Productivity Dispersion, Misallocation, and Reallocation Frictions: Theory and Evidence from Policy Reforms

Abstract

Recent research maintains that the observed productivity variation across firms reflects resource misallocation and concludes that large GDP gains may be obtained from market-liberalizing polices. Our theoretical analysis examines the impact on productivity dispersion of reallocation frictions in the form of costs of entry, operation, and restructuring, and shows that reforms reducing these frictions may raise dispersion of productivity across firms. Contrary to conventional wisdom, the model does not imply a negative relationship between aggregate productivity and productivity dispersion. Our empirical analysis focuses on episodes of liberalizing policy reforms in the US and six East European transition economies. We find that deregulation of US telecommunications equipment manufacturing is associated with increased, not reduced, productivity dispersion, and that every transition economy in our sample shows a sharp rise in dispersion after liberalization. Productivity dispersion under communist central planning is similar to that in the US, and it rises faster in countries liberalizing more quickly. We also find that lagged productivity dispersion predicts higher future productivity growth, likely because dispersion reflects experimentation by both entering and incumbent firms. The analysis suggests there is no simple relationship between the policy environment and productivity dispersion.

Appendices

Appendix A

Proofs

Existence and uniqueness of stationary equilibrium. Let $R = \left( {1 - G\left( {\phi _{e} } \right)} \right)N$ be the total mass of entrants corresponding to a given entry threshold $\phi _{e}$. Note that for any given $\phi _{e}$ there exists a unique corresponding $R$ by the assumption that $N$ is exogenously given and the fact that $G$ is monotonic. Therefore, $R$ and $\phi _{e}$ can be used interchangeably to denote the extent of entry. Let $\mu \equiv \mu \left( {R,x,\theta _{r} } \right)$ be an invariant measure that corresponds to a given triplet $\left\{ {R,x,\theta _{r} } \right\}.$ Consider now the pair $\left\{ {R\left( {\theta _{r} } \right),x\left( {\theta _{r} } \right)} \right\}$ such that given $\theta _{r} \in U \equiv \left[ {0,1} \right]$, $\left\{ {R\left( {\theta _{r} } \right),x\left( {\theta _{r} } \right)} \right\}$ satisfies the free entry condition (3) and the exit condition (2) with equality for the associated invariant measure $\mu$. Denote by ${\mathcal{T}}_{1} :U \to \left[ {0,N} \right] \times U$ the mapping that yields a pair $\left\{ {R\left( {\theta _{r} } \right),x\left( {\theta _{r} } \right)} \right\}$ for any given $\theta _{r} \in U.$ Next, let $\theta _{r} \left( {R,x} \right)$ be the value that satisfies the restructuring condition (4) with equality for a given pair $\left\{ {R,x} \right\}$ and the associated invariant measure $\mu$. Denote by ${\mathcal{T}}_{2} :$ $\left[ {0,N} \right] \times U \to U$ the mapping that associates a given pair $\left\{ {R,x} \right\}$ with some $\theta _{r} \in U$ that satisfies the restructuring condition with equality. The proof of existence and uniqueness then amounts to showing that the composite mapping ${\mathcal{T}} = {\mathcal{T}}_{1} \circ {\mathcal{T}}_{2}$ possesses a unique fixed point that lies in the interior of $U.$ Some of the arguments in the proofs below follow closely the related arguments in Hopenhayn (1992). Note that the model satisfies all the basic assumptions A1-A5 in Hopenhayn (1992) and, in addition, the conditions U1 and U2 therein. In particular, the model reduces to Hopenhayn’s (1992) framework when the restructuring cost, $c_{r} ,$ is prohibitively high, the mass of potential entrants, $N,$ is infinite, and the distribution of entrants’ priors, $G,$ is does not vary by $\phi .$

Existence. First, note that the invariant measure $\mu$ is defined by

$$\mu \left( \theta \right) = \left( {{\mathcal{L}}\mu } \right)\left( \theta \right) + N\mathop \int \limits_{{\phi _{e} }}^{1} \left( {\int _{0}^{\theta } h_{e} (z|\phi ){\text{d}}z} \right)g\left( \phi \right){\text{d}}\phi ,$$

where ${\mathcal{L}}$ is the operator such that for any set $S \subset U,$

$${\mathcal{L}}\left( S \right) = \left\{ {\begin{array}{ll} \mathop \int \limits_{{y \in S}} h(y|z)\mu \left( {{\text{d}}z} \right),&{\text{for }}z \ge x. \\ 0, &{\text{otherwise}}. \\ \end{array} } \right.$$

