Capitalist mechanism and cycles

1. Atomized allocation and possibility of error.
2. Profit maximization and growth.
3. Economic mechanisms act like mathematical algorithms that can be unstable .
4. Negative feedback.

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Important. 1. Fragmented allocation and possibility of error

• Allocation structure in capitalist systems (and other systems)
• Dividing the problem into small allocation units (firms, families, etc.)
• The coordination of allocation units through the pricing mechanism
• Other systems follow this same pattern, such as cooperatives linked to the market, or anarchism. What we say about cycles also applies to these systems.
• The overall decision-making process can be highly flawed because the allocation process is independent of other companies. The entrepreneur must decide without knowing what other companies will do . Error is very likely even if the allocation were established in a system where decisions only had to be made about a present value.
• Furthermore, since the allocation involves production processes that take time , the entrepreneur must decide on the future they will need to anticipate based on available information from the present and the past. And the future will be very difficult or impossible to predict.
• Thus in our game, in certain sessions when there is a shortage of wheat, a lot of wheat is produced, sometimes too much, because when you decide on production you don't know what the other students are going to decide and you don't know what will happen in the next session.
• You're acting without knowing what the other players are going to do,
• And you act based on a future you cannot know.

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2. Profit maximization and growth

• Profit maximization drives companies and the economy as a whole to grow as much as possible.
• (Although capitalist systems are subject to the limitations imposed by the environment.)
• In our game this is also the case, and growth tends; although in our game it is designed without resource limitations and exponential growth is possible.
• Biological populations also tend to grow exponentially if they have resources.
• (But with environmental limitations, they exhibit logistical or cyclical behavior.)

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3. Economic mechanisms act like mathematical algorithms

• The economic problem of establishing an allocation in reality corresponds to the mathematical problem of solving economic models and theories.
• The mathematical algorithms we use to solve models (like Divide et impera) often operate in parallel to how economic mechanisms work in capitalist systems.
• Thus, the need to divide the allocation into companies and coordinate them with the market has its parallel in mathematical algorithms, where the global problem is also divided into subproblems and an attempt is made to coordinate the solutions of the independent subproblems.
• In these mathematical algorithms, the maximum conditions and Lagrange multipliers operate in parallel to how accounting and values do in economic mechanisms.
• Capitalisms operate like real algorithms ; they are economic mechanisms that solve the real allocation problem in a way that is parallel to how mathematical algorithms solve the economic problem on the logical plane, using accounting and values, and dividing the general problem into companies and coordinating them with the market to solve it.

3. Economic mechanisms and mathematical algorithms can be unstable

• But mathematical algorithms can be unstable and exhibit cycles or chaotic behavior.
• And if mathematical algorithms can be unstable, the real algorithms that we call economic mechanisms, and which operate in parallel to mathematical algorithms, can also be unstable.
• Studying the instability conditions of mathematical algorithms gives us clues about the instabilities of economic mechanisms.
• For example:
• To try to solve some economic models, such as the Von Neumann model if it's large, we need algorithms like the simplex method or divide and conquer. But these mathematical algorithms can be unstable and exhibit cycles, which is a clear indication that the economic mechanisms of capitalist systems can also be unstable.
• To solve the general equilibrium equations, we also need algorithms, which are similar to economic mechanisms to some extent, and which can also be unstable. See Herbert Scarf, Some Examples of Global Instability of the Competitive Equilibrium , https://www.jstor.org/stable/2556215
• Algorithms that also allow for an economic interpretation, such as the simplex method of linear programming, can also enter into cyclical patterns.

