February 3, 2017 10:1 WSPC/S0218-1274 1750016

International Journal of Bifurcation and Chaos, Vol. 27, No. 1 (2017) 1750016 (11 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S021812741750016X

A Mathematical Model of Demand-Supply Dynamicswith Collectability and Saturation Factors

Y. Charles LiDepartment of Mathematics, University of Missouri,

Columbia, MO 65211, USAliyan@missouri.edu

Hong YangMathematics of Networks and Communications Research Department,

Bell Laboratories, 600 Mountain Avenue,Murray Hill, NJ 07974, USAh.yang@research.bell-labs.com

Received October 10, 2016

We introduce a mathematical model on the dynamics of demand and supply incorporatingcollectability and saturation factors. Our analysis shows that when the fluctuation of the deter-minants of demand and supply is strong enough, there is chaos in the demand-supply dynamics.Our numerical simulation shows that such a chaos is not an attractor (i.e. dynamics is notapproaching the chaos), instead a periodic attractor (of period-3 under the Poincare periodmap) exists near the chaos, and coexists with another periodic attractor (of period-1 under thePoincare period map) near the market equilibrium. Outside the basins of attraction of the twoperiodic attractors, the dynamics approaches infinity indicating market irrational exuberance orflash crash. The period-3 attractor represents the product’s market cycle of growth and recession,while period-1 attractor near the market equilibrium represents the regular fluctuation of theproduct’s market. Thus our model captures more market phenomena besides Marshall’s marketequilibrium. When the fluctuation of the determinants of demand and supply is strong enough,a three leaf danger zone exists where the basins of attraction of all attractors intertwine andfractal basin boundaries are formed. Small perturbations in the danger zone can lead to verydifferent attractors. That is, small perturbations in the danger zone can cause the market toexperience oscillation near market equilibrium, large growth and recession cycle, and irrationalexuberance or flash crash.

Keywords : Law of demand; law of supply; competitive market; market equilibrium; chaos;heteroclinic cycle; Melnikov function.

1. Introduction

The dynamics of demand and supply is the keyfor a market. One can observe the demand andsupply dynamics in action from common commodi-ties like houses [Dorofeenko et al., 2014] and gaso-line [Aleksandrov et al., 2013; Plante, 2014]. Dueto the huge surplus in supply, gasoline price has

sharply dropped recently [Plante, 2014]. Gasolineprice and housing price affect people’s daily life. Itis paramount to build better mathematical modelson demand and supply dynamics. Various math-ematical tools have been developed in studyingdemand and supply (see e.g. [Heo, 2014; Tsitsiklis &Xu, 2014; Weinrich, 2007]). Here we are employing

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dynamical system tools to study the dynamics ofdemand and supply generalizing the classical Mar-shall model.

Our model shall describe the demand-supplydynamics of the global market on one product. Thisis the topic of Microeconomics [Stone, 2012; Olney,2009]. We are not dealing with aggregate demandand aggregate supply which belong to Macroeco-nomics [Karlan et al., 2014]. The one product globalmarket that we are modeling is close to a compet-itive market where the market dynamics is moreobjective in contrast to monopolistic, oligopolis-tic, and monopolistic competitive markets. In anideal competitive market, no individual buyer orseller can influence the price, the product feature isstandardized, buyers and sellers are well informed,and firms can enter and leave with no significantbarrier [Stone, 2012]. The law of demand statesthat as the price increases, the quantity demandeddecreases, ceteris paribus (i.e. holding all other fac-tors constant). On the other hand, the law of sup-ply states that as the price increases, the supplyquantity increases, ceteris paribus. According toAlfred Marshall, the demand curve and supplycurve intersect at a market equilibrium (Fig. 1).The market dynamics approaches the equilibrium,a phenomenon that Adam Smith called an“invisiblehand” leading the market dynamics to the equilib-rium. This is the classical theory on demand-supplydynamics. According to this theory, an individualfirm in a competitive market is a price taker (theprice of its product is set by the market equilib-rium). Such a demand-supply model is very ideal.In reality, other factors (that are held constant inthe statements of the laws of demand and supply)change dramatically and have significant effect ondemand-supply dynamics. The prices of most prod-ucts do not stay close to their equilibrium values.

price

quan ty

Supply Curve

Demand Curve

Fig. 1. The demand curve and supply curve intersecting atthe market equilibrium.

