Time and Equilibrium : 2 Important , But Invisible , Concepts of Economics , with Application to Shipping Industry

As the World built, time established. Economists, however, put the “time” in the “ceteris paribus” basket, i.e. outside demand, supply and price. Moreover, Newton was mistaken in assuming that time flows independently. We saw that since the establishment of analysis, one science borrowed from the other, and economics borrowed from Physics: equilibrium, continuity—where nature does not make leaps—as well as Adam Smith’s invisible hand; in addition, management borrowed negative feedback from mechanical engineering; Newton, unwillingly, however, made harm to management by giving ground to managers to consider “humans as machines”. A whole array of theories and concepts-mentioned-followed from this. But our research passed from surprise to surprise: time in finance has 3 types: clock, trading (investors) and fractal (fractions). Given the difficult concept of “fractality”, we gave a mathematical and a simple geometrical exposition. Moreover, time... in time series is distinguished in further 3 types: random (white noise), persistent (black noise) and antipersistent (pink noise). So far 8 types of time... Einstein added another one: time as the 4 dimension of the Universe... Mathematics in its role in presenting reality-par excellence expressed by “Marginalism” in 1870 in economics—and by using the 1938 “logistic equation” (re-discovered in 1971)—we saw what a “control coefficient” changing in time can achieve by leading the system from stability to chaos. Equilibrium is only a special case when the degree of chaos is low. Economists (Hicks, Joan Robinson) attributed to equilibrium subjective interpretations; we agree that equilibrium is not technical, mathematical or belonging to markets, but psychological. Be happy when accepting a price to be in equilibrium with seller. Samuelson, before modern theory of chaos (after 1968) appeared, he dethroned equilibrium and proved that equilibrium is when firms “maximize profits”. Poincare H was the first to see “chaos” in 1889 and Samuelson P saw “nonlineariHow to cite this paper: Goulielmos, A.M. (2018) Time and Equilibrium: 2 Important, But Invisible, Concepts of Economics, with Application to Shipping Industry. Modern Economy, 9, 536-561. https://doi.org/10.4236/me.2018.93035 Received: January 24, 2018 Accepted: March 24, 2018 Published: March 27, 2018 Copyright © 2018 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/

ty" in 1947.Chaos revolution, however, re-enthroned time.Smale (1961)-a chaos pioneer, who proved the existence of chaos, argued that chaos is a characteristic of dynamics and dynamics is the time evolution of a set of nature's (and economy's) states (words in italics added).We applied the following 3 models: {1} ( ) .All models lead to chaos and all models embody time as consisting of difference or differential equations.These now are our tools if time returns into our analysis.

Introduction
As God built the world, the concept of calendar time established.The World built in 7 days (one week).During the "day" Sun is present, and during "night" Moon (earth's satellite) shines.Also, 4 seasons are established by Earth approaching Sun at 4 different distances in 12 Moon rounds (336 days-a year).A month is also introduced by a complete orbit of Moon round the Earth (28 days).

Time in Economics
Economists do not display time during the determination of price by supply and demand, which are Marshall's blades of a pair of scissors.Economists work their analyses under the assumption of "ceteris paribus 1 " (Latin).Time is locked-in in ceteris paribus.

Time in Physics and Thermodynamics
Newton (1642-1727) thought time to be reversible.Thermodynamics, however, proved that time is irreversible… By mixing 2 fluids, e.g., red and blue, one gets purple; here the "entropy" is high, and "uncertainty", related to it, is also high; these are time-dependent, and un-mixing is impossible.Moreover, the "molecular structure of universe" can only be described by probabilities…

Equilibrium in Economics
The condition of equilibrium economists tried-unsuccessfully, we reckon-to accommodate into economic analysis.Adam Smith (1723-1790) believed that selfishness provides a powerful fuel in a commercial society, a prior idea of "workable competition"; private interests are harmonized with social interests by 1 It means that all other (relevant) factors, with the exception of those presented in stasis, are constant.This expression means in English: "other things being equal".an "invisible hand".Obviously, Smith conceived a model of competition, not as it works, but how it should work-given also his engagement with his "theory of moral sentiments", though no reference was made to this.