Dantzig, Linear Programming and Extensions , https://www.rand.org/pubs/reports/R366.html

Important. 4. Negative feedback

• With feedback , the system responds to a stimulus. It can be of two types:
• Positive feedback :
• The system's response to the stimulus amplifies the stimulus.
• “A positive feedback loop comes into play during childbirth. During labor, the baby’s head presses against the cervix—the lower part of the uterus through which the baby must pass—and activates neurons that send signals to the brain. These neurons send a signal that causes the pituitary gland to release the hormone oxytocin. �Oxytocin increases uterine contractions and, therefore, the pressure on the cervix. This causes the release of more oxytocin and leads to even stronger contractions. This positive feedback loop continues until the baby is born.” https://es.khanacademy.org/science/ap-biology/cell-communication-and-cell-cycle/feedback/a/homeostasis
• Negative feedback :
• The system's response to the stimulus inhibits the stimulus.
• For example, if the body's sensors detect a high temperature, the sweat glands activate to cool the body. If the sensors detect a low temperature, we shiver. Both responses tend to bring the temperature back to a normal level.

Important. Negative feedback and cycles.

• Not all cycles are the result of feedback. The day-night cycle or the cycle of the seasons are not the result of feedback, but of a periodic phenomenon that causes them: the Earth's rotation on its axis and around the Sun.
• But the systems we have identified as similar cycles are characterized by being regulated by negative feedback:
• If the measured information is large, the regulator tends to reduce it.
• If the measured information is small, the regulator tends to increase it.
• This is the case in Watt's regulator, in thermostats, in Cepheid variable stars, or in the cycles of biological populations. For example:
• If the machine's speed is high, the Watt regulator reduces it, and if the speed is low, it increases it.
• If the temperature is low, the thermostat turns on the heating to raise it, and if it is high, it turns it off to lower it.
• If the size of the cepheid variable is small, the opacity of the gas increases, and with it the temperature, which causes it to increase in size; and if it is large, the opacity decreases, and with it the temperature, which causes it to decrease in size.
• If the population (of lemmings, or hares, or humans) is large, the lack of food causes its reduction, and if it is small, the abundance of food causes its increase.

Important. Negative feedback and cycles

• The existence of negative feedback sometimes leads to stable behavior.
• Watt's regulator was stable with old machines, most stars do not oscillate like Cepheids, and many biological populations show a stable size.
• The cycle appears in these systems with negative feedback when the systems respond too strongly to the regulators.
• The transition from stable to cyclical behavior is so common that it has a name: Hopf bifurcation, https://www.youtube.com/watch?v=CwGPPza9Ulk
• See Norbert Wiener, Cybernetics , chap. IV, Feedback and Oscillation, https://archive.org/details/wiener-1938/Cybernetics%20or%20Communication%20and%20Control%20in%20the%20Animal%20and%20the%20Machine%20-%20Norbert%20Wiene_OCR/page/95/mode/2up
• It is therefore possible that a regulator may exhibit stable behavior under certain circumstances, and that the same regulator may exhibit unstable behavior if the intensity with which the system responds increases.
• This is what happened with Watt's regulator. With older machines, the regulator resulted in a stable machine speed. But with modern, more powerful machines that responded much more forcefully, oscillations appeared.
• It is also possible for a biological population to be stable when it has a low reproduction rate, while a similar species with a higher reproduction rate may exhibit cycles like those described by Condorcet. It is the species with a high reproduction rate that show cycles.
• (We will see later that the economic cycle is very similar to the demographic cycles of humans or biological populations, as described by Condorcet, but also to the cycles caused by Watt's regulator, thermostats, etc.)

Important. Negative feedback: capitalisms

• The mercantile-capitalist mechanism is based on negative feedback loops, in several senses:
• At the production level of a commodity in relation to others.
• If the quantity produced of a commodity is high (its price will decrease, and with it) the quantity to be produced will decrease;
• If the quantity produced of a commodity is low (its price will increase, and with it) the quantity to be produced will increase.
• At the overall production level with respect to the limits imposed by the environment.
• If the economy grows beyond the limits of the environment (costs increase, and with them) it tends to decrease;
• If the economy is below the limits of the environment (costs decrease, and with them) it tends to grow.
• The classical economists already saw this negative feedback, although they did not call it that, between overall production and the limits of the environment and thought that the result was a steady state.
• The intensity of the response will be determined by the intensity with which the capitalist system impacts the limits imposed by the environment, which is determined by technology among other aspects.
• If the mercantile-capitalist regulator responds with little intensity, we will have stable behavior , as the classical economists expected.
• But if the market-capitalist regulator responds too intensely, we will have unstable behavior , cycles, or even aperiodic behavior.

