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mhjensen
2018-05-11 16:58:55 -04:00
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doc/pub/How2ReadData/html/How2ReadData-solarized.html
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@@ -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
$$