Studies in the Dynamics of Economic Systems
N. Tran1, W. Weyerman1, C. Giraud- Carrier1, K. Seppi1, S. Warnick1, and R. Johnson2
1 Information Dynamics and Intelligent Systems Group
Computer Science Department, Brigham Young University, Provo, UT 84602
2 Charles River Associates, Inc.
Salt Lake City, UT 84101
Abstract— This paper demonstrates the utility of systems
and control theory in the analysis of economic systems. Two
applications demonstrate how the analysis of simple dynamic
models sheds light on important practical problems. The first
problem considers the design of a retail laboratory, where the
small gain theorem enables the falsification of pricing policies.
The second problem explores industrial organization using
the equilibria of profit- maximizing dynamics to quantify the
percentage of a firm’s profits due strictly to the cooperative
effects among its products. This ” Value of Cooperation”
suggests an important measure for both organizational and
antitrust applications.
I. INTRODUCTION
In today’s society, economic systems are among the most
heavily instrumented of any engineered system. Point- of-sale
data- capturing mechanisms, such as scanners, are ubiq-uitous,
and businesses invest heavily in data warehousing
and analysis technologies. With the advent of the internet,
it is not uncommon for firms to warehouse terabytes of
data. Various disciplines, such as statistics, data mining,
machine learning, and operations research have postured
to help managers transform this data into useful decisions.
In spite of these efforts, control theoretic approaches to the
problem still add unique value by emphasizing dynamic
behavior and the effects of feedback.
This paper demonstrates the point by exploring the
implications of simple dynamic models of economic phe-nomenon.
Section II. highlights the first example, discussing
recent work in the design of a live retail laboratory at
Brigham Young University. Retail is modeled as a feedback
process. Optimal control formulations of various pricing
mechanisms have recently gained renewed popularity in
the field of Revenue Management [ 8], but little work has
studied the verification of such mechanisms. Our work on
the design of a retial laboratory reveals the importance of
robust control formulations for yielding pricing strategies
capable of being conclusively invalidated.
Section III. develops an entirely different view of market
dynamics emphasizing the competitive interaction between
firms and their subsequent industrial organization. Here
This work is supported by grants from the Rollins Center for eBusiness,
the Office of Research and Creative Activities at BYU, the BYU Bookstore,
and Sandia National Laboratories. Please direct all comments and questions
to Sean Warnick at sean@ cs. byu. edu.
we show that a game- theoretic measure on the value of
cooperation emerges naturally by studying how equilibria of
the market system change under different coalition models.
The point is made on a simple example.
In both of these cases, simple dynamic models facilitate
insight into the nature of economic behavior. Although
economics and control have an established history, it is
our view that economic systems continue to provide a rich
application domain for control theory to contribute to a truly
interdisciplinary area [ 4], [ 9].
II. DESIGN OF A RETAIL LABORATORY
A. Retail as a Feedback Process
We begin our discussion with the idea of a firm. Our
firm will begin its life with a fixed amount of capital,
C( 0), ( or credit available, but our assumption is that there
is a fixed bound on the purchase power available to the
firm), and a list of n wholesale goods it can choose to
purchase. There may be many potential suppliers of these
goods, so the actual information the firm receives is a matrix
of wholesale prices ( effective costs) we denote as a p × n
matrix PW( k), where k = 0, 1, 2, ... is the number of days
since the firm began operation. Thus, PW( k) are the prices
the firm receives on the morning of day k, and we assume
they will not change until the next morning.
The firm purchases wholesale goods on day k at price
PW( k) by specifying a matrix XW( k), which identifies
how many of the n goods from each of the p suppliers the
firm will purchase at the current price, PW( k). Of course,
the firm has limits on its spending characterized by the
constraint that i, j xw( i, j) pw( i, j) = C( k), where xw( i, j)
and pw( i, j) are the ( i, j) th elements of XW( k) and PW( k),
respectively.
The firm then makes offers to its retail market for a price
of its choosing. Since the firm may offer different effective
prices to its m customers ( or different store locations, or
other type of channel), say through coupons or special
offers, etc., the pricing decision of the firm is represented
by a m×n matrix PR( k), which means the prices available
to the m retail customers ( or channels, or segments) of the
n goods on the morning of day k. Note that goods that the
firm has not purchased wholesale are listed as available for
very high prices.
Proceedings of the
2005 IEEE Conference on Control Applications
Toronto, Canada, August 28- 31, 2005
TB4.2
0- 7803- 9354- 6/ 05/$ 20.00 © 2005 IEEE 861
W
Retail
Market
Wholesale
Market
Firm
P
P X
X R R
W
Fig. 1. A retail firm interacts with wholesale and retail markets.
