An exposition of economic dynamics based on Lagrange multipliers without the use of utility or game theory.

Mitch Leonards

English

Research Papers in Economics

System-based Economics
(second edition)
Mitchell Leonards  (nom de plume)
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
An exposition of economics using linear algebra and Lagrange multipliers to describe production and
consumption.  The structure of this work is based on the framework presented in Systems.
ML (August 2004)Table of Contents
Economic Network Structure
First-Order Metrics
Second-Order Metrics
First-Order Dynamics
Second-Order Dynamics
4
6
10
14
154
Economic Network Structure
Section 1:  Composition
Let an economic network be defined as a set of agents and the exchanges of goods they execute.
Let At be the set of agents encompassed by an economic network at time t, such that
At = [A1...Am] (2.1)
Within At, let Bt be the set of agents that can act as producers at time t, such that
Bt = [B1...Bi] (2.2)
where i < m.
Within At, let Ct be the set of agents that can act as consumers at time t, such that
Ct = [C1...Cr] (2.3)
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 the set of types of goods that can be produced by the agents in Bt, such that
Nt = [N1...NI] (2.4)
Let nt be the set of types of goods that can be consumed by the agents in Ct, such that
nt = [n1...nR] (2.5)
Let bt be the set of goods produced by the agents in Bt, such that
bt = [Bt][NB,t] (2.6)
where Bt is a 1 x i matrix and NB,t is an i x I matrix.  From these definitions, the number of units of goods produced is
given by
N gh
h
I
g
i
= = å å
1 1
(2.7)
and the number of units of goods of the E
th type produced is given by
N gE
g
i
= å
1
(2.8)
Let ct be the set of goods consumed by the agents in Ct, such that
ct = [Ct][nC,t] (2.9)
where Ct is a 1 x r matrix and nC,t is a r x R matrix.  From these definitions, the number of units of goods consumed is
given by
n jk
k
R
j
r
= = å å
1 1
(2.10)
and the number of units of goods of the E
th type consumed is given by
n jE
j
r
= å
1
(2.11)
Section 2:  Architecture
Given the structure of economic networks, there are four possible production configurations:
a) protocol-based production determination,
b) consumption-based production determination,5
c) consumption-regulated protocol-based production determination, and
d) protocol regulated returns-based production determination.
Similarly, there are four possible consumption configurations:
a) protocol-based consumption determination,
b) production-based consumption determination,
c) production-regulated protocol-based consumption determination, and
d) protocol regulated returns-based consumption determination.
Protocol-based determination configurations can be characterized as vertical streams, while consumption- and
productions-based determinations can be considered horizontal streams.  Since these streams are not inherently
exclusive, any set of production and consumption configurations can exist in an economic network.
Section 3:  Change
Let an agent reaction be defined as an agent’s response to a given set of information.  Let an environmental
response be defined as a set of changes in the set of information available to agents.  Given these definitions, there are
four types of responses:
a) agent responses to other agent responses (1),
b) agent responses to environmental responses (2),
c) environmental responses to agent responses (3), and
d) environmental responses to other environmental responses (4).
Responses of either type can be initiated by
a) past agent responses (AP),
b) expected future agent responses (AF),
c) past environmental responses (EP), and
d) expected future environmental responses (EF).
Accordingly, there are eight possible causality types in economic networks:
a) 1-AP,
b) 1-AF,
c) 2-EP,
d) 2-EF,
e) 3-AP,
f) 3-AF,
g) 4-EP, and
h) 4-EF.
Since no combinations are inherently exclusive, any set of causality types can exist in a given process.
These changes in economic networks can be recorded in a two dimensional chart.  Let there be two row types,
one for environmental components and one for agents or agent types.  Let columns represent time frames.
The time frame each column represents can be selected to correspond to the initiation and termination of
process stages.  Where parallel tracks converge and diverge at different points in time, columns can be constructed
according to a common denominator.
