Market Structure and Price Cost Margins

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Market Structure and Price-Cost Margins Author(s): Roger Clarke and Stephen W. Davies Source: Economica, New Series, Vol. 49, No. 195 (Aug., 1982), pp. 277-287Published by: on behalf of and Wiley The London School of Economics and Political Science The

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Economica, 49, 277-287

Market Structure and Price-Cost Margins By ROGER CLARKE and STEPHEN W. DAVIES

University of Sheffield and University of East Anglia

In a paper that has become a standard reference in the structure-conduct- performance approach to industrial economics, Cowling and Waterson (1976) developed a theoretical rationale for expecting industry profitability to be positively correlated with the level of concentration. Their model, based on a simple algebraic description of oligopoly behaviour, has been interpreted as providing a justification for the common empirical practice of cross-industry regression analysis of price-cost margins in terms of concentration levels.

In the present paper we show that, on further examination, this model implies a joint determination of margins and concentration. This provides some interesting comparative static results of relevance to inter-industry differences in concentration levels. Equally, however, it leads to various (cautionary) insights concerning the meaning of price-cost margin versus concentration regressions.

I. THE COWLING-WATERSON MODEL

Following Cowling and Waterson, we envisage an industry of N firms1 producing a homogeneous good; marginal costs are constant for each firm over the relevant output range but vary across firms, with mean /4, and variance 2 -cc. The ith firm sets output, Xi, to maximize profits, Iri = pXi - ciXi where p

is price and ci is marginal cost. Noting that p=p(Z=1Xe) the equilibrium condition for i is:

(1) p l- 1Xi (1+Ai) =ci

where p dX (X=NX)

~Xdp_I and

Ai=d Z Xjl/dX j$1 i.e. the ith firm's conjecture as to its rivals' responses to a change in its ouput.

As shown by Cowling and Waterson (p. 268), multiplication of (1) by Xi, summing over the N firms and re-arranging, we have:

(2) 7= H(1+ ) R where

R pX H 2 ad E X X.


278 ECONOMICA [AUGUST

Thus, the industry average price-cost margin (or profit-revenue ratio, if there are no fixed costs) is positively related to the Herfindahl index of concentration, inversely related to the elasticity of demand, and positively related to a weighted conjectural variation term. In the special case of Cournot conjectures, Ai 0 O for all i and Ir/R = H/n-.

II. A MODEL OF CONCENTRATION

The Cournot equilibrium Let us now proceed in a different direction from Cowling and Waterson,

initially on the presumption that firm i expects no responses to its output change; i.e. Ai = 0 for all i.

We first re-express (1) as xi C (3)

and summed over the N firms, we have:

(4) 1 =1qN -q r E p

Squaring (3), substituting for p from (4) and summing yields C2

(5) H = ZX/X2 = -_q2N + 2 (E ci2

Note that E c2/(Z ci)2, which might be termed the "Herfindahl index of costs", can be re-written as (1 + -2c/,2)/N or (1 + v2)/N where vc is the coefficient of variation of marginal costs.

Substituting this into (5), we derive: (6) H = (1/IN) + (1 - rqN) c/ Thus the Herfindahl index exceeds its lower limit (1/N) by an amount that depends on 17 and v C.

Referring back to Cowling and Waterson's expression for 7-/R, (2), this might also be written as:

(7) R= 1 (1-r1N)2vc R N'q Nrj In other words, we have a "theory" of concentration (and of the price-cost margin) in terms of demand and cost conditions, as represented by -q, N and v2. Of particular interest, equations (6) and (7) (and (13) and (10) below) show that both H and 7-/R will be larger the greater are cost (efficiency) differentials between firms, as measured by v2. They therefore support an argument advanced by Demsetz (1973) that part of any positive relationship between industry profitability and market concentration may be explained by the superior efficiency of the largest firms.

Partially collusive equilibria Moving away from the special Cournot case, we need to develop a slightly

more refined picture of jt than that provided in the original Cowling-Waterson


1982] MARKET STRUCTURE AND PRICE-COST MARGINS 279

paper. To that end we shall suggest a specification of Ai which allows us to "parametize" a range of behaviour including Cournot as a special case. Our approach is most easily explained by first considering the polar extreme to Cournot, i.e. perfect collusion. This we interpret to mean that each firm believes that all others will react to output changes so as to maintain market shares. In other words, firm i conjectures:

dX, dXi -=- for all j.

