a sunk cost that is required to set up the firm, the profitability of which is determined ex-post from a random productivity 23See, for instance, Veracierto (2001), Buera and Shin (2013), Peters (2016).

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draw. The two set-ups share the property that not any “idea” is profitable enough to become a firm, although they differ in how costly it is to set it up. Further evidence allowing to more concretely identify the properties of the process of firm creation would be very valuable for all models where the number of producers is determined endogenously.

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# **A Calibrating the Distribution of Product Attributes**

This appendix shows how to use the data about the distribution of output shares in multi-product firms to calibrate the shape parameter of the Pareto distribution of product attributes. We restrict attention to the case were firms are in the interior of the product space.

In the model, the output of a firm with productivity _e_<sup>_ω_</sup> producing a good with product attribute _e_<sup>_λ_</sup> is given by where the factor of proportionality is a function of prices and other parameters in the model.

Under Pareto distribution of product attributes, the total output of the firm is: Therefore, the output share of a product _e_<sup>_ω_+</sup><sup>_λ_</sup> is The determine the shape of the distribution of this share, its cumulative distribu36
~~FP~~ G ~~E~~ ) PC <u>(</u> ~~<u>—)</u> JEP~~ O ( ~~)~~