Analytical Problems

Herfindahl index of market concentration

One way of measuring market concentration is through the use of the Herfindahl index, which is defined as

H = _i=1∑ⁿ s_i²,

where s_t = q_i/Q is firm i'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 cornpetitive pressure. We will assess the validity of this intuition using several models.

a. 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, where Cournot duopolists face different marginal costs. Use your answer to Problem to compute equilibrium firm outputs, market output, price, consumer surplus, industry profit, and total welfare, substituting the particular cost parameters c₁ = c₂ = 1/4. Also com¬pute the Herfindahl index.

d. Repeat your calculations in part (c) while assuming that firm l's marginal cost c₁ falls to 0 but c₂ 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?