- Email: [email protected]
Bilinear, dynamic single-ports and bond graphs of economic systems
Bilinear, Dynamic Single-ports and Bond Grafihs of Economic Systems by JOHNW.BREWER Professor CA 95616,
of Mechanical U.S.A.
Engineering,
University
of
California,
Davis,
and PAULP.CRAIG Professor U.S.A.
of Applied
Science,
University of California,
Davis,
It is shown that reinuestment and inventory effects can bond-graph terms. The associated diferential equations in price variables are bilinear forms. These effects are the fundamental inertia of economic bond graph theory. Properties of the components and graphs are discussed.
ABSTRACT;
CA 95616,
be described in and order-flow and compliance example market
I. Introduction Bond graphs are well known graphical descriptions of dynamic physical systems (1). These descriptions are sufficiently precise to serve as immediate guides to the derivation of system equations and simulations (2). The extension to heat transfer systems is well known (1, 3). Inherent in bond graphs are component descriptions (“constitutive relations”), system topology and power preserving transformationst (“junction structures”). Some effort has been made to extend bond graph theory to economic systems (4) and to demographic systemsS (5). The extension to economic systems is made most easily since a “value rate” (cash flow, for example) plays the same role in these systems as is played by power in physical systems. The common junction structure elements of physical system bond graphs (0, 1, TF) can also be used to describe the topology of many economic systems. Among the challenges to the economic bond graph theorist is the discernment of the constitutive relations of all common components of economic theory. The supply and demand curves are quite easily described in this way (4). The contribution of this paper is the statement of constitutive relations for tThe power preserving transformations, represented in bond graphs of most physical systems, have no meaning in the bond graphs of heat transfer. For this reason, some authors refer to the bond graphs of heat transfer as “pseudo” bond graphs (1,3). SAs it turns out, bond graphs of demographic systems are “pseudo” in the sense described in footnote t above. See the basic reference (5). The
Franklin
Institute
001~32/82/04018S-12$03.00/O
185
John W. Brewer and Paul P. Craig
two very important dynamic effects: the buildup of production from reinvestment of retained earnings and the buildup of inventories due to price disequilibrium. These effects are represented, respectively, by inertia (I) and compliance (C). There are many interesting aspects to the constitutive relations presented here: they are nonlinear but are associated with differential equations whose right hand sides are bilinear forms (hence the name “bilinear dynamic components”). Other interesting features will be highlighted in the discussion. In what follows, a summary of basic economic-bond-graph terminology is followed by a detailed discussion of the bilinear components. An important property and two simple but interesting examples are then provided. II. The Bond Graphs of Economic
Systems
In this section, the terminology of economic bond graphs is presented. In general, the concepts involved are nearly identical to those which form the basis of physical bond graphs. However, some changes in notation and terminology make the translation of theory from the conventional economic literature to bond graph symbols a bit easier. First, the conjugate variables, which define a port, are p = price per unit commodity,
(1)
f = time rate of flow of orders.
(2)
The price is taken to be the dollar price here, although the labor price or energy price is worthy of study. The flow of orders is a slightly fictitious variable which is numerically equal to the flow rate of commodity but is taken to be in the opposite direction (from buyer to seller) hence in the same direction as the flow of money.t The flow of money is denoted v and is easily shown to be V=pf.
(3)
Indefinite integrals are impulse Y=
I pdt,
(4)
and orders
tThis sets up an analogy between commodity and negative material flow in most electrical currents). sterile from a theoretical point of view.
186
It is likely,
however,
electrons (the true that the analogy is
Journal
of the Franklin Institute Pergamon Press Ltd.
Bilinear, Dynamic Single-ports
and Bond Graphs
Positive values of q represent accumulations of unfilled orders while negative values represent accumulation of commodity, commonly called inventory or stocks. The basic single ports are associated with nonlinear functional relations of the form P = d%(f),
(revenue source-sinks),
(6)
4 = #v(P),
(compliances),
(7)
y = 41(f).
(inertia).
(8)
Most of the above notions are summarized in the tetrahedron of state of Fig. 1. The sign conventions associated with revenue source-sinks make them more interesting than mere analogs of physical resistors. More on this later. The signal sources S,, S, have the same meaning as the corresponding symbols in the bond graphs of physical systems (1). 1. Sign conaention and elementary junction structures The sign convention is the assigned positive direction for the flow of cash, I? This sign convention is illustrated in Fig. 2. It must be understood from the outset that sign convention is part of the specification of the graph: it will likely never be aut0mated.t
Order (commodity) flow
FIG. 1. Tetrahedron
rate
of state for economic
systems.
