Central European University
Department of Economics
EVOLUTIONARY MODELING
OF INSTITUTIONALIZED NORMS
IN DEVELOPMENT CONTEXT
(CORRUPTION AND POLICY
CONSISTENCY MODELS).
By
Rashid Maksumov
In partial fulfilment of the requirements
for the degree of Master of Arts
Supervisor: Professor Jacek Rostowski
Budapest, Hungary
2005
For My Friends
1
Abstract.
Thesis is intended to make an overview of factors important for the development problems
of nowadays focusing on the norms of behavior and on the reasons for certain norms to appear.
Certain norms of behavior can lead to different equilibriums to persist. Construction of evolutionary
interaction models is used. Evolutionary low utility switching model is describing the intrageneration dynamics of interaction of agents based on the mechanism of threshold switching.
Conclusion is that welfare improving and inequality reduction policies can make the situation with
corruption less pronounced to certain extent. Following evolutionary models concentrate on the role
of state in the development context. State in itself is producing set of beliefs and expectations
based on which economic agents engage in certain type of interactions which considered to be best
in the current setup made by the state. Basic conclusion made is that implementation by the state of
consistent and sustainable long-term oriented policy embodied in the everyday norms of actions of
state employees and institutions will contribute to the sustainable cooperative equilibrium with
citizens and foreign investors to arise.

I express my gratitude to prof. Rostowski for the reasonable criticism and to prof. Gintis for the introduction to
powerful analytical methods.
2
Table of Contents
Introduction ………………………………………………………………….4
A. Theoretical overview ……………………………………..………………5
B. Low utility switching model ……………………………………………...12
B1. Setup and dynamics of the model …………………………………….…12
B2. Interpretation of the model’s results ………………….….……………...23
C. Policy consistency models ………….…………………………………….27
C.1. Government Norms Inertia model ……………………………….……..27
C.2. Length of Property Rights protection model ……………………………33
Concluding remarks ………………………………………………………….35
Appendix ……………………………………………………………………..36
Bibliography …………………………..……………………………………..39
3
Introduction.
Question of development always is on the agenda and always leads to controversial debates.
There are different approaches of dealing with these issues. Some time before development theories
were highly influenced by “stage” way of thinking which supposed that all the countries should
pass the same stages of development. In the recent years more broad approach to development had
place: it included issues of human capital formation, population growth, inequality, urbanization,
agricultural transformation, education, health, unemployment, etc. The approach of institutional
economics seems to be more appropriate for the analysis of development issues. Such factors as
society and dynamics of development, historical prerequisites, existing institutional norms and
country’s situation are to be taken into consideration by this approach. Certain norms and certain
types of economic order (or disorder) can constitute institutional complementarities that correspond
to certain equilibrium states. Certain norms and “the rules of the game” which are accepted by the
majority lead to more efficient equilibrium that is stable in the certain conditions particularly if
motivations and personal utility levels are high enough, this situation is described in the low utility
switching model of the work. Some “low level of efficiency” equilibriums can be defined as so
called poverty traps. In this type of equilibriums countries can be locked in the circle of poor
incentives and norms, which do not contribute to the efficiency of economic system, to spread of
innovations and to the increase in social welfare. This work intended to describe some of the
mechanisms that can lead to Pareto-inferior non-cooperative equilibriums in the society.
I start with theoretical overview of the relevant development aspects and of some literature
related to modeling approach used. The presented then model describes in evolutionary setup the
situation of multiple equilibriums introducing threshold switching mechanism and considers the
initial conditions that can lead to one or the other type of equilibrium. The role of the state sector is
considered afterwards, highlighting the importance of consistency with certain norms and
procedures.
4
A. Theoretical overview.
Development issues touched in this work are related to the reasons why certain norms
become institutionalized in certain institutional design and lead to the appearance of poverty traps
in development. Poverty traps suppose the possibility of high and low equilibriums of economic
growth and productivity. Multiple equilibriums may result from market failures, non-market
coordination problems, legal system inconsistencies, institutional deficiencies and many other
reasons.
There are 3 basic directions of explanations, which were highlighted by the recent research
as the reasons for poverty traps. These are critical thresholds, institutions, neighborhood effects1.
They describe the whole scope of possible reasons for the appearance of poverty at the scale of
economies and on the level of failed subgroups within rich countries.
Critical thresholds explanation emphasizes levels of wealth or human capital – certain level
have to be achieved before market forces will present the opportunity to the country or individual to
get out of the poverty trap. Capital market imperfections literature present one of the examples of
this direction of research. At the country level there can be investment thresholds, human capital
thresholds, general education thresholds, etc. Scale effects also can be the reason for the threshold
to appear. So basically main reason for the development traps in this approach is the scarcity of
productive opportunities for those not achieving certain threshold level. But still there is a need
rather not for one-time opportunity but rather a continuous appearance of productive possibilities
for long period of time. It can be achieved only if environment and institutional arrangements allow
for it.
1
“Poverty traps”. Edited by Samuel Bowles, Steven N. Durlauf and Karla Hoff. World Bank, 2000. P. 4
5
Institutional explanation concentrates on the social, legal and political structures to explain
how they change economic and social interactions and what is their role in the development related
issues. Some institutions can induce high inequality and corruption to become part of sustainable
equilibrium. Poor protection of property rights leads to poor performance of the markets and low
investment. Basically institutional arrangements affect the way the players of the economy interact
with each other. So if predatory behavior pays off and has advantage over the productive economic
actions in some institutional framework then it will prevail there since those trying to implement
productive economic activities will be simply driven out as their actions are not supported by the
complementary behavior of others and by certain institutional arrangements. Coordination failure
can arise if certain institutions will not support certain norms of behavior. Which way the society
will go can depend on initial conditions and on institutional design. For markets to do their job
efficiently for the prosperity of the society specific institutions are needed to support incentives for
investment and production. Otherwise, country can fall into the situation of low level of growth and
investment.
Many examples of modeling of the possible institutional processes can be found in the book
of S. Bowles2 (2004). Model of Residential Segregation3 describes the appearance of long term
Pareto inferior outcome because of small discriminatory tastes of the 2 different types of inhabitants
– tastes for small domination of one type in the population can lead to complete segregation of
neighborhoods, although by the agents’ preferences shared neighborhood is of higher utility.
