A Model of Discovery

Transcription

1 Moel of Discovery Michele olrin an Davi K. Levine 1 his version: December 26, 2008 First version: December 13, 2008 Prepare for the 2008 merican Economic ssociation Meetings. Session itle: Intellectual Property Rights an Economic Growth in the Long-Run Session hair: Nathan Rosenberg Session Discussants: ronwyn Hall an. Zorina Khan bstract: Empirical research has reache the puzzling conclusion that stronger patents o little or nothing to encourage innovation. We show that the facts that have le to the assumption of fixe cost in the iscovery process can be equally well explaine by a stanar moel of iminishing returns. his may explain much of the misunerstaning of the (supposely positive) role of monopoly in innovation an growth, thereby accounting for the empirical puzzle. 1 Department of Economics, Washington niversity in St. Louis, St Louis, MO 63130;

2 1 1. Introuction here is an emerging boy of evience (essen an Meurer [2008], Lerner [2008] an Mokyr [2008], for example) that patents o not have much impact on innovation. It is true that stanar moels of capital laers such as Scotchmer [1991], olrin an Levine [200], an Llanes an rento [2008] allow for the possibility of patents iscouraging innovation. However this can only be a long-run consequence: innovation is iscourage when so many patents have been create that aitional innovation becomes heavily epenent on them. he evience in Lerner [2008], among other, shows that even in the short-run there is no increase in innovation from strengthening patent protection. From the perspective of existing theory this is a puzzle. he stanar theory of innovation is essentially a static theory: incurring a fixe cost creates a unit of knowlege that is then available to all for free. So-calle Schumpeterian moels of knowlege creation, such as those of ghion an Howitt [1992], while ostensibly ynamic, simply stack a sequence of these static moels en-to-en. his stanar approach rests on two technological assumptions: fixe cost of creation an free availability of newly create knowlege. In earlier work (olrin an Levine [2002, 2008b]) we challenge the latter assumption, an we iscusse the evience against it extensively in olrin an Levine [2008a]. However, like the rest of the profession, until recently we accepte the conventional wisom that, following the expense of a fixe cost, knowlege springs full-grown in the form of an eureka moment. he goal of this paper is to challenge such wisom an to argue that iminishing returns in knowlege creation is at least as plausible an has profounly ifferent implications for the impact of monopoly on innovation. he implications for the facts that motivate the stanar fixe cost theory are the same as for that theory. However, from the perspective of a theory of iminishing returns in knowlege creation the failure of government grants of monopoly to increase innovations is not a puzzle at all it is one of the main preictions of the theory. What is the evience that knowlege arises in eureka moments? It is true that innovators work on new ieas for some perio of time before marketing them or implementing them in new proucts. It is also true that, after ieas are brought to market,

3 2 the process of creating original knowlege is replace by the process of making cheap copies. Superficially this suggests that ieas have little value until they reach fruition, after which they are reveale an cheaply copie. Our goal is to show that these same facts are consistent with a iminishing returns technology for the creation of new ieas. he key intuition is that even with iminishing returns an perfect ivisibility, the first few shars of new knowlege the unfinishe notes, the ea-ens that have been encountere, the computer program with many bugs have enormous value in the process of further knowlege creation. his means that even if incomplete or imperfectly polishe proucts are value positively by consumers, it is optimal to keep them off the market for an initial perio of time. Our story is this. Knowlege is encapsulate in perfectly ivisible blueprints. In general, original knowlege creation generates new blueprints from existing blueprints an labor. Initially, though, labor alone is use as no (copies of) original knowlege actually exist; implying that the technology for proucing original knowlege must allow for positive output from labor alone. It is also natural to think that initially the first few bits of knowlege are much more useful in the prouction of aitional new knowlege than in consumption or in being sprea aroun by making copies of themselves. In such a worl, the social optimum is to use labor an all the knowlege acquire so far to increase the amount of new knowlege for a time. Eventually iminishing returns to the prouction of new knowlege sets in: it becomes optimal to prouce cheap copies an consume them; at this point the original knowlege creation process is phase out as the iea is complete an it comes to fruition. If we treate new knowlege creation as a black box an thought of a static moel in which usable knowlege jumpe full-grown out of the box, this woul look like a fixe cost plus cheap reprouction moel. Yet probing the black box, we see that, in fact, this is a stanar iminishing returns economy an has no increasing returns to scale. In the fixe cost plus cheap reprouction story, there is no cumulative process of knowlege creation: there are eureka moments in which a fully usable piece of knowlege appears. However, knowlege creation oes not generally procee in that way. We o not generally write a finishe first chapter first: we write an outline of the whole book, a sketch of each chapter, an so forth. hen we begin the process of revision an polishing until we get to the complete prouct. In oing this we go back to what we

