A framework for economic analysis based on matrix algebra.

Mitch Leonards

English

Research Papers in Economics

1
Economic Analysis (2
nd Edition)
Mitchell Leonards (nom de plume)
email:  mitchleonards@yahoo.ca
web site:  www.geocities.com/mitchleonards
This work is licensed under the Creative Commons Attribution-NoDerivs-NonCommercial License. To
view a copy of this license, visit http://creativecommons.org/licenses/by-nd-nc/1.0/ or send a letter to
Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.Abstract
A framework for economic analysis based on matrix algebra.  In this edition, the mathematical model
introduced in earlier works has been extended.
ML (January 2005)
PS:
In conventional economic analysis, the physical aspects of a given set of processes are captured by economic
variables such as cost and returns to scale, which are in turn used to construct sets of equations to model the structure
of the market where the processes are executed.  While this approach is sufficient for applied research, it is not ideal
for theoretical purposes because technological and physical constraints are treated as endogenous.
Utility theory and other techniques to model agent behavior in markets are only relevant where agents can
execute more than one action, so they constitute an application of theory, not a core principle.  Accordingly, a
foundational theoretical model should not include any decision-making construct, but must contain the principles
required to construct one.
This paper presents a generalized model that explicitly describes the physical nature of processes and uses
that description to derive economic variables and relationships.  The result of this approach is a systematic, compact
framework that can be adapted to analyze a variety of micro and macro phenomena.
The introduction presents the definitions the model is constructed from, and chapters 1-3 present the
structural model.  Chapters 4-7 present the dynamics that define the operation of the model.Table of Contents
Introduction
Processes
Economic Processes
Economic Networks
First-Order Dynamics
Second-Order Dynamics
Third-Order Dynamics
Fourth-Order Dynamics
4
5
6
7
9
10
11
124
Introduction
Section 1:  Observation
Let structure be defined as a set of elements and its organization.
Let transformation be defined as the alteration of a structure.
Let observation be defined as the identification of a structure.
Let qualification be defined as the identification of a structure’s attributes.
Let data be defined as observations and qualifications.
Section 2:  Analysis
Let composition be defined as the derivation of a structure’s attributes based on a given set of data.
Let decomposition be defined as the derivation of an element’s attributes based on a given set of data.
Let a principle be defined as a concept that defines the state of a structure.
Let a protocol be defined as an implementation of a principle in a particular context.
Let induction be defined as the derivation of a principle based on a given set of data.
Let deduction be defined as the derivation of a protocol based on a given set of data.5
Processes
Section 1:  Composition
For a given transformation,
let a reactant be defined as an element that is added, removed, or reorganized;
let an accelerant be defined as an element that accelerates the transformation or allows it to occur but is not a
reactant;
let a decelerant be defined as an element that decelerates the transformation or prevents it from occurring but
is not a reactant;
let a regulator be defined as an element that acts as an accelerant or decelerant;
let an input be defined as an element that acts as a reactant or regulator;
let a product be defined as a structure formed by a transformation for a given purpose;
