2.1 Set Up

Economic environment: Consider a world with two potentially asymmetric countries i = 1, 2 and

free firm entry into production.⁷ In each country, a measure L_i of consumers inelastically supply a unit of labor, and aggregate expenditure is E_i. A representative consumer derives utility U_i from consuming a homogenous good H_i and differentiated varieties z ∈ Ω_i:

Demand q_i(z) for variety z with price P_i(z) in country i is thus $$q_i\left(z\right)=\beta E_i{P}_{iQ}^{\sigma -1}{p}_i{\left(z\right)}^{-\sigma }$$, where βE_i is total expenditure on differentiated goods, $$P_{iQ}={\left[{\int }_{z\in {\Omega }_i}{p}_i{\left(z\right)}^{\alpha }dz\right]}^{1/(1-\alpha )}$$ is the ideal price index in the differentiated sector, and α = 1/(1 α) > 1 is the elasticity of substitution across varieties.

The homogeneous good is freely tradeable and produced under CRS technology that converts one unit of labor into one unit of output. When β is sufficiently low, both countries produce the homogeneous good, such that it serves as the numeraire, P_iH = 1, and fixes wages to unity, w_i = 1. We will refer to this case simply as β< 1. When β = 1 by contrast, only differentiated goods are consumed, and wages are endogenously determined in equilibrium. The aggregate consumer price index is thus $$P_i=P_{iQ}^{\beta }$$.

In each country, a continuum of monopolistically competitive firms produce horizontally differentiated varieties that they can sell at home and potentially export. Firms pay a sunk entry cost $w_i{f}_i^E$ and, should they commence production, fixed operation costs w_if_ii and constant marginal costs. Exporting from i to j requires fixed overhead costs w_if_ij and iceberg trade costs such that τ_ij units of a good need to be shipped for 1 unit to arrive, where τ_ii = 1 and τ_ij 1 if i ≠ j. We allow for τ_ij ≠ τ_ij, and analyze symmetric and asymmetric reductions in τ_ij to assess the impact of different trade reforms.

Firm productivity and resource misallocation: In the absence of misallocation, firms in country i draw productivity φ upon entry from a known Pareto distribution $$G_i\left(\phi \right)=1-{({\phi }_i^m/\phi )}^{\theta }$$, where $$\theta >\sigma -1and{\phi }_i^m>0$$.⁸ This fixes firms’ constant marginal cost to w_i/φ. Under resource misallocation on the other hand, firms draw both productivity φ and distortion δ from a known joint distribution H_i(φ,τ). Firms’ marginal cost is now determined by their distorted productivity φ = φδ and equals w_i/φ = w_i/(φδ).⁹ For comparability with the case of no misallocation, we assume that φ is Pareto distributed with scale parameter $${\underset{¯}{\phi }}_i^m$$ and shape parameter θ.

Conceptually, τ captures any distortion that creates a wedge between the social marginal cost of an input bundle and the private marginal cost to the firm. Formally, this implies a firm-specific wedge in the first-order condition for profit maximization. Such a wedge may result from frictions in capital or labor markets or from generally weak contractual institutions that support inefficient practices like corruption and nepotism.¹⁰ Distortions τ will lead to deviations from the first-best allocation of productive resources across firms: If a firm can access “too much” labor “too cheaply”, this would be equivalent to a subsidy of τ > 1. Conversely, capacity constraints, hiring and firing costs would correspond to a tax of τ < 1.

Modeling misallocation in this way has several appealing features. First, it permits a transparent comparison of firm and economy-wide outcomes with and without misallocation. Under misallocation, firm selection, production and export activity depend on φ and τ only through distorted productivity φ = φτ, while optimal resource allocation in the first best depends on φ alone. Thus two parameters regulate the degree of misallocation: the dispersion of the distortion draw, σ_τ, and the correlation between the distortion and productivity draws, ρ(φ,τ).¹¹ Misallocation occurs if and only if ρ_τ > 0, but its severity need not vary monotonically in the ρ_τ ρ(φ,τ) space.¹²

Second, introducing distortions on the input side is qualitatively isomorphic to allowing for distortions in output markets, such as firm-specific sales taxes.¹³ Our theoretical formulation thus ensures tractability without loss of generality. In the empirical analysis, we correspondingly exploit different measures of broad institutional quality, capital and labor market frictions, and product market regulations.

