- Email: [email protected]
Durable goods monopoly, learning by doing and the Coase conjecture
European Economic Review 36 ( I992) I 57-Z 77. tiorrh-Hoknd
_Norwgian Center firr Research irt Organizcution ad
Management,
N-5015 Bergen;, iVorv;c~
Received December 1989, final verson received March 2991
The Coaae conjecture states that if a durable-goods monopohst can make oks to sell arbitrariiy frcqtientiy, then in equihbrium she will charge the competitive price, and the market will be saturated qtiickly. This conjecture is shown to be refuted when production of the durable good is subject to learning by doing: For the linear case it is shown that there exist a stationarystrategy perfect equilibrium in which the monopolist charges competitive prices, but the market is saturated slowlg. An interesting analogy between bargaining and durable-goods monopoly with learning by doing is also pointed out.
In his seminal article on durable-goods monopoly, Cease (1972) confined his discussion to production technologies characterized by constant or decreasing returns to scale. For the constant returns case, he conjectured that if a durable-goods monopolist can make offers to sell arbitrarily frequently, then in equilibrium she will charge the competitive price and the market will be saturated quickly. [‘With complete durability, the price becomes independent of the number of suppliers and is thus always equal to the competitive ith constant marginal costs, production would price’, Cease ( 1972, p. 144). take place in the first perio and would then cease’, Coast ( 1972, p. 147).] With some stationarity restrictions on buyer s’ strategies, Gul et al. (1986) hereafter GS W - proved this conjecture in a framework of a discrete time, infinite horizon, extensive f0rm game with a continuum 0f buyers. For a reduction technology exhibiting increasing short run marginal costs, Kahn ( 1986) identified an equilibrium for which as the time between offers shrinks, “Parts of this work were done during an enjoyable visit at the IDepartment of Economics and the Center for Economic Policy Research at Stanford University. I am grateful to Bob Wilson for comments on an earher version of this article, to Paul Davki for extensive discussions on the topics with which this paper is concerned, anyr to two anonymous referees [or constructive criticism. A.ny errors are solely mint. Ffna~5l srr;~port fro25i the Norwegian Research CounciI (NAVF) is gratefully acknowiedged.
158
T.E. Olsen, Durahlc goods monopoly
and the Coase conjecture
the monopolist’s production path does not converge to the efficient path, but in fact yields positive profits. Although this result contradicts ‘the Coase conjecture’ stated above, it is in line with Coase’s brief discussion of the increasing marginal costs case [Coase (1972, pp. 147~148)]. Malueg and Solow (1990) have recently obtained similar results for a case where marginal costs besides being increasing in the rate of output, are also increasing in the cumulative stock of produced goods. ’ * 2 The main purpose of this article is to show that ‘the Coase conjecture’ is not generally valid when the production technology exhibits increasing returns to scale of the type associated with learning by doing. Modifying the GSW model such that marginal costs are supposed to be constant in the short run (at each production date), but decreasing with cumulative production, we show the following: In the linear case there is an equilibrivrm with the property that, as the time between offers shrinks to zero. (i) price converges to marginal cost at each date, and (ii) the rate of prc.! let supply becomes in the limit proportional to that part of the total m ;*kct which remains to be served. Property (i) implies that monopoly pro!?: ;;nish, while property (ii) implies that in the limit, the time required tc ser ..e a fraction y Thus, all consuniels whose of the market is proportional to -In (1-q). valuations exceed the minimum unit cost eventually buy the good, but for any finite date there is a lower positive bound on the mass of consumers which remain to be served. We make the interesting observation that the limiting equilibrium path can be interpreted as a competitive equilibrium path for a continuous-time version of the model. The equilibrium described above clearly contradicts ‘the Coase conjecture’, interpreted as a statement that if a durable-goods monopolist can make offers arbitrarily frequently, then in equilibrium she will (a) price at marginal cost (charge the competitive price), and (b) saturate the market quickly. More precisely, it is part (b) of this conjecture which is contradicted, part (a) is indeed verified also in this equilibrium. Intuitively, these results are due to the fact that in this &!i$rium the monopolist cannot internalize the learning benefits generated in pcoduction. If the monopolist had the ability to commit her future actions, she would produce at a fast pace in order to quickly exploit the learning benefits. (See section 5 below.) She would also cease prod! :tion before the market %,,vas saturated. Following Cease, we may think o1’ the monopolist lacking the commitment ability as a sequence of short-term monopolists, ea:h of whom is competing with her successors. A producer who is early in this sequence
‘It is not explicitly shown in the article that prcfits are positive, but it is easily verified that this is indeed the case. See also the comments in footnote 11 below. ‘Departing from the assumption of continuous demand, Bagnoli et a!. (1989) have shown that Coase’s conjecture is not valid when demand is discrete.
T.E. Olsen, Durable goods monopoly
and the Coase conjecture
159
cannot fully appropriate the learning benefits generated by her own production, hence she ha? less of an incentive to produce at a fast pace than has a producer with cor?.mrtment power. This effect tends to slow down the rate of production r.el&ive to the commitment case. On the other hand there is the Coasian dvnamics of successively declining price offers due to the monopolists’ inability to internalize the capital losses on the stock of produced goods [Bulow (1982)]. This effect tends to speed up the production process and to extend the ultimate degree of market penetration. Regarding the speed of the process, it seems a priori not clear which effect will dominate. But note that, since an early producer can influence the cost structure of her successor, she tran to some degree control the latter’s price-cutting behavior. Through this link there is an added incentive for the early producer to produce at a slow pace. Thus one might expect that the tendency to produce slowly will dominate. This is indeed verified in the model below.3 One of the parameters characterizing the production technology in the present model is related to the number of feasible learning adjustments within the time span defined by a sales period. All results regarding perfect equilibria described above are shown to hold for any value of this parameter - provided the number of feasible adjustments within each period is at least one. This observation is of some independent interest - for two distinct reasons. First, since the parameter in question alternatively can be interpreted as a measure of the degree of static increasing returns in production, it follows that such static increasing returns are (within some range) shown to have no impact on the asymptotic properties of the equilibria considered here. Second, for one particular value of the parameter, the model is formally equivalent to a model of a ‘common values’ bargaining situation. The paper thus extends the familiar set of analogies between durable goods monopoly and bargaining. Following this introduction, section 2 contains a description of the model, and a derivation of the equilibrium in stationary and continuous strategies. In section 3 it is shown that this equilibrium does not obey the Cease conjecture. A brief analysis of efficient production paths in the presence of learning effects and durability is presented in section 4, while section 5 derives the pricing policy of a monopolist with commitment power. The
3The equilibrium having these properties is the one for which buyers’ strategies are specified by reservation prices depending continuously on the stock of the durable good. The equilibrium strategies are stationary, and thus belong to the class of strategies for which GSW proved the Coase conjecture in the standard model. For the same model, Ausubel and Deneckere ( 1989) derived a class of non-stationary (‘reputational’) equilibria which don’t obey the Coase conjecture. For sufficiently short sales periods the payoff set for these equilibria were shown to include payoffs for the seller arbitrarily close to the maximal profits achievable by a seller with commitment power. For linear specifications it can be shown that such Folk results generalize to the learning-by-doing case considered here.
160
bargaining concludes.
T.E. Olsen, Durmhlegoods monopoly crd the Coase conjecture
interpretation
is considered
in section
6, while
section
7
The specification used here is an adaption of the framework in GSW to account for learning by doin,* in production. There is a unit mass of consumers indexed by YE [O, 11, each of whom demands one unit of the good. The function F(q) and the discount factor 6 define the preferences of the consumers: If consumer 4 buys the good in period rr at the price Pn, he obtains benefits (F(q) - P,)6”. Production and sales occur at discrete dates. The production technology is assumed to exhibit constant returns to scale in the short run (at each production date), but to become more favorable as experience is accumulated. Following standard formulations, e.g. Spence f !981), Fudenberg and Tirole (1984) the stock of accumulated production is he;d used as a measure of experience. Let C(y,) denote the unit cost of production at date m, given >O has been produced and sold in the past. that a total volume qrn= In the simplest version of the model presented in the following, the dates of production and sales coincide. I-Iaving made a price offer at some date, the seller then supplies (at constant cost) whatever is demanded at that price. Note that in the standard model for the constant returns to scale technology, the period length only determines the frequency with which offers can be made during any interval of real time. Under the assumption that the dates of sales and production coincide, the period length in our model also determines the number of ‘learning adjustments’ that can be made over such an interval. This simple formulation thus assumes that learning is time consuming (experience gained during production at date n has a bearing on costs only at dates n + k, kz l), and that the ‘learning lag’ is equal to the time period between offers/sales. In general these parameters could be varied independently. In the following we formally analyse the case where learning adjustments take place more frequently than sales, and show that the main conct .,sions for the case of equal frequencies remain valid under this assumption. This case is of some independent interest, since it is formally equivalent to a case where the dates of production and sales coincide, but where the technology exhibits increasing returns to scale in the short run. The case where sales occur more frequently than learning adjustments is not analysed formally in this paper, but we strongly suspect that the main conclusions for the simple model are valid in this case as well.4
T.E. Sisrvr, Drirahk gooiis
vvwvzcvpo~~ avrd fiveCome
c'onjec'ture
161
The model in this paper is further restrricted so as to consider only the linear case, Le. both the demand function F(q) and the learning function C(q)
are taken to be linear. We assume that there exist buyers whose valuation of the good is no larger than the minimum achievable cost? It is then not restrictive to assume that the two functions have the following form: F(q)= l-q, C(q)=c[l
osqs -q-J,
1,
oiqs
(1) 1, o
1.
(2)
All agents have the same discount factors. The monopolist offers at each (sales) date a price, and consumers then decide whether or not to buy in that period. Based on the results for the non-learning case (see GSW), it is to be expected that this model will have many subgame perfect equilibria, even in strategies which are stationary for the consumers. We restrict attention to the stationary strategy combination for which the buyers’ strategies are specified by a continuous reservation-price function. If P(q) dsaotes this function, then after a history with total sales qE [0, 11, the strategy for a buyer of type q’> q is to buy the good at the current date if and only ir’ the quoted price does not exceed P(q’). The seller’s strategy is specified by a continuous price-oiler function, say K( ), such that K(q) denotes the price she will offer for the current period if she has observed a history with total sales q E [0, l]. Consider first the case where the dates of production and sales coincide. For a stock qE [0, 11, let W(q) denote the monopolist’s present value of remaining profits. If the monopolist offers a price p, demand for that period will be q’- q, where q’ is given by P(q’) =p. Optimization on the part of the monopolist requires
Wd=max {cP(ql)-wmq’
(3)
--q)+Wq’)}.
