Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue9.Aug2000:

Journal of Economic Dynamics & Control
24 (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 uncertainty about the e$ciency of R&D. A central premise of the paper is that designing
appropriate technology policies is more di$cult when one is at the cutting edge of
technology. In technological laggards, information gleaned from observations of more
advanced countries provides noisy signals about the e!ort required to develop a speci"c
product generation, and aids in the design of policy. The paper shows how these signals
in#uence growth rates and technology policies. ( 2000 Elsevier Science B.V. All rights
reserved.
JEL classixcation: O3
Keywords: Growth; Quality ladders; Information

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

* Corresponding author. Tel.: #1-713-743-3798; fax: #1-713-743-3798.
E-mail address: pthompso@bayou.uh.edu (P. Thompson).
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

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

about those parameters. A central premise of this paper is that designing
appropriate technology policies is more di$cult when one is at the cutting edge
of technology. In technological laggards, information gleaned from observations
of more advanced countries provides noisy signals about the e!ort required to
develop a speci"c product generation, and aids in the design of policy. Firms in
less advanced countries are therefore guided more accurately to undertake
research of appropriate intensity, and planners can set more precise policies. In
short, there is a greater correlation in technological laggards between ex ante

and ex post optimal behavior in both competitive and e$cient equilibria.
In order to highlight the e!ect of information on comparative growth,
I construct a model which strips away &traditional' sources of international
linkages that have been extensively analyzed elsewhere. Thus the model economy contains no physical capital, no trade, no foreign direct investment, and no
technology transfer in the traditional sense of imitation. What is left is a novel
source of international growth linkages in which di!erences in average growth
rates are driven only by the ability to observe parameters in more advanced
countries. The model explicitly acknowledges the conventional wisdom that
technology policies are somehow easier to design when one is not at the frontier,
a feature of comparative growth that has not previously been explored.
I develop a version of the quality-ladder model due to Aghion and Howitt
(1992), Grossman and Helpman (1991) and Segerstrom et al. (1990). The main
extension in this paper is that the e$ciency of R&D in the development of each
product generation depends on a random variable that is assumed to be
correlated across countries, and in contrast to previous work I focus primarily
on the non-stationary equilibria that arise among technological laggards. The
learning mechanism by which technological laggards update their beliefs about
parameters governing the e$ciency of R&D e!ort is very general and may be of
independent interest. It allows for imperfectly observable R&D e$ciencies in
more advanced countries as well as imperfect correlation of R&D e$ciencies

across 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 technological laggards that adopt optimal policies, and those that follow a laissez faire
policy. First, technological laggards that observe no signals about the e$ciency
of R&D grow at the same rate, on average, as countries at the world technological frontier. Signals need not raise the expected instantaneous growth rate.
However, su$cient conditions are derived for signals to raise the expected
growth rate of a laggard adopting an optimal policy, and these conditions seem
plausible. Speci"cally, signals enable lagging countries to grow more rapidly on
average when returns to scale in R&D e!ort do not diminish too rapidly and
parameters yield R&D employment that is a small fraction of the total labor
force. In fact, under these conditions signals generate higher growth even if the
signals turn out to be completely uninformative. The more informative the

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

1287

signals are, the greater is the parameter space for which signals raise growth in
the 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 raise
the expected instantaneous growth rate, nor have I been able to generate
a counterexample. Nonetheless, while the implications of signals for catch-up
remain an open question under laissez faire, there are some interesting "ndings.
Prominent among these is a contrast between the e$cient and laissez faire
models in the e!ect that news about the e$ciency of R&D in the next race has
on R&D e!ort in the current race. News suggesting that R&D will be particularly e$cient in the next race encourages the social planner to raise R&D
e!ort in the current race: the news makes it more desirable to complete the
current race. In contrast, the same news lowers R&D e!ort in the laissez faire
equilibrium, because it implies that the winner of the current race is likely to
enjoy monopoly pro"ts for only a short period of time. That is, signals about the
e$ciency of R&D in future generations that raise the expected intensity of R&D
in the next race are good news from the perspective of the social planner, but bad
news for "rms participating in the current R&D race.
Finally, the model also has something to say about policy variability. At the
world technological frontier, the optimal intervention is a subsidy to R&D that
remains constant from one product generation to the next. However, among
technologically lagging countries the optimal intervention may be a subsidy or
a tax, and it is a random variable that varies across product generations. Among
countries that adopt e$cient policies, therefore, one should expect technology

policy in laggards to become more similar over time to those adopted at the
frontier. Indeed, just this process seems to have been taking place in the rapidly
growing Asian economies. Knowing when and where to support R&D and
technology development seems to have been closely related to growth performance. Yet, as these countries' technological capabilities have advanced toward
the world technological frontier, many distinctive features of their technology
policies have been abandoned. There are of course too many ingredients to
technology policy to allow a succinct summary of the secular trends here; but it
seems clear that Singapore and Japan e!ectively began to abandon targeted
technology support more than a decade ago, and that more recently South
Korea and Taiwan have been following suit. To many observers, the decline of
targeting in the advanced Asian economies is a symptom of their success. As
industries advance toward the technological frontier, the argument goes, it
becomes more di$cult to identify promising technological avenues, and consequently to decide which technologies merit government support. It is just these
features of policy intervention that the present paper attempts to formalize.
While readers may conclude that the pure information e!ect studied in this
paper is less important for growth than the traditional prescriptions of more
saving and more education, it does, I think, matter.

