1
A GENERAL THEORY OF SUSTAINED GROWTH
WITH MULTIPLE GPTs
By
Kenneth I. Carlaw
Associate Professor
Department of Economics
University of British Columbia - Okanagan
Kelowna, British Columbia
Canada
Email: kenneth.carlaw@ubc.ca
And
Professor Emeritus Richard G. Lipsey
Department of Economics
Simon Fraser University
Vancouver, British Columbia
Canada
Email: rlipsey@sfu.ca
2
I. INTRODUCTION
Our evolutionary vision of long term growth is of an historical, irregular,
path dependent process in which heterogeneous agents operate in a nonstationary environment of uncertainty. In our version, this environment is
occasionally punctuated by the arrival of General Purpose Technologies
(GPTs), which transform economic, social and political relations.1 This
evolutionary view contrasts with the Newtonian approach found in more
traditional neoclassical and endogenous growth models.
One of the most important problems in evolutionary growth theory is to
bridge the gap between (i) the vast amount that is known about micro
behaviour of agents concerned with innovation and technological change
and (ii) the aggregate growth behaviour of economies, which we seek to
understand.
Most evolutionary theories of technological change take a ‘bottom–up’
approach. They start with micro behaviour, rejecting the concept of
representative agents and instead build on diversity and selection. However,
these micro-based evolutionary approaches encounter the difficult, and as
yet unresolved, challenge of aggregating from realistic, heterogeneous,
micro behaviour to observed macro relations.
Macro growth theories that seek to incorporate evolutionary concepts
encounter two basic problems. First, they need to get beyond the flat concept
of technology inherent in any aggregate production function. The early
models of GPTs took an important step in this direction. However, whenever
these models employ such neoclassical maximising techniques as
dynamically stationary equilibrium concepts, the attempt to model
technology in an increasingly structured way rapidly makes them
analytically intractable. This in one important reason why all the first
generation models of GPTs in the Helpman (1998) volume omit most of
what our historical studies suggest are important characteristics of GPTs. We
have circumvented this modelling problem by building non-conventional,
dynamic, non-stationary equilibrium models of GPT-driven growth. In them,
1
Lipsey Carlaw and Bekar (2005: 98) (hereafter LC&B) define this concept as follows: “A GPT is a single
generic technology, recognisable as such over its whole lifetime, that initially has much scope for
improvement and eventually comes to be widely used, to have many uses, and to have many spillover
effects.”
3
agents face an uncertain future and so do the best they can with limited
knowledge of current relations.2
The second problem is to bridge the micro-macro gap. In spite of such
works as John Foster’s (1987) Evolutionary Macroeconomics, the top-down
approach still seems a long way from bridging this gap. We are approaching
this problem by building a simple non-equilibrium macro model and then
amending it incrementally in order to make it approach the kinds of micro
behaviour that are observed in our historical studies and that are
incorporated in many micro-evolutionary models of innovation and
technological change. 3
In our original three-sector model, newly invented GPTs immediately
had their full effects felt in the sector that used them. 4 Subsequent versions
have introduced the following complications, all of which are found in
Chapters 14 and 15 of Lipsey, Carlaw and Bekar (hereafter LC&B):
allowing the effects of a new GPT to diffuse through the rest of the system
according to a logistic time path; varying the types of expectations on which
agents base their resource-allocation decisions allowing for heterogeneity in
the effects of a new GPT by adding a second consumption sector that can be
affected by a new GPT in a different way than the first sector; allowing for
the effects that new GPTs have in requiring alterations to the structure of the
economy (what we call the ‘facilitating structure’), which is a first
approximation to modelling what Freeman and Perez (1988) and Freeman
and Louca (2001) call the ‘crisis of adjustment’.
We have used various versions of our single-GPT model for several
applications. We refuted by counter-example the Alchian-Friedman
hypothesis that any reasonable economy would behave as if all its agent
were maximisers, even if they were not. We found some interesting
conditions under which rational and adaptive expectations produced
identical behaviour of the model and other conditions where they produced
radically different behaviour. We were able to show in Monte Carlo
simulations that changes in total factor productivity (TFP) do not come even
near to tracking changes in technology, thus supporting the more general
arguments to this effect given in Lipsey and Carlaw (2004). We were also
2
3
See LC&B Chapters 14 and 15, Carlaw and Lipsey (2006a).
For our fuller justification of this approach, see LC&B, 367-9. For the historical evidence, see especially
LC&B, Chapters 5 and 6.
4
This model was first given in Carlaw and Lipsey (2001) and finally published in Carlaw and Lipsey
(2006a)
4
able to fit one version our model to Australian data, revealing some
interesting possibilities.
We are now tackling the next two major tasks in our research program.
The first is to incorporate more than one GPT whose effects differ from each
other, thus allowing for relations of complementarity and competitiveness
among existing GPTs. This is an important task since one of the limitations
of the first generation of GPT models in Helpman (1998) was that they
contained only one GPT whose behaviour determined the macro behaviour
of the whole economy, when in fact that behaviour is determined by the
interaction of a number of GPTs that always exist simultaneously, some in
competition with each other and some that cooperate with each other. In
LC&B (2005) we placed great emphasis on this criticism but were not able,
in the time available, to build more than a very crude two-GPT model in
which each operated in a sector that was independent of the other. This
barely scratched the surface of the problem.
Modelling more than one GPT simultaneously operating in an economy
will allow us to refine further the possibilities for applying the model to
issues such as those just mentioned, as well as applying it to a range of
issues that could not even be considered within the confines of single-GPT
models. Although many applications are possible, each requires a separate
paper. This, primarily theoretical paper, lays the groundwork for further
applied work by building a model that is general in that it can handle as
many GPTs, as many sectors, as many industries within each sector, as many
different relations of competitiveness and complementarity among GPTs,
and as many variations in the behaviour and expectations of the agents
operating in the model’s various sectors and industries, as is desired.
The second task is to incorporate one of the key aspects of evolutionary
models: heterogeneous behaviour within any one activity and a selection
mechanism that determines survival and demise. This extension will come
next.
II. THE SINGLE- GPT MODEL5
What we call our base-line model contains three sectors, each containing
a single industry that produces (i) a consumption good, c, (ii) applied
research, a, that improves productivity in the other two sectors and (iii) pure
5
The model described in this section is based on the ‘base-line’ model in LC&B (2005) Chapter 14. We
repeat it here because we must amend it substantially to allow it to develop into the multi-GPT model of
Part III.
5
knowledge, g, that occasionally leads to the discovery of a new GPT. Each
sector has its own distinct aggregate production function. Thus the intraindustry technology is flat, while the technological structure is imposed
through the inter-industry relations among these three different production
functions.
The index n counts the number of times that a new GPT has been
invented and adopted; if a GPT is invented but not adopted, n does not
change. We use subscripts n, n-1,… to indicate the most recently invented
and the previously invented GPTswe speak of different vintages of the
one generic type of GPT. When a new GPT is invented we call it the
‘challenger’, while the GPT that was already in place is called the
‘incumbent’.
The term t always refers to the sequence of time periods. When
subscripted, it refers to one specific time For examples, tn and (t-1)n refer
respectively to the time at which the incumbent GPT was invented and
adopted and to one time period before that.
The Resource Constraint
There is a single exogenously determined resource supply, R, that is
allocated each period among the three sectors:
(1)
R = rc,t + ra,t + rg,t .
The Consumption Sector
