Learning from the experience of others: Parameter uncertainty and economic growth in a model of...

*Corresponding author. Tel.: #1-713-743-3798; fax: #1-713-743-3798.E-mail address: [email protected] (P. Thompson).

Journal of Economic Dynamics & Control24 (2000) 1285}1313

Learning from the experience of others:Parameter uncertainty and economic growth

in a model of creative destruction

Peter Thompson*

Department of Economics, University of Houston, Houston, TX 77204-5882, USA

Received 1 April 1998; accepted 29 March 1999

Abstract

This paper analyzes a quality-ladder model of economic growth incorporating uncer-tainty about the e$ciency of R&D. A central premise of the paper is that designingappropriate technology policies is more di$cult when one is at the cutting edge oftechnology. In technological laggards, information gleaned from observations of moreadvanced countries provides noisy signals about the e!ort required to develop a speci"cproduct generation, and aids in the design of policy. The paper shows how these signalsin#uence growth rates and technology policies. ( 2000 Elsevier Science B.V. All rightsreserved.

JEL classixcation: O3

Keywords: Growth; Quality ladders; Information

1. Introduction

This paper analyzes a quality-ladder model of economic growth incorporat-ing uncertainty about parameters governing the e$ciency of R&D, and learning

0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 1 8 - 4

about those parameters. A central premise of this paper is that designingappropriate technology policies is more di$cult when one is at the cutting edgeof technology. In technological laggards, information gleaned from observationsof more advanced countries provides noisy signals about the e!ort required todevelop a speci"c product generation, and aids in the design of policy. Firms inless advanced countries are therefore guided more accurately to undertakeresearch of appropriate intensity, and planners can set more precise policies. Inshort, there is a greater correlation in technological laggards between ex anteand ex post optimal behavior in both competitive and e$cient equilibria.

In order to highlight the e!ect of information on comparative growth,I construct a model which strips away &traditional' sources of internationallinkages that have been extensively analyzed elsewhere. Thus the model econ-omy contains no physical capital, no trade, no foreign direct investment, and notechnology transfer in the traditional sense of imitation. What is left is a novelsource of international growth linkages in which di!erences in average growthrates are driven only by the ability to observe parameters in more advancedcountries. The model explicitly acknowledges the conventional wisdom thattechnology policies are somehow easier to design when one is not at the frontier,a feature of comparative growth that has not previously been explored.

I develop a version of the quality-ladder model due to Aghion and Howitt(1992), Grossman and Helpman (1991) and Segerstrom et al. (1990). The mainextension in this paper is that the e$ciency of R&D in the development of eachproduct generation depends on a random variable that is assumed to becorrelated across countries, and in contrast to previous work I focus primarilyon the non-stationary equilibria that arise among technological laggards. Thelearning mechanism by which technological laggards update their beliefs aboutparameters governing the e$ciency of R&D e!ort is very general and may be ofindependent interest. It allows for imperfectly observable R&D e$ciencies inmore advanced countries as well as imperfect correlation of R&D e$cienciesacross countries. Moreover, learning need not be optimal in any statistical sense,although two examples with Bayesian updating of prior beliefs are provided.

The main results of the paper concern the properties of the model in technolo-gical laggards that adopt optimal policies, and those that follow a laissez fairepolicy. First, technological laggards that observe no signals about the e$ciencyof R&D grow at the same rate, on average, as countries at the world technolo-gical frontier. Signals need not raise the expected instantaneous growth rate.However, su$cient conditions are derived for signals to raise the expectedgrowth rate of a laggard adopting an optimal policy, and these conditions seemplausible. Speci"cally, signals enable lagging countries to grow more rapidly onaverage when returns to scale in R&D e!ort do not diminish too rapidly andparameters yield R&D employment that is a small fraction of the total laborforce. In fact, under these conditions signals generate higher growth even if thesignals turn out to be completely uninformative. The more informative the

1286 P. Thompson / Journal of Economic Dynamics & Control 24 (2000) 1285}1313

signals are, the greater is the parameter space for which signals raise growth inthe e$cient equilibrium.

The model is less informative about expected growth under laissez faire.I have not been able to establish analogous conditions under which signals raisethe expected instantaneous growth rate, nor have I been able to generatea counterexample. Nonetheless, while the implications of signals for catch-upremain an open question under laissez faire, there are some interesting "ndings.Prominent among these is a contrast between the e$cient and laissez fairemodels in the e!ect that news about the e$ciency of R&D in the next race hason R&D e!ort in the current race. News suggesting that R&D will be parti-cularly e$cient in the next race encourages the social planner to raise R&De!ort in the current race: the news makes it more desirable to complete thecurrent race. In contrast, the same news lowers R&D e!ort in the laissez faireequilibrium, because it implies that the winner of the current race is likely toenjoy monopoly pro"ts for only a short period of time. That is, signals about thee$ciency of R&D in future generations that raise the expected intensity of R&Din the next race are good news from the perspective of the social planner, but badnews for "rms participating in the current R&D race.

Finally, the model also has something to say about policy variability. At theworld technological frontier, the optimal intervention is a subsidy to R&D thatremains constant from one product generation to the next. However, amongtechnologically lagging countries the optimal intervention may be a subsidy ora tax, and it is a random variable that varies across product generations. Amongcountries that adopt e$cient policies, therefore, one should expect technologypolicy in laggards to become more similar over time to those adopted at thefrontier. Indeed, just this process seems to have been taking place in the rapidlygrowing Asian economies. Knowing when and where to support R&D andtechnology development seems to have been closely related to growth perfor-mance. Yet, as these countries' technological capabilities have advanced towardthe world technological frontier, many distinctive features of their technologypolicies have been abandoned. There are of course too many ingredients totechnology policy to allow a succinct summary of the secular trends here; but itseems clear that Singapore and Japan e!ectively began to abandon targetedtechnology support more than a decade ago, and that more recently SouthKorea and Taiwan have been following suit. To many observers, the decline oftargeting in the advanced Asian economies is a symptom of their success. Asindustries advance toward the technological frontier, the argument goes, itbecomes more di$cult to identify promising technological avenues, and conse-quently to decide which technologies merit government support. It is just thesefeatures of policy intervention that the present paper attempts to formalize.While readers may conclude that the pure information e!ect studied in thispaper is less important for growth than the traditional prescriptions of moresaving and more education, it does, I think, matter.

P. Thompson / Journal of Economic Dynamics & Control 24 (2000) 1285}1313 1287

The layout of the paper is as follows. Section 2 presents the model,The competitive equilibrium is characterized and contrasted with the solutionto the social planner's problem. Section 3 describes how signals frommore advanced countries inform technological laggards about the productivityof their own R&D programs. Section 4 presents the main results on com-parative growth, and Section 5 concludes. All proofs are provided inAppendix A.

2. The model

I employ a simple quality ladder framework, based on the seminal work ofAghion and Howitt (1992), Grossman and Helpman (1991), and Segerstromet al. (1990). The world consists of a number of countries, each consisting ofa single manufacturing sector and a research sector. The research sector in eachcountry is characterized by a sequence of patent races, each aimed at improvingthe quality of the country's existing state-of-the-art product. The duration ofa patent race depends in part on a random variable that governs the e$ciency ofR&D, and whose value is not known with certainty during the race. The winnerof a patent race employs limit pricing to secure a temporary monopoly that lastsuntil the next innovation. Only the highest-quality product developed withina country at any point in time is manufactured. There is no trade, and there areno transfers of technology across countries. That is, all knowledge required tomanufacture a given product generation must be developed locally. The onlysense, therefore, in which the model might be interpreted as a multi-countrymodel is that countries may be able to learn something from other countriesabout the value of the R&D e$ciency parameter. The remainder of this sec-tion considers an arbitrary country currently engaged in developing productgeneration q.

2.1. Consumers

A representative consumer maximizes the expected present value of lifetimeutility,

max EtP

=

t

e~o(q~t) ln u(q) dq, (1)

subject to an intertemporal budget constraint,

AQ (t)"r(t) A(t)#w(t)¸#n(t)!m(t), (2)

where A(t) is consumer's wealth, n(t) is "rm pro"ts which accrue to the represen-tative agent, and m(t) is the #ow of consumption expenditure. The consumer is


1Throughout the paper, superscripts denote powers of variables and subscripts are used forindexing purposes.

endowed with ¸ units of labor earning a wage rate of w(t). The consumptionindex satis"es1

u(t)"Cq(t)~1+j/1

jjxj(t)D, (3)

where xj(t) denotes consumption of the jth generation of the good, j'1 is the

proportional improvement in quality between consecutive generations, andq(t)!1 denotes the number of product improvements that have been madeavailable in the country by time t.