For example, the watch market is quite close to acompetitive market. But individual firms are notprice takers. Rolex watch price is much higher thatthose of less known brand watches. Even the aver-age price of watches change substantially in time.Those factors that are held constant in the state-ments of the laws of demand and supply are calleddeterminants [Stone, 2012]. The main determinantsof demand are [Stone, 2012]:

(1) taste and preference,(2) income level,(3) prices of related goods,(4) the number of potential buyers,(5) future expectation on the product and income.

The main determinants of supply are [Stone,2012]:

(1) production technology,(2) costs of resources,(3) prices of other commodities,(4) future expectations on the product,(5) the number of potential sellers,(6) taxes and subsidies.

In the statements of the laws of demand and sup-ply, the amounts of demand and supply are amountsduring a time period. The demand and supplycurves do not depend on time, and they are staticcurves. In our demand-supply model, the amountsof demand and supply depend on time (they are theamounts at that time, not during a time period),and the price also depends on time. The demand-supply dynamics is represented by the temporalevolution of the price and the amounts of demandand supply. During this evolution, the determi-nants of demand and supply constantly influencethe dynamics. Such models are closer to the reality.Due to the variation of price from firm to firm, theprice of the product’s entire market is defined to bethe average price.

Definition 1.1. The price P (t) of a product attime t is defined to be the average price over theproduct’s global market at time t. The amount ofdemand D(t) is defined to be the total amountof demand in the product’s global market. Theamount of supply S(t) is defined to be the totalamount of supply in the product’s global market.

Our model will be represented by a system ofordinary differential equations involving P (t), D(t)and S(t). Earlier studies on differential equation

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A Mathematical Model of Demand-Supply Dynamics

models [Kobayashi, 1996; Soltes et al., 2012] focusedon the following equation

dP

dt= f(D(P ) − S(P ))

where f(x) satisfies xf(x) > 0 when x = 0, andtime delay may be involved. Convergence of thedynamics to market equilibrium was the main inter-est [Kobayashi, 1996; Soltes et al., 2012]. Our modeltakes the general form

dP

dt= fp(D − S),

dD

dt= fd(Pd − P ) + Fd(t), (1)

dS

dt= −fs(Ps − P ) + gs(D − S) + Fs(t),

where (fp, fd, fs, gs, Fd, Fs) are general functionsfor now, and Pd and Ps are threshold prices ofdemand and supply. The price equation states thatprice change is determined solely by D − S. Priceincreases when D − S > 0, and decreases whenD−S < 0. The change in the amount of demand Ddepends on the price relative to the threshold priceof demand. Usually the difference D − S has littleinfluence on buyers, and buyers just buy wheneverthey need and can afford the product. So dD

dt haslittle dependence on D − S. On the other hand,D − S has more significant influence on produc-ers, and producers need to know D − S to projectthe future trend of price and profit. So dS

dt dependson D − S next to Ps − P (the functions gs andfs). The determinants of demand also influence dD

dt .Taste and preference can change with time. Theycan also change with price. When the price getsvery high, the product may turn into a collectableproduct fd > 0 (collectability factor). When theprice gets extremely low, the market is over satu-rated, and the product may become less desirablefd < 0 (saturation factor). The income level and theprices of related goods can fluctuate in time, andthey can be represented by a function of time Fd(t)that is independent of the three variables (P,D, S).The determinants of supply also influence dS

dt . Pro-duction technology, costs of resources, and price ofother commodities, and taxes and subsidies changewith time, and they can be represented by a func-tion of time Fs(t) that is independent of the threevariables (P,D, S).

Collectability and saturation factors can alsobe used to model certain stocks. Some stock’s price

may get much higher than its true value, and moreand more investors continue to buy them. On theother hand, when the stock’s price gets much lowerthan its true value, still less and less investors wantto buy them.