4’ 2:4
Write the solution of (3) as q1 = q + A(q). The monopolist’s can then be written as
~(q)=~(q+4q)L Optimization
The latter ‘But see valuation
asp=1.
(4
on the part of consumers requires
F(q’) - P(y’) =S[F(q’)
equality
optimal price offer
-
says that
wtn
that
(5)
qi =q+4?).
the marginal
buyer
Secton 5 for some comments regarding equilibria strictly exceeds the minimum achievable cost.
(type q’)
should
be
in the case where every buyer’s
162
T.E. Olsen, Durable gotids monopoly and the Coase conjecture
indifferent between immediate purchase at price p= IQ’), and purchase next period, when the price is expected to be K(4’). Following GSW and Stokey (1982) we hypothesize that the solutions are linear functions and solve for the coeficients. Thus, assuming
the value function will be quadratic; W(4) =( l/2)0$1 - 412, and (3 - 5) yield the following equations for the coefficients: -$a+[fl(l--a)--cl-&0(1-x)=0,
(6)
-o=ca-(j?[l
(7)
-r-J-c),
(1 -S)=j3[1-6(1
-r)].
(8)
Eq. (6) follows from the first-order condition for (3), while (7) follows from the envelope theorem applied to (3). Eq. (8) follows from (5). It is easily verified that (6-8) has a unique solution for the parameters (/?,a,cu) with 0 c a c 1, and that this solution satisfies o>O, c
-a2)]=[1
-6(1--a)2-2a](1
-S),/[l4+6a].
(9)
In order to verify that the path generated by the system (3-5) is the outcome path of a subgame perfect equilibrium, it is sunicient to specify strategies which (a) prescribe behavior that is optimal for each player on all histories that result from no simultaneous deviations, and (b) generate the given path (GSW p. 159). For any history characterized by some state 4 E [0, 1) - such that almost all buyers with valuations exceeding l-4 have bought in the past” - let the strategies of the monopolist and the remaining buyers be given by the price offer function K( ) and the reservation-price function P( ), respectively. Note that any history resulting from no simul#taneous deviations from these strategies can be characterized by some such state 4. It follows from (3) that for any such history, the seller’s strategy is a best response to the strategies so specified for the consumers. Similarly it follows from (5) that each consumer’s strategy is a best response to the strategies thus prescribed for the seller and for the other consumers, respectively. These ‘That is, all such buyers except possibly a subset of measure zerro have bought in the past. The strategies are thus constant on histAes which differ only by different actions on the part of a set of consumers of measure zero. The latter restriction was imposed on the set of strategies for which CSW proved the Cease conjecture.
T.E. Olsen, Duruhle goods monopoly
und the Come conjecture
163
strategies thus prescribe behavior that is optimal for each player on all histories that result from no simultanc=ous deviations. We may then conclude that the associated outcome path is the equilibrium path of a subgame perfect equilibrium. Consider now the case where learning adjustments take place more often than sales. Let n be the number of times there is learning within a period, and let Qi denote the volume produced in the ith of these sub-periods. Assume for the sake of simplicity that there is no discounting within the sales period. [In Appendix A we show that with respect to the asymptotic properties (as S-4) of the equilibrium path, this is an innocuous assumption.] The total cost of producing a quantity Q = Q1 +a - + Qn In one period is then C(q)Q, +C(q+Q,)Q,+***+C(q+Q, +.*+Q,+)Q,, where y is the volume produced in the past. For the linear cost function (2) it is easily verified that the cost of producing Q units in this manner is minimal when the quantities produced within the sub-periods are equal’; Qi=Q/n.The minimal cost is then easily seen to be n
c( I- @Q - c( 1 - l/n)Q2/2. With a slight abuse of notation, define C(q,#; i) by C(q,q’;i,)=c[(
1 -y)--i.(q*
-4)/2],
0si.s
1.
(10)
Given an initial volume of sales 4, the (minimal) cost of producing q1 -4 units in a period with n learning adjustments is thus C(q’, q;ii)(q’ -q), where il= I- (ht. Notice that this cost function exhibits increasing returns to scale with respect to the within-period output quantity y’-y, and that i, can be interpreted as a measure of the degree of such increasing returns.’ Replacing C(g) in (3) with C(y, 4’; A) then leads to the following modifications of eqs. (6-7): -/&i-[p(1-0!)-+&_0(1 -cn=ccx-(/?[i
-x]-c)-2m.
-@+Rca=O,
(6’)
(7’)
amounts to and cumuk~tive effect of a margir,al increase ;:, Qi, iln-I, where cl’= y + Q. A corresponding marginal rrlauc!:on in Q, reduces costs by c’[ 1- 4’ + Qi]v C[ 1 -(q’-- QII)J. Ur.iess Q, .: Q,, it would therefore be pr.,rltable to shift production between subperiods. HWe have included 2.= 1 as an admissible vak in (lo), even though there is no feasible production plcn which actually attains the averar.i cost value C(q,q’; I), no! even if we altow for an infinite r.umber (n= x) of adjustments wi:irin the period. (For any such pian, ccsts can be reduced by shifting production from a hi;;&volume to a low-volume subperiod.) The value is attained. however, if we allow for contiritious production agd iearning within the periorl. ‘The
direct
TX. OPserr,DrrrahCe goods monopoly and
154
Eq. (8)
remains unchanged. while
takes the
ihe Coase conjecture
the eq. (9) defining the parameter 2
z!GW
fcm2-P
These modifications require that aP1saEes take place at the beghing of each period. This may be an unrealistic assum tion when learning adjust-
ments are supposed to take place during each ales period. It is, however, s even if we assume that ail. sales take place at easily seen that (9’) still this modification aEPpayoffs will be reduced by the end of each period. the factor 6 relative to the payoffs given in the text. The relations (3)-(5) must therefore stiII hold, rovided W(q) is interpreted as the value accming to the rmnopolist as of the end of the period, given accumulated sales q in p~&Ql~s
peri&.
We now examine the properties of the above equilibrium when the discount factor 6 approaches 1. An increase in 6 can be interpreted either as agents becoming !ess impatient, or as a shortening of the time period between sales. Recall that in the modei variants given above, a change of the period length dso aflects the ‘learning lag’ in production. A reduction of the period length impEies that both learning adjustments and sales can take place more frequently. in the iimit, as the period length shrinks to zero, learning ad.ustments become in eflect instantaneous. Since the cost structure associated with the simple case of the learning lag being eqiuai to the period length is a special case of (IQ), we consider the equilibrium associated with the latter cost function. From (9’) it is dear that cx= &?+-4l as a---+I. Define
16.5
a*=iEma(S)/(l -is)= l/c-
1.
(11)
b+l
It fdows
from(8)
converges
to
/3 in the reservatbn-pricefuncticsn P&f) c, the coefkierntof Eke learning cwve. T , both P(q) and the
that the coefkient
rice-offer function M(4) = P(q + PA( f - q)) converge to cosE function C(q). ence, for 6 close to I, the price offers along the equilibrium: path are only slightly above unit costs in each period, and in every period only a small fraction of the remaining buyers accept the offer. As 6-s I, the sefler’s profits vanish. It is worth noting that these results hold true for all admissible values of the parameter 2 (02 25 1). Also recall from (IQ) that t.his parameter can be interpreted as a measure of the degree of short run - or, for fixed period length, static - increasing returns in production. In GSW it was suggested that such increasing returns may provide means for credible commitments on the part of the monopolist (GSW, p. 173) The interesting issue is then whether such possible commitments ahow the monopolist is escape from the vanishing profits predicted by the Gase conjecture. The analysis above shows that for any degree of increasing returns represented by 2. being in the range [0, I] in our model, no such commitment effect will be present? Along the equilibrium path, the stock of the durable good evolves according to LJ~+1 = q, + A(y,) = q,[ 1 - r] + 31, 212 0. Starting from 4o = 0, after n periods the stock is qn= 1 -(l
-ct)“,
nz0.
The number of periods required to reach market penetration given by the integer value of In(l -y)/ln(
y~(::(o,1) is then
1 --A),
(12)
which obviously goes to infinity as &+ 1. Now consider the case where the time between ofTen, cal! it E, shrinks. Let r- lim,,,( 1 -&))/1: be the instantaneous rate of discorrnt. The average supply rate per period (r[l - y]/&) then corrverges to rcP[t -4-j as ~40. Again
it
is
WOFfh
notEng
that,
akhough
ah5
kxtson
2
of
the
l%x-miining
166
TX. Olsen, Durable goods monopoly and the Coase conjecture
market that is being served in each period depends on the learning adjustment parameter i. - and is in fact a strictly increasing function of i. .the asymptotic rate of market penetration is independent of this parameter. Since now ( I/E) In ( 1 -o+ - ra* as E+O, it follows from (12) that the time ?E-- tt(q) required to reach market penetration 4 E [O, 1) converges to
t(4)= -(l/ra*)ln(l-@,
a*= l/c-!.
(13)
Thus, in equilibrium all consumers eventually buy the good, but for every finite date there is a lower positive bound on the mass of consumers which remain to be served. Contrary to the standard constant returns case, rhe market is not saturated quickly when there are learning economics in production.’ ’ The slow market penetration illustrated by (13) shows that the full Coase conjecture does not generalize to a technological environment characterized by learning-by-doing. However, recall that we did show that equilibrium prices converge to marginal cost each period. [In equilibrium, the price, the marginal cost and the unit cost in period n are ail multiples of (1 -q,,), and If the coeflicients are given by, respectively; /3(1 -cc), c( 1 - i.or>and c( 1 -h/2).] it is the case that ‘competitive pricing’ yields prices equal to marginal costs in this model, then Coase’s conjecture regarding convergence of monopoly prices to competitive prices remains literally true also for this environment. We shall now give two such interpretations of ‘competitive pricing’. Assuming complete learn’ng Tpiiiovers Between firms (each firm’s ccsts being a function of industry-wide rather than firm-specific experience), St can be seen that unit-cost pricing is sustainable as a subgame perfect equilibrium outcome in Gigopoiy. For expositional reasons we verify this assertion here only for the simple case i, =O, for which marginal and unit costs of course coincide each period. Specifically, suppose there are several firms in the industry. In every period n 20, each firm has a constant-cost production technology characterized by unit costs [C’(q,J], depending only on the total industry-wide volume (4”) produced and sold in the past.’ 2 Let each firm’s strategy be to charge a price equal to the current marginal cost in each “The path q(r) delined by ( 13) is formally similzr to the time-consisient path derived in Malueg and Solow (1990) for a case where there i% a ~~.sitiz~ stock effect on costs. It is easily seen however, that the shape of the time-consistent path in their model critically depends on their assumption th::t marginal costs are increasing In the nrle of output. Thus. although the positive stock effect in!luences the shape of :he path. it is not in itself sufficient to generAte a path exhibiting delay. “This assumption of Iarnin~~, @lovers being complete is unarguably an extreme one, but perhaps not more extreme than ;!:I 2~..amption to the effect that learning is completely firmspecific. i.e. no spillovers. In the cor!‘.zxt of oligopolistic markets for non-durable goods, Stoke); ( 1986) used the former assumption. while Space ( 198 1) and Fudenberg and Tirole ( 1984) used the !at ter.
