Economies of Scope

Economies of scope are said to be presented whenever it is less costly to produce a set of goods in one firm than it is to produce it in two or more firms. For the two-product case, this amounts to determining the sign of C(Q1,0) + C(0,Q2) – C(Q1,Q2). The first two terms in this equation are the total costs of producing product 1 in one firm and product 2 in another. The third term is the total cost of having these products produced by the same firm. If the difference is positive then economies of scope exist; if it is negative there are diseconomies of scope; and if it is zero then there are neither economies nor diseconomies of scope.

The degree of scope economies for the n-products case is given by the ratio below:

SC = [ C(Q1,0,0,…,0) + C(0,Q2,0,…,0) + … + C(0,0,0,…Qn) – C(Q1,Q2,Q3,…,Qn) ] / C(Q1,Q2,Q3,…,Qn)

Whenever SC exceeds zero, scope economies are present, which tell us that it is cheaper for one firm to produce both goods jointly.

RAC and a measure of scale economy based on a multiproduct environment

The Ray Average Costs (RAC) are defined as total costs divided by q. That is, RAC(q) = C(rq,(1-r)q)/q where q1=rq, q2=(1-r)q and r is the proportion in which product 1 is produced.

We know that RAC(q) falls iff dRAC(q)/dq <0.

But dRAC(q)/dq = (1/q)[r(dC/dq1)+(1-r)(dC/dq2)]-(1/q)(1/q)C(q)
= (1/q)(1/q) [rq(dC/dq1)+(1-r)q(dC/dq2)-C(q)]
= (1/q)(1/q) [q1(dC/dq1)+q2(dC/dq2)-C(q)]

So we have economics of scale if q1(dC/dq1)+q2(dC/dq2)-C(q) is negative or q1(dC/dq1)+q2(dC/dq2) is less than C(q)

The measure of scale economy is the following: S= C(q)/ [q1(dC/dq1)+q2(dC/dq2)].
We can see that q1(dC/dq1)+q2(dC/dq2) is less than C(q) only if S is greater than 1.

Consequently, RAC(q) falls, rises or is constant as q increases depending on whether S is above, below or equal to 1. Note that this S is the multiproduct analogue of the ratio of average to marginal cost (See Tuesday, May 2, 2006).

Product-Specific Economies of Scale based on a multiproduct environment

Suppose that instead of a C(q) cost function, we have a C(q1,q2) cost function. This new function represents the cost of a firm that produces q1 units of product 1 and q2 units of product 2.

The incremental costs of increasing product 2 from 0 to q2 holding product 1 constant is IC2 = C(q1,q2) – C(q1,0)

The average incremental costs of increasing product 2 from 0 to q2 holding product 1 constant is AIC2 = IC2/q2

So the product-specific economies of scale (PS2) of q2 holding the other output, q1, constant is: PS2=AIC2/MC2 , where MC2 is the marginal cost of product 2, which is the partial derivative of C(q1,q2) with respect to q2.

Note that in case with more than two products, we have PSi = AICi/MCi as before, but all outputs except qi must be held fixed (not only q1 as in the example above).

Single-Product Economies of Scale

The firm is said to have economies of scale if its average cost falls as output increases. But if average cost rises with output, the firm is said to have diseconomies of scale. Finally, if average cost does not vary with output, the firm has constant returns to scale.

There are a lot of reasons to expect a firm’s average cost to decline as its output increases (at least initially). Two of them are the followings: One is that fixed costs do not vary with output. Secondly, as output rises, a firm can use its labor in more specialized tasks (more efficient).

As I mentioned, scale economies exist if average costs (AC) falls as output expands. This can happen only if average cost (AC) is above marginal cost (AC > MC). This relationship suggest that a natural measure of scale economies (S) is the ratio of average to marginal cost (S=AC/MC). So economies of scale exist if S>1, diseconomies of scale exist if S<1 and constant returns to scale exist if S=1.

U.S. Horizontal Merger Guidelines

According to the Horizontal Merger Guidelines that the U.S. Department of Justice and the Federal Trade Commission issued in 1992 (revised in 1997), concentration is measured using the HHI, as follows:

a) Post-Merger HHI Below 1000. The Agency regards markets in this region to be unconcentrated. Mergers resulting in unconcentrated markets are unlikely to have adverse competitive effects and ordinarily require no further analysis.

b) Post-Merger HHI Between 1000 and 1800. The Agency regards markets in this region to be moderately concentrated. Mergers producing an increase in the HHI of less than 100 points in moderately concentrated markets post-merger are unlikely to have adverse competitive consequences and ordinarily require no further analysis. Mergers producing an increase in the HHI of more than 100 points in moderately concentrated markets post-merger potentially raise significant competitive concerns.

c) Post-Merger HHI Above 1800. The Agency regards markets in this region to be highly concentrated. Mergers producing an increase in the HHI of less than 50 points, even in highly concentrated markets post-merger, are unlikely to have adverse competitive consequences and ordinarily require no further analysis. Mergers producing an increase in the HHI of more than 50 points in highly concentrated markets post-merger potentially raise significant competitive concerns.

In our example with the nine firms (See Friday, March 10, 2006), suppose that the two smaller firms want to merge. The initial HHI is 1540 and the postmerger HHI is 1580 (8 firms with market shares 25, 20, 15, 10, 9, 8, 7 and 6). Because the postmerger HHI is between 1000 and 1800, and the change does not exceed 100, the merger does not raise concerns about competition. Now think the possibility that the 2 largest firms want to merge. The postmerger HHI is 2540 (8 firms with market shares 45, 15, 10, 8, 7, 6, 5 and 4). Because the postmerger HHI exceeds 1800, and the change in the HHI exceeds 50, concerns about competition are raised.

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)

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)

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