the measurement of monopoly power in dynamic markets

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THE MEASUREMENT OF MONOPOLY POWERIN YNAMIC MARKETS*byRobert S. Pindyck

March 1984Sloan School of Management Working Paper No. 1540-84

Support from the National Science Foundation, under Grant No. SES-8012667is ratefully acknowledged.


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ABSTRACT

In arkets in hich price and output are determined intertemporally,the standard Lerner index is a biased and sometimes misleading measureof actual or potential monopoly power. This paper shows how the Lernerindex can be modified to provide a meaningful instantaneous measure ofmonopoly power applicable to dynamic markets, and discusses the aggre-gation of that instantaneous measure across time. The importance ofaccounting for intertemporal constraints in ntitrust and related appli-cations is llustrated by the analysis of four examples: an exhaustible.resource, the "learning curve," costs of adjustment, and dynamic adjust-ment of demand. An analogous index of monopsony power applicable todynamic markets is lso suggested.


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1. IntroductionThe measurement and analysis of monopoly and monopsony power have been a

prime concern of industrial organization, and are of obvious importance in hedesign and application of antitrust policy. Much of the recent literature hasfocused on the structural and behavioral determinants of monopoly and monopsonypower, including the characteristics of costs and demand, and the ways in hichfirms in he market interact with each other. There has been less concern,however, with the development of a measure of monopoly (or monopsony) power.Instead, the Lerner index, first introduced in 1934, has for years been acceptedas the standard measure of monopoly power, and is ften used as a summary sta-tistic in ntitrust applications.1

The Lerner index is ust the margin between price and marginal cost, i.e. L =(P MC)/P. In a static market, D l/rnf, where f is he elasticity of demandfacing the firm, so that the firm's elasticity of demand completely determinesits monopoly power.2 However, this need not be the case in a dynamic market.When price and output are determined intertemporally, the Lerner index need notequal the inverse of the firm's elasticity of demand, and neither L nor l/nf willnecessarily provide a meaningful measure of monopoly power.

By a dynamic market, I ean one in hich price and production are inter-temporally determined. Examples include markets for exhaustible or renewableresources, markets in hich supply is ffected by a learning curve or by thepresence of adjustment costs for quasi-fixed factor inputs, and markets in hichfirms' demands respond over time (as opposed to instantaneously) to changes inprice. (Almost all real-world markets are dynamic.) In ll of these examples,the Lerner index is insufficient (if ot misleading) as a measure of monopolypower. In ome cases (e.g. an exhaustible resource) it verstates the extent ofmonopoly power, and in ther cases (e.g. when there are learning curve effects)it nderstates it. In very case it pplies only to an instant of time, whilethe impact of monopoly power always applies to some interval of time.


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-2-As an extreme example of the inapplicability of the Lerner index, even as an

instantaneous measure of monopoly power, consider the case of a onopolist pro-ducer of an exhaustible resource who faces an isoelastic demand curve and haszero extraction costs. It is ell known that the price and production trajectoriesin his case are identical to those for a perfectly competitive market, i.e. themonopolist has absolutely no monopoly power.3 The Lerner index, however, willequal 1 t every instant of time. As a second example, consider a monopolist whois tarting up production using a technology in hich the learning curve is mpor-tant. Because current production reduces future production costs, current pricecan be below current marginal cost. Thus even though output will be less thanthat in a ompetitive market, the Lerner index can turn out to be negative.

Even if he Lerner index sufficed as an instantaneous measure, it ould havelittle to tell us about the potential impact of monopoly power unless it ereproperly aggregated across time. Simply calculating "short-run" and "long-run"degrees of monopoly power (based on short-run and long-run demand curves) may notbe very informative for two reasons. First, the short-run monopoly price willdepend on more than the firm's short-run demand curve, so that the use of thatdemand curve to calculate a "short-run" degree of monopoly power is isleading.(Consider a firm whose demand is nelastic in he short run but more elastic inthe long run as consumers' "habits" adjust or as new firms enter the market. Ifthe firm optimizes, it ill initially produce above the point at which marginalcost equals short-run marginal revenue, so that the short-run demand elasticitywill overstate the firm's short-run monopoly power.) Second, the firm's gainand consumers' loss from monopoly power depends on the rate at which demandadjusts, which in urn depends on the firm's entire (optimal) price trajectory.Given that the objective is a prescriptive statistic that can be applied to anti-trust and related problems, one therefore needs a measure that reflects thetrajectory of monopoly power over time, weighted by the firm's revenues (andconsumers' expenditures).4


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-3-In his paper I suggest a simple generalization of the Lerner Index that pro-

vides a eaningful instantaneous measure of monopoly power for markets in hichprice and production are intertemporally determined, and I discuss the aggregationof that measure across time. This generalization is uite straightforward, andsimply involves incorporating any relevant "user costs" -- i.e. the sum of (dis-counted) future costs or benefits that result from current production decisions --in arginal cost. Although the generalization is traightforward, its applicationto specific examples of dynamic markets -- natural resources, learning curveeffects, adjustment costs, and dynamic demand functions -- provides useful insights

into the ways in hich intertemporal constraints on production and price affectmonopoly power, and why quasi-static analyses (e.g. comparisons of short-run andlong-run demand conditions) can be misleading. Much of this paper is hereforedevoted to the series of examples listed above.

This paper ignores the general question of how oligopolistic firms interactwith each other, even though that will often be a major determinant of the actualmonopoly power that prevails in a market. In ffect, I am concerned with themeasurement of potential monopoly power, which may be much larger than actualmonopoly power if irms compete aggressively. If hat makes the scope of thispaper seem limited, remember that measures of potential monopoly power often playan important role in he evaluation of mergers and acquisitions, the assessmentof demages resulting from actual or potential collusive activity, and other aspectsof antitrust. 5

The next section of this paper shows how the Lerner index can be generalizedto provide an instantaneous measure of monopoly power that properly accounts forintertemporal constraints on production and price, and discusses the aggregationof that instantaneous measure across time. The following four sections use thatmeasure to examine how monopoly power is ffected by resource depletion, thelearning curve, costs of adjusting factor inputs, and dynamic adjustment of demand.Finally, I present and discuss a measure of monopsony power applicable to dynamic


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markets. As one would expect, this measure is losely related to our measure ofmonopoly power.

2. A Measure of Monopoly PowerIn his section we begin by putting aside the problem that monopoly power --

i.e. a firm's ability to earn "excess profits" -- only has operational meaningover a time interval, and we compare the price, output, and profit flows for amonopolist 6 to those for a competitive market at a particular time t, ut recog-nizing that these variables are intertemporally determined. After adapting theLerner Index so that it rovides a meaningful instantaneous measure of monopolypower, we discuss the problem of aggregating across time.

A. An Instantaneous MeasureThe Lerner index assumes static profit maximization, so that the firm produces

where marginal revenue equals marginal cost. In a competitive market price willequal marginal cost, and the resulting output is welfare maximizing," i.e. the sumof consumer plus producer surplus is a maximum. That is he basis for measuringmonopoly power in erms of the extent to which price deviates from marginal cost.

