Margins, Concentration and Oligopolistic Interdependence*

ROBERT DIXON and ALAN GUNTHER University of Melbourne, Parkville, Victoria 3052

In this paper we examine the relationship between margins, elasticities of demand and the Hirschman-Herfindahl index of concentration for Australian manufacturing. Our main aim is io estimate the degree of collusion in various manufacturing sectors in the yearq,1968-69, 1972-73 and 1977-78. There appear to be marked differences in the apparent degree of collusion between indusfria. It i s argued that once sub-opfimization is allowed the Cowling- Waterson model will yield biased estimates of the degree of collusion.

This paper examines, theoretically and empirically, the determination of industry mark- up of price on cost in terms of own-price elasticity of demand, seller concentration and the extent to which firms in an industry recognize their inter- dependence. Data relating to the manufacture of (predominantly) consumer goods in Australia are used in an attempt to estimate the degree of apparent collusion in those industries. Important aspects of the paper include the use of econometric estimates of elasticities of demand, the derivation and use of Hirschman-Herfindahl indices for the industries concerned, and an extension of the Cowling-Waterson model of oligopoly to allow for departures from profit maximization. Section I presents a standard model of profit-maximizing oligopoly. Section I1 indicates the data sources and presents estimates of the ‘apparent’ collusion in Australian manufacturing. Section 111 extends the model to make some allowance for departures from profit maximization. The final section draws some

We would like to thank Keith Cowling, Guay Lim, Alan Powell, Philip Williams and two referees for helpful comments. Any errors or omissions are our responsibility.

conclusions and suggests some avenues for further research.

I A Model of Profit-Maximizing Oligopoly In this section we present the (now) standard

economic model‘ of the level of margins in an industry which is oligopolistic and where individual firms are assumed to maximize profits.

Consider a n industry which consists of n firms producing a homogeneous commodity which is sold at a single, industry-wide price.* We will assume that the marginal costs for each firm are constant over the relevant range of output, whilst allowing for the possibility that cost functions may differ between firms. Each frrm sets output (X,) so as to maximize profits (ni = p X , - c,X, - F,, where c = marginal cost = average variable cost and F= fmed cost). Let 7 be the (constant) industry- wide elasticity of demand (7 = (dX/X) / (dp /p ) ) .

I An excellent exposition of modern models of oligopoly will be found in Waterson (1984).

2 The model is also applicable to the case where we have a cluster o f prices for (slightly) differentiated products provided that price relativities are maintained. (See footnote 3 below).

199

200 THE ECONOMIC RECORD JUNE

The profit-maximizing condition for the r'th firm may be written as3

p - p ( (I/q)(dX/dXj)(X;/x)) = c;. (1)

The firm's gross profit (i.e. output multiplied by the difference between price and average variable cost) will equal

(p - c;) x; = X p ( ( 1 /q)(dX/dX;)(X;/X)) . (2)

Let us define 8, to be the (conjectural4) elasticity of industry output with respect to the output of the r'th firm (i.e. 8; = (dX/X) / (dX, /X, ) ) . Equation (2) may be rewritten as

Summing the above over'all firms in the industry and assuming, for simplicity, that 8 is the same for all firms yields an expression for aggregate gross profits. Dividing that sum by the value of industry sales (Xp = CXp) yields an expression for the industry gross margin (w5

M = ( e l q ) . (4)

This is to say that an industry's gross margin will reflect the magnitudes of the industry elasticity of demand and a conjectural elasticity (8) which indicates the degree of recognition of mutual interdependence in the industry. The elasticity B may be seen as measuring the extent to which the industry margin approaches, or diverges from, the joint profit-maximizing outcome. To see this, we rearrange (4) to yield a n expression for B

8 = Mq. ( 5 )

The term in brackets will equal the inverse of the elasticity of demand for the products of the r'th firm (i.e. [(dXJXj)/(dp/p)] - 1). Where an array of prices prevails in the industry the inverse of the elasticity of demand for the products of the r'th firm may be written as ( ( l / r l ) @ / p , ) ( d p i / d p ) ( d X / ~ X j ) ( X l / ~ ) . Provided that price relativities are maintained, i.e. (dp/p) = (dp;/p;), this expression reduces to that given in the text.