Consumption output depends on the resources devoted to it and the
productivity coefficient, which is the stock of applied knowledge, At
multiplied by parameter, µ, which divides this knowledge between the
proportions that are useful in the consumption sector and in the pure
research sector, all raised to a positive power that does not exceed unity:6
6
LC&B define technology as follows.
“Technological knowledge, technology for short, is the set of ideas specifying all activities
that create economic value. It comprises: (1) knowledge about product technologies, the
specifications of everything that is produced; (2) knowledge about process technologies, the
specifications of all processes by which goods and services are produced; (3) knowledge
about organisational technologies, the specification of how productive activity is organised in
productive and administrative units for producing present and future goods and services
(which thus includes knowledge about how to conduct R&D).”
What matters for the present model is that, at our present level of abstraction, no distinction is needed
between technological knowledge and the technological artefacts that embody it. So when we use the A
6
(2)
ct = ( µ At )α1 rcα,t2
α1 ∈ (0,1], α 2 ∈ (0,1)
The restriction on α1 implies non-increasing returns to the new knowledge
created by the applied R&D sector. The restriction on α2 implies decreasing
returns to the resources applied to this sector.7
The term µ reflects our decision on how to handle the complementarity
between applied knowledge and the other two sectors. First, there is an
obvious complementarity between applied R&D and the consumption sector
because applied knowledge increases productivity directly by shifting the
production function for the consumption good. Second, the spillovers from
applied to pure research are well attested by many studies.8 There are two
basic ways in which we could model this latter type of spillover. We could
assume that the whole stock of applied knowledge was useful in the
consumption sector, while some fraction of it spilled over as an externality
to increase the efficiency of pure research. We have done this in other
models, but here we assume that the stock of applied knowledge is divided
between what is useful in the consumption sector and what is useful in the
pure research sector. This division ensures that we do not introduce any
permanent increasing returns to accumulating knowledge, nor any
externalities. It has two advantages. First, it allows us to produce a model of
sustained endogenous growth without some of the characteristics that are
often needed to sustain growth in models of endogenous growth. Second, it
creates a model suited to studying the relation between technological change
and standard measures of changes in TFP, which usually assume constant
returns to scale.
Note that there are no lagged relations in (2). Appendix 1 provides a
possibly more intuitively appealing description of the discrete time structure
with an inter-temporal trade-off, where the A term is lagged by one period
and the maximization problem is articulated using a Bellman equation. In
the text, we assume that applied R&D knowledge has an immediate impact
produced by the applied R&D sector in the consumption goods industry’s productivity parameter, this can
represent applied knowledge and the capital goods that embody that knowledge.
7
We assume diminishing returns to the resources allocated to production in each sector for both empirical
and theoretical reasons. Empirically, we argue that even in the long term there are fixed resources, such as
were broadly included under the category ‘land’ by earlier economistse.g., natural resources, sites for
cities and harbours and the carrying capacity of the environment. Thus, growth in the very long term is only
possible if technology changes. Theoretically, we wish to show that sustained growth is possible without
the assumption of either the increasing or constant returns to resource inputs that is required by many other
models of sustained growth.
8
See, for example, LC&B, Chapter 4 and Rosenberg 1982: Chapter 7.
7
on the productivity of consumption. Doing so does not change any of the
model’s qualitative behaviour.9
Because the results of applied research in the current period affect the
productivity of consumption in the same period, there is no inter-temporal
substitution in the model. Thus the maximizing trade off is between the
amount of the consumption good directly produced by the resources devoted
to its production with a given production function and amount of the
consumption good indirectly produced by the resources that go into applied
research to improve the consumption sector’s productivity. A marginal reallocation of resources from the consumption sector to applied research
directly reduces the production of the consumption good, while indirectly
increasing its production by raising the productivity of those resources that
remain in the consumption sector. Maximisation of consumption output in
any one period requires that no reallocation of resources between the two
sectors can increase consumption output. Similar considerations apply to the
pure research sector, as detailed later in the paper.
In the text, we do not use the Bellman equation notation shown in
Appendix 1 for two reasons. First is simplicity. 10 Second, we do not wish to
model a closed form, stationary, dynamic equilibrium. So, we will not
satisfy the maximum principle and the transversality condition normally
applied to such maximization procedures. Instead, the agents generating the
behaviour in the model have insufficient foresight and information to be able
to solve for the infinite horizon stationary equilibrium. This models the welldocumented, genuine uncertainty that accompanies technological change.11
This approach turns out be beneficial for our modeling. It frees us from
the technical feasibility constraints encountered by the first generation GPT
models discussed above and allows us to model the long-term growth
process as a dynamically non-stationary phenomenon. This seems to us to be
consistent with the data of growth and change seen in most economies
before such things as filtering and structural break modifications are made to
the data.
9
We checked for this by removing all of the model’s many uncertainties and determining that it then
behaves exactly like an endogenous balanced growth model
10
There are no technical reasons why we could not introduce lags in this relation, as well as in the later one
concerning the payoff from pure research and we show this model in Appendix 1. But it would greatly
complicate the calculations, requiring much time discounting, while, as far as we can see, adding nothing
new to the results of the model.
11
For a full discussion of this point se LC&B (29-33).
8
The Applied Research Sector
The production of new applied knowledge, at, depends on the resources
devoted to that sector and on its productivity coefficient which is the useful
stock of GPT knowledge, G:
(3)
at = (Gt ) β1 raβ,t2
β1 ∈ (0,1], β 2 ∈ (0,1)
Once again, we make a simplifying assumption about the time-lag structure
in the model, allowing the stock of useful pure knowledge to have an
immediate impact on the applied R&D sector as soon as it is produced.
Appendix 1 provides a possibly more intuitively appealing version where the
stock of pure knowledge is lagged one period.
The stock of applied knowledge, At, is the current flow of produced
applied knowledge plus the depreciated stock of such knowledge
accumulated in the past:
(4)
At = at + (1-ε)At-1
(Although this sector only produces at at time t, we speak of At as being
produced by this sector.)
The Diffusion of the Effect of an Existing GPT
In this section, we investigate the working of the model with exogenously
determined GPTs. To do this we need two GPTs, the incumbent and the
GPT that it replaced. This change over occurred in the past at time tn. The
realised productivity of the incumbent GPT is given at each time period by
Gt, which enters the productivity coefficient of the applied R&D sector.
In our earlier model (Carlaw and Lipsey 2006), the full potential
productivity of the new GPT was felt immediately giving rise to unrealistic
spikes in growth when a new GPT was discovered. To model the observed
slow diffusion of a new GPT, we use a logistic diffusion process similar to
that used in LC&B, Chapters 14 and 15. We let ( Gn−1 )(t −1) be the productivity
n
of the first GPT to be adopted (and numbered n – 1) evaluated in the final
period before it was displaced by the challenger, (t-1)n. We let ϑt Pt be the
full potential productivity of the incumbent GPT when it was first adopted at
time tn. and, for now, we let ξ t = 1 . (The separate meanings of θ and P are
explained in the next section.)
n
n
n
9
 eτ +γ (t −tn )  
Gt = ξtn ( Gn −1 )( t −1) + 
ϑtn Ptn − ( Gn −1 )(t −1) 
τ + γ ( t − tn )  
n
n 