Ina dynamic setting, risk-neutral firms maximize the sum of their (expected)discounted profits.7 In a competitive market, this results in he same outputpath as that which maximizes the sum of (expected) discounted consumer plus pro-ducer surplus. This, of course, does not imply that marginal revenue equalsmarginal cost at every instant, nor that price equals marginal cost under competition.An exhaustible resource market is he obvious textbook example, and as explainedearlier, it llustrates the failure of the Lerner index as a easure of monopolypower,

However, the Lerner index can be rescued as an instantaneous measure of mono-poly power by altering it s follows:

(1)L*(t (Pt FMCt)/Pt = 1 - (FMCt/Pt)


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-5-where FMCt is he full marginal social cost at time t, valuated at the monopolyoutput level. By this I ean that first, FMCt includes any (positive or negative)"user costs" that result from the intertemporal nature of the firm's optimizationproblem. Second, those user costs are to be calculated under the assumption thatthe firm is a price taker (i.e. competitive), so that they reflect the change inthe present value of future discounted consumer plus producer surplus resultingfrom an increment in urrent production. Third, as with simple marginal cost inthe standard Lerner index, those user costs are to be evaluated at the monopolyoutput level. With FMCt calculated in his way, 0 L*(t) < 1 or all t, ndL*(t) = 0 in a perfectly competitive market.

The rationale for this measure can be made clear by going back to the exampleof an exhaustible resource market. We again assume that demand is ixed and iso-elastic, and marginal cost MC = O, o that the competitive and monopoly outputpaths are identical. Figure 1 hows the average and marginal revenue curves, andoutput and price at a particular time t. Note that the monopolist output levelis uch that

MR = MC + Xm = Xm (2)where Xm is he user cost to the monopolist of one extra unit of cumulative pro-duction, or equivalently, the value to the monopolist of the marginal in ituunit of reserves. In a competitive industry, on the other hand, the output levelis uch that

P = MC + Xc Xc (3)where Xc is ompetitive user cost (often referred to as "Hotelling rent").8 Inthis case the competitive and monopoly output levels are the same, so that Xc =Xm + MR.

It is mportant to distinguish between the monopoly and competitive usercosts, as they are derived from two different objective functions. The monopolyoutput path is hat which maximizes the sum of discounted profits, and monopoly


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-6-user cost is he reduction in hat sum resulting from one extra unit of cumulativeproduction (i.e. one less unit of in itu reserves). The competitive output pathis hat which maximizes the sum of discounted consumer plus producer surplus (thelatter is ust aggregate profit), with competitive user cost the reduction in hatsum from one extra unit of cumulative production. Competitive user cost exceedsmonopoly user cost because consumer plus producer surplus exceeds the monopolist'sprofit.9 In he case shown in igure 1, he difference between the two usercosts is ust equal to the difference between average and marginal revenue, sothat the output levels are the same.

Since a easure of monopoly power should make a comparison with competitiveconditions, or equivalently (in he absence of externalities), with the socialwelfare maximum, itis ompetitive user cost that should be added to marginalcost in alculating FMC. In he case shown in igure 1, rice is hen equal toFMC, so that L*(t) = 0 for all t, s we would expect.

In eality extraction cost is arely if ver zero, so that in eneral a monop-olist would produce less than a competitive industry initially, but more later(i.e. the monopolist "overconserves").10 We must then be careful when calculatingcompetitive user cost. The value of competitive user cost at time t ill dependon output at t, s well as the path of (expected) future output. When measuringmonopoly power, competitive user cost must be calculated using the monopolist'soutput path, just as marginal cost is easured at the monopoly output level whencalculating the Lerner index in a static model. In his way we obtain the fullmarginal social cost of producing the monopoly output.

This is illustrated in igure 2. Xc ,m is he value of competitive user costat time t evaluated for the monopoly output path {Om(t)}, where time t is omepoint after production has commenced. In he Figure, Xc m is hown smaller thanXcc' competitive user cost evaluated over {QC(t)}, because monopoly output is(initially) smaller, so that monopoly reserves at time t are larger. Monopolyoutput at each instant is t the point where marginal revenue equals marginal


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cost plus monopoly user cost, Xm At that output the "excess" average profitattributable to monopoly power, EP, is rice minus full marginal cost, FMC,where FMC = MC + Xcm The instantaneous degree of monopoly power at time t isthen given by: L*(t) = 1 - [MC + c m(t)]/p(t) (4)

Observe from eqn. (4) nd Figure 2 that the presence of a positive user costreduces monopoly power. In ther words, given a emand curve and some marginalextraction cost function, the firm's monopoly power will be lower than it ouldbe if ts reserves were limited so that its user cost were zero. (Because itstotal volume of sales is ixed, the monopolist is imited to choosing the allo-cation of those sales across time.) Use of the standard Lerner index wouldclearly overstate the firm's true degree of monopoly power. Of course one mightask how important user cost is, nd to what extent one would be misled by applyingthe standard Lerner index. This question is xplored in ection 3 f the paper.

There are other examples of intertemporal pricing and production where usercost is egative, and monopoly power is ncreased. Consider a arket in hichfirms move down a "learning curve," i.e. as they produce, learning by doing re-duces their average and marginal costs. As Spence (1981) has shown, the fullmarginal cost of current production is hen less than current marginal productioncost. The reason is hat an incremental unit of current production reduces futureproduction costs by moving the firm further down the learning curve, so that pro-

duction of the unit brings a benefit (a egative user cost) that partly offsetsits cost.

This is llustrated in igure 3, here marginal cost MCt is onstant withrespect to the instantaneous rate of output Qt' but declines (from an initialvalue of MCO, asymptotically to MC) as cumulative output increases. Thus ifthe industry were competitive and there were no externalities (e.g. resultingfrom the diffusion of experience-based information across firms), price would beless than current marginal cost because of the negative user cost associated


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-8-with learning. Price and marginal cost would both fall over time, with priceasymptotically approaching marginal cost as the industry matured, learningeffects were exhausted, and user cost approached zero.

Similarly, for a onopolist, production will be at the point where MR = MCt+ Am, where Am is he (negative) user cost to the monopolist (i.e. the changein he sum of discounted future profits from an extra unit of cumulative pro-duction today). However, the user cost relevant to calculating the firm's"excess" average profit EP and its instantaneous degree of monopoly power isAcm', he competitive user cost (i.e. the change in he sum of discounted futuresurplus from an extra unit of cumulative production), again evaluated for themonopolist's output path. (Figure 3 applies to a point in ime after productionhas commenced; since the monopoly output is ess than the competitive output,Ac m is reater in agnitude than cc' As Figure 3 shows, because Ac m < 0,the effect of learning is o increase the firm's monopoly power, and the use ofthe standard Lerner index would underestimate that monopoly power. The extentto which the standard Lerner index is iased is xamined in ection 4.

In ddition to examining resource depletion and the learning curve in etail,later sections of this paper also examine the effects of adjustments costs for afirm's factor inputs, and the effects of dynamic demand adjustment. As thosesections show, eqn. (1), with FMCt = MCt + Ac m(t), always provides a correctinstantaneous measure of monopoly power. We now turn to the aggregation of thatinstantaneous measure across time.

B. Aggregation Across TimeAn instantaneous measure of monopoly power is learly insufficient as a

statistic for antitrust and related analyses. A firm's instantaneous degree ofmonopoly power might be high initially, and then fall as consumer demand adjustsor as new firms enter the industry. Clearly a complete assessment of a firm'smonopoly power requires a ethod of aggregating the instantaneous index across time.

