## Friday, March 24, 2006

### The HHI as a function of the weighted average of the firms’ price-cost margins and the market demand elasticity

HHI = ∑ Si. Si as it equals the sum of the squared market shares of each firm in the industry. (See Friday, March 10, 2006) = -ε. (∑ Si. Si )/(-ε) (multiply and divide with -ε) = -ε.∑Si.L, where L is the Lerner Index of market power in an oligopoly environment (See Friday, March 17, 2006). That is, the HHI equals the absolute value of the market demand elasticity multiply with the weighted average of the firms’ price-cost margins.

Note that in case of monopoly L= -Si/ε=-100/ε (Share equals 100%) So HHI=-ε.100.L=10000 (See Friday, March 10, 2006)

## Friday, March 17, 2006

### The Lerner Index of market power in an oligopoly environment and its relation with elasticity of demand

An oligopoly consists of n identical firms that produce a homogeneous product. The firm 1 chooses its output q1 to maximize its profits (All the firms do the same). So we have Π1 = p(Q).q1 – m.q1 where m is the constant marginal cost for each firm and p, the price, is a function of total output. By the identical firms assumption S1 = q1/Q = 1/n or Q=n.q1 (Equation 1) where S1 is the output share of firm 1. The first order condition is the following: (Equation 2) MR = p + q1.p´ = m where p´ is the derivative of price with respect to Q. The second order condition holds so we have profit maximization.

The Lerner Index (L) is the difference between the price and marginal cost as a function of price. It is also called price-cost margin or price-cost markup. So (Equation 3) L = (p – m) / p = - q1.p´ /p (By Equation 2)

The elasticity of demand is a characteristic of the demand curve and is defined as the percentage change in quantity that results from 1 percent change in price. If the elasticity of demand is very high (a large negative number), then the curve is said to be elastic. With a very elastic demand, a small price change induces a very large change in the quantity demanded. If the elasticity is low (a small negative number), the demand curve is inelastic, and a price change of 1 percent has relatively little effect on the quantity demanded. So the demand elasticity ε equals Q´.p/Q where Q´ is the derivative of total output with respect to p. So the reciprocal of the elasticity of demand 1/ε equals p´.Q/p (Equation 4)

So L equals - q1.Q.p´ / p.Q (By Equation 3 & multiply and divide with Q) = - (q1/Q) . (1/ ε) (By Equation 4) = -1/(n.ε) (By Equation 1) or –S1/ ε (By Equation 1)

## Friday, March 10, 2006

### Measures of Industry Concentration

The most common measures of industry concentration are the CR4, CR8 and HHI. The four-firm Concentration Ratio (CR4) is the share of industry sales accounted for by the four largest firms. The eight-firm Concentration Ratio (CR8) is the share of industry sales accounted for by the eight largest firms. The Herfinhahl-Hirschman Index (HHI) equals the sum of the squared market shares of each firm in the industry. Note that this index is a function of all firms’ market shares. HHI has lower limit zero in case of perfect competition (firms→∞ and shares→0) and upper limit 10000 in case of monopoly (1 firm with share 100).

Example: An industry has nine firms with the following market shares: 25, 20, 15, 10, 8, 7, 6, 5 and 4. CR4 = 25 + 20 + 15 + 10 = 70, CR8 = 25 + 20 + 15 + 10 + 8 + 7 + 6 + 5 = 96 and HHI = 625 + 400 + 225 + 100 + 64 + 49 + 36 + 25 + 16 = 1540