Deadweight Loss in Oligopoly: A New Approach
Author(s): Alan J. Daskin
Source: Southern Economic Journal, Vol. 58, No. 1, (Jul., 1991), pp. 171-185
Published by: Southern Economic Association
Stable URL: http://www.jstor.org/stable/1060041
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DeadweightLoss in Oligopoly:A New Approach*
ALAN J. DASKIN
Boston University
Boston, Massachusetts
I. Introduction
Ever since Harberger's[18] pioneeringarticle, researchershave attemptedto estimatethe welfare
loss resultingfrom the exercise of marketpower. Despite nearlyfourdecadesof work in this area,
however, surprisinglyfew researchersdisagree with Harberger'sfindingthat deadweightloss is
quite small. While many disputehis claim thatdeadweightloss in manufacturingamountsto only
about 0.1% of GNP, few estimates exceed more than 1-3% of the value of output.' Some of the
higherestimates, moreover,include losses of a broadernaturethanthe purelyallocativelosses on
which Harbergerfocuses.
In this paper I use a generalizationof a recentmodel of oligopoly to estimatethe magnitude
of deadweight loss in the U.S. manufacturingsector. While the exact magnitudeof the losses
depends on several parameters,the estimates indicatethat deadweightloss may be considerably
higher than earlier work suggests, ranging from roughly 6-10% of the value of shipments if
demandis inelastic to over 20% if demandis elastic.
The theoretical model used to derive the empiricalestimatesbelow is an explicit model of
oligopoly, ratherthan an extension of the basic model of monopoly on which Harbergerbases
his work. While much of the existing work in this areaextends Harberger'sbasic framework,the
predominanceof oligopolistic marketstructuresin many industriesjustifiesan approachbased on
a model of oligopoly. The particularmodel I use is flexibleenoughto allow for the exercise of different degrees of marketpower in differentindustries.Moreover,unlike most previousempirical
work in this area, the model allows for differencesin both costs and conduct among firms within
any particularindustry.
Section II provides a brief (and deliberatelyselective) review of the literaturedesigned to
motivate the particularapproachI have taken. (For a more complete review, see Scherer and
Ross [25].) In section III, I develop the theoreticalmodel used to estimatedeadweightloss in an
industry.The theoreticalmodel extends and generalizesthe oligopoly models of Dixit and Stern
[14] and Clarkeand Davies [4], which trace their lineage to work of Cowling and Waterson[8].
Section IV adapts the theoreticalmodel for use with availabledata on U.S. manufacturing,and
section V presents and discusses empiricalestimatesof the model. A brief conclusionfollows.
*I would like to thankMyong-HunChang, StephenRhoades, MarthaSchary,JohnWolken, an anonymousreferee,
the editor, and seminar participantsat the Departmentof Justice, the SouthernEconomic Association meetings, and
Boston University for helpful discussions on earlier versions of this paper. I, of course, bear full responsibilityfor any
errors.
1. Kamerschen[20] and Cowling and Mueller [5] are two notableexceptions.
171
172
Alan J. Daskin
II. A Brief (and Selective) Review of the Literature
Much of the work in this area stems from Harberger'swell-knowndeadweightloss triangle. In
its simplest textbook form, the intuitionbehind Harberger'sanalysis is straightforward:Firms'
decisions to set price above marginalcost reduce consumersurplusand increase firms' profits
relativeto their levels in a competitiveenvironment.The differencebetween the increase in producer surplusand the reductionin consumersurplusrepresentspure deadweightloss ratherthan
redistributionfrom consumersto producers.
If demand is linear and long-runmarginalcosts are constant-and equal for the monopolist and the hypothetical competitive industry-the deadweightloss for a monopolist (DWL) is
given by
DWL = (1/2)(Ap)(Aq),
where Ap = Pm - Pc and Aq = qm -q denote the deviations from competitive pricing and
output that result from the monopolist's exercise of marketpower. (Subscriptsnt and c denote
monopoly and competition, respectively.)
Harbergerand others then rewriteDWL in terms of observablevariables.Letting r/ denote
the absolute value of demandelasticity, we can rewriteDWLas
DWL = (1/2)>r2rl/R,
where -r is the monopolist's excess profits and R is the firm's revenue. Using this framework
and several empirical assumptions, Harberger[18] estimates that deadweight loss in the U.S.
manufacturingsector was on the orderof 0.1 percentof GNP in the late 1920s.
Many later researchers,while using the same basic model, criticize Harberger'sempirical
assumptions.Much of their criticism focuses on his assumptionthatr7 = 1 for all industries,his
failureto account for the interdependenceof Ap and Aq (throughthe demandfunction), and his
estimationof excess profits. (For summariesof some of this work, see Waterson[33], Clarke[3],
and Schererand Ross [25].)
Cowling and Mueller [5; 6] and Cowling, Stonemanet al. [7] attemptto correct for these
problems-and others-and suggest that welfare losses are considerablyhigher than had been
suggested by Harberger.2Although Cowling and Mueller estimate welfare losses attributableto
individualfirms ratherthan industries,they ultimatelyderive their estimatesof allocative losses
by startingfrom the same basic theoreticalstructureas Harberger;i.e., they look at a "welfare
loss triangle" whose area is (1/2)(Api)(Aqi) for an individualfirm i [5, 729]. Given their other
assumptions,such estimatesof welfareloss reduceto simple functionsof the firm'sexcess profits.