4 3 i before an change it, proucing aitional bits of usable knowlege in an almost seamless path. So it is far from clear that two copies of an incomplete iscovery the notes containing the intuition, the first few experiments, an so forth woul be worth less to consumers than one copy of the complete one. ut regarless, if we take the extreme view that there is an inivisibility that the iscovery is worthless until it is available as an entire unit its inivisibility may not matter simply because the iea was not going to be use anyway until that point. In terms of patents: the implication is that for those proucts an inustries for which iminishing returns is the relevant moel, government aware monopolies, although they will be value by those getting the monopolies, will strictly reuce welfare. Worse, while monopolists may bring their prouct to market earlier, they will o so by skimping on research an evelopment. In this setting iniviual patents will reuce, rather than increase, innovation. 2. Moel of reation an opying We take time b to be continuous. We consier the market for a new prouct that i not previously exist. onsumers erive utility from consuming X units of the prouct, an provie labor,, accoring to the iscounte present value S E ; X W = D. We assume that is strictly increasing, finite, strictly concave, an twice continuously ifferentiable for X SP. Since the goo is not aggregate consumption, but rather a new prouct, we think it is natural to assume that with LIMX l X. Labor is in limite supply with b,. In the moel, knowlege is encapsulate in blueprints K p. Initially there are no blueprints, so K. However, there is an original knowlege creation technology for creating blueprints from labor an blueprints. he simplest an most traitional metho of combining capital an labor might seem the obb-douglas prouction function. However, the obb-douglas cannot prouce any output if the capital input is zero, so we use a perturbe obb-douglas technology to express the fact that new knowlege can be create from labor alone K!K I O O

5 where! I, an K p p are the blueprints an O labor use in creating original knowlege. Notice that once labor is allocate to the original creation process, the marginal prouct of blueprints is very high (infinite at KO ). his is the opposite of the usual assumption. In stanar theory it is assume that nothing of value is prouce until a threshol is reache. y contrast we assume that the very few first bits of knowlege incomplete sketches, intuitions, an so forth are extremely valuable in the prouction of aitional original knowlege. he apple contribute great value to Newton s unerstaning of gravitation. We assume there is also a technology for the inexpensive copying, or imitation, of blueprints. his is given by K "K, where " S an K are the blueprints use for making copies. his is like a copying machine, or the competitive market: put in an original or copy, an some number " of new copies are prouce. s it is often the case, original knowlege an its imitations are perfect substitutes. Lastly, blueprints can be use to prouce a flow of consumption. It is convenient to assume that the units are chosen such that X K X where K X are the blueprints use to prouce consumption. We can summarize the accumulation technology by the equation of motion for the total amount of blueprints available K!K I " K X K O O along with the constraints K X p K X K p. Notice that the rate of O O increase of knowlege capital is boune by K MX[!K b I, "K ], hence there is a function the sense that 2 + such that K b + S E + D.. We assume that utility is finite in X n example of a function satisfying our assumptions is X E. 3. haracterizing the Optimal Plan We first give a technical characterization of the optimal plan. 2 Note that this forces LIMX l X so that assumption is reunant.