See Appelbaum (1982, p. 289). In the event that 0; was not identical for all firms,

equation (4) would appear as:

(4') M = (E(X;/Xy;)(l/q). In other words, we could interpret the 0 in equation (4) as a weighted average of Bi.

Recall that if the firms in the industry were maximizing joint profits the gross margin would equal the inverse of the elasticity of demand. If we divide the actual amount of the margin in the industry (M) by the value which would be obtained for the margin under conditions of joint profit maximization ( l /q ) we obtain an index (Mq) which indicates the degree to which observed margins approach the level which would obtain under a perfect cartel. Clearly, if the aim of the exercise is to predict ' . . . the extent of the departure of price (or alternatively, of rate of return) from the competitive level' (Stigler, 1968 p. 30) indices such as 0 are likely to be of considerable interest, both empirically and theoretically.

It is of particular interest to enquire into the relationship between our 8 index (and margins) and the level of industrial concentration. To do this, we follow an approach akin to that suggested by Cowling and Waterson (1976) and Dickson (1982).

We define industry output to be

X = X I + EX,, where i f j.

I t follows from the above that

d X / X = (X, /X)(dX,/X,) + (1 - (X, /X)) (dc XJ/EX,) . (6)

Let 9, be a measure of interdependence, defined as

+, = (dcX, /CX,) / (dX, /X, ) . (7)

Given o u r definition of 8, (i.e. 8, = (dX/X) / (dX, /X, ) ) . the indices 8, and 9, are related. Dividing both sides of equation (6) by (dX,/X,) gives

Substitution of (8) into (3) yields an expression for the profits of the r'th firm as

Summing the above over all the firms in the industry, and assuming that is the same for all firms, yields an expression for aggregate gross profits. Dividing that sum by the value of industry sales (pa yields an expression for the industry

1986 OLIGOPOLY BEHAVIOUR 20 I

(gross) margin

M = (H/q) ( 1 - 4') + ( * / q ) (9)

where H = C ( X , / m 2 , i.e. the Hirschman- Herfindahl index of concentration.

Like 0, the parameter 4' may be used as an indicator of the extent of collusion in the industry and as a measure of the degree to which the industry approaches the joint profit-maximizing result. If CP equals unity we have the completely collusive result (i.e. M = ( l /q) ) , whilst if + equals zero we have the Cournot result.

Given certain underlying assumptions', namely all firms maximize short-run profits, the firms in each industry are producing a single homogeneous commodity or a range of slightly differentiated products whose relative prices remain constant8, and profit margins within each industry are determined by competition ' among existing producersg, we have been able to establish a relationship, within any industry, between the gross margin, the Hirschman-Herfindahl index, the industry elasticity of demand and the parameter 4'. In the following section of this paper we concern ourselves with the modest exercise of using information on the values of M, q and H in order to estimate the value of 4' for various Australian manufacturing industries.1° In doing this we differ from earlier studies of the relationship between margins and concentration for Australia in that we make explicit use of the Cowling-Waterson model, we use (econometric) estimates o f industry elasticities of demand and our own estimate of the Hirschman-Herfindahl indices.

6 In the event that a, was not the same for all firms, equation (9) would appear as:

( 9 ' ) M = I/V(H+ z *J,/X- z IPAVxr). These expressions (9 and 9 ' ) may be compared with similar expressions to be found on p. 198 of Cowling (1981) and p. 40 of Dickson (1982).

7 We are grateful to a referee for the concise formulation which follows.

* In effect this means that each input-output industry is assumed to be a single economic market.

9 We ignore potential competition and competition from imported products.

10 A later study will attempt a time-series analysis of a for various industries. We do however anticipate severe problems in obtaining adequate data on the Hirschman- Herfindahl index across time.