Next we set up a simple specific model of thedemand-supply dynamics starting from (1). In gen-eral, we expect (fp, fd, fs, gs) to be linear only nearzero, but for fp we believe that linear approximationshould perform very well based on the general prin-ciple that price increases (decreases) when demandis more (less) than supply. We choose

fp(D − S) = α(D − S)

where α > 0 is a parameter. That is, the changerate in price is proportional to D − S. We choose

fd(Pd − P ) = β(Pd − P )[1 − β1(Pd − P )2],

where β > 0 and β1 > 0 are parameters, and we takeinto account collectability and saturation factors.When the price is too low, the market is alreadysaturated, and the demand will not increase any-more. Since the lowest price is zero, we have

1 < β1P2d. (2)

When the price gets too high, the product maybecome a collectable item, and the demand canincrease. For the supply equation, we choose

fs(Ps − P ) = γ(Ps − P ),

gs(D − S) = δ(D − S),

where γ > 0 and δ > 0 are parameters. Highprice and high demand are positive factors for sup-ply increase. In summary, we arrive at the follow-ing simple specific model for the demand-supplydynamics incorporating collectability and satura-tion factors:

dP

dt= α(D − S),

dD

dt= β(Pd − P )[1 − β1(Pd − P )2] + Fd(t), (3)

dS

dt= −γ(Ps − P ) + δ(D − S) + Fs(t),

where again (α, β, β1, γ, δ, Pd, Ps) are positiveparameters. For the functions Fd(t) and Fs(t), wecan choose

Fd(t) = a sin(ω1t), (4)

Fs(t) = c + b sin(ω2t), (5)

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Y. C. Li & H. Yang

where the income level and prices of related goodsoften fluctuate, and these factors lead to the oscil-latory nature of Fd(t) (a and ω1 are parameters).Production technology is represented by the con-stant term c in Fs(t). Costs of resources, prices ofother commodities, and taxes and subsidies oftenfluctuate, and they are represented by the oscilla-tory sine term in Fs(t) (c, b and ω2 are parameters).

In the third equation of (3),

Ps − P = Pd − P + (Ps − Pd)

and the (Ps−Pd) term can be incorporated into thec term in Fs(t). We will set c = b = 0, and Fd(t)term is enough to represent the fluctuation factorof determinants of supply and demand. Let

p = P − Pd, q = D − S,

where p ≥ −Pd, then the system (3)–(5) takes theform

dp

dt= αq, (6)

dq

dt= −βp(1 − β1p

2) − γp − δq + a sin(ω1t), (7)

where (p, q) = (0, 0) represents the market equilib-rium.

2. Analysis of the Simple SpecificModel

2.1. Integrable dynamics

When δ = a = 0, the system (6)–(7) is an integrableHamiltonian system

dp

dt=

∂H

∂q, (8)

dq

dt= −∂H

∂p, (9)

where

H =12αq2 +

12(β + γ)p2 − 1

4ββ1p

4.

There are three fixed points in the system (8)–(9)when

β + γ

β< β1P

2d (10)

which is stronger than (2). The three fixed pointsare

(0, 0),

(±√

β + γ

ββ1, 0

). (11)

p

p=bp=-bp=-aq

Fig. 2. The heteroclinic cycle. a = Pd and the dynamicscannot go beyond p = −a to the left since the price P cannot

be negative. b =q

1β1

and p = ±b are the lines across which

the changes of demand switch signs.

The first fixed point represents the market equi-librium, the negative p-value fixed point is namedthe saturation fixed point, and the positive p-valuefixed point is named the collectability fixed point.The market equilibrium is a neutrally stable center.The eigenvalues of the market equilibrium under thelinearized dynamics of (8)–(9) are

λ = ±i√

α(β + γ).

The saturation and collectability fixed points areunstable saddles with eigenvalues

λ = ±√

2α(β + γ).

