Minor update

2018-05-11 16:58:55 -04:00
parent c4865d0ada
commit 7b054b0218
10 changed files with 263 additions and 71 deletions
@@ -102,11 +102,7 @@ Automatically generated HTML file from DocOnce source
               None,
               '___sec35'),
              ('Simulating financial transcations', 2, None, '___sec36'),
-              ('Project 4a): Simulation of Transactions', 3, None, '___sec37'),
-              ('Project 4b): Recognizing the distribution',
-               3,
-               None,
-               '___sec38')]}
+              ('Simulation of Transactions', 3, None, '___sec37')]}
 end of tocinfo -->

 <body>
@@ -181,8 +177,7 @@ MathJax.Hub.Config({
     <!-- navigation toc: --> <li><a href="#___sec34" style="font-size: 80%;"><b>Parameter estimation</b></a></li>
     <!-- navigation toc: --> <li><a href="#___sec35" style="font-size: 80%;"><b>A program relevant for the biological problem</b></a></li>
     <!-- navigation toc: --> <li><a href="#___sec36" style="font-size: 80%;"><b>Simulating financial transcations</b></a></li>
-     <!-- navigation toc: --> <li><a href="#___sec37" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Project 4a): Simulation of Transactions</a></li>
-     <!-- navigation toc: --> <li><a href="#___sec38" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Project 4b): Recognizing the distribution</a></li>
+     <!-- navigation toc: --> <li><a href="#___sec37" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Simulation of Transactions</a></li>

        </ul>
      </li>
@@ -216,7 +211,7 @@ MathJax.Hub.Config({
 <center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
 <br>
 <p>
-<center><h4>May 9, 2018</h4></center> <!-- date -->
+<center><h4>May 11, 2018</h4></center> <!-- date -->
 <br>
 <p>
 <!-- potential-jumbotron-button -->
@@ -1412,7 +1407,7 @@ Change <code>r</code> in the program and play around to make a better fit!
 <h2 id="___sec36" class="anchor">Simulating financial transcations </h2>

 <p>
-The aim of this project is to simulate financial transactions among financial agents
+The aim here is to simulate financial transactions among financial agents
 using Monte Carlo methods. The final goal is to extract a distribution of income  as function
 of the income \( m \).   From Pareto's work (<a href="http://www.institutcoppet.org/2012/05/08/cours-deconomie-politique-1896-de-vilfredo-pareto" target="_self">V.&nbsp;Pareto, 1897</a>) it is known from empirical studies
 that the higher end of the distribution of money follows a distribution 
@@ -1483,9 +1478,9 @@ exponentially decreases with \( m' \).
 We assume that we have \( N=500 \) agents.   In each simulation, we need a sufficiently large number of transactions, say \( 10^7 \). Our aim is find the final equilibrium distribution \( w_m \). In order to do that we would need
 several runs of the above simulations, at least \( 10^3-10^4 \) runs (experiments).

-<h3 id="___sec37" class="anchor">Project 4a): Simulation of Transactions   </h3>
+<h3 id="___sec37" class="anchor">Simulation of Transactions   </h3>

-Your task is to first set up an algorithm which simulates the above transactions with an initial
+Our task is to first set up an algorithm which simulates the above transactions with an initial
  amount \( m_0 \).
  The challenge here is to figure out a Monte Carlo  simulation  based on the
  above equations.
@@ -1496,12 +1491,45 @@ Your task is to first set up an algorithm which simulates the above transactions
  \( m,m+\Delta m \). The number of times you register this income, represents the value that enters the histogram.
  You will also need to find a criterion for when the equilibrium situation has been reached.

-<h3 id="___sec38" class="anchor">Project 4b): Recognizing the distribution   </h3>
+<p>

-Make thereafter a plot of  \( \log{(w_m)} \) as function of \( m \)
-  and see if you get a straight line.
-  Comment the result.
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic">#!/usr/bin/env python</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.mlab</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mlab</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">random</span>