-IThere are those who believe that the sign convention should be part of the specification of a bond graph of a physical system; that is, the specification of the sign convention cannot be automated (programmed). Vol. 313, No. 4, pp. 185-l%. Printed in Great Britain
April 1982
187
John W. Brewer
and Paul P. Craig
f, =f2
P, = P2 +
+ fj
FIG. 2. (a) The zero junction
as an order (commodity) one-junction as a cost accumulation
P,
distribution point.
point.
(b) The
The zero- and one-junctions have the same mathematical meaning here as in physical bond graphs. The interpretation is different, however: as illustrated in Fig. 2, zero-junctions represent the distribution of orders (and goods) at a common price while one-junctions represent the accumulation of costs on a common flow. For both junctions in Fig. 2, the requirement PIfI
=
Pzf2
+
(9)
P3f3
is known as Walrus’s Law to economists and is simply a statement of proper cash accounting. Obviously, this statement is the exact mathematical equivalent of power conservation in physical systems. It is Walras’s Law which makes the use of bond graph descriptions of economic systems SO appealing. A possible interpretation of Fig. 2(a) is that a single demand 0 is bonded to two suppliers 0, 0. Figure 2(b) might represent a demand @ bonded to a supplier 0 through a middle man 0. The relation pI = p2 + p3 means the sales price is the transportation cost plus the production cost.
2. Bidding assignments The notation and concepts associated with causality in the discussions of physical bond graphs are crucial to the derivation of state equations (6). Whether or not causality is truly involved here is a question of great interest to the philosophically minded and, yet, somehow irrelevant to many important matters. A conventional economic concept which seems to be equally well described by the same notation is that of bidding. For this reason and perhaps because of our own philosophical orientation, the causal strokes of physical bond graphs are called bidding strokes in our discussions of economics. The mathematical meaning is the same in the two cases.t Figure 3(a) is a fully augmented bond graph.
tin our personal conversations with physical bond graphers, we note a certain tendency to think of “causality” assignments more as an assignment of signal flow. Journal 188
of the FrankIin Institute Pergamon Press Ltd.
Bilinear, Dynamic Single-ports
and Bond Graphs
FIG. 3.
4 SUPPLY
Rs’
f
x
Ro
Demand
FIG. 4. FIG. 3. Sign convention
and bidding strokes on the bond graph of a simple economic system.
FIG. 4. Supply-demand
curves and the bond graph representation market.
of the elementary
3. Supply and demand One encounters supply and demand curves in every elementary text on economics (7). Examples of these curves are shown in Fig. 4 in terms of the notation of this paper. Clearly, both curves are represented by general case (6) and so the market system is represented by the bond graph of Fig. 4. Notice that, unlike the usual case of physical systems, R-components can be sources of revenue as well as sinks.t Revenue sources, of course, do not share the properties one commonly associates with most passive physical resistances. This example underscores our assertion that the sign convention must be considered as part of the specification of the graph. 4. Multiports For sake of completeness in this introductory section, we should mention that economic systems contain interesting and important multiports. Two such multiports are illustrated in Fig. 5. Constitutive relations for these multiports are obtained by applying Lagrange multiplier theory to certain constrained optimization problems (4). The results sometimes define lossless R-fields (non-linear junction structures). The process is interesting but not relevant to the present discussion. tA reviewer pointed out that both sinks and sources could be given the convention and the differences accounted for in the constitutive relations. alternative view is valid but less natural. Vol. 313, No. 4, pp. 185-l%, Printed in Great Britain
April
same This
1982
189
John W. Brewer and Paul P. Craig (al
(b)
Labor demand
Demand
1
d
Multtport
Multiport
demand
Food
Energy
SUPPlY
suPPlY
Labor SUPPlY
FIG. 5. Two examples
III. Bilinear Dynamic
of economic
multiports.
Components
In this section, the constitutive relation for certain dynamic, economic effects are presented. Begin with the situation illustrated in Fig. 6. Here some of the revenue obtained by the supplier is reinvested to increase production. The situation is well described by
fl = f* = f3 PI = P2 + P3
(sales price is production cost plus right-hand side of (10) is the revenue interpret KI = cost of increasing
charge for reinvestment). flowing into reinvestment,
f
by one unit/year.
Since the one may
(11)
(a) Retnvestment
SUPPlY
(b)
FIG. 6. Dynamics
of
reinvestment (inertia): (a) word bond graph representation.
bond
graph,
Journalof 190
(b) complete
the Franklin Institute f’ergamon Press Ltd.