This may also be written as:

d Y. XI Y. XI A si isi X A ==dX- XI X

In this case, then,

E(XiX-X2) -1 /= x~2 =H -

and substituting this into (2) gives the usual monopoly result: ir/R =1/r. For all intermediate cases, however, let us suppose:

dX, dXi j cdX, for all j # i and for all i, and 0 < a


in this sense a parametizes a whole spectrum of non-cooperative behaviour. Interestingly, this result generalizes the derivation of an identical result by Cowling (1980) which applies to the equal sized firm case.3

To derive an expression for H, we can re-write (9) as:

(11) X +1 1- (1-i

Summing over N and re-arranging gives the following expression for price: (12) p = (Z ci)q I{N(1 - ) - (1 - )}-1. Substituting this into (11), which is then squared and summed, yields:

(13) H = + 1- NC

As can be seen, if this is compared with (6), the introduction of "partial collusion" has complicated things only slightly: brief inspection will show that, for a > 0, concentration is now higher than in the Cournot case. Note also that substitution of (13) into (10) would enable us to express II/R in terms of the "structural" parameters of the model: v2, N, q and a (such an expression is a counterpart to (7) in the Cournot model).

Before we proceed to a more detailed examination of (13), it is perhaps worth digressing very briefly to question our assumption that a does not vary between firms in a given industry. For example, smaller firms may feel more able to get away with output changes undetected than would large firms: if so, ai would be positively related to Xi/X. Alternatively, given a small collusive group of large firms and a competitive fringe, ai would vary across i and j. Given the aims of the present paper, we shall not pursue such possibilities in much detail here; in any case, much of the argument of the following section can be established by venturing no further than the simple Cournot model. Nevertheless, with one eye on future developments, the following findings are of some interest. Allowing a to vary across i (but not j), we find that ll/R remains unchanged in (10) if ai is independent of market share. On the other hand, a tendency for larger firms to adopt larger values for ai means that (10) will tend to understate Il/R while the opposite is the case if ai and Xi/X are inversely related.4 Turning to the effects on H, things are somewhat more complicated. From (11) it is clear that, where a is constant across firms, the only source of firm size inequalities (leading to increased concentration) is variations between firms in ci. Once we allow a to vary between firms, this is another potential source of inter-firm size differences (basically, large a for firm i presents it with a low perceived firm demand elasticity which mitigates against a large optimal output). But obviously, the variance in Xi/X depends not only on the variance of ai but also its covariance with ci. So without further specific assumptions, no hard conclusions are possible.

III. IMPLICATIONS

Since (6) is merely a special case of (13) with a = 0, we consider only the latter in developing the implications. Simple differentiation shows that, according to this model concentration is higher



(a) the greater is v', the inequalities between firms in costs (and, by implica- tion, the greater is the cost advantage of the leading firms);

(b) the larger is 71, the elasticity of demand; (c) the larger is a, the degree of collusion; (d) on the other hand, the effect of N is indeterminate. if, for example, vc

is held constant,

dH_ V2C(, _ )2 (1+ V2C) dN (1-a )2 N 2

Thus dH/dN has the sign of

N r-a) _+c)

(1-Oa)2 2

Determinants of concentration If we relate these findings to the existing literature on the determinants

of concentration, a number of interesting parallels and insights emerge. First, the positive relation between H and v c is intuitively obvious since, in this simple model, variations between firms in costs are the only reason for unequal market shares (and, as can be seen from (11), the extent to which cost inequalitites will be reflected in market share inequalities depends on the magnitudes of a and r). An explanation of cost inequalities might draw fruitfully on the stochastic theories of firm growth, which have traditionally worked in terms of various forms of the Law of Proportionate Effect (see, for example, Engwall's (1973) discussion of these models). It does not seem unreasonable, for instance, to envisage the host of factors that determine any firm's costs (e.g. process innovation, increases or reductions in X-efficiency and control loss) as giving rise to some sort of random walk, with consequent costs inequaltities emerging.5 This might signal an interesting departure from the conventional explanations of market share inequalities, which have tended to emphasize demand uncertainties (e.g. Weiss, 1963).

Second, the positive relation between H and a provides a different viewpoint on the conventional argument that the Herfindahl index serves as a potential indicator of the strength of collusion. Stigler (1968) employs a co-operative model to suggest that higher values of H increase the likelihood of effective collusion. In the present model, however, the causal direction is reversed, with a determining the magnitude of H. (In fact, the two viewpoints could be reconciled by making a endogenous in this model, with or without time lags.)