6 5 Proposition 1: unique continuous optimal plan [ K K X ] exists an is O characterize by Lagrange multipliers M p that evolve accoring to M M S! K ± O an satisfy the transversality conition E S M K l, an by first orer conitions M! K I p W, with equality if, O X! K O b M ±, with equality if X b O ±, with equality if K X K "! K O Proof: See the ppenix. ; Our primary goal is to unerstan how the technology for original creation interacts with the copying technology an with consumption. Our key result is that, initially, only the original creation technology is use an, for a while, blueprints are not use either for copying or consumption. Proposition 2: here is a time has K K. X such that if b b then in the optimal plan Proof: From the first orer conitions an X b, if (*) M! K! K " O O then the optimal plan is KKX. In particular, since the optimal plan is continuous, it suffices to prove that these inequalities hol as K l. Notice that, by continuity, M l M. If, the result follows immeiately. Otherwise we may solve the first orer conition for the optimal use of labor, O M! K I W ± O an substitute it into (*) to fin the conitions! K I O M! K O W ±

7 6! K O! K " O I M W ± from which the result again follows as K l an M l M O. ; Our secon result shows that the use of the original creation technology is temporary in the sense that asymptotically it is not use at all, an after a point in time, some capital is always use for consumption. Proposition 3: In the optimal plan: M is ecreasing, LIM l M, X is increasing, LIM l X, an LIM l K l. O Proof: Observe that M M S! K O ± an " b! K O ± imply M M b S ". Notice that M for. Hence not only is M ecreasing, but by integrating both sies of the inequality, it satisfies a boun of the form È" SÉÈ É M b M E, so certainly LIM l M. Next suppose for S that X S X. he labor supply can be solve from the first orer conition as ² M! K I O M K MIN, O», W ± ¼, from which! K M K O O ² È ÉÈ É! K IK O O ž žÿ O» W ± ¼! MIN K, M his is increasing in M an ecreasing in K O. Since X from the first orer conitions X! K M M K O O ±. Since X X an since M is ecreasing it S follows from X! K b M M K S S OS S OS ± that! K M K! K M K OS S OS O O an so K S K p. his implies

8 7, "! K M K! K M K p " OS S OS O O a contraiction. Next suppose that X is boune away from zero, say by. From M M S! K O ± an X b M! K O ± we have M b SM. Since LIM l M, eventually M S implying that M after a certain time, which is impossible. Next observe that " b! K, implies an upper boun on K O. Since O M l an K is boune above we see that this implies l. hen the first orer conition " b! K also implies K l. O O ; he role of the copying technology is less straightforwar, in accorance with available evience. If marginal utility, hence eman, falls rapily to zero, it may be that the copying technology is never use an knowlege goes irectly from the original creator to the consumers, without imitators stepping in to copy. We woul expect copying to be relevant when there is strong eman for a large number of copies of the consumption goo. Specifically, let us efine strong asymptotic eman. WX Definition : We say that eman is strong asymptotically if X p WE. X n example of a utility function with asymptotically strong eman is X E. We can then show that, while asymptotically the original creation technology is abanone, the copying technology is use that is, when eman is strong, after a time copying takes over the iscovery process an imitators enter the market. Proposition 5: If eman is asymptotically strong, then in the optimal plan LIM SPK. Proof: s before, we observe from the first orer conitions that M p S " M an that, for H " S an ' M E H, this implies that M b 'E H. Next observe that since X for large we have, for all sufficiently large, that M SM X. Define Since 5 M E SÈS É X DS S w.

9 8 an M l b E SÈ S É X DS b S, we see that 5 is boune asymptotically. Since 5 S5, it follows that 5. ombining these two steps, we see that for large enough we can write M b as 'E H onsequently, for any SÈ É H E X DS 'E b S. SS E X DS b 'E H S S ES XS S DS H b 'E H XS DS S H b 'E S E X DS b 'E 'E MINbb S XS b 'E MX bb S XS b S H 'E K b y assumption K WK WE p, so WE 'E S H WK S H b. S H aking logs LOG' W S H K p. W W Since K l we must have K l, hence LIM SP K, implying O LIM SPK. O S ;