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

The layout of the paper is as follows. Section 2 presents the model,
The competitive equilibrium is characterized and contrasted with the solution
to the social planner's problem. Section 3 describes how signals from
more advanced countries inform technological laggards about the productivity
of their own R&D programs. Section 4 presents the main results on comparative growth, and Section 5 concludes. All proofs are provided in
Appendix A.

2. The model
I employ a simple quality ladder framework, based on the seminal work of
Aghion and Howitt (1992), Grossman and Helpman (1991), and Segerstrom
et al. (1990). The world consists of a number of countries, each consisting of
a single manufacturing sector and a research sector. The research sector in each
country is characterized by a sequence of patent races, each aimed at improving
the quality of the country's existing state-of-the-art product. The duration of
a patent race depends in part on a random variable that governs the e$ciency of
R&D, and whose value is not known with certainty during the race. The winner
of a patent race employs limit pricing to secure a temporary monopoly that lasts
until the next innovation. Only the highest-quality product developed within

a country at any point in time is manufactured. There is no trade, and there are
no transfers of technology across countries. That is, all knowledge required to
manufacture a given product generation must be developed locally. The only
sense, therefore, in which the model might be interpreted as a multi-country
model is that countries may be able to learn something from other countries
about the value of the R&D e$ciency parameter. The remainder of this section considers an arbitrary country currently engaged in developing product
generation q.
2.1. Consumers
A representative consumer maximizes the expected present value of lifetime
utility,

P

=
e~o(q~t) ln u(q) dq,
t
subject to an intertemporal budget constraint,
max E

t

AQ (t)"r(t) A(t)#w(t)¸#n(t)!m(t),

(1)

(2)

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

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

1289

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

C

u(t)"

D

q(t)~1
+ jjx (t) ,
j
j/1

(3)

where x (t) denotes consumption of the jth generation of the good, j'1 is the
j
proportional improvement in quality between consecutive generations, and
q(t)!1 denotes the number of product improvements that have been made
available in the country by time t.
Homothetic preferences ensure separability of expenditure and prices, and the
consumer's problem can be solved in two stages. In the "rst stage, the consumer
allocates expenditure, m(t) across products. It will be assumed below that
marginal cost is equal to w(t) for all "rms, and that each generation of goods can
only be produced by the single "rm that holds the relevant patent. The preferences given by Eq. (3) imply that quantity and quality are perfect substitutes.
The consumer therefore purchases only the single good with the lowest qualityadjusted price. Following standard practice, I assume that when qualityadjusted prices are equal, the highest quality product is consumed. Bertrand

competition in this setting induces the holder of the national state-of-the-art
patent to set a limit price of jw(t). Only the state-of-the-art product is consumed
and the monopoly leader captures revenue of m(t). In the second stage, the
consumer allocates expenditure over time, which satis"es the familiar Euler
equation, m5 (t)/m(t)"r(t)!o. I let expenditure be the numeraire so that
m(t)"1. It then follows that the market rate of interest, r(t) is always equal to the
discount rate, o.
2.2. Firms
Manufacturing requires only labor, ¸ (t), and one unit is required to produce
x
one unit of output of any good. The owner of the state-of-the-art patent sets
a markup of j over marginal cost, which secures a temporary monopoly and
yields instantaneous pro"ts n(t)"(j!1)m(t)/j"(j!1)w(t)¸ (t). All "rms
x
owning patents on earlier product generations earn zero pro"ts. As m(t)"1, it
follows that w(t)"1/j¸ (t) and n(t)"(j!1)/j.
x
Denote with v (t) the discounted pro"ts of the successful innovator of the qth
q
generation product. R&D also requires only labor. Let ¸ (i, t) denote the labor

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

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

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

aggregate labor devoted to the race. The expected discounted pro"ts of "rm i are
¸ (i, t)
!w(t) ¸ (i, t) dt,
(4)
E[v (t)a ] ¸ (t)b dt q
q
q q q
¸ (t)
q
which equals zero under the assumption of free entry to the patent race. The
term a ¸ (t)b dt, b(1, is the probability2 that the patent race will be won in the
q q
next momentary interval dt, and ¸ (i, t)/¸ (t) is the probability that, if the race is
q
q
won, "rm i will be the winner. Innovations are Poisson events with a timevarying intensity a ¸ (t)b dt, and the duration, q, of the race for product generaq q
tion q is given by the time-varying exponential distribution expM:q a ¸ (t)b dtN,
0 q q
where t"0 denotes the time at which the race began. For any "xed ¸ , a is
q q
proportional to the arrival intensity, and may naturally be interpreted as
a measure of R&D e$ciency.3
The R&D e$ciency parameter a is not known with certainty during the
q
patent race.4 Before and during a race, "rms assign to a a subjective distribuq
tion F (a ). The main features of a are: (i) it is independent across product
n q
q
generations, and (ii) it is correlated across countries.5 Thus, while a country at
the technological frontier must make decisions based only on the prior distribution of a , countries which have the opportunity to look ahead to other
q
countries' experiences with the same generation of technology will have a better
idea about the value that a is likely to take. The subscript n denotes the number
q
of observations available on a and at any point in time it may vary across
q
product generations.6 These characteristics of F will be explored in detail in
n
Section 3. However, it is useful at this stage to make some assumptions about
the prior distribution (or, equivalently, the distribution of a for a country at the
technological frontier):