The steps similar to Lemma 4 in Hopenhayn (1992) guarantee the existence of $\mu .$ Also, following Lemma 5 in Hopenhayn (1992), $\mu$ is jointly continuous in its arguments, strictly decreasing in $x$, and strictly increasing in $R$ (strictly decreasing in $\phi _{e}$). Now, let $R_{1} \left( {\theta _{r} } \right)$ be the entry mass that satisfies (3) with equality for a given exit rule $x\left( {\theta _{r} } \right).$ Similarly, let $R_{2} \left( {\theta _{r} } \right)$ be the entry mass such that the exit rule $x\left( {\theta _{r} } \right)$ satisfies (2) with equality. The properties of $R_{1}$ and $R_{2}$ follow from Lemmas 6 and 7 in Hopenhayn (1992). Theorems 2 and 3 in Hopenhayn (1992) then imply the existence of a pair $\left\{ {R\left( {\theta _{r} } \right),x\left( {\theta _{r} } \right)} \right\}$ such that $R\left( {\theta _{r} } \right) > 0$ and $x\left( {\theta _{r} } \right) \in \left( {0,1} \right)$, for any given $\theta _{r}$, as long as $c_{e}$ is not too high. Therefore, ${\mathcal{T}}_{1}$ is a well-defined, continuous operator that maps $\theta _{r}$ into a pair $\left\{ {R\left( {\theta _{r} } \right),x\left( {\theta _{r} } \right)} \right\}$ that satisfies (2) and (3) with equality. Next consider the mapping ${\mathcal{T}}_{2} .$ Given a pair $\left\{ {R,x} \right\}$, the left-hand side of (4) is continuous and strictly decreasing in $\theta _{r}$ by the assumptions of the model. Therefore, there exists a unique value $\theta _{r}$ that satisfies (4) with equality, as long as $c_{r} < E_{r} [V\left( {\theta ^{\prime}} \right)|0]$, i.e., the benefit from restructuring exceeds the cost of doing so for the least productive firm. Thus, ${\mathcal{T}}_{2}$ is a well-defined, continuous function that maps any $\left\{ {R,x} \right\}$ into a $\theta _{r}$ that satisfies (4) with equality. Given the continuity of ${\mathcal{T}}_{1}$ and ${\mathcal{T}}_{2} ,$ the composition ${\mathcal{T}} = {\mathcal{T}}_{1} \circ {\mathcal{T}}_{2}$ is then a continuous function that maps $U$ onto itself. The existence of a fixed point $\theta _{r}^{*}$ then follows from the Brouwer fixed point theorem. This fixed point is in the interior of $U$ and satisfies $\theta _{r}^{*} >$ $x^{*} ,$ as long as $c_{r} < E_{r} [V\left( {\theta ^{\prime}} \right)|0].$ Consequently, there exists a triplet $\left\{ {\theta _{r}^{*} ,x^{*} ,R^{*} } \right\}$ and the associated invariant measure $\mu ^{*} ,$ that constitute a stationary equilibrium with positive entry, exit, and restructuring.

