How to Deal with the Flaws of the Demand-Supply Model?

For economic science, the Demand-Supply Model is a fundamental truth, describing the behavior of both consumer and producer in the markets. It would be irresponsible to deny that, yet the model has several serious flaws.

1. We cannot use it to predict future prices and amounts of goods produced and consumed. At best we can see the direction of the changes, which is important, no doubt about it. Yet it is not enough to make the model a tool for practical calculations.

2. It shows only static market situations. We cannot track down the path of the price, while it is moving (as we simply postulate) to the equilibrium point. Because of that, we just ignore the possibility of not reaching the equilibrium (in fluctuations, for example).

3. We are inclined to make an absolute of the point that demand and supply curves intersect. The Demand-Supply Model looks so universal, so true regardless of market nature (monopolistic markets theory only modifies it, without radical changes).

Yet in the markets with a speculative component it may be not true. From the teaching standpoint, it is critical to formulate the limits of the basic classical model in the very beginning of the economics course to prevent creation of dogmas in the minds of students.

Point 1 is too big a problem to even come close to resolving it. Still, there is no real science without quantitative predictions. Yes, we use extrapolations for that purpose. Unfortunately, they often fail due to misunderstanding of the process they describe (being nowadays extremely sophisticated).

What about points 2 and 3?

Dealing with point 3, I am showing the Demand-Supply Model without intersection of curves (except zero point). It is the asset bubble model, presented in my previous posts. It does make sense to show it again in this piece with some simplifications. Here it is.

Dem Suppl of a bubble

As I have already explained it, D1 is the demand curve for the boom phase of a bubble, S is the supply curve, and D2 is the demand curve for the bust phase of a bubble.

This picture does not tell us anything about how an asset bubble develops; it shows no dynamics at all. This is where dynamic equations (finite differential equations, to be exact) must step out to show the path of the price, and there is no better way to put the path on a graph than using spider diagram. This way I am dealing with point 2 of my agenda too.

Let us denote:

P(t) – price at moment t, $M,

S(t) – quantity supplied at t, units,

D(t) – quantity demanded at t, units

∆P(t) – change in price at t, $M,

The equation for supply curve S is:

P(t) = S(t)                                         (1)

The equation for demand curve D1 is:

P(t) = 0.5D(t)                                  (2)

The equation for demand curve D2 is:

P(t) = 2D(t)                                      (3)

In my post “Asset Bubble Microeconomics: A Peculiar Demand and Supply Diagram” I assumed (for illustration purposes only) that the price change was equal to the difference between demand and supply. I admit that it is too strong an abstraction. There must be, of course, some elasticity coefficient in this equation, depending on how the price reacts to shortages or surpluses in the markets. So

∆P(t) = k(D(t) – S(t)),                                                   (4),

where k is the elasticity coefficient.

Let us take the boom phase first. Transform (4) using (1) and (2), which are the boom phase equations.

What we get is

∆P(t) = k(P(t)/0.5 – P(t))                                             (5)

Suppose k = 0.2. This strong is the reaction of the price to the difference between supply and demand. Then the equation for the booming price is

∆P(t) = 0.2P(t)

With starting price = 1, the following graph shows price changes until the supposed crash point at approximately double price is reached.

We can see also the spider diagram of the booming price.

Spider for boom

Now I am going to show price dynamics after the price crashes at the level=2 and goes subsequently down (the bust phase).

In this case we need to transform (4) using equations (1) and (3), describing the bust phase.

The result is

∆P(t) = k(P(t)/2 – P(t))                                               (6)

Elasticity coefficient is still k=0.2. Then the equation for price dynamics when the bubble bursts is

∆P(t) = -0.1P(t)

Starting from high point 2, the price goes down as the following graph shows.

Spider for bust

The spider diagram on the graph shows the path of the price during the bust.

Both phases of the bubble should have been put on the same graph, of course. I separated them to avoid overloading it with details.

Conclusion: equations and spider diagrams are a big advantage as far as teaching Demand–Supply Model (theory) in Economics is concerned.

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Are Low Interest Rates the Major Reason for Asset Bubbles?

The low interest rates policy in the U.S. led to the housing bubble of the 2000s – this opinion is shared by many US economists and even by president of the Dallas Fed Richard Fisher  (http://en.wikipedia.org/wiki/Causes_of_the_United_States_housing_bubble#Historically_

low_interest_rates).