The phase plane diagram of (8)–(9) is shown inFig. 2. The minimal value of p is −Pd (since theprice P cannot be negative). As p decreases across−√

1β1

, demand switches from increasing to decreas-ing [cf. (3)], and the dynamics of (8)–(9) reaches a

(unstable) equilibrium at p = −√

β+γββ1

and D = S.

As p increases across√

1β1

, demand switches fromdecreasing to increasing [cf. (3)], and the dynam-ics of (8)–(9) reaches a (unstable) equilibrium at

p =√

β+γββ1

and D = S. In terms of the originalvariables, at the market equilibrium (11),

P = Pd, D = S = const

At the saturation fixed point (11),

P = Pd −√

β + γ

ββ1, D = S,

dD

dt=

dS

dt= negative const,

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A Mathematical Model of Demand-Supply Dynamics

thus, the amounts of demand and supply are equaland decrease at the same rate in time. Such a stateis unstable. At the collectability fixed point (11),

P = Pd +

√β + γ

ββ1, D = S,

dD

dt=

dS

dt= positive const,

thus, the amounts of demand and supply are equaland increase at the same rate in time. Such a stateis unstable.

On the phase plane (Fig. 2), connecting thesaturation and collectability fixed points is a het-eroclinic cycle of two heteroclinic orbits. The upperheteroclinic orbit has the expression

p = A tanh(Ωt + t0), (12)

q =AΩα

sech2(Ωt + t0), (13)

where t0 is a parameter,

A =

√β + γ

ββ1,

Ω =

√12α(β + γ).

The lower heteroclinic orbit is given by (−p,−q)where (p, q) is the upper heteroclinic orbit (12)and (13).

2.2. Chaotic dynamics

When δ = 0 and a = 0, the dynamics of (6)–(7) isnot integrable, and we will show that it is chaoticvia a Melnikov integral and Shadowing Lemma. TheMelnikov integral is given by [Li, 2004],

M =∫ +∞

−∞dH

evaluated along the heteroclinic orbit (12)–(13).Thus

M = α

∫ +∞

−∞[−δq2 + aq sin(ω1t)]dt

= −δA2Ωα

∫ +∞

−∞sech4τdτ

+ aA

∫ +∞

−∞sech2τ sin

(ω1

Ωτ − ω1

Ωt0

)dτ

p

q

Fig. 3. The transversal intersection between the broken het-eroclinic orbits, forming a Poincare net.

= −4δA2Ω3α

− 2aA sinω1

Ωt0

×∫ +∞

0sech2τ cos

ω1

Ωτdτ.

Setting M = 0, we get

sinω1

Ωt0 = −2δAΩ

3αa

[∫ +∞

0sech2τ cos

ω1

Ωτdτ

]−1

.

(14)

When

|a| >2δAΩ3α

[∫ +∞

0sech2τ cos

ω1

Ωτdτ

]−1

, (15)

the Melnikov integral M has infinitely many sim-ple roots given by (14) which imply that the brokenpieces of the heteroclinic orbit reintersect transver-sally under the Poincare map F of (6)–(7) (a factproven mathematically and rigorously when δ anda are small [Li, 2004]) (Fig. 3). The intersectionpoints form a transversal heteroclinic cycle underthe Poincare map F of (6)–(7). Then via Shadow-ing Lemma approach, it is rigorously proved thatthere is chaos in the dynamics of (6)–(7) [Li, 2004].The next key question is whether or not the chaos isan attractor, and this will be answered by numericalsimulations.

3. Numerical Simulation of theSimple Specific Model

Here we are going to numerically simulate thedynamics of (6)–(7) in terms of attractors and theirbasins of attraction. We choose the parameters as

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Y. C. Li & H. Yang

follows:

α = 1, β = 1, β1 = 0.25, γ = 1, ω1 = π.

We leave the other two parameters δ and aadjustable for different numerical simulations, andrecall that δ = a = 0 corresponds to the integrabledynamics. When δ = 0.01, the Melnikov integralpredicts that when |a| > 0.2656, there exists chaos.But this chaos may not be an attractor. Our numeri-cal simulations show that this is indeed the case: thechaos is not an attractor, instead a period-3 attrac-tor (under the Poincare map) exists near the chaos.