2 As dt is arbitrarily close to zero, a (t)¸ (t)b dt will also be arbitrarily close to zero almost
q
q
everywhere as long as E(a ) is "nite and ¸ (t)b is bounded. The latter is bounded by the resource
q
q
constraint, while the former is assumed below.
3 Note also that this formulation of the R&D technology implies that individual "rms face
constant returns to scale in R&D, leaving their size indeterminate, but returns to scale in aggregate
R&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.
4 Aghion and Howitt (1992) consider an example in which a is random but its value is known
q
during the race.
5 The assumption of zero correlation across product generations is made for analytical convenience. The key insights of the model are not sensitive to the introduction of serial correlation across
product generations. What is central to the paper is the assumption that countries have something to
learn about a from other countries.
q
6 More precisely, one should write F
(a ) to re#ect the fact that the number of signals varies
n(q,t) q
with calendar time and product generation. I shall use the shorthand notation to avoid unnecessary
clutter.

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

1291

Assumption 2.1. Let F(a),F (a) denote the prior distribution of a, let E(a)(R
0
denote its prior mean, and let ¹ denote the product generation being developed at
the world technological frontier. (i) F(0)"0, (ii) F(a) is diwerentiable, and (iii)
F (a )"F(a), ∀j50.
0 T`j
Assumption 2.1 restricts a to the positive half line (thereby ensuring that no
country enters a no-growth trap), and states that no observations are available
for any product generation not yet developed somewhere.
The memoryless property of the R&D production function implies that
innovations are independent Poisson events. As long as no new information
about the value of a is received, the intensity of R&D is constant for the
q
duration of any race and the time to the next innovation is exponentially
distributed. Note also that v (t) } the value of winning the qth race } depends on
q
the duration of the (q#1)th race. Thus, conditional on a , the expected value
q`1
of winning the qth race can be written as

P CP

E (v (t)Da )"
n q
q`1

=

x

D

e~oqn(t) dt [a ¸ (t)b
q`1 q`1

0
0
]expM!a ¸ (t)bN] dx
q`1 q`1
(j!1)
.
(5)
"
j[o#a ¸ (t)b]
q`1 q`1
Integrating Eq. (5) over all possible values of a
yields the unconditional
q`1
expected value of winning the qth race, E (v (t)). Then, combining with the zero
n q
pro"t condition Eq. (4), the equilibrium intensity of aggregate R&D satis"es

P

(j!1)E (a ) =
1
n q
dF (a ).
(6)
¸ (t)1~b"
q
w(t)j
o#a ¸ (t)b n q`1
0
q`1 q`1
The intensity of R&D in the race for the qth monopoly depends not only on
the expected value of a , but also on the outcome of the race for the next
q
generation. The term a ¸ (t)b dt denotes the probability that the race for
q`1 q`1
the (q#1)th monopoly ends in the interval dt, where ¸ (t) denotes the
q`1
intensity of R&D in the (q#1)th race that is believed to be optimal given the
information available at time t; F (a ) denotes the current subjective distribun q`1
tion for a . The instantaneous probability that the qth monopoly ends its
q`1
tenure enters as an addition to the rate of discount. The expected payo! to the
winner 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 ¸ (t)b.
q`1 q`1
2.3. The competitive equilibrium
Substituting for w(t) in Eq. (6) and using the full-employment constraint,
¸ (t)#¸ (t)"¸, the competitive equilibrium R&D intensity during the race for
x
q

1292

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

generation q is de"ned by the "xed point expression,
¸ (t)1~b"(j!1)(¸!¸ (t))E (a )
q
q
n q

P

=
dF (a )
n q`1
.
o#a ¸ (t)b
0
q`1 q`1

(7)

For technological laggards, laissez faire equilibria must satisfy the nonlinear "rst-order di!erence equation written in implicit form in Eq. (7). Boundary conditions are given by equilibria that apply at the world technological
frontier.
The "rst lemma establishes that there is a unique stationary equilibrium at the
frontier.
Lemma 2.2. A country at the technological frontier has a unique stationary competitive equilibrium with positive R&D intensity, ¸c , satisfying
T

P

= dF(a)
(¸c )1~b
T
"(j!1)E(a)
.
(o#a(¸c )b)
¸!¸c
0
T
T

(8)

