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Electronic copy available at: http://ssrn.com/abstract=1588462

1

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

Abstract

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 mikemakowsky@gmail.com.

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Electronic copy available at: http://ssrn.com/abstract=1588462

2

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

Verspage 2007).2

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).

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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

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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.

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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

Turnovsky (1999)

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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.

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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

(1)

i

[1,2,... ]

Hn

H

and firms

(2)

[1,2,... ]

F

F

j

m

where each household (i) purchases

j

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,

j

Q , using inputs of labor, Lj, capital, Kj, and

knowledge , Aj, where knowledge is composed of public, G, and private, Rj, knowledge:

(3)

jjjj

jj

QA K Lj

AGR

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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.

(4)

jjjjj

jjjj

C w LrKSj

p QC

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

pAC

. 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 t

MPL

p

1):

(5)

1

,, 1

j t

, 1

j t

, 1

j tj t

wKLpj

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

,1

(,,,, , ,)

jjjj tjj

C K L QA w r

such that

1

max

j

L

, 1

j tj

QArw

.

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

stocks,

1

,,

0

j t j t

Ry

. 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:

(6)

,,1

,1

log( )

j

j tj t

tj t

S

R

yRj

G

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

wage.

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9

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,

, 1

j tt

GR

,

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

(7)

,, 1

j t

,

, , 1

j t

,

, , 1

j t

12

, , 1

j t

12

whereif

1

j tj t

j tj t

j t

tt

j t

tt

STRj

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,

0

t

, 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,

1 1,,

max,...

tttm t

GGyy

.

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

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product of the said fraction in the previous time step. The profit maximizing quantity to

be sold is

1

max

j

11

()

jjj

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

wage,

*

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,

iw

, 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) .

(12)

,1

()(1)( )

w

i

w

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) .

(13)

,1, 1

i t

q

() (1( ))

F p

+c

p

i

p

E Cpdpi

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

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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.

(8)

1

,

*

,

...,

max

mNTS

i

jjj

iiii

www

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

firms.

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.

(9)

1

,

*

,

...

min

m

i

jjj

iiii

pp

pp pi

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

function,

(10)

1/

()

iii

Uqx

.

For a given wage rate and price, the optimal quantities of q and x are

(11)

11

*

i

11

11

*

i

11

1

1

qpMp

xMp

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

(1)

iii

Mwd

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.

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12

1

11

nz

NTS

thz

hb

Cxx

. The NTS pays a wage to its employees equal to the marginal

product of labor from the previous step,

1

, , 1, 1, 1

NTS t

w

NTS tNTS t

L

NTS t

Kp

.

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

B,

(14)

,0,

max(...)

j t j t

B

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

primary goods.

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.

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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

produce.

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

max

jj

qq

, 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

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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

j

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

Parameter

M

N

α, β

Context/Related Function

Starting number of firms

Number of customers

(

j

QGR

(

j

QGR

(

ii

Uqx

Initial public stock

Loanable funds multiplier

r(Qt-1, Ψ)

Value

100, 200

2000, 4000

)

jjj

K L

0.5

γ

λ

)

jjj

K L

0.15

1/

)

i

-0.1

1

5

0.05

Gt=0

Γ

Ψ

,0

j t

ς†

, , 1

j t

,

j tj t

0.002

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

towards monopoly.

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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