The Influence of Newton and Descartes
Most sciences and economics-as well management-were heavily influenced by the scientific principles of Newton and Descartes (1596-1650).They argued that the "natural state of a system" is the equilibrium, and departure from it will be damped out.Economists by adopting the concept of equilibrium were astonished by the two depressions in Black Monday and Tuesday (1929 and 2008).
In traditional management , equilibrium was a core principle!Fayol H (in 1916 in French; 1949 in English) and other early management writers (Taylor F in 1911) invented management control mechanisms based on the perception: "firms as machines 2 ", meaning "humans as machines".Moreover, Weber M. (1946 in English) conceived "firms as bureaus".Firms-as a result-functioned by drafting plans (planning), budgets and applying "management by objectives" (1965).This considered being a "command and control" system, where control is done through "monetary rewards" and "punishments".
Moreover, "reductionism" (i.e.making complex matters less complex), introduced the systems: division of labor, "task", standardized procedures, quality control (1991), cost accounting, and organizational charts.Also, the budget-used in shipping companies par excellence-performance reviews, audits, standards, etc. all applied controls.In fact, the negative feedback used i.e. a mechanism for maintaining a desirable equilibrium… No body, however, proved so far the benefits of equilibrium in management acting in a volatile world… The classical system sought-out for amore "certain" reality.This was promised by the "command and control" pattern and this is what managers prefer on the basis of the principle: "achieve most with a minimum effort".But everybody witnessed that business world became more complex-day after day (especially by Citibank and Xerox).Complexity increases over time, and in the case of living systems, like economy, this is called evolution.Some suggest correctly to absorb complexity instead of trying to reduce it.This is a theory derived from "Control theory" and "Cybernetics"-systems dynamics.The idea was to reduce variety.

Aim and Organization of Paper
The purposes of this paper is: (1) to show the role that time plays in Economics, Finance, Chaos Theory, Physics and Shipping; (2) to state what exactly equilibrium means in Economics, Physics, and Complexity Theory; (3) to use the "lo- gistic equation", the Henon's and Lorenz's attractors in applying chaos to shipping markets and (4) to show the relationship between equilibrium and profit maximization due to Samuelson.
The paper is organized as follows: next is a literature review followed by methodology; then time in maritime economy and finance is presented.Next, the equilibrium concept in economics, Physics and Complexity Theory is showed.
Then chaos theory is applied to shipping markets; finally we conclude.In Appendix we deal especially with the concept of time in Physics.

Time in Marshall
Time 3 preoccupied Marshall ((1920) [1] pp. 92; 274; 287), whose "time" is characterized as "operational", i.e. near the "clock" time.Blaug (1997 [2] p. 354) argued correctly that time "periods" in Marshall are short or long, according to the "partial or complete" adaptation of producers and consumers to changing circumstances.The actual clock-time periods in Marshall, however, left undefined.A matter which is very important 4 , especially when one wishes to pass from theory to reality.Moreover, in the preface of the 8 th edition of his book ( [1] (preface, pp.xii-xii)), Marshall was apologetic about the largely static character of his analysis with the frequent use of the "ceteris paribus" assumption.
3 Except the "time preference" issue.In shipping, "short run" is defined as the period when a company cannot change its capital: capital in shipping consists of…the "number of ships".But to sell or buy a used ship is a matter of 1-3 months.This "long" run is really… very short in actual time.To build a ship is surely a matter may be of even 2 years on average.This is really "long run"... So, long run has 2 different durations, depending on whether we use new or used capital…Also firm's long run with used capital is different than industry's long run with new capital… Moreover, growth is not realized if industries use used capital…, but this is growth for an individual firm...

Marshall and Biology
Marshall ([1] p. xii) argued that important for economics 8 is "economic biology9 "... What Marshall meant-we believe-is that economy has to be studied at best as a living "organism'" (Blaug [2] p. 404) (italics added))."Economic biology" means to introduce "continuous or discrete changes" into economic analysis.A "dynamic system" has the time as independent variable and a small number of dependent variables, the movement of which is controlled by rules.Time may be continuous (+∞ to −∞), a case where differential equations are used, or discrete, like in time series, a case where difference equations are used.The mass of a fluid is the same.If a water pipe-with a diameter 2 cm-is connected with a pipe of 1 cm-the flow speed of the fluid-not compressible-will be 4 times higher.The speed of the flow of a moving fluid is changing (Young and Freedman, (2000) [3] p. 439)).

Hicks' Economic Expectations
7 Kratylos, pupil of Plato, said: "Heraclitus argued that everything moves, and nothing is immobile, and humans-in front of the flow of a river-cannot enter twice in the same river" (waters).Aetios said that Heraclitus dismissed the state of rest and of immobility from everything, and everyone, as this is a characteristic of the dead.He attached to all a movement: eternal to eternals and temporary to temporaries.See: Heraclitus-all his work-collected by T Falkos-Arvanitakis, Zitros editions, 1999, ISBN 960-7760-36-0 (in Greek).A "dynamic economic system" has a set of variables-acting one on another-evolving in time, and following certain specific laws or rules.