Third, the indeterminate effect of N on H arises because it matters what size an additional firm is before one can say that concentration rises or falls. In the case noted above, where vc is held constant, it is clear that the expected inverse relation between N and H materializes only given low values of vC. Consider, for example, the following arbitrarily chosen values: N = 30, r9 = 2, X = 2; in these circumstances, for dH/dN < 0, we require:

1+V2 2 >8,100 or v2 < 00001

vc



i.e. virtually no inequalities in costs. More generally, where v2 can also change, positive values of dv /dN (as in the case where the new firm is highly efficient) also contribute to raising H. This makes us sceptical of the view, sometimes expressed, that firm numbers are a reasonable inverse proxy for concentration. On the other hand, we must strongly qualify our results by underscoring our implict assumption that N is an exogenous parameter in the model. We return to this assumption in the appendix.

Empirical studies of the relation between price-cost margins and concentration Let us now return to the original raison d'etre for Cowling and Waterson's

paper: the construction of a sound theoretical underpinning for cross-industry regression analysis of price-cost margins in terms of concentration. Here we are able to draw various cautionary implications from our Section II which are of direct relevance to their empirical work, but also, for that matter, to any study of price-cost margins.

First, let us stylize past regression analysis in this area by the following hypothetical simple regression:

(14) Y, = a + bZ + u j = 1 ..r where Y1 is a measure of the price-cost margin, Zi is some index of concentration in industry j, and ui is a disturbance term. Such an equation has formed the basis of scores of regression studies over the years.

Now, Cowling and Waterson's work helped to establish three points of relevance to the specification of such a regression. First, Z might be sensibly measured by the Herfindahl index of concentration. Second, a logarithmic functional form might be most appropriate. In terms of their basic equation (our (2) above), in the Cournot case (A = 0), a relation linear in logs implies a = 0 and b = 1 in equation (14). (More generally, it is possible that b '-1 but >0 if 1 +Ai is related to concentration Hi.) Third, the industry elasticity of demand, i-j, should appear in the equation.

In the light of our finding (b) above, we wish to re-emphasize their third point. Not only does the omission of iq leave an under-specified estimating equation, as pointed out by Cowling and Waterson; but, since H is positively related to i,, the omission of the latter provides a classic textbook case of an omitted variable correlated with an included explanatory variable. In the logarithmic version of (2), the omitted variable is

-logqi-, and as this is inversely related to log Hi, we should expect ordinary least squares (OLS) to generate estimates of b in (14) that are biased downwards (in simple regression, at least).

Given that r1 is generally unobservable, this result is, of course, extremely inconvenient. In fact, Cowling and Waterson themselves attempt to avoid this problem by estimating (14) in what amounts to a logarithmic first difference form. Thus, if demand elasticities are constant over time, they drop out of the estimating equation. Frankly, our intuition offers neither support nor grounds for rejecting this "solution". Nevertheless, it is somewhat asymmetric to assume rj constant over time while H is not. Indeed, according to our equation (13), changes in iq might well be the source of changes in H. In the absence of such changes, we must assume by irnplication that all changes in H are cost-related (i.e. are due to changes in v2 and/or N).



Our second worry is perhaps even more fundamental. By establishing that the Cowling-Waterson model also generates an explanation of concentration, we have illustrated what is ex post a fairly obvious point: both Ir/R and H are determined by the cost and demand conditions in the industry concerned. Indeed, this is already clear from (6) and (7) in the Cournot case, and in the more general model of Section II substitution of (13) into (10) shows again that both I/R and H depend on -q, v 2, N and a. In other words, 7r/R and H are jointly determined in the equilibrium solution to the model, so that it would be misleading to view (2) as supporting a causal relationship from

6 concentration to the price-cost margin or, indeed, vice versa.

The essential endogeneity of i/R and H suggests considerable doubts about OLS estimation of equations such as (14). As one example of how the argument might proceed, let us suppose that the stochastic disturbance term in (14) represents, at least in part, random deviations from strict optimizing behaviour at the firm level. We shall characterize this by the addition of a random term E in equation (1) and for brevity, we consider the Cournot special case (Ai = 0).

Thus,

(la) p(1-t)-ci+Ei.