Cells in the first column of this chart show the initial state of an economic network, and all subsequent cell
show the changes.  Agents can enter and exit an economic network during various time frames, so the set of agents
present in a given time frame is not necessarily identical to the set of agents in any other time frame.6
First-Order Metrics
Section 1:  Ratios
Given the structure of an economic network, a set of ratios can be constructed:
it / mt (r - 1.1)
rt / mt (r - 1.2)
it / rt (r - 1.3)
It / Rt (r - 1.4)
N i gh
h
I
g
i
= = å å
1 1
/ (r - 1.5)
N i gE
g
i
= å
1
/ (r - 1.6)
n r jk
k
R
j
r
= = å å
1 1
/ (r - 1.7)
n r jE
j
r
= å
1
/ (r - 1.8)
N N gE
g
i
gh
h
I
g
i
= = = å å å
1 1 1
/ (r - 1.9)
n n jE
j
r
jk
k
R
j
r
= = = å å å
1 1 1
/ (r - 1.10)
N n gE
g
i
jE
j
r
= = å å
1 1
/ (r - 1.11)
N n gh
h
I
g
i
jk
k
R
j
r
= = = = å å å å
1 1 1 1
/ (r - 1.12)
Section 2:  Properties
Let z be a given matrix of any given agent metrics.  Let Z be a given matrix of any given exchange metrics.
A:  Category 1
Let category 1 level 1 property 1 (C1-L1-P1) be Atz.
Let category 1 level 1 property 2 (C1-L1-P2) be Btz.
Let category 1 level 1 property 3 (C1-L1-P3) be Ctz.
Let category 1 level 2 property 1 (C1-L2-P1) be NtZ.
Let category 1 level 2 property 2 (C1-L2-P2) be ntZ.
Let category 1 level 3 property 1 (C1-L3-P1) be btZ.
Let category 1 level 3 property 2 (C1-L3-P2) be ctZ.
B:  Category 2
Let category 2 level 1 property 1 (C2-L1-P1) be itz1 / mtz2.7
Let category 2 level 1 property 2 (C2-L1-P2) be rtz3 / mtz2.
Let category 2 level 1 property 3 (C2-L1-P3) be itz1 / rtz3.
Let category 2 level 1 property 4 (C2-L1-P4) be ItZ1 / RtZ2.
Let category 2 level 2 property 1 (C2-L2-P1) be [ ][ ]/ N Z i gh
h
I
g
i
= = å å
1 1
1 .
Let category 2 level 2 property 2 (C2-L2-P2) be [ ][ ]/ N Z i gE E
g
i
1
1 = å .
Let category 2 level 2 property 3 (C2-L2-P3) be [ ][ ]/ n Z r jk
k
R
j
r
2
1 1 = = å å .
Let category 2 level 2 property 4 (C2-L2-P4) be [ ][ ]/ n Z r jE E
j
r
2
1 = å .
Let category 2 level 3 property 1 (C2-L3-P1) be ([ ][ ]) / ([ ][ ]) N Z N Z gE
g
i
E gh
h
I
g
i
= = = å å å
1
1
1 1
1 .
Let category 2 level 3 property 2 (C2-L3-P2) be ([ ][ ]) / ([ ][ ]) n Z n Z jE
j
r
E jk
k
R
j
r
= = = å å å
1
2
1 1
2 .
Let category 2 level 3 property 3 (C2-L3-P3) be ([ ][ ]) / ([ ][ ]) N Z n Z gE E
g
i
jE
j
r
E 1
1 1
2
= = å å .
Let category 2 level 3 property 4 (C2-L3-P4) be ([ ][ ]) / ([ ][ ]) N Z n Z gh
h
I
g
i
jk
k
R
j
r
1
1 1 1 1
2
= = = = å å å å .
C:  Category 3
Let category 3 property 1 (C3-P1) be the relationship between exchange types.  C3-P1 can be determined by
identifying rows in bt and ct that correspond to the same agent.