Homothetic preferences ensure separability of expenditure and prices, and theconsumer's problem can be solved in two stages. In the "rst stage, the consumerallocates expenditure, m(t) across products. It will be assumed below thatmarginal cost is equal to w(t) for all "rms, and that each generation of goods canonly be produced by the single "rm that holds the relevant patent. The prefer-ences given by Eq. (3) imply that quantity and quality are perfect substitutes.The consumer therefore purchases only the single good with the lowest quality-adjusted price. Following standard practice, I assume that when quality-adjusted prices are equal, the highest quality product is consumed. Bertrandcompetition in this setting induces the holder of the national state-of-the-artpatent to set a limit price of jw(t). Only the state-of-the-art product is consumedand the monopoly leader captures revenue of m(t). In the second stage, theconsumer allocates expenditure over time, which satis"es the familiar Eulerequation, m5 (t)/m(t)"r(t)!o. I let expenditure be the numeraire so thatm(t)"1. It then follows that the market rate of interest, r(t) is always equal to thediscount rate, o.

2.2. Firms

Manufacturing requires only labor, ¸x(t), and one unit is required to produce

one unit of output of any good. The owner of the state-of-the-art patent setsa markup of j over marginal cost, which secures a temporary monopoly andyields instantaneous pro"ts n(t)"(j!1)m(t)/j"(j!1)w(t)¸

x(t). All "rms

owning patents on earlier product generations earn zero pro"ts. As m(t)"1, itfollows that w(t)"1/j¸

x(t) and n(t)"(j!1)/j.

Denote with vq(t) the discounted pro"ts of the successful innovator of the qth

generation product. R&D also requires only labor. Let ¸q(i, t) denote the labor

"rm i devotes to the race for the qth patent, and let ¸q(t)":¸

q(i, t) di denote


2As dt is arbitrarily close to zero, aq(t)¸

q(t)b dt will also be arbitrarily close to zero almost

everywhere as long as E(aq) is "nite and ¸

q(t)b is bounded. The latter is bounded by the resource

constraint, while the former is assumed below.

3Note also that this formulation of the R&D technology implies that individual "rms faceconstant returns to scale in R&D, leaving their size indeterminate, but returns to scale in aggregateR&D are diminishing. The R&D technology allows us to restrict attention to a representative "rm,irrespective of aggregate returns to scale in R&D.

4Aghion and Howitt (1992) consider an example in which aq

is random but its value is knownduring the race.

5The assumption of zero correlation across product generations is made for analytical conveni-ence. The key insights of the model are not sensitive to the introduction of serial correlation acrossproduct generations. What is central to the paper is the assumption that countries have something tolearn about a

qfrom other countries.

6More precisely, one should write Fn(q,t)

(aq) to re#ect the fact that the number of signals varies

with calendar time and product generation. I shall use the shorthand notation to avoid unnecessaryclutter.

aggregate labor devoted to the race. The expected discounted pro"ts of "rm i are

E[vq(t)a

q] ¸

q(t)b dt

¸q(i, t)

¸q(t)

!w(t) ¸q(i, t) dt, (4)

which equals zero under the assumption of free entry to the patent race. Theterm a

q¸q(t)b dt, b(1, is the probability2 that the patent race will be won in the

next momentary interval dt, and ¸q(i, t)/¸

q(t) is the probability that, if the race is

won, "rm i will be the winner. Innovations are Poisson events with a time-varying intensity a

q¸q(t)b dt, and the duration, q, of the race for product genera-

tion q is given by the time-varying exponential distribution expM:q0aq¸q(t)b dtN,

where t"0 denotes the time at which the race began. For any "xed ¸q, a

qis

proportional to the arrival intensity, and may naturally be interpreted asa measure of R&D e$ciency.3

The R&D e$ciency parameter aq

is not known with certainty during thepatent race.4 Before and during a race, "rms assign to a

qa subjective distribu-

tion Fn(a

q). The main features of a

qare: (i) it is independent across product

generations, and (ii) it is correlated across countries.5 Thus, while a country atthe technological frontier must make decisions based only on the prior distribu-tion of a

q, countries which have the opportunity to look ahead to other

countries' experiences with the same generation of technology will have a betteridea about the value that a

qis likely to take. The subscript n denotes the number

of observations available on aq

and at any point in time it may vary acrossproduct generations.6 These characteristics of F

nwill be explored in detail in

Section 3. However, it is useful at this stage to make some assumptions aboutthe prior distribution (or, equivalently, the distribution of a for a country at thetechnological frontier):


Assumption 2.1. Let F(a),F0(a) denote the prior distribution of a, let E(a)(R

denote its prior mean, and let ¹ denote the product generation being developed atthe world technological frontier. (i) F(0)"0, (ii) F(a) is diwerentiable, and (iii)F0(a

T`j)"F(a), ∀j50.

Assumption 2.1 restricts a to the positive half line (thereby ensuring that nocountry enters a no-growth trap), and states that no observations are availablefor any product generation not yet developed somewhere.

The memoryless property of the R&D production function implies thatinnovations are independent Poisson events. As long as no new informationabout the value of a

qis received, the intensity of R&D is constant for the

duration of any race and the time to the next innovation is exponentiallydistributed. Note also that v

q(t) } the value of winning the qth race } depends on

the duration of the (q#1)th race. Thus, conditional on aq`1

, the expected valueof winning the qth race can be written as

En(v

q(t)Da

q`1)"P

=

0CP

x

0

e~oqn(t) dtD[aq`1¸q`1

(t)b

]expM!aq`1

¸q`1

(t)bN] dx

"

(j!1)

j[o#aq`1

¸q`1

(t)b]. (5)

Integrating Eq. (5) over all possible values of aq`1

yields the unconditionalexpected value of winning the qth race, E

n(v

q(t)). Then, combining with the zero

pro"t condition Eq. (4), the equilibrium intensity of aggregate R&D satis"es

¸q(t)1~b"

(j!1)En(a

q)

w(t)j P=

0

1

o#aq`1

¸q`1

(t)bdF

n(a

q`1). (6)

The intensity of R&D in the race for the qth monopoly depends not only onthe expected value of a

q, but also on the outcome of the race for the next

generation. The term aq`1

¸q`1

(t)b dt denotes the probability that the race forthe (q#1)th monopoly ends in the interval dt, where ¸

q`1(t) denotes the

intensity of R&D in the (q#1)th race that is believed to be optimal given theinformation available at time t; F

n(a

q`1) denotes the current subjective distribu-

tion for aq`1

. The instantaneous probability that the qth monopoly ends itstenure enters as an addition to the rate of discount. The expected payo! to thewinner of the qth race is therefore equivalent to an instantaneous pro"t #ow,(j!1)/j, earned in perpetuity with an interest rate of o#a

q`1¸q`1

(t)b.

2.3. The competitive equilibrium

Substituting for w(t) in Eq. (6) and using the full-employment constraint,¸x(t)#¸

q(t)"¸, the competitive equilibrium R&D intensity during the race for


generation q is de"ned by the "xed point expression,

¸q(t)1~b"(j!1)(¸!¸

q(t))E

n(a

q)P

=

0

dFn(a

q`1)

o#aq`1

¸q`1

(t)b. (7)

For technological laggards, laissez faire equilibria must satisfy the non-linear "rst-order di!erence equation written in implicit form in Eq. (7). Bound-ary conditions are given by equilibria that apply at the world technologicalfrontier.

The "rst lemma establishes that there is a unique stationary equilibrium at thefrontier.

Lemma 2.2. A country at the technological frontier has a unique stationary com-petitive equilibrium with positive R&D intensity, ¸c

T, satisfying

(¸cT)1~b

¸!¸cT

"(j!1)E(a)P=

0

dF(a)

(o#a(¸cT)b)

. (8)

Periodic equilibria at the world technological frontier cannot always be ruledout in this model. One can prove their existence by following the analysis inAghion and Howitt (1992), (Section 3A). However, I will follow Aghion andHowitt (1992), and others, in restricting attention to the unique stationaryequilibrium at the world technological frontier.

