Chapter 7, Problem 7CACQ

To determine

The general rule for impact on HHI when two firms in market merge.

Explanation of Solution

Sales of firms are termed in industry as: $S_1,S_{2,},S_3\mathrm{........}{S}_n$ respectively such total sales in the industry is $S_r$

When firms are not merged, then HHI is calculated as:

  $10,000{\displaystyle \sum _{i=1}^n{\left(\frac{S_i}{S_T}\right)}^2=10,000\left[{\left(\frac{S_1}{S_T}\right)}^2+{\left(\frac{S_2}{S_T}\right)}^2+\mathrm{...}+{\left(\frac{S_i}{S_T}\right)}^2+{\left(\frac{S_{i+1}}{S_T}\right)}^2+\mathrm{...}+{\left(\frac{S_n}{S_T}\right)}^2\right]}$

When there is merger between two firms: HHI is calculated as follows:

  $\begin{array}{l}10,000\left[{\left(\frac{S_1}{S_T}\right)}^2+{\left(\frac{S_2}{S_T}\right)}^2+\mathrm{...}+{\left(\frac{S_i+S_{i+1}}{S_T}\right)}^2+\mathrm{...}+{\left(\frac{S_n}{S_T}\right)}^2\right]\\ \end{array}$

Difference between above two:

  $10,000\left[{\left(\frac{S_i+S_{i+1}}{S_r}\right)}^2-{\left(\frac{S_i}{S_T}\right)}^2-{\left(\frac{S_{i+1}}{S_T}\right)}^2\right]$

  $=10,000\left[\frac{2S_i+S_{i+1}}{S_T{}^2}\right]$

Since,

  $\frac{S_i}{S_T}=\frac{S_{i+1}}{S_T}=0.3$

Comparison of HHI when firms are merged and not merged:

  $10,000\left[\frac{2S_i{S}_{i+1}}{S_T{}^2}\right]$

  $=10,000\times 2\times 0.3\times 0.3$

  $=1800$

Value of HHI increases by 1800.

For the purpose of generalization, if the two firms merge, the increase in HHI post-merger is given by:

  $10,000\left[{\left(\frac{S_i+S_{i+1}}{S_T}\right)}^2-{\left(\frac{S_i}{S_T}\right)}^2-{\left(\frac{S_{i+1}}{S_T}\right)}^2\right]$

  $=10,000\left[\frac{2S_i{S}_{i+1}}{S_T{}^2}\right]$

  $=2\times 10,000\times \frac{S_i}{S_T}\times \frac{S_{i+1}}{S_T}$

Where, $\frac{S_i}{S_T}$ and $\frac{S_{i+1}}{S_T}$ are the pre-merger market share of the two merging firms.

Economics Concept Introduction

Introduction:HHI is a measure to know concentration of market power in an industry. Higher the value of HHI, higher is the firm’s market power.