T.E. Olsen, Durable goods monopoly
and the Coase conjecture
period. Let the buyers’ strategies be given by a reservation PO( ) defined by F(q)[l -S] = P,(q) - SC(q), i.e.
&m=(l
-~+W[l-43,
qE[O,
167
price function
1-J.
By the same type of argument that was given in section 2 it is seen that this strategy combination constitutes (part of) a subgame perfect equilibrium, and marginal-cost pricing is of tours e the equilibrium outcome. Note that the consumers’ strategies are stationary and specified by a reservation price function which is continuous in the state variable q. In this equilibrium, the stock of the durable good evolves according to Yn=
1-(c/Cl -S+Sc])“,
n>,O.
It is easily verified that, as the period length shrinks to zero, the equilibrium path converges to the path specified by (13). Thus, if we interpret the Bertrand-type pricing scheme given here as ‘competitive pricing’, it remains true also for this technological environment that the monopoly equilibrium path will become arbitrarily close to a competitive path if the time period between sales is sufficiently short. The path characterized by (13) can also be interpreted as the Walrasian competitive equilibrium path for a continuous-time version of our model [David and Olsen (1986)]: Suppose there is continuum of firms on the supply side, and that there are complete learning spillovers among these firms. Let P,, t 10, denote the competitive price path. If P, is differentiable and decreasing, a consumer whose valuation is F(q) buys the good at the date t = 7(q) given by r[F(q) - P,] + dP,/dt = 0. Since instantaneous marginal costs are assumed to be constant, at every date we must have price equal to marginal costs. Assuming complete learning spillovers, this condition takes the form
where qr is the stock of accumulated production at date I’. Thus, in equilibrium the stock qt and the price P, must satisfy F(q!) = P, -(dP,/dt)/r and condition (14). Hence, the stock q, must evolve according to
Under suitable regularity conditions (linearity is certainly sufficient), this differential equation will have a unique solution, and the price path defined by (15) is the uique competitive equilibrium path. (Due to the continuum of iQISigrlQfiCantiy to the supplying firms, each firm’S OWIQ prGdQ.QCtiOn COntributeS
Fatal st0ck of accumulated sales, hence each firm can at most obtain
pr&s
of zero under the price path given by ( I a)-( 15). There exist individual produLntion prans which in 1 e aggregate fulfiik~ (15) and yield zero p-ofits, hence these production schemes are individuaily optimal.) Pt then follows from (IS) that the time T= T(q) requited to reach a stock q E [O, I] is given
bY
For the linear specifications
(1 -2) this function clearly coincides with the function given in (13), proving that the eqtdibrium path under monopoly akasian competitive converges to a path whfch Es the continuous-time equihbrium. in this article the c0nvergence resuit has been established only for the Einear case. It seems reasonable to conjecture that a simiiar result holds for more general forms of demand and 1earnEng functions. This suggests the following generalization of the Cease conjecture: For every stationary equilibrium, as the period beiween offers vanishes, the time required to reach market penetration :I ~(0,1) converges to T(q) given by (16). If this holds true, then clearly the behavior of C’(q)/[F(q) -. “‘!rjr)] near y = I will determine wherher the market wiII be saturated in finite time.
The anaIysis of perfect eqrrilibria above
has focused on the case where learnring adjustments :in production occur at leasi 2s frequently as sales. %ntuition seems to suggest FhaF the two main characteristics of the equilibria derived above - i.e. asymptoficaiiy sbw market penetration and vanishing profiis - h0id true than sales. 13
Since
also when iearing
the monopolist
obviously
dees
adjustmenFs take place fess fieqtccniiy
not properly
internalize
the IearCg
Let V(4) denote the optimal value function for the sociail planner. This function satisfies the following dynamic programming equation:
Proceeding as in section 2 we find that W(4) is quadratic: V(q)= (W)vC~ -4]“, and that the optimal quantity to produce in any period is proportional to that fraction of the market which remains to be served; 4l- 4 = y( I - 4), 4 E CO1I]. The parameters v and y can be determined from e first-order and envelope conditions for the social value function V(4). These conditions take the following forms, respectively:
(1 -y)-c-6v(l v = 1 - c - cy +
(17)
-p)+ci,y=@
(w
ci;y.
Two cases must be distinguished. Consider first A< 8. Combining (18) the (second-order) equation defining “Jthen takes the form (1 -c)(l
-6)+“J[G(P -c)-
(I
7) and
1+6(1 ---r)(ll -E.)c+c;l]=O.
It is easily verified that OC;I < P and vr 0. Clearly *;-AI as 6+ 1. Thus, as S+ 1, the eficient fraction of remaining consumers which are being served each period vanishes. owever, since evidently y/( 1 -S)-+ a as ii-, 1, this efficient fraction is asymptotically much larger than the fraction being served under monopoly. Also notice that, since v-+1 -c, the optimal value V(4) converges to the ‘static’ net consumers surplus:
jEF(Q)-c(QPIdQ=(1/2)(1 Y
-C)(P -4)2,
C(4)=C(B -4).
(1%
Nt-xl consider the case A= 1. IRecaP that this can be interpreted as a case where learning adjustments take place infinitely often within each S&X period. From ( i 7) and (l&) we then obtain ;‘= 1 and v = “, -cc’. Tire efficient solution in this case is thus to sup;:+ the ultimate quantity (= 1) in t period. PIotice that the value functnon V(q) is then exactly of the form ( 39). This is an intuitiveiy reasonable outca;xe, since the garticujar learning process associated with L= 1 aFlows for 2 gradual movement “down the
IeartLg cwve !konYi ay q 20 to iq= 1 within one period. The associated. total pr~$~xtio~ costs ..ire then recisely equal to rhe tataS costs obtained in the nisit as the periot.Iength v nishes (and ths learning takes more 5.~4 mcii-e often)for tke case where there are only a finite number OFlearning adjustments within exh period. (2.< 1). ofeach period shrink as in section 2, we see that forany q E [O, 11, the eRicierJ timeto reach market penetration q converges to zero. This ihlustrates aga ..n that the monopolist’s rate of production is too ;low in the presence OFleaxing by doing.
@ace
provides an analysis This sectio~~t mor,opclis”, Y ith commitmerit power,
of
OF a given a technology characterized by learning etieds of the type captured by (10). The optimal policy is found to depend critically on the ‘learning adjustment’ parameter i.. For A< 1 we find that the committed monopolist will spread her sales over an infinite number of periods, ad hence commit herself to intertemporal price-dixrimination. However, although production and sales will be spread over time in the . . commltmcnl-SoEislzG~, the rate of product supply is much larger, and the total vdume of sales is distinctly smaller. than those of the non-committed mGncpoIist andysed in section 2. As the time period between sales vanishes, the rat;’ of Saks for the commitment solution becomes infinite, such that contrary to the nen-commitment case in section 2 - almost all sales take place irl; a vanishgly sma”:I intcrvai of (real) time. f%r th@ skuxkud constant CGSP case it is Well known that the optimal commitment policy invoks no intertemporal price-discrimintion [Stokey ( 1979)]. The intuition fcx why it occurs in the presence of learning enects is that this pricing poky diGWS the monopolist to take advantage of the dynam-c idcreasing returris represented by the learning curve. However, if the parameter 2. tkx the value 1 we know from the previous section that all these learning dfects can “be achieved with one period. It is therefore well in ,XXYX”d Wit{?&ui”,ion thpt for /;;= I the optimal policy is for the monopolist to sell in only OrK period. the
0
ricing
policy
171
[(i -iQF(y,,+Jqr+2 -CEy,,y,,.,)(q,+,-q,)lr’i” c
?i
=
.
5
6
&=O
”
0
Defining v, = 1- qn, the optimal solution for the specifications defined by ( 1) and ( 10) is givenby I)-2(1
tS(l-I,)cv,+I+[ic(G+
-di+6c)~c,,+c(1-i.)v,_,=-(1-6).
(20)
For Rc 1 the solution is of the form 1 -(I/“= _= 1’0.
