Electronic copy available at: http://ssrn.com/abstract=1588462
Emergent Pareto-Levy Distributed Returns to Research in
an Agent-based Model of Endogenous Technical Change
Michael D. Makowsky*
David M. Levy
November 19, 2010
We build an agent-based model of endogenous technical change in which heterogeneous
investments in patented knowledge generate Pareto-Levy distributed returns to
investment in research from very weak distributional assumptions. Firms produce a
homogenous good and a public stock of knowledge accumulates from the expired patents
of privately produced knowledge. Increasing returns to scale are derivative of
endogenously produced technology, but the market remains competitive due to imperfect
information and costly household search. The interaction of heterogeneous knowledge,
research investment, revenues, and search outcomes across agents endogenously
generates the empirically observed mixture distribution of market returns that are Pareto-
Levy distributed in the upper tails. These distributional characteristics have ramifications
for endogenous growth models given the importance of extreme values and market
leaders in technological advancement. Average growth rates demonstrate early positive
returns to patent length that diminish rapidly but never become negative in the parameter
space tested. The variance of growth rates increases with patent length. The distribution
of growth rates is characterized by “fat tails,” and kurtosis increasing with patent length
when search costs are sufficiently high.
JEL Codes: C63, L11, O33, D83
Keywords: patents; endogenous growth; increasing returns to scale; price
dispersion; search; heterogeneous agents
*Makowsky: Department of Economics, Towson University. Levy: Department of Economics,
George Mason University. We would like to thank Robert Axtell and seminar participants in the
Department of Social Complexity, George Mason University. We thank Omar Al-Ubaydli for a
detailed list of comments. Makowsky thanks the Towson College of Business and Economics for
summer financial support. Please send correspondence to email@example.com.
Electronic copy available at: http://ssrn.com/abstract=1588462
Theories of endogenous technical change built with knowledge serving as a non-rival
input into productivity and, in turn, as a source of increasing returns to scale, have served
to model exponential growth and offer a better understanding of disparate of rates of
growth observed across countries (Grossman and Helpman 1994; Romer 1994). The
capacity to cope with increasing returns to scale, however, motivated the abandonment of
price taking perfect competition, and the allowance of market power within firms (Romer
1990; Grossman and Helpman 1991; Aghion and Howitt 1992). It should not be
surprising that, given this reliance on knowledge inputs and market power, that
intellectual property rights, or patents, have become a major topic of exploration in
theories of endogenous growth (Horowitz and Lai 1996; Futagami and Iwaisako 2003;
O'Donoghue and Zweimüller 2004; Iwaisako and Futagami 2007).1
Models incorporating patents into theories of endogenous growth, however, have
not accounted for the peculiar distributional properties of the returns to innovation. We
offer an alternative modeling strategy that allows for endogenous technical change, is
characterized by long run increasing returns to scale, and emerges a distribution of
revenues across firms that is best characterized as a mixture distribution that is Pareto-
Levy in the upper quartiles and is often dominated by a small number of extreme values.
This peculiar mixture distribution is similar to those observed in patent revenue return
research (Epstein and Wang 1994; Scherer, Harhoff et al. 2000; Silverberga and
Patents bring the necessary market power to firms that seek to obtain monopoly
rents from their excludable private knowledge. This excludable private knowledge,
however, also engenders heterogeneity across firms that are all producing with differing
knowledge inputs. Heterogeneous knowledge quickly leads to heterogeneity in
productive capacity, marginal products of standard (rival) inputs, and prices. Such a
world is considerably less tractable for traditional modeling, and is typically inhospitable
1 To varying degrees, the models proposed in this literature are built using the foundations laid out by
Aghion and Howitt (1992), Grossman and Helpman (1991), and Judd (1985).
2 Mixed distributions with extreme values have also been offered as a tractable representation of Knightian
uncertainty and a challenging environment for policy (Epstein and Wang 1994).
to decentralized competition. The structural imposition of monopolistic competition in
the form of a continuum of goods produced by firms returns us to more tractable territory,
but comes at a cost. With a continuum of goods in demand, and each firm producing a
unique good that cannot be perfectly substituted for by goods produced by competing
firms, our potentially Schumpeterian landscape looks considerably less destructive.
Imperfect substitution, long thought to be necessary to allow many firms to exist in an
industry with increasing returns, attenuates the consequence of discoveries which would
be explosive in a world with perfect substitutes. The monopolistic competition model,
governed by the Law of One Price, retains the representative firm by allowing for
heterogeneous goods. We provide a model using the exact opposite: a set of
heterogeneous firms competing to produce and sell a single homogenous good, each
offering the good to consumers at their own distinct price.
There is considerable evidence that the returns to research are highly skewed,
with distributions dominated by extreme values. Research into these returns has used a
variety of creative datasets, including citation records, initial public stock offerings
(IPOs), and self-reported revenue returns to patents (Harhoff et al. 1998; Harhoff et al.
1999). The most appropriate statistical distribution for the characterization of the returns
seems to be some combination of the lognormal and Pareto-Levy distributions (Scherer,
Harhoff et al. 2000; Silverberga and Verspage 2007). The overall distribution within the
empirical work is best characterized by a lognormal distribution with outliers in the upper
tail. However, the upper tail of the distribution, particularly when looking at IPO data, is
better characterized by the Pareto-Levy power law distribution. Such power law
distributions are not unheard of in market competition and concentration data. Axtell
(2001) finds that the size of firms, in terms of individuals employed, is Zipf distributed in
the United States. Within power law distributions, the upper tail accounts for an
extraordinary share of the distribution’s value. Models that account for growth derivative
of technical innovation that leverage some form of market power stand to benefit from
either including such features of the returns to research or, preferably, generating them
endogenously (Luttmer 2007). Concerns about the importance of the distribution of
research outcomes, in particular of the upper tail and outliers have been recently
expressed (Silverberga and Verspage 2007). As they note such Pareto power law
distributions might not even have first moments, something which has severe
implications for risk analysis.