II Data and Estimation Our data on industry elasticities of demand were

derived for use in one of the large-scale models of the Australian economy. This model (the ORANI model, which is a Johansen-type multisectoral modell') identifies 65 manufacturing industries, this being the level of aggregation used in the 1968-69 Austral ian Input -Output Tables (Australian Bureau of Statistics, 1976). For each industry, household demand elasticities were estimated using the approach pioneered by Frisch

Since the elasticities refer to household demand we have only included in our study those industries where sales to private consumers make up a signi- ficant part of total gross output." Also, we chose to exclude the industry 'Milk Products' (ASIC Code 212) as we felt its activities were regulated t o an extent that would invalidate its appearance in our study. As a result we are left with 20 manufacturing industries in our sample. A list of these industries, the ratios of consumer demand to gross output and the estimated household demand elasticities is given in Table I . We assume, in what follows, that the elasticity of demand for each industry is constant over time.

It is possible to calculate (minimum) values of the Hirschman-Herfindahl index from data given in the Australian Bureau of Statistics publications on industrial concentration. These bulletins give estimates of the share in industry turnover (amongst other things) of each of the top five groups of four firms and on the share taken up by the remainder. We are able, using this data, to calculate the Hirschman-Herfindahl indices for the years 1968-69, 1972-73 and 1977-78. In calculating the index for any industry we assume that within each group the shares of the firms are equal and so we

(1959).'2

1 1 Details of the model may be found in Dixon el ol. (1982).

12 The procedure used is described in detail in Tulpule and Powell (1978) and Dixon er ol. (1982). We are grateful to Alan Powell for providing us with data o n the calculated own-price elasticities.

13 Screening was achieved with the aid of the 1968-69 Input-Output tables. Industries included in the study are (i) those industries (20 in all) where the ratio of consumption demand to gross output exceeds 50 per cent, and (ii) industries which are not eligible under (i) above, but where consumption demand is the single most important destination for gross output (only one industry, Household Appliances n.e.c., qualified under this heading).

202 THE ECONOMIC RECORD JUNE

TABLE 1 ‘Consumer Goo& Industries

21.01 21.03 21.04 21.06 21.07 21.09 21.10 21.11 22.01 23.06

24.01 24.02 24.03 25.04

27.04

27.05 27.06 33.03 34.01

34.05

Input-Output Code and Industry(a)

Meat products Fruit and vegetable products Margarine, oils and fats Bread, cakes and biscuits Confectionery and cocoa products Soft drinks, cordials and syrups Beer and malt Alcoholic beverages n.e.c. Tobacco products Textile floor covering, felt and felt products Knitting mills Clothing Footwear Furniture, mattresses, brooms and brushes ’

Pharmaceutical and veterinary products, agricultural chemicals Soap and other detergents Cosmetic and toilet preparations Household appliances n.e.c. Leather tanning, leather and leather substitute products n.e.c. Ophthalmic articles, jewellery, silverware and other manufacturing

ASIC69(b) ASIC78(b)

21 I 213 214 216

2181 2191 2192,3 2194,5 2210

233 1.2 24 1 242 243

252 & 3443

2723,4 2725 2726

3322,3

34 I

3441,2 597

21 1 213 214 216

2173 2185 2186,7 2188.9 219

2352.3 244 245 246

254 & 3483

2163.4 2765 2766 3352.3

345

348 1.2 5.7

.60

.77

.53

.98

.89

.9 1

.82

.88

.93

.64

.66 .87 .93

.67

s o .so .69 .44

.50

.73

DE(d)

.294

.246

.244

.072

.230

.232

.316

.305

.308

.835

.I57

.165

.I56

.829

.636

.635

.635

.838

.265

.659

(a) Code number in 1968-69 Input-Output Tables. Source: Input-Output Tables 1968169. (b) Australian Standard Industrial Clasification Code. (c) Proportion of gross output sold directly to domestic consumers. Source: Input-Output Tables 1968169. (d) Estimated household demand elasticity. Source: see footnote 12.

can only establish a minimum value (or lower bound) for the Hirschman-her findah index for that industry. The value of the indices for the three years are given in Table Li4