1 + e

(5)
The equation shows actually useful general purpose knowledge, Gt ,
becoming available for use in applied research according to a logistic
diffusion process in which tn is the adoption date of the incumbent GPT,
[ ϑt Pt - ( Gn −1 )(t −1) ] is the difference between the full potential productivity of
n
n
n
the incumbent GPT and the productivity of the GPT it replaced evaluated at
the time at which that GPT was last used, (t – 1)n. γ and τ are calibration
parameters that control the rate of diffusion.12
To see how this diffusion process works, note that when the new GPT
was first discovered, the value of the diffusion coefficient in equation (5)
was a very small number (since t = tn and τ is set equal to 0.06 for our
simulations). Thus, the initial productivity of producing with the new GPT
was only slightly more that the current productivity of producing with the
previous GPT.
Now assume that the diffusion process is complete so that the value of
the diffusion coefficient is effectively unity. Then (with ξt set at unity), the
productivity of the incumbent GPT, Gt, is approximately equal to the
productivity of the previous GPT, (Gn −1 )(t −1) in equation (5) plus the full
potential productivity increment conferred by the existing GPT ( i.e.,
ϑt Pt − (Gn −1 )(t −1) ) in equation (5)) now turned into actually useful knowledge.
Between these two dates, the diffusion process feeds the potential
productivity increment of the new GPT into its actual productivity
logistically.
n
n
n
n
n
Now consider the effect of ξ. It is a random number, which alters the
lower asymptote of the logistic curve. The probability function that
determines ξ is such that there is a higher likelihood of producing small
upward than downward shifts.13 This random variable models two things.
The first is the possibility of what LC&B call ‘historical increasing returns’
that exist at the early stages of a new GPT but are exhausted once the new
12
Because we are dealing with discrete time periods rather than continuous time, there are several ways in
which the transition from one GPT to another can be handled. Since the subsequent behaviour is unaffected
by how we handle the transition, we adopted what was computationally the easiest method. When a new
GPT is invented at period tn, the actual value of the G that is used in the applied R&D sector is the value of
the replaced GPT’s G at period tn-1, plus one period’s diffusion of the increment in potential productivity
brought about by the new GPT.
13
Depending on what simulations we are running, we draw ξ from different probability distributions, but
in all casesξ has a mean slightly greater than unity and a small variance.
10
technology is fully embodied.14 The second is surprises that affect how the
new GPT operates in practice rather than how it looks on the drawing board.
The Pure Knowledge Sector and the Endogenous Production of New
GPTs
To make the GPTs endogenous, we introduce our model’s third sector. It
produces pure knowledge, gt, according to the production function:
gt = ( (1 − µ ) At ) θt (rg ,t )σ 2
σ1
(6)
σ 1 ∈ (0,1], σ 2 ∈ (0,1) .
The restrictions on the σ’s ensure diminishing returns to resources used in
producing pure knowledge, rg,t, and either constant or diminishing returns to
that portion, 1-µ , of applied knowledge, At, used in the pure knowledge
sector.15 To model the uncertainly that no one knows in advance exactly how
much useful pure knowledge will be generated by a given amount of
resources devoted to pure research, we multiply rg,t by θt, which is a random
variable distributed uniformly with support [0.8, 1.2], mean 1, and variance
(0.4)2/12.
This period’s current stock of potentially useful pure knowledge, Pt, is
last period’s stock, suitably depreciated by δ, plus the flow produced this
period:
Pt = gt + (1 − δ ) Pt −1
(7)
A new GPT is invented when the random viable, λt, equals or surpasses a
threshold value λ*. The model is calibrated such that this occurs only
infrequently. (See below under the heading “Simulation”.)
If the new GPT is adopted, its potential productivity is determined by the
amount of accumulated pure knowledge at that date, Pt , multiplied by a
random variable, ϑt , whose values typically produce small variations around
unity. Thus the productivity potential of the newly invented nth GPT is ϑt Pt .
The random variable models the observation that the actual potential of any
GPT to raise productivity cannot be precisely predicted when it is originally
n
n
n
14
n
See LC&B Chapter 12, especially 397-401
Once again the stock of applied knowledge is not lagged for simplicity and the more intuitively appealing
lag structure is discussed in Appendix 1.
15
11
being developed.16 It does this by altering the upper asymptote of the logistic
diffusion curve shown in equation (5), thus varying the time taken for a GPT
to reach the final stages of its development and how much productivity
increase it will produce over that time.
The slow diffusion of the full potential of a new GPT, as described in
equation (5), creates a problem that we did not encounter in Carlaw and
Lipsey (2006) where all of a new GPT’s potentially useful knowledge
became available to the applied R&D sector at once. In that case, the new
GPT was always adopted since the passage of time ensured that it embodied
much more knowledge than the previously invented GPT. In the present
case, however, the potentially useful new knowledge only becomes actually
useful over time and, as shown below in equations (8) and (9) below, the
initial impact of the new GPT may be less than that of the incumbent.
Furthermore, the lower asymptote of the newly adopted GPT is shifted either
up or down by the random variable ξt whose one-time effect is felt during
the first period of the new GPTs life. Before making their adoption decision
agents, form an expectation, ξt , about the value that this variable will take
after adoption. If ξt is less than unity, indicating that agents are sceptical
about how well the new technology will work in practice, this increases the
chances that the challenging GPT will not initially be expected to be as
productive as the incumbent. In this case, it will be rejected.
n
n
n
The adoption decision requires some type of comparison of the
productivity of the new GPT with that of the incumbent. LC&B consider
three adoption criteria in which the new GPT is adopted: (i) always without
comparison with the incumbent, (ii) if its expected initial level of the
productivity exceeds that of the incumbent and (iii) if its expected initial
level of the productivity is at least as high as the incumbent’s and its
expected rate of growth is higher in the near future.17
For illustrative purposes in this paper, we use the second criteria and
adopt the new GPT if its expected initial level of productivity exceeds that of
16
Both λ and ϑ are derived from beta distributions, where each distribution is defined as
νη
x (ν −1) x (η −1)
with support [0,1], mean (ν/(ν+η)) and variance
.
beta( x | ν ,η ) =
2
Beta(ν ,η )
(ν + η ) (ν + η + 1)
Beta(ν,η) is the Beta function, and ν and η are parameters which take on positive integer values. The value
of ϑtn is set with support [0,1.2] by multiplying all draws by 1.2. The parameters of the Beta distribution
are set such that the mass is centered on one.
17
See LC&B, 452-3. There they define five cases because they divide each of our cases (ii) and (iii) in the
present text into two sub-cases depending on what happens to the rejected GPT.
12
the incumbent. If the GPT fails that test, it is sent back to the pure research
sector for further development. It will again be considered for adoption
when a new technological breakthrough occurs as signalled by another
favourable drawing of λt > λ*.
To apply this adoption test, we need to compare the level of productivity
under the incumbent GPT with that expected of the challenger if it is
adopted. Here we compare what the incumbent’s G value would be at time tn
if it were not replaced at that time (shown in the RHS of equation (9)), with
the G value that would be produced in the first period of diffusion with the
new GPT (as shown in equation (8) and the LHS of (9). This is a simple
matter when we have only one GPT in use at a time and only one applied
R&D sector. However, the comparisons become more complex in Part III
when we have several different types of GPTs and several different vintages
of each type, with the possibility that these different vintages are being used
at the same time in different applied R&D sectors. Because we will need it
in the multi-GPT model, we use an accounting variable, H, to make the
relevant comparison here. H t measures what the actual realised productivity
would be at time tn if the new GPT were adopted.
n
H tn = ξtn ( Gn −1 )(t −1)
(8)
n
(
 eτ +γ ( t −tn ) 
+
ϑtn Ptn − ( Gn −1 )(t −1)
τ + γ ( t − tn ) 
n
1 + e

)
Equation (8) has the same structure as equation (5) with ξt replaced by ξt .
But this time it is evaluated at one specific time, the time when the new GPT
is invented. Note that since we are calculating the diffusion over only the
first period, the gamma terms disappear from the logistic diffusion
expression, leaving only the value of τ to determine how much of the
potential increment associated with the challenger accrues in the first period.
n
n
We then compare this value with the productivity that the incumbent
GPT would have if it were used instead of the challenger at period tn
H tn > (Gn −1 )tn
(9)
If the new GDP is adopted, its logistic diffusion continues as modelled in
equation (5) with ξt replaced by ξt , which is determined by another random
drawing that alters the value of ξt somewhat from that of ξt . This alteration
models the surprise that occurs after the GPT has been adopted, as e.g.,
when many of the first offices to introduce computers into a structure
designed for hard copy procedures found their efficiency actually declining,
only much later to recover and then rise rapidly. So if the test in (9) is
n
n
n
n
13
passed, the new GPT begins its actual logistic evolution with a surprise
drawing of ξt . (With the adoption, tn is reset at the current value of t and the
index n is increased by 1.)
n
If this test is not passed, the GPT is returned to the pure knowledge
sector to be worked on and to await another break through, modelled as λ ≥
λ* at some later date.
Resource Allocation
In this paper, we could use a resource allocation mechanism based either
on maximising behaviour or behaviour that more closely follows the
assumptions of evolutionary models. We describe both here, however,
because one of our goals is to build an evolutionary model of growth driven
by technological change that allows us to move from macro to micro
behaviour, we use evolutionary behaviour in our simulations. In a later
paper, we compare behaviour under this evolutionary modelling with that
which follows from a maximisation assumption.
What really matters for our model is that agents respond to relative intersectoral differences in perceived rates of returns, which they do in both the
maximising and the evolutionary versions. These are just two of the ways in
which we can model agents behaving as groping into an uncertain future in a
profit oriented way.
Maximization: To model maximising behaviour with respect to resource
allocation, we need some further assumptions. First, we need to know how
agents form expectations of the consumption payoff from resources devoted
to pure research. In LC&B and in Carlaw and Lipsey (2006), we assume that
agents estimate the payoff on the assumption that pure knowledge is
immediately useful in the applied R&D sector, rather than being useful only
after a new GPT is discovered. This is equivalent to assuming that they have
a zero time discount. We model this assumption in the last two lines of the
maximization procedure presented below.
Second, we assume, as in LC&B and in Carlaw and Lipsey (2006), that
agents seek to maximise their returns measured in consumption units. We
assume that agents do not know the future consumption payoffs from any
line of expenditure because they do not know the probability distributions
that are generating the disturbances on the outcomes. In the LC&B
formulation, agents form expectations of the payoffs to expenditures based
on their perceptions of the current period’s marginal productivities. Also,
14
since agents cannot anticipate surprises, they make decisions on the
assumption that the expected value of the surprises is zero. Thus they
allocate resources among the three sectors according to their expected
current marginal products measured in consumption units. Under certain
assumptions, this is equivalent to perfect competition.18
We could assume that a social planner makes the entire allocation over
all three sectors. Alternatively, we could assume that the allocation is made
by private price-taking agents in the consumption and applied research
sectors19 and by a government that taxes agents in these two sectors to pay
for pure research, which has the assumed payoff just described.
Recursive substitution of the constraints into the objective function yields
the following reduced form:
α1
β1
σ
σ
β