W
Retail
Market
Firm
X P R R
Wholesale
Market
P W X
W
Firm
Market
( P , X ) W R
( P , X ) R
Fig. 2. By rearranging Figure 1 ( left), we note that the distinction between
a retail and wholesale market simply imposes structure on the more general
feedback relationship ( right).
The retail market then responds with purchases through-out
day k characterized by a matrix XR( k). This m × n
matrix describes how much of each of the n goods each
of the m retail customers or segments purchased that
day. At this point, money from the day’s transactions are
collected, and the next morning’s available capital becomes
C( k+ 1) = C( k)- i, j xw( i, j) pw( i, j)+ i, j xR( i, j) pR( i, j).
Although real prices are quantized by the fundamental
monetary unit, and the unavailability of a good may be
considered ontologically different than associating a very
high price with the good, we will ignore these details
for simplicity and consider that PW( k) ? Rp×n and
PR( k) ? Rm×n. We thus define the firm as an operator
F : ( Rm×n × Rp×n) ? ( Rm×n × Rp×n), its wholesale
market as an operator MW : Rp×n ? Rp×n, and its
retail market as an operator MR : Rm×n ? Rm×n, which
yields the picture shown in Figure 1. More generally, we
could consider cases where the wholesale and retail markets
are interrelated, yielding a combined market operator M :
( Rm×n × Rp×n) ? ( Rm×n × Rp×n). This results in a very
general feedback structure as shown in Figure 2, where
C is an internal state of the firm. We assume that such
interconnection is well- posed, in that solutions of the system
u and y exist and are unique.
Notice that in this view of retail, a firm’s interaction
with the market is to buy and sell. Coupons, markdowns,
special promotions, advertising, etc. are all modeled as
equivalent modes of communicating offers to the firm’s
customers. Likewise, all modes of communication with a
firm’s suppliers are considered equivalent. Although this
may not be precise for real economic systems, it is a
reasonable simplification for the purposes of our analysis.
Furthermore, note that usual classifications of the firm
as a price- setter or price- taker, or of the market as being
competitive, oligopolistic, or monopolistic are unnecessary
here. The firm makes its procurement and pricing decisions
with the best information it has at any particular time, and
information about whether it is operating in a competitive
environment or has some special market power will be
reflected in how the market responds.
B. Validation of a Learned System
With this view of the feedback process, where all market
influences and disturbances have been lumped together into
one operator M : U ? Y, we want to explore fundamental
limits on validating a learning process. We may suppose
that M has a state space realization of the form
x( k + 1) = f( x( k), u( k), k) x( 0) = xo
y( k) = g( x( k), u( k), k) ( 1)
where u is a vector of size p × n × m × n ( the number
of distinct decisions the firm can make at time k), x is
a presumably very large vector of internal market states
( that we, nevertheless, assume to be finite), and y a vector
of market observations ( same length as u). We assume
that the firm has no special information about its market,
and thus only knows [ u( k), y( k)] over a finite duration
k = 0, 1, 2, ..., t. In particular, the firm does not observe
the internal market states x, nor does it have complete
information of f or g.
Once in operation, however, an intelligent firm will begin
to consider its observations [ u( k), y( k)], k = 0, 1, ..., t and
use them to develop some understanding of how its market
behaves. It may accomplish this by hiring experts who retain
some partial information of the market dynamics f and g, or
by employing novel learning processes that analyze its data
in effective ways. However the firm operates, though, it will,
explicitly or implicitly, develop a model, or theory, of how
the market behaves and attempt to use this understanding
to its advantage.
Assuming that the firm has explicitly developed a model
of its market, consider the model as an operator T : V ?
U ? Y. That is, the understanding a firm obtains about
its market enables it to establish expectations about what it
will observe for at least some of the actions it might take
( u ? V). Regardless of how the firm developed its particular
model T, however, it will need to determine how well the
model approximates the true market before it will have the
confidence to widely use the model to develop pricing or
other policies. Restricting the market to the situations where
the model applies, one approach the firm could take would
be to measure some notion of distance between M and T.
862
Ideally, one would measure this error using the gain, or
induced norm, of an operator, given as
where | · | indicates an appropriate signal norm, and E =
M - T is the prediction error. In practice, however, this
quantity is difficult to compute. The first challenge arises
from the fact that each signal u or y is a time function
of infinite duration. Next, a search over all possible inputs
u ? V to find the supremum is impractical, especially when
most firms typically have market response data for only a
few different pricing functions ( e. g. often only the ” normal”
price followed by a sale price).