The aims of this paper are three-fold. First, we seek to build a model characterized
by the long run increasing returns to scale and exponential growth properties of existing
models of endogenous technical change and growth. Second, we abandon the traditional
monopolistic competition model, and replace it with a model of competitive firms
producing a homogenous good in a market characterized by price dispersion. Third, we
simulate the model under a variety of parameterizations and examine the distributional
properties of returns to investment in research. In doing so, we find that the distributions
of returns to research in our model take exhibit Pareto-Levy power law properties,
particular in the upper quartiles. We also test the impact of the key parameters of the
model, patent length and search costs, on the distribution of average growth rates across
large batches of simulation experiments.
2 An Agent-based Computational Model
Agent-based models allow for the explicit construction and active decision
making of unique individual agents whose direct and indirect interactions emerge
macroscopic outcomes (Epstein et al. 1996; Epstein 2006). Within our model we create a
market composed of heterogeneous, individually autonomous households and firms that
make decisions in accordance with their type, unique information set and personal
history, and the rules that govern their behavior. This methodology offers two
overarching advantages. First, it allows for knowledge to be truly dispersed, with each
agent holding a unique subset of the information available in the market. Second, the
deep population heterogeneity3 of the model allows for the exploration of emergent
3 By “deep heterogeneity” we mean that each agent is in principle unique. Economic models will often have
a handful of agent “types,” but thousands of unique agents would not be mathematically tractable.
distributions whose properties we can observe as not just in terms of first and second
moments, but as entire populations of values. This is especially important given our
interest in the upper tails and extreme values.
Our model is built using elements prominent in O’Donoghue and Zweimüller
(2004) and Iwaisako and Futagami (2007). Like the model presented in O’Donoghue and
Zweimüller (2004), our model is composed of two sectors, one in which technology
investment and innovation are possible and one in which innovation is not possible, with
inputs of only labor and capital. Individual, technology enabled, firms produce a
homogenous quality primary good (q) while an aggregated non-technical sector (NTS)
produces a secondary good (x). Households supply labor to both sectors, collect wages,
earn uniform returns to shares of rents paid to capital, and maximize a universal utility
function by purchasing a combination of x and q. The model is always composed of a
fixed quantity of households, and as such growth within the model is not dependent on
the exogenous increase in labor.4
Time in the model occurs in discrete steps and substeps. Sets of agents (organized
by type) are activated in a fixed schedule, but within each set, agents are activated in a
randomized order. While firms are effectively acting simultaneously, households are not.
A household may purchase the last of a firm’s inventory or fill its final hiring slot.
Potential order effects add to the complexity of model outcomes, but constant
randomizing of activation order prevents model artifacts (Axtell 2001).
While agents, within their types, are homogenous in capacity, exogenous
parameterization, and behavioral rules, they each face a world with costly, imperfect, and
heterogeneous information. Households search for both lower prices and higher wages,
seeking to maximize their consumptive bundle, while being constrained by a finite
amount of time to be split between wage earning labor and search, and the ensuing time
expenditures associated with searching the market. Firms, on the other hand, face the
uncertainty of a research process that may or may not yield a competitive increase in
4 For a discussion of scale dependent vs. scale independent models of endogenous growth, see Eicher and
excludable knowledge as well as a marketplace of consumers that may or may not
discover them as a low price provider of goods. They respond to these uncertainties by
making decisions regarding research investment predicated on simple heuristics and
limited information. Given the complexity of the relationships between households and
firms, the non-technology sector (NTS) is governed by a number of simplifying
assumptions that grant the model additional tractability. The NTS operates as a single
agent in the model, hires all who are willing to work for its offered wage, and always
meets the sum of its market orders.
The labor supply in the model is fixed, but capital grows as a set fraction of the
total productivity from the previous time step. Growth, nonetheless, is driven by technical
innovation. As within a Schumpeterian model of creative destruction (Aghion and Howitt
1992), innovation is motivated by desire to both gain monopoly rents and avoid
bankruptcy. In this manner both the carrot and the stick are applied every step of the
model: success in research and development leads to lower production costs, greater
rents, and more customers, whereas failure leads to higher prices, fewer customers, and
brings the firm one step closer to closing its doors. The prospect of permanent failure is
one of the salient features of working with a competitive market for a homogenous good.
In a monopolistic competition model, where goods exist along a continuum, there is no
prospect for complete failure to attract customers. This is where creative destruction is
tamed in models of monopolistic competition. In our model, on the other hand, with
agents searching over a set of producers offering a homogenous good, a firm with inferior
productive technology will be unable to offer a competitive price and will be more likely
to be passed over by potential customers. This market remains competitive,5 as opposed
to collapsing to monopoly, because of price dispersion and costly search, which allows
second-best firms to attract sufficient customers to retain positive profits, or at the very
least manageable losses that can endured in the short run (Levy and Makowsky 2010).