Model of Cultural Evolution of Preferences4 introduces the mechanism of conformism and it allows
for the stable interior equilibrium where both types of followers of some norms are present.
Modeling approach used in the present thesis follows the approach of these models. S.Bowles
advocates the view that certain institutional evolutionary processes can be too complicated to solve
mathematically – that is why simulations should be used. With respect to institutional poverty traps
2
Bowles, Samuel (2004) Microeconomics. Behavior, Institutions, and Evolution. Princeton University Press.
Bowles, Samuel (2004) Microeconomics. Behavior, Institutions, and Evolution. Princeton University Press. P. 66
4
Bowles, Samuel (2004) Microeconomics. Behavior, Institutions, and Evolution. Princeton University Press. P. 374
3
6
Bowles (2004) presents set of the models concerning the appearance of inequality and the ways of
reducing it. Unequal conventions are modeled as self-reinforcing and Pareto inferior - they can be
changed by idiosyncratic non-best response actions of sub-population not abiding with persistent
status quo.
Neighborhood effects present one more possible explanation for appearance and persistence
of poverty. It mostly concentrates on the reasons for the individual poverty to persist emphasizing
the role of social environment and interactions in it. Certain patterns of behavior can become
persistent if concentrated in certain neighborhoods and are passed to the next generation. New
economic geography approach allows for such effects to be described with respect to geographical
neighborhoods too.
Institutional setup and social environment are changing the way people interact through the
mediation of the norms that are known and are implemented in practice by the majority members of
the society. Economic interactions also include substantial normative field. Basu (2003) emphasizes
that market process includes a lot of formal and informal rules that stipulate the actions of the
participants. For example, norms that are necessary for standard economics models to correspond to
the real processes are divided by Basu (2003) into 3 categories: rationality limiting norms,
preference changing norms and equilibrium selection norms5. K.Basu also considers the
development issues in the framework of dual economy consisting of a small industrialized sector
(“modern”, “urban” sector which operates in the same way as modern industrial economy) and an
agricultural sector. Usually large in developing countries agricultural sector has competition that
works within certain limitations created by “custom, norms, barriers to entry and law”6. Spread of
innovations in such dual economy is not high enough for sustainable economic growth.
So normative aspect is very important in the functioning of the effective economic system.
Weak norms of protecting property rights, of enforcement of economic transactions and of state
5
Basu K.(2003) Prelude to political economy. A study of the social and political foundations of economics. Oxford
University Press. P. 72-73
6
Basu K.(1997) Analytical Development Economics. MIT press. P. 249
7
sector functioning will lead to appearance of persistent institutional equilibrium that will prevent
the economic system to produce effective and social welfare-increasing outcome. One of the main
sources of the spread of harmful norms is corruption. This phenomenon exists in every country but
in some countries it is much more spread than ion others. Broad cross-national study conducted by
Treisman (1999) states that “causation runs from economic development to lower corruption as
well as from corruption to slower development”7. Economic development supposes higher levels of
personal incomes. This fact can be used as one of the justifications for the model in the section B.
Another finding relevant for the present work is provided by Treisman (1999) fact that corruption
appeared to be lower in the countries with common law based system, which is highly oriented on
the procedural aspects of conduct of the law. This orientation on the procedures and consistency in
implementing them presents good ground for the state inertia model presented in part C. One more
interesting fact stated in the same article is that federal states considered to be more corrupt than
unitary ones: one of possible explanation for that “there is a greater intimacy and frequency of
interactions between private individuals and at more decentralized levels (Tanzi 1995, Prud’homme
1995)”8 – everyday interaction with state employees is an explanation here – thus making the
models of interaction can be the right approach for the description of the spread of such
phenomenon as corruption.
As justification of the way of modeling used in the low utility switching model of present
work (see part B) concept of goals as reference points is used (Heath, Chip et al (1999)). This
concept supposes that once setting the goal at some level person’s performance can be described
with the properties of value function. Value function has 3 main properties: division of “the space
of outcomes into the regions of gain and loss”9 (reference point), loss aversion, diminishing
sensitivity. Reference point encompasses the psychological effect of setting the goal – if defined
goal (for example some level of income) is not achieved then this is subjectively reflected as loss. If
Treisman, David (1999) “The Causes of Corruption: A Cross-National Study ”, Journal of Public Economics, 76, P. 3
Treisman, David (1999) “The Causes of Corruption: A Cross-National Study ”, Journal of Public Economics,76, P. 11
9
Heath Chip, Larrick Richard, Wu George (1999). Goals as reference points, Cognitive psychology, 38, P. 82
7
8
8
goal is achieved – region higher than reference point is considered as gain region. Loss aversion
characterizes the property of value function that losses are more costly than gains – so marginal
cost of the decrease of one point in the loss area is higher than marginal cost of decrease in the gain
region. Diminishing sensitivity – is the fact that the more distant we are from reference points the
less is the marginal effect.
We can see these properties on the following chart:
Chart A.1. Value function.
Proposed in the work method of switching supposes that agents switch from law-observing
citizens to law-violating ones if their level of utility is lower than certain value  (this value of
course can be different for different individuals but in the model this detail is neglected). Making
the analogy with reference points – we can suppose that this value  have properties of inversed
reference point – particularly subjective perception of the cost of being punished marginally is
larger in the area higher than the switching point  , and smaller in the area lower than .
9
We can summarize it with the following diagram:
Chart A.2. Goal setting procedure and value function.
So we can consider the region lying higher than the point  - region where setting of the
goals happens with taking into consideration legal norms (case A in the chart A.2) – here marginal
utility of money is lower than the subjective perception of the marginal cost of being punished. In
the region lower than  (case B) goals are set without taking norms of law as something to be
strictly obeyed, marginal utility of earnings here is higher than subjectively percept marginal cost of
being punished for non-observing of laws. After the goal is set normal properties of value function
have place.
Rizzo& Zeckhauser (2003) provide econometric evidence for the effect of relative position
to reference income targets on the behavior of young physicians in US. Below income target
physicians produced stronger income growth than those higher relative to reference value. Those
below reference value try to boost their earnings by non-adequately expensive recommendations to
patients. So we see that for those below reference value marginal utility of money is rather high so
they are less sensitive to higher marginal cost of being punished for non-appealing actions.