Finally, even if these other challenges could be resolved,
just measuring the prediction error for a single, particular
pricing function for even a limited duration is a difficult
task. Although we may know how many widgets actually
sold for three months in response to a particular, even
a constant price, it is difficult to generate a comparable
response from a market model because it is unclear how to
set the model’s initial conditions. If the market is assumed
to be asymptotically stable, then the effects of an incor-rect
choice of initial conditions eventually would become
negligible. Nevertheless, it would be unclear how long one
should wait for this to occur. Moreover, the assumption of
market stability in the first place seems questionable in an
application domain bent on unbounded growth, ever- present
competition, and ubiquitous ” business cycles”.
These difficulties cause various modes of approximate
validation heuristics to be used in practice. The most
sophisticated of these use cross validation techniques to
attempt to approximate a measure of the modeling error.
Ultimately, this approach of trying to validate a learned
system by measuring the model error is attractive in that
the process can be conducted entirely from cached data.
As a result, the data warehousing industry has emerged,
offering firms the chance to store terabytes of data to drive
both learning and the subsequent validation processes.
Nevertheless, the link between these practical heuristics
and real validation is not firmly established. An alternate
validation process addresses the above difficulties, but at
the cost of demanding more involvement from the firm in
the scientific process than merely caching its data.
C. The Retail Laboratory and the Small Gain Theorem
Instead of trying to measure a comprehensive error of a
market model, another approach to validation focuses on
whether the proposed model is sufficient for an intended
purpose. This approach uses a given model to design
feedback policies that accomplish a specified objective.
The effects of these policies on the true market are then
compared with the results expected from the model, and
the small gain theorem is employed to falsify models that
are insufficient for the stated objective.
u
M
F
y
R
F
z w
M
y T u
Fig. 3. The operator M can be factored as the feedback interconnection
of R and T.
G
z R w
T
F
y u
z
R
G
w
Fig. 4. Regrouping operators, the small gain theorem guarantees that F
stabilizes M provided < 1.
In particular, the true market M can be factored into the
feedback interconnection of a theory of the market T with
the unknown remainder R. This feedback factorization often
allows an unstable operator M to be decomposed into the
feedback interconnection of two stable operators R and T.
Thus, although the conditions of the small gain theorem
require that the unknown part of the market, R, be stable,
this is a weaker condition than demanding that the market
itself, M, be stable.
The novelty of the approach arises when we consider
actually implementing a specific policy F that translates
observations from the market, y into specific actions u.
Although F may be designed only with information of
T, the small gain theorem says it will stabilize the real
market M provided < 1, where G is the feedback
interconnection of F and T. Note that we expect G to be
stable since we design F such that it stabilizes T.
Thus, the scientist can test the quality of a given market
theory T by designing a policy F that robustly and asymp-totically
stabilizes it with a desired rate of convergence.
Analysis or simulation of this ” ideal” closed- loop system,
G, then indicates how long it will take the feedback
interconnection of F on any system M ” close” to T ( in the
sense that is small enough) to stabilize. This suggests
a finite duration experiment that can be tested empirically
by implementing the policy F ” live” on the true market M.
Observing the closed- loop results over the required duration
863
either results in success, meaning the closed- loop market
is behaving as designed ( at least so far), or failure, which
implies that the conditions of the theorem were not met and
T is not close toM in that. A model can thus
be conclusively invalidated, provided that a mechanism for
implementing a closed- loop policy ” live” on the true market
exists. This is the value of a true economic laboratory,
where experiments can be conducted and observed, even
without perfect control over the internal variables or outside
influences of the market.
III. INDUSTRIAL ORGANIZATION
A. Profit Maximizing Dynamics
Now, consider a market, M, of N products. Without loss
of generality, give these products an arbitrary order and
integer label so that M= { 1, 2, ..., n}. Let p ? RN be the
vector of ( non- negative) prices for these N products, and
let q : RN ? RN be the ( non- negative) demand for these
products at prices p.
We now consider a firm, F to be a subset of the N
products in the market, F ? 2M. This implies that the
firm controls the production and distribution of the products
assigned to it. Most importantly for our analysis, since we
consider a Bertrand market model, this implies that the firm
may set the prices of the n = | F| products assigned to it.
We suppose that the products of the market are parti-tioned
between m firms. This implies that no two firms
control the same product, Fi Fj = Ø ? i = j, and that
the union of all products assigned to the m firms compose
the entire market, m
i= 1 Fi = M.