Remember that neither L(t) nor the Lerner index L(t) provide information


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so will the firm's monopoly power. This is mportant in he context of antitrustpolicy; the time-aggregated degree of monopoly power might be high initially(at t = 0), so that antitrust action seems warranted, but such action will be oflittle value if it akes considerable time to implement, and Im(t) will fallrapidly over time. Finally, observe that 0 I(t) < 1, nd I (t) 0 only ifL*(T) = 0 for all T t.

The index Im(t) has value as a summary statistic for antitrust and relatedapplications. Furthermore, it rovides a means of identifying those factors thatare important determinants of monopoly power, and quantifying their effects.This is est demonstrated through some examples of markets in hich the inter-temporal aspect of production is mportant.3. Exhaustible Resources

Let us return to the case of an exhaustible resource market. Besides illus-trating how the index of monopoly power given by eqn. (5) s alculated, we wishto examine the way in hich actual or potential monopoly power depends on usercost relative to price, and the extent to which the traditional Lerner index isa biased statistic for real-world resource markets.

The approach is o construct a simple model that can be solved analyticallyfor the competitive and monopoly production rates, and that lets one:relatemonopoly power to user cost in a straightforward way. Such a model is iven bythe following assumptions: (i) he reserve base is ixed,12 (ii) arket demandis soelastic, i.e. q(p) = bp'a, ith r > 1, nd (iii) marginal and average ex-tracion cost are constant with respect to the rate of extraction, but an iso-elastic function of reserves, with an elasticity equal to the inverse of thedemand elasticity, i.e. MC = AC = cR-1/.1 3 To deal with problems of commonaccess and aggregation in he competitive case, we will assume that individualfirms own shares of a unitized resource stock, or equivalently that firms owntheir own reserves, but with identical costs, so that competitive productiondecisions lead to identical output paths for individual firms, and the competi-


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tive equilibrium is ocially optimal.l4Aside from analytical tractability, an important advantage of this model is

that it elps to clarify the connection between monopoly power and user cost.In articular, as shown below, in his model L and L are functions only of n andp, here p is he ratio of competitive user cost to competitive price. Also,observe that price and cost rise asymptotically in his model as the reservebase dwindles, but reserves are never exhausted. When c = 0 we have the specialcase in hich the competitive and monopoly output paths are identical, but as cincreases, the two will diverge.

The competitive price and production paths are given by the solution of:15

max Jf [ qp(v)dv - C(q,R) ertdt (6)q 0 0subject to dR/dt = - q, R(O) = R0 (7)For a monopolist, price and production are given by the solution of:

max fO [ (q)q - C(q,R) ] e-rtdt , (8)q 0

also subject to (7). I have shown elsewhere (1983) that with p(q) = (q/b)-l/ nand C(q,R) = cR 1 /nq, the competitive rate of production is iven by the rule:

qc(R) = b [(n ) Ac + c R = R (9)where A c = Ac(n,c,r,b) is he solution to:

(a-l)An/(n-1 ) cA1 /(l) = [b/(n-l)r]l/(n-) (10)Ti c cAlso, the monopoly rate of production is iven by:

q(R) = b nL-- ) [(- ) A + c ] R mR (11)

where Am = A (,c,r,b) is he solution to:1 6


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(n-l )An/(n-l)+ cAl/(nl) = (n-l) [b/nqr] (-) (12)rI m mthe reader can check that as c + 0,

qc(R) , qm(R) + nrR (13)and that for any given R, qc(R) > qm(R) for c > 0.

Calculating the degree of monopoly power requires an expression for competitive17user cost. As shown in y earlier paper, that user cost is iven by:

Xc = ( ) AcR !14)Recall that this user cost is o be evaluated over the monopoly output path, andtherefore the monopoly path for reserves. The latter is iven by Rm(t) = Re- mtso that Ac m(t) is btained by substituting this into (14).

We can now compute the instantaneous degree of monopoly power. Substituting(14) into eqn. (4) nd noting that the monopoly price is m = (qm/b)I/(mR/b) l/n, we have:

L*(t) = 1 - (%m/b)l/' [c + Ac(n-l)/ ]=1- (b Ac ( )/1 (15)n c + Am(n- l)/J

Observe that in his model, L*(t) is onstant over time, so that Im= L*. Also,the reader can check first, that L* = 0 when c = 0, an d second, that Ac, Am - 0when r + , so that c 0 and L + L, where L is the standard Lerner index,and is given by L = 1 - c(qm/b)1/ q. Observe that the standard index isbiased by an amount

A (n-12 2*LCa (16)c + A (-1)/qm

Note that A -+ 1 as c 0.


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Clearly the bias in he standard Lerner index depends on the relative impor-tance of user cost. To explore this dependence, it is seful to define theparameter p = IXc c/Pc, i.e. the ratio of competitive user cost to competitiveprice. Then 0 < p < 1, ith p = 0 if reserves are infinite (i.e. no resourceconstraint), and p = 1 hen c O. Observe that in his model p and Ac c areboth proportional to R-1 / , so that p is onstant and independent of R. Substi-tuting for Ac,c and Pc

A (n-l)/np c (17)c + Ac(n-l)/n

Numerical solutions of the model can now be utilized to examine the dependenceof monopoly power on user cost, and the extent of the bias in he standard Lernerindex. The model is articularly convenient for this because L , L, nd thereforeA are functions only of p and n.18 In ther words, once p and n have been specified,it is nnecessary to specify c, b, r r. (Given p and n, c is etermined givenb/r.) We therefore want to examine how L , L, nd A vary as p increases from zeroto one. This dependence is hown in able 1 and Figures 4A-4E for = 1.5, 3,5,and 10.19

For many exhaustible resource markets, values of p in he vicinity of .1 o.4 re not unreasonable.20 Observe that for this range of values, the percentagebias in he standard Lerner index becomes quite pronounced, especially when demandis lastic. For example, when r = 5, L is in he range .23 to .44, but L is inthe range of only .15 to .06. This suggests that the intertemporal constraintsimposed by resource depletion can be quantitatively important determinants ofactual or potential monopoly power.


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-14-TABLE 1 - EXHAUSTIBLE RESOURCE

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4. The Learning CurveThe presence of learning by doing will also cause the standard Lerner index

to be a biased measure of monopoly power, but now the bias is in he oppositedirection. As explained earlier, user cost is egative, so that a onopolistproduces at a point where marginal revenue is elow marginal cost.

In pecifying a model of learning by doing, assumptions about the diffusionof learning across firms and the strategic interaction of firms are particularlyimportant. For example, Fudenberg and Tirole (1983) have shown that both thelevel and rate of change of output for oligopolistic firms will depend criticallyon the way in hich they operate strategically. In articular, output is igherif irms follow "closed-loop" strategies (i.e. change their output patterns overtime in esponse to unanticipated changes in heir competitors' outputs) thanwill be the case if irms assume their rivals are pre-committed to fixed outputpaths. Also, in he first case the diffusion of learning across firms willdecrease output, but can increase it in he second case.