10
Acemoglu& Verdier (2000) use cost-benefit analysis to the analysis of corruption: from one
side government intervention is needed to deal with arising market failures, but from the other side
more intervention leads to more possibilities of corruption since more bureaucrats should be hired
to gather the information about regulated areas. So any government intervention will result in some
amount of corruption – it still will be optimal until levels of corruption are low and market failures
to be solved are of high significance. Acemoglu& Verdier suggest that in growing economies
optimal level of intervention is higher because of more market failure situations, whereas in mature
developed economies optimal government intervention should be small otherwise there will appear
additional possibilities for corruption. One of their conclusions is that in developing countries
optimality analysis can be non-suitable and corruption there is harder to rationalize with optimal
government intervention considerations.
Next sections present pieces of modeling of spread of corruption due to low utility switching
and of state employees actions’ consistency as a source of Pareto improving cooperative
equilibrium.
11
B. Low utility switching model.
This model describes the situations in the normative sphere of the society those either are
more predisposed to the spread of corruption (levels of incentives for law observing are not high
enough – X-type of agents prevail), or are more favorable to the diminishing of corruption (when
levels of individual welfare are high enough than certain threshold value (reference point) – Y-type
of agents is the majority). Although majority of X or Y-type of agents do not affect the level of
social welfare – just some redistribution happens – presence of high share of X-type agents can lead
to worse economic performance and poverty increase. If for example we consider open economy
then there can be obvious reputation effect and country will be excluded from gains of openness
because of undermined reputation and wide-spread corruption. One more argument deserving
consideration is that corrupted agents do not prefer economically productive activities but rather are
involved in half-criminal and criminal transactions. Agents not “playing by the rules” create
informational noise, which prevents market forces from making their main function - most effective
distribution of resources. So such a country with higher probability can fall into the situation of
poverty trap.
B.1. Setup and dynamics of the model
Numerous agents interact in the economy. 2 types of agents are present: X and Y.
n – number of agents
p – fraction of X-type agents (we define this type of agents as more predisposed to the corruption)
1-p – fraction of Y-type agents (we assume this type of agents as less disposed to corruption –
more law observing);
p=1 can be entitled the name of “full corruption” state;
p=0 is “no corruption” state
12
Each period agents are randomly paired and interact by the following form of interaction:
X
x
( wx ; wx )
y
( wx   ; wy   )
y
( wy   ; wx   )
( wy ; wy )
where  >0, Initial state in the distribution of agents is p=0.5
We suppose that in the economic interaction X-type of agents have some advantages over
Y-type of agents (since they are less limited by some law prescriptions) so X-type gains some
amount  at the expense of Y-type if paired with Y-type. So level of wy is decreasing if this
happens and level of wx is increasing by  . We can think of wx and wy as of levels of wealth or
utility (more sophisticated view can consider them as levels of abeyance of legal norms but it
doesn’t seems to be proven real enough).
Switching from X-type to Y-type happens if wx of the agent exceeds threshold value  (>0)
– this switching mechanism is based on the theory of goals as reference values described in the
Theoretical overview.
Switching from Y-type to X-type happens if wy of the agent becomes less than threshold
value  (>0).
Calculation of the switching effects is based on the average levels of wealth:
n
pn
wx 
w
i 1
pn
xi
; wy 

i  pn 1
wyi
(1  p )n
Model developed further supposes within-generation dynamics of norm switching – agents
switch on the basis of the result of their interactions, so initial state defines the way the system will
move. Modeling of within generation dynamics of certain norms allows to consider short-run
changes of the behavior and the possible actions that will lead to change in the dynamical path of
the system from one direction to another.
Formulating the replicator dynamics equation:
p '  p  vp(1  p) by   (  by )  vp(1  p) bx   (bx   )
13
Where:
bx  pwx  (1  p)( wx   )  wx    p : expected average payoff for X-type agents from interaction
by  p(wy  )  (1  p)wy  wy  p : expected average payoff for Y-type agents from interaction
p ' p  p  vp (1  p )[ by   (  by )  bx   (bx   )]
 - fraction of population in the switching mode
 - measure of responsiveness to payoff and threshold difference
( - by ) – probability with that the Y-type agent will switch to X–type if by < 
( bx - ) – probability with that the X-type agent will switch to Y–type if bx > 
by    by   1
by    by   0
bx    b   1
x
bx    b   0
x
There are 4 main cases of the dynamical development of the system depending on the initial values
of expected payoffs. The underlying description does not present a strict mathematical solution
since there is a lot of uncertainty in the presented model about the levels of individual wealth – they
can change the way system behaves. The model is not supposed to be strictly closed in a sense that
some wealth can come exogenously from outside in the form of state financed programs – that is
why some implications about desired policies are stated. The proposed description identifies the
main directions in which system can move, it allows for the description of the possible measures
that can change the dynamics towards more favorable “non-corruption“ case.
1-st case:
bx   ; by   Situation favoring diminishing of the corruption in the society
b   1; b   0
x
y
14
p  vp(1  p)  (  bx )  vp (1  p )  (   (1  p)  wx )
bx  wx    p       (1  p)  wx  0  p  0  p
by  wy  p    what happens with inequality wy  p   as p
Proposition B.1.1. wy is falling with p
?
given that by   .
Proof of proposition B.1 is in the Appendix.
Let’ now consider the pace of fall of wy and of fall in p  to find out what effect will
prevail in the inequality by  wy  p   as p will fall. Let’s consider the case when p fall by
1
n
that is one person switched from x to y .
w y = wy - wy
 w y ? p  
w y =
1
1
( wy  wymin ) ? p  = 
n
(1  p)n  1
n
( wy -wymin ) ? 
(1  p)n  1
k=
n
>1
(1  p)n  1
wymin   as n is large enough
k ( wy -  ) ? 
2 possible conditions:
a) k ( wy - ) <   w y < p 
- wy falls with lower speed than p.
b) k ( wy -  ) >   w y > p  - wy falls with higher speed than p.