Within the differentiated sector, misallocation stems from the inefficient allocation of production resources and consequently market shares across firms. Since CES preferences and monopolistic competition will imply a constant mark-up μ = 1/α > 1, there is no additional misallocation due to variable mark-ups across firms as in Dhingra and Morrow (2016). When β < 1, however, there will also be markup driven misallocation across sectors: Because perfectly competitive producers of the CRS homogeneous good do not charge a mark-up, the differentiated sector will be “too small”.

2.2 Economy Equilibrium

Firm behavior: Producers choose their price P_ij(φ) and quantity q_ij(φ) to maximize profits π_ij(φ) separately in each market j. With no distortions, the optimal behavior of a firm with productivity φ is:

where l_ij(φ), c_ij(φ) and r_ij (φ) are the employment, costs and revenues associated with sales in j.

Since profits are monotonically increasing in productivity, firms in country i sell in market j only if their productivity exceeds threshold $${\phi }_{ij}^{*}$$. The domestic and export cut-offs are implicitly defined by:

Upon entry, firms commence production if their productivity is above $${\phi }_{ii}^{*}$$, and exit otherwise. We assume as standard that the parameter space guarantees selection into exporting, $${\phi }_{ii}^{*}>{\phi }_{ii}^{*}$$, for any τ_ij > 1. In the case of misallocation, the profit-maximization problem of a firm with distorted productivity φ = φτ generates the following second-best outcomes:

While it would be socially optimal to allocate input factors and output sales based on true firm productivity φ, in the market equilibrium this allocation is instead pinned down by distorted productivity φ. Along the intensive margin, firms with low (high) distortions φ produce and earn less (more) than in the first best, and set higher (lower) prices than efficient. Along the extensive margin, a highly productive firm might be forced to exit if it faces prohibitively high taxes, while a less productive firm might be able to operate or export if it benefits from especially high subsidies. Firms thus sell in the domestic and foreign market if their distorted productivity exceeds cut-offs ${\underset{¯}{\phi }}_{ii}^{*}and{\underset{¯}{\phi }}_{ij}^{*}$, respectively:

General equilibrium: The general equilibrium is characterized by conditions that ensure free entry, labor market clearing, income-expenditure balance, and trade balance in each country.

Consider first the case of no misallocation. With free entry, ex-ante expected profits must be zero:

where E_i[·] is the expectation operator and I(·) is the indicator function.¹⁴

A key implication of the free-entry condition is that the productivity cut-offs in country i for production and exporting must always move in opposite directions following trade reforms that affect τ_ij or τ_ij. Intuitively, any force that lowers ${\phi }_{ij}^{*}$ tends to increase expected export profits conditional on production. For free entry to continue to hold, ${\phi }_{ii}^{*}$ must therefore rise, such that the probability of survival conditional on entry falls and overall expected profits from entry remain unchanged.

Let L_iH and L_iQ denote respectively total labor employed in the homogeneous and differentiated sectors. Labor market clearing in country i requires:

where M_i is the mass of entering firms in the differentiated sector. When β < 1, we restrict the parameter space to ensure LIH > 0, such that the wage is determined by productivity in the homogenous-good sector. When β = 1 and L_iH = 0, by contrast, wages are flexible and determined by L_iH = L_iQ.