At issue, then, is hat to compare the monopolist's output to. I hoose tocompare the monopoly output path to that of a social planner, and it is helatter that I ill refer to as "competitive." The reasons for this choice areas follows. First, as Fudenberg and Tirole have shown, if ost is ot a suffi-ciently convex function of instantaneous output, a competitive (i.e. price-taking)equilibrium will not exist. Second, in n oligopolistic context, the strategicinteractions of firms can be described by a number of reasonable alternativebehavioral modes, and it is ot clear how much rationality and sophistication toassign to firms in eal world markets. (Economists find closed-loop strategiesdifficult if ot impossible to calculate, and there is ittle reason to expectthat firms are any better at calculating them.) Finally, my interest iscomparison of the monopoly output path with the socially optimal path. Dependingon the nature of firms' strategic interactions, the monopoly equilibrium may bebetter than the "second-best" oligopoly equilibrium,21 but the issue here is ow


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monopoly compares to a first-best equilibrium, and most important, how that com-parison depends on the intertemporal constraints imposed by the learning curve.

I therefore specify a model that is xactly analogous to the exhaustibleresource model developed in he last section. Demand is soelastic, and marginalcost is n isoelastic function of cumulative production, so that the competitiveand monopoly production rates are linear functions of cumulative production, andL and L are both functions of only n and p. This helps to illustrate the depen-dence of monopoly power on user cost, and the close relationship between theeffects of resource depletion and learning by doing.

As before, demand is iven by q = bp . Marginal and average cost are equaland given by MC = AC = c(a + x)1/r where a > 0 and x is umulative production:

tx(t) = ; q(T)dT (18)0For simplicity, we write MC = cxl/, and set x(O) = 1. As shown in he Appendix,this model has a solution only if c, /r, and rn satisfy the following constraint:

c >)( -( (19)This constraint implies an upper limit on the magnitude of user cost relative toprice. If it is ot met, the present discounted value of the flow of net surplusis nfinite.

In he Appendix it is hown that the competitive rate of production is ivenby the rule: 1-n

qc(x) = b [(n-) Bc + c x cX (20)where Bc = Bc(n,c,r,b) is he solution to:

(-n-l Bn/(nl) + cBl/(nfl) = [b/(n-l)r]l/ (21)(21)and competitive user cost is iven by:

Ac ( cx-l/n (22)


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Also, the monopoly rate of production is iven by:

q (x)= b l( )[ -l Bm x = X (23)

where Bm = B (n,c,r,b) is he solution to:_( ) B n/(- ) + cBm l/(-l) = (ril)[b/n r]l/(n-1 ) (24)

The close connection between this model and the previous one for an exhaustibleresource should be clear. The monopoly and competitive prices of the exhaustibleresource rise exponentially as both marginal extraction cost and user cost rise,and production asymptotically approaches zero, while in his example monopoly andcompetitive production rise exponentially as production cost, user cost, and priceall approach zero.

Using the solutions above for production and user cost, our instantaneousmeasure of monopoly power can be written as:

L*(t) = 1 (n-l ) c (25)c - Bm(n-l)/n

Again, L (t) is constant over time, so that Im L . Also, L = 1 - (em/b)l/c,so the bias in he standard Lerner index is

-B (n-1)2/n2L L* = c (26)c - Bm(Ti-1 )/ni.e. the standard Lerner index underestimates the true degree of monopoly power.As with an exhaustible resource, as r , -+ 0 and L - L.

We will again use the index p = Xc c['/pc to measure the relative importanceof user cost, and thus the relative importance of the learning curve effect. Inthis model, p is iven by:


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-18-

Bc(n-l)/np= (27)c- c(n-l)/n

The constraint of eqn. (19) implies a corresponding constraint on p; the modelhas a solution only ifp 1/(-1).

Once again we would like to examine how the degree of monopoly power L andthe bias in he standard Lerner index vary with n nd p. This dependence isillustrated numerically in able 2 and Figure 5. As shown, the standard Lernerindex underestimates the true degree of monopoly power, and the bias can be quitesignificant, especially for larger values of n. For example, when n is3 r 5and p is 20, the bias exceeds .10 in agnitude. 22

Just as a positive user cost (an exhaustible resource) reduces a firm'smonopoly power, a negative user cost (a earning curve) increases it. The reasonis hat Ic > IXml, so the incentives associated with learning push a competitiveindustry further down the market demand curve than they push a onopolist downits marginal revenue curve. Put another way, since the monopolist produces lessanyway over the long run, he receives less benefit (in erms of reduced futurecosts) from increasing current output. Learning therefore leads a monopolistto initially increase output less than it oes a competitive industry, andincreases the spread between the monopoly and competitive prices. This is ustthe opposite of what happens with an exhaustible resource. There a onopolistsees a lower cost from the depletion of his reserves, and so has less incentiveto conserve relative to a competitive industry; this reduces the spread betweenthe monopoly and competitive prices.


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-19-

TABLE 2 - LEARNING CURVEA. = 1.5 E. n = 3.0

L.. *

(.) 7:? :

'. 4]'I66C) 4"V.... 4.1C). 443':7C. 4 ,22) 50C)0()

U. 5169(). 5329

...

C) C.') .,

( '.2994..: , I 40.. 29950). 2994

C. rl = 5.01. *

0.2(:)(.)5: . .226). .562). 2848). 3129

L

C) . 19970. 189 )). 118:C).775

C). 1 7 =':0. 1 7 4.9

o d &1 t a rh oC). :),) . t I2

0. IC(:C). 15(. 2)0.250.30(. 350.4(':0. 50.50

del t a

-.. ()898... 3115.. . 1. 1 '-- . 1812'-. 2174-. 2335

0010

30

455560707580905

C)0C)()

035

50

7)

909500

0) . 6669). 679'7() . 6/: :1., '7 .6

Ui. 76,1(,. 7-721Q. 1 E3 .

1). 79i3)0. 7964.) 68'). 7.:40. 811.7081 6:3(d. 2C)C). 82890. 8365(), 84()0. 8340) . 84670. 8498(,). 3528(). 557C). 3585

0. 8687:). 871..87.3'). 8755(),..8776(). 8796

L0. 666 ,C,. 66-1) . b C:1:. 6 6 I

ic).. 6546

0. 6477C). 46.65:),,) 4 ICU . 64 1:,..5tJ(). 642(J. 4 1.7

, 64140.6/.1.2:). 64iC)

C) . 61('170. 64C52(). 64)2

C..) h(2 4). 6 ._; 9 * ;). 639

( . 6: .91C). 63940. 6389.C). 6289C , 8.30.6 (. '0. 6 80. 6 3 5

--' () ) ,. ;..... ,~) (,,)..... j 1'- .

.C)573

, .. ; 7

-. 1642

- 1695.. * 9 '"I.-.';8 (1 t(l--1. 1962

-.. 1 464J-". 184

-194

-- 13 9

-. 2 2.4r- ..366. ,,408-. 240:)3

rho

C). 000.050.100. 15C). 20

. 25

de]. t:._)3( )()1--. 0: 3;86

-. 0744. -,,)7.-- . :, :


22/45

-20-5. Costs of Adjustment

A firm's long-run cost structure differs from that in he short run becauseit akes time to alter the capital stock and change the firm's production capacity.As Lucas (1967) and Gould (1968) have shown, one way to capture this is y assumingthere are convex costs of adjustment associated with changes in he capital stock(and/or changes in ther factor input levels). When adjustment costs are present,a firm will experience an (internal) capital gain or loss when it djusts to anew long-run equilibrium position, and those capital gains (which occur as capacityis educed) or losses (occurring as capacity is ncreased) are part of full marginalcost. As shown below, this reduces monopoly power during periods of industry expan-sion, and increases it uring periods of contraction.