Bilinear,
If one differentiates
the constitutive YZ =
then (10) will be obtained.
Dynamic
Single-ports
and Bond Graphs
relation
(12)
KI ln f~ + KO
Since (12) is a special case of (8), it is clear that
reinvestment
effects are well described as inertia components. This inertia is termed bilinear because the right-hand side of (10) is a
bilinear form. There are, of course, other ways to obtain capital to increase production: loans or sale of partial ownership (sale of stocks). The second case to be investigated is illustrated in Fig. 7. Here the producer bids the market price (p, = p3) based upon his inventory. The demand orders are sent either to the warehouse (inventory) or directly to the factory, hence f, = f2 + f3. The zero-junction nicely describes this situation. The price that is bid is likely to have the form illustrated in Fig. 7(b): the asking price monotonically diminishes with buildup of inventory and monotonically increases with the buildup of unfilled orders. It is probable that this relation can be well approximated in certain limited and important regions by p2 - exp (q2) or, equivalently, q2=KClnp2+K,.
(a)
(13)
1 I ,i
0
$1
Demand
E\ \ (b)
I
Stored commodity
--G---D
I
Unfilled orders
(c)
FIG.
7.
Dynamics of inventory (compliance): (a) word bond graph, (b) constitutive relation of inventory effect, (c) complete bond graph representation.
Vol. 313, No. 4, pp. 185-l%. Printed in Great Britain
April
1982
191
John W. Brewer and Paul P. Craig When (13) and (7) are compared, it is clear that inventory effects are well described as compliances. Certainly, alternative constitutive relations could be proposed. When (13) is differentiated with respect to time, the result is
dpz
dt
so that a bilinear compliance
forms
& P2f2
(14)
is the result.
1. Interpretation of Kr, K, Denote the steady state values define I, and C, by
Generic
=
of conjugate
variables
by
f* and p* and
Kr = fJ,,
(1%
Kc = P*C,.
(16)
of (10) and (14) are
!Lf_L
(f >p
(17)
dt- I, f*
and
dp_l p f; dt - C, 0P* thus, I, conjugate
and C, variables
(18)
are analogous to physical inertias remain near steady state values.
IV. A Basic Property of Bilinear Components, the Basic Bilinear I-R-C Market
Logistic
and
Growth,
compliances
if
and
In this section, we describe an interesting property of the bilinear components and treat two fundamental examples in some detail.
dynamic
1. Bilinear components maintain the sign of state variables Generic forms of (12) and (13) can be rewritten as f = exp ([Y -
&IIK~)
(19)
and P =
(20)
exp ([q - KJIK).
Thus, f and p will always be positive. The same conclusion from (17) and (18) for which p = 0 or f = 0 represent capture Journal
192
can be obtained states! of the Franklin Institute Pergamon Press Ltd.
Bilinear, Dynamic Single-ports
and Bond Graphs
This property is unexpected but very welcome: there is no good interpretation of a negative price and any mathematical model that would allow the flow of orders to reverse on certain bonds (go from seller to buyer, for instance) would be an embarrassment. 2. Market penetration by logistic growtrr (6) The first example to be studied is illustrated in Fig. 8. Here a supplier builds up its output by reinvesting part of its revenue. If the supplier supplies a relatively small share of the market, bilateral effects are greatly diminished and the entire rest of the market may be thought of as a price-signal source. It is an easy matter to show that df
5
where 4R(.) represents
=
& rrsp- 4JRCf)l
(21)
the supply curve. The steady state condition is 4R(f*)
If one linearizes the supply curvet 4R(f)
=
R =
=
(22)
s,.
about this steady state 4R(f*)+R(f
(23)
-f*),
4Kf*)
where the prime denotes differentiation
with respect to
f.
It follows that
(24) This is the famous logistic growth equation (8) and is an obvious implication of bilinear components with linearized supply curves.
FIG. 8. Bond
graph
of market
penetration of a relatively growth).
tObviously, linearization of the supply tion of the differential equation. Vol. 313, No. 4, pp. 185-l%, Printed in Great Britain
April
curve
should
small
producer
not be confused
(logistic
with lineariza-
1982
193
John W. Brewer and Paul P. Craig
The approximation of market penetration by logistic growth has considerable empirical verification (6) and it is interesting to find it as a result of our analysis.