I am now in a position to provide conditions for the existence of a uniquecompetitive equilibrium during the race for product generation q(¹ in atechnological laggard:

Theorem 2.3. Assume there is a unique sequence, Fn(a

q), F

n(a

q`1),2, F

n(a

T~1), of

subjective distributions with positive support. Then, there is a unique competitiveequilibrium intensity of R&D, ¸c

q, during the race for generation q(¹.

Theorem 2.3 states that if the intensity of R&D at the world technologicalfrontier is uniquely de"ned then so is the intensity of R&D during the race forthe current product generation in a technological laggard. The intuition behindthe absence of multiple, periodic equilibria in laggards is straightforward. Thepossibility of cyclical equilibria at the frontier arises because there is no bound-ary condition to pin down the value of ¸

T`jfor any j'0. In contrast, restricting

attention to the unique stationary equilibrium at the technological frontier pegsthe value of ¸c

Tprecisely and provides a unique boundary condition that de"nes

unique values for R&D intensities in races to develop less advanced technolo-gies in lagging countries.


2.4. The social planner

This section characterizes the e$cient intensities of R&D. It is well knownthat laissez faire equilibria in this class of models are not typically e$cient.Aghion and Howitt (1992), Dinopoulos (1994), and Grossman and Helpman(1991), among others, discuss the opposing externalities that lead to an ambigu-ous relationship between competitive and e$cient R&D intensities. In this andthe following subsections, the externalities that have been identi"ed previouslyin steady-state analysis will be derived for technological laggards. In addition,however, there are some important di!erences between the competitive ande$cient intensities of R&D that do not arise when attention is restricted tocountries at the technological frontier.

Indirect utility in this model is given by

C(t)"P=

t

e~o(q~t)[(q(q)!1) ln j#ln(¸!¸q(q))] dq, (9)

where q(t) denotes the product generation for which researchers are racing attime t. Thus, the consumption good at time t is indexed by q(t)!1.

The social planner maximizes the expected value of Eq. (9) subject to thetechnological conditions governing the rates of innovation; q(t) is an integer-valued step function where the steps are formed from a Poisson process withmagnitude one and intensity a

q¸

q(t)b. The Bellman equation for this problem is

maxLq(t)

o<q(t)"(q!1) ln j#ln(¸!¸

q(t))

#[<q`1

(t)!<q(t)]P

=

0

aq¸

q(t)b dF

n(a

q), (10)

where <q(t) denotes the value function at time t when researchers are racing for

product generation q. The right-hand side of the Bellman equation consists ofthree terms. The "rst two terms describe the #ow of consumption bene"tsreceived during the race for product generation q. The third term is the jump inthe value function that occurs when generation q is developed, multiplied by theexpected probability that the innovation takes place in the next instant. Themaximized sum of these three terms equals the interest o<

q(t) that can be earned

on a risk-free bond of size <q(t).

Di!erentiating Eq. (10) yields the "rst-order condition

¸Hq(t)1~b"bE

n(a

q)(¸!¸H

q(t))[<

q`1(t)!<

q(t)]. (11)

Although the social planner's problem appears simple, its solution is not easilyobtained. In general, the subjective distributions, F

n(a

q), are not stationary over

product generations, and standard dynamic programming techniques for


7The comparative statics in Lemma 2.4 are familiar from analogous results in Aghion and Howitt(1992) and Grossman and Helpman (1991).

8The clearest prior statement of this result is given by Grossman and Helpman (1991), (pp. 104,105) for the case where b"1 and a is known.

stationary problems apply only for countries at the technological frontier.Following the analysis of Section 2.3, I "rst characterize the solution that appliesat the frontier (Lemma 2.4). E$cient and laissez faire rates of growth at thefrontier are compared in Proposition 2.5. Existence and uniqueness of thesolution to the planner's problem is then established for technological laggardsby backward induction (Theorem 2.7). In the following subsection, I comparethe e$cient and laissez faire R&D intensities for technological laggards, andcharacterize the optimal subsidy to R&D (Proposition 2.8).

Recall that ¹ is the lowest index of undiscovered technologies in the world.For a country attempting to develop generation T the solution to Eq. (9) takesthe form o<

T"A#(¹!1) ln j, where A is a coe$cient to be determined.

Thus, <T`1

!<T"o~1 ln j and ¸H

Tsatis"es the "xed-point expression

¸H1~bT

"

b ln j E(a)(¸!¸HT)

o. (12)

It is easy to see that a unique interior solution to Eq. (12) exists, in which case<

Tsatis"es

o<T"(¹!1) lnj#ln(¸!¸H

T)#

E(a)¸HT

ln jo

, (13)

where ¸HT

is the increasing function of E(a) de"ned in Eq. (12). Lemma 2.4 followsimmediately.7

Lemma 2.4. At the technological frontier, there is a unique solution to the socialplanner's problem, with R&D intensity increasing in b, j and E(a), and decreasingin o.

Let EgcT

denote the expected growth rate of quality at the frontier underlaissez faire and let EgH

Tdenote the corresponding expected growth rate in the

e$cient equilibrium. The following proposition is readily established.8

Proposition 2.5. If ¸ is suzciently large, there exists a pair of values,1(j

0(j0(R, such that iw j3[j

0, j0], then EgH

T'Egc

T; j

0[j0] is decreasing

[increasing] in ¸.


Proposition 2.5 highlights the fact that economy size is critical in determiningwhether or not the competitive equilibrium generates a faster growth rate than issocially optimal. Jones and Williams (1997) have calibrated a simple R&D-based growth model to US data. They concluded that the optimal intensity ofR&D may be as much as four times larger than the current intensity. Followingtheir lead, it will be assumed throughout the remainder of this paper thatj0(j(j0.

Assumption 2.6. EgHT'Egc

T.

I turn now to the social planner's problem for technological laggards. First, itcan be con"rmed that a unique solution exists:

Theorem 2.7. Consider a country currently racing for product generation q(¹,that has posterior means E

n(a

q), E

n(a

q`1),2, E

n(a

T~1) of the R&D productivity

parameters. Then, (i) there is a unique solution to the social planner's problem thatdepends on the sequence MME

n(a

j)NT~1

j/q, E(a)N; (ii) R&D intensity is increasing in

each element of the sequence.

Theorem 2.7 highlights the forward-looking nature of the social planner intechnological laggards. If the planner expects R&D e!ort in future generationsto be especially productive, then he will accelerate the current R&D racein order to bring forward the expected arrival date of those future generations.Conversely, a pessimistic outlook for future R&D races retards the currentgrowth race. Even though technological leapfrogging is excluded fromthis model, the planner's current R&D policies always depend on all technolo-gies that have been developed elsewhere but that remain to be developedat home.

Note also that the solution to the social planner's problem depends only onthe sequence of subjective means, while the competitive equilibrium intensity ofR&D depends on all moments of the subjective distribution F

n(a

q`1). The

intuition behind this result is straightfoward. Firms engaging in the race togenerate product generation q care about the expected present value of themonopoly pro"ts that accrue to the winner. As the expected present value ofmonopoly pro"ts is a nonlinear function of a

q, all moments of F

n(a

q`1) enter into

the solution. In contrast, the social bene"ts of any innovation planner lastinde"nitely, so that the duration of the monopoly for product generation q is notof direct concern to the social planner.

2.5. Comparing ezcient and competitive equilibria in technological laggards

There are several important di!erences between R&D intensities in thee$cient and competitive regimes, which can conveniently be illustrated by


9Use Eq. (11) in Eq. (10) to remove <q(t). This yields an expression relating ¸H

qto <

q`1. Then

update (11) and (10) by one product generation and combine them to remove <q`2

!<q`1

. Thisgenerates an expression relating ¸H

q`1to <

q`1. The two new expressions so obtained can then be

combined to eliminate <q`1

.

10The substitution of ln j for (j!1) confounds two market failures. The "rst is the proxtdestruction e!ect: a new monopoly earns pro"ts, (j!1), only by destroying the pro"ts earned by theprevious monopoly. As the social planner is not concerned with the identity of the "rm that currentlyearns pro"ts, this private value of innovation is ignored. The second is the consumer apppropriabilitye!ect: a social planner values the increment to consumer surplus, ln j, but this value is notappropriable by "rms who therefore ignore it.

writing the optimal intensity of R&D, ¸Hq(t) as a function9 of ¸H

q`1(t),

¸Hq(t)1~b

¸!¸Hq(t)"

lnjbEn(a

q)

o#

bEn(a

q)

o C¸H

q`1(t)

b(¸!¸Hq`1

(t))

!