Let cj( qj), j = 1, ..., N be the cost of production of qj
units of product j. The profit of the ith firm then is given
by
pi =
j? Fi
[ qj( p) pj - cj( qj( p))]
A profit- maximizing firm under the Bertrand model of
market behavior will tend to change its prices to maximize
its short- term profit [ 7]. We model this behavior by assum-ing
that the firm will evolve the prices of its products in
the direction of maximally improving its profits. That is,
if product j belongs to firm i, then we expect the firm to
evolve the price of product j as
dpj( t)
dt
= ? pi( p)
? pj
p( t)
where p( t) is the pricing vector for the entire market at time
t.
Notice that these dynamics suggest that if the partial
derivative of profits is negative with respect to the price of
product j, that the firm should decrease the price of product
j. This is in the direction of improving profits. Likewise, if
the partial derivative were positive, the firm would increase
the price of product j to improve profits. When the partial
derivative is zero, the motivation is to hold the price at this
locally profit- maximizing position.
Reordering the N market products so that each firm’s
products are grouped together, and letting ni be the number
of products controlled by firm i, we then can partition the
pricing vector into components associated with each firm. If
every firm in the market is assumed to be profit maximizing,
this yields the following market dynamics:
( 3)
where the dot notation p ? ( t) is used to represent dp( t)/ dt.
Notice that if the market system ( 3) has an equilibrium,
such a pricing vector peq would represent prices from which
no firm can improve its profits by unilaterally changing the
prices over which it has control. Under certain technical
conditions such an equilibrium can be shown to exist.
Moreover, this equilibrium can often be shown to be asymp-totically
stable, in the sense that any pricing vector p( 0) will
converge to the equilibrium peq as t?8.
B. The Firm as a Coalition
Under the assumption that the market dynamics are
stabilizing, we expect price perturbations to re- equilibrate.
In this context, it is convenient to simplify the problem
by only considering the profits of the firms at equilibrium.
These profits define a payoff function reminiscent of those
used to define coalition games.
Let v( Fi) = pi| p= peq
be the payoff or profit of firm i at
the market equilibrium prices peq. In this way the firm may
be thought of as a coalition of ni players in an N- player
cooperative game. Each player is a one- product company
that completely manages the production, distribution, and
pricing decisions for its product. The firm, then, is a
confederacy of these one- product companies that works
together to maximize their combined profits or payoffs.
The theory of coalition games studies the behavior of
such coalitions once the payoff function is defined for
every possible coalition [ 1], [ 5]. The idea is that any given
coalition Fi yields a well- defined payoff v( Fi), and then
a number of questions can be explored regarding how to
distribute the payoff among the members of the coalition,
etc.
Our situation is different because the payoff to a given
firm doesn’t just depend on the products it controls, but
864
also on the market structure of the products outside the
firm. For example, consider a 10- product market and a three
product firm in the market. The payoff to the firm does not
just depend on the prices of the three products it controls,
but also on the prices of the other seven products. The
profit- maximizing equilibrium prices of these other seven
products, however, may be set differently depending on
whether they belong to a single firm or whether they are
controlled by seven different companies. Thus, the payoff
to the three- product firm depends on the entire market
structure.
Coalition game theory addresses such situations by con-sidering
partition systems and restricted games. For our
purposes, it is sufficient to partition the N products of M into m firms and then assume that this structure is fixed
outside of the particular firm that we are studying. This
enables us to work with a well defined payoff function
induced by the profit- maximizing dynamics of firms within
the market without eliminating the multiple- coalition ( i. e.
multiple firm) cases of interest.
C. Value of Cooperation
To quantify the value of organizing a group of one-product
companies into a single firm, we need to compare
the profits the firm receives if it sets its prices as if each
of its products were independent companies with those it
realizes by fully capitalizing on cooperation between the
products. More precisely, let peq be the profit- maximizing
equilibrium prices for the given market structure. In con-trast,
consider the new profit maximizing equilibrium prices
achieved without cooperation if Fi were divided into its
constituent one- product companies and each independently
optimized their prices. Let this second set of equilibrium
prices serve as a basis for comparison, or reference, and
be denoted pref . The relative value of cooperation ( VC)
of a given firm Fi in market M with structure S =
F1, F2, ..., Fm is then given by
RVCref ( Fi, S) =
pi| peq - pi| pref
pi| peq
( 4)
This measure is interpreted as the percentage of profits
due to cooperation within the organization. It is bounded
between zero and one, and it facilitates direct comparison
between firms of different sizes.
Sometimes we may be interested in measuring the value
of cooperation between structures other than the current
market structure and the reference structure. This could be
the case when considering mergers between firms, or when
management is considering selling off a piece of the firm.
