One way of measuring market concentration is through the use of the Herfindahl
index, which is defined as
\\[
H=\sum_{i=1}^{n} s_{i}^{2}
\\]
where \(s_{t}=q_{i} / Q\) is firm \(i^{\prime}\) s market share. The higher is
\(H\), the more concentrated the industry is said to be. Intuitively, more
concentrated markets are thought to be less competitive because dominant firms
in concentrated markets face little competitive pressure. We will assess the
validity of this intuition using several models.
a. If you have not already done so, answer Problem \(15.2 \mathrm{d}\) by
computing the Nash equilibrium of this \(n\) -firm Cournot game. Also compute
market output, market price, consumer surplus, industry profit, and total
welfare. Compute the Herfindahl index for this equilibrium.
b. Suppose two of the \(n\) firms merge, leaving the market with \(n-1\) firms.
Recalculate the Nash equilibrium and the rest of the items requested in part
(a). How does the merger affect price, output, profit, consumer surplus, total
welfare, and the Herfindahl index?
c. Put the model used in parts (a) and (b) aside and turn to a different
setup: that of Problem \(15.3,\) where Cournot duopolists face different
marginal costs. Use your answer to Problem 15.3 to compute equilibrium firm
outputs, market output, price, consumer surplus, industry profit, and total
welfare, substituting the particular cost parameters \(c_{1}=c_{2}=1 / 4 .\)
Also compute the Herfindahl index.
d. Repeat your calculations in part (c) while assuming that firm 1's marginal
cost \(c_{1}\) falls to 0 but \(c_{2}\) stays at 1/4. How does the cost change
affect price, output, profit, consumer surplus, total welfare, and the
Herfindahl index?
e. Given your results from parts (a)-(d), can we draw any general conclusions
about the relationship between market concentration on the one hand and price,
profit, or total welfare on the other?