Holt [19], however, points out that such "Lernerequationloss estimates,"as he calls them,
are appropriateonly for a monopoly with lineardemandand constantcosts. They are inappropriate estimatesfor the welfareloss from the exercise of marketpowerin an oligopoly. In particular,
he criticizes Cowling and Mueller's focus on individualfirms and shows that the summationof
2. Cowling and Mueller also look at a broaderrange of losses than those consideredin most of the earlier literature, including, e.g., losses due to advertising.In this section, I focus on theirapproachto allocativeloss only. In a recent
paper, Shinjo and Doi [29] apply Cowling and Mueller'sapproachto estimatewelfarelosses in Japan.Among their other
findings, they conclude [29, 252] that "aggregateWL [welfareloss] as a percentof nationalincome is found similar to
the orderof magnitudereportedfor the U.S. and Europeancountries."
DEADWEIGHTLOSS IN OLIGOPOLY:A NEW APPROACH
173
such estimatesacross all firmsin an industrydoes not lead to an accurateestimateof welfarelosses
for the industry.3In general, Holt notes [19, 289] that "[t]hepreciserelationshipbetween industry
profits and welfare losses depends on the numberof firms, the natureof cost asymmetries, and
the source of monopoly power in the industry."(See also Schmalensee[26].)4
Masson and Shaanan[21] also arguethat it is more appropriateto model the industryrather
than the firm.5They acknowledge, however,that they do so at the cost of losing individualfirm
differences.In particular,their analysis precludesconsiderationof differencesin firms' costs.
Recent papersby Gisser [15], Dickson [11], Daskin [10], and Willner[35] model the industry ratherthan individualfirms using a dominant-firmsmodel, while Dickson and Yu [13] use a
more general model to estimate welfarelosses in Canadianmanufacturing.
The model outlined below also considers welfare loss at the industrylevel, thereby avoiding aggregationproblems inherentin a firm-levelapproach.The model is more general than the
dominant-firmsmodels noted above, however, allowing a broaderrange of oligopoly solutions.
In addition, the model is flexible enough to allow for cost asymmetriesamong firms within an
industry.Finally, unlike Dickson and Yu's model, the model below allows for differentbehavior
among firms within an industry.
III. The Theoretical Model
In this section I develop the theoreticalmodel to be used to estimate deadweightloss in an industry. Since the model derives from a model in Dixit and Stem [14] (henceforthD-S), I first
summarizethe salient featuresof their model.6To do so, firstdefine the following notation:
X
n
xi
si
ci
=
=
=
=
=
total industryoutput;
the numberof firms in the industry;
outputof firm i (i = 1, 2, ..., n);
xi/X = the marketshareof firm i; and
marginalcost of firm i.
Any particularfirm's marginalcosts are assumedto be constantat some level, but the level may
vary across firms. Industrydemandis assumedto be isoelastic:X = Kp -7, whereK is a constant
and r/ is the absolute value of demandelasticity.7
3. For a symmetricn-firm Cournotoligopoly with constantcosts and lineardemand, for example, he shows that
the summationof firm-level welfare loss estimates of the sort suggested by Cowling and Mueller will overstatewelfare
loss estimates for the industryby a factorof n.
4. Cowling and Mueller might well acknowledgethese points, includingthe difficultyin aggregatingwelfarecosts
across firms; see Cowling and Mueller [5], especially pp. 745-6. They argue, however,that their estimates of "relative
cost of monopoly for each firm" [5, 739] are still meaningful.
5. For an interestingdiscussion of the aggregationproblems inherentin a firm-levelapproach,see Masson and
Shaanan [21, 521]. They reiterate some of the points made by Holt [19] and Schmalensee [26] and provide a clear
comparisonof their work and that of Cowling and Mueller.
6. Clarkeand Davies [4] presentthe same model. Since Dixit and Ster explicitly suggest the applicationto deadweight loss, I refer to the model as the D-S model. The two pairsof authorsapparentlydevelopedthe same basic model
concurrently.
7. For much of the theoretical developmentin Clarke and Davies [4] and Dixit and Stem [14], marginal cost
curves need not be horizontaland demand need not be isoelastic. FollowingDixit and Stem, I make these assumptions,
relativelycommon ones in this area, to get tractableexpressionsfor consumersurplus,producersurplus,and welfareloss.
Dickson and Yu [13], whose work also derives from the same basic model, do not assume constantmarginalcosts. As
174
Alan J. Daskin
Firm i chooses its optimal output to maximize its profits, given its conjectures about the
change in output by other firms in response to a change in xi. Dixit and Stem-as well as Clarke
and Davies-parameterize firm i's conjectures as follows:
dxk/dxi = a(xk/Xi),
k # i;
i.e., firm i assumes that the percentage change in other firms' outputs will be directly proportional to the percentage change in its own output.8 The absence of any subscript on a reflects the
Dixit-Stern/Clarke-Davies assumption that all firms have the same proportional conjectures about
all other firms in their industry.9 Although a is constant for all firms in an industry, it may vary
across industries. While D-S focus their attention on values of a between a = 0, the Courot
conjecture, and a = 1, perfectly collusive conjectures, there is no inherent need to constrain a to
exceed 0.10
Given this structure, D-S solve for the equilibrium values of the endogenous variables,
including market price, p, and the firms' market shares, si (i = 1,2, .... n). Graphically, the
equilibrium is depicted in their Figure 1, which I have reproduced below, where
c* = min(ci)
denotes the marginal cost of the lowest-cost firm. Total industry output is OX, and the progressively higher "steps" correspond to the lower price-cost margins of higher-cost firms.