Further, the expiring of patents and the subsequent sharing of previously private
5 In contrast to the bulk of the existing literature, Hellwig and Irme (2001) build a general equilibrium
model of endogenous technical change that includes competitive markets, though their unique equilibrium
is characterized by a low steady-state growth rate.
knowledge allows for turnover in who stands as the technology leader (Grossman and
Helpman 1991). In reality, it is not just profit, but the prospect of losses and bankruptcy
that motivates investment in research and development.
In contrast to traditional general equilibrium models, there is no social planner
maximizing agent utility, nor a Walrasian auctioneer finding market clearing prices. Each
agent, governed by type (firm, household) specific rules, is autonomous. From the
thousands of interacting, decision-making agents emerge aggregate trends in research
investment, technology, growth, wages, profits, and market concentration. Agents are
myopic, backward looking, and absent any sophisticated strategy. They are governed by a
strictly bounded rationality and costly information, but nonetheless manage to prosper in
what are often rapidly growing economies.
The model is composed of two vectors of agents, households
where each household (i) purchases
iq units from the firm,
ij , offering the lowest price
known to her during time step t. All variables that are not exogenously set vary across
time steps. For ease of explication, we will not include t as a subscript except when
previous time steps (t - 1) are relevant.
Firms produce the primary good,
Q , using inputs of labor, Lj, capital, Kj, and
knowledge , Aj, where knowledge is composed of public, G, and private, Rj, knowledge:
Q A K Lj
subject to the costs of production, Cj, including the wages, wj, paid to employees; rent
paid to capital, r, and the investment in research and development, Sj.
C w LrKSj
Profits, π, are a function of Qj sold at price, pj, and Cj. Firms post unique prices in
the market equal to lagged AC, such that
, , 1
j t j t
. Each firm also posts its own
wage in the labor market in the hopes of attracting prospective employees. Firms set their
wages equal to the monetized marginal product of labor from the previous turn (
, , 1
j tj t
Given this wage rate, firms establish a maximum number of employees they are willing
to employ by engaging in standard cost minimization of the production function given6
(,,, , , ,)
jjj j tjj
C K L QA w r
During each step firms engage in research from which knowledge returns are
uncertain, generating a quantity of private knowledge, or patent,
, j t
y that is temporarily
excludable for Φ time steps, and contributes to a summed portfolio of private knowledge
j tj t
. The process of research and development is modeled as an
exponential probability function, dependent on the firm’s investment, Sj , its current
portfolio of private knowledge, Rj,t, and the existing stock of public knowledge, Gt:
j t j t
t j t
6 If a firm fails to sell a single unit of q, but does not go out of business, they use the mean qj among firms
still in business and use that to establish an average cost and marginal product for setting their price and
where Z is a unit rectangular variate. The ratio of investment
j S to private knowledge
created with each patent,
, j t
y , is declining as the existing stock of knowledge,
grows. This choice to model the costs of innovation as increasing with the existing stock
of knowledge is based on the empirical observation that the costs of patents have been
increasing over time (Kortum 1993). Firms choose unique research investments
j S equal
to their investment from the previous turn adjusted by factor χ, where
, , 1
, , 1
, , 1
, , 1
j tj t
j t j t
This research investment adjustment rule entails a simple profit seeking heuristic on
behalf of the firm, with which each individual firm gropes towards an investment
procedure that increases profits. The increment of change,
, is exogenously set
parameter uniform across firms. Firms myopically grope towards greater profits,
switching directions whenever their previous turn resulted in reduced profits.
Each firm’s stock of private knowledge, Rj,t, is a rolling portfolio of patented
knowledge. Each step, the oldest patent, yj,t-Φ, expires. The expired patent of greatest
magnitude is added to the public knowledge stock,
ttt m t
Research results in more efficient production that is rewarded by greater profits and
greater prospects for long run survival in the marketplace. This, in turn, incentivizes the
long run contribution to the public stock of knowledge and ideas in the form of expired
patents which lead to long run growth. At the same time, the rolling expiration of patents
allows for turnover in private knowledge leadership at any given time step.
Once a firm has conducted its research, set its price and wage, and hired its
employees, it can establish a profit maximizing quantity to produce and sell. Capital, K,
in the model is available from exogenous pool at price r(Qt-1, Ψ), where Ψ is a fraction of
total productivity in the model from the previous time step, and r() is the marginal
product of the said fraction in the previous time step. The profit maximizing quantity to
be sold is
q A L L p A r
Agent search occurs within each time step t, in sub-steps τ=1...m where each
increment of τ represents an act of search by the agent.7 Households first search over the
set of wages offered by firms, then search over the set of prices posted for q. Their search
activities are governed by simple income maximizing and cost minimizing search functions
based on a desire to continue searching so long as the expected increase in the highest known
iw , or decreases in the lowest known price,
ip , will result in a net increase in purchasing
capacity given the cost of an additional sub-step of search,
, where ς is the amount of an
agent’s time endowment expended by an act of search. In both wage and price search, the
decision variable is the number of search actions, τ, that constitute the fixed sample size
that households decide prior to the first discovered price. Both wage and price search
result in a fixed sample size. Households assume a non-degenerate exponential distribution of
wages F(w) on [ , ]
w w and maximize the expected total income (highest found wage earned over
the time remaining after search) .
() (1) ( )
E Mcw F w dwi
Households similarly assume a non-degenerate lognormal distribution of prices F(p) on
[ , ]
p p and minimize the expected total cost (cost of purchasing qt-1 plus cost of search) .