The logic is as ironclad as it is simple: cheap credit facilitates speculation in the asset markets. It sounds so obvious that should definitely obtain the status of universality, the status of a theory.

Or should it? Speculation, despite all its frenzy, is still based on benefit-cost considerations of individuals and businesses. It is not impossible that with relatively high borrowing cost the returns from speculation would be so big that this business could be justified financially.

This sounds very abstract, of course.  Much better way to deal with the issue is either to verify statistically the point of great impact of interest rates on developing asset bubbles or falsify it statistically.

The latter, according to Karl Popper, is a quite effective way to reach the scientific truth.

What I am going to do now is to falsify the point that low interest rates are the defining factor in developing asset bubbles.

Barry Ritholtz gives us the following picture of 10 yr US treasury yields since 1790.

Interest rates since 1790

The modern times downward trend is dramatic. In December 2013 it all but reached 3% (this figure is not from the chart). Is this trend (and the underlying trend of declining and keeping low Federal Funds rate) is the main culprit as far as the asset bubble of the 2000s is concerned? Is it a harbinger of future asset bubbles?

Take a look at the range the US 10 year treasury yield reached in 1820-1855 as the above chart shows us. In 1825 the yield is approximately 5%. It later increases peaking in 1840 to reach almost 6%, then falls abruptly to 5% in 1845, to rise again until 1860. In this year, it hits almost 7%.

Within this period, there were two booms with ensuing busts, one in 1836, another in 1857.

The famous Canal boom of 1836 involved a lot of speculation with the land, surrounding the Great Lakes. Mason Gaffney describes this period of US economic history in his work titled “The U.S. Canal Boom and Bust, 1820 – 1842” ( http://www.masongaffney.org/workpapers/The_US_Canal_Boom_and_Bust_1820-42_WP01.pdf0).

There are also works of prof. E.Glaeser on the issue of historical land speculation in this area.

Both 1836 and 1857 booms coincide with the increase of 10 year treasury yield.

During the rest of 19th century we can see the downward trend of the yield, with 1873 and 1893 booms.

Definitely, this complex picture cannot work well for the verification of the point at issue, but it works pretty well for the falsification of it. Some very strong asset bubbles, booms and busts in the US economic history were not accompanied by the periods of declining interest rates.

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The Phenomenon of Price Threshold in US Housing Bubble of the 2000s

So there was a drastic change in the speculation volume ratio to all sales of homes (a threshold) in the course of US housing bubble of the 2000s (see post  “Critical Mass of Speculation Triggers Asset bubbles”). I have hypothesized that crossing this  threshold triggered the bubble, the positive feedback loop of booming prices of the period.

Let us see if the adequate  threshold could be detected in the Case-Shiller index of home prices of the time. Threshold is a twist, may be a kink, a drastic change in price dynamics.

Can we see something like that in the dynamics of Case-Shiller index for the period 2000- 2006? I added two tangent lines to the picture of the index to catch the approximate region of the supposed price threshold (see below).

Twist in price dynamicsThe graph shows two distinctive periods, with different slopes of the curve. The approximate slope (the first derivative of the price function) in the beginning of the period until, perhaps, the year 2003 is much flatter than afterwards. The change is obvious and significant (I am skipping numbers). It looks like the price threshold was reached in the year 2003.

Mathematical modeling ( see post “Modeling the US Housing Bubble of the 2000s”) reveals this drastic change too. Let us see it again with some simplifications.

AproxIn Model 1 the price grows with constant rate (in the traditional economic interpretation).  It is a typical exponential growth. The first derivative of the price function is changing, of course, yet the change is gradual, nothing suggests a twist or some drastic change of speed.

In Model 2 the price grows with accelerating rate. As we can see, it is much closer to the dynamics of the Case-Shiller index of the period. Moreover, it reflects better the essence of a bubble with its explosive character. The price in Model 2 at first grows slowly, and after some point explodes. Still, the price threshold is not very much pronounced, so this model may be regarded as a rough approximation of bubble dynamics. It smooths out the threshold-type change, yet it is there, in about 2003, which is also the year of drastic change in the ratio of flipping activities to all sales in the housing market ( http://libertystreeteconomics.newyorkfed.org/2011/12/flip-this-house-investor-speculation-and-the-housing-bubble.html?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+LibertyStreetEconomics+%28Liberty+Street+Economics%29).