Let category 3 property 2 (C3-P2) be the relative value of exchanges.  C3-P2 can be determined by
comparing elements in a given Z.
Section 3:  Accounting
A:  Money Supply
Let the domestic currency of a given economic network be good ND in Nt, and good nD in nt.  Where interest
exists, the interest rate charged on loans influences the value of ND, and the interest rate paid on investments influences
the value of nD.
Where financial institutions can loan deposits, the multiplier effect on the supply of currency is given by
([ ][ ]) / ([ ][ ]) N Z n Z gD D
g
i
jD
j
r
D 1
1 1
2
= = å å (3.1)
where Z1D is the value of loans and Z2D is the value of deposits, measured in units of the domestic currency.  (1.1) is a
form of C2-L3-P3.
Where financial institutions must retain a given percentage of the deposits they hold,
0 1 1
1 1
2 £ £
= = å å ([ ][ ]) / ([ ][ ]) / N Z n Z rr gD D
g
i
jD
j
r
D (3.2)
where rr is the reserve ratio.8
Where the value of goods measured in units of the domestic currency changes, the price level is given by
N N gD
g
i
gh
h
I
g
i
= = = å å å
1 1 1
/ (3.3)
(1.3) is a form of (r - 2.9).
Accordingly, the inflation rate is given by
[( / ) ( / )]/ ( / ) , , , , , , N N N N N N gD t
g
i
gh t gD t gh t gD t gh t
h
I
g
i
g
i
h
I
g
i
g
i
h
I
g
i
=
- - - -
= = = = = = = = å å å å å å å å å -
1
1 1 1 1
1 1 1 1 1 1 1 1
(3.4)
B:  Balance of Payments
Let E be the set of goods exported by the agents in an economy, and let e be the set of goods imported.  Using
these definitions, the current account balance is given by
[ ][ ] [ ][ ] N Z n Z gE E
g
i
je e
j
r
1
1
2
1 = = å å - (3.5)
where Z1E is an index of the prices of the goods in E and Z2e is an index of the prices of the goods in e, measured in
units of the domestic currency.
Let F be the set of domestically owned foreign assets and let f be the set of foreign owned domestic assets.
Using these definitions, the capital account balance is given by
[ ][ ] [ ][ ] N Z n Z gF F
g
i
jf f
j
r
1
1
2
1 = = å å - (3.6)
where Z1F is an index of the market values of the assets in F and Z2f is an index of the market values of the assets in f,
measured in units of the domestic currency.
C:  Income
Let p be price and q be quantity.  Where R is revenue,
R = (p1)(q1)
   = [bt][Z1t] (3.7)
where Z1t is a I x 1 matrix.
Similarly, where C is cost,
C = (p2)(q2)
   = [ct][Z2t] (3.8)
where Z2t is a R x 1 matrix.
For a given economic network,
pt = R - C
    = [bt][Z1t] - [ct][Z2t] (3.9)
Accordingly,
p p T t
t
T
=
= å
1
(3.10)
for t = 1...T periods.
Where loans exist, the values of any loans are stated in Z1t, and the costs of loans are stated in Z2t.  The
revenues generated by investments are stated in Z1t, and the costs of investments are stated in Z2t.9
Once interest rates, price levels, and inflation are determined, their effects on the value of loans and
investments can be calculated using conventional equations.  The results can be incorporated into Z1t and Z2t as
required.10
Second-Order Metrics
Section 1:  Type 1 Statistics
Let v be velocity.
V
b c
t
xt
u w
w
R
u
I
=
+
= = å å ( ) ( ) 1 1
1 1 (1.1)
V
b Z c Z
t
yt
t t =
+ 1 2 (1.2)
where Z1 is a I x 1 matrix and Z2 is a R x 1 matrix.
Let M be mass.