In this model, Figure 1 is the oligopoly counterpart of the standard textbook graph illustrating
Harberger's deadweight loss for a monopoly. Using the usual Marshallian measures of surplus,
the welfare loss for the industry is the difference between the loss in consumer surplus and the
gain in producer surplus resulting from the exercise of market power. As Dixit and Stern point
out, however, the firms' unequal costs make it necessary to specify a standard against which to
compare the oligopoly equilibrium. If we take p = c* (and the associated output on the industry
demand curve) as our standard for comparison, the area marked "DWL" in Figure 1 gives the
welfare loss due to oligopoly."
noted in the text below, however, their approachimplies a monotonic(positive) relationshipbetween firms' marginsand
marketshares within an industry,which is inconsistentwith the data for many industries.
8. Similar parameterizationsof conjecturesare common in this and related literature.See, e.g., Dickson [12],
Cubbin[9], Sleuwaegen [31], and Dickson and Yu [13].
9. Dixit and Ster and Clarke and Davies make this assumptionin large part for the sake of mathematicalsimplicity and tractability,especially in deriving some of their comparativestatics results. Clarkeand Davies briefly discuss
some of the issues and problems involved in a more generalparameterization
of conjectures.In the theoreticalextension
that I develop below-and in its empirical application-I weaken the assumptionthata is constantfor all firms in the
industry.
10. Dixit and Ster note [14, 126] that "[c]ases of a < 0 are also conceivable, workingtowards 'accommodating' behaviouron part of other firms, an example being where they adjusttheir outputsto keep price constant." For a
discussion of the relationshipsamong values of the conjecturalvariationelasticity, the natureof competition, and the
appropriatemeasure of concentration,see Dickson [12]. (Dickson [12] defines firm i's conjecturalvariationelasticity
as (dX/X)/(dxi/xi); i.e., the proportionalchange in the numeratorincludes the change in firm i's output. Given that
definition, each firm's conjecturalvariationelasticity would be 0 in perfect competition,e.g., since each firm assumes
thata change in its own outputhas no effect on total marketoutput.)
11. Although Cowling and Mueller [5] focus on the estimationof welfareloss for individualdominantfirms, they
mentionin passing an approachthat seems strikinglysimilarto the one in Dixit and Ster, which I modify below: "If a
policy were adopted forcing the most efficient size or organisationalstructureupon the entire industry,the welfare loss
underthe existing structurewould have to be calculatedusing the profitmarginof the most efficientfirm and the output
of the entire industry,ratherthan the profit marginsof the individualfirms and their outputs." ([5, 744], emphasis in
original)They also point out [5, 729] that "we shall obtainwelfareloss estimatesby individualfirmsfrom their price/cost
DEADWEIGHTLOSSIN OLIGOPOLY:A NEW APPROACH
175
Quantity
Figure 1.
The Dixit-Stern/Clarke-Daviesmodel, however, has an unfortunateimplication. Their
assumptionthat a is the same for all firms in the industryimplies a monotonicrelationshipbetween price-cost margins and marketshares [14, 128]. (See also the discussion of my equation
(2) below.) Low-cost firms necessarilyhave high marketshares (and high price-cost margins)
in the model, and vice versa for high-cost firms. In fact, we rarelyobserve such a monotonic
relationshipbetween price-cost marginsand marketsharesfor all firmsin any given industry.'2
To remedy that discrepancybetween theory and fact, I extend the D-S model by allowing
firms in an industryto have differentconjectures;a, therefore,is indexed by i.13Formally,the
industry'sinverse demandfunctionis given by p (X), and firmi 's conjecturescan be writtenas
dxk/dxi
= a i (Xk/Xi)
or
dXk/Xk = ai(dxi/xi)
for all k #- i.'4
margins. These estimates indicate the amountof welfare loss associated with a single firm's decision to set price above
marginalcost. ... To the extent other firms also charge higherprices, because firm i sets its price above marginalcost,
the total welfare loss associated with firm i's marketpowerexceeds the welfareloss we estimate."
12. For a relatedobservation,see Stylized Fact 4.12 (and supportingreferences)in Schmalensee[28, 984]: "Within
particularmanufacturingindustries,profitabilityis not generallystronglyrelatedto marketshare."
13. For empirical evidence on the variationof conjectureswithin an industry,see, e.g., Gollop and Roberts [17],
Slade [30], or Rogers [24].
14. Although this parameterizationof firm i's conjecturalvariationelasticity is sufficientto remove the monotonic
relationshipbetween firms' marketsharesand price-cost margins, an anonymousrefereehas pointed out that the conjecturalvariationelasticity could be even more general. Specifically,firm i might have differentconjecturesabout different
is not necfirms, so we would have to write the conjecturalvariationelasticity as aik, where aik -(dxk/Xk)/(dxi/xi)
essarily equal to aij for j = k. From an empiricalpoint of view, that more general formulationof conjecturalvariation
elasticities necessitates estimation of many more parameters;as noted below, the availabledata are already quite limited. For a brief discussion of some of the theoreticalissues involved in such a parameterization
of conjecturalvariation
elasticities, see Clarkeand Davies [4, 280].