,1 , 1
() (1( ))
E Cp dpi
For the sake of simplicity, households assume exponential price and lognormal wage
distributions built around the correct mean and standard deviations in the population of
firms.8 Additionally, the wage paid by the NTS is known to each household without cost.
7 There are m firms, and thus m prices over which to potentially search. If the cost of a unit of search, ∆h,
equaled zero, all agents would continue search until τ equaled m.
8 We tested the model with multiple distributional assumptions and search rules, including uniform
distributions of varying precision and simple “mean value as expected return” heuristics. All search rules
produced qualitatively similar results. The exponential and lognormal distributions were chosen for their
Each household i searches over the wage set Θ, where Θi,τ is the subset of wages
known to household i after τ search efforts.
w w wi
In addition to their wage, each household receives a uniform dividend, d, of the rent
outlaid by firms and the NTS to capital inputs and any positive profits accumulated by
After searching for a wage, each household i then searches over the price set Ω,
where Ωi,τ is the subset of prices known to household i after τ search efforts.
Once households have executed their searches and found a lowest known price and
highest known wage, they maximize a constant elasticity of substitution (CES) utility
For a given wage rate and price, the optimal quantities of q and x are
Where the total income of the household, Mi, is a function of the household’s wage, the number
of sub steps spent searching, and the costs of search, ς, and its dividend from capital rents and
firm profits, d, such that
The non-technical sector (NTS) acts as a single agent. It sets the price for x, η,
price based on the average cost of production from the previous time step
fidelity to observed results in the model and mathematical tractability. We will provide documentation of
these results upon request.
. The NTS pays a wage to its employees equal to the marginal
product of labor from the previous step,
, , 1, 1 , 1
NTS t NTS t
At the end of each step, all firms are evaluated for potential bankruptcy. All firms
for which costs exceed revenues (πj < 0) must borrow funds to remain solvent. This debt
accumulates across steps. Bankruptcy occurs when accumulated debt exceeds the limit of
j tj t
B is a function of the greatest profits previously realized by any firm in a single step,
adjusted by an exogenous multiplier, Γ.
2.1 Simulation Steps and Sub-step Ordering
Our model is characterized by a schedule of agent decisions and model events. This
schedule plays out in a series of steps and sub-steps.9 A run of the model is constituted by
an initialization (t = 0) followed by a set number of model steps (t=1…T), during which
every agent is activated in random order, as arranged by the model sub-steps. The sub-
steps are ordered as follows:
1) Each firm, j=1…m, sets its offered wage (see Equation 5) and its offered price for
2) All expired patents are made public; the largest patent value is added to the
cumulative stock of public knowledge. All sub-superior knowledge disappears.
3) Each firm conducts research (see Equation 6).
4) The NTS sets both its offered wage and the price for secondary goods.
5) Households, i = 1…n, are activated in random order and execute τ searches over
the set of all available wages. Households are always aware (without cost) of the
9 The model is written in Java using the MASON agent modeling library (Luke et al. 2005). The step/sub-
step construct is built into the MASON model scheduling system.
NTS wage. Once they have decided on their fixed sample size, the first wage in
their discovery set is their employer from the previous time step (see Equation 8).
6) Given the fruits of their research investment, their posted price and wages, the
price of capital, and the number of employees they were able to hire, firms
establish a profit-maximizing limit to the amount of the primary good they will
7) Households are activated in random order and execute τ searches over the set of
all available prices. Once they have decided on their fixed sample size, the first
price in their discovery set is their seller from the previous time step (see Equation
9). Once search is concluded, the household maximizes its utility function,
choosing an optimal bundle of q and x. If the firm offering the lowest known price
to the household is unable to fulfill the entire desired quantity of q, the household
purchases the remaining amount from the firm with the second lowest price firm.
For tractability, the household will not seek out a third firm if the quantity desired
is still not met. Once a firm has orders for
, it is withdrawn from the set
of unknown prices Ω.
8) Having received all of their market orders, firms will acquire the amount of
capital necessary to produce
jq and fulfill all existing market orders.
9) If a firm is unable to procure any market orders, it may go bankrupt. Bankruptcy
results when a firm’s outstanding debt is greater than the quantity that is available
in the commercial loan market (see equation 14). In the model simulations
executed in this paper, firms were exempt from bankruptcy rules during the first
ten simulation steps, allowing firms to adapt to initialized conditions.
3 Simulation Results
We ran the model under a variety of patent length and search cost parameterizations, with
400 time steps constituting a run. In experiments where we simulate the model for a
single run, we ran it with 4000 households and 200 firms. For larger batches where we
made comparisons across runs, we ran it with 2000 households and 100 firms. The key
exogenously set parameters are summarized in Table 1.10
Our emphasis, in this paper, is on the distributional properties that are observable
across firms. These properties, however, are of limited interest if they do not occur in a
model of endogenous, exponential growth. Figure 1 plots log Q, where
Qq , over
time in a single run of the model. All firms produce the same good, price differences at
any time tick the model are simply the consequences of positive search costs and
technology differences, so there is no particular merit to working with “real” output.
Table 1 Model Parameters
Starting number of firms
Number of customers
Initial public stock
Loanable funds multiplier
j tj t
Search cost [0.00001, 0.0001]
† The total sub step time endowment for an agent is 1. As such when search
costs, ς, equal 0.00005, that is equivalent of 0.005% of their time endowment,
meaning it takes 0.005% of an agent’s sub step time endowment to engage in
another act of search.