(10) ct =  µ v ( (1 − µ ) E[ At ]) 1 ( rg ,t ) 2 + (1 − δ )Gt −1  ( ra ,t ) 2 + (1 − ε ) At −1 

 

(r )
α2
c ,t
.
The expectations operator is applied to the stock of applied knowledge in
this equation because there is a problem of simultaneous determination. We
adopt the simplest of assumptions about expectations by setting E[At] = At-1.
The maximization problem is:
max ct = ( µ At )α1 (rc ,t )α 2
{rc ,t ,ra ,t , rg ,t }
s.t.
Rt = rc ,t + ra ,t + rg ,t
(11)
At = at + (1 − ε ) At −1
at = ( Gt ) raβ,t2
β1
gt = ( (1 − µ ) At ) rgβ,2t
β1
Gt = gt + (1 − ε )Gt −1
where the upper bars indicate expectations of the variables.
Resource allocation with mutation and selection: This is the version we use
in this paper. It is designed to bring the model closer to evolutionary
behaviour. We seed the model with an arbitrary allocation of resources and
18
In all treatments by other authors, agents are modelled as having perfect foresight about the future
evolution of new GPTs. Our assumption of no foresight seems closer to what we observe than the
assumption that agents are sufficiently foresighted to maximise over the whole of a GPT’s lifetime—a
lifetime that can easily extend over more than a century.
19
Digit-Stiglitz style monopolistic competition can be introduced into these two sectors with an increase in
complexity and no change in the qualitative behaviour of the model.
15
use a random mutation to change this allocation period by period. The
resource allocation behaviour is completed by a selection mechanism. It
keeps the mutated allocation if it results in an increase in some objective,
that in various experiments is either consumption or knowledge stocks. If the
increase does not occur, the previous period’s allocations are retained.
The model is flexible enough to accommodate many other assumptions
and we have examined some of these elsewhere.
III. THE MULTI-GPT MODEL
Key Characteristics of the New Model
We now move to a model with multiple GPTs. We maintain the same
three sectors: pure knowledge, applied knowledge and consumption goods,
but now allow for many distinct industries within each sector. For example,
each pure knowledge industry produces knowledge relevant to one generic
type of GPT.
There are several generic types of GPTs, such as materials, information
and communication technologies (ICTs), power delivery systems, modes of
transportation, and tools.20 Within each generic type, there are several
different kinds. We distinguish these by the sequence of their invention,
which we refer to as vintages, such as the steam engine, the internal
combustion engine, and the electric motor, within the power delivery
category.
In our model, GPTs are invented by each of the pure knowledge
industries at widely separated and randomly determined times. Also many
surprises occur in such things as the payoff from pure research, the initial
adaptation of new GPTs, the length of life and eventual productivity of new
GPTs, how fast they eliminate competitors, and how well they cooperate
with other incumbent GPTs. These mirror the observations that with
radically new technologies, there is much that cannot be known in advance
and much learning by doing and using must occur as agents grope into an
uncertain future in a profit-oriented but not profit-maximising manner.21
When a new vintage of some generic type of GPTs is developed, it may
drive out previous vintages quickly or only spread slowly from industry to
industry while one or more existing GPTs of earlier vintages co-exist for a
20
See LC&B (131-6) for a discussion of various types of GPTs.
There is a mass of evidence supporting these statements. For a summary and evaluation see LC&B,
especially Chapter 2
21
16
long time. This mirrors the various different performances of GPTs that have
been observed throughout history.22
New GPTs feed their knowledge into applied R&D industries, These use
the knowledge to produce new applied knowledge that increases the
productivity of the consumption goods industries and the pure knowledge
industries. This models the mutual causation loops that are observed to exist
among applied and pure research.23
Unlike models that contain a single GPT, the macro behaviour of the
model is not governed by the behaviour of one dominant GPT and the arrival
of new GPTs can rarely if ever be inferred from observing macro behaviour.
This suggests that one should be cautious in accepting the arguments of
some economic historians that the study of 19th century English macro
behaviour can be used to determine whether or not GPTs were responsible
for the observed growth.
Some Assumptions and Definitions
There are J industries, each producing a different consumption good,
j∈(1,J), and making use of some or all of the stocks of applied knowledge.
There are K applied R&D industries, each producing a distinct type of
applied knowledge, k∈(1,K), and making use of all of the existing types of
GPTs.24 There are X industries that produce pure knowledge and make use
of some applied knowledge. Each of these occasionally invents a new GPT
of type x∈( 1, X). One of each type of GPT is initially seeded in each applied
industry and new GPTs are invented periodically. They are adopted for use
in the various applied industries at various times. There will thus be a
sequence of GPTs of each type (vintage) and different vintages of any one
generic type may be employed in different applied R&D industries at any
one time.
These characteristics require us to keep separate track of the vintages and
the sequence of adoptions of each type of GPT. We use the index nx to
identify the vintages of GPTs of any one type. Every time a GPT of type x is
invented and adopted in at least one industry, nx increments by one. If the
22
For evidence on the varied impacts of new GPTs on existing technologies, including other GPTs, see
LC&B, Chapters 5 and 6.
23
For an excellent study of such feedback loops see Rosenberg (1982) Chapters 3 and 7.
24
It is a trivial matter to alter the model to make some of the applied R&D industries not use some of the
various types of GPTs, although typically in practice all GPTs are widely used.
17
GPT is rejected by all industries, it is returned to its pure research industry
for further development and nx does not change. At any one time, we refer to
the Gs that have been produced by the latest and the previously invented
GPTs of type x as Gnx and G(xn −1) .
x
x
In this model we must also keep track of the adoption of specific GPTs in
specific industries since a new GPT does not have to be adopted at once by
all industries. We use the index d k , z to count the sequence of adoptions of
GPT type z (= x) by applied R&D industry k. Note that x ∈(1,…X) and
z∈(1,…X) are two different designations for the same set of pure knowledge
producing industries and the GPTs that they produce.25 Later we need to
show the sequence of adoptions of a GPT of any type in industry k, which
we indicate by dk .
We also need to keep track of the date of each GPT’s invention (and
adoption in at least one applied R&D industry). We use tn and
(t − 1) n identify invention date of the latest vintage of the type-x GPT and one
period before that, while t( n −1) is the invention date of the GPT one vintage
earlier than the latest vintage. The time stamps are reset every time a new
vintage of type-x GPT is invented (and adopted by at least one applied R&D
industry).
x
x
x
The Resource Constraint
Resources must now be allocated among J consumption, K applied
research and X pure research industries. So the single-GPT model’s resource
constraint in (1) above now becomes:
(12)
J
K
X
j =1
k =1
x =1
Rt = ∑ rt j + ∑ rt k + ∑ rt x
The Consumption Industries
25
We have used x to identify each type of GPT that is discovered by its corresponding pure knowledge
industry. However, different GPTs of type-x may be operating simultaneously in the K different applied
R&D industries. We distinguish these by their different adoption dates and for this purpose we use z to
identify the type of each GPT. The need for these two separate identifications of the same GPT will be
clarified later, when we consider the interaction among GPT operating in different industries within the
same sector.
18
The J consumption industries are defined in equation (13), which
replaces the single-GPT model’s equation (2).
1
(13)
 K
K
ctj = ∏ ( µ Atk )α k  (rt j )α K +1 , α k ∈ (0,1] ∀k ∈ K , α K +1 ∈ (0,1)
 k =1