Uniqueness Suppose that the stationary equilibrium is not unique, i.e., the mapping ${\mathcal{T}}$ has more than one fixed point. Let $x_{1}^{*} < x_{2}^{*}$ denote the two exit thresholds for two different stationary equilibria with the corresponding measures $\mu _{1}^{*}$ and $\mu _{2}^{*} .$ By the definitions of $x_{1}^{*}$ and $x_{2}^{*} ,$ $V\left( {x_{1}^{*} ;\mu _{2}^{*} } \right) < V\left( {x_{2}^{*} ;\mu _{2}^{*} } \right) = 0,$ and $V\left( {x_{1}^{*} ;\mu _{1}^{*} } \right) = 0.$ Thus, there must be some firm type $\theta$ such that $\tilde{\pi }\left( {\theta ;\mu _{2}^{*} } \right) < \tilde{\pi }\left( {\theta ;\mu _{1}^{*} } \right).$ However, the free entry condition implies that $V^{e} \left( {\phi _{1}^{*} ;\mu _{1}^{*} } \right) = V^{e} \left( {\phi _{2}^{*} ;\mu _{2}^{*} } \right) = c_{e} .$ Therefore, while profits $\tilde{\pi }$ are lower for some firm type $\theta$ under $\mu _{2}^{*} ,$ they cannot be lower for all $\theta$, for otherwise $V^{e} \left( {\phi _{2}^{*} ;\mu _{2}^{*} } \right) < c_{e} .$ This argument implies that if profits move in the same direction for all $\theta$ going from one equilibrium to another, then the free entry condition cannot be satisfied for both equilibria—a contradiction. Assumptions U1 and U2 in Hopenhayn (1992), both of which are also assumed here, ensure that profits for all firm types move in the same direction and hence, the equilibrium is unique.

Proposition 1

As a result of a decline in $c_{r}$ , $x^{*}$ and $\phi _{e}^{*}$ either both increase or both decrease.

Proof

Let

$$\tilde{V}\left( {\theta ;\mu } \right) = \frac{{V\left( {\theta ;\mu } \right)}}{{c_{r} }},$$

be the rescaled value function for a firm, where the dependence on the measure $\mu$ is made explicit. One can then rewrite (1) as

$$\tilde{V}\left( {\theta ;\mu } \right) = \frac{{u\left( \theta \right)m\left( {p\left( \mu \right),w,r} \right)}}{{c_{r} }} - \frac{{c_{f} }}{{c_{r} }} + \beta {\text{max}}\left\{ {0,E_{r} \left[ {\tilde{V}\left( {\theta ^{\prime}} \right)\left| {\theta \left] { - 1,E_{n} } \right[\tilde{V}\left( {\theta ^{\prime}} \right)} \right|\theta } \right]} \right\},$$

where we used the assumption that the profit function is separable in productivity and prices, i.e., $\tilde{\pi }\left( {\theta ;\mu } \right) = u\left( \theta \right)m\left( {p\left( \mu \right),w,r} \right),$ for some functions $u$ and $m.$ Now consider two industries such that $c_{r}$ is lower in the second industry: $c_{r}^{2} < c_{r}^{1} .$ The aim is to show that if $x_{2} \ge x_{1}$ ($x_{2} < x_{1}$) then it must be the case that $\phi _{e}^{2} \ge \phi _{e}^{1}$ ($\phi _{e}^{2} < \phi _{e}^{1}$). Toward that end, let the measures of firms be $\mu ^{1}$ and $\mu ^{2}$. If $x_{2} \ge x_{1}$

$$\int \tilde{V}\left( {\theta ^{\prime};\mu _{1} } \right)h_{n} (\theta ^{\prime}|x_{2} ) \ge \int \tilde{V}\left( {\theta ^{\prime};\mu _{1} } \right)h_{n} (\theta ^{\prime}|x_{1} ) = 0 = \int \tilde{V}\left( {\theta ^{\prime};\mu _{2} } \right)h_{n} (\theta ^{\prime}|x_{2} ),$$

where the first inequality follows from the fact that $H_{n} (\theta ^{\prime}|\theta )$ is strictly decreasing in $\theta ,$ and the equalities from the fact that $x_{1}$ and $x_{2}$ are the marginal firm types so they must have zero expected value from continuing. But the inequality $\int \tilde{V}\left( {\theta ^{\prime};\mu _{1} } \right)h_{n} (\theta ^{\prime}|x_{2} ) \ge$ $\int {\tilde{V}\left( {\theta ^{\prime};\mu _{2} } \right)h_{n} (\theta ^{\prime}|x_{2} )}$ can hold can only when

$$\frac{{m\left( {p\left( {\mu _{1} } \right),w,r} \right) - c_{f} }}{{c_{r}^{1} }} \ge \frac{{m\left( {p\left( {\mu _{2} } \right),w,r} \right) - c_{f} }}{{c_{r}^{2} }},$$

which implies

$$m\left( {p\left( {\mu _{1} } \right),w,r} \right) - c_{f} \ge \frac{{c_{r}^{1} }}{{c_{r}^{2} }}\left( {m\left( {p\left( {\mu _{2} } \right),w,r} \right) - c_{f} } \right) > m\left( {p\left( {\mu _{2} } \right),w,r} \right) - c_{f} ,$$

where the last inequality follows because $c_{r}^{1} > c_{r}^{2} .$ Therefore, period profit for each firm type, $\tilde{\pi }\left( {\theta ;\mu } \right),$ is higher in industry 1, and so is the value of each firm type