let a by-product be defined as a structure formed by a transformation for no given purpose;
let an output be defined as a structure that is a product or by-product;
let a resource be defined as an object that is an input or output.
Section 2:  Organization
Let a track be defined as a set of a chronologically sequential transformations related by a common set of
inputs.  From this definition, a given track can converge with other tracks as well as diverge.
Let a process be defined as a set of b tracks related by a common set of outputs.  Where b > 1, tracks can
occur simultaneously or sequentially.
For analytical purposes, a process can be divided into subsets of transformations according to some set of
criteria.  Let these subsets be called stages.
Let a process descriptor be of the form
[I]/[O]
t
where [I] is an n x n matrix of inputs, [O] is an N x N matrix of outputs, and t is the time required to execute the
process.
Let each element in [I] and [O] be of the form
( ) g Q
q
S
T
where g is the identifies of the resource, q is the number of units of the resource involved in one execution of the
process, Q is the number of units of the resource involved in ten executions of the process, T is the set of chronological
constraints the resource must satisfy, and S is the set of spatial constraints the resource must satisfy.
Identifiers can be technical or common names.6
Economic Processes
Section 1:  Composition
Let a good be defined as a phenomenon defined by a given set of specifications.
Let a type be defined as a set of goods defined by a given set of specifications.
Let production be defined as the process of transforming a set of inputs into a given set of outputs.
Let consumption be defined as the process of transforming a set of outputs into a given set of inputs.
Let search be defined as the process of locating a set of goods that satisfy a given set of criteria.
Let acquisition be defined as the process of obtaining a set of goods located in a search.
Let evaluation be defined as the process of determining the performance of a set of goods.
Let notification be defined as the process of presenting a set of information about a set of goods.
Let distribution be defined as the process of transferring a set of goods to a set of consumers.
Let development be defined as the process of creating a good that satisfies a given set of criteria.
Section 2:  Organization
Let supply be defined as the sequence of processes that results in the production of a given set of goods.
Let demand be defined as the sequence of processes that results in the consumption of a given set of goods.
Let a vertex be defined as a type where production or consumption occurs.
Let a vector be defined as a type that connects vertices.
Let an agents be defined as a type that can act as a vertex or vector.
Let a lattice be defined as a given set of vertices, vectors, and agents.7
Economic Networks
Section 1:  Composition
Let an economic network be defined as a set of lattices.
Let At be the set of agents encompassed by an economic network at time t, such that
At = [A1...Am] (1.1)
Let Bt be the set of agents in At that are producers at time t, such that
Bt = [B1...Bi] (1.2)
where i < m.
Let Ct be the set of agents in At that are consumers at time t, such that
Ct = [C1...Cr] (1.3)
where r < m.  From these definitions, it is possible for a given agent to belong to subsets Bt and Ct at the same point in
time.
Let Nt be an i x I matrix of the set of goods produced by the agents in Bt, such that
Nt = [N1...NI] (1.4)
for I types of goods.
Let nt be an r x R matrix of the set of goods consumed by the agents in Ct, such that
nt = [n1...nR] (1.5)
for R types of goods.
Let bt be a 1 x I matrix of the set of goods produced by the agents in Bt, such that
bt = [Bt][Nt] (1.6)
where Bt is a 1 x i matrix.