M b c xt u w
w
R
u
I
= +
= = å å ( ) ( ) 1 1
1 1
(1.3)
M b Z c Z yt t t = + 1 2 (1.4)
Section 2:  Type 2 Statistics
Let a be acceleration.
a
V V
t
xt
xt xt =
- -1 (2.1)
a
V V
t
yt
yt yt =
- -1
(2.2)
Let g be growth.
g
M M
t
xt
xt xt =
- -1 (2.3)
g
M M
t
yt
yt yt =
- -1
(2.4)
Section 3:  Type 3 Statistics
Let P be momentum.
Pxt = (Mxt)(Vxt)
    =
+
= = å å [( ) ( )] b c
t
u w
w
R
u
I
1 1
2
1 1 (3.1)
Pyt = (Myt)(Vyt)
    =
+ ( ) b Z c Z
t
t t 1 2
2
(3.2)
Let F be force.
Fxt = (gxt)(axt)11
    = + - +
= =
-
=
-
= å å å å
1
3 1 1
1 1
1 1
1
1 1
1
2
t
b c b c ut wt
w
R
u
I
ut
u
I
wt
w
R
[( ) ( )] (3.3)
Fyt = (gyt)(ayt)
    = + - + - - - -
1
3 1 2 1 1 1 1 2 1
2
t
b Z c Z b Z c Z t t t t t t t t [( ) ( )] (3.4)
Section 4:  Type 4 Statistics
A:  Lower Thresholds for Type 1 Statistics
For a given At, assume a minimum set of reciprocators, Ct(min), and a minimum set of goods they require,
nt(min), where
nt(min) = [n1(min) ... na(min)] (4.1)
From these definitions,
ct(min) = [Ct(min)][nt(min)] (4.2)
At minimum velocity bt is identical to ct(min), so
bt(min) = ct(min)  (4.3)
and
I = R (4.4)
If i > r, then  N n gE jE
j
r
g
i
<
= = å å
1 1
, for all E goods.
If i = r, then  N n gE jE
j
r
g
i
=
= = å å
1 1
, for all E goods.
If i < r, then  N n gE jE
j
r
g
i
>
= = å å
1 1
, for all E goods.
V
b c
t
c
t
xt
u w
w u
w
w
(min)
( ) ( )
=
+
=
= = = å å å 1 1
1 1
1
1
2
a a a
(4.5)
V
b Z c Z
t
c Z
t
yt
t t t
(min)
(min) (min) (min) =
+
=
1 2 2 2
(4.6)
M b c c xt u w
w u
w
w
(min) ( ) ( ) = + =
= = = å å å 1 1
1 1
1
1
2
a a a
(4.7)
M b Z c Z c Z yt t t t (min) (min) (min) (min) = + = 1 2 2 2 (4.8)
B:  Upper Thresholds for Type 1 Statistics
For a given At, assume a maximum set of initiators, Bt(max), and a maximum set of goods they produce, Nt(max),
where
Nt(max) = [N1(max) ... NW(max)] (4.9)
From these definitions,
bt(max) = [Bt(max)][Nt(max)] (4.10)
At maximum velocity ct is identical to bt(max), so12
bt(max) = ct(max)  (4.11)
and
I = R (4.12)
If i > r, then  N n gE jE
j
r
g
i
<
= = å å
1 1
, for all E goods.
If i = r, then  N n gE jE
j
r
g
i
=
= = å å
1 1
, for all E goods.
If i < r, then  N n gE jE
j
r
g
i
>
= = å å
1 1
, for all E goods.
