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 paribus¹” (Latin). Time is locked-in in ceteris paribus.

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…

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 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.

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 (1911-1947), 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 machines²”, 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.

Do we need equilibrium? Yes, if we believe in determinism, predictability and balance. No, if we believe in structure, pattern, self-organization, life cycle (ideas from biology). Yes, if we believe that there is a single accessible end-point of balance. No, if external effects and differences are key-drivers; no, if any economic system is constantly “unfolding”. Yes, if there are no real dynamics and everything is in equilibrium. No, if economy is constantly on the edge of time; it rushes forward, structures constantly coalescing, decaying, and changing…

Time³ 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 important⁴, 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.

Marshall [1] believed that nature is smooth: “Nature makes no jumps” (leaps) (“Natura non facit saltum”) he stated, based on “principle of continuity” (Blaug (1997) [2] p. 379), an idea from Physics.⁶ However, unlike rivers, markets… jump (Mandelbrot and Hudson, (2006) [4] p. 85-86; 227-229)! Heraclitus (~460 BC) “said”⁷ that “one cannot enter into a river and encounter the same waters”, meaning that “everything is constantly changing”… in a process of “becoming”; something resembling emergence in the complex adaptive systems (Battram, (1998) [5] p. 13).

Marshall ( [1] p. xii) argued that important for economics⁸ 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.

Hicks (1946) [6] , p. 120) argued that Marshall refers to a (final) supply for sale, determined by past expectations (taking into account preferences and incomes). Traders do not know what exactly suppliers will bring, or what buyers will demand. Prices thus are fixed by trial and error. Marshall argued that supply and demand must be equal: the buyers to buy what they desire (at the market price) and sellers to sell what they desire. Thus, equilibrium is a balance of desires… Hicks rejected the “full foresight” assumption, which Joan Robinson (1965; [7] ) restored by a new concept that of “lucidity”. Hicks (p. 140) said: “economists have often toyed with the idea of a system where all trading persons have ‘perfect foresight’―leading to awkward logical difficulties…”

Joan Robinson (1965) [7] ) argued indirectly that equilibrium (pp. 57-59), as a balance, comes from “celestial mechanics”, and it has to be applied with great caution in the “economics (of a stationary state)”. This metaphor is treacherous, she argued. Robinson transferred the concept of equilibrium from markets to people, by inventing 2 terms: tranquility and lucidity… Tranquility is when economy is in smooth regular markets… where expectations-based upon past experience―are very confidently held… constantly fulfilled and renewed over time, and lucidity, where people are fully aware of the situation in all markets etc.

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 R₂ and R₃ do matter; supply has a negative slope. R₃ is stable from below and R₂ indicates a price in an unstable range, as demand’s slope is positive. R₄ shows potential instability. Prices could be at rest―given able time―at either R_s or R₃, but not at unstable R₂. 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.

Source: modified from Zannetos (1966, [8] p. 18).

Graphic 1. Demand and Supply affected by price-elastic expectations determining freight rates a la Zannetos-Multiple equilibria a la Marshall.

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 equilibrium¹⁰ 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 Cournot¹¹ (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 econometrics¹²).

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 mentioned¹³ Birkhoff¹⁴ 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 order¹⁵.

West and Goldberger (1987) [10] ) argued that the “physical fractal¹⁶ structures” are the ones generated by nature. They are more error-tolerant and more stable. One market can absorb shocks as long as it retains its fractal structure. Peters (1994) [11] p. 8) recognized that a 4σ change in time series is a crash (Goulielmos, (2017) [12] ). When market’s investment horizon shortens, the market becomes erratic and unstable. Do the real world and markets show local randomness and global determinism (Peters, (1994) [11] pp. 10-11)? Farmer and Lo (1996) [13] ) advanced a theory dealing with the “adaptive markets hypothesis”.

Abraham (2000) ( [14] p. 81-90) argued that Chaos revolution, 1968-1998, begun with the work of “Takens and Ruele” on “turbulence” ( [15] 1971)―mentioned also by Goulielmos (2017)―consisting the turning point in Science. To this “Thom” contributed (1973 [16] ) as well “Li and Yorke” (1975 [17] ), who brought the word of “chaos” out.

