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.¹1 The code that generates the accompanying plot for this example and checks for the correct condition can be found at https://github.com/crisla/Capital-Investment-in-Assortative-Matching-with-large-Firms.
The correct condition is

(1)

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 , 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 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)

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 , which is concave in where k is a generic capital good. In a small open economy, the price i per unit of capital is exogeneous. Using , 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 , or more precisely, in terms of the total output 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.

该文件纠正错误的Eeckhout和基歇尔（2018中为那些允许通用资本投资扩展内积极选型匹配（此后PAM）的underived条件符号）：它发生在主要理论到应用的技术溢价与通用资本投资和分配不当的争论；请参见第 104 页。此注释提供了正确的条件，证明了这一点，并调整了随附的示例。¹1可以在 https://github.com/crisla/Capital-Investment-in-Assortative-Matching-with-large-Firms 中找到为该示例生成随附图并检查条件是否正确的代码。
正确的条件是

(1)

背景：Eeckhout 和 Kircher ( 2018 ) 考虑了具有给定分布的公司类型y和工人类型x的竞争性经济。企业根据生产函数生产产出，其中x是企业y雇用的工人类型，l是此类工人的数量。输出在l 中严格凹入。定义为r 个这样的公司的产出，这些公司总共雇用l 个这样的工人。该论文推导出了一个条件，即较高的公司类型雇用较高的工人类型。这种积极的分类匹配需要

(2)

其中下标表示交叉偏导数，这里省略了参数。

在Eeckhout 和 Kircher ( 2018 ) 的第 4 节第 104-106 页中，引入了额外的资本投资。企业现在可以根据 进行生产，其中k是一般资本品。在小型开放经济体中，每单位资本的价格i是外生的。使用，可以使用上一段中的条件来获得排序条件。

104页上和附录B不等式Eeckhout和基歇尔（2018）直接表示该分类条件的情况来讲，或更精确地说，在总输出方面，可以通过下述方法制造ř这些公司雇用总共升这样工人和k 个单位的资本。这有一个符号错误，并且不等式具有相反（因此不正确）的符号。在这种情况下，正分类匹配的正确条件是 (1)。

本笔记的其余部分证明了这一结果，并重新审视了数值说明。为这个说明使用适当的函数形式，产生与正确排序条件的正分类匹配，揭示了与原始论文中讨论的相同的定性特征。