The spider web model

Stability and fluctuations in the production level of a commodity relative to others

Remembering. The spider web model

• The cobweb model has been suggested as an illustration of how the price mechanism operates when production takes time .
• The price, let's say of wheat, at time t decreases with the quantity of wheat available at time t (demand curve).
• The quantity of wheat to be produced at time t+1 increases with the price of wheat at time t (supply curve).
• Consequently, the amount of wheat to be produced at time t+1 decreases with the amount of wheat available at time t .
• We have negative feedback (we will explain this in other classes)
• The dynamic results in:
• At time t, for a quantity of wheat, the price of wheat is formed, and for that price, the quantity of wheat to be produced at the next time t+1 is established.
• In time step t+1 for the existing quantity of wheat, the price of wheat is formed, and for that price the quantity of wheat to be produced in the next time t+2 is established.
• And so on for the rest of the time steps.
• The system can be stable and converge to an equilibrium, or unstable and diverge from equilibrium .

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Remembering. The spider web model

• Let's take an example. Let's start in year 0, in a situation where there is little wheat.
• Year 0: as there is little wheat, its price will rise.
• Year 0: as the price of wheat is high, its planting will increase, because producing wheat seems to be very profitable.
• Year 1: Therefore, there will be a large wheat production.
• Year 1: as there is a lot of wheat, its price will go down.
• Year 1: as its price is low, its planting will be reduced, because producing wheat seems to be unprofitable.
• Year 2: Therefore, there will be a low wheat production.
• Year 2: as there is little wheat, its price will rise.
• We are in the same situation as in year 0, only two years later, and the same thing is happening again.
• Therefore, if in one year there is little wheat (and the price is high) the following year there will be a lot of wheat (and the price will be low), and if there is a lot of wheat (and the price is low) the following year there will be little wheat (and the price will be high).
• This dynamic can lead to:
• Or a tendency towards equilibrium, where the quantity of wheat and its price do not change over time;
• Or a tendency towards imbalance, where the quantity of wheat and its price evolve cyclically.
• In general, the stronger the system's reaction, the more unstable it is .

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Optional. The spider web pattern

• We start with an offered quantity of Q1 (let's say wheat), which is below the equilibrium point determined by the intersection of the supply curve S and the demand curve D.
• For the relatively low quantity Q1, the demand is such that the price paid for wheat is high, P1. A high price is paid because there is little wheat available.
• Given this high price P1, wheat producers decide to produce a large quantity Q2 the following year. A lot is produced because the price is high.
• For the high quantity Q2, demand is relatively low, and the price falls to P2. Little is paid because there is a lot of wheat.
• Due to the low price P2, wheat producers decide to produce less the following year, the quantity Q3. Little is produced because the price is low.
• For the low quantity Q3, demand is relatively high, and the price rises to P3, etc. A high price is paid because there is little wheat available.
• In the case of the left, for those specific supply and demand curves, the dynamics are such that the system tends towards the equilibrium position, towards the point where the supply and demand curves intersect.
• But if the slopes are different, the system can be unstable, as in the case on the right, and move away from the point where the curves intersect.

The convergent case: each new result successively approaches the intersection of supply S and demand D.