176
Alan J. Daskin
Firm i then chooses xi to maximize its profits, given by
T7i
-
p (X)xi - cixi - fixed costs.
The firm's first-orderconditionis then
d7ri/dxi = p + xip'[l + E (dxk/dxi)] - ci = 0,
k$i
which, after some algebraicmanipulation,can be writtenas15
p{l - [ai + (1 - ai)si]/l}
(1)
ci.
In terms of firm i's price-cost margin, mi = (p - ci)/p, we can write (1) as
mi = (p - ci)/p = (l/rl)[ai + (1 - ai)si].
(2)
Ceterisparibus, therefore,a firmwith a relativelylow ai will tend to producea higherquanand
have a higher marketshare than a firm with identicalcosts but higher ai. Once ai is
tity
allowed to vary across firms in the industry,there need not be a monotonicrelationshipbetween
firms' price-cost margins (mi) and their marketshares(si).'6The largestfirms in an industryare
not always the most efficient.
Figure 1 is now helpful in adaptingDixit and Stem's algebraiccalculationof consumersurplus, producers' profits, and the resulting deadweightloss. Calculationof producersurplus is
straightforward:
ZE i =
(p - ci)xi
= R E[(p - ci)/p](xi/X)
=R
since R = pX and X =
(3)
misi
xi .17
15. Omittingthe explicit index k $ i for conveniencein all the summationsbelow, the firstorderconditionimplies
p + xip'[1 + E(dxk/dxi)] = ci. We can then write the left side of the latterequalityas
p + xi(p/p)(/X)p'[l
+ >(dxk/dXi)]
=p + p(xi/X)(X/p)p'[l
+
(dxk/dxi)]
= p{ -(si/n7)[l
+-
= p{
+ ai EXk/Xi]} = p{1 -(si/rl)[l
-(si/rl)[l
(dxk/dxi)]} = p{l - (si/q)[l +
= p{ - (si/r)[
+ ai(l - si)/si]} =p{
=p{l - [ai +
-a i)si]/7},
-
sil/r
a i(xk/xi)]}
+ a,i(CEXk /X)/(xi/X)]}
- ai(l - si)/lr}
the left side of equation(1) in the text.
16. In terms of Figure 1, as we "climb the steps" from c* to p, the horizontallength of the steps may get shorter
or longer. For the model in which a is the same for all firms in the industry,Dixit and Ster derive an equationthat is
identical to my equation(2) except that they have no subscripton a. (See their equation(9).) In that case, mi and si do
rise and fall together.
17. Using equation(2) in the text, we can rewritethe rightside of (3) as
DEADWEIGHTLOSSIN OLIGOPOLY:A NEW APPROACH
177
The reductionin consumersurplusis given by the areaunderthe demandcurve between c*
andp. For r7 ? 1, that area is given by
Ku-
du = [K/(
- r)]u l
= [Kp' 7/(1 -r)][1
-(c*/p)l
1]
= [R/(1 - r)][1 - (c*/p) -'] = [R/(1 - r)][1 - (1 - m*)1-'7]
(4a)
where m* is the price-cost marginof the lowest-cost firm. For r1= 1, the lost consumersurplus
is given by
f
Ku-'du = K ln(u)P: = K ln(p/c*)
=R *ln(p/c*)
since, if r = 1,R =pX =pKp-
=K
= R ln[1/(1 - m*)].
(4b)
Deadweightloss is then given by the differencebetween the appropriateexpressionin either
or
(4a) (4b) and the expressionin (3). The next section explainshow I use (3) and (4) to estimate
deadweightloss in an industryfrom data given in the Censusof Manufactures.
IV. Empirical Adaptation of the Theoretical Model: Sample and Data Considerations
In an attemptto apply the theoreticalmodel above to a wide rangeof industries,I use the 1977
Censusof Manufactures[32] (henceforththe Census)as the basic sourceof dataon U.S. manufacturingindustriesat the four-digit S.I.C. level. Unfortunately,the more recent 1982 Census does
not providethe data that are necessaryto calculateprice-costmarginsfor firmsof differentsize.'8
Even the 1977 Census does not providedataon individualfirms' marketsharesor price-cost
margins, as apparentlyrequiredby equations (3) and (4). Instead, the Census aggregates data
for five groups of firms. Ratherthan provide individualmarketshares, the Census provides data
on four concentrationratios:the four-firm(CR4), eight-firm(CR8), 20-firm (CR20), and 50-firm
(CR50) concentrationratios. Thus we can extractinformationon combinedmarketsharesof various "ranks"of firms: CR4 = I4= si = the combinedshareof the first four firms, or first rank;
CR8 - CR4 = 8i=5si = the combined share of the next four firms (firms 5-8), or rank;and so
on for rank 3 (firms 9-20), rank4 (firms21-50), and rank5 (firms51-n).