3. The bilinear I-R-C-market Consider the market illustrated in Fig. 8: the supplier bids the market price, p, based upon his inventory, and the revenue in excess of his production cost is reinvested to increase production, f. Proceeding from (10) and (14) with the fully augmented bond graph as a guide, state variable equations are found to be
S=&P(*,(P)-f)
(25)
E
and
where &.s(‘) is the constitutive function of supply (that is, the supply curve) and &,(e) is 4&(a) (the inverse constitutive function of demand). Clearly, the steady state conditions (p*, f*) are given by the intersection of the supply and demand curves: +D(P,)
=
f*v
(27)
W-9
P* = 4RS(fd*
If (25) and (26) are linearized about this point, the eigenvalues, system are roots of
A, of the linear
A2+a,h+az=0
(2%
where al = +-
*
4kdfd
-&
+btPd
(31)
and
a2
=
&*
*
[1 - vUp,Midf*)l,
where the prime denotes differentiation of a function argument and I, and C, are defined in (15) and (16). 194
with respect Journal of the
to its
Franklin Institute Pergamon Press Ltd.
Bilinear, Dynamic Single-ports
and Bond Graphs
Investment
c. Inventory
FIG. 9. A basic I-R-C market oscillator. It is a trivial task to show that the A are complex if (33)
We can expect this condition to occur if the intersection of the supply and demand curves is in a region where the slopes 4hS and I,G~are small. This is, of course, the condition for oscillation (other than limit cycles). The linearized system is stable if al, a2 > 0 [refer to (31)l. This will be the usual case since (of, is most often negative (less is demanded as the market price increases). There are, however, certain exotic situations where +f, is positive in some limited range: markets for certain elitist goods where price increases make the commodity seem more desirable and the rate demanded actually increases. In this situation instability of the equilibrium point can occur. It is probable that limit cycles will occur in this situation. V. Concluding Remarks
It is unlikely that linear compliances and inertias (4) will play much of a role in dynamic economic-system analysis. However, the bilinear component effects described in this paper do occur in many markets and so the bilinear compliance and inertia are as fundamental to economic bond graph theory as are the supply and demand curves. The logarithmic constitutive relations also appear in physical system theory; for instance, in thermodynamics (1). In this paper, we use p and f as state variables. This choice is also used in physical system theory (9), although, more commonly, y and q are chosen there (1,2). It is only with the choice of p, f that the bilinearity of the components is apparent. A fact suggested by the examples of Section IV is that quadratic differential equation theory will be as important to the study of economic state equations as the linear theory is to the study of physical state equations. Quadratic equations appear quite commonly in the mathematical ecology literature (IO, 11). Vol. 313. No. 4. pp. 185-l%, Printed in Great Britain
April 1982
195
John
W. Brewer
and Paul P. Craig
Real markets are very complicated: many suppliers and many demanders acting through nonlinear multiports (refer to Fig. 5). The complexity of economic systems is an important motivation for the development of bond graph theory; bond graphs are unambiguous and succinct statements of models and are the basis for automated analysis procedures. References (1) D. C. Karnopp and R. C. Rosenberg, “System dynamics: a unified approach.” John Wiley, New York, 1975. (2) R. C. Rosenberg, “State space formulation for bond graph models of multiport systems”, J. Dynamic Syst., Meas., Control, Series G., V. 93, pp. 35-40, 1971. (3) M. Hubbard and J. W. Brewer, “Pseudo bond graphs of circulating fluids with application to solar heating design”. J. Franklin Inst., Vol. 311, pp. 339-354, 1981. (4) J. W. Brewer, “Structure and cause and effect relations in social system simulations”, IEEE Trans. Syst., Man, Cybernet., SMC-7, pp. 468-474, 1977. (5) J. W. Brewer, “Bond graphs of age dependent renewal”, ASME Winter annual Meeting, 80-WA/DSC-4, Chicago, II., 1980. (6) V. Peterka, “Macrodynamics of technological change: market penetration by new technologies”, Intern. Inst. Appl. Syst. Anal., RR-77-22, A-2361, Laxenberg, Austria, 1977. (7) P. A. Samuelson, “Economics”, McGraw-Hill, New York, 1970. (8) P. A. Samuelson, “Foundations of Economic Analysis”, Atheneum, New York, 1970. (9) D. C. Karnopp and R. C. Rosenberg, “Analysis and simulation of multiport systems”, M.I.T. Press, Cambridge, Ma., 1968. “Some simple nonlinear dynamic models of (10) A. V. Quinlan and H. M. Paynter, interacting element cycles in aquatic ecosystems”, J. Dynam. Syst., Meas., Co&r. (ASME), Vol. 98, Series G, N.1, pp. 6-19, 1976. “Generalized predator-prey oscillation in ecological and (11) P. A. Samuelson, economic systems”. Proc. Natn. Acad. Sci. U.S.A., Vol. 68, pp. 980-983, 1971.