The divergent case: each new outcome moves successively away from the intersection of supply S and demand D.

https://en.wikipedia.org/wiki/Cobweb_model

Dispensable. Historical origin

• Cobweb price formation was described independently in 1930 by Henry Schultz, Jan Tinbergen, and Umberto Ricci. Four years later, Nicolas Kaldor drew attention to these analyses, which he termed "cobweb" analysis, and showed their relationship to equilibrium determination in cases where adjustments are completely discontinuous. Wassily Leontief demonstrated that when supply or demand curves have an erratic shape, a convergent or divergent series can be obtained from the same set of curves. The formulation of the theorem in 1938 was by Mordecai Ezekiel.
• According to Ezekiel, "classical economic theory rests on the assumption that price and production will always tend towards an equilibrium position if this is altered (see Walras's Law); on the contrary, the cobweb theory demonstrates that even under static conditions, the process does not necessarily occur, but rather the prices and production of some goods can fluctuate indefinitely and move further and further away from the equilibrium point."
• Wikipedia: https://es.wikipedia.org/wiki/Teorema_de_la_telara%C3%B1a
• https://www.economicshelp.org/blog/glossary/cobweb-theory/

Pedro Caldentey, The pig cycle in Spain in the period 1959-1977 , https://www.mapa.gob.es/ministerio/pags/biblioteca/revistas/pdf_ays%2Fa014_04.pdf

Optional. The spider web pattern

Double-click on the spreadsheet to modify the parameters of the lines in the yellow cells and check the resulting dynamics .

Optional. The spider web pattern

• Depending on the parameters of the curves, we will have different behaviors:
• Stable equilibrium (the system tends towards constant production and prices),
• Neutral equilibrium (the system remains in a cycle that does not vary in amplitude),
• Unstable equilibrium (the system oscillates increasingly).
• In this model, the price mechanism controls production through negative feedback:
• It reacts to abundance by reducing production.
• It reacts to scarcity by increasing production.
• The stronger the system reacts, the more unstable it will be .
• The more intensely the price falls with abundance, or the more intensely production increases with the price, the more unstable the system will be.
• In terms of the model, the steeper the curves, the more instability there will be.
• The more sharply the price decreases at time t with the quantity at time t, the more instability we will have;
• The stronger the quantity produced at time t+1 grows with the price at t, the more instability we will have;
• In the three examples on this slide, we have only increased the strength of that last growth.

Stable equilibrium Neutral equilibrium Unstable equilibrium

Optional. The spider web pattern

Using the scissors example that Marshall includes in his Principles , the dynamics turn out to be divergent!

https://archive.org/details/in.ernet.dli.2015.149776/page/n360

Remembering: Growth and limits of the environment

Stability in the overall production level with respect to the limits imposed by the environment. (We will see the fluctuations later)

Remember: Exponential growth

If a magnitude has an annual growth rate of

• 1% doubles every 71 years, multiplies by 10 every 233 years, by 100 every 464 years, and by 1000 every 696 years.
• 2% doubles every 36 years, multiplies by 10 every 118 years, by 100 every 234 years, and by 1000 every 350 years.
• 3% doubles every 25 years, multiplies by 10 every 79 years, by 100 every 157 years, and by 1000 every 235 years.
• 4% doubles every 19 years, multiplies by 10 every 60 years, by 100 every 119 years, and by 1000 every 178 years.
• 5% doubles every 16 years, multiplies by 10 every 49 years, by 100 every 96 years, and by 1000 every 143 years.
• 10% doubles every 9 years, multiplies by 10 every 26 years, by 100 every 50 years, and by 1000 every 74 years.
• 20% doubles every 5 years, multiplies by 10 every 14 years, by 100 every 27 years, and by 1000 every 39 years.

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Approximate formula (best for low rates)�

years to double ≈ 70 / growth rate�

The years to double are approximately equal to 70 divided by the growth rate

It is advisable to learn this equation or its meaning in words.

Remembering. Long-term logistics behavior

• If k is less than or equal to 1 then decay to extinction .
• If k is greater than 1 and less than 2 then logistic growth to steady state .
• If k is greater than 2 and less than 3 then logistic growth with decreasing oscillations to steady state .
• If k is between 3 and 4, very interesting things happen (cycles and chaos) that we will see at another time; those who are impatient can consult Veritasium: This equation will change how you see the world (the logistic map), https://youtu.be/ovJcsL7vyrk?t=27

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You can modify k, P, and X0 in the yellow cells of the spreadsheet by double-clicking on them to visualize the different behaviors.