In measuring gross margins, we must accept that data on fixed costs are unavailable. We use as our

l4 To calculate the Herfindahl index from grouped data, we calculate the mean share per firm in each group, square this figure and then multiply this by the number of firms in each group. We then sum this figure over the number of groups we have available. Simple correlation coefficients between the four-fm concentration ratio and the Herfindahl index in each year are: for 1968-69,0.943; for 1972-73.0.986; and for 1977-78, 0.941. Whilst there were some changes to the system of industrial classification in 1977-78 our data for that year do appear to be comparable with earlier periods. The source of the data used in these calculations is ABS Industry Concentration Statistics (various years).

measure of margins for each industry the ratio of gross operating surplusis to the value of turnover. These data are obtained from the censuses of manufacturing for the years 1968-69, 1972-73 and 1977-78 and are reported in Table 3. From equation (9) it is possible to estimate the

value of a for any industry with the aid of data on margins, demand elasticities and the Hirschman- Herfindahl indices of concentration. A simple rearrangement of (9) yields

+=(Mq-H)/(l -H). (10)

In Table 4 we report the results of these calculations for our 20 industries over the three years for which we have comparable data. Three

15 That is, the ratio: (Value Added minus Wages and Salaries)/Turnover.

1986 OLIGOPOLY BEHAWOUR 203

TABLE 2 Hirschman-Herfindahl Indicedo)

Input-Output Code and Industry 1968-69 1972-73 1977-78

21.01 21.03 21.04 21.06 21.07 21.09 21.10 21.11 22.01 23.06

24.01 24.02 24.03 25.04

27.04

27.05 27.06 33.03 34.01

34.05

Meat products Fruit and vegetable products Margarine, oils and fats Bread, cakes and biscuits Confectionery and cocoa products Soft drinks, cordials and syrups Beer and maltcb) Alcoholic beverages n.e.c.(b) Tobacco products Textile floor covering, felt and felt productdb) Knitting mills Clothing Footwear Furniture, mattresses, brooms and bru s hes(b) Pharmaceutical and veterinary products, agricultural Soap and other detergents Cosmetic and toilet preparations Household appliances n.e.c.(b) Leather tanning, leather and leather substitute products n.e.c. Ophthalmic articles, jewellery, silverware and other manufacturing(b)

.0221

.0475

.I412 ,0443 .0927 .0572 ,1681 ,0597 .I667

. I 166

.0196

.w

.0325

,0102

.0524

.0654 ,0453 .0371

.0252

.0667

,0243 .0418 .I167 .0509 .I195 .0882 ,1618 ,0465 .2OOo

,0802 ,0208 . m 7 .0326

.0111

.0377 ,0642 ,0502 .0526

.0173

,0402

.0188 ,0584 .I310 .0555 . I 155 .0853 .I634 .0528 .3333

,0750 .0154 ,0095 .0465

.0118

.0359

.0694

.0438

.0519

,0244

.0386

(a) Calculated as described in footnote 14. (b) The turnover ratio for the group is estimated as a weighted average of turnover ratios for the individual

industries which make up the group.

things stand out. First, the considerable variety of the estimates obtained in any one year. Second, an apparent tendency for the value of 9, within any one industry, to rise over the three years. Third, the presence of a number of negative values, especially in the industries producing food, drink and tobacco products.

We begin our discussion of the results by focusing upon industries where the estimate of @ is positive. The highest values of 9 and thus, by implication, the greatest degree of recognized mutual interdependenceI6 (which, following Waterson, 1984, p. 24, we shall refer to as ‘apparent collusion’) occur in the following industries - cosmetics and toilet preparations; pharmaceutical and veterinary products and agricultural chemicals; furniture, mattresses, brooms and brushes; and

I6 The presence of a large (in absolute terms) negative value of may also be indicative of a high degree of recognition of mutual interdependence, as we shall see in a moment.

soap and other detergents. Over time it appears to be the case that the degree of collusion is rising in most industries. The increasing trend is most marked in the meat products, textile floor covering (etc.), knitting mills, furniture, household appliances (n.e.c) and leather industries. At the same time there appears to have been a steady decline in + in the pharmaceuticals and clothing industries.

The appearance of a large number of negative estimates for 9 warrants some consideration. If we accept the model presented in Section I as valid, we have to account for the negative values for 9 either by accepting them as resulting from true observations (that is, we rule out measurement error) in which case we are obliged to provide a satisfactory explanation of the implied behaviour of firms, or by regarding them as the result of false observations (that is, the values simply result from measurement error). Alternatively, we might regard the negative values as evidence the model presented is invalid (that the model is mis-specified).