Periodic equilibria at the world technological frontier cannot always be ruled
out in this model. One can prove their existence by following the analysis in
Aghion and Howitt (1992), (Section 3A). However, I will follow Aghion and
Howitt (1992), and others, in restricting attention to the unique stationary
equilibrium at the world technological frontier.
I am now in a position to provide conditions for the existence of a unique
competitive equilibrium during the race for product generation q(¹ in a
technological laggard:
Theorem 2.3. Assume there is a unique sequence, F (a ), F (a ),2, F (a
), of
n q n q`1
n T~1
subjective distributions with positive support. Then, there is a unique competitive
equilibrium intensity of R&D, ¸c , during the race for generation q(¹.
q
Theorem 2.3 states that if the intensity of R&D at the world technological
frontier is uniquely de"ned then so is the intensity of R&D during the race for
the current product generation in a technological laggard. The intuition behind
the absence of multiple, periodic equilibria in laggards is straightforward. The
possibility of cyclical equilibria at the frontier arises because there is no boundary condition to pin down the value of ¸
for any j'0. In contrast, restricting
T`j
attention to the unique stationary equilibrium at the technological frontier pegs
the value of ¸c precisely and provides a unique boundary condition that de"nes
T
unique values for R&D intensities in races to develop less advanced technologies in lagging countries.

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

1293

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

P

=
e~o(q~t)[(q(q)!1) ln j#ln(¸!¸ (q))] dq,
(9)
q
t
where q(t) denotes the product generation for which researchers are racing at
time 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 the
technological conditions governing the rates of innovation; q(t) is an integervalued step function where the steps are formed from a Poisson process with
magnitude one and intensity a ¸ (t)b. The Bellman equation for this problem is
q q
C(t)"

max o< (t)"(q!1) ln j#ln(¸!¸ (t))
q
q
Lq(t)
=
#[< (t)!< (t)] a ¸ (t)b dF (a ),
(10)
q`1
q
q q
n q
0
where < (t) denotes the value function at time t when researchers are racing for
q
product generation q. The right-hand side of the Bellman equation consists of
three terms. The "rst two terms describe the #ow of consumption bene"ts
received during the race for product generation q. The third term is the jump in
the value function that occurs when generation q is developed, multiplied by the
expected probability that the innovation takes place in the next instant. The
maximized sum of these three terms equals the interest o< (t) that can be earned
q
on a risk-free bond of size < (t).
q
Di!erentiating Eq. (10) yields the "rst-order condition

P

(11)
¸H(t)1~b"bE (a )(¸!¸H(t))[< (t)!< (t)].
q
q`1
q
q
n q
Although the social planner's problem appears simple, its solution is not easily
obtained. In general, the subjective distributions, F (a ), are not stationary over
n q
product generations, and standard dynamic programming techniques for

1294

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

stationary problems apply only for countries at the technological frontier.
Following the analysis of Section 2.3, I "rst characterize the solution that applies
at the frontier (Lemma 2.4). E$cient and laissez faire rates of growth at the
frontier are compared in Proposition 2.5. Existence and uniqueness of the
solution to the planner's problem is then established for technological laggards
by backward induction (Theorem 2.7). In the following subsection, I compare
the e$cient and laissez faire R&D intensities for technological laggards, and
characterize 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) takes
the form o< "A#(¹!1) ln j, where A is a coe$cient to be determined.
T
Thus, <
!< "o~1 ln j and ¸H satis"es the "xed-point expression
T`1
T
T
b ln j E(a)(¸!¸H)
T.
¸H1~b"
T
o

(12)

It is easy to see that a unique interior solution to Eq. (12) exists, in which case
< satis"es
T
E(a)¸H ln j
T
,
o< "(¹!1) lnj#ln(¸!¸H)#
T
T
o

(13)

where ¸H is the increasing function of E(a) de"ned in Eq. (12). Lemma 2.4 follows
T
immediately.7
Lemma 2.4. At the technological frontier, there is a unique solution to the social
planner's problem, with R&D intensity increasing in b, j and E(a), and decreasing
in o.
Let Egc denote the expected growth rate of quality at the frontier under
T
laissez faire and let EgH denote the corresponding expected growth rate in the
T
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 (j0(R, such that iw j3[j , j0], then EgH'Egc ; j [j0] is decreasing
T 0
T
0
0
[increasing] in ¸.