In fact, there was no housing bubble in the US before 2003. It is the year when the price threshold of speculation in the US housing market was crossed, creating the explosive market frenzy of 2003-2006.

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Critical Mass of Speculation Triggers Asset Bubbles

The author of the article “Not fully inflated” (“The Economist”, December 7, 2013) accepts so easily the definition of an asset bubble as “an increase in the price of an asset of more than two standard deviations above the trend, taking inflation into account”, developed by GMO, a fund management group.

This purely statistical approach, though, has nothing to do with the essence of a bubble.

In my opinion, the positive feedback loop of speculation is the essence of a bubble*, creating a specific trajectory of interdependent price changes better described by some differential equation. Random deviations from this trajectory (not from the trend or from the mean) do matter but constitute a secondary component of the bubble dynamics.

This is what I suggest as a definition of an asset bubble.

It is a positive feedback loop of precipitously rising and ultimately falling asset prices, caused by speculation, the volume of which has crossed some threshold.

  This perception stems from my previous posts. The notion of threshold (or critical mass of speculation) is crucial to this definition. As I showed earlier, the demand curve of a bubble is upward sloping. The transition from classical downward sloping demand curve to upward sloping one is a leap, a drastic change of the market process. There is no continuity in this transition. The market crosses some threshold in the sense that the volume of speculation (flipping houses, for example) in relation to all market transactions becomes so high, that it triggers market frenzy. Below the threshold, the speculation is more or less sound, not disrupting the self-regulating market dynamics. The threshold crossed, a bubble has born. Still a hypothesis, the notion of threshold of speculation has some statistical support, at least in case of US housing bubble of the 2000s.

* It is a common misperception that low interest rates constitute the main cause of an economic bubble. It turns out, though, that under current economic circumstances in the US, with interest rates near historical lows, there is no housing bubble. So the cause is without effect?

Low interest rate, in my view, is not a necessary condition either; it is just a contributing factor, the fertile ground for bubbles. (This metaphor is quite appropriate. What is the main cause of a plant? It is a seed. No seed, no plant. A plant can grow on poor soil, yet rich soil contributes a lot to its growth. Rich soil alone does not produce a plant). By the way, was the interest rate near zero in case of the famous Tulip bubble?

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One Way to Define If There Is a Housing Bubble

In this post I focus on the assumption of a critical point or a threshold on the way to a housing bubble, made before. I believe that at some point the critical mass (volume) of flipping operations emerges, producing explosion-type market speculation.

The knowledge of the threshold value for an individual market creates new analytical possibilities. When prices and sales of homes are up, everybody asks is it a bubble or not?

To answer this question we can compare actual volume of flipping with its threshold value. If actual volume of flipping is less than threshold value, there is no bubble. In case flipping exceeds (or equals) threshold value, there is a bubble.

Two questions may arise out of this.

First, why such a hypothesis?

Let us take a look again at the demand-supply diagram of market transformation into a bubble. I simplified it a bit.

Threshold

The demand curve in case of a bubble is upward sloping. In a pre-bubble situation, it is downward sloping. One cannot explain this drastic change by gradual changing of the angle of the downward sloping curve. There must be something not unlike the accumulation of critical mass of radioactive material to produce atomic explosion.

It is possible that the volume of flipping in relation to all sales of homes must reach some threshold to ignite a positive feedback loop of a bubble. There may be psychological reasons for that. If too many investors become involved in flipping, why not to jump on the bandwagon?

The second question: do the facts on the ground support the hypothesis?

At this point I am turning to the article “ Flip This House”: Investor Speculation and the Housing Bubble” by Andrew Haughwout, Donghoon Lee, Joseph Tracy, and Wilbert van der Klaauw (  http://libertystreeteconomics.newyorkfed.org/2011/12/flip-this-house-investor-speculation-and-the-housing-bubble.html?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+LibertyStreetEconomics+%28Liberty+Street+Economics%29 ).

There are graphs in the article depicting the share of purchases by investors willing to flip houses in all purchases of houses in the US for the period 1999-2010. They deal separately with investors owning one, two, three, four houses. I am showing these graphs here.

share of flippingSource: Andrew Haughwout, Donghoon Lee, Joseph Tracy, and Wilbert van der Klaauw. “ Flip This House”: Investor Speculation and the Housing Bubble”, Federal Reserve Bank of New   York, 2011.