Before considering the effect of these a) and b) conditions, let’s look on the effect of p falling on
pn 1
pn
wx : wx 
w
i 1
pn
xi
; wx 
w
x
1
pn  1
15
Using the same logic as in the proof of proposition B.1.1. we can write:
wx 
1
pnwx  wxmax
[ wx ( pn  1)  wxmax ]  wx 
pn
pn  1
Let’s compare wx and wx :
wx ? wx
wx ?
pnwx  wxmax
pn  1
( pn  1) wx ? pnwx  wxmax
wxmax ? wx
wxmax  
as n is large enough
 ? wx
bx  wx    p   - we cannot make exact conclusion whether  > wx or  < wx .
Basically 2 combinations of conditions are possible here:
i) With  < wx we will have wx < wx - so wx is growing with falling p : inequality
bx  wx    p   will be sustained. So the only change can occur here from conditions a) and b):
a) by   will be sustained with falling p . So this case will lead to convergence p  0 (all
individuals switch to Y-type).
b) at some point 1-st case converges to 3-rd case ( bx   ; by   ) since by   will switch
to by   (although it can happen that with falling wy 1-b case will converge to 1-a case and p 
0 – we see positive effect on the society of more equal distribution of incomes).
ii) If  > wx we will have wx > wx - so wx is falling with falling p - it is impossible with
simultaneous fall in wy given that total sum of welfare is constant.
So summarizing the 1-st case:
1-i-a) either p  0 ;
16
1-i-b) or at some point 1-st case converges to 3-rd case where bx   ; by   (considered later).
2-nd case:
bx   ; by   Situation favoring spread of corruption in the society
b   0; b   1
x
y
p  vp(1  p)  (  by )  vp(1  p)  (   p  wy )
by  wy   p <      p  wy  0  p  0  p
bx  wx    p < 
Proposition B.1.2. wx is growing as p
given that bx   .
Proof of proposition B.2 is in the Appendix.
Let’ now consider pace of the growth of wx and of p  to find out what effect will prevail
in the inequality bx  wx    p   as p will grow. Let’s consider the case when p grow by
1
that is one person switched from y to x .
pn  1
wx = wx - wx
wx ? p 
1
1
( wxmax - wx ) ? 
n
pn  1
n
wxmax  wx ? 
pn  1
m=
n
>1
pn  1
wxmax   as n is large enough
m (  - wx ) ? 
17
2 possible conditions here:
a) m ( - wx ) <   wx < p 
b) m ( - wx ) >   wx > p 
Before considering the effect of these conditions let’s first consider the effect of p
on wy :
Following the same logic as in the proof of proposition B.1.1 we can write:
n
wy 

i  pn 1
n
wyi
n  pn
; wy 

i  pn  2
wyi
n  pn  1
wy (1  p)n  wymin
1
min
wy =
[ wy  (n  pn  1)  wy ]  wy 
(1  p)n  1
(1  p) n
We should compare wy and wy :
wy ? wy
wy ?
wy (1  p)n  wymin
(1  p)n  1
[(1  p)n 1]wy  wymin ? wy (1  p )n
wymin ? wy
wymin   as n is large enough
 ? wy
by  wy   p <  - we cannot make exact conclusion whether  > wy or  < wy .
2 basic combinations of conditions are possible here:
i) With  > wy we will have wy > wy so wy is falling with growing p. The condition
by  wy   p <  is sustained in this case. So the process here can be described from the effect of
the conditions a) and b):
18
a) bx   will be sustained with growing p . So this case will lead to convergence p  1
(all individuals switch to X-type; although it can happen that with growing wx 2-a case will
converge to 2-b case – so again as in the 1-i-b) case we see that less inequality creates condition for
more social solidarity and less corruption).
b) at some point 2-nd case converges to 3-rd case ( bx   ; by   ) since bx   will switch
to bx   .
ii) In the case with  < wy ( wy < wy ), so wy is growing it is impossible with
simultaneous growth in wx given that total sum of welfare is constant..
We have 2 possible scenarios here: 2-i-a) bx  wx    p <  and by  wy   p <  are sustained
p1
2-i-b) At some point 2-nd case converges to 3-rd case where bx   , by   (considered next) since
bx   will switch to bx   . It will happen at the level of wy and p for which wy -  p -  = 0
3-rd case:
bx   ; by  
b   1; b   1
x
y
p  vp(1  p)  [(  by )  (bx   )] = vp(1  p)  [2   (2 p  1)  wx  wy ]
bx  wx    p >      (1  p)  wx < 0
by  wy   p <      p  wy > 0
 p = vp(1  p)  [(   (1  p)  wx )  (   p  wy )]
Multiple equilibriums’ situation:
19
3a) if     (1  p)  wx >    p  wy
( bx -  >  - by ) then  p < 0  p
(Situation
favoring the diminishing of corruption in the society.)
We know from the analysis in the 1-st case that with falling p wx will either grow or will
fall but with lower speed than fall of p . So condition bx  wx    p >  will not be broken.
Now let’s consider what happens with wy : it can either fall or grow.
If wy grows with falling p then condition by  wy   p <  can be broken and 3-rd case
will converge to 1-st one. But if wx will fall with higher speed than wy will grow then 3a case can
go to 3b case.
If wy falls then if it falls with lower pace than p we will have condition by  wy   p < 
to be broken and 3a) case will converge to 1-st case. If wy is falling with the speed lower than those
of p but at the same time wy is falling with higher speed than wx is growing then it is possible that
condition     (1  p)  wx >    p  wy will change its sign at some point and case 3a will go to
case 3b.
20
Summarizing possible paths in the case 3a:
p  0 (if condition     (1  p)  wx >    p  wy is not broken: when wy will fall with
3-a-i)
lower speed than wx will grow; when wx will fall with lower speed than wy will grow; when both
wx and wy are growing – this can happen if we introduce some income growth mechanism in the
model).
3-a-ii) 3a  3b (condition     (1  p)  wx >    p  wy is broken: when wy will fall with
higher speed than wx will grow; when wx will fall with faster speed than wy will grow; when both
wx and wy are falling - this can happen if we introduce some recession effect on incomes in the
model)
3-a-iii) 3-rd case  1-st case (when wy will grow with falling p; when wy falls with lower speed
than p)
3b) if     (1  p)  wx <    p  wy ( bx -  <  - by ) then  p > 0  p
(Situation
favoring the spread of corruption in the society).
then by  wy  p <  is supported because p
but can change sign because of wy
.