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

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

1295

Proposition 2.5 highlights the fact that economy size is critical in determining
whether or not the competitive equilibrium generates a faster growth rate than is
socially optimal. Jones and Williams (1997) have calibrated a simple R&Dbased growth model to US data. They concluded that the optimal intensity of
R&D may be as much as four times larger than the current intensity. Following
their lead, it will be assumed throughout the remainder of this paper that
j (j(j0.
0
Assumption 2.6. EgH'Egc .
T
T
I turn now to the social planner's problem for technological laggards. First, it
can 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 (a ), E (a ),2, E (a
) of the R&D productivity
n q n q`1
n T~1
parameters. Then, (i) there is a unique solution to the social planner's problem that
depends on the sequence MME (a )NT~1, E(a)N; (ii) R&D intensity is increasing in
n j j/q
each element of the sequence.
Theorem 2.7 highlights the forward-looking nature of the social planner in
technological laggards. If the planner expects R&D e!ort in future generations
to be especially productive, then he will accelerate the current R&D race
in order to bring forward the expected arrival date of those future generations.
Conversely, a pessimistic outlook for future R&D races retards the current
growth race. Even though technological leapfrogging is excluded from
this model, the planner's current R&D policies always depend on all technologies that have been developed elsewhere but that remain to be developed
at home.
Note also that the solution to the social planner's problem depends only on
the sequence of subjective means, while the competitive equilibrium intensity of
R&D depends on all moments of the subjective distribution F (a ). The
n q`1
intuition behind this result is straightfoward. Firms engaging in the race to
generate product generation q care about the expected present value of the
monopoly pro"ts that accrue to the winner. As the expected present value of
monopoly pro"ts is a nonlinear function of a , all moments of F (a ) enter into
q
n q`1
the solution. In contrast, the social bene"ts of any innovation planner last
inde"nitely, so that the duration of the monopoly for product generation q is not
of 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 the
e$cient and competitive regimes, which can conveniently be illustrated by

1296

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

writing the optimal intensity of R&D, ¸H(t) as a function9 of ¸H (t),
q
q`1
¸H(t)1~b
¸H (t)
lnjbE (a )
bE (a )
q`1
q
n q# n q
"
b(¸!¸H (t))
¸!¸H(t)
o
o
q`1
q
¸H(t)
¸!¸H (t)
q
q`1
,
!
#ln
b(¸!¸H(t))
¸!¸H(t)
q
q
and which can be compared with the laissez faire equilibrium,

C

A

BD

(14)

P

=
¸c (t)1~b
dF (a )
q
n q`1
"(j!1) E (a )
.
(15)
n
q
¸!¸c (t)
o#a ¸c (t)b
q
0
q`1 q`1
Note that when q"¹, the second term on the RHS of Eq. (14) vanishes. The
di!erences between the "rst terms on the RHS of Eqs. (14) and (15) re#ect
market failures that are now familiar in quality ladder models. First, the solution
to the planner's problem substitutes ln j for (j!1) because the planner cares
about 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 the
competitive equilibrium. This di!erence is a congestion externality that arises
because the planner recognizes that each "rm contributes to aggregate diminishing 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 "rm
discounts the pro"t #ow at the rate o#a ¸c (t)b. This di!erence arises
q`1 q`1
because "rms survive only to the next innovation while the social value of an
innovation lasts forever. These features of the model are well known from
steady-state analyses.
Among technological laggards, however, there is a fourth divergence between
the e$cient and competitive outcomes. The planner's problem includes a term
that depends on the di!erence between the optimal intensity of R&D in the
current race and the intensity of R&D that is currently expected to be optimal in
the next race. This additional term in Eq. (14) re#ects the forward-looking
nature of the social planner: ¸H (t) is in fact a summary statistic for the current
q`1
subjective expectations of all future R&D e$ciencies.

9 Use Eq. (11) in Eq. (10) to remove < (t). This yields an expression relating ¸H to < . Then
q
q
q`1
update (11) and (10) by one product generation and combine them to remove < !< . This
q`2
q`1
H
generates an expression relating ¸
to < . The two new expressions so obtained can then be
q`1
q`1
combined to eliminate < .
q`1
10 The substitution of ln j for (j!1) confounds two market failures. The "rst is the proxt
destruction e!ect: a new monopoly earns pro"ts, (j!1), only by destroying the pro"ts earned by the
previous monopoly. As the social planner is not concerned with the identity of the "rm that currently
earns pro"ts, this private value of innovation is ignored. The second is the consumer apppropriability
e!ect: a social planner values the increment to consumer surplus, ln j, but this value is not
appropriable by "rms who therefore ignore it.

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

1297

A subsidy, s , to R&D can be employed to equate ¸H and ¸c . Naturally,
q
q
q
s increases with the distance between the e$cient and competitive intensities of
q
R&D. At the technological frontier, ¸H and ¸c are constant, and so it follows
T
T
that the optimal subsidy to R&D, sH, is also constant. Substituting for the unit
T
cost of R&D in Eq. (6) and then comparing Eq. (8) with Eq. (12), the optimal
subsidy at the frontier is given by

P

o(j!1) = dF(a)
,
sH"1!
T
o#a¸Hb
b ln j
0
T

(16)

which, by Assumption 2.6, is positive.
In contrast, for q(¹, sH is a random variable that varies across product
q
generations, and within a product generation whenever a new signal is received.
Moreover, one cannot even sign sH; bad news about the prospects for developing
q
product generation q#1 increases ¸c but reduces ¸H, and in the face of
q
q
su$ciently bad news the former may be larger.
Proposition 2.8. The optimal subsidy, sH, q(¹, for technological laggards is
q
a random variable, not necessarily with strictly positive support. The optimal
subsidy at the world technological frontier, sH, is constant and positive across
T
product generations and countries.