In equilibrium, aggregate income must equal aggregate expenditure. With free entry, aggregate corporate profits net of entry costs are 0, such that total income corresponds to the total wage bill. Consumers’utility maximization implies the following income-expenditure balance:¹⁵

Consider next the case of misallocation. The free entry and labor market clearing conditions are analogous to those above after replacing productivity φ with distorted productivity φ = φτ. The income-expenditure balance, however, has to be amended. While firm (φ, τ]) incurs production costs $c_{ij}(\phi ,\eta )=w_i(f_{ij}+\frac{{\tau }_{ij}{q}_{ij}(\phi ,\eta )}{\phi \eta })$, the payment received by workers is $c_{ij}^i(\phi ,\eta )=w_i(f_{ij}+\frac{{\tau }_{ij}{q}_{ij}(\phi ,\eta )}{\phi })$. The gap $c_{ij}^{\text{'}}(\phi ,\eta )-c_{ij}(\phi ,\eta )$ is the social cost of distortionary firm-specific taxes or subsidies, which we assume are covered through lump-sum taxation T_i of consumers in i. When a firm is subsidized and $c_{ij}(\phi ,\eta )<c_{ij}^{\text{'}}(\phi ,\eta )$ for example, it pays its employees less than what it would have without the subsidy, and consumers pay the difference. The new equilibrium conditions become:

Welfare: Welfare in country i is given by real consumption per capita and can be expressed as:

Welfare is thus proportional to the real wage, w_i/P_i, and the ratio of disposable income to gross income, χ_i. In the absence of misallocation, all income accrues to worker-consumers, such that E_i = w_iL_i and χ_i = 1. In the presence of misallocation, by contrast, some income is not available to consumers due to the tax burden of distortions, such that E_i = w_iL_i T_i and χ_i < 1; albeit less realistic, it is in principle possible that χ_i > 1. Misallocation also affects the price index P_i through distortions to firm selection on the extensive margin and to firm prices and market shares on the intensive margin.

One can show that the real wage, and therefore also welfare, is a function of two equilibrium outcomes: the (distorted) productivity cut-off for production, ${\phi }_{ii}^{*}or{\underset{¯}{\phi }}_{ii}^{*}$, and the share of disposable income, χ_i:¹⁶

Lemma 1 Without misallocation, welfare increases with the domestic productivity cut-off, $\frac{d{W}_i}{d{\phi }_{ii}^{*}}>0$. With misallocation, welfare increases with the distorted domestic productivity cut-off (holding x% fixed), $\frac{\partial W_i}{\partial {\underset{¯}{\phi }}_{ii}^{*}}>0$, and with the share of disposable income in gross income (holding $${\underset{¯}{\phi }}_{ii}^{*}$$ fixed), $$\frac{\partial W_i}{\partial {\chi }_i}>0$$.

With efficient resource allocation, a higher productivity cut-off $${\phi }_{ii}^{*}$$ implies a shift in economic activity towards more productive firms, which tends to lower the aggregate price index and increase consumers’ real income. With misallocation, distortions affect welfare through the reduction in disposable income χ_i and through the sub-optimal selection and size of active firms based on distorted productivity φ rather than true productivity φ. One direct implication of Lemma 1 is that welfare is proportional to the domestic productivity cut-off if and only if there are no allocative frictions. Another implication is that the welfare impact of trade liberalization depends on how a reduction in τ_ij affects $${\phi }_{ii}^{*},{\underset{¯}{\phi }}_{ii}^{*}$$, and χ_i.

Note that in the two-sector general equilibrium, welfare reflects both distortion-driven misallocation across firms within the differentiated sector and markup-driven misallocation across sectors, both of which are reflected in the economy-wide price index P_i. One cannot analytically decompose these two sources of misallocation, and their relative contribution is state-dependent.¹⁷

2.3 From Theory to Empirics

A key challenge in evaluating the gains from trade is that productivity and welfare are not directly observable. Here we characterize the mapping between these theoretical objects and their empirical counterparts. We focus on firm and aggregate productivity in the differentiated sector, which are the objects of interest in both the single- and two-sector models.

Theoretical vs. measured firm productivity: The theoretical concept of firm productivity is quantity-based, while empirical measures are generally revenue-based. For our purposes, real value added per worker is a valid proxy for effective firm productivity inclusive of any distortions.