According to (13), the productivity coefficient in each of the j consumption
industries at time t is the geometric mean of the amount of each of the k
stocks of applied knowledge that is useful in consumption production (each
raised to the appropriate power αk) and the amount of resources devoted to
producing in that industry (raised to the power αK+1).
In the version of the model used in this paper, we make applied research
industry-specific. Thus we have the same number of applied research and
consumption industries, J = K, and we make the knowledge produced in
each of the applied industries, j, useful only in the one corresponding
consumption industry, k. This simplifies the production functions for
consumption goods industries as follows:
(13)’
ctj = ( µ Atk )α k (rt j )α K +1 , α k ∈ (0,1] ∀k ∈ K , α K +1 ∈ (0,1) and j = k
Note that in both (13) and (13)’ the proportion of applied knowledge that
is useful in consumption and in pure research, µ, is assumed to be the same
for all types of applied knowledge.26
The Applied Research Industries
Each applied R&D industry utilizes some vintage of each of the X GPTs.
Their production functions now have X + 1 arguments, X for all the types of
GPT knowledge, and one for the resource input (the time subscript is needed
on the Gs because as time passes, the efficiency of each GPT evolves in a
manner analogous to equations (7), (8) and (9) for the single-GPT model).
Equation (3) for the single-GPT model becomes:
1
(14)
 X
X
atk = ∏ (ν dkk,,xz (Gdxk ,z )t ) β x  (rt k ) β X +1
 x =1

Atk = atk + (1 − ε ) Atk−1
β x ∈ (0,1) ∀ x ∈ X , β X +1 ∈ (0,1)
26
It is a simple matter to make these proportions differ across industries (µj) if there was reason to do so.
19
where the industry specific parameters, ν dk , x , will be used later to allow for
surprises about the productivity of specific new GPTs and for
complementarities among new and existing GPTs, but for now are all set at
unity.27 The first line of (14) tells us is that the output at time t of the kth
R&D industry (which produces R&D useful in the jth consumption industry)
is determined by the amount of resources devoted to that industry raised to
the appropriate power βX+1, and a productivity coefficient, which is the
geometric mean of the current knowledge provided by the X GPTs of the
particular vintages used in that industry, each raised to its own coefficient βx
(and to be modified by its corresponding ν term when these are no longer all
set at unity). The second line tells us that the productivity parameter of the
kth goods production industry, Atk , is the past accumulated stock of the
applied knowledge that is relevant to that industry (suitably depreciated)
plus the current production of that knowledge.
k ,z
The Pure Knowledge Industries
There are X industries each producing a different type of pure knowledge.
The productivity coefficient in each is the geometric mean of the various
amounts of the K different kinds of applied knowledge that are useful in
further pure research (one for each applied R&D industry and each raised to
its power σk). The output of pure knowledge in industry x is a function of
this productivity coefficient and the amount of resources devoted to that
industry. Equation (6) for the single-GPT model is now amended as follows:
(15)
 K
gtx = ∏ (1 − µ ) Atk
 k =1
(
1
)
σk
K x x
 θt rt

(
)
σ K +1
,
σ k ∈ (0,1], ∀ k ∈ K and σ K +1 ∈ (0,1) .
As in the single-GPT model, θtx , which models the uncertainty about the
actual results of any given amount of resources devoted to pure research, is a
random variable distributed uniformly with support [0.8, 1.2], mean 1, and
variance (0.4)2/12.)
The stock of potentially useful knowledge in each of the X lines of pure
research is accumulated according to:
27
Each ν multiplies a specific vintage of general purpose knowledge identified by superscript x, which is
operating in a given applied sector identified by superscript k. The subscript on each ν identifies the
adoption sequence of the latest type z GPT to be adopted in applied R&D sector k. In the later section,
entitled “relations among GPTs”, the meaning and application of these νs are explained in some detail.
20
Pt x = gtx + (1 − δ ) Pt −x1
(16)
which is equation (7) of the single-GPT model amended to allow for the X
separate pure knowledge industries.
New GPTs are invented in each of the X lines of pure research in the
same way as in the single-GPT model, when the drawing of the λ for that
industry equals or exceeds the critical value of λ, i.e., λtx ≥ λ * x . For
simplicity, we let the critical value of lambda for each of the X industries in
the pure knowledge sector be the same: λ * x = λ * ∀ x ∈ X .
Now let a new GPT of type x be invented at time tn because λtx ≥ λ * . The
index tn , which indicates the time of the latest invention of a type-x GPT
(with at least one adoption), is reset to equal the current t. The index n,
which indicates the number of inventions of type-x GPTs (with at least one
adoption of each) is augmented by one.
x
x
The stock of potentially useful pure knowledge that goes into the new
GPT, and that is available to be transformed into actually useful pure
knowledge, is ϑt Pt x where, as in Part II, ϑt models the observation that the
nx
nx
nx
applied potential of a GPT cannot be precisely predicted when it is originally
being developed. (The sets of λ’s and ϑ’s are drawn from beta distributions
as in the single-GPT model.)
We now need to determine if and when the new GPT will be adopted in
each of the applied R&D industries. To do so, we alter the accounting
variable for the single-GPT model given in equation (8) to allow for multiple
GPTs. This new equation shows the productivity that a newly adopted GPT
is expected to have if it is adopted at time t in some particular sector.
(
(17)
H nxx
) (
t
= G(xn −1) x
)
( t −1)nx
 eτ +γ ( t −tnx )  x x
+
ϑ P − (G(xn −1) x )(t −1)nx ,
 1 + eτ +γ (t −tnx )  tnx tnx


(
)
The first term on the RHS is the Gx associated with the incumbent, now
numbered (n-1)x, evaluated one period before the new GPT is
invented, (t − 1)n . The last term is the gap between the Gx just referred to and
the full potential of the new GPT. In the initial set of comparisons, the new
GPT has just been invented so that tn in the logistic diffusion term is set
equal to t.
x
x
Each of the k applied research industries considers whether or not to
adopt the new GPT. In each, the productivity of that applied research
21
industry using the original set of GPTs is compared with the expected
productivity if the new GPT of type-x were to replace the existing GPT.
However, this test is more complex than in the single-GPT model because,
as shown in equation (14), the productivity of each of the X GPTs operating
in each of the K applied R&D industries is affected by a set of νs. Up to now
we have held theνs constant at unity. We now allow the νs in any one
applied R&D industry to change whenever a new GPT is adopted in that
industry, either increasing or decreasing in a manner specified below.
The new GPT of vintage xn comes with a vector of new expected values
ν
that influence its own productivity in each of the K applied R&D
industries. It may also cause theνs associated with the other GPTs in
industry j to alter when it is adopted in that industry. If it does so, this
indicates a change in the complementarity relations between the type-x GPT
and the other types of GPTs with which it cooperates in industry j. We say
much more about the determination and interpretation of the νs in the
following section.
k , xn
When considering adopting a new GPT of type x in applied R&D
industry k, agents must form expectations on theνs that will be attached to
each of the GPTs operating in that industry if the new GPT is adopted. We
assume that they expect the ν attached to GPT of type x to change if they
adopt the new vintage of that GPT. They do not, however, expect those νs
attached to the other (i.e., non-x-type) GPTs used in industry j to change.
Thus agents are assuming that the direct productivity of the challenging GPT
operating on it own will change from that of the incumbent, but that the
challenger will cooperate with the other GPTs operating in that industry
precisely as well as did the incumbent. (We will see in the next section that
this assumption is a source of surprises for agents when they employ a new
GPT.)
Since in each R&D industry the only ν that agents expect to change is the
one associated with the challenging x-type GPT, we can compare the
productivities for any of the k industries by comparing the νGx that would be
produced if the incumbent were left in place with the ν G x that is expected to
be produced if the challenger were adopted.28 This comparison is made in
28
If the νs attached to the other GPTs operating in industry k were also expected to change, we would have
k
to calculate the complete value of the at term from (12) as it would be under the existing GPTs and set of
νs and then compare it with what that a term would be if the challenger replaced the incumbent in that
industry with the new set of ν s.
22
each of the K applied sectors at time t = tn so the test, stated generally for all
industries, is:
x
(
( )
)
ν (kd,+x1) H nx  ≥ ν dk , x Gdx  for each k∈(1,K)
k ,z
x t 
k ,z t