Zannetos Applied Hicks' Model in Tanker Shipping
Zannetos (1966; [8]) adopted Hick's theory of expectations ([6] p. 117), where the current supply of a good depends mainly on the price that firms expected to be.This implies that if current prices are high, then future prices are expected to be higher, and vice versa.Zannetos (p.239) argued that static economic analysis was unable to explain how rates were formed in tanker markets.There are substantial price movements-away from equilibrium-creating expectations that future rates will increase at a higher speed than hitherto.So, short term (spot) rates are formed by demand, as a function of "price expectations" and static supply (Graphic 1).
As shown, a number of partial equilibria are possible-outside R s -the region of strict static relevance (and inelastic).E.g. the slopes of demand and supply in regions R 2 and R 3 do matter; supply has a negative slope.R 3 is stable from below and R 2 indicates a price in an unstable range, as demand's slope is positive.R 4 shows potential instability.Prices could be at rest-given able time-at either R s or R 3 , but not at unstable R 2 .Markets are prone to excess capacity and depressions.
Zannetos further argues (op.cit., [8] p. 21) that it is not unreasonable to assume that expectations alternate between elasticity and inelasticity, if the market stays long enough at an equilibrium point, before it goes into a spin…; as the memory of those operating in the market may not last long enough to recall how the market came at rest...
He applied the nonlinear model of "cobweb theorem", where he found its adjustment paths to be similar with those of spot tanker rates in a plethora of webs.
The fluctuations have the static long run equilibrium as a central focus (p.244) (unstable).An empirical investigation using a questionnaire among shipping tanker companies is required, we believe, to confirm Zannetos' theory today.Samuelson (Nobel winner) spoke with the language of Mathematics.Samuelson (1967) [9]) re-published his "Foundations of Economic Analysis" on his ideas since 1947.Samuelson argued that economics is a softer, and less exact, science than conventional Physics (preface ix) and for Marshall, stability of equilibrium requires that the supply curve cuts the demand curve from below (p.18 [9]).
The idea of equilibrium 10 Samuelson (p.21) said is a matter of the equations involved in maximizing (minimizing) conditions."All economic results emerge from maximizing assumptions" he argued (p.22).Important are the slopes of the curves at equilibrium.Statical is the equilibrium of the intersection of a pair of curves (=Marshall's case), which it may be stationary, and timeless, but holding over time (p.313).
Samuelson devoted 8 chapters to analyzing comparative-statics.In 3 chapters, he introduced the "correspondence principle" relating comparative statics with dynamics (& stability).The last 2 chapters devoted to "dynamical systems" (also to stability etc.).He argued (p.351) that Walras (1834-1910) provided the notion of the "determinateness of equilibrium" on a statical level, which Pareto (1848-1923) further elaborated.Pareto, however, laid the basis for a theory of comparative-statics, pioneered by Cournot 11 (1801-1877).Pareto failed to use the secondary conditions for maxima.Samuelson suggested "comparative-dynamics" (p.351-2).A system is dynamical if its behavior over time is determined by functional equations in which "variables at different points of time" are involved in an "essential" way (p.314) (the term "essential" comes from econometrics 12 ).