V
b c
t
b
t
xt
u w
w u
u
u
(max)
( ) ( )
=
+
=
= = = å å å 1 1
1 1
1
1
2
W W W
(4.13)
V
b Z c Z
t
b Z
t
yt
t t t
(max)
(max) (max) (max) =
+
=
1 2 1 2
(4.14)
M b c b xt u w
w u
u
u
(max) ( ) ( ) = + =
= = = å å å 1 1
1 1
1
1
2
W W W
(4.15)
M b Z c Z b Z yt t t t (max) (max) (max) (max) = + = 1 2 1 2 (4.16)
Section 5:  Type 5 Statistics
A:  Lower Thresholds for Type 2 Statistics
a
V V
t
xt
xt xt
(min)
(min) =
- -1
(5.1)
a
V V
t
yt
yt yt
(min)
(min) =
- -1
(5.2)
g
M M
t
xt
xt xt
(min)
(min) =
- -1
(5.3)
g
M M
t
yt
yt yt
(min)
(min) =
- -1
(5.4)
B:  Upper Thresholds for Type 2 Statistics
a
V V
t
xt
xt xt
(max)
(max) =
- -1
(5.5)
a
V V
t
yt
yt yt
(max)
(max) =
- -1
(5.6)
g
M M
t
xt
xt xt
(max)
(max) =
- -1
(5.7)13
g
M M
t
yt
yt yt
(max)
(max) =
- -1
(5.8)
Section 6:  Type 6 Statistics
A:  Lower Thresholds for Type 3 Statistics
Pxt(min) = (Mxt(min))(Vxt(min))
          =
= å [ ] 2 1
2
1
c
t
w
w
a
(6.1)
Pyt(min) = (Myt(min))(Vyt(min))
          =
( ) (min) 2 2
2 c Z
t
t
(6.2)
Fxt(min) = (gxt(min))(axt(min))
          = - +
=
-
=
-
= å å å
1
2 3 1
1
1 1
1
1 1
1
2
t
c b c wt
w
ut
u
I
wt
w
R
[( ) ( )]
a
(6.3)
Fyt(min) = (gyt(min))(ayt(min))
          = - + - - - -
1
2 3 2 1 1 1 1 2 1
2
t
c Z b Z c Z t t t t t t [( ) ( )] (min) (6.4)
B:  Upper Thresholds for Type 3 Statistics
Pxt(max) = (Mxt(max))(Vxt(max))
          =
= å [ ] 2 1
2
1
b
t
u
u
W
(6.5)
Pyt(max) = (Myt(max))(Vyt(max))
          =
( ) (max) 2 1
2 b Z
t
t
(6.6)
Fxt(max) = (gxt(max))(axt(max))
          = - +
=
-
=
-
= å å å
1
2 3 1
1
1 1
1
1 1
1
2
t
b b c ut
u
ut
u
I
wt
w
R
[( ) ( )]
W
(6.7)
Fyt(max) = (gyt(max))(ayt(max))
          = - + - - - -
1
2 3 1 1 1 1 1 2 1
2
t
b Z b Z c Z t t t t t t [( ) ( )] (max) (6.8)14
First-Order Dynamics
Section 1:  Elasticity
Let elasticity be defined as the relationship between the price of a good and quantity produced or consumed,
or
¶
¶
Q
P
x
t
t
t = (1.1)
Let consumer determined elasticity (CDE) be defined as an elasticity where xt < -1.  Let neutrally determined
elasticity (NDE) be defined as an elasticity where xt = -1.  Let producer determined elasticity (PDE) be defined as an
elasticity where xt > -1.
Section 2:  Returns to Scale
Let returns to scale (RS) be defined as the relationship between the quantity of a good produced or consumed
and its price, or
¶
¶
P
Q
x
t
t
t = (1.2)
Let increasing returns to scale (IRS) be defined as an RS where xt < 0.  Let constant returns to scale (CRS)
be defined as an RS where xt = 0.  Let decreasing returns to scale (DRS) be defined as an RS where xt > 0.
