_{1}

^{*}

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^{th} 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”...

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).

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.

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 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.

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…

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 “logistic 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^{3} preoccupied Marshall ((1920) [^{4}, especially when one wishes to pass from theory to reality. Moreover, in the preface of the 8^{th} edition of his book ( [

Marshall [^{6} However, unlike rivers, markets… jump (Mandelbrot and Hudson, (2006) [^{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) [

Marshall ( [^{8} is “economic biology^{9} “... What Marshall meant―we believe―is that economy has to be studied at best as a living “organism’” (Blaug [

Hicks (1946) [

Joan Robinson (1965) [

Zannetos (1966; [

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., [

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, [

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) [

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}).

Nonlinearity opened a door nearer to reality. Samuelson (1967) [^{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) [^{16} 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) [

Abraham (2000) ( [

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 problem^{17}. 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 (Smale^{18} [^{19} (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) [

Farmer (2002) [

Stopford (2009 [

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 [^{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…

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).

We must mention that Ruelle^{24} D. [^{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 Physics is presented in Appendix. In

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,

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.

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?

In finance exist 3 times...: “Clock”, “Trading”, and “Fractal” (Mandelbrot and Hudson (2006; [

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) [

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), [^{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) [^{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) [

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 (

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: P m 1 ( n ) as n à ∞, and m = m_{1}, where N m 1 ( n ) P m 1 ( n ) , is at maximum as n à ∞. Skipping the proof^{29}, the final result is m_{1} = 2p_{2}n.

^{29}Shown in Mpountis (2004) [

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).

Another distinction of time is given below (

As shown, “time”… in time series can be: (1) random, (2) persistent and (3) anti-persistent. As argued by Peters (1994 [

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) [^{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.

^{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.

Marshallian ((1920) [^{30}. 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…

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, 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.

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

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 (

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…

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) [

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 ( [

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, [^{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) [

Let laid-up^{35} tonnage follow the quadratic equation^{36}: 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 (

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 (_{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

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% (

In 1971, May^{37} 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 model^{38}―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 (

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 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.

As shown,

Economists locked in time in “ceteris paribus”, unable to work in a 3-dimensional space, where demand, supply, price and time could be presented. Marshall did not explain the content of ceteris paribus. Economic theory is mainly static. Economics made a progress by shifting the curves achieving the “comparative-statics”, thus obtaining 2 photographs at 2 different times to compare.

Reality obliged economists, and Marshall, to introduce 4 periods, where time is loosely and unrealistically defined: market period, short run, long run and very long run. Time is best represented by a differential equation.

While “marginalism” “saved” economists giving mathematics a prominent role in economics, after 1870, Marshall (in 1890) wrote that mathematics is scaffolding, which should be removed. Marshall innovated for placing price on Y axis and quantity on X for the first time.

One is surprised to find out that time has so many different expressions among sciences. Physics has one time for every area; maritime economics ignored time, but Japanese Shimojo; finance has 3 concepts for time, but “fractal” time is quite realistic followed by “trading” time. Chaos theory re-introduced time into economic analysis using differential and difference equations.

Equilibrium condition is adopted from Physics, appears nowhere in management, obscures the fact that maximization of profit is the true and unique end of enterprises. Enterprises are satisfied if they maximize their profits. Another obscure point is the belief that equilibrium―if disturbed, it returns to it―a hope that has been proved false crisis after crisis and at the end of 2008 meltdown par excellence. Equilibrium is a technical condition to say that what is demanded is

supplied, at a price securing the profits/utilities of both sides. Being in disequilibrium is more interesting.

Stability is a function of time. If we want a constancy, and stability, we need a low-level chaos adhering to a simple “attractor”. But no one excludes high-order chaos, which follows a more complex “attractor” with periodic oscillations.

Logistic equation is a well-celebrated tool applied here to represent laid-up tonnage. One is surprised to find out that laid-up can produce 4 different behaviors over time, where at a = 3 tonnage varies from 63% to 70%, and back to 63% in a continuous pattern. Similarly the model Demand = X_{n}_{+1} = 1 − aX_{n} + Y_{n} and Supply = Y_{n}_{+1} = bX_{n}, can determine the freight rate using 2 difference equations.

Finally, Demand and Supply determined price over time using 3 differential equations to describe shipping market like… weather. In dynamical systems, the state of the economy is represented by a set of variables and a system of difference equations, or differential equations, which describe how these variables change over time. Because new niches, new potentials, new possibilities, are continually created, the economy operates far from any optimum or global equilibrium. Systems are the adaptive nonlinear networks a la John Holland (in 1988).

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

Newton (in 1687) argued wrongly that “time flows equally without relation to anything external”. Let place a mirror in a box’s top and a light source at bottom (^{40}: the hypotenuse. The Pythagorean theorem gives its length = [(1/2vt)^{2} + L^{2}] {2}. Light returns to its source through the other hypotenuse down, and thus the total distance is 2 times equation {2}. Raising {2} to square power and taking the square root, we have: [(1/4v^{2}t^{2} + L^{2})2] {3}, equal to ct.

Subtracting v^{2}t^{2} from both sides of {3}, divide by c^{2} and take the square root of both sides, we get: 2 L / c = t ′ = t 1 − v 2 / c 2 {4} (where v now is a fraction of c). Here, we have 2 times: t and t ′ ! Equation {4} states that time t ′ (between 2 events^{41} measured in a reference frame at the same place) = to time t (between 2 events measured in a reference frame at a different place), multiplied by the square root of 1 − v^{2} ((where v is the (relative) speed of the 2 reference frames (as a % of light’s speed c)). So, time t between 2 events―at a different place―is greater of t ′ ―by 1 − v 2 ―where t ′ is the time between the 2 events at the same place… Example: let v = 0.8^{42}, then t ′ is 60% of t.^{ }

[*] Wolfson, R. (2003). Simply Einstein: Relativity Demystified, W W Norton & Co, NY, ISBN 0-393-05154-4.