Let ct be a 1 x R matrix of the set of goods consumed by the agents in Ct, such that
ct = [Ct][nt] (1.7)
where Ct is a 1 x r matrix.
From equation 1.6, the number of units of goods produced is given by
N b gh h
h
I
h
I
g
i
=
= = = å å å 1
1 1 1
(1.8)
and the number of units of goods of the E
th type produced is given by
N b gE
g
i
E
= å =
1
1 (1.9)
From equation 1.7, the number of units of goods consumed is given by
n c jk
k
R
j
r
k
k
R
= = = å å å =
1 1
1
1
(1.10)
and the number of units of goods of the E
th type consumed is given by8
n c jE
j
r
E
= å =
1
1 (1.11)
Section 2:  Organization
Given the structure of an economic network, a set of ratios can be constructed:
it / mt (r - 2.1)
rt / mt (r - 2.2)
it / rt (r - 2.3)
It / Rt (r - 2.4)
b i h
h
i
1
1 = å / (r - 2.5)
b1E / i (r - 2.6)
c r k
k
R
1
1
/
= å (r - 2.7)
c1E / r (r - 2.8)
b b E h
h
I
1 1
1
/
= å (r - 2.9)
c c E k
k
R
1 1
1
/
= å (r - 2.10)
b1E / c1E (r - 2.11)
b c h
h
I
k
k
R
1
1
1
1 = = å å / (r - 2.12)9
First-Order Dynamics
Section 1:  Accounting
A:  Processes
A process value descriptor can be constructed by isolating the number of units of each resource involved in a
process and the per unit value of each resource, where value is expressed in terms of a given resource.
Accordingly,
 C Q
q
S
T
Q
q
S
T = ( ) ( ) d e (1.1)
where C is the cost of executing a process, d is a 1x R matrix of inputs, and e is an R x 1 matrix of per unit input prices.
Similarly,
R Q
q
S
T
Q
q
S
T = ( ) ( ) z h (1.2)
where R is the revenue generated by a process, z is a 1 x I matrix of outputs, and h is an I x 1 matrix of per unit output
prices.
Isolating the q or Q terms from 1.1 gives
C = qd pe (1.3)
where q denotes quantity and p denotes price.
Isolating the q or Q terms from 1.2 gives
R = qz ph (1.4)
where q denotes quantity and p denotes price.
B:  Economic Processes
For the set of economic processes executed by a set of agents,
Rt = qz ph
    = Rproduction 
   = btYt (1.5)
where Yt is an I x 1 matrix of prices, and
Ct = qd pe
    = Cconsumption
    = ctZt (1.6)
where Zt is an R x 1 matrix of prices.
Where pt is profit at time t,
pt = R t - C t
    = btYt - ctZt (1.11)
Accordingly,
Pt t
t
T
=
= åp
1
(1.12)
for t = 1...T periods.10
Second-Order Dynamics
Section 1:  Scale
Let scale be defined as the relationship between the price and quantity of a good, such that
Pq t = xq tQ1t + x1t (1.1)
where x1t is a Q1t independent component of Pq t.
Let increasing returns to scale (IRS) be defined as a scale where xq t < 0.  Let constant returns to scale (CRS)
be defined as a scale where xq t  = 0.  Let decreasing returns to scale (DRS) be defined as a scale where xq t > 0.
Section 2:  Elasticity
Let elasticity be defined as the relationship between the quantity and price of a good, such that
Qit = xitPq t + x2t (2.1)
where x2t is a Pq t independent component of Qit.
Let inelastic elasticity (IE) be defined as an elasticity where xit  < -1.  Let neutral elasticity (NE) be defined as
an elasticity where xit = -1.  Let elastic elasticity (EE) be defined as an elasticity where xit  > -1.
Section 3:  Symmetry
Let symmetry be defined as the relationship between the profits of separate economic processes, such that
pkt = xktpit + x3t (3.1)
where x3t is a pit independent component of pkt.
Let increasing symmetry (IS) be defined as a symmetry where xkt > 0.  Let neutral symmetry (NS) be defined
as a symmetry where xkt = 0.  Let decreasing symmetry (DS) be defined as a symmetry where xkt < 0.
Section 4:  Conclusion
Equation 3.1 can be expressed as
pk k zi hq di eq zi hq di eq t t t t t t t t t t x q p q p q p q p = - + - ( )
       = - + - x b Y c Z b Y c Z t t t t t t t t t k i q i q i q i q ( ) (4.1)
Let Pkt be defined as
Pk k p t t
t
T
=
= å
1
(4.2)
for t = 1...T periods.11
Third-Order Dynamics