Section 3:  Externality
Let externality be defined as the relationship between the returns to different economic activities, or
¶p
¶p
1
2
,
,
$
t
t
t x = (1.3)
Let increasing externality (IE) be defined as an externality where  $ xt > 0.  Let neutral externality (NE) be
defined as an externality where  $ xt = 0.  Let decreasing externality (DE) be defined as an externality where  $ xt < 0.15
Second-Order Dynamics
Section 1:  Production Constraints
Let the elasticity constraint be expressed as
bt = xbtZ1t (1.1)
Let the RS constraint be expressed as
Z1t =xbt bt (1.2)
Let the externality constraint be expressed as
p1t = $ xbt p2t (1.3)
Section 2:  Consumption Constraints
Let the elasticity constraint be expressed as
ct = xctZ2t (2.1)
Let the RS constraint be expressed as
Z2t =xct ct (2.2)
Let the externality constraint be expressed as
p1t = $ xct p3t (2.3)
Section 3:  Economic Network Constraints
Given the production and consumption constraints, an economic network’s income equation can be expressed
as
pt = [btZ1t - l1(bt - xbtZ1t) - l2(Z1t -xbt bt) - l3(p1t - $ xbt p2t)] -
[ctZ2t - l4(ct - xctZ2t) - l5(Z2t -xct ct) - l6(p1t - $ xct p3t)]
   = [btZ1t - l1(bt - xbtZ1t) - l2(Z1t -xbt bt) - l3(btZ1t - ctZ2t - $ xbt (b2tZ21t - c2tZ22t))] -
[ctZ2t - l4(ct - xctZ2t) - l5(Z2t -xct ct) - l6(btZ1t - ctZ2t - $ xct ( b3tZ31t - c3tZ32t))]
(3.1)
Accordingly,
¶p
¶
l l l l
b
Z x Z Z
t
t t t t = - + - + 1 1 2 3 1 6 1 (3.2),
¶p
¶
l l l l
Z
b x b b
t
t t t t
1
1 2 3 6 = - + - + (3.3),
¶p
¶
l l l l
c
Z Z x Z
t
t t t t = - + - - 3 2 2 4 5 6 2 (3.4), and
¶p
¶
l l l l
Z
c c x c
t
t t t t
2
3 4 5 6 = - - + - (3.5).16
A:  Production-Specific Constraints
¶p
¶
l
x
Z
bt
t = 1 1 (3.6)
¶p
¶
l
x
b
bt
t = 2 (3.7)
¶p
¶
l
$
( )
x
b Z c Z
bt
t t t t = - 3 2 21 2 22 (3.8)
¶p
¶
l
b
x Z
t
bt t
2
3 21 = $ (3.9)
¶p
¶
l
Z
x b
t
bt t
21
3 2 = $ (3.10)
¶p
¶
l
c
x Z
t
bt t
2
3 22 = - $ (3.11)
¶p
¶
l
Z
x c
t
bt t
22
3 2 = - $ (3.12)
¶p
¶l1
1 = - + b x Z t bt t (3.13)
¶p
¶l2
1 = - + Z x b t bt t (3.14)
¶p
¶l3
1 2 2 21 2 22 = - + + - b Z c Z x b Z c Z t t t t bt t t t t $ ( ) (3.15)17
B:  Consumption-Specific Constraints
¶p
¶
l
x
Z
ct
t = - 4 2 (3.16)
¶p
¶
l
x
c
ct
t = - 5 (3.17)
¶p
¶
l
$
( )
x
b Z c Z
ct
t t t t = - - 6 3 31 3 32 (3.18)
¶p
¶
l
b
x Z
t
ct t
3
6 31 = - $ (3.19)
¶p
¶
l
Z
x b
t
ct t
31
6 3 = - $ (3.20)
¶p
¶
l
c
x Z
t
ct t
3
6 32 = $ (3.21)
¶p
¶
l
Z
x c
t
ct t
32
6 3 = $ (3.22)
¶p
¶l4
2 = - c x Z t ct t (3.23)
¶p
¶l5
2 = - Z x c t ct t (3.24)
¶p
¶l6
1 2 3 31 3 32 = - - - b Z c Z x b Z c Z t t t t ct t t t t $ ( ) (3.25)

System-based Economics

http://128.118.178.162/eps/mic/papers/0408/0408005.pdf

System-based Economics

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