In fact, there are two coexisting periodic attractors(period-1 and period-3 under the Poincare map) asshown in Fig. 4 where two values of a are cho-sen as a = 0.25 (below the critical value 0.2656)and a = 0.35 (above the critical value 0.2656).The solid dot is the period-1 attractor under thePoincare map, while the small loop above the dot isthe continuous periodic attractor under the dynam-ics of (6)–(7). The three stars form the period-3attractor under the Poincare map, while the loopconnecting the three stars is the continuous peri-odic attractor under the dynamics of (6)–(7). For

p-3 -2 -1 0 1 2 3

q

-3

-2

-1

0

1

2

3Periodic Orbit 1 ( = 0.01, a = 0.25)

PO-1Separatrix

δ

p-3 -2 -1 0 1 2 3

q

-3

-2

-1

0

1

2

3Periodic Orbit 1 ( = 0.01, a = 0.35)

PO-1Separatrix

δ

(a) δ = 0.01, a = 0.25 (b) δ = 0.01, a = 0.35

p-3 -2 -1 0 1 2 3

q

-3

-2

-1

0

1

2

3Periodic Orbit 3 ( = 0.01, a = 0.25)

PO-3Separatrix

δ

p-3 -2 -1 0 1 2 3

q

-3

-2

-1

0

1

2

3Periodic Orbit 3 ( = 0.01, a = 0.35)

PO-3Separatrix

δ

(c) δ = 0.01, a = 0.25 (d) δ = 0.01, a = 0.35

Fig. 4. Periodic orbit attractors for values of a below and above its critical value 0.2656 predicted by Melnikov integral, thedot and stars are images under the Poincare period map. The separatrix frame is the one in Fig. 2.

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A Mathematical Model of Demand-Supply Dynamics

p-3 -2 -1 0 1 2 3

q

-3

-2

-1

0

1

2

3

PO-1Separatrix

p-3 -2 -1 0 1 2 3

q

-3

-2

-1

0

1

2

3

PO-1Separatrix

(a) δ = 0.1, a = 2.6 (b) δ = 0.1, a = 3.5

p-3 -2 -1 0 1 2 3

q

-4

-3

-2

-1

0

1

2

3

4

PO-3Separatrix

p-3 -2 -1 0 1 2 3

q

-4

-3

-2

-1

0

1

2

3

4

PO-3Separatrix

(c) δ = 0.1, a = 2.6 (d) δ = 0.1, a = 3.5

Fig. 5. Periodic orbit attractors for values of a below and above its critical value 2.656 predicted by Melnikov integral, thedot and stars are images under the Poincare period map. The separatrix frame is the one in Fig. 2.

reference, we also plot the separatrix in Fig. 2. Onecan see clearly that the attractor structure is thesame for both cases a = 0.25 and a = 0.35. Thisshows that when the value of a crosses the criticalvalue a = 0.2656, the structure of the attractorsdoes not change, and chaos is not an attractor. Thesame conclusion holds in Fig. 5 where δ = 0.1 andthe critical value is a = 2.656. The period-1 attrac-tor represents the market regular fluctuation nearthe market equilibrium, while the period-3 attrac-tor represents the market recession (depression) andlarge growth cycle. Next we will focus on the entire

phase space and study the basins of attraction ofall the attractors. For δ = 0.1, when 2.4 < a < 6.5,the period-3 and period-1 attractors coexist, and inthis case there are a total of four attractors, positiveinfinity and negative infinity besides the period-3and period-1 attractors. Positive infinity and nega-tive infinity represent market irrational exuberanceand flash crash. When a ≤ 2.4 or a ≥ 6.5, period-3attractor disappears, and in this case there are atotal of three attractors, positive infinity and nega-tive infinity besides the period-1 attractor. Whena ≤ 2.4, the basin of attraction of the period-1

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Y. C. Li & H. Yang

p-3 -2 -1 0 1 2 3

q

-4

-3

-2

-1

0

1

2

3Attractors (δ = 0.1, a = 2.6)