204 THE ECONOMIC RECORD JUNE

TABLE 3 Gross Marginsla)

Input-Output Code and Industry 1968-69 1972-73 1977-78

21.01 Meat products .041 .119 .I58 21.03 Fruit and vegetable products .I80 .22 1 .206 21.04 Margarine, oils and fats .I72 .172 .I35 21.06 Bread, cakes and biscuits .204 .191 .I99 21.07 Confectionery and cocoa products .I56 .2S6 .241 21.09 Soft drinks, cordials and syrups .239 .I97 .203 21.10 Beer and malt .278 .304 .215 21.11 Alcoholic beverages n.e.c. .272 .286 .265 22.01 Tobacco products .302 .372 .306 23.06 Textile floor covering, felt and felt products .I89 .I79 . I77 24.01 Knitting mills .I57 .I71 .I81 24.02 Clothing .I57 .I70 .I73 24.03 Footwear .I53 .179 .I87 25.04 Furniture, mattresses, broom and brushes .I76 .I74 .209 27.04 Pharmaceutical and veterinary products,

agricultural chemicals .355 .323 .303 27.05 Soap and other detergents .291 .289 .275 27.06 Cosmetic and toilet preparations .385 .363 .356 33.03 Household appliances n.e.c. .155 .17S .I85 34.01 Leather tanning, leather and leather

substitute products n.e.c. .155 .I65 .185 34.05 Ophthalmic articles, jewellery. silverware

and other manufacturing .223 .230 .239

(a) Defined as ((Value Added - Wages and Salaries)/Turnover). Source: ABS, Manufacturing Establishments: Derails of Operations by Industry Class (various years).

TABLE 4 Estimates of ( M 'I - HI/( 1 - H)'

Input-Output Code and Industry ~~~ ~~~~~

1968-69 1972-73 1977-78

21.01 Meat products .009 .011 .028 21.03 Fruit and vegetable products - .ooQ ,013 - .008 21.04 Margarine, oils and fats - .I16 - .085 - ,113 21.06 Bread, cakes and biscuits - .031 - .039 - .044 21.07 Confectionery and cocoa products - .063 - .069 - .068 21.09 Soft drinks, cordials and syrups - .002 - .047 - .042 21.10 Beer and malt - .096 - .078 - .I14 21.11 Alcoholic beverages n.e.c. ,025 .043 ,030 22.01 Tobacco products - .088 - . I 0 6 - .358 23.06 Textile floor covering, felt and felt products .047 .075 .079 24.01 Knitting mills .005 .006 .014 24.02 Clothing .022 .021 .019

25.04 Furniture, mattresses, brooms and brushes .I37 .I35 .I63 27.04 Pharmaceutical and veterinary products,

agricultural chemicals . I 8 3 .I74 .I63

24.03 Footwear - .099 - .00s - .018

27.05 Soap and other detergents .I28 .I28 ,113 27.06 Cosmetic and toilet preparations .209 .I90 .I91 33.03 Household appliances n.e.c. .096 .099 . I 0 9 34.01

34.05 Ophthalmic articles, jewellery, silverware

(a) Calculated by utilizing the data reported in Tables 1-3.

Leather tanning, leather and leather substitute products n.e.c. .016 .027 .025

and other manufacturing ,086 .I16 ,124

1986 . OLIGOPOLY BEHAVIOUR 205

If we accept the model as valid and we disregard the possibility of measurement error, the negative values for CP have a straightforward interpretation: they indicate ‘accommodating behaviour on the part of other firms, an example being where they [the other firms] adjust their outputs to keep price constant’ (Dixit and Stem, 1982, p. 126). In this sense large negative values for 9 are (like large positive values) indicative of apparent collusion, that is, of a high degree of recognized inter- dependence. It is tempting to rely upon this inter- pretation of the negative values of CP and thus to reject the notion that they reflect model mis- specification or measurement error. However, we suspect that one or both of these latter possibilities may be the case. The industries which exhibit negative values of CP differ from the others in that they have a combination of relatively low margins and relatively low demand elasticities. Indeed, all of the demand elasticities in our sample lie within the inelastic range. These facts lead us to question the appropriateness of the modelI7 since they