 k ,z

(18)
where ν (kd,+x1) is the expected ν for the GPT of type x (=z) used in industry k.
k ,z
Because the index on n was incremented when the λ test was passed, but no
adoption has yet taken place (so the d index was not incremented), we must
identify the expected ν with the (d+1)th adoption in applied R&D industry
k. ( H nx )t is as defined in equation (17). ν dk , x is the actual value of ν that was
established at the time of the dth adoption of a vintage of type-x GPT.
(Gdx )t is the incumbent GPT of type x (adoption number d) in industry k
evaluated at time t. If the test is passed, the new GPT is adopted in industry k
and the dk index for that industry increments by one.
k ,z
x
k ,z
If none of the k applied industries adopts the GPT, it is returned to its
pure knowledge industry. The indexes dk and n are returned to their previous
values (i.e., incremented downwards by one) and td is returned to its
k ,z
previous valueit is as if the favourable drawing of λt >λ* had never
occurred. Pure research then continues on the GPT and the improved GPT
will not again be considered for adoption until there is another major
breakthrough (signalled by a λt > λ*).
If at least one industry, k, adopts the new GPT, the diffusion process for
that GPT starts. Also a new set of νs is established for all of the GPTs
operating in that industry. We explain how these are determined in the next
section.
Equation (5) for the diffusion of a single GPT is now amended to allow
for multiple GPTs. In it, we remove the random term, ξ since we are now
able to decompose this surprise term on the lower asymptote of the logistic
diffusion of each GPT into a more complex set of surprises by use of the ν
terms as described in the next section.
(19)
(G ) = (G
x
nx
t
x
( n −1) x
)
( t −1)nx
 eτ +γ (t −tnx )  x x
+
ϑ P − (G(xn −1) x )(t −1)nx ,
 1 + eτ +γ (t −tnx )  tnx tnx


(
)
23
where ( G( n −1)
x
)
t
is evolving according to its own logistic diffusion process
with a starting date of tn .29 Notice that the RHSs of (17) and (19) are
identical. Equation (19) gives the evolving Gxs in the industries that have
actually adopted the latest vintage of that type of GPT. Equation (17) tracks
the same evolution in the productivity variable to be available for
comparisons in the future when industries have than not already adopted the
latest vintage wish to compare their productivities with the evolving
productivity of any later vintage of that type of GPT.
x
To illustrate the adoption process, suppose that applied industry k = 1
adopts GPT nx, but the others do not. The evolution of GPT knowledge
useful in the applied R&D industry 1 then follows the path outlined in
equation (19). The evolution in those industries that are using the previous
vintage (n-1) of the type-x GPT follows that equation with the subscripts
altered as follows:
(19)’ ( G
x
( n −1) x
) = (G
t
x
( n − 2) x
)
( t − 2)( n−1)
 eτ +γ ( t −t( n−1)x )  x
P x − (G(xn − 2) x )(t − 2)( n−1) ,
+
ϑ
 1 + eτ +γ (t −t( n−1) x )  t( n−1) x t( n−1)x
x


(
x
)
Similar alterations to (19) will show the evolution of the Gs in those
industries that are using the vintage (n-2) of the type-x GPT, and so on.
For those industries that are not using the latest vintage of type-x GPT,
the comparative test of equation (18) is applied each period taking into
account the improvements in efficiency of both the incumbent GPTs and that
of the challenger due to the logistic diffusion process (through the evolution
of the H term in (17)). This comparison is made with each newer vintage of
the GPT than the one currently being used in the industry in question. If any
of these test are passed, the appropriate newer vintage is adopted. The
subsequent flow of useful GPT knowledge is then switched to the evolving
flow of the newly adopted GPT. The production function of the applied
R&D industry in question then becomes equation (14) whose d replaces
whatever earlier d adoption vintage had been in use.
Pure researchers in industry x continue to amass knowledge relevant to
inventing another new GPT, which will come along at some future date
according to the random drawings of λx. It is quite possible that a new GPT
of type x will be invented before the previously invented GPT has replaced
its predecessor in all industries, just as, for example, the jet engine replaced
29
This is a general formulation that will allow for many vintages of GPTs of one type to be in use in the
use in the various industries: G( n − 2) x , G( n −3) x etc.
24
the turbo prop long before the turbo prop had replaced the piston engine in
all types of aircraft. Thus there may be several vintages of type-x GPTs
operating in the various applied R&D industries. The need to keep track of
these possibilities is one of the reasons that our notation is so cumbersome.
The Relation Among GPTs
Substitutability between an incumbent GPT and its challenger: One of
each type of GPT is always used in each applied R&D industry. But, a later
vintage of type x may be used in some industries while one or more earlier
vintages of that same type are still used in other industriesmirroring what
we see in reality when a new GPT spreads slowly through the economy,
displacing the incumbent GPT industry by industry. Thus, all GPTs of any
one type are (more or less good) substitutes for each other. How good a
substitute for the incumbent the challenger is expected to be in each
industry, k, depends on how the challenger’s expected productivity
(modified by its associated expected ν coefficient) compares with the
incumbent’s productivity (modified by its associated actual ν coefficient).
How good it actually turns out to be initially depends on the realised value
of its own ν.
The various vintages of any one type of GPT compete with each other in
the sense that in each period the industries using the earlier vintages
compare their evolving efficiencies with their expected efficiency if they
adopt a later vintage. A switch to the later vintage is made whenever it raises
expected productivity.
One major source of surprise in our model is when the realisedν, deviates
from the value it was expected to take, ν , when the adoption decision was
made. This is done using a random deviation from the initial expected value.
This difference models the surprises when a GPT turns out in practice to be
either somewhat less efficient than it was expected to beas, for example,
when the first generation of computers introduced into offices organized for
hard copy information were initially less productive than expected  or
more efficient than expected  as, for example, when the laser turned out to
be far more useful than its original developers thought it would be.
Complementarity among different types of GPTs: The second source of
surprises models the complementarity among the different types of the
GPTs. In any given industry, these GPTs cooperate with each other to a
greater or lesser degree. In practice, no one can be quite sure just how
25
complementary with the other GPTs used in that industry a new GPT
introduced into industry k will turn out to be. Whether it is better or worse
than the incumbent at cooperating with them is typically something that can
only be discovered in practicea case of learning by using.
According to the test given by inequality (18), the adoption decision on
the challenging GPT of type x in the kth applied R&D industry is made in the
expectation that the ν coefficients for all of the non-x type GPTs currently
being used in that industry will remain unchanged. However, if the GPT is
adopted, these νs are altered. These changes model the alterations in the
degree of complementarity between the new type-x GPT and all the other
non-x type GPTs with which it is now cooperating.
An example: Perhaps an example will help to illustrate these points. Let
there be three types of GPT (developed by three pure research industries)
and three types of applied R&D industries (producing knowledge useful in
three types of consumption industries). Thus we need nine individual νs, one
for each GPT that is used in each applied R&D industry. Initial values for
these are shown in Table 1. (We use superscripts to identify each element in
the matrix, rather than the more usual subscripts in order to conform to the
notation in the earlier part of the paper.)
TABLE 1
DISTRIBUTION OF νs FOR A MODEL WITH THREE-GPTS,
AND THREE APPLIED R&D INDUSTRIES
1
K
2
3
1 ν
ν
ν 3,1
X
2 ν 1,2 ν 2,2 ν 3,2
3 ν 1,3 ν 2,3 ν 3,3
1,1
2,1
The initial seed values of the elements in this matrix can each be set at
any desired values to model any desired set of complementarities among the
different GPTs operating in each industry. For illustrative purposes, we set
them all at unity.
Now let a new type-2 GPT be invented. When agents are considering
adopting the challenger in applied R&D industry 1, they need to form
26
expectations on the three ν values in the first column vector that contains the
νs for that industry. We assume they expect ν1,1 and ν1,3 to be unchanged by
the introduction of the new GPT, but that ν1,2 will now take on the value
ν 1,2 . Extrapolating this across all of the applied industries, we see that the
new GPT brings with it a row vector of expected new values of the νs that
will be associated with its operation in each of the three applied R&D
industries if it is adopted there, ν 1,2 , ν 2,2 and ν 3,2 , and the expectations of all
otherνs in the matrix are set to their previous period’s actual value.
If the test shown in inequality (18) is passed, the type-2 challenging GPT
is adopted in applied R&D industry 1. As soon as that happens, all three of
the νs in the first column change. ν1,2 alters from ν 1,2 . This is the first source
of surprise discussed above. ν1,1 and ν1.3 alter from the values shown in
Table 2, which were their actual values when cooperating with the
incumbent type-2 GPT and their expected values when cooperating with the
challenging type-2 GPT. This is the second source of surprises discussed
above.
It follows that at any one time each column vector of existingνs was
determined when the most recent entry of a GPT occurred in that industry.
We call that GPT the ‘source GPT’ for the νs in that industry.
Similar tests are made for applied R&D industries 2 and 3 and the
challenger is accepted or rejected in each. Where it is accepted, the
appropriate column of νs is altered; where it is rejected the column of νs
stays as it was in Table 1.
Next we need to explain how the several distinct sets of random
variations of the νs are determined. First, we need to determine the row
vector (x = 2) of expected coefficients for the challenging GPT if it were to
be adopted in each of the three applied R&D sectors. These expected ν s are
calculated as the actual νs for the incumbent GPT of type x modified by the
addition of a random variable π. For our general modelling purposes these
random variables can be drawn from a number of different probability
distributions and for illustrative purposes we chose a uniform distribution
with support [-0.05, 0.05]. For the example discussed above the expected
ν s are:
27
ν k ,2
 ν k ,2 + π if (ν k ,2 + π ) ∈ [0.5,1.5]

