Selection and Sorting of Heterogeneous Firms through Competitive Pressures

To understand theoretically how competitive pressures affect selection and sorting of firms with different productivity, we study the Melitz (2003) model under the H.S.A. (Homothetic with a Single Aggregator) class of demand systems. H.S.A. is tractable due to its homotheticity and to its single aggregator that serves as a sufficient statistic for competitive pressures, which acts as a magnifier of firm heterogeneity. It is also flexible enough to allow for the choke price, the 2nd law of demand--``a higher price leads to a higher price elasticity''--, and the 3rd law of demand--``a higher price leads to a smaller rate of change in the price elasticity.'' We show, among others: i) More productive firms have higher profits and revenues; they have higher markup rates under the 2nd law and lower pass-through rates under the 3rd law. Employments are not monotone in firm productivity; they are hump-shaped under the 2nd and 3rd laws. The 2nd law also implies the procompetitive effect and strategic complementarity in pricing. ii) A lower entry cost leads to more competitive pressures, which reduces the markup rates of all firms under the 2nd law and raises the pass-through rates of all firms under the 3rd law. The profits of all firms decline (at faster rates among less productive firms under the 2nd law), which leads to a tougher selection. The revenues of all firms also decline (at faster rates among less productive firms under the 3rd law). A lower overhead cost has similar effects when the employment is decreasing in firm productivity, which occurs under the 2nd and the 3rd laws for a sufficiently high overhead cost. iii) Larger market size also leads to more competitive pressures, reducing the markup rates of all firms under the 2nd law and raises the pass-through rates of all firms under the 3rd law. The profits among more productive firms increase, while those among less productive decline under the 2nd law, which leads to a tougher selection. The revenues among more productive firms also increase, while those among less productive decline under the 3rd law at least when the overhead cost is not too large. iv) The impacts on the masses of entrants and of active firms depend, often crucially, on whether the elasticity of the distribution of the marginal cost is increasing or decreasing with Pareto-distributed productivity being the knife-edge case. v) Both a lower entry cost and larger market size may cause an increase in the average markup rate under the 2nd law and a decline in the average pass-through under the 3rd law due to the composition effect, since they also lead to a tougher selection, forcing less productive firms with lower markup rates and higher pass-through rates to shrink and to exit. This suggests that a rise of the markup may occur due to increased competitive pressures, causing a shift from the less productive/smaller to the more productive/larger. vi) In a multi-market setting, competitive pressures are stronger in larger markets. And more productive firms sort themselves into larger markets under the 2nd law. Due to this composition effect, the average markup (pass-through) rates can be higher (lower under the 3rd Law) in larger (thus more competitive) markets. This result suggests a caution when interpreting the evidence that compares the average markup and pass-through rates across markets with different sizes.