17 We cannot, of course, rule out measurement error. For example the indices of structure do not take into account differentiation (including spatial differentiation), the role of importation and exportation etc. Further our cost data are inadequate as a measure of true marginal costs. We would be surprised, however, if these deficiencies were such as to yield the systematic differences in the sign of the estimates as between the Food, Drink and Tobacco industries and the others. In addition, we could question the estimates of the demand elasticities. However, they are estimated as a set and. whilst subject to scaling, the likely adjustment to one (say, Tobacco products, which at the outside may have an elasticity perhaps 1.25-1.50 times that used here) when applied to all is still going to leave the majority of the estimates well within the inelastic range. Again, it is unlikely that this can explain the differential results between the various groups of industries.

It may be noted that where the demand for an industry’s output is inelastic this does not necessarily imply that each firm in the industry is not maximizing its profits, given its perception of the elasticity of demand for its own output. Each firm may believe that a small price cut will increase its total revenue even if such a price cut would reduce the revenue accruing to the industry. Each firm’s perception of the elasticity of demand for its output will depend upon the elasticity of demand for the industry’s output and the firm’s conjecture concerning its rivals’ reactions to a change in its output.

Let q,, where qr = (dX,/X,)/(dp/p), be the r’th firm’s perception of the elasticity of demand for its own output. It follows, from the definition of 0, that

q/q l = ((d~/x)/(d~/p)/(dx~,)/(dp/~)) = e.

suggest that departures from profit maximization must be allowed for.l* In the next section we modify the model t o allow for errors in optimization.

III A Model with Sub-optimal Behaviour In the previous section we noted that there may

be a case for re-casting the model presented in Section I in such a way as to allow for systematic departures from profit maximization. One reason for this is that the estimated demand elasticities are well into the inelastic range in most of the industries in the study. A second reason, and one which has force independent of the first, arises from the existence of phenomena such as barriers to entry, tariffs, and levels of production which are below minimum efficient scale in many industries. Given this, it seems desirable to allow for sub-optimal outcomes. In modifying the model to allow for departures from profit maximization our main aim will be to show the effects which the relaxation of that assumption will have upon our interpretation of the estimates presented in Table 4. We proceed along analogous lines to those traversed in Section 1.

We will allow for errors in optimization or, more precisely, persistent sub-opt imizat ion, by postulating an inequality between marginal revenue ( m ) and marginal cost (c) at the output level set by the r’th firm (X,) such that19

m, + el = c, (1 1) where el = error in optimization.

Hence, if 0<8< I then ~ , > q . Where the demand for the industry’s output is inelastic the demand for a firm’s output may, or may not, be inelastic depending on the value of 0.

A similar argument may be advanced in relation to Chamberlin’s theory of monopolistic competition. Here the firm’s dd curve will be more elastic than the firm’s DD curve and the ‘demand curve for the general class of product’ (Chamberlin, 1950, p. 90 and the diagram on p. 91). A further, and most forceful, illustration of the argument is provided by the theory of perfect competition where the perfectly competitive industry is in equilibrium. Here there is no presumption that the industry’s elasticity of demand cannot be less than one even though the elasticity of demand for each firm’s output is perceived to be infinite.

I * It could be that other assumptions are systematically violated, e.g. our (implicit) assumption that we are dealing with undifferentiated oligopoly.