196
Journal of the Franklin Institute Pergamon Press Ltd.
of Mechanical U.S.A.
Engineering,
University
of
California,
Davis,
and PAULP.CRAIG Professor U.S.A.
of Applied
Science,
University of California,
Davis,
It is shown that reinuestment and inventory effects can bond-graph terms. The associated diferential equations in price variables are bilinear forms. These effects are the fundamental inertia of economic bond graph theory. Properties of the components and graphs are discussed.
ABSTRACT;
CA 95616,
be described in and order-flow and compliance example market
I. Introduction Bond graphs are well known graphical descriptions of dynamic physical systems (1). These descriptions are sufficiently precise to serve as immediate guides to the derivation of system equations and simulations (2). The extension to heat transfer systems is well known (1, 3). Inherent in bond graphs are component descriptions (“constitutive relations”), system topology and power preserving transformationst (“junction structures”). Some effort has been made to extend bond graph theory to economic systems (4) and to demographic systemsS (5). The extension to economic systems is made most easily since a “value rate” (cash flow, for example) plays the same role in these systems as is played by power in physical systems. The common junction structure elements of physical system bond graphs (0, 1, TF) can also be used to describe the topology of many economic systems. Among the challenges to the economic bond graph theorist is the discernment of the constitutive relations of all common components of economic theory. The supply and demand curves are quite easily described in this way (4). The contribution of this paper is the statement of constitutive relations for tThe power preserving transformations, represented in bond graphs of most physical systems, have no meaning in the bond graphs of heat transfer. For this reason, some authors refer to the bond graphs of heat transfer as “pseudo” bond graphs (1,3). SAs it turns out, bond graphs of demographic systems are “pseudo” in the sense described in footnote t above. See the basic reference (5). The
Franklin
Institute
001~32/82/04018S-12$03.00/O
185
John W. Brewer and Paul P. Craig
two very important dynamic effects: the buildup of production from reinvestment of retained earnings and the buildup of inventories due to price disequilibrium. These effects are represented, respectively, by inertia (I) and compliance (C). There are many interesting aspects to the constitutive relations presented here: they are nonlinear but are associated with differential equations whose right hand sides are bilinear forms (hence the name “bilinear dynamic components”). Other interesting features will be highlighted in the discussion. In what follows, a summary of basic economic-bond-graph terminology is followed by a detailed discussion of the bilinear components. An important property and two simple but interesting examples are then provided. II. The Bond Graphs of Economic
Systems
In this section, the terminology of economic bond graphs is presented. In general, the concepts involved are nearly identical to those which form the basis of physical bond graphs. However, some changes in notation and terminology make the translation of theory from the conventional economic literature to bond graph symbols a bit easier. First, the conjugate variables, which define a port, are p = price per unit commodity,
(1)
f = time rate of flow of orders.
(2)
The price is taken to be the dollar price here, although the labor price or energy price is worthy of study. The flow of orders is a slightly fictitious variable which is numerically equal to the flow rate of commodity but is taken to be in the opposite direction (from buyer to seller) hence in the same direction as the flow of money.t The flow of money is denoted v and is easily shown to be V=pf.
(3)
Indefinite integrals are impulse Y=
I pdt,
(4)
and orders
tThis sets up an analogy between commodity and negative material flow in most electrical currents). sterile from a theoretical point of view.
186
It is likely,
however,
electrons (the true that the analogy is
Journal
of the Franklin Institute Pergamon Press Ltd.
Bilinear, Dynamic Single-ports
and Bond Graphs
Positive values of q represent accumulations of unfilled orders while negative values represent accumulation of commodity, commonly called inventory or stocks. The basic single ports are associated with nonlinear functional relations of the form P = d%(f),
(revenue source-sinks),
(6)
4 = #v(P),
(compliances),
(7)
y = 41(f).
(inertia).
(8)
Most of the above notions are summarized in the tetrahedron of state of Fig. 1. The sign conventions associated with revenue source-sinks make them more interesting than mere analogs of physical resistors. More on this later. The signal sources S,, S, have the same meaning as the corresponding symbols in the bond graphs of physical systems (1). 1. Sign conaention and elementary junction structures The sign convention is the assigned positive direction for the flow of cash, I? This sign convention is illustrated in Fig. 2. It must be understood from the outset that sign convention is part of the specification of the graph: it will likely never be aut0mated.t
Order (commodity) flow
FIG. 1. Tetrahedron
rate
of state for economic
systems.