Tracking the growth of log Q over time, we observe two distinct periods. In the
early time steps of the run, we see an “organizational” period in the model, which
typically (but not always) concludes within the first 50 steps, within which growth is
erratic, often characterized by large swings up and down, as firms grope towards
10Model results are, unsurprisingly, sensitive to the specification of γ (the output elasticity of Aj) and Γ (the
maximum debt firms can take on). This sensitivity is the result of their influence on market concentration.
Specifically, large values of γ and small values of Γ result in faster rates of attrition, driving the model
profitable strategies and unprofitable firms go bankrupt and exit the model. Eventually
the model settles into steady growth trend, which is ostensibly a random walk, but does
sometimes exhibit small, semi-regular cycles. Growth observed in the model is
exponential and consistent, and largely parameter insensitive (given minimal returns to
research and elasticities of output). Given this type of growth, we can proceed to focus on
the distributional properties observed within the set of active firms.
Figure 1 Log Q over Time. Patent length = 12, Search Costs = 0.01%. The model was initiated with
4000 households and 200 firms.
3.2 Outcomes Across Firms Within a Single Run
Firms in our model are homogeneous ex ante and heterogeneous ex post. Given the
randomness of research outcomes and cost constraints faced by searching households,
each firm experiences its own unique history of research outcomes, sales, and
profitability. Firms are confronted with two levels of uncertainty: research uncertainty
and commercial uncertainty.11 They do not know the stock of excludable knowledge that
their research investment will bear, nor do they know whether customers will
successfully find them even if they are able offer a relatively low price. These
uncertainties result in differing research investments, private knowledge stocks, posted
prices, qj sold, and revenues generated.
Research investment (as a percentage of revenues), χj, is the manner in which the
individual firms most directly respond to their own unique history. In Figures 2 and 3 we
present box and leaf plots of χj across all 200 firms at time step 400 in several runs of the
model, pooled over different patent length (Figure 2) and search cost (Figure 3)
parameterizations. It is interesting to note that though the mean investment is not
increasing with longer patent lengths, the spread of investment rates is. We observe both
a greater variety of investment rates, larger maximums, and smaller minimums with the
longest patent lengths. One possible inference is that when patent lengths are longer,
firms that have dominated the market operate with a history of large returns to R&D
investment and act accordingly, while firms that have not dominated operate with a
history that advises a strategy of minimizing costs to maximize profits as a “second best”
firm. In Figure 3 we see no noticeable change in mean investment rates over search
costs, but again an increase in the spread of investment rates when search costs are larger.
When search costs are very low, households sample a larger fraction of the offered prices,
and their purchasing of goods will more directly follow the distribution of knowledge and
prices. When search costs are higher, however, there is greater randomness in the model.
The correlation between sales and prices is far murkier when search costs are high, and in
turn firms have vastly different experiences in their personally observed connection
between research investment and profit. Greater variety of experienced histories leads to
a greater variety of firm behavior.
11 This is not unlike the three types of uncertainty (technical, commercial, and financial) laid out in Scherer
et al. (2000)
Figure 2. Box plots of research investment percentage across firms, organized by Patent Length (4 to
40) and Search Cost (0.01% to and 0.2%), at t = 400.
Figure 3. Box plots of research investment percentage across firms, organized by Patent Length (4 to
40) and Search Cost (0.001% to and 0.02%), at t = 400.
Figure 4 presents histograms of logged total revenue, TRj, across all firms at time
step 400 in several runs of the model, parameterized with different patent lengths and
search costs. The symmetry of log TRj in some of the histograms in Figure 4 gives the
appearance of a distribution that is potentially lognormal. None of the distributions of log
TRj, however, pass the Shapiro-Wilk test for normality. While the bulk of the
distributions of log TRj are dominated by a central mode, many of the distributions have
prominent outliers, especially in the upper tail, and exhibit fat tails more generally.
Thirteen of the fifteen histograms qualify as leptokurtotic, each with raw kurtosis greater
than 5.0 (see Table 2, column 1).
Figure 4 Distribution of Total Revenue, organized by Patent Length (4 to 20, vertically) and Search
Cost (0.001%, 0.01%, and 0.02%, horizontally), at t = 400. Each subfigure is from a single run of the
Search Cost = 0.001% Search Cost = 0.01% Search Cost = 0.02%
PL = 4
PL = 8
PL = 12
PL = 16
PL = 20
In Figure 5 we chart the rank, N, of each observation, where the rank can be
interpreted as number of other observations within the same model simulation run that
are of equal or greater value than TRj at step t = 400, using data from the single
simulation run with patent length of 16 and search costs of 0.02% . The shape of the
results in Figure 5 bears a strong resemblance to what was found by Scherer et al. (2000)
in their study of the value (in Deutschmarks) of German patents from 1977 to 1995,
duplicated here in Figure 6. In both the simulation and patent data, the lower observations
within the distribution are concave to the origin, but the higher value observations take a
more linear relationship between log value and log rank. A log-linear relationship
between rank and value is indicative of a potential power law nature of the distribution of
values. Both the simulation results and German patent return observations also include
outliers significantly beyond the rest of the distribution.
Fig 5 Log Rank over Log TR, Single Run of the Model. Patent length = 16, Search costs = 0.2%
Figure 6 Rank over Estimated Value (Log-Log scale), from Scherer, Harhoff, et al. (2000)
Again, we simulated the model under a range of parameterizations. In Figure 7 we cluster
similar charts of the Log Nj over Log TRj , with each graph charting the results at time
step 400 of a single run of the model, under different combinations of patent length and
search cost parameterizations (charts of simulations with longer patent lengths can be
found in the appendix). The concave to the origin shape and increased linearity in the
upper tail are fairly consistent. Further, the top ranked observation, and sometimes
several more, is frequently a significant outlier from the rest of the observations.
