Emergent Pareto-Levy Distributed Returns to Research in a Multi-Agent Model of Endogenous Technical Change
ABSTRACT We build a multi-agent model of endogenous technical change in which heterogeneous investments in patented knowledge generate Pareto-Levy and lognormal 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 but seemingly idiosyncratic Pareto- Levy and lognormal mixture distribution of market returns. These distributional characteristics have ramifications for endogenous growth models given the importance of extreme values and market leaders in technological advancement. Average growth rates in the model have a global maximum at a finite, non-zero patent length. The distribution of growth rates is characterized by “fat tails.” The variance of growth rates increases with patent length.
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ABSTRACT: Previous research has shown that the distribution of profit outcomes from technological innovations is highly skew. This paper builds upon those detailed findings to ask: what stochastic processes can plausibly be inferred to have generated the observed distributions? After reviewing the evidence, this paper reports on several stochastic model simulations, including a pure Gibrat random walk with monthly changes approximating those observed for high-technology startup company stocks and a more richly specified model blending internal and external market uncertainties. The most highly specified simulations suggest that the set of profit potentials tapped by innovators is itself skew-distributed and that the number of entrants into innovation races is more likely to be independent of market size than stochastically dependent upon it.Journal of Evolutionary Economics 12/1999; 10(1):175-200. · 1.00 Impact Factor
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ABSTRACT: Analyses of firm sizes have historically used data that included limited samples of small firms, data typically described by lognormal distributions. Using data on the entire population of tax-paying firms in the United States, I show here that the Zipf distribution characterizes firm sizes: the probability a firm is larger than size s is inversely proportional to s. These results hold for data from multiple years and for various definitions of firm size.Science 10/2001; 293(5536):1818-20. · 31.03 Impact Factor
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ABSTRACT: Growth models that incorporate nonrivalry and/or externalities imply that the size of an economy may influence its long-run growth rate. Such implied scale effects run counter to empirical evidence. This paper develops a general growth model to examine conditions under which balanced growth is void of scale effects. The model is general enough to replicate well known exogenous, as well as endogenous, (non-) scale models. The authors derive a series of propositions that show that these conditions for nonscale balanced growth can be grouped into three categories that pertain to (1) functional forms, (2) the production structure, and (3) returns to scale.Economic Journal. 01/1999; 109(457):394-415.
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 firstname.lastname@example.org.
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:
QA 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
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 (
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
(,,,, , ,)
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 t j 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 tj 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
j tj t
j tj 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,
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
qA LL 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) .
E McwF 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( ))
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 tNTS 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 t j 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
, , 1
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