Chaos Theory and Complexity
Nonlinearity opened a door nearer to reality.Samuelson (1967) [9], p. 338) was aware by saying "linear systems lack the qualitative richness of nonlinear systems", which "introduce also a theory about fluctuations".He mentioned 13   Birkhoff 14 G D. He argued that (p.339) in the field of economics only one nonlinear system received complete treatment: the "cobweb" theorem, in which supply lags and equals a nonlinear difference equation of the 1 st order 15 .
West and Goldberger (1987) [10]) argued that the "physical fractal 16 struc-10 In a simple Demand and Supply model, price p and quantity q are the unknowns and equilibrium is: q − D(p, a) = 0 and q − S(p) = 0, where "a" is a shift parameter (e.g. for tastes), and where Dp < 0 and Da > 0 (p.260).Smale (2000) [18] (p.[20][21] proved that if a dynamical system possesses a "homoclinic" point then it also contains a "horseshoe".Time is considered by Smale as a continuous entity, but measured by discrete units.He said that chaos is a characteristic of dynamics, and dynamics is the time evolution of a set of Nature's states ( [18] p. 13).Farmer (2002) [19]) supported the idea of biology in economics-arguing that… "market ecology" shows how financial firms engage in specialized strategies, which can be sorted into groups analogous to species in biology.Kirman (1997) [20] argued that the very process of learning and adaptation and the feedback from the consequences of that adaptation generate highly complicated dynamics, which may well not converge to equilibrium… Stopford (2009 [21], pp.163-168), in shipping economics, copied Marshall in his analysis of the 3 periods: market, short run and long run, though Marshall had and a fourth period: the "very long run", ignored by Stopford, but not by Goulielmos (2017 [22]).
In summary, Marshall introduced time in the form of 4 periods into economic analysis, as a method of exposition, but with a recreation and tool to reality.Moreover, another unreal assumption of Marshall, and not only, was "ceteris paribus".Finally, while Marshall was "flirting" with dynamics, it was 20 Samuelson to write about.Hicks transferred the concept of equilibrium from price formation to human expectations.Hicks removed also one leg from the model of Perfect Competition that of "Perfect Foresight".Joan Robinson kept distances from equilibrium and instead introduced the concepts of "lucidity" (for foresight) and "tranquility" (for equilibrium).
Economics waited Samuelson [9] to upgrade its status by expressing most of it in mathematics: a recreation and tool'... he wrote (p.vii).Samuelson (1967) [9] dethroned equilibrium as the prime end of enterprises (microeconomics) and replaced it with the "maximization of profits".He cleared out concepts like statical 21 , comparative-statics, comparative-dynamics, dynamics 22 etc.He established the "intimate formal dependence between comparative-statics and dynamics".Moreover, Samuelson classified dynamics in 6 different classes…

Methodology
The "logistic equation" {1} will be used here to simulate equilibrium in shipping markets: X next = aX(1 − X) {1} 23 .Important is coefficient "a" or C (0 -4): a parameter describing the characteristics of the system; X next is a variable (0 -1, or 0% -100%) in future (%), 1 − X gives what remains of X over time and X 0 = the initial rate (assumed 50% or 0.5).20 Samuelson argued (p.311, fn.[9]) that in none of his writings Marshall showed more than a passing familiarity…with the biological notions of his time.Marshall influenced by Spencer H, (1820-1903), an English philosopher, who was after the unification of all knowledge on the basis of the single principle of evolution (1860).
We must mention that Ruelle 24 D. [23] supported mathematically the existence of "attractors" 25 -to be used below-showing the behavior of natural systems; the tendency of a system is to move toward some underlying pattern (as energy is lost).We will use two well-known attractors-"Henon's" and "Lorenz's"-which we will apply to find out equilibrium in shipping markets.

Time in Maritime Economy
Time in Physics is presented in Appendix.In Figure 1, the freight rate determination appears in shipping industry and the firm (=vessel), by supply and demand, but with time absent...The above is a "photograph".It is taken of a freight market on a certain date and time; a static picture… of a dynamic market.Alternatively, maritime economists applied the so called "comparative-statics" by allowing shifts in the curves caused by changes assumed to occur in the factors presented-like supply of ship space.This moved to the right (3 shifts, Figure 1, left part).
The time needed for shifts to occur and their extent, are not shown (shifts assumed here are equal, parallel, and positive)."Comparative-statics" determine only the direction in which variables change, as a result of a disturbance to original equilibrium (F 1 ).Only the new equilibria (F 2 , F 3 ) are shown, which can be "compared" with the previous one F 1 .This is a comparison of two photographs taken at different times.In the long term, a general dynamic system is adapted to attractor.The attractor "attracts" the system asymptotically.An attractor is part of its "phase space"-a mathematical space-such that a point from a near position approaches-as time goes-by-increasingly. See Ian Stewart, (1989), Does God play Dice?The new mathematics of chaos, B Blackwell, p. 132, in Greek translation (1998), ISBN 960-7122-01-1.Modern Economy Figure 2 is the only graph in shipping economics-to the best of our knowledge-where time is included in the freight rate determination (Shimojo (1979) [24]); McConville, (1999) [25], pp.253-4).
As shown, time affects freight rates, because the 3 supply and the 3 demand curves differ at the 3 times: T, T 1 and T 2 , where F 1 < F < F 2 and Q < Q 1 < Q 2 .If equilibrium is affected by time, why maritime economists put it out in ceteris paribus?

Clock Time
The clock time is linear; this is in which we think.