Without misallocation, observed value added and employment correspond respectively to total firm revenues, $r_i\left(\phi \right)={\sum }_j{r}_{ij}\left(\phi \right)I(\phi \ge {\phi }_{ij}^{*})$, and total labor hired, $l_i\left(\phi \right)={\sum }_j{l}_{ij}\left(\phi \right)I(\phi \ge {\phi }_{ij}^{*})$. Denoting labor used towards fixed costs as $f_i\left(\phi \right)={\sum }_j{f}_{ij}\left(\phi \right)I(\phi \ge {\phi }_{ij}^{*})$ and normalizing by the price index in the dierentiated industry $P_{iQ}=P_i^{1/\beta }$, real value added per worker Φ_i(φ) is:

One can show that without distortions, real value added per worker increases monotonically with theoretical firm productivity conditional on export status, ${\Phi }_i^{\text{'}}\left(\phi \right|\phi <{\phi }_{ij}^{*})>0and{\Phi }_i^{\text{'}}(\phi |\phi \ge {\phi }_{ij}^{*})>0$.¹⁸

In the case of misallocation, real value added per worker reflects firms’ effective productive capacity given distortions, and can thus be labeled Φ_i(φ,τ]). It is now monotonic in theoretical distorted productivity conditional on export status, $${\underset{¯}{\Phi }}_i^{\text{'}}\left(\phi \eta \right|\phi \eta <{\underset{¯}{\Phi }}_{ij}^{*})>0\text{}and{\underset{¯}{\Phi }}_i^{\text{'}}(\phi \eta |\phi \eta \ge {\underset{¯}{\Phi }}_{ij}^{*})>0$$:¹⁹

Measured aggregate productivity and OP decomposition: Let measured aggregate productivity, $${\tilde{\mathrm{\Phi }}}_i$$, be the weighted average of measured firm productivity. Without distortions, $${\tilde{\mathrm{\Phi }}}_i$$ is:

where ${\theta }_i\left(\phi \right)\equiv l_i\left(\phi \right)/\left[\underset{{\phi }_{ii}^{*}}{\overset{\infty }{{\displaystyle \int }}}{l}_i\right(\phi \left)\frac{d{G}_i\left(\phi \right)}{1-G_i\left({\phi }_{ii}^{*}\right)}\right]$, is firm φ's share of aggregate employment.²⁰ This will correspond exactly to employment-weighted average real value added per worker across firms in our CompNet data. Of note, the choice of employment shares as firm weights ensures that ${\tilde{\mathrm{\Phi }}}_i$ equal the ratio of aggregate revenue to aggregate employment in the differentiated sector adjusted by the sectoral price index; this in turn parallels how statistical agencies measure aggregate labor productivity with real GDP per worker deflated by sector price indices.

As an accounting identity, measured aggregate productivity, ${\tilde{\mathrm{\Phi }}}_i$, can be decomposed into the measured unweighted average productivity across firms, ${\overline{\mathrm{\Phi }}}_i$, and the measured covariance of firms’ productivity and share of economic activity, ${\tilde{\mathrm{\Phi }}}_i$, known as the OP gap (Olley and Pakes, 1996):

The OP decomposition reveals how adjustments across and within firms shape aggregate measured productivity. Changes in ${\overline{\mathrm{\Phi }}}_i$ reflect two effects of firm selection: exit/entry into production modifies the set of active firms, and exit/entry into production or exporting impacts measured firm productivity. Changes in $${\ddot{\Phi }}_i$$ indicate reallocation of activity across firms with different productivity levels through changes in their share of production resources and implicitly sales. The OP decomposition remains valid in the case of misallocation, when $$\underset{¯}{\phi },{\phi }_{ii}^{*},{\underset{¯}{\Phi }}_i(\phi ,\eta ),and{H}_i(\phi ,\eta )replace\phi ,{\phi }_{ii}^{*},{\Phi }_i\left(\phi \right)$$, and G_i(φ) in (2.24).