q*y”+v*, y*=
1/(2--C),
112 I!,
WI
I -v*,
G3
where y
is the unique solution - in the interval (0, 1) - to the (second-order) characteristic equation associated with (20). Fdote that the ultimate market penetration for the committed monopolist is 4* = I- v$ =( 1 - c)/(2 -c), and that this quantity is independent of the learning adjustment parameter R. It is easily checked that q.+ is the unique volume of sales for which ‘marginal revenue’ [F(q) + @‘(4)] is equal to marginal cost C(q). The price sequence associated with (2 1) is given by p,=(l -6)
i
cTk-I F(qn+k)=yDq*y”+a’*,
ri~0,
(23)
k=l
where Dr(f
--@/(I -6y)-+O
as 6-L
(24)
(The limit in (24) follows e.g. from L’ ospital’s rule, noting that y-31 and the derivative y’+oo as 6 + 1.) Thus, yr s are decreasing over time along the optimal sequence, and ultimately approach the valuation of the marginal buyer (v* =F(y,)). observe that for 6 :,-lose to 1, the initial price offer pu is he price oflers are only slightly above the valuation (vJ of this buyer. oes find it advan&xlining over time, thus the committed monopdslist ::;~y,eousto price-discriminate, but the degree of Grscrimination is very SmaH ‘S;r S close to ! . Ht is hard not to verify that, if we let the discount fzxt~r go bctvqeen sales shrink to zero - the limit given by
172
T.E. Olsen, Durable goods monopoly and the Coase conjecture
Notice that there are sales in every period, but as the length (E) of the sales vanishes, the time required to reach any level of market penetration 4, q cq*, vanishes as well: Denoting this time by t:(q), we have period
C(s) = (ln C4*- d/ClnY/(I- 4lM I- 44
4 < 4*-
Thus, almost all sales take place in a vanishingly short interval of real time. This outcome is in stark contrast to the outcome for the non-comitment case derived in section 2, and shows that the slow pace of sales in the latter case is due not ortZ~j to the learning effects in production, but rather to a combination of those learning effects, the seller’s inability to commit her future actions, and the buyers’ expecztations about the seller’s future actions. Recall that, when the period length is short, it is socially efficient to sell and produce at a fast pace. The commitment solution has this property, but it is obviously not efficient. It is easily verified that, as 64, the consumers’ commitment path converges to surplus associated vvA A,h +ho cLLW
(VW-dC(1 -c)/(2-c)12=~CF(q)-C(q)lGq. 0
Given the power to commit her future actions, the monopolist restricts the total volume of sales, and that is clearly the source of the inefficiency in this case. Comparing the commitment and non-commitment paths we may note that the social surplus is smaller in the former case. However, this result is likely to be a particular feature of the linear specifications considered here. Finally consider the case d= 1. It then follows from the optimality condition (20) that t’,= 1/(2-c) = v# for all n. Given infinitely rapid learning within sub-periods, the optimal solution under commitment is thus to induce sales q* in the very first period, and no sales thereafter. The associated total production costs are given by (c -cq,/2)q,, which is precisely equal to the total costs obtained in the limit as the period length vanishes (and thus learning takes place more and more often) for the case where there are only a finite number of learning adjustments within each period (ic< 1). The analysis of equilibrium and commitment output paths in this paper focuses on the case where there exists buyers whose valuation of the good is lower than the minimum achievable cost. Based on the analysis in this section we can, however, draw some conclusions regarding perfect equilibria fr the case where every! buyer’s valuation strictly exceeds the lowest possible cost. To this end, suppose the total market were given by [O,q], where q is strictly smaller than the quantity q1 given in (21). Then obviously the monopolist with commitment power would clear the whole market in the first period, charging a price of p= F(q). (This wou!G also be the optimal policy, given that only consumers in [q, a], some q >O, remain.) Then UXHXiCk~ the non-commitment case, an assume consumers fdow a
T.E. Olsen, Duruhlegoods monopoly and t&e Come conjecture
173
reservation-price strategy given by P(q) =p for all 4 in [0, q]. The monopolist’s best response to this strategy clearly entails charging the price p for all histories with state 4 in [O,q]. it follows that the two strategies so defined are mutually best responses on all such histories, hence the associated outcome path is a perfect equilibrium path. Note that the market is saturated in finite time (after one period), and that this result clearly is due to the ‘gap’ between the demand and cost schedules lor the ultimate buyer (@. For the linear model the ‘gap’ case can generically be represented by restricting the total market to be [O,
6. A bargaining interpretation It is well known that the formalism of the standard (zero cost) durable goods model accommodates the case of bilateral bargaining where a seller with a known valuation repeatedly makes offers to a single buyer with a privately known valuation whose probability distribution is common knowledge. Interestingly thc;le is a similar interpretation of the learning-by-doing model studied in this payer. Consider the following parable: A tract of land which possibly contains oil has been surveyed by an oil company. The surveillance technology is sufftciently sophisticated that the exact amount of oil, and therefore its value is known to the firm. The land owner does not have access to the surveillance technology, and her extraction technology is inferior to that of the company. The value of any oil in the ground to her is therefore only a fraction of the value of the same quantity to the firm. Let c denote this fraction, and assume it is common knowledge that the company’s valuation (v) is uniformly distributed on [0,11. Suppose the parties bargain over the tract in such a way that the land owner repeatedly makes offers to the oil company, who either accepts or rejects each offer. Bargaining is concluded the first time the buyer accepts an offer. Let 4 = 1 -v denote the type of a buyer whose valuation is V, and suppose the buyer follows a reservation-price strategy given by the continuous function P(q). The seller’s optimal expected value U(q), conditional on the buyer’s type being at least 4 then satisfies q’
P(q’)(g’-q)- 5 c(l -Q)dQ+iiU(&)(l-4’) 4
(I--@.
TX. Olsen, Durable goods monopoly
174
and the Coase conjecture
The second term on the RHS is (apart from a normalizing factor) the seller’s expected opportunity cost if the buyer accepts the offer. This cost term is easily calculated to be
cC(l-4H41-4m[q’
-q]*
Note that this expression corresponds exactly to the one period cost incurred by the durable-good monopolist if she produces 4l-4 in the period, given that learning occurs infinitely often; see the discussion following (10). It then follows that the seller’s expected value U(q), conditional on all buyer types with valuations exceeding l-4 having rejected in the past, is given by U(4) = W(q)/( 1 - 4), where W(4) is the equilibrium value for the durablegoods monopolist, given that all buyers with valuations exceeding l-4 have bought in the past, and learning being ‘continuous’ (i,= 1) within each period. Moreover, the strategies given by the reservation-price function P(4) and the price-offer function K(4) defined in (3-5) - with the obvious modification for the cost function C( ) - constitute (part of) a sequential equilibrium for the bargaining game, and the associated outcome path is a sequential equilibrium path. The interpretation given here implies that the two agents have ‘common valuations’ of the good in question. Vincent ( 1989) analysed a similar ‘common values’ bargaining situation, in which the seller is assumed to be the informed party, and the (uninformed) buyer is making all the offers. His anaiysis is confined to the case where there is a ‘gap’ between the buyer’s and the seller’s valuations, for all feasible types. Vincent establishes existence and generic uniqueness of a perfect Bayesian equilibrium in this case, and shows that for a broad class of such cases there will be delay to agreement - even as the time between offers vanishes. The similarity between Vincent’s model and ours - the schedules corresponding to F( ) and C( ) are, respectively, both increiising and both decreasing in the two models - suggests that these results should IX valid also for the ‘gap’ case in our framework. On the other hand, th< results developed here for the ‘no gap’ case should similarly be valid for the corresponding case in Vincent’s model. Moreover, the isomorphism established here between ‘durable goods monopoly with learning-bydoing’ and ‘bargaining with common values’ is helpful in providing additional insights into the reasons why delay occurs in such bargaining situations, because the intuition developed for the former interpretation can be applied to the latter.
..
8l;& +s.‘=main contribution of this article was to show that the coasian 1’...;;?2erties(i.e. competitive pricing asd quick markei penetration) cf equi..!
librium paths in durable-goods monopoly are quite sensitive to changes in the technological environment. This was shown by demonstrating that, for an equilibrium *where buyers’ strategies are stationary, the introduction of some element of learning by doing in production can induce significant changes in the equilibrium outcome path: While, as the period between sales shrinks to zero the time to reach market penetration q converges to zero in the standard constant returns case, this time becomes proportional to - ln (l-q) in the presence of learning effects. Thus, although the limiting outcome path in the presence of learning effects was shown to exhibit competitive pricing (price equal to marginal costs), the market is penetrated more slowly than predicted by the Coase conjecture in this equilibrium. The price sequence chosen by a monopolist with commitment power was shown to generate quick market penetration. From this observation it follows that the slow pace of sales in the non-commitment case is due not on!y to the learning effects in production, but rather to a combination of those learning efects, the seller’s inability to commit her future actions, and the buyers’ expectations about the seller’s future actions. Regarding welfare properties of equilibria, it was shown that both the outcome path for the stationary equilibrium and the commitment path were socially ineficient, - but for different reasons. The latter is inefficient because the associated total volume of sales is restricted, the former because the associated speed of sales is too small. The results in this paper were derived within the framework of a simple linear model under the assumption that learning adjustments in production could take place at least as frequently as sales. We noted that the parameter characterizing the relative frequency of learning adjustments to sales had no impact on the asymptotic properties of the stationary-strategy equilibrium discussed above. Another interesting point was the observation that for one particular value of this parameter :he model becomes formally equivalent to a model of bargaining with ‘common values’. The paper thus extends the set of analogies between durable goods monopoly and bargaining.
In section 2 the claim was made that for the case of ‘earning adjustments occurring more frequently than sales it is not critical to abstract from discounting within each sales period. In this appendix the claim is substantiated I?;) showing that the asymptotic properties of the stationary equilibriurii for this case are not affected when such discounting is taken into accoun? Suppose there are n + 1 learning adjustments within each sales period. Let p = #‘(” I) be the discount factor associated with the learning lag. B;fq md q1 denote the s;cl:ks at the beginning and at the end, respectively, of a sales
perio
en the total
Ci+--q,
L-q'),
discounted productidn
ccst over this period is givept. by
where
CT(U,yl)=EnEn(cv(~-c,)-E_Eecv,(c,
-~?Z~~.*".$.~~~~,(~~,-~‘~))
illi 'I 1 -qi,
by
where qi is t e stock at the end of the ith subperisd. the envelope theorem we have
N0te
that
The objective function is strictly convex for p sufficiently close to 1, and the solutions to the minimization problem are then given implicitly by
Note
that the 21;s are linear
functimsof v
and d, and hence the PotA cost
hnction is quadratic in these variables. Solving explicitly we have uk= C, z: -e-czz;, where z:, z2 are the &mnplex conjugate) roots of z2 - 22 + l/p = 0. Define G,(p) by G,(p) = z!J- z:, m = 1,2,. . . , M -+ I. Some algebra then yields
it is straightforward but tedious to show that the coefficients defined by the khtities abave satisfy
A,
--zY
-4
-B(P
-(jo-.-fl(1
-a).
p_jqein$q-jre~
Lsere W(q) to be the vahe as of the end of the saiesperiod when tracsac~bns actually take place - given accumulated sales q before the period.]These
two quations
yield
From the eq. (8) for j3 and the two IIirnits given above we then see that oc/(l -6)+a0, where c1’ is given by c= l/(ll Jra’)--0, hence 6~O=a*=l/~- 1. ?‘his completes the proof. It is worth noting from this proof that it is the linear structure of the marginal cost functions (w.r.t. both the stock q and the period output q1 -4), apn,Gthe asymptotic behavior of the coefficients of these functions which makes the result go through.