Trading Time or θ-Theta
Trading time is the time in which markets operate; investors' time.Θ is a random non-decreasing function of clock time; an intuitive notion of how markets operate over time (Mandelbrot (2010) [26] p. 163)).Both θ(t) and X(θ) are "self-affine" functions, i.e. similar under different scales.Θ intervenes in linear clock: it speeds-it-up in periods of high volatility and lows-it-down in periods of   stability.Market scales 26 .Economics have no intrinsic time scales.Θ is flexible.Another kind of clock is needed to measure θ… The actual implementation of Θ generalizes the generating equation: y 1/H + (2y − 1) 1/H + y 1/H = 1 {1}.The root of this equation, H, when determined, it can define the 3 quantities above in {1}: Δ 1 θ, Δ 2 θ, Δ 3 θ, which all add to 1. Θ as a function of clock time no longer reduces identically to clock time (Mandelbrot, (2002), [27] p. 76-77)).Let Δx be the order of a magnitude of a "price change" over a time increment Δt (small): then, Δx ~ Δt 1/α or Δx ~ Δt H , where 1/α = H: time invariant (but ≠1/2); alpha is a power exponent and H is Hurst's exponent.Then X(θ) stands for a price function of θ, and thus X[θ(t)] = P(t).

Fractal 27 Time-FT
FT is equal to θ, but only when ruled by "devil's staircase" 28  (Mandelbrot (2010) [26] p. 41).FT does not influence the statistical characteristics of the object (or chart) under study (Peters, 1994, [11] p. 5).Self-similarity exists, meaning that the object or process in time is similar at different scales: spatial, temporal, statistical.A fractal is scale-invariant and it lacks a characteristic scale.This is a power law scaling too.The feature, which turns out to be the 2 nd characteristic of fractals, is dimension.A random walk has a fractal dimension of 2. A shipping freight rates index for dry cargoes since 1741 has a fractal dimension 1.30.

26
Moreover, each time-scale, each holding-period, for a stock or a bond, has its own risk.Price variations scale with time.27 The term "fractal" comes from the Latin verb "frangere", meaning to break something; it has been introduced in 1975 by B Mandelbrot in French: "Fractal Geometry of Nature".In fact this means to deal with fractions.28 The dependence ρ(Ω) is devil's staircase, where each rational ρ = p/q, is represented by an interval of Ω values (the p/q-locking intervals).If r = 0, then ρ = Ω, where ρ is a rotation number.Αcircle map is Xt + 1 = f(Xt) or Xt + Ω -r/2πsin(2πΧt), which is a transformation of the phase of one oscillator through a period of the 2 nd one.Ω describes the ratio of undistributed frequencies and r governs the strength of the nonlinear interaction [28] [29].Let the number of high and low prices in a market [0 1] be N.Let the % of high prices be: p 1 = n 1 /N, and of low prices-distributed in two groups-be: p 2 = n 2 /N, where n 2 < n 1 , so that: p 1 + 2p 2 = (n 1 + 2n 2 )/N = 1 {1}.One histogram has 0p 1 peak-in the middle-and a base extending from 1/3 to 2/3, and 2 other histograms (left and right), which have a lower peak 0p 2 , and a base extending from 0 to 1/3 for the left and from 2/3 to 1 for the right (Figure 5).
Each of the above 3 distributions are divided into 3 further distributions, the heights of which are multiplied by p 1 , for high prices, and by p 2 , for low, in n steps.The result is that low prices eventually become numerous.The group that plays the more serious role (high prices) can also be found, with probability: ( ) n m P as n  ∞, and m = m 1 , where N P , is at maximum as n  ∞.Skip- ping the proof 29 , the final result is m 1 = 2p 2 n.Now, if p 1 = p 2 = 1/3 or p 1 = 0 and p 2 = 1/2, this is the simple fractal (of a single dimension).This (dimension) is given by: D 1 = p 1 logp 1 + 2p 2 logp 2 /log(1/3) = 1 {2} if p 1 = p 2 = 1/3.And ~0.63, if p 1 = 0 and p 2 = 0.5.This indicates that dimensions are ∞, and this market is multifractal.D 1 can indeed be generalized for all real numbers q, denoted by D q (not shown).Figure 4 shows the peaks of 1 3 P and 1 1 2 P P histograms of high prices and low ones.The graph is a mathematical fractal construction of histograms so similar to those of… an "actual" market!

Time… in Time Series
Another distinction of time is given below (Figure 5):     anti-persistent.As argued by Peters (1994 [11] p. 5), most people support the idea that time is deterministic.But, we have random catastrophic events, like natural and economic disasters (Goulielmos, 2017 [12])...