¸Hq(t)

b(¸!¸Hq(t))

#lnA¸!¸H

q`1(t)

¸!¸Hq(t) BD , (14)

and which can be compared with the laissez faire equilibrium,

¸cq(t)1~b

¸!¸cq(t)"(j!1) E

n(a

q)P

=

0

dFn(a

q`1)

o#aq`1

¸cq`1

(t)b. (15)

Note that when q"¹, the second term on the RHS of Eq. (14) vanishes. Thedi!erences between the "rst terms on the RHS of Eqs. (14) and (15) re#ectmarket failures that are now familiar in quality ladder models. First, the solutionto the planner's problem substitutes ln j for (j!1) because the planner caresabout consumer surplus while what matters in the competitive equilibrium is"rm pro"ts.10 Second, the planner's solution includes b, which is absent from thecompetitive equilibrium. This di!erence is a congestion externality that arisesbecause the planner recognizes that each "rm contributes to aggregate diminish-ing returns to scale in R&D, while returns to scale are constant for the individual"rm. Third, the planner discounts the future at the rate o, while the "rmdiscounts the pro"t #ow at the rate o#a

q`1¸c

q`1(t)b. This di!erence arises

because "rms survive only to the next innovation while the social value of aninnovation lasts forever. These features of the model are well known fromsteady-state analyses.

Among technological laggards, however, there is a fourth divergence betweenthe e$cient and competitive outcomes. The planner's problem includes a termthat depends on the di!erence between the optimal intensity of R&D in thecurrent race and the intensity of R&D that is currently expected to be optimal inthe next race. This additional term in Eq. (14) re#ects the forward-lookingnature of the social planner: ¸H

q`1(t) is in fact a summary statistic for the current

subjective expectations of all future R&D e$ciencies.


A subsidy, sq, to R&D can be employed to equate ¸H

qand ¸c

q. Naturally,

sqincreases with the distance between the e$cient and competitive intensities of

R&D. At the technological frontier, ¸HT

and ¸cT

are constant, and so it followsthat the optimal subsidy to R&D, sH

T, is also constant. Substituting for the unit

cost of R&D in Eq. (6) and then comparing Eq. (8) with Eq. (12), the optimalsubsidy at the frontier is given by

sHT"1!

o(j!1)

b ln j P=

0

dF(a)

o#a¸HbT

, (16)

which, by Assumption 2.6, is positive.In contrast, for q(¹, sH

qis a random variable that varies across product

generations, and within a product generation whenever a new signal is received.Moreover, one cannot even sign sH

q; bad news about the prospects for developing

product generation q#1 increases ¸cq

but reduces ¸Hq, and in the face of

su$ciently bad news the former may be larger.

Proposition 2.8. The optimal subsidy, sHq, q(¹, for technological laggards is

a random variable, not necessarily with strictly positive support. The optimalsubsidy at the world technological frontier, sH

T, is constant and positive across

product generations and countries.

3. International signals of R&D e7ciency

It is necessary at this stage to impose some minimal structure on the relation-ship between the prior distribution, F(a), a technologically lagging country'sposterior distribution of a

q, F

n(a

q), and the n realizations of a

qthat a country has

observed. In this section I make two assumptions that are su$cient to enable meto say something useful in Section 4 about comparative growth.

The "rst assumption is that the signals received and the rules used totransform the signals into an expectation generate values of E

n(a

q) that are

unconditionally unbiased.

Assumption 3.1. Let H(En(a

q)Da

q) denote the conditional distribution of E

n(a

q) when

the unobserved ezciency of R&D is aq, and let E(E

n(a

qDa

q))":E

n(a

q) dH denote its

conditional expectation. Then :E(En(a

q)Da

q) dF"E(a).

The second requirement is that observations of the values of aq

realized inmore advanced countries provide useful information about the value of a

q. That

is, large values of En(a

q) should be more likely when the true (unobserved) value

of aq

is large. This requirement is made precise in the sense of "rst-orderstochastic dominance.


Assumption 3.2. For any aAq'a@

q, H(E

n(a

q)DaA

q)4H(E

n(a

q)Da@

q). If this holds as

a strict inequality for some En(a

q)'0, then the signals are informative.

These two assumptions can accommodate a large variety of stochastic envi-ronments. At one extreme, the parameter a

qis perfectly observed after the

completion of an innovation race, but it is imperfectly correlated across coun-tries. By observing realizations of a

qin advanced countries, a technological

laggard can learn something about the distribution from which its own value ofaqwill be drawn. At the other extreme, a

qis identical across countries but it is

imperfectly observed. For example, a laggard may observe the time or costrequired to produce an innovation, and from this it can infer something aboutaq. Assumptions 3.1 and 3.2 can accommodate these extremes as well as a combi-

nation of imperfect correlation and imperfect signals.The two assumptions also do not require that signal processing be optimal.

However, two parametric examples in which the updating rule is Bayesian areprovided here.

Example 3.3 (Perfect information, imperfect correlation). Assume that thecountry-speci"c e$ciency parameter, a

qis a random variable drawn from an

exponential distribution with unknown parameter vq. The parameter of the

distribution is speci"c to the product generation, but not to the country. Assumefurther that v

qis itself a random variable that has a prior gamma distribution

with parameters a"2 and b'0. The prior density for a is given by

f (a)"P=

0

b2v2e~(a`b)vC(2)

dv"2b2

(a#b)3. (17)

Assume now that a country has observed n realizations of aqwith mean k, and

that prior beliefs are updated by Bayes' rule. The 2-tuple Mk, nN is a su$cientstatistic for a

qand the posterior density function, derived in Appendix B, is

given by

fn(a

qDk)"

(n#2)( b#nk)n`2

(a#b#nk)n`3. (18)

Of course, Eq. (18) contains the prior density as the special case in which n"0.It is shown in Appendix B that Eq. (18) satis"es Assumptions 3.1 and 3.2.

Example 3.4 (Imperfect information, perfect correlation). Suppose that aq

is thesame for all countries. The technological laggard observes the durationst1, t

2,2, t

n, of patent races in n countries at the technological frontier. Given the

technology of innovation described in Section 2, the durations are exponentiallydistributed with unknown parameter a

qRb. The intensity of R&D is the same in


11Taking limits of Eq. (18) one obtains f"ea@k/k, and by the law of large numbers k converges onto 1/v

q.

all countries at the frontier, so one can choose units such that Rb"1. Assumefurther that the prior distribution of a

qis gamma with parameters a'0 and

b'0. Then the posterior distribution of aqis gamma with parameters a#n and

b#+ni/1

ti. This result is standard (e.g. De Groot, 1970), and is not analyzed in

the appendix.

While both examples are special cases of Assumptions 3.1 and 3.2, there is animportant di!erence between them. In Example 3.4 a su$cient number ofobservations allows a country to know the value of a

qprecisely. In Example 3.3,

in contrast, the technological laggard can never know its value of aqprecisely. As

nPR, the limiting posterior distribution in Example 3.3 does not becomedegenerate, but rather converges with probability one to an exponential distri-bution with known parameter v

q.11

4. Comparative growth

This section focuses on the impact of signals on expected instantaneousgrowth rates. The main results are as follows. Growth rates for e$cient techno-logical laggards observing signals are more variable than they are for countriesat the frontier. Signals do not always increase the expected growth rates ofe$cient laggards, but they are more likely to do so when aggregate returns toscale in R&D do not diminish too rapidly and the equilibrium level of R&Demployment is a small fraction of the labor force. Even under these conditions,I cannot establish that signals raise the average instantaneous growth rate oftechnological laggards adopting a policy of laissez faire. There is an intuitivereason, explained below, why one might not expect a laissez faire equilibrium toexhibit a clear comparative growth result. However, this observation should beviewed with caution. I have also been unable to "nd an empirical counter-example in which the expected growth rate of a technological laggard is lowerunder laissez faire than at the frontier, and so the e!ect of signals on growthunder laissez faire remains an open question.

There are other important ways in which these results have limited scope.Most important, comparative analysis is made awkward by the fact that thee!ect of signals on technological laggards depends very much on the questionthat is asked. One could ask, as I do in this section, whether signals raise theaverage instantaneous expected growth rate of a technological laggard. Butthere are other, equally valid questions. For example, one could ask whether theexpected length of time required to develop product generation q is reduced by


signals. The answers to the two questions need not be the same in a stochasticenvironment, because the relationship between the two measures of growth isnon-linear. Too see this, consider a country that lags one generation behind thefrontier and observes precisely either of two e$ciency parameters, a

0and a

1,

where a1'a

0and each of which could have occurred with probability one half.