Welfare vs. measured aggregate productivity: From a policy perspective, welfare and domestic aggregate productivity matter for different objectives: The former captures consumer utility at a point in time, while the latter indicates a country’s productive capacity, improvements in which drive growth over time. However, these two objects can differ, even under allocative efficiency: Welfare in country i depends on the price index P_i faced by consumers in i, which reflects the prices of all varieties sold in i. Intuitively, W_i is related to the weighted average productivity of all domestic and foreign firms supplying i, using their activity in i as weights. By contrast, $${\tilde{\mathrm{\Phi }}}_i$$ is the weighted average productivity of domestic firms, using their total employment as weights. This distinction is irrelevant in special cases, such as symmetric countries and bilateral trade costs, when the measure, productivity, prices and market shares of firms exporting from i to j are identical to those of firms exporting from j to i.²¹

One can express measured aggregate productivity as a function of the real wage, w_i/P_i, and the size-weighted average distortion across firms, $${\tilde{\eta }}_i$$, where δ = 1 and $${\tilde{\eta }}_i=1$$ without misallocation:

Together, equations (2.19), (2.20) and (2.25) imply that shocks that move the (distorted) productivity cut-offs for production and exporting will shift $${\tilde{\mathrm{\Phi }}}_i$$ through their effect on the equilibrium wage w_i (if β = 1), the aggregate price index P_i, and the average distortion $${\tilde{\eta }}_i$$. In particular:

Lemma 2 Without misallocation, measured aggregate productivity increases with the domestic productivity cut-off, $$\frac{d{\tilde{\mathrm{\Phi }}}_i}{d{\phi }_{ii}^{*}}>0$$. With misallocation, this relationship becomes ambiguous, $$\frac{d{\widetilde{\text{Φ}}}_i}{d{\underset{¯}{\phi }}_{ii}^{*}}\gtrless 0$$.

Lemmas 1 and 2 imply that measured aggregate productivity can be a sufficient statistic for welfare only without misallocation.²² With misallocation, W_i and $${\tilde{\mathrm{\Phi }}}_i$$ are not closed-form functions of the misallocation parameters, and we therefore simulate the model using standard parameters from the literature (see Section 2.5) to numerically explore their relationship. We assume productivity and distortions are joint log-normal with μ_φ = μ_σ = 1, σ_φ = 1, and vary the levels of distortion dispersion σ_η ∈ [0,0.5] and productivity-distortion correlation ρ(φ,η) ∈ [0.4,0.4].

Figure 1A shows that welfare peaks at σ_η = ρ(φ,η) = 0 and falls as the distortion dispersion widens for the given ρ(φ,η). At low levels of σ_η, Wi rises as the distortion and productivity draws become more positively correlated, but the opposite holds at sufficiently high levels of σ_η. While measured aggregate productivity behaves similarly under this parametrization in Figure 1B, W_i and $${\tilde{\mathrm{\Phi }}}_i$$ need not co-move under alternative assumptions (unreported). For completeness, Figure 1C plots measured average productivity $${\overline{\mathrm{\Phi }}}_i$$ against the misallocation parameters.

Numerical Simulation: Welfare and Measured Aggregate Productivity

Citation: IMF Working Papers 2020, 163; 10.5089/9781513554440.001.A001

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Numerical Simulation: Welfare and Measured Aggregate Productivity

Citation: IMF Working Papers 2020, 163; 10.5089/9781513554440.001.A001

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• Download figure as PowerPoint slide

Numerical Simulation: Welfare and Measured Aggregate Productivity

Citation: IMF Working Papers 2020, 163; 10.5089/9781513554440.001.A001

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• Download figure as PowerPoint slide

OP covariance vs. misallocation: The OP covariance is related to allocative efficiency in that $${\ddot{\Phi }}_i>0$$ in a frictionless economy (when both Φ_i(φ) and θ_i(φ) conditionally increase in φ) but $${\ddot{\Phi }}_i\gtrless 0$$ in the presence of distortions.²³ However, one cannot interpret a rise in $${\ddot{\Phi }}_i$$ as an improvement in allocative efficiency, because the optimal allocation of resources across firms is generally state-dependent and reliant on the economic environment (i.e. demand structure, cost structure, market structure, productivity distribution). Even if the optimal covariance $${\ddot{\Phi }}_i^{*}$$ were known, both values below and above it would indicate deviations from the first best. Moreover, the absolute difference $$|{\ddot{\Phi }}_i^{*}-{\ddot{\Phi }}_i|$$ need not be proportional to or even monotonic in the degree of misallocation and the welfare loss associated with it.