In such cases it is easy to extend the definitions of RVC by
simply replacing the equilibrium and reference prices with
the equilibrated profit- maximizing prices of the two market
structures being compared.
It is instructive to contrast the RVC with other measures
used to characterize cooperative games. Hart and Mas-
Colell [ 3] defined a measure, called the potential, P, that
computes the expected normalized worth of the game i. e.
the per- capita potential, P/ N, equals the average per- capita
worth ( 1/ m) i ( pi)/(| Fi|). Given a market structure, this
measure characterizes the expected profit of an average-sized
firm ( where size is measured with respect to the
number of products the firm controls) in the market, even
if such a firm does not actually exist.
Moreover, the potential has been connected to another
measure, called the Shapley value, Fj , which yields the
marginal contribution of each product in the market [ 6].
This measure characterizes how the payoff of a coalition
should be divided between members of the team. In both
cases, the potential and Shapley value do not suggest
anything about the intrinsic benefit of forming coalitions
in the first place.
The Relative Value of Cooperation, RVC, on the other
hand, captures the natural significance for organizing pro-duction
into multi- product firms. Nevertheless, this measure
does not yield any information about how the profit of
a firm should be efficiently invested into each of the
firm’s constituent production lines. Thus, this measure is
inherently different from the potential or shapely value since
these focus more on the value of a member of a coalition to
the group, rather than the value of the coalition as a whole.
D. Example
To illustrate the point, consider a two product economy
with linear demand given by
100
( 5)
Suppose that the unit production cost of each product is
c1 = 10, c2 = 10. If we consider a market structure where
each product is produced by an independent company, the
profit function for each company becomes
p1( t) = q1( t) ( p1( t) - c1)
= - 3.5p21
- p1p2 + 135p1 + 10p2 - 1000
( 6)
p2( t) = q2( t) ( p2( t) - c2)
= - 2p22
- 3p1p2 + 30p1 + 120p2 - 1000 ( 7)
Taking the partial derivatives of each profit function with
respect to the appropriate pricing variable, we find the
profit- maximizing market dynamics to be:
Figure 5 shows how the two- firm dynamics drive an
initial pricing vector to a profit- maximizing equilibrium.
This equilibrium price is
17.4
.
and the associated equilibrated profits are p1 = 161.84,
p2 = 109.52.
865
Fig. 5. Two firm price trajectory and profit function
Now, consider a market structure where both products are
controlled by the a same firm. In this case, the firm’s profit
function becomes
p( t) = q1( t) ( p1( t) - c1) + q2( t) ( p2( t) - c2) ,
= - 3.5p21
+ 165p1 - 4p1p2 + 130p2 - 2p22
- 2000.
( 9)
With this market structure, the firm adjusts the prices of
both products to optimize the same objective. These new
dynamics become:
Fig. 6. One firm price trajectory and profit function
Figure 6 shows how the single- firm dynamics drive an
initial pricing vector to a profit- maximizing equilibrium.
The new equilibrium price is given by
20.83
.
and the associated equilibrated profits are peq = 316.667.
The relative value of cooperation in this example thus
becomes
RV C = 0.1431
This suggests that in this market, just under 15% of the
profits of the two- product firm are due to cooperation.
IV. CONCLUSION
This paper explored two applications of dynamic analysis
in economic systems. The first considered a firm to be
an operator in feedback with its associated market. The
firm is characterized by its pricing policy mapping sales
data into prices. We discussed problems associated with
the verification of such policies, and showed how robustness
results from the small gain theorem suggest that a true retail
laboratory could be feasible.
The second problem quantified measures to calibrate the
value of cooperation within a specific firm in a given market.
The idea is to assume profit- maximizing dynamics among
the firms within the market and compare equilibrium profits
in two different scenarios. The first scenario considers the
firm as it is, as a single economic entity with a unified
objective and exhibiting full cooperation between its various
economic units. The second scenario considers splitting
the firm into its constituent economic units and computing
market equilibrium prices if these units maximized their
individual profits. Normalizing the difference between the
cooperative profits of the first scenario and the aggregate
profits of the independent units of the second scenario define
a measure we call the Relative Value of Cooperation, RVC,
of the firm in its current market environment. This measure
reveals the percentage of profits derived from cooperation
within the firm. Quantifying the value of cooperation is a
first step in understanding how firms exert market power in
their respective environments. This information is important
for both managers, who hope to leverage the information to
better lead their organizations, and regulators, who monitor
the impact of corporate decisions on social welfare [ 2].
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