For simplicity, assume output is a function only f capital K, .e. q = F(K),with F'(K) > 0 and F"(K) < O, o that there are diseconomies of scale. Thecapital stock is ssumed to be "quasi-fixed," so that the purchase and installationof "usable" capital at a rate I nvolves a cost vI + C(I), where v is he purchaseprice of a unit of capital, and C(I) is he full adjustment cost, with C(O) = 0,C'(I) >(( 0,i.e. it is ore costly to increase capacity quickly than slowly.23 Firms areassumed to maximize:

Max fI [p(q)q - vI - C(I)e rtdt (28)I(t) Osubject to K = I K (29)where 6 is he depreciation rate, and a dot denotes a time derivative, i.e.K = dK/dt. Note that competitive firms perform this maximization with p(q) =ptaken as given.

It is asily shown that the optimal level of investment satisfies:

I=C ){(r + 6)[ v + C'(I) - MR FK} (30)C 711-) K~~~~~~~~~~~~(0where MR is arginal revenue. (In competitive market, MR = AR = p.) The


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-21-

behavior of investment and the capital stock in ompetitive and monopolisticmarkets is haracterized by the phase diagram of Figure 6. In hat figure, Kmmand K are the steady-state equilibrium capital stocks in onopolistic andccompetitive markets. Because there are diseconomies of scale, in oth marketsI (>) if (t) ) K .

In steady-state equilibrium,MR F (r+6 [ v + C'(I) ] (31)

where the right-hand side of (31) is he cost of a arginal unit of capital.The marginal cost of an additional unit of output is hen

MC = (r 6) v + C'(I) ]/FK (32)In quilibrium, the use of this "direct" marginal cost in he standard Lernerindex would give an unbiased measure of the degree of monopoly power. Furthermore,this "direct" marginal cost can itself be measured -- it is imply the amortizedcapital outlay required to increase production capacity by one unit.

In isequilibrium, however, full marginal cost is ot equal to direct mar-ginal cost. Equating full marginal cost with marginal revenue, observe fromeqn. (30) that

FMC = MC - C"(I)I/FK (33)The value to the firm of the marginal unit of capital is ts total purchase andinstallation cost, v + C'(I), so that C"(I)I is he rate of capital gain on theunit, and C"(I)I/FK is he corresponding capital gain in erms of a arginal unitof production capacity. If a firm is growing so that K(t) < K , I < 0 and FMC> MC.The reason is that as K(t) - K , the marginal profit rate is falling, so that thevalue to the firm of a arginal unit of capital is alling. This capital lossraises the full marginal cost of additional production capacity. Conversely,suppose the firms in a competitive market cartelize, agreeing to reduce theiraggregate production capacity to a monopoly level (path ABC in igure 6). Eachincremental reduction in apacity raises the marginal profit rate and bestows a


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-22-

capital gain; I > 0 and FMC < MC.In eriods of adjustment, the standard Lerner index is herefore a biased

measure of monopoly power. The actual degree of monopoly power will depend onwhether output is rowing or contracting, and cannot be determined simply froman elasticity of demand. The bias is ositive during periods of industry expan-sion, and negative when industry output is ontracting. The latter case is mpor-tant, and the effect is ften ignored in nalyses of the potential monopoly powerfrom collusion. If a cartel forms in n industry that had been competitive, itsmonopoly power will exceed the value that would be inferred from the marketdemand curve (even if hat demand curve is tatic). The bias is argest in heearly periods (I O0 s K K), but depending on the discount rate and the sizeof marginal adjustment costs C'(I) relative to v,it ay be sufficient to signi-ficantly affect the time-aggregated index of monopoly power, I. 24

6. Dynamic Demand FunctionsIn ost markets demand responds dynamically to price changes. From the point

of view of a firm or group of firms with monopoly power, the response of demandcan occur as consumers adjust their spending patterns, or as other (competitive)firms expand their production capacity. If onopoly power is o be exercisedrationally, that dynamic response of demand must be taken into account.

The fact that demand is ynamic will not by itself cause the standard LernerIndex to be biased as an instantaneous measure of monopoly power. The reason isthat the full marginal social cost of production is qual to direct marginal cost,i.e. competitive producers will produce so that price is qual to marginal cost atevery instant (although price will be changing over time). As long as the optimalmonopoly price is sed to compute the Lerner Index (at each instant), it ill notbe biased.

A problem arises, however, when the short-run demand curve (or a short-runelasticity) is sed to infer a short-run degree of monopoly power. Aside fromthe obvious problem of time aggregation, the short-run demand curve (taken by


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-23-

itself) can be misleading as an indicator of a monopolist's short-run monopolypower. To see why, suppose the monopolist's demand curve is ore elastic inthe long run than the short run. Then itis ptimal for the monopolist toinitially set output above the point where marginal cost equals short-run marginalrevenue; doing so creates a benefit by retarding the response of demand and theadjustment to long-run equilibrium. Just the opposite is he case if he monop-olist's demand curve is ess elastic in he long run.

The following simple model illustrates how the short-run demand curve canoverstate a onopolist's short-run degree of monopoly power. Suppose a group offirms cartelize and gain monopoly power, but if hey increase price, the outputof a set of "competitive fringe" firms will gradually increase, so that carteldemand is ore elastic in he long run that the short. Then the cartel's demandcan be written as:

q(t) = a - alP(t) - u(t) (34)where u(t) is ompetitive supply, and itself depends on price as follows: 25

= blP(t) - b2u(t) (35)Here b2 determines the speed at which competitive production responds to price,and the cartel's demand adjusts over time.

Given some initial (equilibrium) price p and quantity q, and letting C(q)be the cartel's (aggregate) cost function, the cartel sets output to maximize:

o00max [ (t)q(t) - C(q) e rtdt (36)q 0subject to (35). It is traightforward to show that the cartel's optimal outputtrajectory must satisfy:

2[ a + C"(q)] q =- (abla + a) + 2aq + SC'(q) + a(28 - r)u (37)where ac /al and = b2 + r bl/a1. Then q ) 0 if q(t) >(


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-24-

but instead reduces it nly part of the way, and then slowly reduces it he restof the way. The reason is hat for the cartel there is a negative user costassociated with current production; an incremental unit of production retards ex-pansion by the competitive fringe, and tnereby increases future potential profits. 26

This is llustrated by a phase diagram in igure 7, nd supply and demandcurves in igure 8, or the case C(q) = cq. In hose figures, the market isinitially competitive, and q is he total output of the colluding firms. Oncethe cartel forms, its optimal output drops to ql, and then gradually falls tothe long-run equilibrium level q as competitive output expands', and price fallsfrom P1 to P* path ABC in igures 7 and 8).27 Until long-run equilibrium isreached, the cartel's output is bove the point where marginal cost equals short-run marginal revenue (again, because of the negative user cost associated withcurrent production). Clearly the cartel's short-run average revenue curve takenby itself overstates the cartel's short-run degree of monopoly power.