+<span style="color: #408080; font-style: italic"># initialize the rng with a seed</span>
+random<span style="color: #666666">.</span>seed()
+<span style="color: #408080; font-style: italic"># Hard coding of input parameters</span>
+Agents  <span style="color: #666666">=</span> <span style="color: #666666">500</span>
+MCcounts <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+Transactions <span style="color: #666666">=</span> <span style="color: #666666">100000</span>
+startMoney <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
+Lambda <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+FinancialAgents <span style="color: #666666">=</span> startMoney<span style="color: #666666">*</span>np<span style="color: #666666">.</span>ones(Agents)
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span> (<span style="color: #666666">1</span>, MCcounts, <span style="color: #666666">1</span>):
+    <span style="color: #008000; font-weight: bold">for</span> j <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span> (<span style="color: #666666">1</span>, Transactions, <span style="color: #666666">1</span>):
+        agent_i <span style="color: #666666">=</span> <span style="color: #008000">int</span>(Agents<span style="color: #666666">*</span>random<span style="color: #666666">.</span>random())
+        agent_j <span style="color: #666666">=</span> <span style="color: #008000">int</span>(Agents<span style="color: #666666">*</span>random<span style="color: #666666">.</span>random())
+        epsilon <span style="color: #666666">=</span> random<span style="color: #666666">.</span>random()
+        <span style="color: #008000; font-weight: bold">if</span> agent_i <span style="color: #666666">!=</span> agent_j:
+           m1 <span style="color: #666666">=</span> Lambda<span style="color: #666666">*</span>FinancialAgents[agent_i] <span style="color: #666666">+</span> (<span style="color: #666666">1-</span>Lambda)<span style="color: #666666">*</span>epsilon<span style="color: #666666">*</span>(FinancialAgents[agent_i] <span style="color: #666666">+</span> FinancialAgents[agent_j])
+           m2 <span style="color: #666666">=</span> Lambda<span style="color: #666666">*</span>FinancialAgents[agent_j] <span style="color: #666666">+</span> (<span style="color: #666666">1-</span>Lambda)<span style="color: #666666">*</span>(<span style="color: #666666">1-</span>epsilon)<span style="color: #666666">*</span>(FinancialAgents[agent_i] <span style="color: #666666">+</span> FinancialAgents[agent_j])
+           FinancialAgents[agent_i] <span style="color: #666666">=</span> m1
+           FinancialAgents[agent_j] <span style="color: #666666">=</span> m2
+
+<span style="color: #408080; font-style: italic"># the histogram of the data</span>
+n, bins, patches <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>hist(FinancialAgents, <span style="color: #666666">50</span>, facecolor<span style="color: #666666">=</span><span style="color: #BA2121">&#39;green&#39;</span>)
+
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;$x$&#39;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;Distribution of wealth&#39;</span>)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r&#39;Money&#39;</span>)
+plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>, <span style="color: #666666">10</span>, <span style="color: #666666">0</span>, <span style="color: #666666">500</span>])
+plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
+plt<span style="color: #666666">.</span>show()
+</pre></div>
 <p>
 We can then change our model to allow for a saving criterion, meaning that the agents save
  a fraction \( \lambda \) of the money they have before the transaction is made. The final distribution will then no longer be given by Gibbs distribution. It could also include a taxation on financial transactions.
@@ -1554,7 +1582,6 @@ $$
 Extract a parametrization of the above curves, see for example <a href="http://www.sciencedirect.com/science/article/pii/S0378437104004327" target="_self">Patriarca and collaborators</a> and see if you can parametrize the high-end tails of the distributions in terms of power laws. Comment your results.

 <p>
-In the rest of this project we will follow the work of <a href="http://www.sciencedirect.com/science/article/pii/S0378437114006967" target="_self">Goswami and Sen</a>. 
 In the studies above the agents were selected randomly, irrespective of whether we allowed for
 saving or not during a transaction. What is often observed is that various agents tend to make preferences for for whom to interact with. We will now study the evolution of the distribution of wealth \( w_m \) by assuming that there is a likelihood 
 $$
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
 <center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
 <br>
 <p>&nbsp;<br>
-<center><h4>May 9, 2018</h4></center> <!-- date -->
+<center><h4>May 11, 2018</h4></center> <!-- date -->
 <br>
 <p>

@@ -1310,7 +1310,7 @@ Change <code>r</code> in the program and play around to make a better fit!
 <h2 id="___sec36">Simulating financial transcations </h2>

 <p>
-The aim of this project is to simulate financial transactions among financial agents
+The aim here is to simulate financial transactions among financial agents
 using Monte Carlo methods. The final goal is to extract a distribution of income  as function
 of the income \( m \).   From Pareto's work (<a href="http://www.institutcoppet.org/2012/05/08/cours-deconomie-politique-1896-de-vilfredo-pareto" target="_blank">V.&nbsp;Pareto, 1897</a>) it is known from empirical studies
 that the higher end of the distribution of money follows a distribution 
@@ -1393,9 +1393,9 @@ exponentially decreases with \( m' \).
 We assume that we have \( N=500 \) agents.   In each simulation, we need a sufficiently large number of transactions, say \( 10^7 \). Our aim is find the final equilibrium distribution \( w_m \). In order to do that we would need
 several runs of the above simulations, at least \( 10^3-10^4 \) runs (experiments).