Let (j!1)a0¸b0

and (j!1)a1¸b

1denote the growth rates that are realized when

each parameter is observed, and note that ¸1'¸

0. The unconditional expected

growth rate is then 0.5(j!1)[a0¸b

0#a

1¸b1]. The unconditional expected dura-

tion of the patent race is given by 0.5[(a0¸b0)~1#(a

1¸b1)~1]. Let ¸

Tdenote the

R&D intensity at the frontier. Then the signals raise the expected instantaneousgrowth rate only if (a

0¸b0#a

1¸b1)/(a

0#a

1)'¸b

Tand they reduce the expected

duration of the patent race only if (a0¸b

0#a

1¸b1)/(a

0#a

1)((¸

0¸

1/¸

T)b. Given

parameter values b"1, a0"1, a

1"2, ¸

0"1 and ¸

1"2, the "rst inequality

requires that ¸T(1.6, while the second inequality requires that ¸

T(1.8. There

is a window in which it is possible that signals increase the expected instan-taneous growth rate while raising the expected duration of the patent race.

4.1. Ezcient technological laggards

The analysis of this section begins with a useful lemma.

Lemma 4.1. For any q(¹, E(aq¸Hq(t)b)5E(a) E(¸H

q(t)b) with a strict inequality

[equality] if signals are informative [uninformative].

E$cient laggards will tend to raise their R&D intensity when the (unobser-ved) value of the R&D e$ciency parameter is high, and they will tend to reduceintensity when e$ciency is low. As all countries face the same unconditionaldistribution for the e$ciency parameter, the positive correlation between a

qand

¸Hq(t)b immediately yields the following result:

Proposition 4.2. Growth rates are more variable across product generations inezcient laggards than they are in countries at the technological frontier.

At the technological frontier, a social planner chooses a constant intensity ofR&D, ¸H

T, and so the expected growth rate is given by EgH

T"(j!1)E(a)¸Hb

T.

The social planner in a technological laggard, in contrast, chooses R&D inresponse to signals received, and the expected growth rate in this case isEgH

q(t)"(j!1)::a¸H

q(t)b dH dF"(j!1)E(a

q¸H

q(t)b)*(j!1)E(a)E(¸H

q(t)b).

Thus E(¸Hq(t)b)*¸Hb

Tis a su$cient condition for technological laggards to grow

more rapidly on average than countries at the frontier. The di$culty in estab-lishing this inequality is that ¸H

q(t)b depends (non-randomly) on E

n(a

q), which is

a random variable that in turn depends on the unobserved value of aq.

Moreover, ¸Hq(t)b also depends on ¸H

q`1which is itself a random variable.


12The latest available percentages from the OECD are: France, 2.0%; Germany 2.7%; Japan,2.8%; United States, 1.9%.

13Bound et al. (1984) have suggested that returns to scale in R&D are approximately constant upto $100 million of expenditure, with decreasing returns setting in thereafter. Thompson (1996)exploited the relationship between equity price and R&D to obtain estimates of b at the two-digitlevel ranging from 0.53 to 1.28, with a mean of 0.84.

Whether one can rank E(¸Hq(t)b) and ¸Hb

Tturns in large part on whether one can

show that ¸Hq(t)b is a convex function of E

n(a

q) and of ¸H

q`1(t). The required

convexities do not always hold. However, the following lemma provides condi-tions under which they do.

Lemma 4.3. If b3(1/2, 1), there exists an e3(0, 1) such that for all ¸Hq(t)3(0, e¸)

and ¸Hq`1

(t)3(0, e¸), ¸Hq(t)b, is a locally convex function of E

n(a

q) and of ¸H

q`1(t).

The lemma requires that aggregate returns to scale in R&D do not diminishtoo rapidly, and that in equilibrium R&D labor is a small enough fraction of thetotal labor force. Although one must be careful in comparing empirical assump-tions of an abstract model with data, the conditions seem plausible. First, R&Dexpenditures among even the most R&D-intensive countries are less than threepercent of GDP.12 Second, industry evidence suggests only weakly diminishingreturns to scale.13 Note also that the conditions of Lemma 4.3 are su$cient butnot necessary. They become necessary conditions only in the limiting case thatthe signals to which the social planner is responding turn out to be completelyuninformative.

Proposition 4.4. If the conditions of Lemma 4.3 hold, then for any q(¹, thefollowing expected growth rates can be ranked: EgH

q'EgH

T'Egc

T.

Proposition 4.4 predicts that quality in e$cient technological laggards willeventually catch up to the world technological frontier, but it does not predictthat convergence will be monotonic. In fact, there are three forces ensuringnon-monotonicity. The "rst is, of course, that R&D e$ciency varies acrosscountries and product generations. At any point in time, a laggard may beengaged in research that is more di$cult than development of the productgeneration currently occupying researchers at the frontier. Second, the intensityof current research e!ort in the e$cient laggard depends positively on thee$ciency of R&D in all its undeveloped product generations. Even if signalsindicate that R&D in the current race is more e$cient than average, bad newsabout future generations can induce an o!setting reduction in current R&D.Third, the technological laggard may occasionally receive misleading signalsabout its R&D parameter, causing it to adjust R&D in the wrong direction.


A numerical example may provide some more intuition about these results.Assume that a

qmay take either of two values, HIGH or LOW, with equal

probability. The social planner observes a signal about the e$ciency of R&D foreach product generation behind the world technological frontier, and assumesthat the signal is a precise predictor of its own country's R&D e$ciency. Table1 reports the social planner's choice of R&D intensity, ¸H

q, for each possible set

of realized signals. The conditional expected growth rates, gHq, and the uncondi-

tional expected growth rates, EgHq, are given for four di!erent degrees of signal

accuracy. Note that the choice of R&D e!ort depends only on the signalsreceived. The expected growth rates, in contrast, also depend on the accuracy ofthose signals.

There are two channels through which signals increase the unconditionalexpected growth rate. The "rst is that signals direct the planner to devote moree!ort to R&D when signals suggest that it will be particularly e!ective. Thesecond channel results from the convexity of the function ¸Hb

q(E

n(a

q)) established

in Lemma 4.3. In the case where signals are completely uninformative, the "rstchannel does nothing to raise the expected growth rate. Hence, Table 1 providesa useful decomposition of the sources of enhanced growth. At the frontier, theunconditional expected growth rate is 2.2%. When signals are completelyuninformative, a social planner developing product generation ¹!1 attains anexpected growth rate of 3.1%, while for generation ¹!2 it is 3.5%. Hence, thegrowth rate rises by 0.9% and 1.3%, respectively, simply as a result of theconvexity. Informative signals raise the expected growth rate further. Forexample, on moving from uninformative to perfectly informative signals theunconditional expected growth rates increases from 3.1% to 4.2% in generation¹!1, and from 3.5% to 4.6% in generation ¹!2.

Of course, these observations do not imply that a social planner shouldrandomly alter the R&D intensity to raise the expected growth rate. If signalsare uninformative and the social planner knows this, the optimal policy isa constant intensity of R&D equal to the rate chosen at the world technologicalfrontier. In contrast, when signals are precise the welfare-maximizing policy is tochoose the R&D intensities indicated in Table 1. In this example, the planneralters R&D intensities under the possibly mistaken belief that the signals areperfectly informative. The welfare e!ect of signals therefore depends on thecorrespondence between the accuracy of signals and the social planner's evalu-ation of their accuracy. To say more, however, would require making furtherassumptions about the properties of H(E

n(a

q)Da

q).

4.2. Technological laggards

When technological laggards adopt a laissez faire approach to R&D, I cannotproduce a ranking of expected growth rates analogous to Proposition 4.4.Signals have two opposing e!ects on the competitive intensity of R&D. Signals


Table 1Numerical example: Social planners's problem

Signal observed Expected growth rate by signal accuracy!