The size of the cartel's degree of monopoly power, Im, depends on how fastthe instantaneous index L*(t) declines over time (as q(t) -* q*), and the timepattern of expenditure. An estimate o Im therefore requires the calculationof the cartel's entire optimal production trajectory. Even if uch a calculationis ot feasible, an estimate of Im based on rough "guessestimate" of q(t) maystill be preferable to use of short-run and long-run demand curves.

In ome markets, short-run demand is ore elastic than long-run demand.This is he case when there is a "stock adjustment effect" -- for example,the product in uestion is a durable good (say copper or aluminum) with a sourceof secondary (or "scrap") supply. In his case a onopolist's short-run demandcurve understates its true short-run degree of monopoly power. Now the monopolist'sshort-run output is elow the point where marginal cost is qual to short-runmarginal revenue, because there is a positive user cost associated with currentproduction (an incremental unit of production retards the reduction of aninitially large source of secondary supply from competitive fringe firms). The

I


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-25-

implications of this might be easiest to see in he context of the 1945 Alcoacase; a defense of Alcoa based on the argument that its monopoly power waslimited in t least the short run is lawed because it gnores the fact that thedynamic adjustment of demand gives the firm an incentive to reduce short-runproduction.28

7. A Measure of Monopsony PowerMonopsony power refers to the ability to purchase at a price below marginal

value, or equivalently (since utility maximization implies purchasing up to thepoint that marginal value is qual to marginal expenditure), at a price belowmarginal expenditure. A static index of monopsony power exactly analogous to theLerner index is herefore:

M = 1 - (P/ME) = 1 - (P/MV) (38)M is imply the percentage difference between marginal value and price, and instatic market is ounded by zero and one. Just as the Lerner Index compares themonopoly price to the marginal social cost of producing the monopoly output level,this index compares the marginal social value of the monopsony level of sales withthe monopsony price.

In a dynamic market, however, M suffers from the same deficiences as theLerner index. First, when price and quantity are determined intertemporally, abuyer will not necessarily equate current marginal value with current marginalexpenditure, so that M may be biased evan as an instantaneous measure. For example,ifa onopsonist's current consumption of a good results in a sufficiently largefuture flow of value (in ddition to current value), the current price can exceedcurrent marginal value, and even though less is urchased then ina competitivemarket, M ill be negative. Second, the value of M at an instant of time sayslittle about the potential impact of monopsony. As with monopoly power, a time-aggregated index is eeded ifit is o be useful as a prescriptive statistic forantitrust and regulatory policy.


28/45


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-27-

shadow value is he increase in hat sum resulting from one extra unit of cumu-lative sales. The competitive sales path is hat which maximizes the sum ofthe discounted flow of total surplus (the area between the demand and supplycurves in a static market), and the competitive shadow value is he increase inthat sum from an extra unit of cumulative sales. The competitive shadow valueis arger in agnitude than the monopsony shadow value because total surplusexceeds the monopsonist's net value.

Since a easure of monopsony power should make a comparison with competitiveconditions, it is he competitive shadow value that should be used in alculatingfull marginal value. Furthermore, that shadow value must be calculated for themonopsony sales path, just as marginal value is easured at the monopsony saleslevel when calculating the index M in a static model. This yields the full marginalsocial value of sales at the monopsony level, i.e. FMVt = MVt + X ,m(t), and theinstantaneous degree of monopsony power is:

M*(t) = 1 - Pm(t)/ [MVt + Xc m(t)] (40)In he exhaustible resource example illustrated in igure 9, X, (t) < O, o thatthe static index M would overestimate the degree of monopsony power. The exhaust-ibility of the firm's reserves reduces its monopsony power in he labor market,just as it educes any monopoly power in he output market.

As a second example, consider a firm that buys a raw material ("steel"), andwhose costs fall as it oves down a learning curve. In his case, each unit ofsteel the firm buys provides current marginal value, but also increases the futureflow of value to the firm by accelerating its movement down the learning curve.Now there is a positive shadow value c m(t) that must be included in alculatingthe firm's instantaneous degree of monopsony power in he steel market; the staticindex M will underestimate that monopsony power.

Once M*(t) has been calculated, it an be aggregated over time in a annerdirectly analogous to eqn. (5) or monopoly power. The expenditures P(t)Q(t) usedas the weights in qn. (5) epresent the gross income flow to the monopolist. The


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-28-

analogous gross value flow to the monopsonist is FMV(t)Q(t). Itis his, and notexpenditures P(t)Q(t) that should be used to weight M (t) hen aggregating overtime. To see this, observe that if xpenditures were used, the weights would bethe lowest in eriods when the degree of monopsony power M (t) s he highest(and would approach zero if M (t) pproached one).29 Then weighting M*(t) byFMV(t)Q(t) and discounting, we obtain the following index of monopsony power:

cof P(T)Q(T)e-r(T-t )dr

I (t) 1 t (41)f FMV(T)Q(T)e r(Tt)dTt

Observe that 0 Is(t) < 1, nd Is(t) = 0 only if M (T)= 0 for all T t.Is(t) isa summary statistics that measures the monopsony power of a buyer lockinginto the future at a particular point in ime.

8. Concluding RemarksThis paper has shown how static measures of monopoly and monopsony power can

be inadequate and possibly misleading when applied to dynamic markets. Suchstatic measures include the standard Lerner index, but also short-run and long-runelasticities of demand. We have suggested alternative measures that properlyaccount for intertemporal constraints, and would be applicable as summary statis-tics in ntitrust and related regulatory problems. Also, the application ofthese measures to a number of examples has shown how a firm's (or cartel's) degreeof monopoly power is ffected by various forms of intertemporal constraints,including those associated with resource depletion, learning by doing, costs ofadjustment, and dynamic adjustment of demand.

We have not attempted to suggest a statistical methodology for estimating thecomponents of these measures of monopoly and monopsony power, and the estimation

problems can indeed be formidable. For example, the estimation of user costs ina competitive market can be difficult enough, and here one needs competitive user


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-29-

cost evaluated over the monopolist's output path, a quantity that is everdirectly observable.

On the other hand, even though user costs and shadow values may be difficult

to estimate, we have seen that ignoring them altogether and simply inferring thedegree of monopoly power from a short- or long-run demand curve can be seriouslymisleading. In ome cases it ay be preferable to assess monopoly power -- andthe resulting damages to consumers -- by constructing dynamic market models tosimulate competitive price and output trajectories. Alternatively, some methodsdo exist for at least roughly estimating the user costs.and shadow values dis-

cussed in his paper and needed to calculate Im and IS.30 The use of roughestimates may be better than no estimates, and is ertainly better than simplyignoring the bias in a static index.