-<h3 id="___sec37">Project 4a): Simulation of Transactions   </h3>
+<h3 id="___sec37">Simulation of Transactions   </h3>

-Your task is to first set up an algorithm which simulates the above transactions with an initial
+Our task is to first set up an algorithm which simulates the above transactions with an initial
  amount \( m_0 \).
  The challenge here is to figure out a Monte Carlo  simulation  based on the
  above equations.
@@ -1406,12 +1406,45 @@ Your task is to first set up an algorithm which simulates the above transactions
  \( m,m+\Delta m \). The number of times you register this income, represents the value that enters the histogram.
  You will also need to find a criterion for when the equilibrium situation has been reached.

-<h3 id="___sec38">Project 4b): Recognizing the distribution   </h3>
+<p>

-Make thereafter a plot of  \( \log{(w_m)} \) as function of \( m \)
-  and see if you get a straight line.
-  Comment the result.
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #228B22">#!/usr/bin/env python</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.mlab</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">mlab</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">random</span>

+<span style="color: #228B22"># initialize the rng with a seed</span>
+random.seed()
+<span style="color: #228B22"># Hard coding of input parameters</span>
+Agents  = <span style="color: #B452CD">500</span>
+MCcounts = <span style="color: #B452CD">1000</span>
+Transactions = <span style="color: #B452CD">100000</span>
+startMoney = <span style="color: #B452CD">1.0</span>
+Lambda = <span style="color: #B452CD">0.0</span>
+FinancialAgents = startMoney*np.ones(Agents)
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span> (<span style="color: #B452CD">1</span>, MCcounts, <span style="color: #B452CD">1</span>):
+    <span style="color: #8B008B; font-weight: bold">for</span> j <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span> (<span style="color: #B452CD">1</span>, Transactions, <span style="color: #B452CD">1</span>):
+        agent_i = <span style="color: #658b00">int</span>(Agents*random.random())
+        agent_j = <span style="color: #658b00">int</span>(Agents*random.random())
+        epsilon = random.random()
+        <span style="color: #8B008B; font-weight: bold">if</span> agent_i != agent_j:
+           m1 = Lambda*FinancialAgents[agent_i] + (<span style="color: #B452CD">1</span>-Lambda)*epsilon*(FinancialAgents[agent_i] + FinancialAgents[agent_j])
+           m2 = Lambda*FinancialAgents[agent_j] + (<span style="color: #B452CD">1</span>-Lambda)*(<span style="color: #B452CD">1</span>-epsilon)*(FinancialAgents[agent_i] + FinancialAgents[agent_j])
+           FinancialAgents[agent_i] = m1
+           FinancialAgents[agent_j] = m2
+
+<span style="color: #228B22"># the histogram of the data</span>
+n, bins, patches = plt.hist(FinancialAgents, <span style="color: #B452CD">50</span>, facecolor=<span style="color: #CD5555">&#39;green&#39;</span>)
+
+plt.xlabel(<span style="color: #CD5555">&#39;$x$&#39;</span>)
+plt.ylabel(<span style="color: #CD5555">&#39;Distribution of wealth&#39;</span>)
+plt.title(<span style="color: #CD5555">r&#39;Money&#39;</span>)
+plt.axis([<span style="color: #B452CD">0</span>, <span style="color: #B452CD">10</span>, <span style="color: #B452CD">0</span>, <span style="color: #B452CD">500</span>])
+plt.grid(<span style="color: #658b00">True</span>)
+plt.show()
+</pre></div>
 <p>
 We can then change our model to allow for a saving criterion, meaning that the agents save
  a fraction \( \lambda \) of the money they have before the transaction is made. The final distribution will then no longer be given by Gibbs distribution. It could also include a taxation on financial transactions.
@@ -1474,7 +1507,6 @@ $$
 Extract a parametrization of the above curves, see for example <a href="http://www.sciencedirect.com/science/article/pii/S0378437104004327" target="_blank">Patriarca and collaborators</a> and see if you can parametrize the high-end tails of the distributions in terms of power laws. Comment your results.