In fact, equation(3) does not requireinformationon individualsharesor price-cost margins;
it requiresa share-weightedaverageof individualfirms' price-cost margins, which can be computedfrom Census data. For the firstrank, for example, we can computea marketshare-weighted
averageof the four firms' price-cost margins, MI, directly (more on the actual definitionof the
price-cost marginbelow.) We can write Ml as
(R/r)[o aiSi + H- oaiS,],
where H = s2 is the Herfindahlindex of concentration.For the model in which ai is the same for all firms in the
industry,the latterexpression simplifiesto (R/tl)[H + a(1 - H)]. (Dixit and Ster [14], equation(15)). Dickson and Yu
[13] use a clever adaptationof the latterexpressionin theirestimationof welfareloss in Canadianmanufacturing,but their
model implies a (positive) monotonicrelationshipbetween firms' marketsharesand price-cost margins.
18. That omission is particularlycurious in light of the fact that the 1982 Censusdoes, for the first time, provide
data on the Herfindahlindex for differentindustries.
178
Alan J. Daskin
4
4
4
i=1
i=1
i=1
Ml = i sim,/ E s = (1/CR4)(> sim,),
so we can use Census data on CR4 and Ml to calculate 4=I simi as Ml *CR4. Similarly,we can
use Census data for the other fourranksto calculatetheircontributionto producersurpluswithout
knowing individualfirm sharesor margins.
Equation(4), which requiresknowledgeof m*, the highestprice-costmarginamongfirmsin
the industry,does requiresome empiricalassumptions.For all five ranks,I assumeequal margins
within each rank, although margins do typically differ across ranks. For ranks 1 and 2, e.g., I
assume that
mi = m2 = m3 = m4 = M
m5
=
m6
and
m7 = ms = M2,
where M1 and M2 can be computed from Census data.'9For the sake of estimating equation
(4) above for any industry,thatassumptionis sufficient,given any particularvalue of rq.For some
of the discussion in the next section, furtherassumptionsabout r/ are required;I discuss those
assumptionsin the text below.
For each rank, I use Census data to calculateMi as
[VSi -payrolli
- (cost of materials)i
- (cost of capital)(averagebook value of depreciableassets)i
- depreciationi - (rental payments)i]/VSi
where VS = value of shipments.The Censusprovidesdatafor VS, payroll, and cost of materials
for each rankwithin each four-digitindustry.Since the Censusdoes not disaggregatebook value
of assets, depreciation, or rental paymentsby rank-it just provides totals for the industry-I
proratethose variablesaccordingto their marketshares. In the empiricalestimationof equations
(3) and (4), I try three alternativevalues for the cost of capital:5%, 10%,and 15%.20
The sample includes all four-digit industriesfor which the 1977 Census reportsdata, with
two generalexceptions:(i) I exclude all industriesthatinclude "nec" (not elsewhereclassified) or
"miscellaneous"in their titles, and (ii) I exclude 10 industriesfor which the Census uses different
rankingsfor data on concentrationand data on payrolland cost of materials.The "full" sample
thatremainshas 363 four-digitindustries.
19. In preliminarywork, I also consideredan alternativeset of assumptions;viz., thatthe top four firmshave equal
conjecturalvariationelasticities and unequal marketshares given by the one of the distributionsof shares suggested in
Schmalensee[27] and Michelini and Pickford[22]. Those alternativeassumptions,however,imply thatthe top four firms
have differentprice-cost margins. Moreover,equation(2) in the text indicatesthatthose marginsthen depend criticallyon
our assumptionabout the value of r1:Differentvalues of 77,combinedwith Censusdata that make it possible to calculate
M,, may then cause the identity of the lowest-cost firm-and with it, the estimatesof estimatesof deadweightloss-to
change. (From (2), a low value of r1 implies a large dispersionof mi, m2, m3, and m4, whose weighted averageequals
Ml; the value of Mi is unaffectedby assumptionsaboutr1, since we calculateit directlyfromCensusdata. For low enough
assumed values of r7, therefore, the implied price-cost marginfor the largest firm, mi, may become the highest in the
industry-and, therefore, the benchmarkfor calculationsof deadweightloss-even if Ml is considerablylower thanM2,
M3, M4, and M5.)
20. In principle, we might like to use differentrisk-adjustedcosts of capital for each industry.For present purposes, however,I am primarilyinterestedin the orderof magnitudeof the estimatesof deadweightloss, so this estimation
approachshould suffice.
DEADWEIGHTLOSSIN OLIGOPOLY:A NEW APPROACH
179
Table I. Deadweight Loss as Percentageof Value of Shipments-No Restrictionson tj
Table IB
Sample Censored Using
Weiss/Pascoe Screen
(257 Industries)
Table IA
Full Sample: 363 Industries
Cost of Capital
Cost of Capital
r/
5%
10%
15%
r
5%
10%
15%
0.25
0.50
1.00
1.50
2.00
2.50
3.00
5.00
6.33%
7.59%
10.43%
13.77%
17.77%
22.63%
28.60%
74.69%
6.22%
7.35%
9.90%
12.89%
16.45%
20.75%
26.02%
66.19%
6.12%
7.14%
9.42%
12.10%
15.28%
19.10%
23.77%
58.93%
0.25
0.50
1.00
1.50
2.00
2.50
3.00
5.00
6.61%
7.80%
10.44%
13.46%
16.96%
21.02%
25.76%
55.25%
6.50%
7.56%
9.90%
12.58%
15.66%
19.21%
23.33%
48.41%
6.39%
7.34%
9.43%
11.80%
14.51%
17.62%
21.21%
42.66%
V. Empirical Results
Tables I and II report deadweight loss in U.S. manufacturing as a percentage of value of shipments
for alternative values of demand elasticity (r1), weighting each industry by its value of shipments.