Productgeneration a

T~2aT~1

R&De!ort

0.50" 0.60" 0.75" 1.00"

¹ } } 1.52 0.022 0.022 0.022 0.022

¹!1 } HIGH 4.83 0.062 0.066 0.072 0.083} LOW 0.05 0.001 0.001 0.001 0.001

MEANS: 2.44 0.031 0.034 0.037 0.042

¹!2 HIGH HIGH 6.58 0.082 0.087 0.095 0.109LOW HIGH 0.15 0.003 0.003 0.002 0.002HIGH LOW 4.20 0.055 0.058 0.064 0.073LOW LOW 0.03 0.001 0.001 0.005 0.000

MEANS: 2.74 0.035 0.037 0.041 0.046

Parameter values used in the example are: ¸"100, b"0.7, o"0.05, j"1.04. The R&D e$ciencyparameters are a"M0.01, 0.02N with probabilities M0.5, 0.5N.!Numbers in bold type are the unconditional expected growth rates; the remaining numbers are the

expected growth rates conditional on each realization of the signals."Signal accuracy is de"ned as follows. The number refers to the probability that a

qis HIGH [LOW]

when the signal observed is HIGH [LOW]. Thus, 0.50 in the "rst column de"nes a completelyuninformative signal. The second and third columns indicate the signals are informative by thede"nition of Assumption 3.2, but are not precise. The fourth column indicates precise signals.

about the e$ciency of R&D in the current race induce more research whenR&D is believed to be e!ective and less when it is believed to be relativelyine!ective. Under the same conditions as laid out in lemma 4.3, this e!ectpromotes growth over the long run. On the other hand, if signals raise theaverage expected growth rate in, say, generation ¹!1, the expected duration ofthe monopoly attained by the winner of the race for product generation ¹!2may be reduced. This channel will have a negative e!ect on R&D e!ort in therace for generation ¹!2. Thus, while it is easy to show that, if the conditions ofLemma 4.3 apply, Egc

T~1'Egc

T, I cannot show that Egc

T~2'Egc

T. However,

I have not been able to produce a counterexample in which EgcT~2

(EgcT. Thus,

the e!ect of signals on the laissez faire growth rate in countries lagging the worldtechnological frontier by at least two generations remains an open question.

Table 2, continuing the earlier example, provides corresponding data for thelaissez faire equilibrium R&D intensities and growth rates. The example doesprovide some interesting insights. First, note that, as in the social planner'sproblem, the unconditional expected growth rate is higher the more informativeare the signals. Second, and this is in direct counterpoint to the previous results,


Table 2Numerical example: laissez faire equilibrium!

Productgeneration

Signal observed Expected growth rate by signal accuracy

aT~2

aT~1

R&D e!ort 0.50 0.60 0.75 1.00

¹ } } 0.75 0.015 0.015 0.015 0.015

¹!1 } HIGH 1.77 0.025 0.027 0.029 0.033LOW 0.08H 0.002 0.001 0.001 0.001MEANS: 0.93 0.013 0.014 0.015 0.017

¹!2 HIGH HIGH 0.61 0.009 0.010 0.011 0.013LOW HIGH 0.00 0.000 0.000 0.000 0.000HIGH LOW 15.90H 0.181 0.193 0.211 0.241LOW LOW 0.09H 0.002 0.001 0.005 0.000

MEANS: 4.15 0.048 0.051 0.056 0.064

!See notes to Table 1. Asterisks denote realizations that lead to a laissez faire intensity of R&D thatexceeds the social planner's choice of R&D.

a high conditional growth expected rate in the race for product generation¹!1 leads to a low expected growth rate in the race for generation ¹!2. Thisre#ects, of course, the fact that observing a

T~1"HIGH is good news for the

social planner developing generation ¹!2, but bad news for "rms engaged inthe race for ¹!2. Third, while the laissez faire intensity of R&D at the frontieris generally less than the e$cient intensity of R&D (calling for a subsidy toR&D), there are three instances in which the social planner in a technologicallaggard would choose to tax R&D. These three instances correspond to theoccasions on which the signal for a

T~1is LOW. In production generation

¹!2, the prospect of a long tenure for the winner of the current R&D raceencourages "rms to conduct more R&D in laissez faire than is socially desirable.

5. Conclusions

This paper explored comparative growth in a quality-ladder model incorpor-ating uncertainty about the e$ciency of R&D in di!erent product generations.In countries at the technological frontier, private and public decisions aboutR&D intensity are made on the basis of a time-invariant prior distributionassigned to the value of a parameter that governs the e$ciency of R&D. Incontrast, technological laggards are able to observe the experience of moreadvanced countries and enjoy an information advantage that a!ects theirexpected rates of growth. Signals indicating the e$ciency of R&D for countriesthat have developed a particular product generation are assumed to be


correlated with the R&D e$ciency in any country that develops the sameproduct generation at a later date. My aim in this paper was to explore theimplications of this information advantage for economic growth.

The principal conclusion is that the e!ects of information depend on whethertechnological laggards adopt e$cient technology policies, or whether they takean approach of laissez faire. When policies are optimally chosen, the modelsuggests that catch-up to the world technological frontier is likely. Su$cientconditions for catch-up are that aggregate returns to scale in R&D do notdiminish too rapidly and that parameters yield equilibria in which R&D e!ortconstitutes a small fraction of the total labor force. The parameter space forwhich catch-up takes place is expanded as signals become more informative.There are now numerous models in which technological catch-up is possiblewithin a framework of endogenous growth. Most of these (e.g. Eaton andKortum, 1997; Barro and Sala-i-Martin, 1995) rely on explicit forms of techno-logy transfer or imitation. This paper describes an alternative mechanism bywhich catch-up may be possible.

The results in this paper contrast with those obtained by Hopenhayn andMuniagurria (1996). They studied a one-sector endogenous growth model inwhich the subsidy to investment may be either positive or zero. They found thatmore frequent regime switches were likely to reduce growth and raise welfare. Inthe present paper, in which regime switching is information driven, the growthrate is likely to increase as a result of policy changes, although welfare need notbe enhanced.

These results were obtained with a minimal set of assumptions on thetransmission and utilization of information that includes, but is not restricted to,optimal updating of prior beliefs by Bayes' rule. I have not been able to replicatethese results for the laissez faire equilibrium. However, the analysis does revealthat, even though the optimal policy at the world technological frontier isalways a subsidy to R&D, there are instances in which technological laggardswould optimally choose a tax. Barro and Sala-i-Martin (1995) observed that wedo not have any theory about convergence or divergence of government policiesacross countries. In a limited sense, this paper provides one. While the papero!ers no insight as to why some countries might choose e$cient policieswhen others do not, it does indicate that, among countries that choose e$-cient policies, the nature of the optimal intervention depends on one's technolo-gical ranking. As countries adopting e$cient policies catch up to the worldtechnological frontier, their policies begin to look like those in other advancedcountries.

In deriving these results, I constructed a model which strips away `tradi-tionala sources of international linkages. The model contains no physical capi-tal, no trade, no foreign direct investment, and no technology transfer.Excluding many of these features is a standard practice that allows one todevelop a tractable model that illuminates the question at hand. However, there


are some simplifying assumptions that might matter in substantial ways.Consider, for example, my choice to exclude international trade. Given themarket structure I have employed, the introduction of international trade to themodel would have important consequences. Under Bertrand competition, qual-ity leaders drive out producers of inferior goods so that in a closed economyonly the state of the art product is consumed. One might, therefore, worry abouthow the model would behave when di!erent qualities of goods coexist in a worldwith international trade. Of course, balanced trade constraints imply that wageswill depend on the highest quality good that each country is capable ofproducing, so that it is not at all obvious that a quality leader in one countrycould drive out producers of inferior goods in other countries. However, thisanalysis remains to be done and, in this sense, the analysis in the present papermight best be interpreted as exploratory.

Acknowledgements

I am extremely grateful for extensive comments from "ve anonymous referees,and helpful suggestions on this and earlier drafts from Margaret Byrne, MichaelPalumbo, and Costas Syropoulos.

Appendix A. Proofs

Proof of Lemma 2.2. By Assumption 2.1, Fn(a

T`j)"F(a) and E

n(a

T`j)"E(a),

∀j50. Hence if a stationary equilibrium with positive R&D e!ort exists, it mustsatisfy Eq. (8). Existence and uniqueness is established as follows. The LHS ofEq. (8) is monotonically increasing in ¸

Tand continuous along the interval

¸T3[0, ¸); the LHS equals zero when ¸

T"0 and approaches #R when

¸TP¸. For E(a)3[0, R) the RHS of Eq. (8) is continuous, monotonically

decreasing in ¸T, and equal to (j!1)E(a)/o when ¸

T"0. Thus a unique

interior stationary equilibrium exists.