Ausubei, L. and R. Deneckere, 1989, Reputation in bargaining and durable goods monopoly, Econometrica 57, 5 1l-53 I. Bagnoii, M.. SW. Saiant and J.E. Swierzbinski, 1989. Dura&e.goods monopoly with discrete demand, Journal of Political Economy 97, 1459-1479. Buiow, J., 1982, Durable-goods monopolists, Journal of Political Economy 90, 314-322. Coase, R&L, 1972, Durability and monopoly, Journal of Law and Economics 15, 133-149. David, P. and T.E. Olsen, 1980, Equilibrium dynamics of diffusion when incremental technological innovations are foreseen, Ricerche Econ”miche 4, 738- ‘i-Tti. Fudenberg, D. and J. Tirole, 1984, Learning-by-doing and market performance, Bell Journal of Economics 14, 522-530. GUI, F., H. Sonnenschein and R. W’r.jui.. 986, :-;;undations of dynamic monopoly and the Coase conjecture, Journal of Economic Trrcory, 39, 155- 190. Kahn, C., 1986, The durable goods monopolist and consistency with jnc~asi;;g costs, Econometrica 54, 275-294. Maiueg, D.A. and J.L. Soiow. i Ok,, iGo,nopoly production of durable exhaustible resources, Economica 57, 29-47. Spence, M., 1981, The learning curve and competition, Be11Journal of Economics I1,49-70. Stokey, N., 1979, Intertemporal price discrimir,ation, Quarterly Journal of Economics, 355-371. Stokey. N., 1982, Rational expectations and durable goods pricing, Bell Journal of Economics 12, 112-128. Stokey, N., The dynamics of industry-wide learning, in: P. Heiier, R.M. Starr and D. Staret, eds., Essays in honor of Kenneth Arrow (Cambridge University Press, Cambridge). Vincent, DR., 1989, Bargaining with common values, Journal of Economic Theory 48,46--62.
_Norwgian Center firr Research irt Organizcution ad
Management,
N-5015 Bergen;, iVorv;c~
Received December 1989, final verson received March 2991
The Coaae conjecture states that if a durable-goods monopohst can make oks to sell arbitrariiy frcqtientiy, then in equihbrium she will charge the competitive price, and the market will be saturated qtiickly. This conjecture is shown to be refuted when production of the durable good is subject to learning by doing: For the linear case it is shown that there exist a stationarystrategy perfect equilibrium in which the monopolist charges competitive prices, but the market is saturated slowlg. An interesting analogy between bargaining and durable-goods monopoly with learning by doing is also pointed out.
In his seminal article on durable-goods monopoly, Cease (1972) confined his discussion to production technologies characterized by constant or decreasing returns to scale. For the constant returns case, he conjectured that if a durable-goods monopolist can make offers to sell arbitrarily frequently, then in equilibrium she will charge the competitive price and the market will be saturated quickly. [‘With complete durability, the price becomes independent of the number of suppliers and is thus always equal to the competitive ith constant marginal costs, production would price’, Cease ( 1972, p. 144). take place in the first perio and would then cease’, Coast ( 1972, p. 147).] With some stationarity restrictions on buyer s’ strategies, Gul et al. (1986) hereafter GS W - proved this conjecture in a framework of a discrete time, infinite horizon, extensive f0rm game with a continuum 0f buyers. For a reduction technology exhibiting increasing short run marginal costs, Kahn ( 1986) identified an equilibrium for which as the time between offers shrinks, “Parts of this work were done during an enjoyable visit at the IDepartment of Economics and the Center for Economic Policy Research at Stanford University. I am grateful to Bob Wilson for comments on an earher version of this article, to Paul Davki for extensive discussions on the topics with which this paper is concerned, anyr to two anonymous referees [or constructive criticism. A.ny errors are solely mint. Ffna~5l srr;~port fro25i the Norwegian Research CounciI (NAVF) is gratefully acknowiedged.
158
T.E. Olsen, Durahlc goods monopoly
and the Coase conjecture
the monopolist’s production path does not converge to the efficient path, but in fact yields positive profits. Although this result contradicts ‘the Coase conjecture’ stated above, it is in line with Coase’s brief discussion of the increasing marginal costs case [Coase (1972, pp. 147~148)]. Malueg and Solow (1990) have recently obtained similar results for a case where marginal costs besides being increasing in the rate of output, are also increasing in the cumulative stock of produced goods. ’ * 2 The main purpose of this article is to show that ‘the Coase conjecture’ is not generally valid when the production technology exhibits increasing returns to scale of the type associated with learning by doing. Modifying the GSW model such that marginal costs are supposed to be constant in the short run (at each production date), but decreasing with cumulative production, we show the following: In the linear case there is an equilibrivrm with the property that, as the time between offers shrinks to zero. (i) price converges to marginal cost at each date, and (ii) the rate of prc.! let supply becomes in the limit proportional to that part of the total m ;*kct which remains to be served. Property (i) implies that monopoly pro!?: ;;nish, while property (ii) implies that in the limit, the time required tc ser ..e a fraction y Thus, all consuniels whose of the market is proportional to -In (1-q). valuations exceed the minimum unit cost eventually buy the good, but for any finite date there is a lower positive bound on the mass of consumers which remain to be served. We make the interesting observation that the limiting equilibrium path can be interpreted as a competitive equilibrium path for a continuous-time version of the model. The equilibrium described above clearly contradicts ‘the Coase conjecture’, interpreted as a statement that if a durable-goods monopolist can make offers arbitrarily frequently, then in equilibrium she will (a) price at marginal cost (charge the competitive price), and (b) saturate the market quickly. More precisely, it is part (b) of this conjecture which is contradicted, part (a) is indeed verified also in this equilibrium. Intuitively, these results are due to the fact that in this &!i$rium the monopolist cannot internalize the learning benefits generated in pcoduction. If the monopolist had the ability to commit her future actions, she would produce at a fast pace in order to quickly exploit the learning benefits. (See section 5 below.) She would also cease prod! :tion before the market %,,vas saturated. Following Cease, we may think o1’ the monopolist lacking the commitment ability as a sequence of short-term monopolists, ea:h of whom is competing with her successors. A producer who is early in this sequence
‘It is not explicitly shown in the article that prcfits are positive, but it is easily verified that this is indeed the case. See also the comments in footnote 11 below. ‘Departing from the assumption of continuous demand, Bagnoli et a!. (1989) have shown that Coase’s conjecture is not valid when demand is discrete.
T.E. Olsen, Durable goods monopoly
and the Coase conjecture
159
cannot fully appropriate the learning benefits generated by her own production, hence she ha? less of an incentive to produce at a fast pace than has a producer with cor?.mrtment power. This effect tends to slow down the rate of production r.el&ive to the commitment case. On the other hand there is the Coasian dvnamics of successively declining price offers due to the monopolists’ inability to internalize the capital losses on the stock of produced goods [Bulow (1982)]. This effect tends to speed up the production process and to extend the ultimate degree of market penetration. Regarding the speed of the process, it seems a priori not clear which effect will dominate. But note that, since an early producer can influence the cost structure of her successor, she tran to some degree control the latter’s price-cutting behavior. Through this link there is an added incentive for the early producer to produce at a slow pace. Thus one might expect that the tendency to produce slowly will dominate. This is indeed verified in the model below.3 One of the parameters characterizing the production technology in the present model is related to the number of feasible learning adjustments within the time span defined by a sales period. All results regarding perfect equilibria described above are shown to hold for any value of this parameter - provided the number of feasible adjustments within each period is at least one. This observation is of some independent interest - for two distinct reasons. First, since the parameter in question alternatively can be interpreted as a measure of the degree of static increasing returns in production, it follows that such static increasing returns are (within some range) shown to have no impact on the asymptotic properties of the equilibria considered here. Second, for one particular value of the parameter, the model is formally equivalent to a model of a ‘common values’ bargaining situation. The paper thus extends the familiar set of analogies between durable goods monopoly and bargaining. Following this introduction, section 2 contains a description of the model, and a derivation of the equilibrium in stationary and continuous strategies. In section 3 it is shown that this equilibrium does not obey the Cease conjecture. A brief analysis of efficient production paths in the presence of learning effects and durability is presented in section 4, while section 5 derives the pricing policy of a monopolist with commitment power. The
3The equilibrium having these properties is the one for which buyers’ strategies are specified by reservation prices depending continuously on the stock of the durable good. The equilibrium strategies are stationary, and thus belong to the class of strategies for which GSW proved the Coase conjecture in the standard model. For the same model, Ausubel and Deneckere ( 1989) derived a class of non-stationary (‘reputational’) equilibria which don’t obey the Coase conjecture. For sufficiently short sales periods the payoff set for these equilibria were shown to include payoffs for the seller arbitrarily close to the maximal profits achievable by a seller with commitment power. For linear specifications it can be shown that such Folk results generalize to the learning-by-doing case considered here.
160
bargaining concludes.
T.E. Olsen, Durmhlegoods monopoly crd the Coase conjecture
interpretation
is considered
in section
6, while
section
7
The specification used here is an adaption of the framework in GSW to account for learning by doin,* in production. There is a unit mass of consumers indexed by YE [O, 11, each of whom demands one unit of the good. The function F(q) and the discount factor 6 define the preferences of the consumers: If consumer 4 buys the good in period rr at the price Pn, he obtains benefits (F(q) - P,)6”. Production and sales occur at discrete dates. The production technology is assumed to exhibit constant returns to scale in the short run (at each production date), but to become more favorable as experience is accumulated. Following standard formulations, e.g. Spence f !981), Fudenberg and Tirole (1984) the stock of accumulated production is he;d used as a measure of experience. Let C(y,) denote the unit cost of production at date m, given >O has been produced and sold in the past. that a total volume qrn= In the simplest version of the model presented in the following, the dates of production and sales coincide. I-Iaving made a price offer at some date, the seller then supplies (at constant cost) whatever is demanded at that price. Note that in the standard model for the constant returns to scale technology, the period length only determines the frequency with which offers can be made during any interval of real time. Under the assumption that the dates of sales and production coincide, the period length in our model also determines the number of ‘learning adjustments’ that can be made over such an interval. This simple formulation thus assumes that learning is time consuming (experience gained during production at date n has a bearing on costs only at dates n + k, kz l), and that the ‘learning lag’ is equal to the time period between offers/sales. In general these parameters could be varied independently. In the following we formally analyse the case where learning adjustments take place more frequently than sales, and show that the main conct .,sions for the case of equal frequencies remain valid under this assumption. This case is of some independent interest, since it is formally equivalent to a case where the dates of production and sales coincide, but where the technology exhibits increasing returns to scale in the short run. The case where sales occur more frequently than learning adjustments is not analysed formally in this paper, but we strongly suspect that the main conclusions for the simple model are valid in this case as well.4
T.E. Sisrvr, Drirahk gooiis
vvwvzcvpo~~ avrd fiveCome
c'onjec'ture
161
The model in this paper is further restrricted so as to consider only the linear case, Le. both the demand function F(q) and the learning function C(q)
are taken to be linear. We assume that there exist buyers whose valuation of the good is no larger than the minimum achievable cost? It is then not restrictive to assume that the two functions have the following form: F(q)= l-q, C(q)=c[l
osqs -q-J,
1,
oiqs
(1) 1, o
1.