Search Cost = 0.001% Search Cost = 0.01% Search Cost = 0.02%
PL = 4
PL = 8
PL = 12
PL = 16
PL = 20
Figure 7 Log Rank over Log TR, organized by Patent Length (4 to 20, vertically) and Search Cost
(0.01%, 0.1%, and 0.2%, horizontally), at t = 400. See Appendix Figure A for longer patent lengths.
Scherer, et al. (2000) analyzes the Pareto-Levy, or power law, distribution
parameters of the patent data with simple Ordinary Least Squares regression analysis
using the log-linear modeling function
where N is the rank of the return on Investment observation, TRj is total revenue, and k
and ω are parameters. Absolute values of ω greater than or equal to one are indicative of
a Pareto-Levy distribution.
In most of their patent data, Scherer et al do not find a log linear fit between TRj
and rank across the full body of observations, instead finding that the bulk of the
distribution is better characterized as lognormal. However, they do observe a much closer
to log-linear fit in the upper tail of the data. We find similar results in our simulation data
across a variety of parameterizations. Simple ordinary least squares analysis of Pareto-
Levy model parameters, when regressed over the full distribution, offers further support,
with values less than one. However, if we isolate the third and fourth quartiles of the
distributions, the resulting coefficients correspond to a Pareto-Levy power law
distribution. The results of the ω slope parameter are included in Table 2 for the overall
distribution (column 2), the third quartile (column 3) and in the fourth quartile (column 4)
for each combination of patent length and search cost parameter combination tested. In
all 15 model simulations, the regression of Pareto-Levy parameters on the full
distribution resulted in slope coefficients less than one. Regression on the third quartile
observations resulted in much larger slope coefficients, with
1 in fourteen of the
specifications. Regression on the fourth quartile observations produced slope coefficients
1 in four of the specifications. All fifteen specifications, however, produced
in at least one of the upper two quartiles.
Examined in tandem, the plots from Figure 7 and the Pareto-Levy slope
coefficients for the upper quartiles (Table 2) of the simulation distribution paint a telling
picture. The plots can be visually broken down into three common components. The first
is the lower-middle portion with a shallow slope and weakly concave shape. The second,
is at steep, flat region usually in the upper-middle portion of the distribution. Third, but
not always, is a small number of extreme values that represent significant outliers from
the rest of the distribution. Even in the simulations whose upper quartiles did not have
1 , we can often visually identify a portion of the distribution characterized by
significant steepness, with the strong possibility of Pareto-Levy characteristics. It is our
view that considered together, the Pareto-Levy coefficients and general visual shape of
the data plots, are evidence of a distributional pattern emergent from the model that is
similar in character to that observed in the Scherer, et al. German patent data –
specifically a mixture distribution of revenues across firms accruing rents from
temporarily excludable knowledge stocks that is Pareto-Levy distributed in the upper
quartiles that is frequently dominated by a small number of extreme values.
Table 2 Log TR: Kurtosis, Shapiro-Wilk Tests, and Pareto-Levy OLS
Regression Coefficients at t=400
4 0.001% 5.21
4 0.01% 6.70
4 0.02% 6.60
8 0.001% 7.05
8 0.01% 10.10
8 0.02% 7.43
12 0.001% 22.01
12 0.01% 10.23
12 0.02% 2.23
16 0.001% 7.45
16 0.01% 3.17
16 0.02% 12.63
20 0.001% 6.79
20 0.01% 8.88
20 0.02% 6.61
*Raw Kurtosis (kurtosis = 3.0 at normality).
Patent Length Kurtosis*
3.3 Growth Rates
In the previous section and simulation experiment, we inspected the distributions of
results across firms within single runs of the model, each with differing patent length and
search cost parameterizations. In this section, we take a different approach, simulating the
model thousands of times, and inspecting the how various outcome properties change. In
turn, we are not looking at individual firms, but rather outcomes that are aggregated
across all firms from each instantiation of the model. In this simulation experiment, we
ran the model 4000 times, with 2000 households and 100 firms in each run, for 400 time
steps in each run. All parameters besides patent length [1,2,…40] and search costs
(0.002% and 0.02%) are held constant and are identical to those reported in Table 1. Each
of the 80 different parameter combinations are run 50 times. Our outcome measure of
concern is the average growth of aggregate Q from step 100 to step 400 in the model.12
We are purposely allowing the model to settle into the post-initialization adjustment
portion of its simulation history before tracking growth rates.
In Figure 8 we have three panels with scatter plots of the first, second, and fourth
moments of the distributions of the average growth rate in each parameter combination.
Observations from the high (0.02%) and low (0.002%) search cost parameter settings are
distinguished in the figure. That said, the patterns exhibited by low and high search cost
observations over patent lengths are very similar. The mean average growth rate is so
similar in high and low search cost runs that the observations are indistinguishable. In
this experiment, we find average growth rates increasing with patent length early on, but
with diminishing returns.13 Despite these diminishing returns, mean average growth never
declines with patent length in the parameter space tested. This is similar, at least
nominally, to Futagami and Iwaisako’s (2003) dynamic analysis of patents in their
endogenous growth model, within which social welfare was maximized by patents with
finite length (see also Horowitz and Lai 1996). While our experiment did not identify a
global maximum, the observation of sharply diminishing returns similarly recommends
patents of finite length.