19 Note that we are, strictly speaking, bereft of a theory of output determination.

206 THE ECONOMIC RECORD JUNE

As before, the firm’s marginal revenue function may be expressed in terms of price and its perceived elasticity of demand

mi = p -p ( ( I/q)(dX/dX;)(X,/X)) (12)

so that (1 1) may be rewritten as

p-p( ( l /q ) (dX/dX,) (X, /X)) +e ,= c,. (13)

The firm’s gross profit, that is output multiplied by the difference between price and average variable cost (where average variable cost is assumed to equal marginal cost), will therefore equal

@ - 4 X, = Xg((I/rl)(dX/dX,)(X,/X)) - X,e,. (14)

As before, we define 0 to be the (conjectural) elasticity of industry output with respect to the output of the r’th fun (i.e. 0, = (dX/X)/(dXi/X,)). We will assume; for simplicity, that 8 is the same for all firms. Equation (14) may be rewritten as

Again, we define 9; to be a measure of inter- dependence, such that

Recall that 0 and 9 are related, such that

Substitution of (16) into (15) yields an expression for the (gross) profits of the r’th firm as

Summing the above over all the firms in the industry, and assuming that 9 is the same for all firms, yields an expression for aggregate gross profits. Dividing that sum by the value of industry sales @X) yields an expression for the industry (gross) margin (M) as:

M = (HIT)( 1 - 9) + ( @ / q ) - C(X,/X)(e , /p) . (17)

If e may be assumed constant across firms, the expression for margins becomesZo

M = H/7(1 - 9) + + / q - e / p . (18)

Thus our model expresses a relationship between gross margins, the Hirschman-Herfindahl index, the industry elasticity of demand, the parameter 9, which is a measure of (or proxy for) inter- dependence, and the extent of departures from profit maximization.

We believe that this model is a simple, yet useful addition to the collection of models of oligopoly. In addition, it may be used to gain insight into the estimates of 9 which are obtained by applying data to the profit-maximizing model, such as equation (10) above.

Notice, from (18), that

(Mq-H)/ ( l - H ) = 9 - ( 7 e / p ) / ( l -H). (19)

This expression may be compared with equation (10) above. What it says is that if the true model is one in which there are systematic departures from profit maximization the s tandard (profit- maximizing) model will yield inaccurate estimates of the degree of collusion. We can go a little further and assert that in the presence of systematic and positive errors of optimization (i.e. marginal revenue less than marginal cost, as might occur if the firms operate in the inelastic region of their demand curves) the usual estimates of the degree of collusion, such as those found in Table 4 of this paper, will understate the true degree of collusion. Notice also that if the true model is that reflected in equation (19). yet we were to interpret the whole of the L.H.S. as reflecting 9 (e.g. by the [false] application of equation (10)). inefficiency (positive errors of optimization) would be masquerading (or interpreted) as a failure to effectively collude.

If equation (19) is a good approximation to the true state of affairs, the indices presented in Table 4 are linear combinations of apparent collusion and optimization errors. Whilst it is tempting to make some comment about the relative weighting of these two factors for some of the industries in our sample (and especially those which exhibit negative values) we think this is unwise, given the absence of an

20 In the event that e is random, with a mean of zero, and provided that ( X , / p X ) is non-stochastic, the expected value of M will equal:

E(M) = ,!?((HI?)( 1 - a) + + / q )

1986 OLIGOPOLY BEHAVIOUR 207

explicit model of inefficiency (or collusion) at present.

I V Concluding Remarks In this paper we have attempted to shed some

light on the relationship between margins on the one hand and the elasticity of demand, concentra- tion and apparent collusion on the other. This quest led us to reformulate the model developed by Cowling and Waterson in such a way as to allow for sub-optimization on the part of firms. We noted amongst other things that the standard model may well yield underestimates of the degree of collusion in industry.

It was beyond the scope of this paper to provide a detailed econometric model of the elements contained in our expression relating margins to concentration (equation (18) above). Such a model would be at this stage largely ad hoc and thus opposed to the (desirable) thrust of the Cowling- Waterson paper, which is aimed at driving ‘ad hocery’ out of these sorts of studies. Any model2’ along these lines would, of necessity, have to take into account matters relating to protection, scale economies and the openness of the Australian economy.

REFERENCES Appelbaum. E. (1982), ‘The Estimation of the Degree of

Oligopoly Power’, Journal of Econometrics, 19, 287-99.

2’ One obvious model of departure from profit maximization, given the matter raised in the previous paragraph, is to relate it to the magnitude of the elasticity of demand (inversely) and to the ratio of variable costs to sala revenue (i.e. the ‘height’ of the marginal cost curve above the horizontal axis.)

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