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Chapter 19: Problem 5

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

Expert verified
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

01

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 \(Q = \sum_{i=1}^n Q_i\).
02

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 \(\Pi_i = (a - b\sum_{i=1}^n Q_i-c_i)Q_i\).
03

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 \(R = \sum_{i=1}^n (a - b\sum_{i=1}^n Q_i)Q_i\).
04

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 \(H = \sum_{i=1}^n (\frac{P_i Q_i}{\sum_{i=1}^n P_i Q_i})^2\).
05

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: \(\frac{\Pi}{R} = \frac{\sum_{i=1}^{n} \Pi_i}{\sum_{i=1}^n (a - b\sum_{i=1}^n Q_i)Q_i}\)
06

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.
07

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.

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Most popular questions from this chapter

Assume for simplicity that a monopolist has no costs of production and faces a demand curve given by \\[ Q=150-P \\] a. Calculate the profit-maximizing price-quantity combination for this monopolist. Also calculate the monopolist's profits. b. Suppose a second firm enters the market. Let \(q_{x}\) be the output of the first firm and \(q_{2}\) the output of the second. Market demand is now given by \\[ q_{x}+q_{2}=150-P \\] Assuming this second firm also has no costs of production, use the Cournot model of duopoly to determine the profit-maximizing level of production for each firm as well as the market price. Also calculate each firm's profits. c. How do the results from parts (a) and (b) compare to the price and quantity that would prevail in a perfectly competitive market? Graph the demand and marginal revenue curves and indicate the three different price-quantity combinations on the demand curve.

S. Salop provides an instructive model of product differentiation. He asks us to conceptualize the demand for a product group as varying along a circular spectrum of characteristics (the model can also be thought of as a spatial model with consumers located around a circle). Demanders are located at each point on this circle and each demands one unit of the good. Demanders incur costs if they must consume a product that does not precisely meet the characteristics they prefer. As in the Hotelling model, these costs are given by \(t x\) (where \(x\) is the "distance" of the consumer's preferred characteristic from the characteristics being offered by the nearest supplier and \(t\) is the cost incurred per unit distance). Initially there are \(n\) firms each with identical cost functions given by \(T Q=/+c q,\) For simplicity we assume also that the circle of characteristics has a circumference of precisely 1 and that the \(n\) firms are located evenly around the circle at intervals of \(1 / n\) a. Each firm is free to choose its own price \((p),\) but is constrained by the price charged by its nearest neighbor \(\left(p^{*}\right) .\) Explain why the extent of any one firm's market \((x)\) is given by the equation \\[ p+t x=p^{*}+t /(l / n)-x J \\] b. Given the pricing decision illustrated in part a, this firm sells \(q,=2 x\) because it has a mar ket on "both sides." Calculate the profit-maximizing price for this firm as a function of \(p^{*}, c,\) and \(t\) c. Assuming symmetry among all firms will require that all prices are equal, show that this results in an equilibrium in which \(p=p^{*}=c+t / n .\) Explain this result intuitively. d. Show that in equilibrium the profits of the typical firm in this situation are \(T T,=t / n^{2}-f\) e. Assuming free entry, what will be the equilibrium level of \(n\) in this model? f. Calculate the optimal level of differentiation in this model-defined as that number of firms (and products) that minimizes the sum of production costs plus demander dis tance costs. Show that this number is precisely half the number calculated in part (e). Hence, this model suffers from "over- differentiation." (For a further exploration of this model, see S. Salop "Monopolistic Competition with Outside Goods," Bell Journal of Eco nomics, Spring \(1979, \text { pp. } 141-156 .)\)

Suppose demand for crude oil is given by \\[ Q=-2,000 P+70,000 \\] where \(Q\) is the quantity of oil in thousands of barrels per year and \(P\) is the dollar price per barrel. Suppose also that there are 1,000 identical small producers of crude oil, each with marginal costs given by \\[ M C=q+5 \\] where \(q\) is the output of the typical firm. a. Assuming each small oil producer acts as a price taker, calculate the market supply curve and the market equilibrium price and quantity. b. Suppose a practically infinite supply of crude oil is discovered in New Jersey by a wouldbe price leader and can be produced at a constant average and marginal cost of \(\$ 15\) per barrel. Assuming the supply behavior of the competitive fringe described in part (a) is not changed by this discovery, how much should the price leader produce in order to maximize profits? What price and quantity will now prevail in the market? c. Graph your results. Does consumer surplus increase as a result of the New Jersey oil dis covery? How does consumer surplus after the discovery compare to what would exist if the New Jersey oil were supplied competitively?

Suppose a firm is considering investing in research that would lead to a cost- saving innovation. Assuming the firm can retain this innovation solely for its own use, will the additional profits from the lower (marginal) costs be greater if the firm is a competitive price taker or if the firm is a monopolist? Develop a careful graphical argument. More generally, develop a verbal analysis to suggest whether competitive or monopoly firms are more likely to adopt cost-saving innovations. (For an early analysis of this issue, see W. Fellner, "The Influence of Market Structure on Technological Progress," Quarterly Journal of Economics [November 1951]: 560-567.)
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