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Exploitation, Exploration and Innovation in a Model of
Endogenous Growth with Locally Interacting Agents
Giorgio FagioloSant’Anna School of Advanced Studies, Pisa (Italy)
giorgio.fagiolo@sssup.ithttps://mail.sssup.it/~fagiolo
DIMETIC Doctoral European Summer SchoolSession 3 – October 8th to 19th , 2007Maastricht, The Netherlands
The Islands Model (Fagiolo and Dosi, 2003)
• Background– Acknowledging with Dick Nelson that there still is a large gap
between:• What we know about the sources of growth, technological
progress, innovation, learning, etc. from empirical research• How we inject this knowledge in standard models of growth
• Theories of Growth vs. Micro/Macro SFs– Empirical analyses of technological change and innovation
(Cf. Stoneman, 1995; Freeman, 1994; Nelson, 1995 and 1998)
– New Empirics of Economic Growth and Development (Cf. Durlauf and Quah, 1998; McGrattan and Schmitz, 1998)
Motivations (1/2)
• ACE/Evolutionary approach– Allows one to be flexible as far as assumptions on individual
behavior, interactions, etc. are concerned– Highly-parameterized model: trade-off between realistic
assumptions, analytical solvability and sharpness of implications
• The role of assumptions– Asking Dick Day’s question:
• “Can one do good science by using models based on assumptions which are clearly at odds with any empirical evidence about micro behavior? ”
– Modeling the economy as a complex evolving system
Motivations (2/2)
• Building a dynamic model of growth that…– Is able (as a plausibility check) to reproduce the fundamental
statistical properties of GDP time series– Allows one to disentangle the role of the basic sources of
growth on the technological side
• Growth as the result of exploration-exploitation trade-off driven by
– Technological opportunities– Path dependency in technological accumulation– Degree of locality / globality of information diffusion– Increasing returns to knowledge base exploitation– Willingness to explore/exploit
The Islands Metaphor
Technological Space Notionally Unbounded Sea Technology Island (‘mine’)
Output Homogeneous Good Firms Stylized Entrepreneurs
Production Mining/Extracting the Good Technological Search Exploration of the Sea
Innovation Discovering a new island Technological Diffusion Spreading knowledge
from islands Imitation Traveling between
already known islands Technological
Difference Distance
between Islands
The Model (1/2)
• Basic ingredients– Time is Discrete– Finite, constant population of stylized firms I={1,2,…,N}– Notionally endless, discrete set of technologies (islands)– Homogeneous good
• Islands– Stochastically distributed on a bi-dimensional lattice– Each node of the lattice can be an ‘island’ with probability π
(‘sea’ with probability 1-π)– Each island (x,y) is characterized by a productivity coefficient
s(x,y)=|x|+|y|
The Model (2/2)
• Initial Conditions– Set of initially known islands (exploited technologies)– All N firms mining on them (randomly allocated)– Each firm working in island (x,y) produces output s.t
α),(),(),( yxnyxsyxq ⋅=
– where
n(x,y) number of firms currently working on (x,y)α>1 increasing returns-to-scale coefficient
Example: 3 initial islands, 10 firms
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Example: 3 initial islands, 10 firms
Dynamics (1/4)
• Exploration– In each t, a “miner” becomes “explorer” with probability ε
• Constant willingness to explore
– Explorers move around randomly in each period
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Dynamics (2/4)
• Innovation– In each exploration period, explorers find a new “island” with a
probability π– The productivity of the newly discovered island is
s* = s(x*,,y*) = (1+W) ⋅ { [|x*| + |y*|] + ϕ qi,τ + ξ }
Poisson (λ) Random Variable
(Low probability high jumps)
Distance from
the Origin Zero-Mean
Random Variable
(High Probability
Low Jumps) Cumulative Learning Effect: Agents
carry with them their previous skills
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Example: Exploration
Example: Exploration
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Example: Exploration
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Example: Exploration
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Dynamics (3/4)
• Imitation– In each t, from any currently exploited island (with at least one
miner on it) a signal about current island’s productivity is released
– Any miner currently working on (x,y) receives and follows the signal with a probability proportional to:
• the productivity of the island the signal comes from• the exp of minus the distance between island and miner
)}minerisland,(exp{),( dyxq ⋅−⋅ ρ
– The higher (smaller) ρ the more global (local) is information and knowledge diffusion
– Imitators move toward the imitated island following the shortest path leading to it (one step per period)
Example: Imitation
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Example: Imitation
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Example: Imitation
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Dynamics (4/4)
• Dynamics of agents’ states: summing up
Reaching an island
In probability, if reached byinformation on a more
productive island
Miners
ExplorersImitators
With prob. εi=ε
In probability, if reached byinformation on a more
productive island
Timing and Aggregate Variables
Miners update output Miners become explorers Explorers look around Imitators approach islands
Information diffusion Miners and explorers collect
signals Imitation decisions
Time t−1 Time t
Given t−1 micro ¯o variables
Update time t micro & macrovariables; next iteration starts
• Focus on– Aggregate output (sum of firms’ output) and growth rates– Number of explorers, imitators, miners
Timing and Aggregate Variables
• Model’s parameters
ρ : globality of information diffusionϕ : path-dependency in learningλ : likelihood of radical innovationsπ : baseline opportunity conditionsα : increasing returns to scale in exploitationε : willingness to exploreN : population sizeT : time horizon
Initial Conditions: ( xi,0 )Micro & Macro Pars: (θi ), Θ
Generate Time-Series through Simulation{( xi,t ), t =1,…,T}{ Xt , t =1,…,T}
Compute a Set of Statistics S= {s1, s2 , … }
on micro/macro Time-Series
Repeat M ind. times
Generate MontecarloDistribution for each
Statistics in S= {s1, s2 , …}
Studying how MontecarloDistributions of Statistics in
S= {s1, s2 , …} behave as initial conditions, micro and macro parameters change
Statistical Tests for difference between moments
Analyzing simulation output
A first question…
Under which general conditions is the economy able to generate
self-sustaining growthas the outcome of the
joint processes of exploitation and exploration ?