-IThere are those who believe that the sign convention should be part of the specification of a bond graph of a physical system; that is, the specification of the sign convention cannot be automated (programmed). Vol. 313, No. 4, pp. 185-l%. Printed in Great Britain
April 1982
187
John W. Brewer
and Paul P. Craig
f, =f2
P, = P2 +
+ fj
FIG. 2. (a) The zero junction
as an order (commodity) one-junction as a cost accumulation
P,
distribution point.
point.
(b) The
The zero- and one-junctions have the same mathematical meaning here as in physical bond graphs. The interpretation is different, however: as illustrated in Fig. 2, zero-junctions represent the distribution of orders (and goods) at a common price while one-junctions represent the accumulation of costs on a common flow. For both junctions in Fig. 2, the requirement PIfI
=
Pzf2
+
(9)
P3f3
is known as Walrus’s Law to economists and is simply a statement of proper cash accounting. Obviously, this statement is the exact mathematical equivalent of power conservation in physical systems. It is Walras’s Law which makes the use of bond graph descriptions of economic systems SO appealing. A possible interpretation of Fig. 2(a) is that a single demand 0 is bonded to two suppliers 0, 0. Figure 2(b) might represent a demand @ bonded to a supplier 0 through a middle man 0. The relation pI = p2 + p3 means the sales price is the transportation cost plus the production cost.
2. Bidding assignments The notation and concepts associated with causality in the discussions of physical bond graphs are crucial to the derivation of state equations (6). Whether or not causality is truly involved here is a question of great interest to the philosophically minded and, yet, somehow irrelevant to many important matters. A conventional economic concept which seems to be equally well described by the same notation is that of bidding. For this reason and perhaps because of our own philosophical orientation, the causal strokes of physical bond graphs are called bidding strokes in our discussions of economics. The mathematical meaning is the same in the two cases.t Figure 3(a) is a fully augmented bond graph.
tin our personal conversations with physical bond graphers, we note a certain tendency to think of “causality” assignments more as an assignment of signal flow. Journal 188
of the FrankIin Institute Pergamon Press Ltd.
Bilinear, Dynamic Single-ports
and Bond Graphs
FIG. 3.
4 SUPPLY
Rs’
f
x
Ro
Demand
FIG. 4. FIG. 3. Sign convention
and bidding strokes on the bond graph of a simple economic system.
FIG. 4. Supply-demand
curves and the bond graph representation market.
of the elementary
3. Supply and demand One encounters supply and demand curves in every elementary text on economics (7). Examples of these curves are shown in Fig. 4 in terms of the notation of this paper. Clearly, both curves are represented by general case (6) and so the market system is represented by the bond graph of Fig. 4. Notice that, unlike the usual case of physical systems, R-components can be sources of revenue as well as sinks.t Revenue sources, of course, do not share the properties one commonly associates with most passive physical resistances. This example underscores our assertion that the sign convention must be considered as part of the specification of the graph. 4. Multiports For sake of completeness in this introductory section, we should mention that economic systems contain interesting and important multiports. Two such multiports are illustrated in Fig. 5. Constitutive relations for these multiports are obtained by applying Lagrange multiplier theory to certain constrained optimization problems (4). The results sometimes define lossless R-fields (non-linear junction structures). The process is interesting but not relevant to the present discussion. tA reviewer pointed out that both sinks and sources could be given the convention and the differences accounted for in the constitutive relations. alternative view is valid but less natural. Vol. 313, No. 4, pp. 185-l%, Printed in Great Britain
April
same This
1982
189
John W. Brewer and Paul P. Craig (al
(b)
Labor demand
Demand
1
d
Multtport
Multiport
demand
Food
Energy
SUPPlY
suPPlY
Labor SUPPlY
FIG. 5. Two examples
III. Bilinear Dynamic
of economic
multiports.
Components
In this section, the constitutive relation for certain dynamic, economic effects are presented. Begin with the situation illustrated in Fig. 6. Here some of the revenue obtained by the supplier is reinvested to increase production. The situation is well described by
fl = f* = f3 PI = P2 + P3
(sales price is production cost plus right-hand side of (10) is the revenue interpret KI = cost of increasing
charge for reinvestment). flowing into reinvestment,
f
by one unit/year.
Since the one may
(11)
(a) Retnvestment
SUPPlY
(b)
FIG. 6. Dynamics
of
reinvestment (inertia): (a) word bond graph representation.
bond
graph,
Journalof 190
(b) complete
the Franklin Institute f’ergamon Press Ltd.
Bilinear,
If one differentiates
the constitutive YZ =
then (10) will be obtained.