We are also interested in the shape of the distribution of outcomes from the model
across multiple runs of the model. Specifically, we were interested in the consistency of
growth outcomes from simulation to simulation under identical model parameterizations.
The second and third panels of Figure 8 present scatter plots of the variance and kurtosis
of the distributions. Again, we find that the variance and kurtosis of average growth rates
exhibit similar patterns in low and high search cost parameterizations. The variance of
average growth rates across simulations is increasing with patent length, and is increasing
12 Average growth rate =
j tj t
13 We can also see in Figure 8 the often truly prodigious rates of growth in the model. It has been our
experience that the model results remain salient and clear, despite our relatively limited size of 100 firms
and 2000 customers, when growth is rapid. When using single runs, such as in earlier sections of the paper,
we have the luxury of a larger agent sets. Running the model thousands of times per experiment is less
practical with larger numbers of agents, however. Larger scale investigation remains for future research.
at a higher rate under the longest patent lengths. Even at its highest levels, however, the
variance remains small relative to the means of the distributions. The raw kurtosis of the
individual distributions of average growth rates is perhaps more interesting. The mean
kurtosis of the distributions was 5.95, considerably higher than a normal distribution.
Kurtosis is increasing with patent length as well, and similar to the variance, is especially
large with the longer patent length runs. The extreme kurtosis of growth rates in longest
patent length regimes suggests an extremely peaked distribution, with the majority of
observations clustered around the middle, but also with outliers exceptional in number
and deviation from the mean. Given that mean growth rate is not increasing significantly
in the longest patent length runs, it would appear that longer patent lengths come with the
possibility of the highest growth outcomes, but with also the possibility of the lowest
possible outcomes. In these simulated histories, growth rates are pulled from fat-tailed
distributions. The longer the patents, the greater the possibility of best and worst case
scenarios In deference to Nicholas Taleb, “black swans” abound when patent length is
longer, and bring with them a risk profile that minimax and similarly risk averse
strategies would argue against.
The distribution of returns to research investment, based on our analysis, appears to be a
mixture distribution dominated by a Pareto-Levy power law distributed in the upper
quartiles. This curious mixture distribution is nowhere assumed in the model. The
economic consequences of perfect substitution combined with positive search costs allow
an occasional innovation to revolutionize the industry without creating a single firm. We
have not attenuated the consequences of such an innovation by the ad hoc assumption of
imperfect substitution. The only distributional shapes assumed in the model are an
exponential distribution of returns to dollars invested in research and the uniform price
and wage distributions assumed by searching households. It is this conspicuous absence
from the model structure that makes the emergence of this mechanically idiosyncratic and
empirically observed distributional shape so compelling. Such a distribution is
Figure 8. Distribution of average growth rates across
runs from Steps 100 through 400, over Patent length.
Clockwise from upper left: a) Mean of Growth Rates
b) Variance of Growth Rates c) Kurtosis of Growth
Rates. Linear fits are included in all subfigures.
Two search costs parameterizations are charted:
0.002% (black/dashed/circles) and 0.02%
(red/solid/triangles). Each observation is the
mean/variance/kurtosis for a set of 50 runs. The
graphs are the results from 4000 model simulations.
characterized both by its skewness and proclivity towards producing outliers that
dominate the rest of the distribution. William Feller (1950) begins the elementary half of
his celebrated pair of volumes on probability theory with distributions without moments.
He demonstrates how an innocent looking process is likely to have a single event which
is as large as the all the other events combined. He expresses his concern that we have to
believe that something can happen before we see it:
In practice such a phenomenon would be attributed to an "experimental
error" or be discarded as "outlier." It is difficult to see what one does not
expect to see. (Feller 1950, p. 91).
In our account of endogenous technical change distributions without moments are not
censored by our intuition. What might such an outlying event look like in technical
change? Perhaps the electronic computer, but economic intuition is uncomfortable with
the naïve question of asking what would 2010 GDP look like if we were to evaluate
computational expenditures in 1950 prices (Nordhaus 2001). We find it more tractable to
simply trim out 1950 computational expenditures by putting them in 2010 prices.
If we take the approach to Knightian uncertainty suggested by Epstein and Wang
(1994) in which familiar distributions are mixed with something strange, then trimming
“outliers” can take Knightian uncertainty and transform it into seemingly well-behaved
risk. Regardless of approach, the point remains that “outliers” are not observations that
can be dismissed. Outliers are driving observed growth and are, in turn, the source of “fat
tailed” distributions of growth observed in Figure 8. When an outlier event occurs, it
changes the entire trajectory of the simulated economy. It is our opinion that models of
endogenous growth would do well to account for such distributional properties, either in
their assumptions or their outcomes (Silverberga and Verspage 2007).
Thinking about models of industrial organization, in 1949 George Stigler saw
something very remarkable in Edward Chamberlin’s theory of monopolistic competition.
