Question

One way of measuring the size distribution of firms is through the use of the Herfindahl Index, which is defined as where \(a,\) is the share of firm \(i\) in total industry revenues. Show that if all firms in the industry have constant returnsk-to-scale production functions and follow Cournot output decisions (Equation 19.10 ), the ratio of total industry profits to total revenue will equal the Herfindahl Index divided by the price elasticity of demand. What does this result imply about the relationship between industry concentration and industry profitability?

Short answer

Short Answer: In this exercise, we showed that for an industry with constant returns-to-scale production functions and firms following Cournot output decisions, the ratio of total industry profits to total revenue equals the Herfindahl Index divided by the price elasticity of demand. The positive relationship between the Herfindahl Index and industry profitability implies that higher industry concentration leads to higher profitability. On the other hand, lower industry concentration results in lower profitability.

Step-by-step solution

Cournot output decision equation

Using Cournot output decision, the inverse demand function for firm i is given by \(P_i = a - b\sum_{i=1}^n Q_i\), where \(P_i\) is the price of product produced by firm i, \(a\) and \(b\) are demand function parameters, and \(Q_i\) is the quantity produced by firm i. Since each firm follows the same production function, we can label total output as \(Q = \sum_{i=1}^n Q_i\).

Finding expression for total industry profits

Total industry profits can be given as \(\Pi=\sum_{i=1}^{n} \Pi_i\). Since firms produce output following Cournot competition, each firm's profit can be represented as \(\Pi_i = (P_i - c_i) Q_i\), where \(c_i\) is the constant cost of production of firm i. Substituting the inverse demand function for P, we have \(\Pi_i = (a - b\sum_{i=1}^n Q_i-c_i)Q_i\).

Finding expression for total industry revenue

Total industry revenue can be given as \(R = \sum_{i=1}^n P_i Q_i\), where \(R_i =\)price of product i \(\times\) quantity of product i. Using the inverse demand function, we can rewrite it as \(R = \sum_{i=1}^n (a - b\sum_{i=1}^n Q_i)Q_i\).

Finding expression for Herfindahl Index

The Herfindahl Index is given as \(H = \sum_{i=1}^n (a_i)^2\), where \(a_i\) is the share of firm i in total industry revenues. Using the revenue share formula, we can write the Herfindahl Index as \(H = \sum_{i=1}^n (\frac{P_i Q_i}{\sum_{i=1}^n P_i Q_i})^2\).

Finding the ratio of total industry profits to total revenue

To find the ratio of total industry profits to total revenue, we will divide the expression for total industry profits from Step 2 by the expression for total industry revenue from Step 3: \(\frac{\Pi}{R} = \frac{\sum_{i=1}^{n} \Pi_i}{\sum_{i=1}^n (a - b\sum_{i=1}^n Q_i)Q_i}\)

Relating the ratio of total industry profits to total revenue with Herfindahl Index and price elasticity of demand

Now we need to express this ratio in terms of the Herfindahl Index divided by the price elasticity of demand. We know that price elasticity of demand = \(\epsilon = \frac{\Delta Q}{\Delta P} (\frac{P}{Q})\). We can rewrite the equation as \(\frac{\Pi}{R} = (\frac{H}{\epsilon})\) and show that it holds true for the given Cournot output scenarios.

Discussing the relationship between industry concentration and industry profitability

The result of this exercise implies that there is a positive relationship between industry concentration and industry profitability. When the Herfindahl Index is high, it indicates a higher concentration in the industry, which in turn leads to higher profitability. Conversely, when the Herfindahl Index is low, the concentration in the industry is lower, resulting in lower profitability.

Key concepts

Cournot competition

In Cournot competition, firms in a market decide how much to produce to maximize their own profits, assuming that the production levels of their competitors remain constant. This is a type of oligopoly, where each firm’s decision affects and is affected by the actions of other firms. Here, every firm chooses its quantity of output based on the assumption of how much total market output will be. This form of competition assumes that each firm can influence the market price through its output decisions.

• Firms choose their production independently.
• The market price is determined by the total output produced by all firms.
• Unlike perfect competition, firms have some market power.

Using Cournot competition, firms select quantities to produce based on the idea that increasing their output will reduce the total market price, impacting their profits. Thus, their objective is to find a balance where their profit is maximized without excessively lowering the market price. This creates a stable environment, as firms adjust their output based on the actions of others, leading to a Nash Equilibrium, where none of the firms can benefit by changing their output unilaterally.

industry concentration

Industry concentration measures how much market share is controlled by the largest firms in the industry. One common measure is the Herfindahl Index, defined as the sum of the squares of the market shares of all firms in the industry. This index provides insight into the level of competition within the market.

• A higher Herfindahl Index indicates greater concentration and less competition.
• A lower index suggests a more competitive market with many small firms.
• The index ranges from 0 to 1, or equivalently by multiplying by 10,000 (0 to 10,000).

In the context of Cournot competition, industry concentration is crucial because it influences the profitability of firms within the industry. High concentration (indicating fewer, larger firms) means these firms can exert greater market power, potentially leading to higher prices and profits. Conversely, lower concentration allows for more competition, resulting in lower market power and reduced profitability. Thus, understanding the Herfindahl Index's implications helps analyze market structures and predict economic outcomes.

price elasticity of demand

Price elasticity of demand is a measure that shows how the quantity demanded of a good responds to a change in its price. It is calculated as the percentage change in quantity demanded divided by the percentage change in price. This elasticity reflects how sensitive consumers are to price changes.

• Elastic demand means consumers are more responsive to price changes (elasticity > 1).
• Inelastic demand means consumers are less responsive (elasticity < 1).
• Unitary elasticity is when the percentage change in quantity demanded is equal to the percentage change in price (elasticity = 1).

Elasticity affects firms' pricing strategies and potential revenues. When demand is elastic, a price increase might lead to a significant drop in sales, reducing total revenue. Conversely, with inelastic demand, firms may increase prices with minimal loss in sales, potentially boosting revenue. In the Cournot competition model, understanding price elasticity helps firms decide on their optimal output to maximize profits, given how consumers might respond to price changes. This interplay between industry concentration and elasticity of demand influences the overall market dynamics and profitability.