 <p>
-In the rest of this project we will follow the work of <a href="http://www.sciencedirect.com/science/article/pii/S0378437114006967" target="_blank">Goswami and Sen</a>. 
 In the studies above the agents were selected randomly, irrespective of whether we allowed for
 saving or not during a transaction. What is often observed is that various agents tend to make preferences for for whom to interact with. We will now study the evolution of the distribution of wealth \( w_m \) by assuming that there is a likelihood 
 <p>&nbsp;<br>
@@ -122,11 +122,7 @@ div { text-align: justify; text-justify: inter-word; }
               None,
               '___sec35'),
              ('Simulating financial transcations', 2, None, '___sec36'),
-              ('Project 4a): Simulation of Transactions', 3, None, '___sec37'),
-              ('Project 4b): Recognizing the distribution',
-               3,
-               None,
-               '___sec38')]}
+              ('Simulation of Transactions', 3, None, '___sec37')]}
 end of tocinfo -->

 <body>
@@ -168,7 +164,7 @@ MathJax.Hub.Config({
 <center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
 <br>
 <p>
-<center><h4>May 9, 2018</h4></center> <!-- date -->
+<center><h4>May 11, 2018</h4></center> <!-- date -->
 <br>
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1330,7 +1326,7 @@ Change <code>r</code> in the program and play around to make a better fit!
 <h2 id="___sec36">Simulating financial transcations </h2>

 <p>
-The aim of this project is to simulate financial transactions among financial agents
+The aim here is to simulate financial transactions among financial agents
 using Monte Carlo methods. The final goal is to extract a distribution of income  as function
 of the income \( m \).   From Pareto's work (<a href="http://www.institutcoppet.org/2012/05/08/cours-deconomie-politique-1896-de-vilfredo-pareto" target="_blank">V.&nbsp;Pareto, 1897</a>) it is known from empirical studies
 that the higher end of the distribution of money follows a distribution 
@@ -1401,9 +1397,9 @@ exponentially decreases with \( m' \).
 We assume that we have \( N=500 \) agents.   In each simulation, we need a sufficiently large number of transactions, say \( 10^7 \). Our aim is find the final equilibrium distribution \( w_m \). In order to do that we would need
 several runs of the above simulations, at least \( 10^3-10^4 \) runs (experiments).

-<h3 id="___sec37">Project 4a): Simulation of Transactions   </h3>
+<h3 id="___sec37">Simulation of Transactions   </h3>

-Your task is to first set up an algorithm which simulates the above transactions with an initial
+Our task is to first set up an algorithm which simulates the above transactions with an initial
  amount \( m_0 \).
  The challenge here is to figure out a Monte Carlo  simulation  based on the
  above equations.
@@ -1414,12 +1410,45 @@ Your task is to first set up an algorithm which simulates the above transactions
  \( m,m+\Delta m \). The number of times you register this income, represents the value that enters the histogram.
  You will also need to find a criterion for when the equilibrium situation has been reached.

-<h3 id="___sec38">Project 4b): Recognizing the distribution   </h3>
+<p>

-Make thereafter a plot of  \( \log{(w_m)} \) as function of \( m \)
-  and see if you get a straight line.
-  Comment the result.
+<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
+<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #228B22">#!/usr/bin/env python</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.mlab</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">mlab</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
+<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">random</span>