With growing p wy will either fall or will grow but with lower speed than rise of p (from
the analysis in the 2-nd case). So condition by  wy  p <  will not be broken.
Let’s describe the possible movements of wx : it can either grow or fall.
If wx is falling with growing p then condition bx  wx    p >  can be broken and 3-rd
case will converge to 2-nd one. But if wy will grow with higher speed than wx will fall then 3b case
can converge to 3a case.
If wx grows then if it grows with lower speed than p we will have condition
bx  wx    p >  to be broken at some point and 3b case will converge to 2-nd case. If wx
21
grows with the speed higher than growth of p than but at the same time wx is growing with higher
speed than wy is falling then it is possible that condition     (1  p)  wx <    p  wy will
change its sign and 3b case will converge to 3a case.
Summarizing possible paths in the case 3b:
3-b-i)
p  1 (if condition     (1  p)  wx <    p  wy is not broken: when wx will grow
with lower speed than wy will fall; when wy will grow with lower speed than wx will fall; when
both wx and wy are falling – this can happen if we introduce some recession effect on incomes in
the model).
3-b-ii) 3b  3a (condition     (1  p)  wx >    p  wy is broken: when wx will grow with
higher speed than wy will fall; when wy will grow with higher speed than wx will fall; when both
wx and wy are growing - this can happen if we introduce some growth effect on incomes in the
model).
3-b-iii) 3-rd case  2-nd case ( wx is falling with growing p; when wx will grow with lower speed
than p).
4-th case:
bx   ; by  
b   0; b   0
x
y
p  vp(1  p)[ by   (  by )  bx   (bx   )]  0 Stable situation here.
Now let’s continue with interpretation and possible conclusions.
22
B.2. Interpretation of the model’s results.
Table B.1 summarizes all the possible trends depending on the initial conditions given by size of
bx and by relative to  :
Table B.1.
4 cases
1-st case: bx   ; by  
1-i-a) p  0 ( bx  wx    p >  and by  wy   p >  are sustained) ;
1-i-b) At some point 1-st case converges to 3-rd case since by   can
switch to by   .
2-nd case: bx   ; by  
2-i-a) p  1 ( bx  wx    p <  and by  wy   p <  are sustained)
2-i-b) At some point 2-nd case converges to 3-rd case since bx   can
switch to bx   .
3-rd case: bx   ; by  
3a)     (1  p)  wx >    p  wy
3-a-i)
p  0 (if condition 3a is not broken: when wy will fall with
lower speed than wx will grow; when wx will fall with lower speed
than wy will grow; when both wx and wy are growing).
3-a-ii) 3a  3b (condition 3a is broken: when wy will fall with higher
speed than wx will grow; when wx will fall with faster speed than wy
will grow; when both wx and wy are falling)
3-a-iii) 3-rd case  1-st case (when wy will grow with falling p;
when wy falls with lower speed than p)
3b)     (1  p)  wx <    p  wy
p  1 (here condition 3b is not broken: when wx will grow
with lower speed than wy will fall; when wy will grow with lower
3-b-i)
speed than wx will fall; when both wx and wy are falling)
3-b-ii) 3b  3a (condition 3b is broken: when wx will grow with
higher speed than wy will fall; when wy will grow with higher speed
than wx will fall; when both wx and wy are growing).
3-b-iii) 3-rd case  2-nd case (when wx is falling with growing p;
4-th case: bx   ; by  
when wx will grow with lower speed than p).
 p =0
Considering case 1 we find here initial conditions favoring diminishing of corruption, still
some precautionary measures can be taken not to allow for case 1-i-b) to have place that can be
23
welfare supporting policies for certain low income fractions of Y-type sub-population (we can
make real world analogy with those segments of population who have fixed income such as
teachers, policemen, etc whose moral condition have strong impact on the society as a whole). In 1i-b case positive effect of more equal distribution of wealth can be traced.
Case 2 presents opposite direction of movement – towards strengthening of the corruption
trends in the society. Low income of both X and Y types drive the society towards “full corruption”
case (p=1). Welfare improving policies here should touch practically every member of society.
Some effect can be achieved by improving welfare of X sub-population because it can lead to more
X-s switching to Y-s and the situation can converge to 3-rd case that is considered next. In case 2-ia it appears that more equal incomes between groups will lead with more probability to situation of
less corruption.
Case 3 presents 2 possible trends:
3a) – situation favoring the diminishing of corruption – when number of X-agents switching
to Y-type is more than Y-agent switching to X. Corruption will diminish substantially if welfare of
Y-s do not fall as sharply as the welfare of X, or if both welfare of X-s and Y-s id growing. It is
obvious that welfare improving policies or growth of the economy, which will increase welfare of
every member of the society will result in “non-corruption” equilibrium to appear (p=0). 3-a case
can converge to 1-st case given that welfare of Y type agents will grow or at least will not fall
sharply – so here welfare improving of the Y-type sub-population seems to be important. 3-a case
can converge to less favorable 3-b case if welfare of Y-type will fall sharply and at the same time
welfare growth of X-type is not sufficient. If welfare of X-type falling sharply and welfare of Ytype is growing slowly or if both welfare of X and Y are falling then 3-a case will go to 3-b case
which is more favorable to the spread of corruption – welfare improving policies are seems to be
important not to allow this.
3b) – situation favoring the spread of the corruption – when number of Y-agents switching
to X-type is more than X-type switching to Y. This situation and thus corruption will prevail (p 
24
1) if welfare of X-type grows with lower speed than welfare of Y type is falling. When both
welfare of X and Y are falling. Welfare improving policies with respect to X-type seems to be of
primer importance here. 3-b case can converge to 2-nd case if welfare of X-type grows with low
speed or if welfare of X is falling through time. Again welfare improvement of X-type is of
importance here. 3-b case can go to 3-a case that is more favorable one – provided that welfare of Y
type will not fall sharply relative to increase of wealth of X-type, or that welfare of X-type will not
fall sharply, or given that both welfare of X and Y are growing. Welfare improving is important to
switch from trend towards “full corruption” equilibrium to trend of towards “no corruption”
equilibrium.