Lemma A.1. Assume Fn(a

q`1) is uniquely dexned with positive support, and that

0(En(a

q)(R. For any xxed ¸

q`1'0, the mapping ¸

q"G(¸

q) dexned by

Eq. (7) has a unique xxed point ¸cq3(0, ¸).

Proof. G(¸q) can be written as

G(¸q)"[(j!1)E

n(a

q)]1@(1~b)/(¸

q`1)(¸!¸

q)1@(1~b),

where /(¸q`1

)"[:(o#aq`1

¸bq`1

)~1 dFn(a

q`1)]1@(1~b). If F

n(a

q`1) is uniquely

de"ned and has positive support, then / is a well-de"ned function mapping[0, ¸] into the closed interval [c, o~1] for some 0(c(o~1. As b(1, G(¸

q) is


Fig. 1.

continuous and decreasing along the closed interval [0, ¸]. Moreover,G(0)"/(¸

q`1)[(j!1)E

n(a

q)¸]1@(1~b)'0, while G(¸)"0. Hence, there is

a unique "xed point ¸cq"G(¸c

q) satisfying 0(¸c

q(¸.

Proof of Theorem 2.3. This is straightforward with the use of Lemma A.1.Lemma 2.2 states that a unique ¸c

T'0 exists. Then Lemma A.1 states that

a unique ¸cT~1

'0 exists if Fn(a

T) is uniquely de"ned with positive support.

Theorem 2.3 then follows by backward induction to product generation q.

Proof of Proposition 2.5. Note that ¸1~bT

/(¸!¸T) is increasing in ¸

T. Combin-

ing Eqs. (8) and (12), yields

¸HT'¸c

Ti!

1

:(o#a(¸cT)b)~1 dF(a)

'

o(j!1)

b ln j. (A.1)

Let /(¸, j) denote the expression on the left hand side of the inequality inEq. (A.1). From Eq. (8) it is easy to see that: ¸c

Tis increasing in j and ¸; for "xed

¸, /(¸, j) is bounded; and /(¸, 1)"o. Let t(j) denote the right-hand side of theinequality, and note that: t(1)"o/b; t@(j)'0; and limj?=

t(j)"R. Thesetwo functions are plotted in Fig. 1. Clearly, for values of j small enough andlarge enough, /(¸, j)(t(j). Furthermore, if ¸ is su$cently small, then/(¸, j)(t(j) for all j. But /(¸, j) is strictly increasing in ¸ for any "xed j.Hence, one can always make ¸ large enough to ensure there is an interval[j

0, j0] for which /(¸, j)'t(j). It is then evident that further increases in ¸

will reduce j0, and increase j0. Finally, if ¸H

T'¸c

T, we have EgH

T!Egc

T"

:(j!1)a(¸HT)b dF!:(j!1)a(¸c

T)bdF"(j!1)E(a)[(¸H

T)b!(¸c

T)b]'0.


Proof of Theorem 2.7. Assume for the moment that <q`1

is uniquely de"ned.Substitute Eq. (11) into Eq. (10) to remove <

q(t):

o¸Hq(t)1~b#¸H

q(t)E

n(a

q)

bEn(a

q)(¸!¸H

q(t))

#ln(¸!¸Hq(t))"o<

q`1(t)!(q!1)lnj. (A.2)

The RHS of Eq. (A.2) is invariant to ¸Hq(t). The LHS of Eq. (A.2) is strictly

increasing in ¸Hq(t) for all ¸H

q(t)3[0, ¸), it equals ln(¸) when ¸H

q(t)"0, and it

approaches #R as ¸Hq(t)P¸. Hence, there is a unique interior solution,

0(¸Hq(t)(¸, as long as o<

q`1(t)!(q!1)lnj'ln(¸). Updating Eq. (10) by

one product generation and setting ¸q`1

"0 (which is not optimal), we obtaino<

q`1(t)5qlnj#ln(¸)'(q!1) ln j#ln ¸. Hence there exists a unique

interior solution for ¸Hq(t).

Given that ¸Hq(t) is uniquely determined, there is a unique maximum for <

q(t)

satisfying

<q(t)"

(q!1)lnj#ln(¸!¸Hq(t))

o#En(a

q)¸H

q(t)b

#

En(a

q)¸H

q(t)b

o#En(a

q)¸H

q(t)b<

q`1(t). (A.3)

From Lemma 2.4, ¸HT(t) and <H

T(t) are uniquely determined. It then follows from

Eq. (A.2) that ¸HT~1

(t) is uniquely determined, and from Eq. (A.3) that <HT~1

(t) isuniquely determined. The uniqueness proof is then completed by backwardinduction to product generation q.

Next, note [from Eq. (A.2)] that ¸Hq(t) is increasing in <

q`1(t) and E

n(a

q), and

that [from Eq. (A.3)] <q(t) is increasing in <

q`1(t) and E

n(a

q). From Lemma 2.4,

<T(t) is increasing in E(a), and so it follows that ¸H

T~1(t) and <

T~1(t) are

increasing in E(a) and En(a

T~1). By backward induction to product generation q,

we can than conclude that ¸Hq(t) and <

q(t) are both increasing in each element of

the sequence MMEn(a

j)NT~1

j/q, E(a)N.

Proof of Lemma 4.1. The conditional expectation is given by

E(¸HqbDa

q)"P

=

0

¸Hqb dH(E

n(a

q)Da

q)

"¸HqbH(E

n(a

q)Da

q)D=0!bP

=

0

¸Hb~1q

d¸Hq

dEn(a

q)H(E

n(a

q)Da

q) dE

n(a

q)

"¸Hbq

(R)!bP=

0

¸Hb~1q

d¸Hq

dEn(a

q)H(E

n(a

q)Da

q) dE

n(a

q),

where the second line was obtained with an integration by parts. As ¸Hbq

(R) hasan upper bound at ¸b, d¸H

q/dE

n(a

q)'0 (Theorem 2.7), and H(E

n(a

q)Da

q) is

decreasing in aq

(Assumption 3.2), E(¸Hbq

Daq) is increasing in a

q. Hence a

qand

¸Hbq

are positively correlated. We then have E(aq¸Hbq

)"cov(aq, ¸Hb

q)#

E(aq)E(¸Hb

q)'E(a

q)E(¸Hb

q)"E(a)E(¸Hb

q). In the limiting case when signals are


uninformative, H is not a function of aqso that a

qand ¸Hb

qare independent, and

cov(aq, ¸Hb

q)"0.

Proof of Lemma 4.3. Di!erentiating Eq. (A.2) with respect to En(a

q), one obtains

d(¸Hq(t)b)

dEn(a

q)"

bo¸Hq(t)b(¸!¸H

q(t))

En(a

q)(b¸H

q(t)#(1!b)¸)(E

n(a

q)¸H

q(t)#o)

. (A.4)

Using Eq. (A.2) to remove En(a

q) from Eq. (A.4) yields, after some rearrangement,

d(¸Hq(t)b)

dEn(a

q)

"

¸Hq(t)2b~1

o G¸2(<

q`1(t)!(q!1) ln j!ln(¸!¸H

q(t)))!2¸¸H

q(t)

¸!b(¸!¸Hq(t))

#ln (¸!¸Hq(t)) (b(¸!¸H

q(t))#¸)#(<

q`1(t)

!(q!1) ln j) (b(¸Hq(t)!¸)#2¸H

q(t)H

#

¸Hq(t)1`2b

o(<q`1

(t)!(q!1)ln j!ln (¸!¸Hq(t)))(<

q`1(t)!(q!1)ln j!b(¸!¸H

q(t)))

.

(A.5)

Eq. (A.5) is strictly positive and di!erentiable with respect to ¸Hq(t) for all

¸Hq(t)3(0, ¸). It is also easy to verify that

limLHq (t)s0

d(¸Hq(t)b)

dEn(a

q)"

1

oC limLHq (t)s0

¸Hq(t)2b~1D[b2¸(<

q`1(t)!(q!1)lnj!ln¸)],

(A.6)

and as <q`1

(t)'<q(t)5(q!1) ln j#ln ¸, the second term on the RHS of

Eq. (A.6) is strictly positive. Hence, limLHq (t)s0

d(¸Hq(t)b)/dE

n(a

q)"0 i! b'1/2.