Remembering. Capitalism, technological advancement, and growth. The great dynamic of the classics

• The classical economists were right to point out that capitalism, like any real system, is subject to the limitations imposed by its environment. Therefore, it cannot grow exponentially in the long term.
• But capitalism has maintained a growth similar to exponential since the classical era, not reaching a steady state.
• Classical thinkers reasoned under the assumption of constant technology .
• Capitalism has changed the way the problem is framed with the continuous development of technology :
• The continuous technological revolution has steadily increased the load capacity (the final steady-state level)
• Discovering new raw materials, or circumventing some limitations when it comes to waste disposal (extensive progress)
• Sources of “energy”: oil, uranium
• New materials
• Chlorofluorocarbons and the ozone hole
• CO2 and climate _change
• Developing more efficient internal metabolisms (intensive breakthrough)
• Substantial improvement in industrial processes (aluminum)
• Improvement in the process of obtaining information and calculation
• We can understand the growth of capitalism as a succession of �logistical behaviors with increasing carrying capacity, through the technological revolution.
• The foundation of economic growth is therefore technological advancement . Without it, capitalism �would eventually stagnate, as it did in ancient Rome.

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A hypothesis about the cycle

Stability and fluctuations in the overall production level with respect to the limits imposed by the environment

A hypothesis about the cycle

• The cycle in capitalisms is the result of the allocation mechanism with negative feedback, which determines the impact of the overall production level on the limits imposed by the environment.
• If the technology is not intense enough, the system shows the stationary state of the classics, perhaps through decreasing oscillations.
• With technology that is not too advanced, the limits imposed by the environment on the system determine that it tends to grow less strongly as it approaches them; we have the classic logistical behavior.
• If the technology is sufficiently advanced, the system exhibits the cycle .
• With advanced technology, the system, when it grows too rapidly, exceeds the limits imposed by its environment; it enters a crisis and falls below those limits; once below those limits, it booms again and grows too rapidly, exceeding those limits once more, and so on. The boom generates the crisis, the crisis generates the boom, and so forth.
• A reduction in the investment rate can have the same effect as a decrease in technological intensity.
• That is why the emergence of the welfare state as well as other unproductive expenditures (such as the military) caused a momentary reduction in the cycle since World War II.
• But as technology continued to grow since 1973, the cycle is evident again.
• This hypothesis needs to be tested .
• It is necessary to start from a theory of allocation and valuation in capitalisms (and not so much a theory of crises), and to verify that this theory effectively explains the facts.

A hypothesis about the cycle

• The cycle is possible because:
• Since each company operates independently of the others, and since the allocation must be made based on a largely unpredictable future, errors in allocation are inevitable and systematic.
• Profit maximization in companies leads to growth maximization, and this in turn puts pressure on the limits imposed by the environment.
• If the system falls below those limits, companies are compelled to invest in ways that tend to push those limits beyond their limits.
• If the system is beyond its limits, companies have no choice but to reduce their activity.
• The physical time separating allocation decisions from the actual production of goods, as well as the influence of other time-related factors such as the lifespan of fixed capital assets, constitute the material basis of the cycle period. The interplay between production times produces a fundamental cycle.

Optional. Illustration of the hypothesis: a logistic model

We used the same logistic model we built to study the influence of the investment rate. Of course, it's not possible to model capitalist systems with such a simplistic model. We used it only as an illustration. However, more advanced and realistic models show analogous behaviors.