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APPENDIX

Here we derive eqns. (2G)-(24). The competitve production rule is hatwhich maximizes:

max J p(v)dv - C(q,x) rt t (A.1)q 0 0 Jsubject to x = q , x(O) = 1 (A.2)with p(v) = bl/n 1 n and C(q,x) = cqx-l / . The solution is obtained via dynamicprogramming. The value function V(x) satisfies the following fundamental equationof optimality:

rV = max {f p(v)dv - C(q,x) + qV } (A.3)q 0Substituting for C(q,x) and maximizing:

- V = p(q) - cxl /n (A.4)

so that production satisfies:q*(x) = p- [ Vx + cx-l/] = b[-Vx + cx l/T]- (A.5)

Now substitute (A.5) for q in qn. (A.3) and rearrange:rV - [-V l cx l-/ (A.6)

(n-l) xIt is asily seen that this equation has the solution

V(x) = B x( l )/n (A.7)where Bc is a constant satisfying eqn. (21). Also Vx [(-l)/l]Bcx-1 n is hemarginal value of an incremental unit of cumulative production, so that competitiveuser cost Ac = - V, as in eqn. (22). Finally, substitute V into (A.5) to obtaineqn. (2) or qc(x).


33/45

-31-* 8 tSince q (x) Ocx, where ec is a constant, x(t) : . Then (A.1) can be

written as:

max f [ (_ T b e - c (-)/e-rtdtq 0 c c dt

- ce ] e[Y(nl )/ - r]tdtC

This integral becomes infinite ifc < (b/e ) n/ (n-1) andthat eqn. (19) must hold for convergence.

The monopolist chooses {qm(t)} to maximize

(A.8)

ec > nr/(n-1), so

Comax [ p(q)q - C(q,x) ] -rtdtq 0 (A.9)

again subject to (A.2). Equations (23) and (24) are obtained by going throughthe same steps as in he competitive solution above.

= o0 [ ( )bl/n (-)/


34/45

-32-FOOTNOTES

1. For example, Landes and Posner (1981) argue that "the Lerner index provides aprecise economic definition of market power." Landes and Posner, and others,refer to "marker power," with "monopoly power" meaning "a igh degree of marketpower." In his paper I se the words "monopoly power" to refer to the abilityto profitably raise price above marginal cost, "monopsony power" to refer to theability to buy at a price below marginal value, and "market power" to mean eithermonopoly or monopsony power.

2. Determining the elasticity of demand for an individual firm in n oligopolisticmarket is o easy matter, even in a static context. Under Cournot assumptions,that elasticity can be related to indices of concentration. See, for example,Encaoua and Jacquemin (1980) and Schmalensee (1982b).

3. See Stiglitz (1976).4. Schmalensee (1982a) discusses the problems associated with aggregating the flow of

social cost of monopoly power across time in ynamic markets. This social cost isusually taken to be the traditional "deadweight loss" triangle (see, e.g., Harberger(1954) and Landes and Posner (1981)), and perhaps in ddition the resources spentto obtain the monopoly power (see Posner (1975)). In ost antitrust applications,the total loss to consumers is he more relevant flow.

5. As Landes and Posner (1981) point out, a finding of successful or attempted monop-olization in iolation of the Sherman Act requires a demonstration that thedefendant has significant monopoly power.

6. In hat follows I take the firm's demand function as given. For simplicity, thefirm can be viewed as a simple monopolist, and I ill refer to its output as the"monopoly output," as opposed to the output that would prevail in a competitivemarket or socially planned economy.

7. This is hat they do, that is, f anagers serve the stockholders' interests.8. In igure 1, X refers to the competitive user cost evaluated at the competitivec,c

output level, and Xc ,m efers to the competitive user cost evaluated at themonopoly output level. The two output levels are the same, so Xc, c = Acm A=


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-33-

his particular case competitive profit (i.e. producer surplus) and the monop-olist's profit are both equal to competitive user cost (i.e. the value of themarginal in itu unit in a competitive market) times the production level.

Stiglitz (1976). This need not always be the case. As Lewis, Matthews, andBurness (1979) have shown, if he elasticity of demand increases with consumption,or if here are quasi-fixed production costs (e.g. leasing fees), the monopolyproduction rate can initially exceed that for a competitive market.Note that this can be written as

Im(t) [f L*(T)PQer(Tt)dT]/[f pQer(T-t dTt ti.e. it is a eighted discounted sum over time of the instantaneous index L,where the weights are expenditures, PQ.That is, e ignore the process of exploration and reserve accumulation. This isactually not a very restrictive assumption, if y "reserves" we mean the potentialresource base (proved reserves plus potentially discoverable reserves), and if weinclude the cost of exploration and discovery as part of production cost.The assumption of an isoelastic marginal cost function is idely accepted as afirst approximation in etroleum engineering, and leads to the well-known"exponential decline curve." Constraining the elasticity to be 1/n may seemartificial, but it is ecessary for an analytical solution, and it oes notcompromise the qualitative realism of the model. Also, note that as n X,MC + c, a constant.In ome resource markets property rights are poorly defined or maintained, so thatthe common access problem arises. In uch cases the competitive equilibrium dis-cussed below should be viewed as an "ideal" and guideline for regulatory policy,and as a comparison to the behavior of a onopolist.Since our assumptions allow us to ignore the common access problems, this isequivalent to each individual producer choosing his production rate qi(t) to

maximieq - C(iR)] rtmaximize [ q - C(qiR)]e dt, where p is aken as exogenous in he maximization.0


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-34-ktl k. Equations (10) and (12) are of the form F(z) = z + alz a2 = O, al, a2, k > 0,

which has one positive solution.Competitive user cost is he marginal social value of an incremental unit ofreserves, i.e. c(t) = VS(t)/3R(t), where Vs(t) is te present value of the flowof consumer plus producer surplus (from time t n) accruing from the extractionand consumption of reserves R(t). In y earlier paper (1983) I erived qc usingdynamic programming, so that Vs(t) is btained explicitly.To see this, first re-write eqn. (17) as:

c : (' ) (1)A c (i)and substitute this for c in qn. (15) for L

1L*= 1- (2 ) [ - (ii)n ( _ p) pAm/A c*k *Thus L = L (n,P,A /Ac). Now. to show that Am/A c is a function only of and p,

divide eqn. (10) by eqn. (12), substitute (i) or c, nd re-arrange:A/(o-1) A/ (n-1)P(Km)+ ( )) - 1) (iii)c c

This equation has a single positive solution (A Acj = p(n,p). The reader canshow that L = L(n,p) as well.

. In hese calculations, r = .05 and b = 1, nd c varies accordingly.