(2)
All agents have the same discount factors. The monopolist offers at each (sales) date a price, and consumers then decide whether or not to buy in that period. Based on the results for the non-learning case (see GSW), it is to be expected that this model will have many subgame perfect equilibria, even in strategies which are stationary for the consumers. We restrict attention to the stationary strategy combination for which the buyers’ strategies are specified by a continuous reservation-price function. If P(q) dsaotes this function, then after a history with total sales qE [0, 11, the strategy for a buyer of type q’> q is to buy the good at the current date if and only ir’ the quoted price does not exceed P(q’). The seller’s strategy is specified by a continuous price-oiler function, say K( ), such that K(q) denotes the price she will offer for the current period if she has observed a history with total sales q E [0, l]. Consider first the case where the dates of production and sales coincide. For a stock qE [0, 11, let W(q) denote the monopolist’s present value of remaining profits. If the monopolist offers a price p, demand for that period will be q’- q, where q’ is given by P(q’) =p. Optimization on the part of the monopolist requires
Wd=max {cP(ql)-wmq’
(3)
--q)+Wq’)}.
4’ 2:4
Write the solution of (3) as q1 = q + A(q). The monopolist’s can then be written as
~(q)=~(q+4q)L Optimization
The latter ‘But see valuation
asp=1.
(4
on the part of consumers requires
F(q’) - P(y’) =S[F(q’)
equality
optimal price offer
-
says that
wtn
that
(5)
qi =q+4?).
the marginal
buyer
Secton 5 for some comments regarding equilibria strictly exceeds the minimum achievable cost.
(type q’)
should
be
in the case where every buyer’s
162
T.E. Olsen, Durable gotids monopoly and the Coase conjecture
indifferent between immediate purchase at price p= IQ’), and purchase next period, when the price is expected to be K(4’). Following GSW and Stokey (1982) we hypothesize that the solutions are linear functions and solve for the coeficients. Thus, assuming
the value function will be quadratic; W(4) =( l/2)0$1 - 412, and (3 - 5) yield the following equations for the coefficients: -$a+[fl(l--a)--cl-&0(1-x)=0,
(6)
-o=ca-(j?[l
(7)
-r-J-c),
(1 -S)=j3[1-6(1
-r)].
(8)
Eq. (6) follows from the first-order condition for (3), while (7) follows from the envelope theorem applied to (3). Eq. (8) follows from (5). It is easily verified that (6-8) has a unique solution for the parameters (/?,a,cu) with 0 c a c 1, and that this solution satisfies o>O, c
-a2)]=[1
-6(1--a)2-2a](1
-S),/[l4+6a].
(9)
In order to verify that the path generated by the system (3-5) is the outcome path of a subgame perfect equilibrium, it is sunicient to specify strategies which (a) prescribe behavior that is optimal for each player on all histories that result from no simultaneous deviations, and (b) generate the given path (GSW p. 159). For any history characterized by some state 4 E [0, 1) - such that almost all buyers with valuations exceeding l-4 have bought in the past” - let the strategies of the monopolist and the remaining buyers be given by the price offer function K( ) and the reservation-price function P( ), respectively. Note that any history resulting from no simul#taneous deviations from these strategies can be characterized by some such state 4. It follows from (3) that for any such history, the seller’s strategy is a best response to the strategies so specified for the consumers. Similarly it follows from (5) that each consumer’s strategy is a best response to the strategies thus prescribed for the seller and for the other consumers, respectively. These ‘That is, all such buyers except possibly a subset of measure zerro have bought in the past. The strategies are thus constant on histAes which differ only by different actions on the part of a set of consumers of measure zero. The latter restriction was imposed on the set of strategies for which CSW proved the Cease conjecture.
T.E. Olsen, Duruhle goods monopoly
und the Come conjecture
163
strategies thus prescribe behavior that is optimal for each player on all histories that result from no simultanc=ous deviations. We may then conclude that the associated outcome path is the equilibrium path of a subgame perfect equilibrium. Consider now the case where learning adjustments take place more often than sales. Let n be the number of times there is learning within a period, and let Qi denote the volume produced in the ith of these sub-periods. Assume for the sake of simplicity that there is no discounting within the sales period. [In Appendix A we show that with respect to the asymptotic properties (as S-4) of the equilibrium path, this is an innocuous assumption.] The total cost of producing a quantity Q = Q1 +a - + Qn In one period is then C(q)Q, +C(q+Q,)Q,+***+C(q+Q, +.*+Q,+)Q,, where y is the volume produced in the past. For the linear cost function (2) it is easily verified that the cost of producing Q units in this manner is minimal when the quantities produced within the sub-periods are equal’; Qi=Q/n.The minimal cost is then easily seen to be n
c( I- @Q - c( 1 - l/n)Q2/2. With a slight abuse of notation, define C(q,#; i) by C(q,q’;i,)=c[(
1 -y)--i.(q*
-4)/2],
0si.s
1.
(10)
Given an initial volume of sales 4, the (minimal) cost of producing q1 -4 units in a period with n learning adjustments is thus C(q’, q;ii)(q’ -q), where il= I- (ht. Notice that this cost function exhibits increasing returns to scale with respect to the within-period output quantity y’-y, and that i, can be interpreted as a measure of the degree of such increasing returns.’ Replacing C(g) in (3) with C(y, 4’; A) then leads to the following modifications of eqs. (6-7): -/&i-[p(1-0!)-+&_0(1 -cn=ccx-(/?[i
-x]-c)-2m.
-@+Rca=O,
(6’)
(7’)
amounts to and cumuk~tive effect of a margir,al increase ;:, Qi, iln-I, where cl’= y + Q. A corresponding marginal rrlauc!:on in Q, reduces costs by c’[ 1- 4’ + Qi]v C[ 1 -(q’-- QII)J. Ur.iess Q, .: Q,, it would therefore be pr.,rltable to shift production between subperiods. HWe have included 2.= 1 as an admissible vak in (lo), even though there is no feasible production plcn which actually attains the averar.i cost value C(q,q’; I), no! even if we altow for an infinite r.umber (n= x) of adjustments wi:irin the period. (For any such pian, ccsts can be reduced by shifting production from a hi;;&volume to a low-volume subperiod.) The value is attained. however, if we allow for contiritious production agd iearning within the periorl. ‘The
direct
TX. OPserr,DrrrahCe goods monopoly and
154
Eq. (8)
remains unchanged. while
takes the
ihe Coase conjecture
the eq. (9) defining the parameter 2
z!GW
fcm2-P
These modifications require that aP1saEes take place at the beghing of each period. This may be an unrealistic assum tion when learning adjust-
ments are supposed to take place during each ales period. It is, however, s even if we assume that ail. sales take place at easily seen that (9’) still this modification aEPpayoffs will be reduced by the end of each period. the factor 6 relative to the payoffs given in the text. The relations (3)-(5) must therefore stiII hold, rovided W(q) is interpreted as the value accming to the rmnopolist as of the end of the period, given accumulated sales q in p~&Ql~s
peri&.
We now examine the properties of the above equilibrium when the discount factor 6 approaches 1. An increase in 6 can be interpreted either as agents becoming !ess impatient, or as a shortening of the time period between sales. Recall that in the modei variants given above, a change of the period length dso aflects the ‘learning lag’ in production. A reduction of the period length impEies that both learning adjustments and sales can take place more frequently. in the iimit, as the period length shrinks to zero, learning ad.ustments become in eflect instantaneous. Since the cost structure associated with the simple case of the learning lag being eqiuai to the period length is a special case of (IQ), we consider the equilibrium associated with the latter cost function. From (9’) it is dear that cx= &?+-4l as a---+I. Define
16.5
a*=iEma(S)/(l -is)= l/c-
1.
(11)
b+l
It fdows
from(8)
converges
to
/3 in the reservatbn-pricefuncticsn P&f) c, the coefkierntof Eke learning cwve. T , both P(q) and the
that the coefkient
rice-offer function M(4) = P(q + PA( f - q)) converge to cosE function C(q). ence, for 6 close to I, the price offers along the equilibrium: path are only slightly above unit costs in each period, and in every period only a small fraction of the remaining buyers accept the offer. As 6-s I, the sefler’s profits vanish. It is worth noting that these results hold true for all admissible values of the parameter 2 (02 25 1). Also recall from (IQ) that t.his parameter can be interpreted as a measure of the degree of short run - or, for fixed period length, static - increasing returns in production. In GSW it was suggested that such increasing returns may provide means for credible commitments on the part of the monopolist (GSW, p. 173) The interesting issue is then whether such possible commitments ahow the monopolist is escape from the vanishing profits predicted by the Gase conjecture. The analysis above shows that for any degree of increasing returns represented by 2. being in the range [0, I] in our model, no such commitment effect will be present? Along the equilibrium path, the stock of the durable good evolves according to LJ~+1 = q, + A(y,) = q,[ 1 - r] + 31, 212 0. Starting from 4o = 0, after n periods the stock is qn= 1 -(l
-ct)“,
nz0.
The number of periods required to reach market penetration given by the integer value of In(l -y)/ln(
y~(::(o,1) is then
1 --A),
(12)
which obviously goes to infinity as &+ 1. Now consider the case where the time between ofTen, cal! it E, shrinks. Let r- lim,,,( 1 -&))/1: be the instantaneous rate of discorrnt. The average supply rate per period (r[l - y]/&) then corrverges to rcP[t -4-j as ~40. Again
it
is
WOFfh
notEng
that,
akhough
ah5
kxtson
2
of
the
l%x-miining
166
TX. Olsen, Durable goods monopoly and the Coase conjecture
market that is being served in each period depends on the learning adjustment parameter i. - and is in fact a strictly increasing function of i. .the asymptotic rate of market penetration is independent of this parameter. Since now ( I/E) In ( 1 -o+ - ra* as E+O, it follows from (12) that the time ?E-- tt(q) required to reach market penetration 4 E [O, 1) converges to
t(4)= -(l/ra*)ln(l-@,
a*= l/c-!.