Time in Einstein (1905) [31]
This is: T = D 2 {1}, where D stands for distance covered by a random particle, and T stands for clock time used to measure it.Equation {1} can also be written as T = D H {2}, and H = 1/2.Equation {2} is a generalization of {1} due to Hurst (1951) [32], where H takes values in the interval [0 1] and D = Range.Time in logs is: log(T) = log(D) + log(c)/H {3}.Einstein added one further dimension in Universe: the 4 th dimension (=time).
Persistent time is T H = D/c {4}, when c is a constant, T stands for time and 0.5 < H ≤ 1. Anti-persistent time is T H = D/c {5}, when 0 ≤ H < 1/2.
Figure 6 shows the log of random time (red line) and the real time (blue line) at which the time series of the "Baltic Panamax Index"-"travelled" from 1999 to 2012. Figure 6 comes from: log(R/S) = log(kT H ) = log(k) + Hlog(T), where R is the range, S the local standard deviation, H is the power law, T = n = time index and k is a constant.Range is the difference between the maximum value of a time series from its minimum value.
As shown, the time series for "Baltic Panamax Index" (blue line) travels faster than random-covering more distance.So, time series is persistent (has a long memory or "black noise").These time series have trends, and if a rise occurred in t − 1, the chances are that it will continue to increase in t + 1.However, they are subject to sudden falls and rises.They have runs of + values that persist for some time, before switching to negative runs, which also persist.The distance  travelled is proportional to a power of the time elapsed, the power taking any value between 0 and 1.
Attention is needed to the "demand" and the "supply" price: we reckon, that before buyers and sellers go to market, they pre-calculate their "prices", including costs and "margins" 31 (profits) at every quantity... Profit is also included in the "demand price" for buyers, and this is something different at each quantity bought (as costs differ).These profits, we believe, are the "forces in action", to copy Physics, and also these are the independent variables…

How We Would Like to Interpret Marshall's Analysis?
Let "supply price" Q si (average) be: P i = (TC i + Π i )/Q i {1}, as a function of profit Π i and of total costs of production TC i , pertaining to quantity produced Q i ; where i is a time index, taking values from 1 to n. Moreover: profits Π 1 < Π 2 < Π 3 , •••, < Π n {2}, as an incentive to produce more, when prices P 1 < P 2 < P 3 < ••• < P n {3}, and if Q di > Q si , and vice versa 32 , where Q di is the quantity demanded at time i; n = 1 stands for start-up time.
We introduced falling costs, as production over time increases, resulting in increasing profits; this is-we reckon-a force for producers to produce more, 30 The concept of "maximization of profits" is not mentioned at this stage… 31 Marshall refers to the "gross earnings of management".

32
This implies decreasing costs of scale; lower costs due to learning; time to advertise; economies of scale; reputation etc. when "asked" by buyers.Marshallian costs can be constant, increasing or falling.
For buyers (merchants rather) Marshall makes no reference to their costs, but only to their (resale) profits.
As a result of the above analysis, a producer is in a pre-equilibrium in any quantity, given his/her "supply price" is paid by buyers.And buyers are in pre-equilibrium in any quantity, if their "demand price" is met.For the market, and for both-sellers and buyers-to be in equilibrium, "supply price" must be equal to "demand price", at a common quantity: the equilibrium quantity.Buyers and sellers go to market prepared with accounting information to find-out there which price will be established.

Equilibrium with Curves
Figure 7 gives Marshall's concept of equilibrium geometrically.Let 0R be the rate of production-actually taking place-and Rd the "demand price" and Rs the "supply price".If Rd > Rs, production 0R is exceptionally profitable, and will increase, where R is the "amount index" (moving right or left).The reverse will happen if Rs > Rd.Equilibrium is achieved when Rd = Rs, and is stable with the shape of the curves shown.

The Laws of Demand and Supply
The movement along a demand or a supply curve, is based 33 on 2 "economic forces": (1) when price increases, supply increases; and vice versa; and (2) when Figure 7. Marshallian stable equilibrium for a good in diminishing return. 33 The "laws" may not work.Suppose one consumer-because of Christmas-stockpiled Coca Cola for 20 days; soon after Christmas, there is a special offer at reduced price; obviously no sale is expected to take place by the consumer (Priesmeyer (1992) [26] pp.70-73).Thus, price falls, but quantity demanded does not rise.price increases, demand falls; and vice versa.This shows that the move along the 2 curves is of the opposite direction, as shown by arrows (Figure 7).Thus, a crossing of the two curves will definitely occur at a certain price and quantity, i.e. the 2 curves will intersect = at the "equilibrium price" and "quantity"; these two forces give the desired stability.As a result, we must be careful that the laws of demand and supply are valid, when equilibrium is examined, something not always certain…

