Corrigendum to: Assortative Matching With Large Firms (Econometrica, 86, 1, (85-132), 10.3982/ECTA14450)

This document corrects an error in Eeckhout and Kircher (2018) in the sign of an underived condition for positive assortative matching (PAM thereafter) within those extensions that allows for generic capital investment: It occurs in the applications of the main theory to The skill premium with generic capital investment and The misallocation debate; see page 104. This note provides the correct condition, proves it, and adjusts the accompanying example.¹ The correct condition is 1 (Formula presented.) Background: Eeckhout and Kircher (2018) considered a competitive economy with a given distribution of firm types y and worker types x. Firms produce output according to production function (Formula presented.), where x is the worker type hired by firm y and l is the number of such workers. Output is strictly concave in l. Define (Formula presented.) as the output of r such firms that employ altogether l such workers. The paper derives a condition when higher firm types employ higher worker types. Such positive assortative matching requires 2 (Formula presented.) where subscripts denote cross-partial derivatives and arguments are omitted here. In Section 4, pages 104–106, of Eeckhout and Kircher (2018), an additional capital investment is introduced. Firms can now produce according to (Formula presented.), which is concave in (Formula presented.) where k is a generic capital good. In a small open economy, the price i per unit of capital is exogeneous. Using (Formula presented.), one can use the condition in the previous paragraph to obtain a sorting condition. An inequality on page 104 and in Appendix B of Eeckhout and Kircher (2018) represents this sorting condition directly in terms of (Formula presented.), or more precisely, in terms of the total output (Formula presented.) that can be produced by r such firms that employ in total l such workers and k units of capital. This had a sign mistake, and the inequality has the opposite (and therefore incorrect) sign. The correct condition for positive assortative matching in this case is (1). The remainder of this note proves this result, and revisits the numerical illustration. Using an appropriate functional form for this illustration that gives rise to positive assortative matching with the correct sorting condition reveals the same qualitative features that were discussed in the original paper. Proof of Condition (1) for PAM With Generic Capital Investment Consider the production function 3 (Formula presented.) We assume that (Formula presented.) is twice differentiable; strictly concave in each of l, r, and k; and displays CRS in l, r, and k (so that F has CRS in r and l). Problem (3) can be rewritten by explicitly solving the maximization problem, that is, 4 (Formula presented.) where (Formula presented.) depends on (Formula presented.) and solves the first-order condition:² (Formula presented.) Calculating the gradient of (Formula presented.) will be useful for further simplifications below. Applying the implicit function theorem, we have 5 (Formula presented.) We know that PAM arises only if (2) holds along the assignment, and PAM arises if (2) holds strictly everywhere. Using the formulation in (4), the second derivatives in (2) can be rewritten as 6 (Formula presented.) Substituting (5) into (6) and then replacing the terms in (2), the PAM condition becomes 7 (Formula presented.) Multiplying both sides of (7) by (Formula presented.), the sign of the inequality changes as (Formula presented.). Simplifying, we get (1). Application to the Misallocation Debate Given this new condition, the production function (Formula presented.) does not satisfy the conditions for PAM. This is the functional form used in the illustration of page 105 and in Appendix B of Eeckhout and Kircher (2018), which constructs the equilibrium using the first-order conditions valid only under PAM. Here, we redo the exercise with a different functional form that does satisfy the PAM condition (1) with the parameter values in Adamopoulos and Restuccia (2014): 8 (Formula presented.) Taking derivatives of this production function and substituting the parameter values for this application reveals that (1) is satisfied.³ Using this new production function, Figure 4 on page 105 in Eeckhout and Kircher (2018) becomes Figure 1. This yields qualitatively simlar results, and the discussion in Eeckhout and Kircher (2018) applies unchanged to this new example. A mean-preserving spread in input heterogeneity reduces heterogeneity in the distribution of land holdings across farms, as better firms buy less but better land. 1 Figure (Figure presented.) Firm size distribution for different dispersion in x (x is log-normally distributed LN(0,0.2), i.e., log(x) is normally distributed with mean 0 and variance 0.2, truncated at the bounds indicated in the legend, with the measure of the truncated distribution normalized to 1).