# Mathematical Economics

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```					Mathematical Economics
Constructive Methods
• So far in this course we have focused a lot on applications
that rely on simultaneous equations or finding the
maximum or minimum of a function.
• Since they rely on mathematical analysis, these are called
“analytic methods.”
• For some purposes, analytic methods are roundabout.
• For showing that something (a market equilibrium, for
example) exists, some people (including a few
mathematicians and economists) would prefer a
“constructive method.”
• That means we show that the object exists, and investigate
it otherwise, by constructing an instance of it.
Constructivism
• A few mathematicians go so far as to reject any
proofs of the existence of an object (an infinite
number, for example) unless the proof constructs
an instance.
• This probably goes too far -- but constructive
methods can contribute something to our
understanding of our models.
• Probably this is why Leon Walras proposed the
“tatonnement” model of market equilibrium.
Market Equilibrium
• As we have seen, we can think of a market equilibrium as
the simultaneous solution of supply and demand equations.
• But -- some economists of Walras’ time wondered -- could
real human beings find their way to a market equilibrium
without sitting down and solving simultaneous equations?
• (After all, we don’t observe them doing that!)
• More generally -- Walras wondered -- is there any process
that would bring buyers and sellers to a market
equilibrium?
• And if so, does it look anything like what real human
beings do in markets?
Tatonnement 1
• Walras proposed the tatonnement process, a
sort of idealized auction process.
• (Tatonnement is French for “groping.”
– An “auctioneer” starts by choosing a price at
random.
– The “auctioneer” then polls the suppliers and
or sell at that price.
Tatonnement 2
– The “auctioneer” computes the excess demand
reported by the suppliers and demanders.
• If excess demand is positive then he raises the price
a little bit.
• If excess demand is negative (there is excess supply)
he reduces the price a little bit.
– The “auctioneer” repeats until excess demand is
zero.
– Trading then takes place at the market-clearing
price.
Constructive Models
• Since the tatonnement process generates a market-
clearing price, it can be thought of as a
constructive model. However, it is a bit
hypothetical.
• Another, somewhat less hypothetical constructive
model is a computer simulation.
• In computer simulation, we construct a program
that (perhaps!) generates the object we are
interested in, such as a market-clearing price.
Agent-Based Computer
Simulation
• In agent-based computer simulation, we write a program
that simulates a population of agents (buyers and sellers,
for instance) and their interactions.
• Research along these lines has been done in economics
since the early 1990’s.
• The economists were following the example of biologists
who constructed models of “artificial life.”
• For example, an entomologist and a computer programmer
constructed a simulation model of an anthill, modelling
each ant as an individual who acted according to very
simple rules. The virtual anthill they created was realistic.
Object-Oriented Programming 1
• While it is not a requirement, agent and artificial
life simulations are often written in object-oriented
programming languages.
• In object-oriented languages much of the program
code is organized into objects.
• Each object has properties (such as numerical
variables) and methods, which are semi-
independent programs that can be run to transform
the properties.
• Each object is an instance of a class.
Object-Oriented Programming 2
• Each instance of a particular class has the same range of
properties and the same methods, although the specific
values of some of the properties may differ from one
instance to another.
• For example, in the simulated anthill,
– The simple rules of ant behavior were methods all
members of the ant class could compute.
– All members of the ant class had properties of relative
location -- near or far from the center of the anthill,
though some were nearer and some further.
– When ants interacted, their methods and their relative
location caused them to move, resulting in different
relative locations (transforming the properties).
Tatonnement Simulation
• In economics, the agents may have methods to
buy, sell, produce, consume, save, invest, attack,
defend, and do similar things we think real
economic agents do.
• Their properties may include endowments of
money and resources, utility, the prices they ask or
are willing to pay, and so on.
• To illustrate the idea, we have an agent-based
simulation of Walras’ tattonement process.
Two Kinds of Agents 1
Two Kinds of Agents 2
• By default there are        • Agents maximize a Cobb-
– 100 buyers with 100        Douglas utility function
dollars each                – Alpha is the exponent
– 100 sellers with 10            for consumer goods
units of consumer good      – Beta for money
each                     • The simulation continues
• The starting price is         until excess demand is less
random by default but can     than 0.0001 of demand, or
be set, as here.              100 steps, whichever is
• All defaults can be set.      first.
An Agent

Here is some of the code for a “buyer” agent.
More Code

Here is the code that uses that method to get
the individual agent’s contribution to demand.
Click “Go”
The Tatonnement
Record of tattonement process:

The random number seed was 13242

The starting price for the tattonement is 2
At round 1 price is 2 and excess demand is 1000.
At round 2 price is 3.472414 and excess demand is 469.9602.
At round 3 price is 3.644472 and excess demand is 435.9703.
At round 4 price is 6.199451 and excess demand is 153.2615.
.....
At round 34 price is 9.99861 and excess demand is 0.0347461.
At round 35 price is 9.999093 and excess demand is 0.0226661.
• Random numbers seem to play a part in
many real economic processes.
• In simulations we usually substitute a series
of pseudorandom numbers.
• A sequence of pseudorandom numbers is a
series generated by a deterministic
algorithm -- not random at all, but …
Pseudorandom Numbers
• A number in a pseudorandom sequence depends on some
previous numbers in the sequence -- but via very complex
rules that make it seem unpredictable.
• A good pseudorandom sequence will closely approximate
the properties of a real random distribution, such as a
normal distribution -- mean, variance, and higher mements
and so on.
algorithm will always give us the same sequence.
• This starting point is called the “random seed.”
Two Sequences
• Here are two sequences of        56023       66066
psedorandom numbers.
• Each is approximately         6.865967     4.079626
normal with mean 5 and        9.287109     9.688747
2.47526   0.5661285
standard deviation 2.5
1.106299    -2.844315
• Their seeds are shown at      5.772239     2.198776
the top.                      1.663438     14.09368
• The seeds 56023 and           10.22548    0.7177999
4.416111    -13.13244
66066 indirectly              4.801075     3.515518
determine the starting      -0.1978917     6.170837
points for the two
sequences.
In the Tattonement
• Random numbers may not play any role in the
tattonement at all.
• In this version, though, the size of the step taken
by the auctioneer depends partly on a random
number.
• As a result, the exact sequence of steps to the
equilibrium depends on the random seed.
• If we experiment with it, we will find that it
doesn’t matter -- but that is an important thing to
know!
The Interface, Again
Try it -- set
the random
seed here
to two
different
numbers and
compare the
simulations.
Comparison
• Comparisons such as that are a way that we
can learn from simulations.
• As you see, the interface for this simulation
allows you to reset a number of parameters,
so that you can explore the difference made
by those changes.
Simple Version
• This is a very simplified simulation. We could
make it more realistic by, for example,
– Allowing different buyers and sellers to have different
endowments and utility functions,
– Allowing for mistakes and/or learning by the agents,
– Matching buyers and sellers at random and letting them
• Many simulations of more complex economic
processes include these phenomena.
Summary
• Constructive methods can contribute to our understanding
of mathematical models.
• Walras’ “tatonnement” provides an example in economics.
• Agent-based computer simulation is a particularly rich and
powerful constructive method.
• We can, for example, construct a computer simulation of
tatonnment that will generate the whole process and
demonstrate how it converges (close) to the equilibrium
price.

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 views: 13 posted: 8/20/2012 language: English pages: 25
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