Dynamic
Single-ports
and Bond Graphs
relation
(12)
KI ln f~ + KO
Since (12) is a special case of (8), it is clear that
reinvestment
effects are well described as inertia components. This inertia is termed bilinear because the right-hand side of (10) is a
bilinear form. There are, of course, other ways to obtain capital to increase production: loans or sale of partial ownership (sale of stocks). The second case to be investigated is illustrated in Fig. 7. Here the producer bids the market price (p, = p3) based upon his inventory. The demand orders are sent either to the warehouse (inventory) or directly to the factory, hence f, = f2 + f3. The zero-junction nicely describes this situation. The price that is bid is likely to have the form illustrated in Fig. 7(b): the asking price monotonically diminishes with buildup of inventory and monotonically increases with the buildup of unfilled orders. It is probable that this relation can be well approximated in certain limited and important regions by p2 - exp (q2) or, equivalently, q2=KClnp2+K,.
(a)
(13)
1 I ,i
0
$1
Demand
E\ \ (b)
I
Stored commodity
--G---D
I
Unfilled orders
(c)
FIG.
7.
Dynamics of inventory (compliance): (a) word bond graph, (b) constitutive relation of inventory effect, (c) complete bond graph representation.
Vol. 313, No. 4, pp. 185-l%. Printed in Great Britain
April
1982
191
John W. Brewer and Paul P. Craig When (13) and (7) are compared, it is clear that inventory effects are well described as compliances. Certainly, alternative constitutive relations could be proposed. When (13) is differentiated with respect to time, the result is
dpz
dt
so that a bilinear compliance
forms
& P2f2
(14)
is the result.
1. Interpretation of Kr, K, Denote the steady state values define I, and C, by
Generic
=
of conjugate
variables
by
f* and p* and
Kr = fJ,,
(1%
Kc = P*C,.
(16)
of (10) and (14) are
!Lf_L
(f >p
(17)
dt- I, f*
and
dp_l p f; dt - C, 0P* thus, I, conjugate
and C, variables
(18)
are analogous to physical inertias remain near steady state values.
IV. A Basic Property of Bilinear Components, the Basic Bilinear I-R-C Market
Logistic
and
Growth,
compliances
if
and
In this section, we describe an interesting property of the bilinear components and treat two fundamental examples in some detail.
dynamic
1. Bilinear components maintain the sign of state variables Generic forms of (12) and (13) can be rewritten as f = exp ([Y -
&IIK~)
(19)
and P =
(20)
exp ([q - KJIK).
Thus, f and p will always be positive. The same conclusion from (17) and (18) for which p = 0 or f = 0 represent capture Journal
192
can be obtained states! of the Franklin Institute Pergamon Press Ltd.
Bilinear, Dynamic Single-ports
and Bond Graphs
This property is unexpected but very welcome: there is no good interpretation of a negative price and any mathematical model that would allow the flow of orders to reverse on certain bonds (go from seller to buyer, for instance) would be an embarrassment. 2. Market penetration by logistic growtrr (6) The first example to be studied is illustrated in Fig. 8. Here a supplier builds up its output by reinvesting part of its revenue. If the supplier supplies a relatively small share of the market, bilateral effects are greatly diminished and the entire rest of the market may be thought of as a price-signal source. It is an easy matter to show that df
5
where 4R(.) represents
=
& rrsp- 4JRCf)l
(21)
the supply curve. The steady state condition is 4R(f*)
If one linearizes the supply curvet 4R(f)
=
R =
=
(22)
s,.
about this steady state 4R(f*)+R(f
(23)
-f*),
4Kf*)
where the prime denotes differentiation
with respect to
f.
It follows that
(24) This is the famous logistic growth equation (8) and is an obvious implication of bilinear components with linearized supply curves.
FIG. 8. Bond
graph
of market
penetration of a relatively growth).
tObviously, linearization of the supply tion of the differential equation. Vol. 313, No. 4, pp. 185-l%, Printed in Great Britain
April
curve
should
small
producer
not be confused
(logistic
with lineariza-
1982
193
John W. Brewer and Paul P. Craig
The approximation of market penetration by logistic growth has considerable empirical verification (6) and it is interesting to find it as a result of our analysis.