Finally, given di!erentiability of Eq. (A.5) and the fact that Eq. (A.5) is strictlypositive for all ¸H

q(t)3(0, ¸), there exists an e

03(0, 1) such that for all

¸Hq(t)3(0, e

0¸), ¸Hb

qis a locally convex function of E

n(a

q).

To show the conditions under which ¸Hq(t)b is a convex function of ¸H

q`1(t),

di!erentiate Eq. (14) with respect to ¸Hq`1

(t) and then use Eq. (14) to eliminateEn(a

q):

d(¸Hq(t)b)

d(¸Hq`1

(t))

"

b¸Hq(t)b(¸!¸H

q(t)) ((1!b)¸#b¸H

q`1(t))

Cb(¸!¸Hq`1

(t)) lnAjA¸!¸H

q`1(t)

¸!¸Hq(t) BB#¸H

q`1(t)D(¸!¸H

q`1(t))((1!b)¸#b¸H

q(t))

.

(A.7)


Note that Eq. (A.7) is di!erentiable in ¸Hq`1

(t) for all ¸Hq`1

(t)3(0, ¸). Hence, evaluateEq. (A.7) at ¸H

q`1(t)"¸H

q(t),

d(¸Hq(t)b)

d(¸Hq`1

(t))KLHq`1(t)/L

Hq (t)

"

b¸Hq(t)b

b(¸!¸Hq(t)) ln j#¸H

q(t)

,

which is strictly increasing in ¸Hq(t) for all ¸H

q(t)/¸((b2 ln j)/(b(1!b)j#(1!b)).

As Eq. (A.7) is di!erentiable in ¸Hq`1

(t), there exists an a'0 and an e1'0 such that

for all ¸Hq(t)3(0, e

1, ¸) and ¸H

q`1(t)3[¸H

q(t)!a, ¸H

q(t)#a], ¸H

q(t)b is a locally convex

function of ¸Hq`1

(t). Note that e1

is strictly increasing in b and is unbounded asbP1. Finally, let e"minMe

0, e

1, a/¸N and the proof is complete.

Proof of Proposition 4.4. Assume that E(¸Hq`1

(t))*¸T

for some q(¹!1. Then,noting the dependence of ¸H

q(t) on E

n(a

q) and ¸H

q`1(t), we can write

EgHq"(j!1) E(a

q¸Hbq

(En(a

q), ¸H

q`1))

(i) '(j!1)E(a)E(¸Hbq

(En(a

q), ¸H

q`1))

(ii) '(j!1)E(a)(¸Hbq

(E(En(a

q)), ¸H

q`1))

(iii) "(j!1) E(a)(¸Hbq

(E(a), E(¸Hq`1

))5(j!1) E (a) ¸Hbq

(E(a), ¸HT)

(iv) "(j!1)E(a)¸HbT"EgH

T

(v) 'EgcT.

Inequality (i) is from Lemma 4.1; (ii) is from the convexity established inLemma 4.3; (iii) is from Assumption 3.1; (iv) is the unique solution to Eq. (14)when E

n(a

q)"E(a); and (v) is a restatement of Assumption 2.6.

The remainder of the proof is by induction. If E(¸Hq`1

(t))5¸HT, we have

E(¸Hq(E

n(a

q), ¸H

q`1))5¸H

q(E(a), ¸H

T)

"¸HT,

which follows from steps (i)}(iv) above, and the fact that¸Hq(t) is a convex function of

¸Hq(t)b. Thus, if E(¸H

q`1(t))5¸H

Tthen E(¸H

q(t))5¸H

T. Finally, it is easy to establish

that E(¸HT~1

(t))5¸HT:

E(¸HT~1

(En(a

T~1), ¸H

T))'¸H

T~1(E(E

n(a

T~1), ¸H

T~1)

"¸HT~1

(E(a), ¸HT)

"¸HT

and this completes the proof.


Appendix B. Characterization of Example 3.3

Derivation of posterior density. As aqis exponentially distributed with parameter v

q,

the sum nk of n realizations of aqhas a gamma distribution with parameters n and v.

The unconditional density for nk is

fn(nk)"P

=

0

fn(nkDv) f (v) dv

"P=

0

b2vn`1(nk)n~1e~(b`nk)vC(n)

dv.

Repeated integration by parts for integer n yields

fn(nk)"

b2(nk)n~1n(n#1)

( b#nk)n`2(B.1)

and Bayes theorem yields the posterior density.

fn(nkDv)"

fn(vDnk) f (v)

fn(nk)

"

vn`1e~(b`nk)v(b#nk)n`2

C(n#2).

The posterior density of aqis then given by

fn(a

q)"P

=

0

ve~aqvfn(vDnk) dv.

Combining the last two equations, and integrating by parts repeatedly for integer n,yields Eq. (18).

Assumption 3.1. Decision makers in technologically lagging countries form expecta-tions as follows:

En(a

q)"P

=

0

aq(n#2)(b#nk)n`2

(aq#b#nk)n`3

daq

"

b#nkn#1

. (B.2)


The density of k conditional on the unobserved true value of aqis given by

fn(kDa

q)"

fn(a

qDk) f

n(k)

f (aq)

"

fn(a

qDk)n f

n(nk)

f (aq)

"

(n#2)(n#1)n2(aq#b)3(nk)n~1

2(aq#b#nk)n`3

. (B.3)

The second line applies the method of transformations to Eq. (B.1); the third linecombines Eqs. (17), (18) and (B.1). From Eq. (B.2), the posterior mean of a

qand the

mean of the signals are related by k"[(n#1)En(a

q)!b]/n. Applying the method

of transformations to Eq. (B.3),

h(En(a

q)Da

q)"

(n#1) fn(kDa

q)

n

"

(n#2) (n#1)2n (aq#b)3((n#1) E

n(a

q)!b)n~1

2(aq#(n#1)E

n(a

q))n`3

which has support [b/(n#1), R). Direct integration gives

E(En(a

q)Da

q)"

naq#(n#2)b2n#2

for all integer n. Hence, using Eq. (17), the unconditional expectation of En(a

q) is

given by

E[En(a

q)]"P

=

0Ana

q#(n#2)

2n#2 BA2b2

(aq#b)3B da

q

"b.

Thus Assumption 3.1 is satis"ed.

Assumption 3.2. The conditional distribution H(En(a

q)Da

q) is given by

H(En(a

q)Da

q)"P

En(aq)

b@(n`1)

(n#2)(n#1)2n(aq#b)3((n#1)v!b)

2(aq#(n#1)v)n`3

dv. (B.4)

Di!erentiating with respect to aqand integrating by parts repeatedly yields

dH(En(a

q)Da

q)

daq

"!

(aq#b)2((n#1)E

n(a

q)!b)n

((n#1)En(a

q)#a

q)n`2

.


Finally, substituting Eq. (B.2) to remove En(a

q) yields

dH(En(a

q)Da

q)

daq

"!

(aq#b)2(nk)n

(aq#b#bk)n`2

(0,

and Assumption 3.2 is satis"ed.

References

Aghion, P., Howitt, P., 1992. A model of growth through creative destruction. Econometrica 60, 323}351.Barro, R.J., Sala-i-Martin, X., 1995. Technological di!usion, convergence and growth. NBER working

paper no. 5151.Bound, J., Cummings, C., Griliches, Z., Hall, B., Ja!e, A., 1984. Who does R&D and who patents?. In:

Griliches, Z. (Ed.), R&D, Patents, and Productivity. University of Chicago Press, Chicago, IL.De Groot, M.H., 1970. Optimal statistical decisions. Wiley, New York.Dinopoulos, E., 1994. Schumpeterian growth: an overview. Osaka City University Economic Review 29,

1}21.Eaton, J., Kortum, S., 1997. Engines of growth: domestic and foreign sources of innovation. Japan and

the World Economy 9, 235}259.Grossman, G.M., Helpman, E., 1991. Innovation and Growth in the Global Economy. MIT Press,

Cambridge, MA.Hopenhayn, H.A., Muniagurria, M.E., 1996. Policy variability and economic growth. Review of

Economic Studies 63, 611}625.Jones, C.I., Williams, J.C., 1997. Measuring the social return to R&D. Mimeo, Stanford University.Segerstrom, P.S., Anant, T.C.A., Dinopoulos, E., 1990. A Schumpeterian model of the product life cycle.

American Economic Review 80, 1077}1091.Thompson, P., 1996. Technological opportunity and the growth of knowledge: a Schumpeterian

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