The share of flipping grows until the year 2006, when the housing bubble peaks. Yet the growth is uneven. There is a distinct area where flipping starts accelerating at a greater rate. It may be the threshold I’ve been talking about. This threshold is more pronounced in case of the states, where the housing bubble was the wildest – in Arizona, California, Florida, and Nevada.

Rough estimation is that the range of 5-10% to all purchases was critical for the US housing market, the average may be around 7% (of course, the rigorous statistical methods of catching the threshold are needed, yet for now the rough estimation is sufficient). Assuming the value of the threshold holds in current circumstances ( 2013), we can compare the current share of flipping with the 7% value of the threshold. To calculate the current share of flipping, I am using the September 2013 RealtyTrac report. Here is its information on flipping in the US.

Flippig trend

Source: September 2013 RealtyTrac report

The number of flipped houses for 4 last quarters is approximately 207,000.  The annualized number of houses sold, according to the report, is 5,673,249 (as of September 2013).

So the annualized share of flipping is 207,000/5,673,249 = 0.036 or 3.6%

It is less than 7% of estimated threshold, and even less than its low-level value of 5%. It is approximately at the level of 1999 share of flipping.

So far so good. If all the hypotheses hold, there is no housing bubble in the US in 2013.

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Can We Discern the Naissance of a New Housing Bubble?

In my post  “Asset  Bubble Microeconomics: a Peculiar Demand and Supply Diagram” I’ve argued that in a housing bubble the demand curve is upward sloping, positioning itself below the supply curve at the boom phase and shifting to the position above the supply curve at the bust phase.

Yet there is always a pre-bubble history of the market functioning “normally”, in a regular mode, with customers buying homes to live in. Investors and speculators exist even in a regular market, of course, but they do not change the nature of the market at this phase. Consumer choice shapes the market, and the demand curve is downward sloping.

The question arises: how does the transformation from a regular market into a housing bubble actually happen?

Do steadily rising home prices necessarily mean a new bubble is already at work?

“Through the roof again?“ asks “The Economist” in June 29th  referring to the rising S&P/Case-Shiller index of house prices in the US in 2013. It underscores the specific combination of rising prices and rising sales of new homes (which means that the demand for homes is growing too). The combination, which may look like the beginning of a new boom because at a boom phase quantity demanded and price rise in tandem too. Based on opinions of some business leaders, though, the unknown author of the article comes to the conclusion that so far so good – there is no newborn housing bubble in the US  economy as of the middle of 2013 (The Economist, June 29th 2013, page 59).

Let us interpret the issue of possible market transition to a bubble in terms of abstract demand-supply model. In a regular, non-bubble market, as I have mentioned earlier, its demand curve must be downward sloping. In a bubble, it is upward sloping. That means at some point a cardinal shift of the demand curve occurs. There is no hope that the transition could be smooth. It is like an explosion (see my posts “Modeling the US Housing Bubble of the 2000s” and “Asset Bubble Patterns: Japan’s Case”).

It would be interesting to detect this tipping point using some economic indicator. Later in this piece, I will hypothesize on the matter.

I think the demand-supply diagram of the mentioned transition to a bubble may look like this:

Market transformations into a bubble

The diagram shows how the transformation process unfolds in time. S denotes supply. Suppose the supply curve does not shift in the process. Demand curves D1, D2, D3, D4, and DB belong to 5 consecutive moments of the transformation.

At the beginning, there is no bubble (diagram D1-S).

From this point forward, there are changes in demand. The number of potential buyers of homes increases, perhaps due to improved economic situation in the labor market, cheap credit, lower mortgage rates, and the like.

These changes shift the demand curve to the right, to position D2.

As a result, some shortage of homes emerges (red section at the initial level of the price).

The market is still regular: buyers have no intention to make money out of their purchases. They have a number of choices; therefore, the demand curve is downward sloping.

Being regular, the market gradually eliminates the shortage, going in the direction of the equilibrium point (follow the arrow), and the shortage diminishes. Note that the price and the quantity demanded grow as well.

There is an important qualification, though.