Chaos, however, glanced first by Poincare H. in 1889. Poincare participated in a mathematical competition, set by King Oskar II of “Sweden and Norway”, for the participants to solve a problem¹⁷. In solving it both Prof. Weirestrass and Poincare-in his 1st attempt-failed. Poincare, as a result of the competition, and due to a mistake he made, discovered accidentally the “homoclinic tangle” or Chaos (Graphic 2), as shown in his paper published in 1890 (by which he won the competition). The case of ∞ solutions of the above problem is indeed related to “homoclinic” behavior (Smale¹⁸ [18] ). A “homoclinic tangle” is found in the manifolds of certain maps (Harding¹⁹ (unknown date)), if manifolds cross transversally.

Source: inspired by Harding (unknown date).

Graphic 2. Let P be a fixed point saddle; then p' is a homoclinic point. f(p') and f⁻¹(p') on stable and unstable manifolds are also homoclinic points, as nà∞.

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 was²⁰ 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 statical²¹, comparative-statics, comparative-dynamics, dynamics²² etc. He established the “intimate formal dependence between comparative-statics and dynamics”. Moreover, Samuelson classified dynamics in 6 different classes…

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

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 scales²⁶. 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}: Δ₁θ, Δ₂θ, Δ₃θ, 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).

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.

A fractal and multifractal model (following Farmer D. in 1980 and Mpountis T (2004) [30] p. 228)).

Let the number of high and low prices in a market [0 1] be N. Let the % of high prices be: p₁ = n₁/N, and of low prices-distributed in two groups-be: p₂ = n₂/N, where n₂ < n₁, so that: p₁ + 2p₂ = (n₁ + 2n₂)/N = 1 {1}. One histogram has 0p₁ 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 0p₂, 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 p₁, for high prices, and by p₂, 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: P m 1 ( n ) as n à ∞, and m = m₁, where N m 1 ( n ) P m 1 ( n ) , is at maximum as n à ∞. Skipping the proof²⁹, the final result is m₁ = 2p₂n.

²⁹Shown in Mpountis (2004) [30] p. 229 (in Greek).

Now, if p₁ = p₂ = 1/3 or p₁ = 0 and p₂ = 1/2, this is the simple fractal (of a single dimension). This (dimension) is given by: D₁ = p₁logp₁ + 2p₂logp₂/log(1/3) = 1 {2} if p₁ = p₂ = 1/3. And ~0.63, if p₁ = 0 and p₂ = 0.5. This indicates that dimensions are ∞, and this market is multifractal. D₁ can indeed be generalized for all real numbers q, denoted by D_q (not shown). Figure 4 shows the peaks of P 1 3 and P 1 2 P 1 histograms of high prices and low ones. The graph is a mathematical fractal construction of histograms so similar to those of… an “actual” market!

Another distinction of time is given below (Figure 5):

As shown, “time”… in time series can be: (1) random, (2) persistent and (3) 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] )...

This is: T = D² {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.

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 Π₁ < Π₂ < Π₃, ・・・, < Π_n {2}, as an incentive to produce more, when prices P₁ < P₂ < P₃ < ・・・ < P_n {3}, and if Q_di > Q_si, and vice versa³², 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, 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.

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 movement along a demand or a supply curve, is based³³ on 2 “economic forces”: (1) when price increases, supply increases; and vice versa; and (2) when

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…

This³⁴ 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…

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.

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 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”.

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.

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.

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...

Let laid-up³⁵ tonnage follow the quadratic equation³⁶: X = aX − aX² {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 X₁ at t₁ is inserted into X₂, 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

oscillates between 63% and 70%. This pattern, between a = 1.5 and 3, gives the 1^st bifurcation. Then at higher a’s, LUT “exists” alternating between 88%, 37%, 83% and 51% (Figure 9).

In 1971, May³⁷ R. re-discovered equation {1}, which shows an inherent tendency 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).

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 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.

Let assume now another model³⁸―with 2 difference equations: Supply for ship space = X n + 1 = 1 − a X n 2 + Y n {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) à ∞. 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.

A more complicated dynamical model³⁹ 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.

As shown, Figure 11 depicts a system of 3 differential equations, which produce a “strange” attractor (right). Although we assume that the freight market is like weather, we have to put conditions so that none of the 3 variables as well time can take negative values. The model, however, does not provide an equilibrium as demand is higher than supply (left) at the intersection of the curves.