3. The bilinear I-R-C-market Consider the market illustrated in Fig. 8: the supplier bids the market price, p, based upon his inventory, and the revenue in excess of his production cost is reinvested to increase production, f. Proceeding from (10) and (14) with the fully augmented bond graph as a guide, state variable equations are found to be
S=&P(*,(P)-f)
(25)
E
and
where &.s(‘) is the constitutive function of supply (that is, the supply curve) and &,(e) is 4&(a) (the inverse constitutive function of demand). Clearly, the steady state conditions (p*, f*) are given by the intersection of the supply and demand curves: +D(P,)
=
f*v
(27)
W-9
P* = 4RS(fd*
If (25) and (26) are linearized about this point, the eigenvalues, system are roots of
A, of the linear
A2+a,h+az=0
(2%
where al = +-
*
4kdfd
-&
+btPd
(31)
and
a2
=
&*
*
[1 - vUp,Midf*)l,
where the prime denotes differentiation of a function argument and I, and C, are defined in (15) and (16). 194
with respect Journal of the
to its
Franklin Institute Pergamon Press Ltd.
Bilinear, Dynamic Single-ports
and Bond Graphs
Investment
c. Inventory
FIG. 9. A basic I-R-C market oscillator. It is a trivial task to show that the A are complex if (33)
We can expect this condition to occur if the intersection of the supply and demand curves is in a region where the slopes 4hS and I,G~are small. This is, of course, the condition for oscillation (other than limit cycles). The linearized system is stable if al, a2 > 0 [refer to (31)l. This will be the usual case since (of, is most often negative (less is demanded as the market price increases). There are, however, certain exotic situations where +f, is positive in some limited range: markets for certain elitist goods where price increases make the commodity seem more desirable and the rate demanded actually increases. In this situation instability of the equilibrium point can occur. It is probable that limit cycles will occur in this situation. V. Concluding Remarks
It is unlikely that linear compliances and inertias (4) will play much of a role in dynamic economic-system analysis. However, the bilinear component effects described in this paper do occur in many markets and so the bilinear compliance and inertia are as fundamental to economic bond graph theory as are the supply and demand curves. The logarithmic constitutive relations also appear in physical system theory; for instance, in thermodynamics (1). In this paper, we use p and f as state variables. This choice is also used in physical system theory (9), although, more commonly, y and q are chosen there (1,2). It is only with the choice of p, f that the bilinearity of the components is apparent. A fact suggested by the examples of Section IV is that quadratic differential equation theory will be as important to the study of economic state equations as the linear theory is to the study of physical state equations. Quadratic equations appear quite commonly in the mathematical ecology literature (IO, 11). Vol. 313. No. 4. pp. 185-l%, Printed in Great Britain
April 1982
195
John
W. Brewer
and Paul P. Craig
Real markets are very complicated: many suppliers and many demanders acting through nonlinear multiports (refer to Fig. 5). The complexity of economic systems is an important motivation for the development of bond graph theory; bond graphs are unambiguous and succinct statements of models and are the basis for automated analysis procedures. References (1) D. C. Karnopp and R. C. Rosenberg, “System dynamics: a unified approach.” John Wiley, New York, 1975. (2) R. C. Rosenberg, “State space formulation for bond graph models of multiport systems”, J. Dynamic Syst., Meas., Control, Series G., V. 93, pp. 35-40, 1971. (3) M. Hubbard and J. W. Brewer, “Pseudo bond graphs of circulating fluids with application to solar heating design”. J. Franklin Inst., Vol. 311, pp. 339-354, 1981. (4) J. W. Brewer, “Structure and cause and effect relations in social system simulations”, IEEE Trans. Syst., Man, Cybernet., SMC-7, pp. 468-474, 1977. (5) J. W. Brewer, “Bond graphs of age dependent renewal”, ASME Winter annual Meeting, 80-WA/DSC-4, Chicago, II., 1980. (6) V. Peterka, “Macrodynamics of technological change: market penetration by new technologies”, Intern. Inst. Appl. Syst. Anal., RR-77-22, A-2361, Laxenberg, Austria, 1977. (7) P. A. Samuelson, “Economics”, McGraw-Hill, New York, 1970. (8) P. A. Samuelson, “Foundations of Economic Analysis”, Atheneum, New York, 1970. (9) D. C. Karnopp and R. C. Rosenberg, “Analysis and simulation of multiport systems”, M.I.T. Press, Cambridge, Ma., 1968. “Some simple nonlinear dynamic models of (10) A. V. Quinlan and H. M. Paynter, interacting element cycles in aquatic ecosystems”, J. Dynam. Syst., Meas., Co&r. (ASME), Vol. 98, Series G, N.1, pp. 6-19, 1976. “Generalized predator-prey oscillation in ecological and (11) P. A. Samuelson, economic systems”. Proc. Natn. Acad. Sci. U.S.A., Vol. 68, pp. 980-983, 1971.
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Journal of the Franklin Institute Pergamon Press Ltd.