ν k ,2 + π < 0.5
if
= 0.5
1.5
ν k ,2 + π > 1.5
if

k∈(1, 3)
If the challenger is adopted in applied R&D industry, k = 1, two types of
changes occur. In both cases, the realisation of the νs that multiply the
GPTs used in that industry differ from the expected values. First, the
challenger’s efficiency differs from its own expected value on which the
adoption decision was made as follows:
ν 1,2
 ν 1,2 + π if (ν 1,2 + π ) ∈ [0.5,1.5]

= 0.5
ν 1,2 + π < 0.5
if
1.5
ν 1,2 + π > 1.5
if

Second, the νs associated with each of the other GPTs used in the given
applied industry also change. This is done by adding a random term π
drawn, again for illustrative purposes only, from a uniform distribution with
support [-0.05. 0.05]. In this example in which a type-2 GPT is adopted in
applied R&D industry 1, the new coefficients on GPTs types x = 1 and x = 3
in industry 1 are:
ν 1, x
ν 1, x + π x if (ν 1, x + π x ) ∈ [0.5,1.5]

ν 1, x + π x < 0.5 ,
= 0.5
if
1.5
ν 1, x + π x > 1.5
if

for x ≠ 2
An increase in one of these νs indicates that the new type-2 GPT
cooperates with one of the non-type-2 incumbents better than did the
replaced type-2 GPT. A decrease indicates the opposite.
It remains to note that when programming our simulations we need a full
identification of the νs in Table 1. Every time a new GPT is introduced in
any one industry, that GPT is a source of changes in all of the νs in that
industry’s column. It follows that the matrix of νs that is operative in any
one period can have up the lesser of K or X different GPT sources. However
the source of the νs in a particular column need not be the latest vintage of
that GPT. A newer vintage of that type might have been invented and
adopted elsewhere but rejected by the industry in question and no other
types of GPTs adopted there in the meantime. Also after several new
vintages have been invented and rejected in sector k, a vintage newer than
28
the incumbent but older than the latest vintage could be adapted. These
possibilities require that we keep track, among other things, of the vintage of
the GPT that is the source of the νs in each column.
So the full identification of each ν that is needed for programming our
computer simulations requires that we identify (i) the applied R&D industry
that is using it, k; (ii) the GPT to which it applies x; (iii) the latest adoption
identification number for that type of GPT, d, and the GPT that is its
“source”, z. Thus, we have a five dimensional array, any element of which
specifies the complete “address” of the given ν as ν dk , x .
k ,z
It is now apparent why we need two different symbols for each type of
GPT. The x indicates the particular G that is being modified by the ν in
question. The z indicates the GPT that is the source of that particular ν, i.e.,
the GPT whose entry caused the industry’s νs to change to their current
values. Only in the one cell where the GPT being modified by the ν is also
source GPT will x=z.
Having defined the νs as a set of random variables for the purposes of
laying out the theory, it is important to note they can also serve as a point of
contact with empirical research. Instead of allowing the νs to be determined
randomly, they could be determined from data on the GPT under
examination. Indeed, we have already done this in a paper applying our
model to Australian data. This capability to calibrate the model by choosing
the νs to approximate from real data suggests a number of policy
experiments that we will conduct in later papers.
Multi-GPT Resource Allocation
The allocation problem for the multi-GPT model now requires the
allocation of resources to J lines of final consumption, K lines of applied
R&D, and X lines of pure research.
Maximization: As in the single GPT model, it is possible allocate resources
by maximizing consumption period by period using current marginal
products (as did LC&B) and Carlaw and Lipsey (2006)). The maximization
problem is altered slightly from the model presented in the single GPT
model because we need to relate the multiple industries in the consumption
sector to each other. We make the simple assumption that the system can be
represented by an additively separable, social (or representative) utility
function defined over the J lines of consumption output. This function
relates the consumption outputs of each industry to each other through their
29
marginal utilities. We assume that the objective function is an additive utility
function defined over the J lines of consumption output.
J
, rt , rt
t
}
j =1
ϕj
( )
max
U (c ) = ∑ ct
j k x
{r
J
t
j
s.t.
(20)
J
K
X
R = ∑ rt j + ∑ rt k + ∑ rt x
j =1
k =1
k αk
t
x =1
j α K +1
ct = ( µ A ) (rt )
j
X
a = ∏ (ν dkk,,xz (Gnxx )t ) β x ](rt k ) β X +1
k
t
x =1
A = atk + (1 − ε ) Atk−1
k
t
Gtx = gtx + (1 − δ )Gtx−1
K
(
g = ∏ (1 − µ ) Atk
x
t
k =1
) (θ r )
σk
x x
t t
σ K +1
Resource allocation with mutation and selection: As in the single GPT
model, we chose an allocation method that is closer to an evolutionary
model of technology-driven economic growth. We model resource allocation
by seeding J + K + X industries with an arbitrary allocation of resources.30
We then use random mutations to change this allocation period by period.
For each type of resource, in each period, we calculate its portion of the total
resources allocated to them as follows:
(21)
qti =
pti rti−1
J +K + X
∑
i =1
∀i ∈ (1, J + K + X ) ,
i i
t t −1
pr
where pti is a random variable drawn from a Beta distribution, calibrated to
have a very small variance around mean 0.5.
We combine this with the selection mechanisms conditioned on either
consumption or improvements in the stock of knowledge. The mechanism
keeps the set of mutations if they result in an increase in whichever of the
objectives is chosen. If the given selection criteria is not met, the previous
period’s values for the resource allocation are retained and mutated once
again in the next iteration.
30
In this case, each industry is arbitrarily allocated an equi-proportionate share of the total resources.
30
Simulation
To simulate the model, we need to put values on the parameters. The
specific numbers shown in Table 2 are for the single GPT model and those
shown in Table 3 are for the multi-GPT model. The values were chosen for
various reasons. The first was to meet the twin criteria of ensuring
diminishing returns to resources and achieving an average annual growth
rate of approximately 2%. GPTs arrive on average every 35 years, but with a
large variance. The second was that several different parameterisations were
tested to check the robustness of the qualitative results and these were found
to be robust to wide ranges of values satisfying our basic assumptions. The
third was that nothing in our subsequent analysis turns on these particular
values. The fourth was that we chose some parameter values to ensure that
knowledge has constant returns. The fifth was that we utilized symmetry
across sectors and across industries within sectors to simplify the
illustrations we wished to make with the model.
TABLE 2
NUMERICAL SIMULATION OF THE SINGLE-GPT MODEL
The following are the parameter values used to simulate the results of the
single GPT model as reported in the text and shown in the figures.
α1 = 1
α2 = 0.3
β1 = 1
β2 = 0.3
σ1 = 1
σ2 = 0.3
v = 0.1
A0 = 1
G0 = 1
R = rc,t + ra,t + rg,t= 1000 ε = 0.01
γ = 0.06
τ = -6
δ = 0.01
µ = 0.5
For λ we choose ν = 5 and η = 10. For ϑ we choose ν = 10 and η = 5. For ξ
we choose ν = 12 and η = 5 and multiply all values of xt by 1.2 so that the
support for the distribution that determines ξ is [0,1.2].
31
TABLE 3
NUMERICAL SIMULATION OF THE MULTI-GPT MODEL
The following are the parameter values used to simulate the results of the
multi-GPT model as reported in the text and shown in the figures.
α j = 1 ∀j ∈ (1, J )
αJ+1 = 0.3
β k = 1 ∀k ∈ (1, K )
βK+1 = 0.3
σ x = 1 ∀x ∈ (1, X )
A0k = 1 ∀k ∈ (1, K )
σX+1 = 0.3
v=1
G0x = 1 ∀x ∈ (1, X ) G0 = 1
ε = 0.01
δ = 0.01
γ = 0.07
τ = -6
µ = 0.5
ϕ j = 1 ∀j ∈ (1, J )
J
K
X
j =1
k =1
x =1
R = ∑ rt j + ∑ rt k + ∑ rt x = 1000
For the λs we choose ν = 5 and η = 10. For the ϑs we choose ν = 10 and η =
5. For ξ we choose ν = 12 and η = 5 and multiply all values of xt by 1.2 so
that the support for the distribution that determines ξ is [0,1.2]. For pi we
