Aggregate industry cost functions and the Herfindahl index.

• Author: Dickson, Vaughan

1. Introduction

   Researchers often use industry data to estimate cost functions or production functions. This presents an aggregation problem since most functional forms do not survive aggregation from the firm to industry level. Although the problem is well-known, it is either accepted as unavoidable because of the paucity of firm data, or a functional form consistent with aggregation is used. For example, [c.sub.i]([y.sub.i], w) = [y.sub.i]f(w) may be imposed, where the subscript denotes firm i, c is total cost, y is firm output and w is the exogenous input price vector faced by all firms. However this solution is itself problematical since, for example, constant-returns-to-scale must be imposed.(1)

   In this paper a procedure for dealing with the aggregation problem is presented that does not mandate the adoption of very restricted functional forms. The context is industry cost functions although the approach is also applicable to production functions. Specifically the Herfindahl (H) index of concentration is introduced into the industry cost function. The incorporation of the H index is derived explicitly from the process of aggregation across firms to obtain the industry function. Using annual industry-level data from 1961 to 1982, cost functions that include the H index are estimated for 16 three and four digit S.I.C. industries drawn from the two digit Canadian food and beverage sector.

   In fact for these 16 industries a Cobb-Douglas form with and without the H index adjustment is estimated, as is a translog form with the H index adjustment. By comparing the H-adjusted form with the conventional unadjusted form, we can measure the bias from either ignoring the aggregation problem or finessing it with a restrictive functional form. We find that constant returns is not a good assumption for Canadian food processing and that the unadjusted form typically overstates economies of scale.

   By estimating cost functions that include H we can also directly measure the effect of industry concentration on industry cost. This is useful because of the debate in the industrial organization literature about the nature of the positive link between industry profits and concentration [8; 10]. This is a debate about the presence and relative strengths of market power effects (higher prices) and efficiency effects (lower costs) associated with higher concentration. Our results show that total costs decline in most of the 16 industries as the H index increases, ceteris paribus. A major conclusion is that efficiency effects are important in Canada's food processing sector.

   In the following we introduce the aggregation procedure that incorporates the H index into the industry cost function. We then estimate for the sample industries a Cobb-Douglas cost function with and without the H adjustment followed by a translog form with the H adjustment. We also compare our translog results with another effort by Cahill and Hazledine [6] to account for the influence of concentration on industry cost in Canada's food processing sector.(2)

2. The Aggregation Procedure

   We begin with a Cobb-Douglas cost function for firm i in a particular industry:(3)

   [Mathematical Expression Omitted].

   In equation (1) [c.sub.i] is total cost for firm i, [q.sub.i] is firm output and w represents exogenous input prices faced by all firms. Since often only annual industry data are available, aggregation is required before estimation can proceed:

   [Mathematical Expression Omitted].

   Clearly this does not in general correspond to the aggregate Cobb-Douglas C = A[w.sup.n][Q.sup.a], Q = [Sigma][q.sub.i], employed by practicioners. A correspondence occurs only if a = 1 (constant returns), or if the concept of a "representative" firm is adopted such that each firm's share of output is invariant from year-to-year. Thus if [q.sub.i] = [k.sub.i]Q where [k.sub.i] is a constant for each firm, then C = A[w.sup.n] [Sigma][([k.sub.i]Q).sup.a] = A[w.sup.n] [Q.sup.a].(4)

   To resolve these aggregation difficulties, we re-write equation (1) as:

   [c.sub.i] = A[w.sup.n][(Q[s.sub.i]).sup.a]. (3)

   In equation (3), [s.sub.i] equals [q.sub.i]/Q which is firm i's share of output. To obtain industry total cost we sum over all firms in the industry:

   [Mathematical Expression Omitted].

   The problem with equation (4) is typically unavailable firm data is needed to measure [Mathematical Expression Omitted]. To deal with this, the concept of the Herfindahl numbers equivalent ([n.sub.e]) is used, this being the number of equal-size firms giving a corresponding H value, i.e., [n.sub.e] = 1/H. Thus if all firms are equal, [s.sub.i] = 1/[n.sub.e] = H and [Mathematical Expression Omitted]. With this substitution, equation (4) can be expressed with industry-level data only:

   C = A[w.sup.n][Q.sup.a][H.sup.a - 1]. (5)

   Of course since firms are not of equal size, [H.sup.a - 1] is only an approximation for [Mathematical Expression Omitted]. However we think it is a useful approximation because it does take into account size distribution effects on industry costs and because it is a sufficiently accurate approximation to be worthwhile. To support this claim, for each industry we took 1982 concentration ratio data giving the share of industry shipments of the largest 4, 8, 12, 16, 20, 50 and remaining firms and, by assuming equal size firms in each of these size classes, we calculated [Mathematical Expression Omitted] and [Mathematical Expression Omitted] for different values of a. For a = (.7, .8, .9, .95) the correlation coefficient for the two terms over...