A Psychological Equilibrium
This 34 is a state when "expectations of the sellers meet the expectations of the buyers".Is it possible for market players to realize their expectations?Yes.If the quantity brought into the market by sellers is all sold at the pre-determined profit.If the price asked from the merchants to buy their expected quantity at their profit is the same.This is equilibrium; this, however, can be achieved only by coincidence... or what Hicks said by trial and error.Here satisfaction in both sides is achieved, where no stocks are created and no demand goes away with empty hands…

The Stability of Equilibrium
What is not so convincing from the above analysis is that if Q d < Q s , price will fall, and if Q d > Q s , price will increase.Surely, if Q d < Q s , stocks will be created and producers will be unsatisfied.Will, however, producers lower their price to sell the entire amount, provided they have brought it to the market and their goods cannot be stored?If Q d > Q s , then producers must have a stock to sell.If there is no stock, then demand will be unsatisfied.It is possible then for the price to increase next time.Thus the whole idea that equilibrium will be achieved, even if disturbed, has to be re-examined, as much confusion has been created and hopes remained unfulfilled.

Equilibrium in Physics
In Physics (Young and Freedman (2000) [3] p. 97)), equilibrium is when a body is acted on by no force; and when a body acted on by several forces, where their vector sum is zero.This body is either at rest, or moving in a straight line, with a constant velocity.Economics has adopted the second, when demand (force) is cancelled by supply (force) and price and quantity are at rest.But can price and quantity also move-as they are-with constant velocity?Theoretically they can.
Equilibrium in Physics is based on Newton's 1 st law, saying that a body acted on by no net force moves with constant velocity, or 0, and a zero acceleration ([3] p. 96).So, economic "static" equilibrium will be attained, if the force of demand is exactly cancelled by the force of supply.How do we know that this is a static equilibrium?Because, price moves at zero speed.If we assume that price 34 After having written this, I read Hicks (1946) [6] saying: "the degree of disequilibrium marks the extent to which expectations are cheated, and plans go astray", p. 132.moves at constant speed-as Physics argued-is exactly what Hicks ((1946) [6] p. 132) said that "prices move constantly into future in a stationary equilibrium over time".

Equilibrium versus Profit Maximization
Given that maximization of profits is the true end of firms, one may ask what is the connection between this end and "equilibrium"?Samuelson (1967, [9] p. 88) placed indeed "profit maximization" prior to "equilibrium", as the former is indeed the 1 st fundamental assumption for him.

The Relationship between Equilibrium and Profit Maximization
Equilibrium needs production with factor combinations so that TC (total cost) is at minimum: the marginal productivity of the last $ is equal everywhere, and the price of each factor of production is proportional to its Physical productivity (marginal), in analogy to marginal cost; the output selected, maximizes net Revenue, and total cost is determined optimally: MC (marginal cost) = MR (marginal revenue)-with a smaller slope than that of MR; the value productivity of each factor (marginal) = its price (MR times marginal physical productivity); and TC ≤ TR.

Equilibrium in Complexity Theory
Battram ((1998) [5] p. 157 and afterwards)) argued that firms can be frequently locked-in to a less than optimal state (meaning < equilibrium).It is strange, but no one in management asks for whether a company achieved equilibrium...