3. International signals of R&D e7ciency
It is necessary at this stage to impose some minimal structure on the relationship between the prior distribution, F(a), a technologically lagging country's
posterior distribution of a , F (a ), and the n realizations of a that a country has
q n q
q
observed. In this section I make two assumptions that are su$cient to enable me
to say something useful in Section 4 about comparative growth.
The "rst assumption is that the signals received and the rules used to
transform the signals into an expectation generate values of E (a ) that are
n q
unconditionally unbiased.
Assumption 3.1. Let H(E (a )Da ) denote the conditional distribution of E (a ) when
n q q
n q
the unobserved ezciency of R&D is a , and let E(E (a Da ))":E (a ) dH denote its
q
n q q
n q
conditional expectation. Then :E(E (a )Da ) dF"E(a).
n q q
The second requirement is that observations of the values of a realized in
q
more advanced countries provide useful information about the value of a . That
q
is, large values of E (a ) should be more likely when the true (unobserved) value
n q
of a is large. This requirement is made precise in the sense of "rst-order
q
stochastic dominance.

1298

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

Assumption 3.2. For any aA'a@ , H(E (a )DaA)4H(E (a )Da@ ). If this holds as
n q q
q
n q q
q
a strict inequality for some E (a )'0, then the signals are informative.
n q
These two assumptions can accommodate a large variety of stochastic environments. At one extreme, the parameter a is perfectly observed after the
q
completion of an innovation race, but it is imperfectly correlated across countries. By observing realizations of a in advanced countries, a technological
q
laggard can learn something about the distribution from which its own value of
a will be drawn. At the other extreme, a is identical across countries but it is
q
q
imperfectly observed. For example, a laggard may observe the time or cost
required to produce an innovation, and from this it can infer something about
a . Assumptions 3.1 and 3.2 can accommodate these extremes as well as a combiq
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 are
provided here.
Example 3.3 (Perfect information, imperfect correlation). Assume that the
country-speci"c e$ciency parameter, a is a random variable drawn from an
q
exponential distribution with unknown parameter v . The parameter of the
q
distribution is speci"c to the product generation, but not to the country. Assume
further that v is itself a random variable that has a prior gamma distribution
q
with parameters a"2 and b'0. The prior density for a is given by

P

=b2v2e~(a`b)v
2b2
dv"
.
(17)
(a#b)3
C(2)
0
Assume now that a country has observed n realizations of a with mean k, and
q
that prior beliefs are updated by Bayes' rule. The 2-tuple Mk, nN is a su$cient
statistic for a and the posterior density function, derived in Appendix B, is
q
given by
f (a)"

(n#2)( b#nk)n`2
f (a Dk)"
.
n q
(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 a is the
q
same for all countries. The technological laggard observes the durations
t , t ,2, t , of patent races in n countries at the technological frontier. Given the
1 2
n
technology of innovation described in Section 2, the durations are exponentially
distributed with unknown parameter a Rb. The intensity of R&D is the same in
q

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

1299

all countries at the frontier, so one can choose units such that Rb"1. Assume
further that the prior distribution of a is gamma with parameters a'0 and
q
b'0. Then the posterior distribution of a is gamma with parameters a#n and
q
b#+n t . This result is standard (e.g. De Groot, 1970), and is not analyzed in
i/1 i
the appendix.
While both examples are special cases of Assumptions 3.1 and 3.2, there is an
important di!erence between them. In Example 3.4 a su$cient number of
observations allows a country to know the value of a precisely. In Example 3.3,
q
in contrast, the technological laggard can never know its value of a precisely. As
q
nPR, the limiting posterior distribution in Example 3.3 does not become
degenerate, but rather converges with probability one to an exponential distribution with known parameter v .11
q
4. Comparative growth
This section focuses on the impact of signals on expected instantaneous
growth rates. The main results are as follows. Growth rates for e$cient technological laggards observing signals are more variable than they are for countries
at the frontier. Signals do not always increase the expected growth rates of
e$cient laggards, but they are more likely to do so when aggregate returns to
scale in R&D do not diminish too rapidly and the equilibrium level of R&D
employment is a small fraction of the labor force. Even under these conditions,
I cannot establish that signals raise the average instantaneous growth rate of
technological laggards adopting a policy of laissez faire. There is an intuitive
reason, explained below, why one might not expect a laissez faire equilibrium to
exhibit a clear comparative growth result. However, this observation should be
viewed with caution. I have also been unable to "nd an empirical counterexample in which the expected growth rate of a technological laggard is lower
under laissez faire than at the frontier, and so the e!ect of signals on growth
under 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 the
e!ect of signals on technological laggards depends very much on the question
that is asked. One could ask, as I do in this section, whether signals raise the
average instantaneous expected growth rate of a technological laggard. But
there are other, equally valid questions. For example, one could ask whether the
expected length of time required to develop product generation q is reduced by