A closed economy without exploration (1/2)
• Diffusion of information drives growth– In this case the model is analytically solvable!– Whenever an island manages to capture all agents the growth
process stops (growth rates are zero)– The process is path-dependent and possibly inefficient
(convergence toward an inefficient level of output is a non-zero probability event)
• Shutting down exploration and innovation– A given initial set of islands (e.g, only 2)– Firms initially mining on them (50%, 50%)– They can only exchange information among the 2 existing
technologies (initial set of islands cannot be expanded)
A closed economy without exploration (2/2)
• Growth is always a transitory phenomenon
Log ofGNP
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Time
Growth
Rates
Time
• Lock-in may occur on the ex-ante less efficient island
A closed economy with exploration (1/4)
• Diffusion of information still drives growth– Process driven by information diffusion– Steady states can be destabilized by ‘irrational’ entrepreneurs
who decide to leave their island even if everyone is there
• Allowing for exploration in a closed box– Initial set of islands cannot be expanded (no innovation)– Explorers are allowed to search only inside initial box – Imitation still occurs as before
A closed economy with exploration (2/4)
• Absorbing states become basins of attraction: growth is a transitory phenomenon but fluctuations can arise
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GN
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A closed economy with exploration (3/4)
• Two ex-ante equally efficient islands
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Min
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A closed economy with exploration (4/4)
• One ex-ante more efficient island: temporary inefficiency may arise
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Min
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Exploration in a Open-Ended Economy
• In the full-fledged model self-sustaining growth can arise!
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Log
of G
NP
A second question…
When the economy does generate self-sustaining growth (full-fledged model), do log(GNP) time-series display empirically
observed statistical properties?
Statistical Properties of Simulated GNP Series
• Yes, if self-sustaining growth does emerge– log(GNP) time series are I(1), i.e. difference-stationary– growth rates are positively correlated over short horizons– persistence of shocks are in line with empirical evidence
• Scale-effects are not present– As in reality, unlike in many endogenous growth models are!
6001600
26003600
4600 50 250 450 650 8500.00%
0.05%
0.10%
0.15%
0.20%AGR
Econometric Sample Size (T)
Population Size (N)
A third question…
When the economy does generate self-sustaining growth (full-fledged model),
what are the roles played by system parameters
(i.e. by the sources of growth)?
The Sources of Growth (1/4)
• Average growth rates (AGRs) increasing in – path-dependency in knowledge accumulation– globality of information diffusion
0,20,4
0,6
1,50,5
-0,5-1,5
-0,05%
0,15%
0,35%
0,55%
0,75%A
GR
ϕlog10 (ρ)
– … as well as in returns-to-scale strength and opportunities
The Sources of Growth (2/4)
• The exploitation-exploration trade-off– AGR are maximized only if there is a balance between resources
devoted to exploration and resources devoted to exploitation
0,0%
0,1%
0,2%
0,0 0,2 0,4 0,6 0,8 ε
AG
R
The Sources of Growth (3/4)
• Emergence of thresholds– I(1) log(GNP) time-series only emerge if increasing returns to scale,
opportunities, path-dependency and globality of information are strong enough!
1,50,5-0,5-1,50
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0.6
Log10 ρ
ϕ
[% o f Acc. AD F(1) Test]
> 90 %
60% < [% of Acc. ADF(1) Test] < 90 %
30% < [% of Acc. AD F(1) Test] < 60 %
The Sources of Growth (4/4)
• Emergence of thresholds– … and if the exploitation-exploration trade-off is solved
0%
20%
40%
60%
80%
100%
0.0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1.0
ε
% o
f Acc
epta
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of A
DF(
1) T
est
A fourth question…
Does the self-sustaining growth process generated by the model
lead to explosive growth patterns? Does the variability of growth rates increase over
time and tends to infinity?
Time Evolution of GNP Growth Rate Variability
• Higher growth is always associated to smaller GR variability!
– Self sustained growth is a self-organized process leading to ordered growth patterns
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No Growt h
Mild Growt h
S S G - Low Opp.
S S G - High Opp.
A final question…
What happens in we inject in the economy more rational firms?
Irrationality as a necessary condition for growth
• Simple setup– CRTS, no info diffusion, no path-dependency– Injecting in the economy a representative rational firm (RRF) who
decides whether to exploit or explore by maximizing expected returns
– RRF knows the structure of the economy and the direction where best islands are (but not where they are)
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Irrational Individuals
Rational Individuals
A laboratory for further research
• Possible extensions – Learning– Multi-layer economies– Demand side and Keynesian cycles– Growth and development– …
• Internet resources – Go to my web-site– https://mail.sssup.it/~fagiolo– Software section– Download “Islands” setup wizard
… and have fun …