Constraints can be imposed by protocols or the structure of an economic network, or constructed from
assumptions.  Given second-order dynamics, there are nine types of constraints:
xkt = ut (1.1)
bit = fbt (1.2)
Yqt = fYt (1.3)
bit = cct (1.4)
Zqt = cZt (1.5)
b t bt i y = (1.6)
Y t Yt q y = (1.7)
c t ct i w = (1.8)
Z t Zt q w = (1.9)
These constraints can be integrated into the equation for Pkt using Lagrange multipliers, producing the
equation
 Pk k k i q i q p l u l f l f l c l c t t t t t t t bt t t Yt t t ct t t Zt
t
T
x b Y c Z = - - - - - - - - - -
= å[ ( ) ( ) ( ) ( ) ( ) 1 2 3 4 5
1
- - - - - - - - l y l y l w l w i q i q 6 7 8 9 t t bt t t Yt t t ct t t Zt b Y c Z ( ) ( ) ( ) ( )]12
Fourth-Order Dynamics
Section 1:  Type 1 Statistics
Let M be mass.
M b c xt h k
k
R
h
I
= +
= = å å ( ) ( ) 1 1
1 1
(1.1)
Myt = btYt + ctZt (1.2)
Let v be velocity.
V b c t xt h k
k
R
h
I
= +
= = å å [( ) ( )]/ 1 1
1 1
       =  Mxt / t  (1.3)
Vyt = (btYt + ctZt) / t
    = Myt / t (1.4)
Section 2:  Type 2 Statistics
Let g be growth.
gxt = (Mxt - Mxt-1) / t  (2.1)
gyt = (Myt - Myt-1) / t (2.2)
Let a be acceleration.
axt = (Vxt - Vxt-1) / t
     = (Mxt - Mxt-1) / t
2  (2.3)
ayt = (Vyt - Vyt-1) / t
     = (Myt - Myt-1) / t
2 (2.4)
Section 3:  Type 3 Statistics
Let P be momentum.
Pxt = (Mxt)(Vxt)
     = (Mxt)
2 / t (3.1)
Pyt = (Myt)(Vyt)
     = (Myt)
2 / t (3.2)
Let F be force.
Fxt = (gxt)(axt)
     = (Mxt - Mxt-1)
2 / t
3 (3.3)
Fyt = (gyt)(ayt)
     = (Myt - Myt-1)
2 / t
3 (3.4)
Section 4:  Type 4 Statistics13
A:  Lower Thresholds for Type 1 Statistics
For a given At, let Ct(min) be the minimum set of consumers at time t and let nt(min) be the minimum set of
goods they consume, where there are a types of goods.
From these definitions,
ct(min) = [Ct(min)][nt(min)] (4.1)
At minimum mass,
bt(min) = [0] (4.2)
Accordingly,
M c c xt k
k
k (min) (min) = =
= å 1
1
1
a
(4.3)
Myt(min) = c1k(min)Zt(min)  (4.4)
Vxt(min) = Mxt(min) / t (4.5)
Vyt(min) = Myt(min) / t (4.6)
B:  Upper Thresholds for Type 1 Statistics
For a given At, let Bt(max) be the maximum set of producers and let Nt(max) be the a maximum set of goods they
produce, where there are W types of goods.
From these definitions,
bt(max) = [Bt(max)][Nt(max)] (4.7)
At maximum velocity ct is identical to bt(max), so
bt(max) = ct(max)  (4.8)
Accordingly,
M b c b xt h k
k h
h
h
(max) ( ) ( ) = + =
= = = å å å 1 1
1 1
1
1
2
W W W
(4.9)
Myt(min) = bt(max)Yt + ct(max)Zt = 2bt(max)Yt (4.10)
Vxt(max) = Mxt(max) / t (4.11)
Vyt(max) = Myt(max) / t (4.12)
Section 5:  Type 5 Statistics
A:  Lower Thresholds for Type 2 Statistics
gxt(min) = (Mxt(min) - Mxt-1) / t  (5.1)
gyt(min) = (Myt(min) - Myt-1) / t (5.2)
axt(min) = (Mxt(min) - Mxt-1) / t
2 (5.3)
ayt(min) = (Myt(min) - Myt-1) / t
2 (5.4)14
B:  Upper Thresholds for Type 2 Statistics
gxt(max) = (Mxt(max) - Mxt-1) / t  (5.5)
gyt(max) = (Myt(max) - Myt-1) / t (5.6)
axt(max) = (Mxt(max) - Mxt-1) / t
2 (5.7)
ayt(max) = (Myt(max) - Myt-1) / t
2 (5.8)
Section 6:  Type 6 Statistics
A:  Lower Thresholds for Type 3 Statistics
Pxt(min) = (Mxt(min))(Vxt(min))
            = [Mxt(min)]
2 / t (6.1)
Pyt(min) = (Myt(min))(Vyt(min))
            = [Myt(min)]
2 / t (6.2)
Fxt(min) = (gxt(min))(axt(min))
            = [Mxt(min) - Mxt-1]
2 / t
3 (6.3)
Fyt(min) = (gyt(min))(ayt(min))
            = [Myt(min) - Myt-1]
2 / t
3 (6.4)
B:  Upper Thresholds for Type 3 Statistics
Pxt(max) = (Mxt(max))(Vxt(max))
            = [Mxt(max)]
2 / t (6.5)
Pyt(max) = (Myt(max))(Vyt(max))
            = [Myt(max)]
2 / t (6.6)
Fxt(max) = (gxt(max))(axt(max))
            = [Mxt(max) - Mxt-1]
2 / t
3 (6.7)
Fyt(max) = (gyt(max))(ayt(max))
            = [Myt(max) - Myt-1]
2 / t
3 (6.8)

Economic Analysis

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Economic Analysis

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