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

1300

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

signals. The answers to the two questions need not be the same in a stochastic
environment, because the relationship between the two measures of growth is
non-linear. Too see this, consider a country that lags one generation behind the
frontier and observes precisely either of two e$ciency parameters, a and a ,
0
1
where a 'a and each of which could have occurred with probability one half.
1
0
Let (j!1)a ¸b and (j!1)a ¸b denote the growth rates that are realized when
1 1
0 0
each parameter is observed, and note that ¸ '¸ . The unconditional expected
1
0
growth rate is then 0.5(j!1)[a ¸b #a ¸b ]. The unconditional expected dura1 1
0 0
tion of the patent race is given by 0.5[(a ¸b )~1#(a ¸b )~1]. Let ¸ denote the
T
1 1
0 0
R&D intensity at the frontier. Then the signals raise the expected instantaneous
growth rate only if (a ¸b #a ¸b )/(a #a )'¸b and they reduce the expected
T
1
1 1 0
0 0
duration of the patent race only if (a ¸b #a ¸b )/(a #a )((¸ ¸ /¸ )b. Given
1
0 1 T
1 1 0
0 0
parameter values b"1, a "1, a "2, ¸ "1 and ¸ "2, the "rst inequality
0
1
0
1
requires that ¸ (1.6, while the second inequality requires that ¸ (1.8. There
T
T
is a window in which it is possible that signals increase the expected instantaneous growth rate while raising the expected duration of the patent race.
4.1. Ezcient technological laggards
The analysis of this section begins with a useful lemma.
Lemma 4.1. For any q(¹, E(a ¸H(t)b)5E(a) E(¸H(t)b) with a strict inequality
q q
q
[equality] if signals are informative [uninformative].
E$cient laggards will tend to raise their R&D intensity when the (unobserved) value of the R&D e$ciency parameter is high, and they will tend to reduce
intensity when e$ciency is low. As all countries face the same unconditional
distribution for the e$ciency parameter, the positive correlation between a and
q
¸H(t)b immediately yields the following result:
q
Proposition 4.2. Growth rates are more variable across product generations in
ezcient laggards than they are in countries at the technological frontier.
At the technological frontier, a social planner chooses a constant intensity of
R&D, ¸H, and so the expected growth rate is given by EgH"(j!1)E(a)¸Hb.
T
T
T
The social planner in a technological laggard, in contrast, chooses R&D in
response to signals received, and the expected growth rate in this case is
EgH(t)"(j!1)::a¸ H(t)b dH dF"(j!1)E(a ¸ H(t)b)*(j!1)E(a)E(¸ H(t)b).
q
q
q q
q
Thus E(¸H(t)b)*¸Hb is a su$cient condition for technological laggards to grow
T
q
more rapidly on average than countries at the frontier. The di$culty in establishing this inequality is that ¸H(t)b depends (non-randomly) on E (a ), which is
q
n q
a random variable that in turn depends on the unobserved value of a .
q
Moreover, ¸H(t)b also depends on ¸H which is itself a random variable.
q`1
q

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

1301

Whether one can rank E(¸H(t)b) and ¸Hb turns in large part on whether one can
T
q
show that ¸H(t)b is a convex function of E (a ) and of ¸H (t). The required
q`1
q
n q
convexities do not always hold. However, the following lemma provides conditions under which they do.
Lemma 4.3. If b3(1/2, 1), there exists an e3(0, 1) such that for all ¸H(t)3(0, e¸)
q
and ¸H (t)3(0, e¸), ¸H(t)b, is a locally convex function of E (a ) and of ¸H (t).
q`1
q
n q
q`1
The lemma requires that aggregate returns to scale in R&D do not diminish
too rapidly, and that in equilibrium R&D labor is a small enough fraction of the
total labor force. Although one must be careful in comparing empirical assumptions of an abstract model with data, the conditions seem plausible. First, R&D
expenditures among even the most R&D-intensive countries are less than three
percent of GDP.12 Second, industry evidence suggests only weakly diminishing
returns to scale.13 Note also that the conditions of Lemma 4.3 are su$cient but
not necessary. They become necessary conditions only in the limiting case that
the signals to which the social planner is responding turn out to be completely
uninformative.
Proposition 4.4. If the conditions of Lemma 4.3 hold, then for any q(¹, the
following expected growth rates can be ranked: EgH'EgH'Egc .
T
T
q
Proposition 4.4 predicts that quality in e$cient technological laggards will
eventually catch up to the world technological frontier, but it does not predict
that convergence will be monotonic. In fact, there are three forces ensuring
non-monotonicity. The "rst is, of course, that R&D e$ciency varies across
countries and product generations. At any point in time, a laggard may be
engaged in research that is more di$cult than development of the product
generation currently occupying researchers at the frontier. Second, the intensity
of current research e!ort in the e$cient laggard depends positively on the
e$ciency of R&D in all its undeveloped product generations. Even if signals
indicate that R&D in the current race is more e$cient than average, bad news
about future generations can induce an o!setting reduction in current R&D.
Third, the technological laggard may occasionally receive misleading signals
about its R&D parameter, causing it to adjust R&D in the wrong direction.