+<span style="color: #228B22"># initialize the rng with a seed</span>
+random.seed()
+<span style="color: #228B22"># Hard coding of input parameters</span>
+Agents  = <span style="color: #B452CD">500</span>
+MCcounts = <span style="color: #B452CD">1000</span>
+Transactions = <span style="color: #B452CD">100000</span>
+startMoney = <span style="color: #B452CD">1.0</span>
+Lambda = <span style="color: #B452CD">0.0</span>
+FinancialAgents = startMoney*np.ones(Agents)
+<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span> (<span style="color: #B452CD">1</span>, MCcounts, <span style="color: #B452CD">1</span>):
+    <span style="color: #8B008B; font-weight: bold">for</span> j <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span> (<span style="color: #B452CD">1</span>, Transactions, <span style="color: #B452CD">1</span>):
+        agent_i = <span style="color: #658b00">int</span>(Agents*random.random())
+        agent_j = <span style="color: #658b00">int</span>(Agents*random.random())
+        epsilon = random.random()
+        <span style="color: #8B008B; font-weight: bold">if</span> agent_i != agent_j:
+           m1 = Lambda*FinancialAgents[agent_i] + (<span style="color: #B452CD">1</span>-Lambda)*epsilon*(FinancialAgents[agent_i] + FinancialAgents[agent_j])
+           m2 = Lambda*FinancialAgents[agent_j] + (<span style="color: #B452CD">1</span>-Lambda)*(<span style="color: #B452CD">1</span>-epsilon)*(FinancialAgents[agent_i] + FinancialAgents[agent_j])
+           FinancialAgents[agent_i] = m1
+           FinancialAgents[agent_j] = m2
+
+<span style="color: #228B22"># the histogram of the data</span>
+n, bins, patches = plt.hist(FinancialAgents, <span style="color: #B452CD">50</span>, facecolor=<span style="color: #CD5555">&#39;green&#39;</span>)
+
+plt.xlabel(<span style="color: #CD5555">&#39;$x$&#39;</span>)
+plt.ylabel(<span style="color: #CD5555">&#39;Distribution of wealth&#39;</span>)
+plt.title(<span style="color: #CD5555">r&#39;Money&#39;</span>)
+plt.axis([<span style="color: #B452CD">0</span>, <span style="color: #B452CD">10</span>, <span style="color: #B452CD">0</span>, <span style="color: #B452CD">500</span>])
+plt.grid(<span style="color: #658b00">True</span>)
+plt.show()
+</pre></div>
 <p>
 We can then change our model to allow for a saving criterion, meaning that the agents save
  a fraction \( \lambda \) of the money they have before the transaction is made. The final distribution will then no longer be given by Gibbs distribution. It could also include a taxation on financial transactions.
@@ -1472,7 +1501,6 @@ $$
 Extract a parametrization of the above curves, see for example <a href="http://www.sciencedirect.com/science/article/pii/S0378437104004327" target="_blank">Patriarca and collaborators</a> and see if you can parametrize the high-end tails of the distributions in terms of power laws. Comment your results.

 <p>
-In the rest of this project we will follow the work of <a href="http://www.sciencedirect.com/science/article/pii/S0378437114006967" target="_blank">Goswami and Sen</a>. 
 In the studies above the agents were selected randomly, irrespective of whether we allowed for
 saving or not during a transaction. What is often observed is that various agents tend to make preferences for for whom to interact with. We will now study the evolution of the distribution of wealth \( w_m \) by assuming that there is a likelihood 
 $$
@@ -127,11 +127,7 @@ div { text-align: justify; text-justify: inter-word; }
               None,
               '___sec35'),
              ('Simulating financial transcations', 2, None, '___sec36'),
-              ('Project 4a): Simulation of Transactions', 3, None, '___sec37'),
-              ('Project 4b): Recognizing the distribution',
-               3,
-               None,
-               '___sec38')]}
+              ('Simulation of Transactions', 3, None, '___sec37')]}
 end of tocinfo -->

 <body>
@@ -173,7 +169,7 @@ MathJax.Hub.Config({
 <center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
 <br>
 <p>
-<center><h4>May 9, 2018</h4></center> <!-- date -->
+<center><h4>May 11, 2018</h4></center> <!-- date -->
 <br>
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1335,7 +1331,7 @@ Change <code>r</code> in the program and play around to make a better fit!
 <h2 id="___sec36">Simulating financial transcations </h2>

 <p>
-The aim of this project is to simulate financial transactions among financial agents
+The aim here is to simulate financial transactions among financial agents
 using Monte Carlo methods. The final goal is to extract a distribution of income  as function
 of the income \( m \).   From Pareto's work (<a href="http://www.institutcoppet.org/2012/05/08/cours-deconomie-politique-1896-de-vilfredo-pareto" target="_blank">V.&nbsp;Pareto, 1897</a>) it is known from empirical studies
 that the higher end of the distribution of money follows a distribution 
@@ -1406,9 +1402,9 @@ exponentially decreases with \( m' \).
 We assume that we have \( N=500 \) agents.   In each simulation, we need a sufficiently large number of transactions, say \( 10^7 \). Our aim is find the final equilibrium distribution \( w_m \). In order to do that we would need
 several runs of the above simulations, at least \( 10^3-10^4 \) runs (experiments).

-<h3 id="___sec37">Project 4a): Simulation of Transactions   </h3>
+<h3 id="___sec37">Simulation of Transactions   </h3>

-Your task is to first set up an algorithm which simulates the above transactions with an initial
+Our task is to first set up an algorithm which simulates the above transactions with an initial
  amount \( m_0 \).
  The challenge here is to figure out a Monte Carlo  simulation  based on the
  above equations.
@@ -1419,12 +1415,45 @@ Your task is to first set up an algorithm which simulates the above transactions
  \( m,m+\Delta m \). The number of times you register this income, represents the value that enters the histogram.
  You will also need to find a criterion for when the equilibrium situation has been reached.