PO-1PO-3Separatrix

p-3 -2 -1 0 1 2 3

q

-4

-3

-2

-1

0

1

2

3Attractors (δ = 0.1, a = 3.5)

PO-1PO-3Separatrix

(a) δ = 0.1, a = 2.6 (b) δ = 0.1, a = 3.5

(c) δ = 0.1, a = 2.6 (d) δ = 0.1, a = 3.5

Fig. 6. Periodic attractors under the Poincare period map for values of a below and above its critical value 2.656 predictedby Melnikov integral, the dot represents the period-1 attractor and the three stars represent the period-3 attractor. Theseparatrix frame is the one in Fig. 2. In the basin of attraction figures, the three leaves form the basin of attraction for theperiod-3 attractor, the central white region is the basin of attraction for the period-1 attractor, the upper region is the basinof attraction for positive infinity (p, q) = (+∞,+∞), and the lower region is the basin of attraction for negative infinity(p, q) = (−∞,−∞).

attractor occupies the central white region in Fig. 7.When a > 2.4, three leaves appear within the basinof attraction of the period-1 attractor, and theyform the basin of attraction of the period-3 attrac-tor (Fig. 6). As a increases, the basins of attractionof all four attractors intertwine into the three leaves,and fractal basin boundaries are formed (Fig. 7).The fractal basin boundaries offer a new kind of

sensitive dependence on initial condition. Whenthe market reaches the three leaf regime, the finalattractor is very sensitive to its initial condition. Asmall perturbation of the initial condition can per-turb into all possible attractors! For instance, oneinitial condition on one of the three leaves leads tothe period-1 attractor (so the market will reach nearthe market equilibrium), a small perturbation of the

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A Mathematical Model of Demand-Supply Dynamics

(a) δ = 0.1, a = 2.4 (b) δ = 0.1, a = 5

(c) δ = 0.1, a = 5 zoomed

Fig. 7. (a) The basins of attraction without period-3 attractor, (b) fractal basin boundaries among all attractors and(c) zoomed in picture of the lower left leaf in (b).

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Y. C. Li & H. Yang

(a) δ = 0.1, a = 6.4 (b) δ = 0.1, a = 6.4 zoomed

(c) δ = 0.1, a = 6.5 (d) δ = 0.1, a = 6.5 zoomed

Fig. 8. (a) Right before the basin of attraction of period-3 attractor disappears, (b) zoomed in picture of the lower left leafin (a), (c) the basin of attraction of period-3 attractor totally disappears, still having fractal basin boundaries and (d) zoomedin picture of the lower left leaf in (c).

initial condition can lead to the period-3 attractor(the market will enter recession and large growthcycle) or one of the infinity attractors (the mar-ket will experience irrational exuberance and flashcrash). So the three leaves are really the market“danger zone”. When a ≥ 6.5, period-3 ceases tobe an attractor and its basin of attraction disap-pears (Fig. 8), but the three leaf region still hasthe fractal basin boundaries of the remaining threeattractors — period-1, positive infinity and negativeinfinity. So when the fluctuation of the determinants

of demand and supply is strong (a is large enough),a danger zone (the three leaves) always exists!

4. Conclusion

We have introduced a dynamical system model onthe dynamics of demand and supply via generalizingthe Marshall model to incorporate collectability andsaturation factors. Collectability and saturationhappen more often than one thought, for instance,many stocks are over-valued (collectability) and

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A Mathematical Model of Demand-Supply Dynamics

under-valued (saturation). Under the Marshallmodel, the dynamics of demand and supply has oneglobal attractor (the market equilibrium). Incor-porating the collectability and saturation factors,the dynamics of demand and supply has as manyas four attractors representing the market regularfluctuation near the market equilibrium, recession(depression) and large growth cycle, and irrationalexuberance and flash crash. So our model capturedmore market phenomena. Our model revealed a“danger zone” where fractal basin boundaries exist.When the market enters the danger zone, small per-turbations can lead to all four attractors, i.e. smallperturbations can cause the market to experiencefluctuation near the market equilibrium, recession(depression) and large growth cycle, and irrationalexuberance or flash crash.

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