12 The latest available percentages from the OECD are: France, 2.0%; Germany 2.7%; Japan,
2.8%; United States, 1.9%.
13 Bound et al. (1984) have suggested that returns to scale in R&D are approximately constant up
to $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-digit
level ranging from 0.53 to 1.28, with a mean of 0.84.

1302

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

A numerical example may provide some more intuition about these results.
Assume that a may take either of two values, HIGH or LOW, with equal
q
probability. The social planner observes a signal about the e$ciency of R&D for
each product generation behind the world technological frontier, and assumes
that the signal is a precise predictor of its own country's R&D e$ciency. Table
1 reports the social planner's choice of R&D intensity, ¸H, for each possible set
q
of realized signals. The conditional expected growth rates, gH, and the uncondiq
tional expected growth rates, EgH, are given for four di!erent degrees of signal
q
accuracy. Note that the choice of R&D e!ort depends only on the signals
received. The expected growth rates, in contrast, also depend on the accuracy of
those signals.
There are two channels through which signals increase the unconditional
expected growth rate. The "rst is that signals direct the planner to devote more
e!ort to R&D when signals suggest that it will be particularly e!ective. The
second channel results from the convexity of the function ¸Hb(E (a )) established
q n q
in Lemma 4.3. In the case where signals are completely uninformative, the "rst
channel does nothing to raise the expected growth rate. Hence, Table 1 provides
a useful decomposition of the sources of enhanced growth. At the frontier, the
unconditional expected growth rate is 2.2%. When signals are completely
uninformative, a social planner developing product generation ¹!1 attains an
expected growth rate of 3.1%, while for generation ¹!2 it is 3.5%. Hence, the
growth rate rises by 0.9% and 1.3%, respectively, simply as a result of the
convexity. Informative signals raise the expected growth rate further. For
example, on moving from uninformative to perfectly informative signals the
unconditional expected growth rates increases from 3.1% to 4.2% in generation
¹!1, and from 3.5% to 4.6% in generation ¹!2.
Of course, these observations do not imply that a social planner should
randomly alter the R&D intensity to raise the expected growth rate. If signals
are uninformative and the social planner knows this, the optimal policy is
a constant intensity of R&D equal to the rate chosen at the world technological
frontier. In contrast, when signals are precise the welfare-maximizing policy is to
choose the R&D intensities indicated in Table 1. In this example, the planner
alters R&D intensities under the possibly mistaken belief that the signals are
perfectly informative. The welfare e!ect of signals therefore depends on the
correspondence between the accuracy of signals and the social planner's evaluation of their accuracy. To say more, however, would require making further
assumptions about the properties of H(E (a )Da ).
n q q
4.2. Technological laggards
When technological laggards adopt a laissez faire approach to R&D, I cannot
produce 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

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

1303

Table 1
Numerical example: Social planners's problem
Signal observed

Expected growth rate by signal accuracy!
R&D
e!ort

0.50"

0.60"

0.75"

1.00"

}

1.52

0.022

0.022

0.022

0.022

}
}

HIGH
LOW
MEANS:

4.83
0.05
2.44

0.062
0.001
0.031

0.066
0.001
0.034

0.072
0.001
0.037

0.083
0.001
0.042

HIGH
LOW
HIGH
LOW

HIGH
HIGH
LOW
LOW
MEANS:

6.58
0.15
4.20
0.03
2.74

0.082
0.003
0.055
0.001
0.035

0.087
0.003
0.058
0.001
0.037

0.095
0.002
0.064
0.005
0.041

0.109
0.002
0.073
0.000
0.046

Product
generation

a

¹

}

¹!1

¹!2

T~2

a

T~1

Parameter values used in the example are: ¸"100, b"0.7, o"0.05, j"1.04. The R&D e$ciency
parameters 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 is HIGH [LOW]
q
when the signal observed is HIGH [LOW]. Thus, 0.50 in the "rst column de"nes a completely
uninformative signal. The second and third columns indicate the signals are informative by the
de"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 when
R&D is believed to be e!ective and less when it is believed to be relatively
ine!ective. Under the same conditions as laid out in lemma 4.3, this e!ect
promotes growth over the long run. On the other hand, if signals raise the
average expected growth rate in, say, generation ¹!1, the expected duration of
the monopoly attained by the winner of the race for product generation ¹!2
may be reduced. This channel will have a negative e!ect on R&D e!ort in the
race for generation ¹!2. Thus, while it is easy to show that, if the conditions of
Lemma 4.3 apply, Egc 'Egc , I cannot show that Egc 'Egc . However,
T
T~2
T
T~1
I have not been able to produce a counterexample in which Egc (Egc . Thus,
T
T~2
the e!ect of signals on the laissez faire growth rate in countries lagging the world
technological frontier by at least two generations remains an open question.
Table 2, continuing the earlier example, provides corresponding data for the
l