Case 4 describes stable situation where no interaction lead to significant change of initial
conditions. Ones being in one of the 2 extreme equilibriums (p=0 or p=1) any idiosyncratic
spontaneous switching will lead for this equilibrium to appear provided that share of idiosyncratic
action will be strong enough to change appropriate initial condition ( bx   to bx   in the case of
p=0 ; by   to by   in the case of p=1).
Presented above dynamics can be modeled using the simulations. Model is flexible enough
for the inclusion of certain externalities for example – government agents who can also be paired
and interact with x and y type. For example let’s describe the setup of the model where state agents
play a role of the creators or destroyers of wealth for non-corruption and corruption variants
respectively:
x ,y - private agents – total number of them is equal to n
z – state agents – total number of them n*q (0<q<1)
Total number of agents: n + n*q
Forms of interaction:
1) Interaction between X and Y is the same as in the Low utility switching model:
x
x
( wx ; wx )
y
( wx   ; wy   )
y
( wy   ; wx   )
( wy ; wy )
25
2)
x
( wx ; wx )
x
z
( wx  a ;z)
(z;z)
(z; wx  a )
z – is just formal payoff here since focus is more on the effect of government agents on X and Y
z
type and on their proportion in the society.
3)
y
y
( wy ; wy )
z
( wy  b ;z)
z
(z; wy  b )
(z;z)
a<1; b<1
i. a>b>0 less corruption
ii.
a<b<0 more corruption
Replicator dynamics equation:
p'
p
vp(1  p)
vp(1  p)


by   (  by ) 
  (bx   )
2
1  q 1  q (1  q)
(1  q) 2 bx 
Description of further dynamics becomes very uncertain – that is why simulation strategy is
appropriate here. Predicted conclusion is that with case of less corruption where state agents are
creators of wealth (a>b>0) - lower levels of wealth are needed to sustain the path towards noncorruption equilibrium, whereas when government agents are destructors of wealth (a<b<0) higher
levels of wealth of X and Y agents are needed for sustaining the path of low corruption. So state
through actions of its employees changes the level of persistence of certain conditions that drive
system to one or other type of equilibrium.
Concluding generally we can state that welfare improving policies can change the path by
which the system will move from “full corruption” one to “no corruption” direction. More equal
26
distribution of wealth also leads to positive effect and with higher probability push to the direction
of diminishing corruption.
C. Policy Consistency models.
State as one of the agents of economic process affects the economic transactions which take
place. Views on the role of the state are quite diversified. In Industrial Organization and Economics
of Regulation state is presented as regulator – antitrust authority to prevent monopolistic situation
on the market. Views in the field of institutional economics touch completely different types of
arguments. For example, Basu (2003) claims that whatever state function should be also enforced
through social norms otherwise all state actions will not bring the desired outcome. This view is
based on the idea that state exists as long as the beliefs and expectations about state actions are
credible. This work describes more activist view of state policies supposing that they can affect the
way the economy works.
C.1. Government Norms Inertia model.
This model presents the adaptation of approach used in d’Artigues & Vignolo (2002).
d’Artigues & Vignolo (2002) consider Barro-Gordon monetary policy game in evolutionary setup.
Numerous private sectors and government are randomly paired and interact in the Barro-Gordon
game. The introduction of inertia in the actions of public sectors and of governments gives the
opportunity to achieve Pareto superior outcome of low inflation set by the government and of low
inflation expectations of the public sectors. Mechanics of the described here model is similar to the
one of d’Artigues & Vignolo (2002) but the sphere of implication is different one, model considers
everyday interactions of state agents (state employees) and private agents – basic conclusion is that
27
introduction of consistently inertial behavior both from the side of state employees and from the
side of private agents makes it possible to achieve Pareto superior cooperative outcome whereas the
result of one staged game suggests the non-cooperation outcome to be the equilibrium one.
Let’s start with the setup of the model. There are 2 type of agents in the model: state agents
and private agents. Paired agents play the following game for some fixed number of periods T:
Private agent action choices
State
agent
actions
C
N
C
(1/2;1/2)
(1;-1)
N
(-1;-1)
(0;0)
C – cooperate
N – not cooperate
This model supposes that cooperation creates some costs due to the efforts made, though if
both sides choose to cooperate then they both get positive utility due to complementary effect.
Formally we can express payoffs in the above normal form of the game with the following utility
functions of state and private agents:
State agent’s utility function: S = {C, N} ; C=1 , N=0
Conditional on private agent’s efforts utility function of state agent:
US =
P=0, US = – S
P=1, US = 1 – 1/2 S
Explanation for such a function of state agent is the following: if private agent is not
cooperating then state agent suffers losses of –S (they can be costs of preventing private agents
from illegal actions, costs of law enforcing, etc.). If private agent cooperates then from utility of
size 1 got by the state agent we subtract effort made but now cost of effort is only half since we
have complementary cooperation effect from the side of the private agent (utility of size 1 obtained
even in the case when state agent do not cooperate and do not put any effort can be explained on the
28
example of the acts of corruption by the state agents – they refuse to cooperate until some type of
reward is given – as the ultimate result of this state agent gets illegal type of income, and private
agent suffers losses (effort losses in our simplified framework), so this type of interactions is not
desirable for the private agents in terms of costs).
Written in one expression state agent’s utility function is : US = P – (1/2)P S
Private agent’s utility function: P = {C, N} ; C=1 , N=0
Conditional on state agent’s efforts utility function of private agent:
UP =
S=0, UP = – P
S=1, UP = (-1)1+P – 1/2 P
Explanation for such a function of private agent is the following: if state agent is not
cooperating then private agent suffers losses of –P. If state agent cooperates then depending on
whether private agent decides to cooperate or not, he/she can get utility of size 1 if decides to
cooperate or disutility of -1 if decides not to cooperate. From the obtained utility or disutility we
subtract effort made but now cost of effort is only half since we account for complementary
cooperation effect because state cooperates in this case. Considered in the setup payoffs of the
game are not symmetric since we suppose here that state agents have certain advantages having the
state authority network in their disposition (simply on average they can have more opportunities in
the considered interaction which seems to be reasonable assumption, so even if state agent chooses
not cooperate option he/she still might have other instruments to influence the private agents – such
situation can happen with high probability in the reality if system of legal enforcement is weak
enough).