(13)
Thus, in equilibrium all consumers eventually buy the good, but for every finite date there is a lower positive bound on the mass of consumers which remain to be served. Contrary to the standard constant returns case, rhe market is not saturated quickly when there are learning economics in production.’ ’ The slow market penetration illustrated by (13) shows that the full Coase conjecture does not generalize to a technological environment characterized by learning-by-doing. However, recall that we did show that equilibrium prices converge to marginal cost each period. [In equilibrium, the price, the marginal cost and the unit cost in period n are ail multiples of (1 -q,,), and If the coeflicients are given by, respectively; /3(1 -cc), c( 1 - i.or>and c( 1 -h/2).] it is the case that ‘competitive pricing’ yields prices equal to marginal costs in this model, then Coase’s conjecture regarding convergence of monopoly prices to competitive prices remains literally true also for this environment. We shall now give two such interpretations of ‘competitive pricing’. Assuming complete learn’ng Tpiiiovers Between firms (each firm’s ccsts being a function of industry-wide rather than firm-specific experience), St can be seen that unit-cost pricing is sustainable as a subgame perfect equilibrium outcome in Gigopoiy. For expositional reasons we verify this assertion here only for the simple case i, =O, for which marginal and unit costs of course coincide each period. Specifically, suppose there are several firms in the industry. In every period n 20, each firm has a constant-cost production technology characterized by unit costs [C’(q,J], depending only on the total industry-wide volume (4”) produced and sold in the past.’ 2 Let each firm’s strategy be to charge a price equal to the current marginal cost in each “The path q(r) delined by ( 13) is formally similzr to the time-consisient path derived in Malueg and Solow (1990) for a case where there i% a ~~.sitiz~ stock effect on costs. It is easily seen however, that the shape of the time-consistent path in their model critically depends on their assumption th::t marginal costs are increasing In the nrle of output. Thus. although the positive stock effect in!luences the shape of :he path. it is not in itself sufficient to generAte a path exhibiting delay. “This assumption of Iarnin~~, @lovers being complete is unarguably an extreme one, but perhaps not more extreme than ;!:I 2~..amption to the effect that learning is completely firmspecific. i.e. no spillovers. In the cor!‘.zxt of oligopolistic markets for non-durable goods, Stoke); ( 1986) used the former assumption. while Space ( 198 1) and Fudenberg and Tirole ( 1984) used the !at ter.
T.E. Olsen, Durable goods monopoly
and the Coase conjecture
period. Let the buyers’ strategies be given by a reservation PO( ) defined by F(q)[l -S] = P,(q) - SC(q), i.e.
&m=(l
-~+W[l-43,
qE[O,
167
price function
1-J.
By the same type of argument that was given in section 2 it is seen that this strategy combination constitutes (part of) a subgame perfect equilibrium, and marginal-cost pricing is of tours e the equilibrium outcome. Note that the consumers’ strategies are stationary and specified by a reservation price function which is continuous in the state variable q. In this equilibrium, the stock of the durable good evolves according to Yn=
1-(c/Cl -S+Sc])“,
n>,O.
It is easily verified that, as the period length shrinks to zero, the equilibrium path converges to the path specified by (13). Thus, if we interpret the Bertrand-type pricing scheme given here as ‘competitive pricing’, it remains true also for this technological environment that the monopoly equilibrium path will become arbitrarily close to a competitive path if the time period between sales is sufficiently short. The path characterized by (13) can also be interpreted as the Walrasian competitive equilibrium path for a continuous-time version of our model [David and Olsen (1986)]: Suppose there is continuum of firms on the supply side, and that there are complete learning spillovers among these firms. Let P,, t 10, denote the competitive price path. If P, is differentiable and decreasing, a consumer whose valuation is F(q) buys the good at the date t = 7(q) given by r[F(q) - P,] + dP,/dt = 0. Since instantaneous marginal costs are assumed to be constant, at every date we must have price equal to marginal costs. Assuming complete learning spillovers, this condition takes the form
where qr is the stock of accumulated production at date I’. Thus, in equilibrium the stock qt and the price P, must satisfy F(q!) = P, -(dP,/dt)/r and condition (14). Hence, the stock q, must evolve according to
Under suitable regularity conditions (linearity is certainly sufficient), this differential equation will have a unique solution, and the price path defined by (15) is the uique competitive equilibrium path. (Due to the continuum of iQISigrlQfiCantiy to the supplying firms, each firm’S OWIQ prGdQ.QCtiOn COntributeS
Fatal st0ck of accumulated sales, hence each firm can at most obtain
pr&s
of zero under the price path given by ( I a)-( 15). There exist individual produLntion prans which in 1 e aggregate fulfiik~ (15) and yield zero p-ofits, hence these production schemes are individuaily optimal.) Pt then follows from (IS) that the time T= T(q) requited to reach a stock q E [O, I] is given
bY
For the linear specifications
(1 -2) this function clearly coincides with the function given in (13), proving that the eqtdibrium path under monopoly akasian competitive converges to a path whfch Es the continuous-time equihbrium. in this article the c0nvergence resuit has been established only for the Einear case. It seems reasonable to conjecture that a simiiar result holds for more general forms of demand and 1earnEng functions. This suggests the following generalization of the Cease conjecture: For every stationary equilibrium, as the period beiween offers vanishes, the time required to reach market penetration :I ~(0,1) converges to T(q) given by (16). If this holds true, then clearly the behavior of C’(q)/[F(q) -. “‘!rjr)] near y = I will determine wherher the market wiII be saturated in finite time.
The anaIysis of perfect eqrrilibria above
has focused on the case where learnring adjustments :in production occur at leasi 2s frequently as sales. %ntuition seems to suggest FhaF the two main characteristics of the equilibria derived above - i.e. asymptoficaiiy sbw market penetration and vanishing profiis - h0id true than sales. 13
Since
also when iearing
the monopolist
obviously
dees
adjustmenFs take place fess fieqtccniiy
not properly
internalize
the IearCg
Let V(4) denote the optimal value function for the sociail planner. This function satisfies the following dynamic programming equation:
Proceeding as in section 2 we find that W(4) is quadratic: V(q)= (W)vC~ -4]“, and that the optimal quantity to produce in any period is proportional to that fraction of the market which remains to be served; 4l- 4 = y( I - 4), 4 E CO1I]. The parameters v and y can be determined from e first-order and envelope conditions for the social value function V(4). These conditions take the following forms, respectively:
(1 -y)-c-6v(l v = 1 - c - cy +
(17)
-p)+ci,y=@
(w
ci;y.
Two cases must be distinguished. Consider first A< 8. Combining (18) the (second-order) equation defining “Jthen takes the form (1 -c)(l
-6)+“J[G(P -c)-
(I
7) and
1+6(1 ---r)(ll -E.)c+c;l]=O.
It is easily verified that OC;I < P and vr 0. Clearly *;-AI as 6+ 1. Thus, as S+ 1, the eficient fraction of remaining consumers which are being served each period vanishes. owever, since evidently y/( 1 -S)-+ a as ii-, 1, this efficient fraction is asymptotically much larger than the fraction being served under monopoly. Also notice that, since v-+1 -c, the optimal value V(4) converges to the ‘static’ net consumers surplus:
jEF(Q)-c(QPIdQ=(1/2)(1 Y
-C)(P -4)2,
C(4)=C(B -4).
(1%
Nt-xl consider the case A= 1. IRecaP that this can be interpreted as a case where learning adjustments take place infinitely often within each S&X period. From ( i 7) and (l&) we then obtain ;‘= 1 and v = “, -cc’. Tire efficient solution in this case is thus to sup;:+ the ultimate quantity (= 1) in t period. PIotice that the value functnon V(q) is then exactly of the form ( 39). This is an intuitiveiy reasonable outca;xe, since the garticujar learning process associated with L= 1 aFlows for 2 gradual movement “down the
IeartLg cwve !konYi ay q 20 to iq= 1 within one period. The associated. total pr~$~xtio~ costs ..ire then recisely equal to rhe tataS costs obtained in the nisit as the periot.Iength v nishes (and ths learning takes more 5.~4 mcii-e often)for tke case where there are only a finite number OFlearning adjustments within exh period. (2.< 1). ofeach period shrink as in section 2, we see that forany q E [O, 11, the eRicierJ timeto reach market penetration q converges to zero. This ihlustrates aga ..n that the monopolist’s rate of production is too ;low in the presence OFleaxing by doing.
@ace
provides an analysis This sectio~~t mor,opclis”, Y ith commitmerit power,
of
OF a given a technology characterized by learning etieds of the type captured by (10). The optimal policy is found to depend critically on the ‘learning adjustment’ parameter i.. For A< 1 we find that the committed monopolist will spread her sales over an infinite number of periods, ad hence commit herself to intertemporal price-dixrimination. However, although production and sales will be spread over time in the . . commltmcnl-SoEislzG~, the rate of product supply is much larger, and the total vdume of sales is distinctly smaller. than those of the non-committed mGncpoIist andysed in section 2. As the time period between sales vanishes, the rat;’ of Saks for the commitment solution becomes infinite, such that contrary to the nen-commitment case in section 2 - almost all sales take place irl; a vanishgly sma”:I intcrvai of (real) time. f%r th@ skuxkud constant CGSP case it is Well known that the optimal commitment policy invoks no intertemporal price-discrimintion [Stokey ( 1979)]. The intuition fcx why it occurs in the presence of learning enects is that this pricing poky diGWS the monopolist to take advantage of the dynam-c idcreasing returris represented by the learning curve. However, if the parameter 2. tkx the value 1 we know from the previous section that all these learning dfects can “be achieved with one period. It is therefore well in ,XXYX”d Wit{?&ui”,ion thpt for /;;= I the optimal policy is for the monopolist to sell in only OrK period. the
0
ricing
policy
171
[(i -iQF(y,,+Jqr+2 -CEy,,y,,.,)(q,+,-q,)lr’i” c
?i
=
.
5
6
&=O
”
0
Defining v, = 1- qn, the optimal solution for the specifications defined by ( 1) and ( 10) is givenby I)-2(1
tS(l-I,)cv,+I+[ic(G+
-di+6c)~c,,+c(1-i.)v,_,=-(1-6).