With the abolition of an “industry” there was no way to keep the “group” from being the
whole of the economy:
It is perfectly possible, on Chamberlin's picture of economic life, that the
group contain only one firm, or, on the contrary, that it include all of the firms in
the economy. This latter possibility can readily follow from the asymmetry of
substitution relationships among firms: taking any one product as our point of
departure, each substitute has in turn its substitutes, so that the adjacent cross-
elasticities may not diminish, and even increase, as we move farther away from
the "base" firm in some technological or geographical sense. (Stigler 1949, p. 15)
This property would suggest that a technological development in one firm could disrupt
firms arbitrarily distant. This explosive general equilibrium property, however, was long
seen as a defect in monopolistic competition which would be later tamed with a
“preference for diversity” which kept firms safely in their niches. It was the tamed
monopolistic competition models which would become the basis for models of increasing
5 Concluding Remarks
It would be useful to extend and test the model in a computing environment that allowed
for larger scale simulations. Given the importance of highly skewed distributions, larger
agent pools could have important ramifications for growth rates and the incentive to
participate in innovation races. Patents in our model are greatly simplified. Future work
would benefit from introducing more sophisticated intellectual property rights, including
both “breadth” and length, as well as a continuum of imitation and obsolescence. This
paper is largely concerned with offering a process to generate the unusual distributional
characteristics of the returns to research and the need for their realization in endogenous
technical change models. More generally, the nature of the “optimal” patent length is
given only cursory attention here, and would benefit from finer grain analysis in future
Aghion, P. and P. Howitt (1992). "A Model of Growth Through Creative Destruction."
Econometrica 60(2): 323-351.
Axtell, R. (2001). Effects of Interaction Topology and Activation Regime in Several Multi-Agent
Systems. Multi-Agent-Based Simulation. G.Goos, J. Hartmanis and J. v. Leeuwen. Berlin /
Heidelberg, Springer: 33-48.
Axtell, R. (2001). "Zipf Distribution of U.S. Firm Sizes." Science.
Eicher, T. S. and S. J. Turnovsky (1999). "Non-Scale Models of Economic Growth." The Economic
Journal 109(457): 394-415.
Epstein, J. M. (2006). Generative social science : studies in agent-based computational modeling.
Princeton, Princeton University Press.
Epstein, J. M., R. Axtell, et al. (1996). Growing artificial societies : social science from the bottom
up. Washington, D.C., Brookings Institution Press.
Epstein, L. G. and T. Wang (1994). "Asset Pricing under Knightian Uncertainty." Econometrica 62:
Feller, W. (1950). An introduction to probability and its applications. New York.
Futagami, K. and T. Iwaisako (2003). "Patent Policy in an Endogenous Growth Model." Journal of
Economics 78(3): 239-258.
Grossman, G. M. and E. Helpman (1991). "Quality Ladders in the Theory of Growth." Review of
Economic Studies 58(1): 43-61.
Grossman, G. M. and E. Helpman (1994). "Endogenous Innovation in the Theory of Growth."
Journal of Economic Perspectives 8(1): 23-44.
Harhoff, D., F. Narin, et al. (1999). "Citation Frequency and the Value of Patented Inventions."
The Review of Economics and Statistics 81(3): 511-515.
Harhoff, D., F. M. Scherer, et al. (1998). "Exploring the tail of patent value distributions."
Working Paper, Center for European Economic Research.
Hellwig, M. and A. Irme (2001). "Endogenous Technical Change in a Competitive Economy."
Journal of Economic Theory 101, 139 (2001) 101: 1-39.
Horowitz, A. W. and E. L. C. Lai (1996). "Patent Length and the Rate of Innovation." International
Economic Review 37(4): 785-801.
Iwaisako, T. and K. Futagami (2007). "Dynamic analysis of patent policy in an endogenous
growth model." Journal of Economic Theory 132(3): 306-334.
Judd, K. L. (1985). "On the Performance of Patents." Econometrica 53(3): 567-586.
Kortum, S. (1993). "Equilibrium R&D and the Patent-R&D Ratio: U.S. Evidence." American
Economic Review 83(2): 450-57.
Levy, D. and M. D. Makowsky (2010). "Price Dispersion and Increasing Returns to Scale." Journal
of Economic Behavior and Organization 73: 406-417.
Luke, S., C. Cioffi-Revilla, et al. (2005). "MASON: A Multiagent Simulation Environment."
SIMULATION 81(7): 517.
Luttmer, E. G. J. (2007). "Selection, Growth, and the Size Distribution of Firms." Quarterly
Journal of Economics 122(3): 1103 -1144.
Nordhaus, W. D. (2001). "The Progress of Computing." Yale Cowles Foundation Discussion Paper
O'Donoghue, T. and J. Zweimüller (2004). "Patents in a Model of Endogenous Growth." Journal
of Economic Growth 9(1): 81-123.
Romer, P. M. (1990). "Endogenous Technological Change." The Journal of Political Economy
98(5, Part 2: The Problem of Development: A Conference of the Institute for the Study
of Free Enterprise Systems): S71-S102.
Romer, P. M. (1994). "The Origins of Endogenous Growth." The Journal of Economic
Perspectives 8(1): 3-22.
Scherer, F. M., D. Harhoff, et al. (2000). "Uncertainty and the size distribution of rewards from
innovation." Journal of Evolutionary Economics 10(1): 175-200.
Silverberga, G. and B. Verspage (2007). "The size distribution of innovations revisited: An
application of extreme value statistics to citation and value measures of patent
significance." Journal of Econometrics 139(2): 318-339
Stigler, G. J. (1949). Five lectures on economic problems. London, New York, London School of
Economics and Political Science; Longmans Green.
32 Download full-text
Figure A1. Log Rank over Log TR, organized by Patent Length (24 to 40, vertically) and Search Cost (0.001%,
0.01%, and 0.02%, horizontally), at t = 400.
Search Cost = 0.001% Search Cost = 0.01% Search Cost = 0.02%
PL = 32
PL = 36
PL = 40