Written in one expression private agent’s utility function is:
UP = (-1)S+P S – (1/2)S P
In this setup we can think of “cooperate” and “not cooperate” options as of choosing to act
consistently according the rules or certain norms in the case of “cooperate”, and choosing to break
the rules, norms or laws in the case of “not cooperate“ option.
29
In one stage game the only Nash equilibrium is NN. Now let’s consider evolutionary setup.
There are number of agents in the economy:
n - number of government agents denoted by set I={1, …, i ,…n}
n - number of private agents denoted by set J={1,…, j , …n}
Each T periods agents are randomly paired and play the game T times. After T periods passed
agents are randomly paired again and new pair again play for T periods and so on.
Private agent action choices
State
agent
C
actions
N
C
(1/2;1/2)
(1;-1)
N
(-1;-1)
(0;0)
Players can choose from two possible actions:
C – cooperate
N – not cooperate
The dynamic process  defines the set of used actions:
 = {z={zI, zJ }: 0 zI  n, 0 zJ  n },
where for each moment of time t , zt ={ztI, ztJ} denotes the number of government agents ztI and
the number of private players ztJ who have used the strategy C at period t. Let’s define state zt’
when all agents in pairs play C as zt’={C, C}, and state zt” when all pairs play N as zt”={N, N}.
The conduct of the evolutionary process happens basically due to two mechanisms: learning
process and mutation. Let’s describe these mechanisms.
Learning process supposes that agents do not use expectations to make their moves but
rather use their past experience, this is quite realistic assumption for the consideration of bounded
rational agents. State agents in their learning follow imitation process: their set of information
supposed to include the performance of other government agents at the current time t . They
compare their performance with the performance of others. The strategy that gave the highest
payoff is then imitated. So at every t+1 period each government finds maximal payoff in Uit which
30
presents the accumulated sum of payoffs from the beginning of the new T period and then imitates
the strategy of government i with highest Ut.
Private agents in their actions follow simple best reply logic based on the action of state
agent at time t-1 with whom they are in pair: strategy is figured out from maximizing utility
Uj
(pj, git-1), where pj is action of private agent j and git-1 is action of state agent i at time t-1 this type
of decision is made at any period but not at first after random pairing was made (first period of the
next T period). At first period after random pairing private agents use maximization of expected
value of utility based on the probability to meet one or the other type of state agents i Ujt (pj, gt-1) =
iI qijUj (pj, gi t-1) where qji is the probability for private agent j to meet state agent i ( qij= 1/n
since pairing is random).
Necessary important assumption here is assumption of inertia in the government agents
learning process and private agents best-reply process: it supposes that agents do not update their
payoffs all the time but rather with certain probabilities: i for government agents and j for the
private agents. Inertia presence can be explained by certain norms of behavior, institutionalized
procedures, habits, it will be crucial for the possibility of sustainable cooperative equilibrium. The
degrees of inertia are measured by (1 - i) and (1 - j).
Proposition C.1.1. The states z’={C, C} and z”={N, N} are the only stationary states of the
outlined evolutionary game. Proof following d’Artigues & Vignolo (2002) is given in the
Appendix.
Process of mutation is introduced to have perturbations in the selection of equilibrium
process –they will allow the surpassing from one stationary state to the other one. With some small
probability  each agent chooses action randomly. If mutation is introduced (>0) we have some
probability transition matrix P()=(pzz’()) over  from state z to z’. Matrix  - is stationary
distribution if  P() =  . Limit distribution in the long run will be:
* = lim 0 
31
Long run equilibriums following d’Artigues & Vignolo (2002) are whose that have positive
value in the limit distribution:
LR ={z | *(z)>0}
When i = j.=1 (no inertia) the long-run equilibrium of the game is z’’= (N,N). Because if
we consider probabilities pz’z’’ and pz’’z’ (pz’z’’ - probability of transition from z’ to z’’, and pz’’z’ –
reverse), the transition from z’=(C, C) to z’’=(N, N) requires only one mutation (pz’z’’= ) and
transition from z’’ to z’ requires at least 2 mutations (pz’’z’= 2) . So pz’z’’ > pz’’z’ as  0 and state
z’’=(N,N) – is the long-run equilibrium.
But with 0<i ,j.<1 result is not directly conclusive. d’Artigues & Vignolo (2002) use the
following proposition:
Proposition C.1.2. In the outlined evolutionary game, state z’ can be reached from z’’ with only
one mutation when inertia is included in the learning process, that is when 0 < i , j < 1.
Proof following d’Artigues & Vignolo (2002) is given in the Appendix.
Proposition C.1.2 makes it possible for z’=(C, C) to be one of the long-run stationary states,
particularly it happens in when transition from z’ to z’’ becomes harder and transition from z’’ to z’
becomes easier. It can take place if when the state agents inertia is much higher than the private
agents inertia ( 1- i >> 1 - j) and T is large enough.
Conclusion here is that provided government agents are consistent in their actions (we can
interpret it as commitment to certain norms of behavior) the cooperative equilibrium can be
achieved in the evolutionary setup where government agents imitate the best strategy played by
state agents and the state agents are highly consistent in their behavior (have high inertia – do not
change their beliefs and actions easily, follow consistently with defined procedures), where private
agents follow the best–reply logic and react flexibly to the actions of the state agents ( low inertia ).
Setup is quite close to reality where private agents are usually more flexible in their actions, and the
result emphasizes importance of the consistency of government agents’ actions – observing certain
32
norms and procedures reliably can be the advice for government agents to achieve the long-run
cooperative equilibrium in the long-run.
C.2. Length of Property Rights protection model.
The logic underlying this model is based on the relation between interests of potential
foreign investors and length of property rights protection that states can provide. If property rights
are protected weakly so long-term investments cannot be properly protected then investors come
with the only aim of “creaming out” of some high profit sectors without long term engagement.
This attitude is not beneficial to states that put efforts in attempt to provide good environment for
investments because short-term “creaming out” do not produce long-term positive effects on the
economy. So government seeing such an attitude of the investors can give providing the best
conditions. The same situation will happen if government do not provide adequate protection of
property rights – investors respond by taking less risky short-term excessive profit extractive
activities. So non-cooperative equilibrium will appear in such situations and will be sustained.