The Laid-Up Tonnage as a "Logistic Equation"
Let laid-up 35 tonnage follow the quadratic equation36 : X = aX − aX 2 {1}, where X stands for laid-up tonnage and "a" is the control coefficient; X(a − aX − 1) = 0 {2}.One of its solution is X = 0, where no laid-up tonnage-LUT, exists; and another solution is X = a − 1/a.Both solutions are interesting.Equation {1} graphically is presented below (Figure 8).
At low levels of "a" there is stability and constancy; as "a" increases, emerge oscillations, complex patterns, disorder, and the whole system "appears" random.Let now take 20 time periods (Figure 9) using the iteration method (i.e. the solution X 1 at t 1 is inserted into X 2 , and so on).At a = 80% (0.8), LUT declines continuously and stabilizes at X = 0.At a = 1.5, LUT = 33%.These cases are examples of low-order chaos, and are stable and predictable.At a = 3, LUT dency of a simple nonlinear system to go through period doubling.If the solutions are constant (a = 1.5) or oscillating with regular, periodic solutions: a = 3 or 3.55, then they are more capable in re-establishing regularity after being drawn away from their regular pattern: this is stability.Laid-up tonnage obeying the logistic equation gives 3 types of curves for t = 1 to t = 20 and for a = 0.80, 1.50 and 3.00.We assume also an initial tonnage in laid-up equal to 50% (0.50), as a starting condition.This means we start when market is in a deep depression, like in 1981-1987 (Graphic 3) (Graphic 3).
As shown at t = 5, 10 and 15 time periods 3 logistic equations appear-with 3 different a's.The conventional Marshallian equilibrium is at t = 15, where S = D, and LUT (%) is zero.This is a "structurally" stable point.The amount (%) of LUT is continuously falling from an initial 50% to 20% and 0% over time.Shipowners are happy.
However, if Market gets worse, if "a" gets higher, the LUP tonnage increases to a (stable) 33% (a deep depression).Here a heavy effort due to extensive scrapping of ships could reduce LUT by 12%, but after 1 -2 periods it will return back.At a higher "a" = 3.00, appears a stable and continuous oscillation ranging from 63% LUT to 70%, i.e. a heavy depression.Finally, the LUT alternates among: 88%, 37%, 83% and 51%, for every 4rth period, for a > 3.00 (not shown here), manifested is the well-known shipping cycle; "a" can get values from 1 to

Shipping Markets as an Attractor of 2 Difference Equations
Let assume now another model 38 -with 2 difference equations: Supply for ship space = − 2 +1 = 1 + n n n X aX Y {1} and demand for ship space = Y n+1 = bX n {2}, where n = 0, 1, 2, ••• If b < 1, X n and Y n are attracted by a set of points-as time (n)  ∞.The following graph (Figure 10) is derived if a = 1.4 and b = 0.3.When a > 0, and small, the attractor is quite simple.
To restrict the above model to have X ≥ 0 and Y ≥ 0, as demand and supply are positive, we excluded all negative values of X and Y (shaded part).We have introduced time (n).When Demand is at maximum, and Supply is zero, price is high.Thereafter, demand and supply move in balance, though dual amounts occur.Coefficients "b" and "a" stand for: "seaborne trade" and "shipbuilding prices" respectively.The supply of ship space is adaptive to demand as most maritime economists assume.

11 Founder of mathematical Economics. 12 Ragnar
Frisch, On the notion of equilibrium and disequilibrium, Review of Economic Studies, III (1935-36), 100-106.13 Cartwright M and Littlewood J L, during World War II, proved mathematically that signs of chaos could exist.14 The ideas of Poincare (Jules Henri, 1854-1912, from France, Prof. of Mathematics in Paris) continued in USA by the American mathematician Birkhoff G D (in 1927; and in 1968)-a Professor at Harvard.But this movement died-out; lived in Russia, Berlin and Holland.In USA Lefshetz S revived the "dynamical systems theory".Samuelson mentioned also Van der Pol (in 1926), Levinson N (in 1942; in 1944)-an MIT mathematician, Smith (in 1942) and Karman (in 1940); he was aware also of Mandelbrot.

Figure 2 .
Figure 2. Freight rate determination where "time" is present.

Figure 3 .
Figure 3. Classification of time in finance.

Figure 4 .
Figure 4.A fractal market price chart created mathematically, using fractions.
and n stand for time; D stands for distance; R stands for range; S stands for (local) standard deviation; H stands for Hurst's exponent or power law; and c is a constant.

Figure 6 .
Figure 6.The speed of BPI time series and that of its Random Walk (1999-2012).

4 .
At a = 3, this is the last time when market is in equilibrium; at a = 3.45 the period is doubling (X(aX − a) = 0) and at a ≥ 3.57, we enter chaos.Massive recent LUT is shown in Graphic 3. The above model is only one-variable model.Of course laid-up tonnage reflects the final result of supply and demand.Source: Greek shipping world (unknown date).

Graphic 3 .
Laid-up tonnage for lack of demand in a recent Shipping depression.

A
more complicated dynamical model 39 with 3 differential equations is: dX/dt = 10(Y − X) {1}, dY/dt = −XZ + 28X − Y {2} and dZ/dt = XZ − 8/3Z {3}, Δt = 0.01, and initial conditions Xo = 6 = Yo and Zo = 27.Where X is the supply of ship space, Y is the demand for ship space and Z is the price or freight rate.In fact we have 4 variables including time.Attractor's dimension is fractal and equal to 2.07.Source: Modified from Manneville[35].