As before, we proceed by multiplying by Xi/X, summing and re-arranging to give the counterpart to (2):

(2a) R=- H Xi. R 77 X p Assuming si to be independent of ci with zero expectation (not crucial to our argument), then substituting for X1/X from (la) allows us to re-write (2a) as:

(2b) =- H ZE R 77 p So the disturbance term in (14) is -(E e2/p), which will vary between industries depending on the extent of intra-industry variance in deviations from optimizing behaviour.

It is easily shown that this is also a determinant of H. Re-expressing (la) as:

Xi / c1 CiE\ X p p

Squaring and summing this expression yields: X ~~~~~~ ~~2 2

H = E 2= N2 (1 _i) _n E?2i x 21~ ~ ~~

2 and then substituting for (1- culp) and re-introducing the v c notation, we can derive the counterpart of our earlier equation (6):

(6a) H =N (1 - N)N_ NN



This differs from (6) only in the addition of the last term, which is, of course, almost identical to the disturbance term in (2b). As such, estimation of an equation relating price-cost margins and concentration by OLS will involve both bias and inconsistency.

Finally, and any estimation problems apart, we must question exactly what the dependent variable IT/R is meant to represent in this model. Fairly obviously, the welfare implications are unclear. This can be seen if we investi- gate the implications of changing a, which is, perhaps, the most suitable candidate as a measure of the exercise of monopoly power in a model such as this. From (12) we have an expression for industry price, and from (10) and (13) we may also derive the following expression for the industry's (weighted average) marginal costs:

N ciX F2(Jrj -a\ (15) MIC1+ = L X c[l?Vcj1lV1a)jJ l=1 x1a where Ac is arithmetic mean marginal costs.

Upon differentiation of (12) and (15) we find that, while price rises with a (as might be expected), MC, falls. This latter result occurs because, as a rises, the market shares of the leading firms (which, in this model, are also the more efficient firms) increase, thus reducing the weighted average marginal cost in the industry. This opens up the possibility of welfare trade-offs of the sort envisaged by Williamson (1968).7 But for present purposes we confine ourselves to the simple statement that it destroys any simple relationship that might otherwise exist (where average industry costs are invariant with respect to the degree of market power) between the size of profits and the deadweight loss.

More positively, our characterization of the degree of market power by the a parameter opens up some potential for developing the work of Demsetz. As is well known, he questions how much positive correlations between profitability and concentration are due to efficiency, and how much they are due to "abuse" of market power (Scherer, 1980; Demsetz, 1973). In principle, our model permits a decomposition of profitability into two component parts. With given efficiency differences (v2), equation (7) shows the value of Il/R that would obtain with a = 0; on one interpretation this might be termed the "efficiency" component of profitability. However, assuming, in addition, some degree of collusion (a > 0), actual profits are given by (10), which can also be expressed in terms of v2, N, r1 and a. The difference between this and (7) could be termed the "abuse" component of profitability. In practice, to effect such a decomposition we would need to estimate a. Such an estimate may be possible by estimating equation (9) using firm level data within each industry on market shares and mark-ups. In principle, then, given industry and firm level profit figures and market shares, numerical estimates of the two com- ponents should be possible.8

IV SUMMARY

By re-examining a well-known theoretical model, which predicts a positive relation between profits and concentration, we have established the co- determination of both variables by underlying cost and demand conditions.



Although it has been expositionally convenient to couch this point within the framework of the Cowling and Waterson model, it is easy to see that much of the discussion of this paper is relevant to any non-cooperative model of this type; for example, Saving (1970), Hause (1977). This intuitively obvious point re-affirms, that, if the structure-conduct-performance approach is to offer insights of any validity, it must acknowledge the essential simultaneity of the three concepts.

ACKNOWLEDGMENTS We would like to thank Mike Waterson, Keith Cowling, John Cubbin, Dick Allard

and an anonymous referee for helpful comments. Any errors or omissions are our responsibility.

APPENDIX: ENDOGENEITY OF N AND NEW ENTRY Along with Cowling-Waterson we have assumed throughout that N (and, for that

matter, v2) are exogenous to the system. In the following respect, however, this is an ambiguous idea.