. Pindyck (1978) has shown that if he world oil market were competitive, the ratioof user cost to price might exceed .4 if he real discount rate were .05, whilefor the world copper market this ratio is in he range of .2 o .3. In is studyof the nickel market, Stollery (1983) finds this ratio to be about .1 o .2.Fudenberg and Tirole shows that the oligopolistic equilibrium can be improved bytaxing output in arly periods and subsidizing it in ater ones.There have been a number of studies that estimate the rate at which cost declinesas cumulative production increases; one of the earliest is y Hirsch (1952), andSahal (1981) provides a more recent survey. Those studies do not, however,


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-35-

provide estilmates of user cost. To determine user cost one would also needinformation about the demand curves for the products in uestion.For a general discussion of adjustment costs and their effects, see Nickell (1978).As in ickell, I ssume adjustment costs are a function of gross investment, sothat the firm has some positive cost even when it is ot expanding its capital stock.For empirical evidence on the size of marginal adjustment costs for U.S. manufac-turing as a whole, see Pindyck and Rotemberg (1983).Alternatively, if emand adjusts over time because of consumers' "habit formation,"we would replace eqns. (34) and (35) with:

q(t) = a - alp(t) + u(t) (34')and u = b0 - blP(t) - b2u(t) (34')Here u(t) is he "habit" component of demand (or alternatively the component ofdemand dependent on a stock of durables that can be adjusted only slowly). Notethat the slope of the short-run demand curve is al, and the slope of the long-runcurve is - (al + bl/b 2).No such user cost would exist in a competitive market, however, because no indi-vidual firm can affect the rate of capacity expansion of other firms. Thereforethe Lerner index is ot biased as an instantaneous measure of monopoly power -- ifthe correct monopoly price is sed in ts calculation.To see that p < 0 as q(t) + q , combine eqns. (34) and (35):

p(t) = [ + ab 2 - b2q / (b + alb 2)Observe that u > O, so that P > .Economic analyses of the Alcoa case tend to focus on long-run monopoly power.See Gaskins (1974) and Swan (1980).Alternatively, if he net value flow (FMV - P)Q were used, the weights would belowest (zero) in eriods where M (t) was lowest (zero).See footnotes 20 and 22.


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-36-REFERENCcS

Encaoua, David, and Alexis Jacquemin, "Degree of Monopoly, Indices of Concentration,and Threat of Entry," International Economic Review, February 1980, 21,pp. 7-105.

Fudenberg, Drew, and Jean Tirole, "Learning-by-Doing and Market Performance," Bell

Journal of Economics, Autumn 1983, 14, Fp. 522-530.Gaskins, Darius W., Jr., "Alcoa Revisited: The Welfare Implications of a Secondhand

Market," Journal of Economic Theory, March 1974, 7, p. 254-271.Gould, John P., "Adjustment Costs in he Theory of Investment of the Firm," Review of

Economic Studies, January 1968, 35, pp. 47-55.Harberger, Arnold C., "Monopoly and Resource Allocation," American Econom-'c Review,

May 1954, 44, pp. 77-87.Hirsch, Werner Z., "Manufacturing Progress Functions," Review of Economics and

Statistics, May 1952, pp. 143-155.Landes, William M., and Richard A. osner, "rket Power in ntitrust Cases,"

Harvard Law Review, March 1981, 94, pp. 937-996.Lerner, Abba P., "The Concept of Monopoly and the Measurement of Monopoly Power,"

Review of Economic Studies, June 974, 1, p. 157-175.Lewis, Tracy R., Steven A. atthews, and H. tuart Burness, "Monopoly and the Rate

of Extraction of Exhaustible Resources: Note," American Economic Review,March 1979, 69, pp. 227-230.

Lucas, Robert E., "Adjustment Costs and the Theory of Supply," Journal of PoliticalEconomy, August 1967, 75, pp. 321-334.

McCray, Arthur W., Petroleum Evaluations and Economic Decisions, Prentice-Hall, 1975.Nickell, Stephen J., The Investment Decisions of Firms, Cambridge University Press,1978.Pindyck, Robert S., "Gains to Producers from the Cartelization of Exhaustible Resource

Markets," Review of Economics and Statistics, May 1978, 60, pp. 238-251.Pindyck, Robert S., "Competitive and Monopoly Resource Production with Stochastic

Reserves," MIT Energy Laboratory Working Paper No. O19!P, July 1983.Pindyck, Robert S., and Julio J. Rotemberg, "Dynamic Factor Demands Under Rational

Expectations," Scandinavian Journal of Econr,omics, July 1983, 85, pp. 223-238.


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-37-

Posner, Richard A., "The Social Costs of Monopoly and Regulation," Journal ofPolitical Economy, August 1975, 83, pp. 807-82/.

Sahal, Devendra, Patterns of Technological Innovation, Addison-Wesley, 1981.Schmalensee, Richard, "Another Look at Market Power," Harvard Law Review, June 1982a,

95, pp. 1789-1816.Schmalensee, Richard, "The New Organization and the Economic Analysis of Modern

Markets," in W. ildenbrand, ed., Advances in conomic Theory, CambridgeUniversity Press, 1982b.

Spence, A. ichael, "The Learning Curve and Competition," Bell Journal of Economics,Spring 1981, 12, pp. 49-70.

Stiglitz, Joseph E., "Monopoly and the Rate of Extraction of Exhaustible Resources,"American Economic Review, September 1976, 66, pp. 655-661.

Stollery, Kenneth R., "Mineral Depletion with Cost as the Extraction Limit: AModel Applied to the Behavior of Prices in he Nickel Industry," Journalof Environmental Economics and Management, June 1983, 10, pp. 151-165.

Swan, Peter L., "Alcoa: The Influence of Recycling on Monopoly Power," Journal ofPolitical Economy, February 1980, 88, pp. 76-99.

111II -a-li------1_-1^_11_


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P

pm pC tt t

Qt

FIGURE I - EXHAUSTIBLEMC=O

p

mPt

PtC

Q. Qf

Q

RESOURCE,

MC + Xc,mI -~ ~ c~

Q

2- EXHAUSTIBLEMC >O

RESOURCE,IGURE


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Qmt

Xm[ c,c+Xc,m

C

FIGURE 3- LEARNING CURVE

c.:0.d

0.6

0.5

0.4

0.30.20.10

0.0 0. f 0.2 0.3 0.4 0.& 0.6 0.7 0.8 0.0 .0RUa

FIGURE 4A - ELASTICITY OF DEMAND = 1.5

P

pt

ptC

0.1 C.2 0.3 0.4 0.6 0.6 0.7 0. 0.9 f.0RHO

p

l

FIGURE 4 - ELASTICITY OF DEMAN = 3.0


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0.1 0.3 0.4 0. 0.6 0.7 0.5 0.8 1.0

4C - ELASTICITY OF DEMAND = 5.0

0.9

40.4 -0.3,O./

0.2 1Or_- U I , , , , I IX Ii tT -- ,- - ,I - I

0.0 0.1 0 0.3 0.4 O. 0.6 0.7 0. O.8 .:RIO

FIGURE 4 - ELASTICITY OF DEMAND = 10.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0RHOFIGURE 4E - EXHAUSTIBLE RESOURCE: BIAS (L- L*)

AS FUNCTION OF RHO

I.J

1:00.90.80.7

* 0.61 0.5- 0.4

0.30.20.1

C

-----ra I. I_____


43/45

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0RHOFIGURE 5 -

I

LEARNINGFUNCTION

CURVEOF RHO BIAS (L-

FIGURE 6-COSTS

U-0.02-0.04-0.06-0.08

* -0.1-0.12-0.16-0.180.

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44/45

u-N

SE' DIAGRAM FOR CARTELCOMPETITIVE FRINGE

Cq

MRLR

FIGURE 8 - CARTELFRINGE AND COMPETITIVE

q

qo

u

FIGURE 7 - PHAAND

p

Pi

_11_2 _ ___


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N I

WC (t)wm(t)

.A r

Lm(t) Lc(t

FIGURE 9- RESOURCEMONOPSONYMARKET

PRODUCERPOWER IN

WITHLABOR

t

)

the measurement of monopoly power in dynamic markets

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