For any particular value of r7, deadweight loss is relatively insensitive to the value chosen for the
cost of capital.
Table IA reports estimates for the full 363-industry sample. For r7 - 1, deadweight loss is
roughly 6-10% of the total value of shipments. As expected, deadweight loss increases with rq.
For large enough values of r7, in fact, deadweight loss becomes a very high percentage of the
value of shipments.21
The estimates in Table IA, however, may be misleading for two reasons. First, S.I.C. codes,
while convenient for empirical purposes, do not necessarily correspond perfectly to true economic
markets [25]. As a consequence, concentration ratios reported by the Census may be too high
or too low. In an attempt to exclude some of the industries for which the definitions are particularly inappropriate, I compared reported (Census) and adjusted values of CR4 given in Weiss and
Pascoe [34], who adjust reported values to correct for several potential sources of over- or underinclusion in the S.I.C. definitions. In Table IB, I exclude all industries for which the reported and
adjusted values of CR4 differ by more than 25% of the adjusted CR4 or five percentage points,
whichever is greater.22Comparing Tables IA and IB suggests that such exclusions increase (re21. There is less than total unanimityaboutthe appropriatevalue of elasticityin such empiricalwork, and there are
surprisinglyfew empiricalstudies that reportestimatesof demandelasticity for a cross section of industries.For a sample
of 46 food and tobacco industriesin the U.S., Pagoulatosand Sorensenreportuniformlyinelasticdemand[23, 241]: "The
values range from a high of .756 for cigars to a low of .008 for flavoringextractsand syrupsin our sample." Shinjoi and
Doi justify their focus on 77 = 1 by noting [29, 248] that "it is based on the findingsthat the price elasticities estimated
from industrytime series data [for Japan]tend to clusteraround1.0 [fn. omitted]."Empiricalestimatesreportedin Allen
[1, 103] suggest thatr1 might be as high as 2 for certainindustriesin the manufacturingsector.
22. Two examples may clarify the rule: If the reportedCR4 is 42% and the adjustedCR4 is 34%, the industry
would not be excluded even thoughthe two numbersdifferby more thanfive percentagepoints, since 42 - 34 = 8 is less
than 25% of 34 (.25 x 34 = 8.5). If the reportedCR4 is 6% and the adjustedCR4 is 10%, the industrywould also not
be excluded even though 16- 101= 4 exceeds 25% of the adjustedCR4, since the differencebetween the two numbersis
less than five percentagepoints.
180
Alan J. Daskin
Table II. DeadweightLoss as Percentageof Value of Shipments-Restrictions Imposedon 7 *
Table IIB
Sample Censored Using
Weiss/Pascoe Screen
(257 Industries)
Table IIA
Full Sample: 363 Industries
Cost of Capital
Cost of Capital
7r*
5%
10%
15%
7*
5%
10%
15%
0.25
0.50
1.00
1.50
2.00
2.50
3.00
6.33%
7.59%
10.43%
13.76%
17.17%
20.12%
22.58%
6.22%
7.35%
9.90%
12.88%
15.94%
18.59%
20.84%
6.12%
7.14%
9.42%
12.10%
14.84%
17.23%
19.28%
0.25
0.50
1.00
1.50
2.00
2.50
3.00
6.61%
7.80%
10.44%
13.46%
16.92%
20.17%
22.82%
6.50%
7.56%
9.90%
12.58%
15.64%
18.59%
21.01%
6.39%
7.34%
9.43%
11.80%
14.50%
17.18%
19.37%
5.00
26.62%
24.92%
23.31%
5.00
27.18%
25.45%
23.82%
*In TablesIIA and IIB, r7was set equal to the minimumof (i) the value indicatedin the table and (ii) the maximum
value possible for each industry,as determinedby restrictionson the conjecturalvariationelasticities. See text for further
discussionof limits on 7 .
duce) our estimates of deadweightloss for r7 ' 1 (r7> 1), but the magnitudeof the change in
the estimates is relatively small for 7r7 3. Even for r7= 5 in TableIB, deadweightloss remains
ratherhigh.
At least some of the estimatesin TablesIA and IB may also be misleadingbecause we have
imposed no restrictionson r. It seems reasonable, however, to impose the restrictionthat no
firm's ai exceed 1; i.e., no firm's conjecturalvariationelasticity is greaterthan a monopolist's
would be. Although the ai's do not appearexplicitly in equation(3) or (4), the link among mi,
1 V i, does lead to a restrictionon 77, which does
a i, and r/, together with the restrictiona i
in
for
a
i,
(2)
(4):
appear
Solving equation
ai = (rlmi - si)/(l
so1ia
- si),
(5)
1 => r7 - 1/mi Vi.