The concept of the market moving towards equilibrium (self-regulating market) is not an absolute. According to an American economist Hyman Minsky (who used the term “coherence” as a proxy for the term “equilibrium”), it is true only if and when market agents believe that the existing prices will hold in the future (Minsky, 2008, pages 116-121).

I think it is possible to somewhat loosen this constraint. In my post “Market Expectations Paradox”  I tried to illustrate the possibility of reaching equilibrium point under the condition that the expectations of the future price are “cautious” or moderate.

I am deeply convinced that the issue of expectations of the future price is critical for our understanding of all modes of market dynamics – be it a regular, self-regulating, coordinating the distribution of the resources market, or a wild, inflating the price bubble.

Let us continue to follow the transformations in our market.  So far, we are in the self-regulating mode.

The demand curve continues to shift to the right, to position D3. This time around it is the result of emerging speculation (like flipping homes practice) and other forms of activities, aiming to extract additional money out of purchases (like using home as collateral to borrow money). This causes demand to grow. Yet the demand curve gets flatter. The reason is that for the same price the quantity of homes demanded grows due to the moneymaking component of the market. The seeds of a bubble are planted.

Still, the market is not a bubble. It moves gradually to the equilibrium point (follow black arrow), not necessarily reaching it. Meanwhile, the demand curve shifts to position D4, due to increasing activities I mentioned earlier. It becomes even flatter than before, but the trend to the equilibrium holds. The price continues to grow.

At some point, the situation changes dramatically. Perhaps, the volume of speculative and other moneymaking activities reaches a critical point. From this point on there is a bubble, and the demand curve jumps to become the upward sloping line (position DB).

The transformation of the market into a bubble is complete.

To my mind, the volume of moneymaking operations in the market in relation to the volume of other purchases of homes may be the indicator we need to monitor with the goal of detecting its critical value.

As Andrew Haughwout and his coauthors mention (“Flip This House”: Investor Speculation and the Housing Bubble, FRBNY), at the peak of US housing boom the investor share of all purchases reached 30%, it roughly doubled from 2000 to 2006. The authors prove statistically that investor share of the housing market is a substantial factor in the development of  2000-2006 US housing bubble.

As to other traditional indicators, like home price to income ratio, or home price to rent ratio, they may be useful, of course, yet they do not go too far, meaning they do not touch the nerve of the transformation process.

Returning to the title of the article in “The Economist”, its question mark is quite appropriate because growing home prices accompanied by rising home sales as such say nothing about the emergence of a housing bubble. As far as the current US situation (year 2013) is concerned, we would rather start looking at moneymaking deals and their relation to the volume of other operations. As analysts have noticed,  the recent growth of home prices is due to massive purchases of foreclosed homes by big investors aiming to resell them with a big profit (flipping homes). Is it the beginning of a new housing bubble? It is possible, but to make an accurate statement we need to know the critical value of the ratio between speculative operations in the market and the regular transactions, which will trigger a housing bubble and radically shift the demand curve from downward  sloping to upward sloping kind.

Literature cited:

Hyman P. Minsky. Stabilizing an Unstable Economy. McGrawHill, 2008.

“The Economist”. “Through the roof again?”, Print edition, June 29th, 2013.

http://economistsview.typepad.com/economist

Mike Whitney. US housing market shifts into reverse: A whirlpool of speculation. http://www.soft.net

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Asset Bubble Microeconomics: a Peculiar Demand and Supply Diagram

There is no doubt that the traditional picture of downward sloping demand curve and upward sloping supply curve does not apply to an asset bubble.

Downward sloping character of the demand curve is inferred from the premises of utility theory.

Someone picks a combination of goods, based on

a) Her budget constraints

b) How many units of one good she is ready to sacrifice to get an additional unit of another good (called marginal rate of substitution in utility theory).

The latter is about her tastes. She maximizes the utility for her at the point where the marginal rate of substitution equals the price ratio of two goods. The utility maximum drops to a lower level when the price of one good rises. As a result, the quantity demanded of this good diminishes. That makes the demand curve for this good downward sloping (see “Consumer choice” chapters in Microeconomics textbooks).

Yet there is no inverse relationship between price and quantity demanded in asset bubble markets. In a perfect bubble market, all participants are for the money. The higher the price at the boom phase of the market, the stronger the incentives to buy more in anticipation of even higher price. That lures more participants to enter the market. In the process, demand grows and the price increases.