From equation (11), if each firm is to achieve its equilibrium at a non-negative market share, we require that the least efficient firm in the industry has costs no greater than c* where:

(Al) 0=- + (1$--) 1-ar 1-a p For simplicity, we assume that fixed costs are sufficiently low that non-negative profits are attainable given marginal costs of c*. Using (12) to substitute for p, and writing E ci = N,Ac, c* may thus be defined as:

(A2) *=1 (1- ) c ( - a)N

To see the implications of this, let us suppose that, in a given industry, costs are distributed according to the following curve:

(A3) N(c) = acb a, b>0 where N(c) denotes the number of firms with costs no more than c. Suppose, in addition, that the distribution is truncated at some upper level c**; thus, N(c**) = N. It may easily be shown that the mean and coefficient of variation are:

(A4) bc=C**b

(A5) v2= 1

This distribution is a distant relative of Pareto's curve (roughly speaking, its comple- ment). Expressions (A4) and (A5) have direct counterparts in the Pareto curve, (A4) corresponds to Van der Wijk's law, and v2 is insensitive to the point of truncation (as in (A5)) for Pareto's curve. (For simplicity we have imposed only a zero lower limit on c, as in Cournot's proverbial mineral spring). Thus, if the equilibrium is to exist, c** must be less than c* in (A2); i.e.

b (1-a) b?1> I j- b + 1 (r -c a)N



or

(A6) N a(b+1). 71 -a

In other words, if the equilibrium is to exist, the number of firms must be no more than some upper limit, where that upper limit is inversely related to a and 'q and positively related to b (and thus, from (A5), inversely related to v2).

Thus, while we might still claim that N is exogenous, we require a constraint on its magnitude, which is dictated by the magnitudes of the other parameters in the system.

Alternatively, we might interpret (A3) as a description of the costs of all potential producers of the good concerned, where only those with costs of c* or less actually operate. In that case, c* is the point of truncation and now

1 -a (A7) N- (b + 1). 7 -a In this case, n is unequivocally endogenous, depending on the demand elasticity, the degree of collusion and the coefficient of variation of costs (via b). On this interpretation, our comparative static results of Section III must be revised. In effect, this interpretation amounts to the assumption that sufficient entry (or exit) will occur in order to enable the equilibrium solution to be achieved. In the Cournot case, for example, this implies that the potential entrants assume no change in imcumbents' outputs following entry. This also allows an easy interpretation of the b parameter as the elasticity of N to c *: large values of b implying large entry rates given a relaxation in the upper limit on costs.

NOTES 1 Throughout, we assume that N is an exogenous parameter of the model. Thus, along with

Cowling-Waterson, we abstract from entry and exit, and it is assumed therefore that all N firms are able to earn non-negative profits in the equilibria considered. We shall argue however, that the number of firms consistent with the equilibrium solution will vary depending on cost and demand conditions. In this sense N is hardly exogenous.

2 Although note that an equilibrium can exist only with a = 1 in the cases of no cost inequalities or a single firm (see (A2) in the Appendix, setting a = 1).

3 Cowling's derivation of equation (10) is based on the assumption that the Ai are invariant with respect to market shares, and it requires, as he himself admits (1980, p. 5), the symmetric case of equal sized firms. In all other cases, Cowling's a is definitionally related to the distribution of market shares and is thus not a parameter. In fact, Cowling employs Cubbin's (1975) earlier measure of apparent collusion, which generates a as a parameter, independent of market shares, only in the special case just mentioned (1975, p. 3, equation (13)). In passing, we should also note that our a measure is quite independent of Cyert and de Groot's (1973) "coefficient of co-operation", which applies to cd-operative solutions in which the profits of firm j enter into i's utility function.

4 To see this, re-write (9) with an i subscript on a as: P-i1

=

-

(si - aisi + ac) Pi 1

where si X/X. Multiplying by si and summing yields

Hi21 1 1 2 =-{H + (aisi - aisi2)} =-{H(1 - a ) + a} +- E (ai - a)(si - s ) R

-a'a'

where c = E(a). Whether or not this exceeds IIR in (10) depends on the sign of the second term. Denoting the standard deviations by a-, and o-,a and the correlation between a and s by r, this term may be written as ro-s,o, - _ s (ai - c}). This forms the basis for the conclusions reported in the text, which are based on the assumption that _ s i(ai - c) will typically approximate to zero.

5 The mean and variance of costs (and thus v2) would then clearly depend on whether low costs for large firms were due to systematic factors (e.g. access to superior factors, faster adoption of new techniques) or whether random effects predominated (i.e. whether or not there is a discernible regression to (or from) the mean).