-<h3 id="___sec38">Project 4b): Recognizing the distribution   </h3>
+<p>

-Make thereafter a plot of  \( \log{(w_m)} \) as function of \( m \)
-  and see if you get a straight line.
-  Comment the result.
+<!-- code=python (!bc pycod) typeset with pygments style "default" -->
+<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic">#!/usr/bin/env python</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.mlab</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mlab</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
+<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">random</span>

+<span style="color: #408080; font-style: italic"># initialize the rng with a seed</span>
+random<span style="color: #666666">.</span>seed()
+<span style="color: #408080; font-style: italic"># Hard coding of input parameters</span>
+Agents  <span style="color: #666666">=</span> <span style="color: #666666">500</span>
+MCcounts <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
+Transactions <span style="color: #666666">=</span> <span style="color: #666666">100000</span>
+startMoney <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
+Lambda <span style="color: #666666">=</span> <span style="color: #666666">0.0</span>
+FinancialAgents <span style="color: #666666">=</span> startMoney<span style="color: #666666">*</span>np<span style="color: #666666">.</span>ones(Agents)
+<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span> (<span style="color: #666666">1</span>, MCcounts, <span style="color: #666666">1</span>):
+    <span style="color: #008000; font-weight: bold">for</span> j <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span> (<span style="color: #666666">1</span>, Transactions, <span style="color: #666666">1</span>):
+        agent_i <span style="color: #666666">=</span> <span style="color: #008000">int</span>(Agents<span style="color: #666666">*</span>random<span style="color: #666666">.</span>random())
+        agent_j <span style="color: #666666">=</span> <span style="color: #008000">int</span>(Agents<span style="color: #666666">*</span>random<span style="color: #666666">.</span>random())
+        epsilon <span style="color: #666666">=</span> random<span style="color: #666666">.</span>random()
+        <span style="color: #008000; font-weight: bold">if</span> agent_i <span style="color: #666666">!=</span> agent_j:
+           m1 <span style="color: #666666">=</span> Lambda<span style="color: #666666">*</span>FinancialAgents[agent_i] <span style="color: #666666">+</span> (<span style="color: #666666">1-</span>Lambda)<span style="color: #666666">*</span>epsilon<span style="color: #666666">*</span>(FinancialAgents[agent_i] <span style="color: #666666">+</span> FinancialAgents[agent_j])
+           m2 <span style="color: #666666">=</span> Lambda<span style="color: #666666">*</span>FinancialAgents[agent_j] <span style="color: #666666">+</span> (<span style="color: #666666">1-</span>Lambda)<span style="color: #666666">*</span>(<span style="color: #666666">1-</span>epsilon)<span style="color: #666666">*</span>(FinancialAgents[agent_i] <span style="color: #666666">+</span> FinancialAgents[agent_j])
+           FinancialAgents[agent_i] <span style="color: #666666">=</span> m1
+           FinancialAgents[agent_j] <span style="color: #666666">=</span> m2
+
+<span style="color: #408080; font-style: italic"># the histogram of the data</span>
+n, bins, patches <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>hist(FinancialAgents, <span style="color: #666666">50</span>, facecolor<span style="color: #666666">=</span><span style="color: #BA2121">&#39;green&#39;</span>)
+
+plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;$x$&#39;</span>)
+plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;Distribution of wealth&#39;</span>)
+plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r&#39;Money&#39;</span>)
+plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>, <span style="color: #666666">10</span>, <span style="color: #666666">0</span>, <span style="color: #666666">500</span>])
+plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
+plt<span style="color: #666666">.</span>show()
+</pre></div>
 <p>
 We can then change our model to allow for a saving criterion, meaning that the agents save
  a fraction \( \lambda \) of the money they have before the transaction is made. The final distribution will then no longer be given by Gibbs distribution. It could also include a taxation on financial transactions.
@@ -1477,7 +1506,6 @@ $$
 Extract a parametrization of the above curves, see for example <a href="http://www.sciencedirect.com/science/article/pii/S0378437104004327" target="_blank">Patriarca and collaborators</a> and see if you can parametrize the high-end tails of the distributions in terms of power laws. Comment your results.

