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

5

3

2

5

3

2

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

.25.25

.25

.25

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

5

3

2

Example: Exploration

Example: Exploration

4

3

2

Example: Exploration

4

3

2

1

Example: Exploration

4

3

2

1

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

4

3

2

1

Example: Imitation

4

3

2

1

Example: Imitation

4

2

2

2

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 &macro 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

-0,4

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0,4

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

100

200

300

400

500

600

1 101 201 301 401Time

GN

P

A closed economy with exploration (3/4)

• Two ex-ante equally efficient islands

0

20

40

60

1 101 201 301 401Time

Min

ers

A closed economy with exploration (4/4)

• One ex-ante more efficient island: temporary inefficiency may arise

0

20

1 101 201 Time

Min

ers

Exploration in a Open-Ended Economy

• In the full-fledged model self-sustaining growth can arise!

0

5

10

15

20

25

30

35

40

1 201 401 601 801 1001Time

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

0.2

0.4

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

nce

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

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 200 400 600 800 990Time

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)

0

100

200

300

400

500

1 400 800 1200 1600Time

GD

P

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 …