choose n = ν = 100. The calibration of the νs is discussed in the section “The
Relation Among GPTs”.
Some Illustrative Simulations
The model has real potential, both to simulate interesting theoretical
possibilities and to track real data by using empirical evidence to select the
many parameters that we have chosen arbitrarily for purposes of illustration.
Here we present two sets of crude simulation results just to illustrate what
can be done with the model. Like most discrete time models, abrupt changes
in the slopes of continuous series for levels cause occasional discontinuous
jumps in growth rates. These do not mirror anything found in reality. In
fully developed models these changes need to be smoothed in the way that
we have smoothed the transition from one GPT to another through our
logistic diffusion that slowly transfers a GPTs potential to raise productivity
into actual increases. This is a modelling exercise that we have not yet
completed, so for the moment, we accept these discrete blips in the growth
rate as artefacts of the discrete-time model and concentrate on the rest of the
time periods.
32
Single GPTs and Macro Behaviour: To demonstrate that the evolution
of a single GPT does not determine the evolution of the entire economy, we
present the growth rate of aggregate consumption output (i.e., the growth
rate of consumption output aggregated across all J consumption industries)
and compare this to the growth rates of pure knowledge in each of the K
applied R&D industries. The simulation results are taken from a model in
which J=3, K=3 and X=3. We look at the GPTs in each applied R&D
industry, rather than in some aggregated way, because not all industries are
using the same vintage of the three types of GPTs at each moment in time
and it is only when these GPTs are adopted in a given applied R&D industry
that they become productively useful.
Figures 1-3 show the growth rate of aggregate consumption output and
the growth rates of the pure knowledge stocks used in each of the three
applied R&D industries k = 1,2,3. In all three figures, the spikes in growth
rates indicate the adoption of a new vintage of the indicated type of GPT. It
is obvious from a comparison of the these figures that new vintages are not
typically adopted at the same time in all industries. For example, GPT type 1
is adopted in industry 1 around period 33, in industry 3 around period 42,
and in industry 2 not until about period 49.
It is obvious from the figures that the growth of aggregate consumption is
influenced in all cases by the discreet jump caused by the arrival of the GPT.
But two other things are also obvious. First, the overall aggregate
consumption growth rate does not track any one of the growth rates of the
three types of pure knowledge stock. Second, leaving the artificial spikes
aside, there is no consistent behaviour in the aggregate consumption data
several periods before and after the entry of a new vintage of any of the
GPTs that would allow one to identify that entry.
This simulation in the multi GPT, multi-applied industry and multiconsumption industry model shows that the arrival and adoption of any
given GPT does not determine the aggregate evolution of growth or output.
So analysis of aggregate variables will not necessarily indicate the arrival
dates nor the impact of important individual GPTs. This is only the
beginning of this type of analysis, but it does suggest caution  at least until
further study is completed  in assessing the attempts by economic
historians to judge the existence and importance of GPTs from a
consideration of aggregate data for GDP or TFP.
33
Figure 1.
Growth rates for aggregate consumption and pure
knowledge stocks used in applied R&D k = 1
2.00E-01
1.50E-01
Aggregate Consumption
GPT Type1
1.00E-01
GPT Type 2
GPT Type 3
5.00E-02
0.00E+00
1 5 9 13 17 21 25 29 33 37 41 45 49
Time
Figure 2.
Growth rates fpr aggregate consumption and pure
knowledge stocks used in applied R&D industry k = 2
2.00E-01
1.50E-01
Aggregate consumption
GPT Type 1
GPT Type 2
GPT Type 3
1.00E-01
5.00E-02
0.00E+00
1 5 9 13 17 21 25 29 33 37 41 45 49
Time
34
Figure 3.
Growth rates for aggregate consumption and pure
knowledge stocks used in applied industry k = 3
2.00E-01
1.80E-01
1.60E-01
1.40E-01
1.20E-01
1.00E-01
8.00E-02
6.00E-02
4.00E-02
2.00E-02
0.00E+00
Aggregate Consumption
GPT Type 1
GPT Type 2
GPT Type 3
1 5 9 13 17 21 25 29 33 37 41 45 49
Different Selection Criteria: Another illustrative simulation shows that
different selection criteria matter for the overall growth performance of the
system. We define two selection criteria for modelling resource allocation
across the various sectors and their industries. The first adopts new
allocations for resources generated by mutating the previous period’s
allocation of resources if a five-period moving average of the aggregate
applied knowledge stock is larger than the same five-period moving average
lagged by one period. The second selection criteria adopts the new allocation
of resources if a five-period moving average of aggregate consumption is
higher than the same five-period moving average lagged by one period. In
order to do comparative dynamics, we use the same initial values and set the
realizations of all random variables to be identical for the two different
simulations runs. The differences in the behaviour of aggregate consumption
is thus due only to differences in the selection criteria.
The following comparisons use the J = 2, K = 2, and X = 2 version of the
multi-GPT, multi-applied and consumption industry model. Figure 4 shows
the aggregate level of consumption output for the two different selection
criteria.
35
Figure 4
Aggregate Consumption
with the indicated selection criteria
4.50E+08
4.00E+08
3.50E+08
3.00E+08
2.50E+08
2.00E+08
1.50E+08
1.00E+08
5.00E+07
0.00E+00
Applied Knowledge
Stock
Consumption Level
1
20 39 58 77 96 115 134 153 172 191
Time
Clearly, the selection criteria that uses a five-period moving average for
aggregate consumption performs better than the selection criteria based on a
five-period moving average of the aggregate applied knowledge stock. Note
that we define “better” as what achieves the higher level of consumption
output because we assume that increases in consumption output are the
desired growth outcomes in the model.
This result is suggestive if we think of the private sector determining the
allocation of resources to the consumption industries and the public sector
determining the allocation to research through direct spending and tax
incentives. If the public sector determines its allocation based on the
behaviour of the applied R&D sector it will do worse that if it uses the
behaviour of total output (GDP) as its guide. Of course, this is a very simple
model and only a crude illustrative application, but does it suggest the kind
of issues that can be investigated with a model as flexible as this one.
36
Appendix 1
With the appropriate time lags in place the Bellman equation for the
single-GPT model is,
V ( At , Gt , t )= max ct + ρ E [V ( At +1 , Gt +1 , t + 1) ] + ρ 2 E [V ( At + 2 , Gt + 2 , t + 2) ]
{rc ,t ,ra ,t ,rg ,t }
s.t.
R = rc ,t + ra ,t + rg ,t
ct = ( µ At −1 )α1 rcα,t2
at = ν ( Gt −1 ) 1 raβ,t2
β
At = at + (1 − ε ) At −1
g t = ( (1 − µ ) At −1 ) 1 rgβ,2t
β
Gt = gt + (1 − ε )Gt −1
where the upper bars indicate expected rather than the actual values of gt and
Gt, And these expectations are adaptive. This is a complicated problem in
two dimensions of state variables. Normally such a system would be solved
to satisfy the maximum principle and the transversality condition (or it
discrete time analogue) which would yield a stationary dynamic equilibrium.
We, however, do not want a stationary equilibrium, nor do we want the
behaviour of the model to act under full information. We want the agents to
be exposed to uncertainty in the form of incomplete information about the
distributions from which our random variables are drawn. As such we
violate the Maximum principle and transversality conditions. We also make
simplifying assumptions about the time lag structure in our model to make
its exposition easier.
37
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