 <p>
-In the rest of this project we will follow the work of <a href="http://www.sciencedirect.com/science/article/pii/S0378437114006967" target="_blank">Goswami and Sen</a>. 
 In the studies above the agents were selected randomly, irrespective of whether we allowed for
 saving or not during a transaction. What is often observed is that various agents tend to make preferences for for whom to interact with. We will now study the evolution of the distribution of wealth \( w_m \) by assuming that there is a likelihood 
 $$
@@ -10,7 +10,7 @@
    "<!-- Author: -->  \n",
    "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
    "\n",
-    "Date: **May 9, 2018**\n",
+    "Date: **May 11, 2018**\n",
    "\n",
    "Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
    "\n",
@@ -1216,7 +1216,7 @@
    "\n",
    "## Simulating financial transcations\n",
    "\n",
-    "The aim of this project is to simulate financial transactions among financial agents\n",
+    "The aim here is to simulate financial transactions among financial agents\n",
    "using Monte Carlo methods. The final goal is to extract a distribution of income  as function\n",
    "of the income $m$.   From Pareto's work ([V. Pareto, 1897](http://www.institutcoppet.org/2012/05/08/cours-deconomie-politique-1896-de-vilfredo-pareto)) it is known from empirical studies\n",
    "that the higher end of the distribution of money follows a distribution"
@@ -1344,9 +1344,9 @@
    "\n",
    "\n",
    "\n",
-    "### Project 4a): Simulation of Transactions\n",
+    "### Simulation of Transactions\n",
    "\n",
-    "Your task is to first set up an algorithm which simulates the above transactions with an initial\n",
+    "Our task is to first set up an algorithm which simulates the above transactions with an initial\n",
    "  amount $m_0$.\n",
    "  The challenge here is to figure out a Monte Carlo  simulation  based on the\n",
    "  above equations.\n",
@@ -1355,14 +1355,58 @@
    "  $w_m\\Delta m$. You will need to set up a value for the interval $\\Delta m$  (typically $0.01-0.05$).\n",
    "  That means you need to account for the number of times you register an income in the interval\n",
    "  $m,m+\\Delta m$. The number of times you register this income, represents the value that enters the histogram.\n",
-    "  You will also need to find a criterion for when the equilibrium situation has been reached.\n",
+    "  You will also need to find a criterion for when the equilibrium situation has been reached."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "#!/usr/bin/env python\n",
+    "import numpy as np\n",
+    "import matplotlib.mlab as mlab\n",
+    "import matplotlib.pyplot as plt\n",
+    "import random\n",
    "\n",
-    "### Project 4b): Recognizing the distribution\n",
+    "# initialize the rng with a seed\n",
+    "random.seed()\n",
+    "# Hard coding of input parameters\n",
+    "Agents  = 500\n",
+    "MCcounts = 1000\n",
+    "Transactions = 100000\n",
+    "startMoney = 1.0\n",
+    "Lambda = 0.0\n",
+    "FinancialAgents = startMoney*np.ones(Agents)\n",
+    "for i in range (1, MCcounts, 1):\n",
+    "    for j in range (1, Transactions, 1):\n",
+    "        agent_i = int(Agents*random.random())\n",
+    "        agent_j = int(Agents*random.random())\n",
+    "        epsilon = random.random()\n",
+    "        if agent_i != agent_j:\n",
+    "           m1 = Lambda*FinancialAgents[agent_i] + (1-Lambda)*epsilon*(FinancialAgents[agent_i] + FinancialAgents[agent_j])\n",
+    "           m2 = Lambda*FinancialAgents[agent_j] + (1-Lambda)*(1-epsilon)*(FinancialAgents[agent_i] + FinancialAgents[agent_j])\n",
+    "           FinancialAgents[agent_i] = m1\n",
+    "           FinancialAgents[agent_j] = m2\n",
    "\n",
-    "Make thereafter a plot of  $\\log{(w_m)}$ as function of $m$\n",
-    "  and see if you get a straight line.\n",
-    "  Comment the result.\n",
+    "# the histogram of the data\n",
+    "n, bins, patches = plt.hist(FinancialAgents, 50, facecolor='green')\n",
    "\n",
+    "plt.xlabel('$x$')\n",
+    "plt.ylabel('Distribution of wealth')\n",
+    "plt.title(r'Money')\n",
+    "plt.axis([0, 10, 0, 500])\n",
+    "plt.grid(True)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
    "We can then change our model to allow for a saving criterion, meaning that the agents save\n",
    "  a fraction $\\lambda$ of the money they have before the transaction is made. The final distribution will then no longer be given by Gibbs distribution. It could also include a taxation on financial transactions.\n",
    "\n",
@@ -1452,7 +1496,6 @@
    "  equilibrium distributions and compare these with the Gibbs distribution. Comment your results.\n",
    "Extract a parametrization of the above curves, see for example [Patriarca and collaborators](http://www.sciencedirect.com/science/article/pii/S0378437104004327) and see if you can parametrize the high-end tails of the distributions in terms of power laws. Comment your results.\n",
    "\n",
-    "In the rest of this project we will follow the work of [Goswami and Sen](http://www.sciencedirect.com/science/article/pii/S0378437114006967). \n",
    "In the studies above the agents were selected randomly, irrespective of whether we allowed for\n",
    "saving or not during a transaction. What is often observed is that various agents tend to make preferences for for whom to interact with. We will now study the evolution of the distribution of wealth $w_m$ by assuming that there is a likelihood"
   ]
@@ -850,7 +850,7 @@ Change `r` in the program and play around to make a better fit!
 !split
 ===== Simulating financial transcations =====