6 The essential simultaneity of this type of non-cooperative model has been noted briefly in various places in the literature; see for example, Encaoua and Jacquemin (1980) and Gollop and Roberts (1979).

7 In fact, Williamson compares the polar cases of perfect competition and monopoly (with the latter experiencing lower marginal costs). Here, we have a spectrum of market structures (defined by the value of a) with prices rising and costs falling as "monopoly" is approached. In this sense the comparisons involved are more general than Williamson's. Also, the present model provides an easy rationale for "costs" falling as market power increases. Williamson, on the other hand, posits scale economies open to the monopolist but not to the competitive industry: an assumption that co-exists rather uncomfortably with that of constant marginal costs.

8 We should, however, qualify this definition of the efficiency component in two respects. First, one might question whether Cournot behaviour is free of market "abuse". Second, as is clear from (15), some part of the "abuse" component is atributable to lower industry costs.

REFERENCES

COWLING, K. (1980). Oligopoly, distribution and the rate of profit. Warwick University (mimeo). and WATERSON, M. (1976). Price cost margins and market structure. Economica, 43,

267-274. CUBBIN, J. (1975). A measure of apparent collusion in oligopoly. Warwick Economic Research

Paper, no. 49. CYERT, R. and DE GROOT, M. (1973). An analysis of co-operation and learning in a duopoly

context. American Economic Review, 63, 24-37. DEMSETZ, J. (1973). Industry structure, market rivalry and public policy. Journal of Law and

Economics, 16, 1-9. ENCAOUA, D. and JACQUEMIN, A. (1980). Degree of monopoly, indices of concentration and

threat of entry. International Economic Review, 21, 87-106. ENGWALL, L. (1973). Models of Industrial Structure. New York: Lexington Books. GOLLOP, F. and ROBERTS, M. (1979). Firm interdependence in oligopolistic markets. Journal

of Econometrics, 10, 313-331. HAUSE, J. (1977). The measurement of concentrated industrial structure and the size distribution

of firms. Annals of Economic and Social Measurement, 6, 73-107. JOHNSON, A. and HELMBERGER, P. (1967). Price elasticity of demand as an element of market

structure. American Economic Review, 57, 1218-1221. SAVING, T. (1970). Concentration ratios and the degree of monopoly. International Economic

Review, 11, 139-145. SCHERER, F. M. (1980). Industrial Market Structure and Economic Performance (2nd ed.),

Chicago: Rand McNally. STIGLER, G. (1968). A theory of oligopoly. Chapter 5 in The Organisation of Industry, Home-

wood, Ill.: Richard Irwin. WEISS, L. (1963). Factors in changing concentration. Review of Economics and Statistics, 45,

70-77. WILLIAMSON, 0. E. (1968). Economies as an antitrust defense: the welfare trade-offs. American

Economic Review, 58, 18-36.


Article Contentsp. [277]p. 278p. 279p. 280p. 281p. 282p. 283p. 284p. 285p. 286p. 287

Issue Table of ContentsEconomica, Vol. 49, No. 195 (Aug., 1982), pp. 219-374Front MatterIs Incomes Policy the Answer to Unemployment? [pp. 219-239]Rational Expectations, Random Parameters and the Non-Neutrality of Money [pp. 241-248]The Simple Economics of Bank Regulation [pp. 249-259]The Monetary Determinants of Real Trade [pp. 261-266]Monetary Expansion and Aggregate Supply in a Small, Open Economy [pp. 267-275]Market Structure and Price-Cost Margins [pp. 277-287]On the Quality of Publicly Supplied Goods [pp. 289-296]Pollution Taxes and Optimal Spatial Location [pp. 297-311]The Relationship between the Origin and Diffusion of Industrial Innovation [pp. 313-323]The Length of a Job in Great Britain [pp. 325-333]Labour Utilization and Statutory Non-Wage Costs [pp. 335-343]On the Firm's Short-Run Quantity Adjustment: "q" Theory of Goods in Process [pp. 345-354]Compensating and Equivalent Variations, and the Deadweight Loss of Taxation [pp. 355-359]Book ReviewsReview: untitled [pp. 361-362]Review: untitled [pp. 362-363]Review: untitled [pp. 363-365]Review: untitled [pp. 365-366]Review: untitled [pp. 366-367]Review: untitled [pp. 367-368]Review: untitled [pp. 368-370]Review: untitled [p. 370]

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