Table IIA reportsestimates of deadweightloss for the entire 363-industrysample when we
impose the restrictionson r7 implied by a i 1 V i. For each industry,I set 77 equal to the
value given in the leftmost column of Table IIA or min(l/mi), whichever was smaller. Since
- 1 is not a binding constraint for
minmi,>(1/mi) exceeds 1 for all industries in the sample, ai
r7 - 1, and the first three rows of TableIIA are identicalto the correspondingrows of Table IA.
For highervalues of r7, however,a i 1 becomes a bindingconstraintfor more and more industries, so the entries in the bottom rows of Table IIA are lower than the correspondingentries in
TableIA. For r7 = 3 and especially for r7 = 5, the entriesin TableIIA are much lower.
Finally, Table IIB reports the results of imposing both the Weiss/Pascoe industry screen
and the restrictionthat all firms' conjecturalvariationelasticities be less than or equal to 1. As
expected, the first three rows of Table IIB are identicalto the correspondingrows in Table IB.
Moreover,there is very little differencebetweenthe entriesin TablesIIA and IIB, which suggests
thatthe restrictionson a i are more importantquantitativelythanthe restrictionson the industries
includedin the sample.
DEADWEIGHTLOSSIN OLIGOPOLY:A NEW APPROACH
181
Table III. Top Five Industries,Rankedby Dollar Value of DeadweightLoss
SIC - IndustryName
3714 - Motor vehicle parts, accessories
3721 - Aircraft
3711 - Motor vehicles and car bodies
3861 - Photographicequip. and supplies
3353 - Aluminumsheet, plate, and foil
CR4
DWLas %
of TotalDWL
Cumulative
% of TotalDWL
62.2%
58.5%
93.4%
72.2%
72.0%
6.33%
6.32%
4.26%
3.46%
3.04%
6.33%
12.65%
16.90%
20.36%
23.40%
Some researchers have noted that firms' tendency to collude is higher the more inelastic is
demand [2, 217]. That tendency might seem to be inconsistent with the results reported in Tables
I and II, in which deadweight loss increases with elasticity. Moreover, equation (5) above might
seem to imply a positive relationship between elasticity and firms' tendency to collude, since
aai/ar = mi/(l - si) > 0.
Recall, however, that r1 and the ai's are exogenous in the theory above-as are r/ and a
in Clarke and Davies [4] and Dixit and Ster [14]-and market shares and price-cost margins
(si and mi) are endogenous. In the empirical application, the endogenous variables si and mi are
observable, while the exogenous variables r} and ai are unobservable. Equation (5), therefore, is
not a behavioral relationship linking firms' tendency to collude to demand elasticity. Instead, it
gives the value of a firm's ai we can infer from the observable values for mi and si along with a
hypothesized value of r/. Since mi and si are "fixed" by the data, the positive sign on aai/lar
has the following interpretation: Given values of mi and si are consistent with a high hypothesized value of demand elasticity only if ai is high. Equation (5), therefore, along with theoretical
restrictions on a i, puts limits on how high the hypothesized value of r7 can be.
To see why deadweight loss increases with r/ in Tables I and II, consider equations (3) and
(4) along with Figure 1. Equation (3) shows that the empirical estimate of producer surplus is not
affected by a change in r/. The invariance of producer surplus is also evident from Figure 1 if
we imagine drawing a demand curve through point A that is more elastic than the demand curve
shown. Such a demand curve reveals, however, that the lost consumer surplus does increase with
r7: A more elastic demand curve would be above the demand curve shown for all quantities to the
right of point A, and the point x, the quantity demanded if price were c*, would be further to the
right in Figure 1. Since producer surplus is invariant with respect to r/ and lost consumer surplus
increases with r/, deadweight loss increases with r .23
While it is not evident from Tables I and II, a small number of industries account for a very
large fraction of the total welfare loss. If 7r = 1 and the cost of capital = 10%, for example,
Table IIB indicates that total deadweight loss for the censored sample of 257 industries is 9.90%
of the total value of shipments. As Table III indicates, however, the top five industries, ranked by
23. Readers who are bothered by this result may object that the optimal markupfor a monopolistdecreases as rl
increases, which is certainlytrue. In the empiricalestimationof deadweightloss above, however,the actual markupsare
given by the data. Higher hypotheticalvalues of r1 have no implicationsfor the actual markupsobserved;they do imply
greater reductions in output from the level that would have been producedat p = c*. The same positive relationship
between estimated deadweightloss and demandelasticity appearsin Harberger'sexpressionfor the deadweightloss from
monopoly: "Thus, the dead-weight welfare loss from monopoly rises as a quadraticfunctionof the relative price distortion . . . and as a linear function of the demand elasticity" [25, 662]. For a given value of the relative price distortion
(p/c* in our case), deadweightloss increases with r1.
182
Alan J. Daskin
40
a
1 35E
30-
.
251
15 20-
,mU.
i 2. :.
10-"'I
9.. 0
0
10
0'
20
.
F.....o
5-
_
?
ME
?
. '
or -..
o 60
30
4o
o0
Four-firmConcentration Ratio (%)
8o
90
100
Figure 2.
dollar value of deadweight loss, account for nearly a quarterof the total deadweightloss in the
sample.