10 9. Impact of Monopoly on Innovation Suppose that a single monopolist controls the market for this prouct, an that she maximizes the present value of her profits. his amounts to replacing X with 2 2 V X X X in the optimization problem. Notice that revenue V X may actually be ecreasing for large X an may fail to be concave. ssuming that it is concave an that X is three times continuously ifferentiable, we may efine 2 VX MXX bx V XS. hen VX S satisfies the properties we have assume of a 2 utility function, an since X will never be chosen so large that VX v V X yiels the solution to the monopoly problem. he key ifference with the original problem is that VX has a lower marginal utility of consumption than X at all levels of X, because V X X X X. We can evelop some simple intuition about the impact of introucing a monopoly on the timing an nature of innovation. se superscripts an V to enote, respectively, the solution to the competitive an the monopolistic problem. Let be the time for which X for an X for b. Notice that M E V X D. V SÈ É V If this remains unchange the monopoly solution, X V, must involve less final prouct sales than uner competition, X for p. herefore, there must be less capital uner V monopoly than competition, K K. here are two ways to prouce less capital an the optimum will require both be use. First, less labor shoul be use in the original creation V V process for b. Secon, X shoul become positive earlier than X. One coul say that, in this sense, innovation takes place more quickly uner monopoly than competition because the new consumption goo is brought to the consumers earlier. However, a smaller quantity is sol an at a higher price; further, less V labor is use in original creation,, meaning that the monopolist also prouces less original knowlege an oes less R&D than uner competition. In this, more relevant, sense the monopolist innovates less than uner competition. One way to think of this is in terms of the public-private partnership uner which universities are encourage to patent ieas evelope using government funing. y awaring a monopoly we woul expect less actual research to be one at universities, but the results of the research that i take place woul be mae available to inustry

11 10 sooner. It is claime that the public-private partnership has been a great success because of the latter. In this moel, that is unambiguously ba, as scientific resources (K ) are misallocate to inustrial applications when it woul be better, from a social point of view, to use them in proucing more original research that woul, optimally, be brought to inustrial fruition somewhat later. 5. onclusion he stanar theory of innovation base on the fixe cost of iscovery plus cheap copying preicts that strengthening patents shoul increase innovation. Overwhelming empirical evience shows that strengthening patents oes little, or nothing, positive to innovation, which is most puzzling for stanar theory. We argue that, even if it looks like there is a fixe cost of creation, in fact there is none. We argue instea that the iscovery activity is best represente by a ecreasing returns technology in which the first few units of new knowlege are so valuable that, for a while, they are optimally investe in proucing further knowlege instea of making copies of themselves or proucing consumption. We evelop a moel that captures this intuition an is consistent with a set of wiely hel facts about innovative activity. his is a moel of competitive iscovery with copying uner ecreasing returns to scale; in this worl, introucing a patent is amaging to welfare: it oes not increase the rate of innovation an may even reuce it. he wiely iscusse puzzle, accoring to which stronger patents o not increase innovative activity, is no longer a puzzle. ppenix: Proposition 1 he result is relatively stanar, an can be erive by efining 6K to the sup of utility achievable with continuous paths of the controls starting at the initial capital K. his is strictly increasing an strictly concave, so quite well behave from a ifferential point of view. We can erive the optimal conitions by solving S E 6 K MXX K K X E S 6K K 6K O LIM ± l subject to K!K I " K X K. he Lagrange multipliers (or costate O O variables) are just M 6 K.

12 11 References ghion an Howitt [1992]: Moel of Growth hrough reative Destruction, Econometrica, 60: Meurer, M. J. an J. E. essen [2008]: Do Patents Perform Like Property?, oston niv. School of Law Working Paper No olrin, M. an D. K. Levine [2002]: he ase gainst Intellectual Property, merican Economic Review Papers an Proceeings, 92: olrin, M. an D. K. Levine [200]: he Economics of Ieas an Intellectual Property, Proceeings of the National caemy of Sciences, 102: olrin, M. an D. K. Levine [2008a]: gainst Intellectual Monopoly, ambrige niversity Press. olrin, M. an D. K. Levine [2008b]: Perfectly ompetitive Innovation, Journal of Monetary Economics, 55: Lerner, J. [2008]: he Empirical Impact of Intellectual Property Rights on Innovation: Puzzles an lues, mimeo, Harvar usiness School. Llanes, G. an S. rento [2007]: nticommons an Optimal Patent Policy in a Moel of Sequential Innovation, mimeo, Harvar usiness School an niversitat utonoma arcelona Mokyr, J. [2008]: Intellectual Property Rights, the Inustrial Revolution an the eginnings of Moern Economic Growth, mimeo, Northwestern niversity Scotchmer, S. [1991]: Staning on the Shoulers of Giants: umulative Research an Patent Laws, Journal of Economic Perspectives, 5: 29-1