Game describing the above considerations is as follows:
Investors action choices
State
action
choices
H
L
L
(2;2)
(0;0)
H
(0;1)
(1;1)
For the State:
H - high degree of property rights protection ;
L - low degree of property rights protection.
For the Investors:
H – high profit margins (“creaming out”) ;
33
L - low profit margins (“stable long-term investments”).
Sustainable equilibrium in one shot game is: LH (Pareto inferior). We see that cooperative
HL equilibrium brings higher payoffs comparing to LH. If we use the framework similar to one in
Government norms inertia model we can find the way how state should act to enforce cooperative
equilibrium.
There is number of governments and foreign investors interacting in the world:
n - number of governments denoted by set I={1, …, i ,…n}
n - number of foreign investors denoted by set J={1,…, j , …n}
Each T periods participants are randomly paired and play the game T times. After T periods
passed participants are randomly paired again and new pair again play for T periods and so on.
Considering the same evolutionary setup as in the model in part C.1 we can make the
following conclusions: State actions consistency, expectations of state consistency by the investors
will make Pareto-superior HL equilibrium sustainable in the long run. Implementation of certain
consistency in state policy makes it possible for Pareto superior outcome to become sustainable in
the long run – it brings more benefits both to the state and to the investors. Thus credible and
sustainable following to adequate norms will result in credibility and credibility is important to
create confidence for investors in order for long-run cooperation to arise and to be supported.
34
Concluding remarks.
Modeling approach used in the low utility switching model presents the phenomenon on
corruption as norm that can spread if level of motivation for law observing behavior is not high
enough. We saw that the way to improve the situation is to implement welfare improving policies
and to reduce levels of inequalities in the society. Of course such an approach do not give the full
picture of the complex phenomenon of corruption, but still model describes the possibility of
environment where corruption can easily occur since norm of weak abeyance of laws can become
prevailing among the majority in the society. Such situation cannot contribute to the appearance of
positive effect of market forces because for efficient functioning market economy there is a need to
have certain normative component in the behavior. Low utility switching model presents the
modeling of the appearance of normative equilibriums through everyday interactions. This type of
models can be one of the prospects in the evolutionary theory.
Government norms inertia model considers the effect of consistent implementation of state
policies and procedures by the state employees in everyday interaction. Consistent following of
some norms and procedures with sufficient inertia will lead to cooperative equilibrium with private
agents to arise. The same approach is used in the Length of property rights protection model in
which state interacts in the game with foreign investors. The conclusion is again that credible and
consistent implementation of policy will make the cooperative equilibrium sustainable in the longrun. This is the way to create positive expectations of citizens and foreign investors about state
activities leading to optimality of conducting the cooperative actions.
The role of incentive aspect and of commitment aspect in the analysis of development
problems were highlighted in the present work. Prospectively many other phenomena can be
modeled with use of evolutionary approach to everyday interactions of people and institutions.
35
Appendix.
given that by   .
Proposition B.1.1. wy is falling with p
Proof:
Let’s consider wy as next period average wealth of y-type agents with change in the number of ytype agents equal to +1:
n
wy 
= wy 

i  pn 1
n
wyi
n  pn
; wy =
n

 wyi
i  pn
n  pn  1
=
wyi
wymin
n  pn
+
=
n  pn (1  p )n  1 (1  p) n  1
i  pn 1

(1  p)n
1
1
+
( wy  (1  p)n  wymin )
 wymin =
(1  p)n  1 (1  p)n  1
(1  p)n  1
Let’s compare wy and wy :
wy ? wy
wy ?
1
( wy  (1  p )n  wymin )
(1  p )n  1
wy (1 
(1  p)n
1
)?
wymin
(1  p)n  1
(1  p )n  1
1
1
wy ?
wymin
(1  p)n  1
(1  p)n  1
wy ? wymin
wymin   as n is large enough
36
wy   (since by   , by  wy  p    wy  p    wy   since p >0 
wy  wy , so wy is falling with falling p. Q.E.D.
Proposition B.1.2. wx is growing with p
given that bx   .
Proof:
pn
wx 
w
xi
i 1
pn
;
Let’s consider wx as next period average wealth of x-type agents with change in the number of xtype agents equal to +1:
pn 1
wx =
pn
 wxi
i 1
pn  1
w
=
i 1
xi
pn
Lets compare wx and

pn
1
1
wmax
pn
+ x = wx 
+
( wx  pn + wxmax )
 wxmax =
pn  1 pn  1
pn  1
pn  1 pn
wx :
wx ?
wx
wx ?
1
( wx  pn  wxmax )
pn  1
wx (1 
pn
1
) ?
wxmax
pn  1
pn  1
wx ? wxmax
wxmax   as n is large enough
wx <  (since bx   , bx  wx    p <   wx + (   p )<  wx <  since   p >0) 
37
wx < wx  wx is growing with growing p .
Q.E.D.
Proposition C.1.1. The states z’={C, C} and z”={N, N} are the only stationary states of the
outlined evolutionary game (part C.1.).
Proof (following d’Artigues & Vignolo (2002)):
Assume that system is in the state z’’’ which is different from z’ and z’’. Then both strategies N and
C are played. In this case, government agents are leaded by the imitative dynamics to the strategy
of the most successful player in I, which can be N or C depending on the opponent strategy
confronted. As the private agent’s best response to each of the two government agent’s strategies is
unique (N if N and C if C), the system yields z’=(C,C) or z’’=(N,N).
Q.E.D.
Proposition C.1.2. In the outlined evolutionary game (part C.1), state z’ can be reached from z’’
with only one mutation when inertia is included in the learning process, that is when 0 < i , j < 1.
Proof (following d’Artigues & Vignolo (2002)):
Assume that the dynamics is in the state z’’=(N,N). A single mutant government agent using C can
generate imitation in I if this player, after the mutation, does not change his/her strategy
(probability 1- i) and at the same time the private agent he/she faces responds rapidly (probability
j). Thus the transition from z’’=(N,N) to z’=(C,C) can be carried out with only one mutation C in
the set of government agents.
Q.E.D.
38
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