Item_​ = s X _t

The investment at time t is equal to the investment rate multiplied by production at time t�

C _t = X _t - Item_​

Consumption at time t equals production at time t minus the investment at time t�

X _t+1 = k It _t ( 1 – I _t / P _t )

Production at time t+1 is equal to growth factor for the investment at time t by ( 1 less the investment at time t divided by the limiting constant at time t )�

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Investment at time _t

s Investment rate (which we assume to be a given constant)

X _t Production at time t

X _t+1 Production at time t+1

k Growth factor (1 plus the growth rate)

P _t Limiting constant at time t

C _t Consumption at time t

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• The first equation tells us that the investment is a given percentage of the pie (just like before).
• The first is that the pie is divided between investment and consumption (just like before).
• The third is that the pie in the next step is generated from the investment, following the logistic equation (we have added the parentheses that limit growth).
• If we know s, k, P _t and X _t we can calculate I _t , C _t and X _t+1

Optional. Logistic model: “extensive” and “intensive” growth, investment rate

Item_​ = s X _t The investment at time t is equal to the investment rate multiplied by production at time t

C _t = X _t - Item_​ Consumption at time t equals production at time t minus the investment at time t

X _t+1 = k Item_​ ( 1 – Item_​ / P _t ) Production at time t+1 is equal to growth factor for the investment at time t times 1 less the investment at time t divided by the limiting constant at time t

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• Our model has three parameters: the growth factor k (which is equal to 1 plus the growth rate), the constraint P _t which can change over time, and the investment rate s.
• With the growth of the time constraint P _t we can model the “extensive” growth, which allows increasing the load capacity (the final steady state of the classics) but without technological improvements in the production processes.
• Such as the discovery of new oil wells.
• In a cow population, doubling the pasture area represents "extensive" growth; we can model it by multiplying Pt by 2.
• With the growth of the growth factor k we can model “intensive” growth, which results from technological advances in production processes.
• Such as an improvement in extraction speed in oil wells.
• Introducing cows that reproduce more quickly implies "intensive" growth; we can model this by increasing k.
• Often, advances are a mix of both categories; the difference between them is not exclusive.
• Fracking is a new technology that allows for greater extraction from existing wells, but it also enables the exploitation of previously inaccessible wells. It thus represents both an "intensive" and "extensive" improvement.
• The investment rate (s) represents the proportion of output that is allocated to production. The remainder is used for consumption, military spending, and other purposes. This has changed significantly throughout history and has had an impact on economic cycles.

Dispensable. Logistics model: �negative feedback

Item_​ = s X _t

X _t+1 = k Item_​ ( 1 – Item_​ / P _t )

By substituting the first expression into the second, the model can be written with a single variable, to highlight the negative feedback around the equilibrium point.

X _t+1 = ks X _t (1 – s X _t / P _t )

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When the slope at the equilibrium point is less than -1, the system is unstable. If we increase the growth factor ko, the investment rate s decreases the slope, and therefore the system tends to become more unstable.

By double-clicking on the spreadsheet, you can modify the investment rate s, the growth factor ko, and the limiting constant in the yellow cells.

Optional. Logistics model

Item_​ = s X _t

The investment at time t is equal to the investment rate multiplied by production at time t

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C _t = X _t - Item_​

Consumption at time t equals production at time t minus the investment at time t

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X _t+1 = k Item_​ ( 1 – Item_​ / P _t )

Production at time t+1 is equal to growth factor for the investment at time t times 1 less the investment at time t divided by the limiting constant at time t

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By double-clicking on the spreadsheet, you can modify the investment rate s, the growth factor ko, and the limiting constants in the yellow cells.

Logistics model: technology and cycles

• In the logistics model, if the economy grows with low intensity (because the "intensive" technology is low), it tends towards a stable behavior, determined by the carrying capacity imposed by the environment (from which we can model its "extensive" growth).
• In the model, if the economy grows very intensely (because the technology is "intensive"), it becomes unstable and shows cycles of booms and busts, or even aperiodic behaviors (chaos).
• When it grows too rapidly, it exceeds the limits imposed by its environment; it enters a crisis and falls below those limits; when it falls below those limits, it booms again and grows too rapidly, exceeding those limits, and so on. Boom generates crisis, crisis generates boom, and so on.
• In the model, a high investment rate tends to increase the instability of the system, because it increases the force with which the system exceeds its limits.
• Consumption, by diverting resources from production and therefore reducing the force with which the system overcomes the limits of the environment, tends to stabilize the economy; see what happened with the increase in public spending -welfare state, armaments, etc.- from the Second World War until the crisis of 1973.