(20)
For Rc 1 the solution is of the form 1 -(I/“= _= 1’0.
q*y”+v*, y*=
1/(2--C),
112 I!,
WI
I -v*,
G3
where y
is the unique solution - in the interval (0, 1) - to the (second-order) characteristic equation associated with (20). Fdote that the ultimate market penetration for the committed monopolist is 4* = I- v$ =( 1 - c)/(2 -c), and that this quantity is independent of the learning adjustment parameter R. It is easily checked that q.+ is the unique volume of sales for which ‘marginal revenue’ [F(q) + @‘(4)] is equal to marginal cost C(q). The price sequence associated with (2 1) is given by p,=(l -6)
i
cTk-I F(qn+k)=yDq*y”+a’*,
ri~0,
(23)
k=l
where Dr(f
--@/(I -6y)-+O
as 6-L
(24)
(The limit in (24) follows e.g. from L’ ospital’s rule, noting that y-31 and the derivative y’+oo as 6 + 1.) Thus, yr s are decreasing over time along the optimal sequence, and ultimately approach the valuation of the marginal buyer (v* =F(y,)). observe that for 6 :,-lose to 1, the initial price offer pu is he price oflers are only slightly above the valuation (vJ of this buyer. oes find it advan&xlining over time, thus the committed monopdslist ::;~y,eousto price-discriminate, but the degree of Grscrimination is very SmaH ‘S;r S close to ! . Ht is hard not to verify that, if we let the discount fzxt~r go bctvqeen sales shrink to zero - the limit given by
172
T.E. Olsen, Durable goods monopoly and the Coase conjecture
Notice that there are sales in every period, but as the length (E) of the sales vanishes, the time required to reach any level of market penetration 4, q cq*, vanishes as well: Denoting this time by t:(q), we have period
C(s) = (ln C4*- d/ClnY/(I- 4lM I- 44
4 < 4*-
Thus, almost all sales take place in a vanishingly short interval of real time. This outcome is in stark contrast to the outcome for the non-comitment case derived in section 2, and shows that the slow pace of sales in the latter case is due not ortZ~j to the learning effects in production, but rather to a combination of those learning effects, the seller’s inability to commit her future actions, and the buyers’ expecztations about the seller’s future actions. Recall that, when the period length is short, it is socially efficient to sell and produce at a fast pace. The commitment solution has this property, but it is obviously not efficient. It is easily verified that, as 64, the consumers’ commitment path converges to surplus associated vvA A,h +ho cLLW
(VW-dC(1 -c)/(2-c)12=~CF(q)-C(q)lGq. 0
Given the power to commit her future actions, the monopolist restricts the total volume of sales, and that is clearly the source of the inefficiency in this case. Comparing the commitment and non-commitment paths we may note that the social surplus is smaller in the former case. However, this result is likely to be a particular feature of the linear specifications considered here. Finally consider the case d= 1. It then follows from the optimality condition (20) that t’,= 1/(2-c) = v# for all n. Given infinitely rapid learning within sub-periods, the optimal solution under commitment is thus to induce sales q* in the very first period, and no sales thereafter. The associated total production costs are given by (c -cq,/2)q,, which is precisely equal to the total costs obtained in the limit as the period length vanishes (and thus learning takes place more and more often) for the case where there are only a finite number of learning adjustments within each period (ic< 1). The analysis of equilibrium and commitment output paths in this paper focuses on the case where there exists buyers whose valuation of the good is lower than the minimum achievable cost. Based on the analysis in this section we can, however, draw some conclusions regarding perfect equilibria fr the case where every! buyer’s valuation strictly exceeds the lowest possible cost. To this end, suppose the total market were given by [O,q], where q is strictly smaller than the quantity q1 given in (21). Then obviously the monopolist with commitment power would clear the whole market in the first period, charging a price of p= F(q). (This wou!G also be the optimal policy, given that only consumers in [q, a], some q >O, remain.) Then UXHXiCk~ the non-commitment case, an assume consumers fdow a
T.E. Olsen, Duruhlegoods monopoly and t&e Come conjecture
173
reservation-price strategy given by P(q) =p for all 4 in [0, q]. The monopolist’s best response to this strategy clearly entails charging the price p for all histories with state 4 in [O,q]. it follows that the two strategies so defined are mutually best responses on all such histories, hence the associated outcome path is a perfect equilibrium path. Note that the market is saturated in finite time (after one period), and that this result clearly is due to the ‘gap’ between the demand and cost schedules lor the ultimate buyer (@. For the linear model the ‘gap’ case can generically be represented by restricting the total market to be [O,
6. A bargaining interpretation It is well known that the formalism of the standard (zero cost) durable goods model accommodates the case of bilateral bargaining where a seller with a known valuation repeatedly makes offers to a single buyer with a privately known valuation whose probability distribution is common knowledge. Interestingly thc;le is a similar interpretation of the learning-by-doing model studied in this payer. Consider the following parable: A tract of land which possibly contains oil has been surveyed by an oil company. The surveillance technology is sufftciently sophisticated that the exact amount of oil, and therefore its value is known to the firm. The land owner does not have access to the surveillance technology, and her extraction technology is inferior to that of the company. The value of any oil in the ground to her is therefore only a fraction of the value of the same quantity to the firm. Let c denote this fraction, and assume it is common knowledge that the company’s valuation (v) is uniformly distributed on [0,11. Suppose the parties bargain over the tract in such a way that the land owner repeatedly makes offers to the oil company, who either accepts or rejects each offer. Bargaining is concluded the first time the buyer accepts an offer. Let 4 = 1 -v denote the type of a buyer whose valuation is V, and suppose the buyer follows a reservation-price strategy given by the continuous function P(q). The seller’s optimal expected value U(q), conditional on the buyer’s type being at least 4 then satisfies q’
P(q’)(g’-q)- 5 c(l -Q)dQ+iiU(&)(l-4’) 4
(I--@.
TX. Olsen, Durable goods monopoly
174
and the Coase conjecture
The second term on the RHS is (apart from a normalizing factor) the seller’s expected opportunity cost if the buyer accepts the offer. This cost term is easily calculated to be
cC(l-4H41-4m[q’
-q]*
Note that this expression corresponds exactly to the one period cost incurred by the durable-good monopolist if she produces 4l-4 in the period, given that learning occurs infinitely often; see the discussion following (10). It then follows that the seller’s expected value U(q), conditional on all buyer types with valuations exceeding l-4 having rejected in the past, is given by U(4) = W(q)/( 1 - 4), where W(4) is the equilibrium value for the durablegoods monopolist, given that all buyers with valuations exceeding l-4 have bought in the past, and learning being ‘continuous’ (i,= 1) within each period. Moreover, the strategies given by the reservation-price function P(4) and the price-offer function K(4) defined in (3-5) - with the obvious modification for the cost function C( ) - constitute (part of) a sequential equilibrium for the bargaining game, and the associated outcome path is a sequential equilibrium path. The interpretation given here implies that the two agents have ‘common valuations’ of the good in question. Vincent ( 1989) analysed a similar ‘common values’ bargaining situation, in which the seller is assumed to be the informed party, and the (uninformed) buyer is making all the offers. His anaiysis is confined to the case where there is a ‘gap’ between the buyer’s and the seller’s valuations, for all feasible types. Vincent establishes existence and generic uniqueness of a perfect Bayesian equilibrium in this case, and shows that for a broad class of such cases there will be delay to agreement - even as the time between offers vanishes. The similarity between Vincent’s model and ours - the schedules corresponding to F( ) and C( ) are, respectively, both increiising and both decreasing in the two models - suggests that these results should IX valid also for the ‘gap’ case in our framework. On the other hand, th< results developed here for the ‘no gap’ case should similarly be valid for the corresponding case in Vincent’s model. Moreover, the isomorphism established here between ‘durable goods monopoly with learning-bydoing’ and ‘bargaining with common values’ is helpful in providing additional insights into the reasons why delay occurs in such bargaining situations, because the intuition developed for the former interpretation can be applied to the latter.
..
8l;& +s.‘=main contribution of this article was to show that the coasian 1’...;;?2erties(i.e. competitive pricing asd quick markei penetration) cf equi..!
librium paths in durable-goods monopoly are quite sensitive to changes in the technological environment. This was shown by demonstrating that, for an equilibrium *where buyers’ strategies are stationary, the introduction of some element of learning by doing in production can induce significant changes in the equilibrium outcome path: While, as the period between sales shrinks to zero the time to reach market penetration q converges to zero in the standard constant returns case, this time becomes proportional to - ln (l-q) in the presence of learning effects. Thus, although the limiting outcome path in the presence of learning effects was shown to exhibit competitive pricing (price equal to marginal costs), the market is penetrated more slowly than predicted by the Coase conjecture in this equilibrium. The price sequence chosen by a monopolist with commitment power was shown to generate quick market penetration. From this observation it follows that the slow pace of sales in the non-commitment case is due not on!y to the learning effects in production, but rather to a combination of those learning efects, the seller’s inability to commit her future actions, and the buyers’ expectations about the seller’s future actions. Regarding welfare properties of equilibria, it was shown that both the outcome path for the stationary equilibrium and the commitment path were socially ineficient, - but for different reasons. The latter is inefficient because the associated total volume of sales is restricted, the former because the associated speed of sales is too small. The results in this paper were derived within the framework of a simple linear model under the assumption that learning adjustments in production could take place at least as frequently as sales. We noted that the parameter characterizing the relative frequency of learning adjustments to sales had no impact on the asymptotic properties of the stationary-strategy equilibrium discussed above. Another interesting point was the observation that for one particular value of this parameter :he model becomes formally equivalent to a model of bargaining with ‘common values’. The paper thus extends the set of analogies between durable goods monopoly and bargaining.
In section 2 the claim was made that for the case of ‘earning adjustments occurring more frequently than sales it is not critical to abstract from discounting within each sales period. In this appendix the claim is substantiated I?;) showing that the asymptotic properties of the stationary equilibriurii for this case are not affected when such discounting is taken into accoun? Suppose there are n + 1 learning adjustments within each sales period. Let p = #‘(” I) be the discount factor associated with the learning lag. B;fq md q1 denote the s;cl:ks at the beginning and at the end, respectively, of a sales
perio
en the total
Ci+--q,
L-q'),
discounted productidn
ccst over this period is givept. by
where
CT(U,yl)=EnEn(cv(~-c,)-E_Eecv,(c,
-~?Z~~.*".$.~~~~,(~~,-~‘~))
illi 'I 1 -qi,
by
where qi is t e stock at the end of the ith subperisd. the envelope theorem we have
N0te
that
The objective function is strictly convex for p sufficiently close to 1, and the solutions to the minimization problem are then given implicitly by
Note
that the 21;s are linear
functimsof v
and d, and hence the PotA cost
hnction is quadratic in these variables. Solving explicitly we have uk= C, z: -e-czz;, where z:, z2 are the &mnplex conjugate) roots of z2 - 22 + l/p = 0. Define G,(p) by G,(p) = z!J- z:, m = 1,2,. . . , M -+ I. Some algebra then yields
it is straightforward but tedious to show that the coefficients defined by the khtities abave satisfy
A,
--zY
-4
-B(P
-(jo-.-fl(1
-a).
p_jqein$q-jre~
Lsere W(q) to be the vahe as of the end of the saiesperiod when tracsac~bns actually take place - given accumulated sales q before the period.]These
two quations
yield
From the eq. (8) for j3 and the two IIirnits given above we then see that oc/(l -6)+a0, where c1’ is given by c= l/(ll Jra’)--0, hence 6~O=a*=l/~- 1. ?‘his completes the proof. It is worth noting from this proof that it is the linear structure of the marginal cost functions (w.r.t. both the stock q and the period output q1 -4), apn,Gthe asymptotic behavior of the coefficients of these functions which makes the result go through.
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