All five industries in Table III have high four-firmconcentrationratios,24so it is natural
to wonder if deadweight loss, expressed as a percentageof an industry'svalue of shipments
(DWL/VS), is correlatedwith the industry'sconcentration.As it turns out, that correlationis
quite small. For r ==1 and a cost of capital of 10%, for example, Figure 2 shows a scatterdiagram of DWL/VS against CR4 for the 257 industriesin the censored sample.25Needless to say,
the graph does not suggest a tight-fittingrelationship.The regressionof DWL/VS against CR4
(with both expressed as percentages)is as follows:
DWL/VS = 9.240 + .0380(CR4),
(1.93)
where 1.93 is the t statistic for the null hypothesis that the true coefficient on CR4 is 0. The
estimatedcoefficient is marginallysignificantin a statisticalsense (5% level), but its magnitudeis
quite small: A ten-percentage-pointincrease in CR4 (from 30% to 40%, e.g.) is associated with
an increase of less than .4 percentagepoints in DWL/VS (from 10.38% to 10.76% when CR4
increases from 30% to 40%, e.g.). These results are not necessarilyinconsistentwith Table III,
which lists the top industriesrankedby dollar value of DWL, not DWL/VS. For industry3711,
e.g., DWL/VS was only 3.57%; the dollar value of DWL was high because the industry'sshipments were so large.
24. The 93.4% four-firmconcentrationratio for industry3711, however,undoubtedlyoverstatesthe true degree of
concentrationin the industry,since the Census reportsdomesticconcentrationratios.
25. Because many points overlap, the graphseems to show fewer than257 points.
DEADWEIGHTLOSS IN OLIGOPOLY:A NEW APPROACH
183
The above estimates of deadweight loss are considerablyhigher than many estimates in
the existing literature.26Part of that differenceundoubtedlystems from the assumptionthat all
firms have constant marginalcost (albeit at differentlevels for differentfirms). That assumption
is particularlysignificant for the firm with the lowest marginalcost (c*), since that finn serves
as a benchmarkin my calculationof deadweightloss.27While the assumptionof constant marginal cost is admittedlya strongone-suggesting perhapsthatthese estimatesof deadweightloss
serve as an upper bound-there are no availableempiricalestimates for the elasticity of marginal cost for a broad range of industries.The assumptionof constantmarginalcost, moreover,
while controversial, has a long history in this literature,extending as far back as Harberger's
original work.28
VI. Summary and Conclusions
Using a generalizationof a recent model of oligopoly, I have estimatedthe magnitudeof welfare
loss for a broad cross section of U.S. manufacturingindustries.Unlike previousempirical work
in this area, the model is flexible enough to allow for both cost asymmetriesand unequal conjectural variationelasticities among firms in an industry.For the sampleconsidered,the evidence
suggests that deadweightloss is roughly6-10% of the value of shipmentsif demandis inelastic
or unit elastic; losses may be considerablyhigher if demandis elastic. Much of the deadweight
loss, however, is accountedfor by just a few industries.
The estimates are subject to all the usual caveats that apply in this area. In particular,the
study relies on S.I.C. classifications,which are imperfectproxiesfor trueeconomic markets,and
Census of Manufacturesaccountingdata. Even the censored sample, which excludes industries
for which adjustedconcentrationratiosdiffersignificantlyfrom the reportedratios, providesonly
rough approximationsof true markets.In relying on Census data, of course, I have lots of company. Indeed there seems to be no reliable, alternative,publicly-availablesource of data on a
broadcross section of industries.With these caveatsin mind, the study indicatesthat deadweight
losses may be considerablyhigher than much of the earlierliteraturewould suggest.
26. Many researchersin this area, however, includingHarberger,reportwelfarelosses in manufacturingas a percentage of GNP, ratherthan as a percentageof the value of shipmentsin manufacturing.Schererand Ross note [25, 664]
that "[d]istortionsattributableto monopoly also exist in sectorsotherthan manufacturing.. . . The manufacturingsector
originatedabout one-fourthof U.S. GNP in the period studiedby Harbergerand one-fifthin the 1980s. To arriveat an
economy-wide welfare loss estimate, figuresderivedfor manufacturingalone must be inflated-perhaps by as much as a
factorof 3 or 4." At least some of the lower estimatesreportedin the literature,therefore,are not directlycomparableto
those reportedin the text above.
27. Cf. Dickson and Yu [13], whose estimates of welfare loss depend on several parameters,including e, the
elasticity of the industry'smarginalcost. For high or infinitevaluesof e, some of theirestimatesof DWL/GNP are comparableto-and in some cases, dramaticallyhigherthan-those reportedin the text above. (Moreover,Dickson and Yu
reportwelfare losses in Canadianmanufacturingas a percentageof CanadianGNP, not as a percentageof manufacturing
shipments. See fn. 26 above.)
28. For two ratherdifferentviews on the subject, cf. Willner[35, 604-605], who claims, "It could even be considered a stylized fact thatlarge corporationshave horizontalmarginalcosts [fn. om.]. We thereforefind Gisser's assumption
of equally elastic and positively sloping supply schedules [for dominantfirms and the competitive fringe] ad hoc and
weakly motivated;"and Gisser [16, 611], who respondsthat "it seems thatthereis neithercompellingempiricalevidence,
nor a priori reasons, supportingthe argumentthat the marginalcosts of the leaders are more elastic than the marginal
costs of the small firms."
184
Alan J. Daskin
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