$V\left( {\theta ,\mu _{1} } \right) \ge V\left( {\theta ,\mu _{2} } \right).$ The expected profit for any potential entrant type $\phi$ must then also be higher in industry 1, implying a lower entry threshold in industry 1, i.e., $\phi _{e}^{2} \ge \phi _{e}^{1}$. The steps of the proof so far can be repeated to show the other combination, $x_{2} < x_{1}$ and $\phi _{e}^{2} < \phi _{e}^{1} .$

Appendix B: The Derivation of Aggregate Productivity

For general production functions, it is not possible to represent the aggregate production function in an industry using the exact same form of the firm-level production function.⁴⁹ To derive an explicit expression for aggregate productivity, TFP, assume, as is common in the literature, that a firm’s production function is of Cobb–Douglas type, $f\left( {k,l} \right) = k^{\lambda } l^{\gamma } ,$ $\lambda + \gamma < 1.$ Suppose also that the fixed cost entails both overhead labor and capital: $c_{f} = rk_{f} + wl_{f} ,$ where $k_{f}$ and $l_{f}$ are the amount of fixed capital and labor per firm, respectively. Let $Q^{*}$ be the aggregate output, and let $K^{*}$ and $L^{*}$ be the total capital and labor used. The industry’s production function can then be written as $Q^{*} = {\text{TFP}} \times K^{{*\lambda }} L^{{*\gamma }} .$ Define $o_{l}^{*} = \frac{{M^{*} l_{f} }}{{M^{*} l_{f} + \mathop \int \nolimits_{0}^{1} l^{*} \left( \theta \right)\mu ^{*} \left( {{\text{d}}\theta } \right)}}$ and $o_{k}^{*} = \frac{{M^{*} k_{f} }}{{M^{*} k_{f} + \mathop \int \nolimits_{0}^{1} k^{*} \left( \theta \right)\mu ^{*} \left( {{\text{d}}\theta } \right)}}$ as the fraction of labor and capital used as overhead, respectively. TFP can then be expressed as

$${\text{TFP}} = \frac{{Q^{*} }}{{K^{{*\lambda }} L^{{*\gamma }} }} = \left( {\frac{{\left( {1 - G\left( {\phi _{e}^{*} } \right)} \right)N}}{{H^{*} \left( {x^{*} } \right)}}} \right)^{{1 - \lambda - \gamma }} \left( {1 - o_{l}^{*} } \right)^{\gamma } \left( {1 - o_{k}^{*} } \right)^{\lambda } \left( {\int _{0}^{1} \theta ^{{1/\left( {1 - \lambda - \gamma } \right)}} h^{*} \left( \theta \right){\text{d}}\theta } \right)^{{1 - \lambda - \gamma }} .$$

(1)

Note that TFP is a geometric average of firm-level TFPQ, $\theta$. TFP is higher when there is a larger mass of potential entrants ($N$), lower entry threshold ($\phi _{e}^{*}$), lower fixed costs ($l_{f}$ and $k_{f}$), lower exit threshold ($x^{*}$), and a higher productivity distribution $H^{*}$, in a first-order stochastic dominance sense. Note that $\theta _{r}^{*}$ also affects TFP through its effect on $H^{*} ,$ implicit in the expression (1). Because $\phi _{e}^{*} ,$ $x^{*} ,$ $\theta _{r}^{*} ,$ $o_{l}^{*} ,$ $o_{k}^{*}$ and $H^{*}$ are all functions of the costs $c_{e} ,c_{f} ,$ and $c_{r} ,$ TFP is also a function of the costs $c_{e} ,c_{f} ,$ and $c_{r} .$