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Logistics model: historical stages (and future)

• We can study the four historical stages we have identified with the model, taking into account that the growth factor k (which represents the intensity with which the technology operates) grows over time, and that the investment rate has also changed:
• Stability in the ancient economy until the beginning of periodic cycles, which became clear in the 19th century with the Industrial Revolution. This situation is characterized by a low growth factor and a high investment rate (k = 3 and ys = 0.8).
• Periodic cycles from the Industrial Revolution to World War II. They arise from the increase in the growth factor due to technological advancement (k = 4 and ys = 0.8).
• Stability from World War II until 1973. This corresponds to an increase in state consumption, with the Welfare State, armament expenditures, etc. (k = 5 ys = 0.6)
• Periodic cycles from 1973 to the present. But technology continued to grow and the system became unstable again, despite the low investment rate (k = 6 and s = 0.6).
• If we continue to increase k, to model the growing “intensive” technology, the economy shows cycles of increasing amplitude first, and then aperiodic behaviors of greater amplitude (chaos).
• This seems to indicate that the mercantile-capitalist mechanism is historically limited, and that in the future (perhaps distant) the economy will have to be regulated in another way.
• But by reducing the investment rate we could decrease instability, by reducing the "intensive" impact of the economy on the limits imposed by the environment (which gives us ample time to look for alternatives).
• This reminds us of John Stuart Mill's positions. We already saw that reducing the investment rate in a logistics-driven world allowed for increased consumption without significantly reducing growth (PPT 10.8). Reducing the investment rate also allows for a decrease in the amplitude of the business cycle.
• Other ways to reduce instability are conceivable, for example, by mimicking the procedures used with unstable mathematical algorithms. However, these procedures appear to be much more complex, requiring detailed knowledge of the system and central control or calculation, and would not be "automatic regulators" as reducing the investment rate would be.
• In any case, let's not forget that "extensive" growth also has its limits.

Optional. Logistics model: �historical stages (and future of capitalism)

Before the Industrial Revolution Industrial Revolution – WWII WWII – 1973 1973 – Today

(Future with constant investment)

(Future with lower investment)

Dispensable. Quadratic applications

• The logistic model we have used to model capitalism is a case of a “quadratic application”, that is, it can be written as a polynomial of degree 2. The dynamics shown by this type of application are common.
• For a small magnitude of the parameter the system is stable
• If we increase the magnitude of the parameter, the system enters a cycle
• If we increase it even further, the system enters into complex cycles and chaos.
• The logistics application is an example

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• The Mandelbroot set is also the result of an iterated quadratic mapping in the complex plane,
• When c is small the system is stable (the large cardioid marked with 1).
• When c is reduced, the system enters a cycle of period 2 (the �larger circle to the left of the cardioid marked with 2).
• When c is reduced further, the system enters complicated cycles �first (the smaller circles on the left) and then chaos.
• https://en.wikipedia.org/wiki/Logistic_map �https://es.wikipedia.org/wiki/Conjunto_de_Mandelbrot �https://www.youtube.com/watch?v=pCpLWbHVNhk

Resources on the economic cycle

• Federal Reserve Economic Data, FRED, https://fred.stlouisfed.org/categories/
• The Conference Board (Business Cycle Indicators) �https://www.conference-board.org/topics/business-cycle-indicators/
• National Bureau of Economic Research, NBER, https://www.nber.org/research/business-cycle-dating
• Introductory bibliography:
• Schumpeter, Business Cycles , https://archive.org/details/businesscycles0001schu , https://archive.org/details/businesscycles0002schu
• Tugan-Baranovskyi, Industrial Crises in England , https://web.archive.org/web/20120206084532/http://www.ucm.es/info/bas/es/tugan/crisis600.pdf
• Burns and Mitchell, Measuring Business Cycles , https://archive.org/details/measuringbusines0000arth