-The aim of this project is to simulate financial transactions among financial agents
+The aim here is to simulate financial transactions among financial agents
 using Monte Carlo methods. The final goal is to extract a distribution of income  as function
 of the income $m$.   From Pareto's work ("V.~Pareto, 1897":"http://www.institutcoppet.org/2012/05/08/cours-deconomie-politique-1896-de-vilfredo-pareto") it is known from empirical studies
 that the higher end of the distribution of money follows a distribution 
@@ -917,8 +917,8 @@ several runs of the above simulations, at least $10^3-10^4$ runs (experiments).



-=== Project 4a): Simulation of Transactions   ===
-Your task is to first set up an algorithm which simulates the above transactions with an initial
+=== Simulation of Transactions   ===
+Our task is to first set up an algorithm which simulates the above transactions with an initial
  amount $m_0$.
  The challenge here is to figure out a Monte Carlo  simulation  based on the
  above equations.
@@ -929,10 +929,45 @@ Your task is to first set up an algorithm which simulates the above transactions
  $m,m+\Delta m$. The number of times you register this income, represents the value that enters the histogram.
  You will also need to find a criterion for when the equilibrium situation has been reached.

-=== Project 4b): Recognizing the distribution   ===
-Make thereafter a plot of  $\log{(w_m)}$ as function of $m$
-  and see if you get a straight line.
-  Comment the result.
+!bc pycod
+#!/usr/bin/env python
+import numpy as np
+import matplotlib.mlab as mlab
+import matplotlib.pyplot as plt
+import random
+
+# initialize the rng with a seed
+random.seed()
+# Hard coding of input parameters
+Agents  = 500
+MCcounts = 1000
+Transactions = 100000
+startMoney = 1.0
+Lambda = 0.0
+FinancialAgents = startMoney*np.ones(Agents)
+for i in range (1, MCcounts, 1):
+    for j in range (1, Transactions, 1):
+        agent_i = int(Agents*random.random())
+        agent_j = int(Agents*random.random())
+        epsilon = random.random()
+        if agent_i != agent_j:
+           m1 = Lambda*FinancialAgents[agent_i] + (1-Lambda)*epsilon*(FinancialAgents[agent_i] + FinancialAgents[agent_j])
+           m2 = Lambda*FinancialAgents[agent_j] + (1-Lambda)*(1-epsilon)*(FinancialAgents[agent_i] + FinancialAgents[agent_j])
+           FinancialAgents[agent_i] = m1
+           FinancialAgents[agent_j] = m2
+
+# the histogram of the data
+n, bins, patches = plt.hist(FinancialAgents, 50, facecolor='green')
+
+plt.xlabel('$x$')
+plt.ylabel('Distribution of wealth')
+plt.title(r'Money')
+plt.axis([0, 10, 0, 500])
+plt.grid(True)
+plt.show()
+
+!ec
+

 We can then change our model to allow for a saving criterion, meaning that the agents save
  a fraction $\lambda$ of the money they have before the transaction is made. The final distribution will then no longer be given by Gibbs distribution. It could also include a taxation on financial transactions.
@@ -978,7 +1013,6 @@ We can then change our model to allow for a saving criterion, meaning that the a
  equilibrium distributions and compare these with the Gibbs distribution. Comment your results.
 Extract a parametrization of the above curves, see for example "Patriarca and collaborators":"http://www.sciencedirect.com/science/article/pii/S0378437104004327" and see if you can parametrize the high-end tails of the distributions in terms of power laws. Comment your results.

-In the rest of this project we will follow the work of "Goswami and Sen":"http://www.sciencedirect.com/science/article/pii/S0378437114006967". 
 In the studies above the agents were selected randomly, irrespective of whether we allowed for
 saving or not during a transaction. What is often observed is that various agents tend to make preferences for for whom to interact with. We will now study the evolution of the distribution of wealth $w_m$ by assuming that there is a likelihood 
 !bt