At the bust phase the price falls. The participants expect the price to go down further, and the disincentives to buy emerge. As more participants withdraw from the market, demand falls and the price continues to decline, in a vicious cycle style.

Clearly, downward sloping demand curve does not hold in this case. In fact, it is upward sloping, albeit the slope, as we will see, is different for each phase of the asset bubble.

As to the supply curve, its upward sloping character holds in the situation of an asset bubble because its bottom line is the construction cost of homes (in case of housing bubble).

The graph below shows the positioning of the demand curves for two phases of an asset bubble in relation to the supply curve.

Demand and Supply curves for an asset bubble

As we can see, at the upward stage of the bubble the demand curve (curve D1) is sloping upward, but the curve is positioned below the supply curve S. The slope of demand curve D1 on the graph is 0.5, and the slope of supply curve is 1 (all curves are linear, a simplification). At the price = $2M (arbitrarily picked), the difference between demand and supply is 2 units (section bc). The price is going to grow, but at the higher price, the difference is even greater. So the boom continues.

The demand curve for the deflating bubble, on the contrary, is positioned above the supply curve, which means that at any price quantity demanded is less than quantity supplied. It is about panic, after all. The slope of demand curve D2 on the graph is 2. At the price = $2M, for example, there is a surplus of units (section ab) and the price will go down. The participants will continue to withdraw from the market and the surplus persists.

If we are going to see the whole process of booming and busting, we must acknowledge that at some point of time demand curve D1 shifts to the left. Its new position is demand curve D2. It happens at the point of growing doubt, which ignites panic. Sometimes it is called Minsky Moment. Demand at Minsky Moment falls abruptly, creating a surplus.

So the demand curve exists in two forms: D1 (boom phase) and D2 (bust phase).

Suppose Minsky Moment happens when the price is $4M (this is not a real price, of course). The initial price is $1M. The slope of demand curve D1 is 0.5, the slope of demand curve D2 is 2, the slope of the supply curve is 1. All curves are linear.

To trace the trajectory of the price under these conditions we need some equations.

Let us denote:

P(t) –  price at moment t, $M,

S(t) – quantity supplied at t, units,

D(t) – quantity demanded at t, units

∆P(t) – change in price at t, $M,

The equation for supply curve S is:

P(t) = S(t)                                         (1)

The equation for demand curve D1 is:

P(t) = 0.5D(t)                                  (2)

The equation for demand curve D2 is:

P(t) = 2D(t)                                      (3)

Equations (1) and (2) combined represent the boom phase, equations (1) and (3), taken together, describe the bust phase of the asset bubble under the above conditions.

Price change depends on the difference between demand and supply. For illustration purposes, we assume that it is equal to the difference between demand and supply, measured in units (at this level of abstraction it does not matter how unit is defined, it might be thousands or millions of homes or condos or it might be land).

For the boom phase we get

D(t) – S(t) = P(t)/0.5 – P(t) and, substituting  ∆P(t) for D(t) – S(t),

∆P(t) = P(t)/0.5 – P(t) or

∆P(t) = P(t)                                                                   (4)

Given the above conditions, the price grows like this:

P(t) $M 1 2 4
∆P(t) $M 1 2

Price = $4M is Minsky Moment. From this point on demand drops and the bust phase begins with shifted demand curve D2. Now we need to take equations (1) and (3).

It goes like this:

D(t) – S(t) = P(t)/2 – P(t) or

∆P(t) = P(t)/2 – P(t) or

∆P(t) = – 0.5P(t)                                                           (5)

The price at the bust phase starts at Minsky moment and falls according to (5), as the table below shows.

P(t) $M 4 2 1
∆P(t) $M -2 -1

Plotting all this data on the same graph, we get:

The asset bubble inflates and deflatesAt the starting point the price = $1M. At this price, the shortage of one unit develops and the price increases to $2M, which spawns even greater shortage of 2 units. This new shortage drives the price to $4M.

Minsky Moment has been reached. The following panic and massive withdrawal from the market crashes demand and the demand curve shifts to position D2. Resulting surplus drives the price down to $2M. Yet shortage persists at this price and the price continues to fall until it reaches $1M, the initial value. The asset bubble has deflated.

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