11.8.1 PRICE BUBBLESBubbles may occur in markets for financial assets becausedemand depends, in part, on expectations about the prices atwhich they may be resold in future. In this Leibniz we develop asimple mathematical model of the market for shares in the FlyingCar Corporation, showing how beliefs about future prices canamplify price shocks and lead to a bubble in which prices rise atan ever-increasing rate.

Suppose there is a competitive market for the shares issued by the FlyingCar Corporation. Let time be divided into weeks. In week , demand for theshares depends on its current price and the amount by which the priceis expected to change in the near future. We assume that, given , thequantity demanded is a decreasing function of , for the usual reasons. Wealso assume that, given , the quantity demanded is an increasing functionof : the more the price is expected to rise, the greater the quantity of theshares that speculators wish to purchase with the expectation of making aprofitable future sale.

To keep things simple, we assume that demand and supply functions arelinear:

where are constants of which , , and (the speculationparameter) are positive. A supply-and-demand diagram, like Figure 11.15 inthe text (reproduced below as Figure 1), then plots different demand curvesfor each value of ; an increase in shifts the demand curve to the right.

Suppose the market clears each week: for all . Then the pricein week is determined by the market-clearing equation:

for all , and we can solve for in terms of :

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Figure 1 The beginning of a bubble in FCC shares.

fundamental value of a share Theshare price based on anticipatedfuture earnings and the level ofsystematic risk, which can beinterpreted as a measure of thebenefit today of holding the assetnow and in the future.

Notice that is what the price would be each week if it were not expectedto change. We can think of as the fundamental value of a share.

To discover what happens to the price over time, we have to make someassumptions about market expectations. Let us see what happens ifexpectations are as accurate as possible from week onward: that is, marketparticipants are able to forecast price changes correctly. Then

for and so on. Substituting this expression for into thepreceding equation, we see that:

Rearranging,

This is the price dynamics curve (PDC) that we discussed in the text. Itdescribes how the price evolves over time. If we begin in period 0 with theprice equal to the fundamental value, , and no shocks occur, then theprice will remain at in period 1, 2, and all subsequent periods.

However, since (which follows from the assumptions that demandcurves slope downward ( ), supply curves slope upward ( ), andspeculation exists ( ), together with our model for expectations), thisequilibrium is unstable.

If a shock occurs, temporarily changing the price in period 1 to ,then the price will rise again in period 2: the PDC tells us that will befurther from the fundamental value than . In the next period it will behigher still.

This is the situation illustrated in Figure 11.18 of the main text,reproduced as Figure 2 below. In the diagram, the market is in equilibrium

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Figure 2 An unstable equilibrium.

in period 0 ( ). The PDC, showing the relationship between and, is represented in the right-hand panel. The PDC cuts the 45 line at

Since the PDC is steeper than 45 line, and after a shock movesthe market away from the equilibrium in period 1, the price rises further inevery period.

You can see in Figure 2 that not only does the price move further fromthe equilibrium, it does so at an increasing rate. That is, the price change ineach period is bigger than the price change in the last period. To show thatour mathematical model predicts this, we can rearrange the equation aboveto obtain:

and hence also

It follows from , and the equation above that

Again since , we can deduce that the price change gets larger eachperiod: that is, the price grows at an increasing rate.

This is the classic example of a bubble: the price rises without boundbecause it is expected to rise, and will do so for as long as expectations arecorrect. Similarly, if , the price falls continually by increasingamounts. In this case the model predicts that the price will eventuallybecoming negative, which does not happen in general. It is possible, how-ever, to construct non-linear versions of this model in which the priceremains positive forever, becoming smaller and smaller but never hittingzero.

For many asset markets, the supply function above is too simple to besensible. For example, one may think of the demand for housing asdepending on the price of houses and its expected rate of change (these,

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together with the interest rate, determine the cost of rentedaccommodation), while the supply of new houses depends on the currentprice. Let represent the housing stock in week . Then the supply func-tion above must be replaced by something like:

where is the rate of depreciation ( ). The mathematics thenbecomes far more complicated, but the conclusion turns out to beessentially the same. For each value of , there is exactly one initial price,say , with the property that if then remains withinbounds for all . Otherwise, explodes or collapses.

Read more: Section 5.1 of Malcolm Pemberton and Nicholas Rau.2015. Mathematics for economists: An introductory textbook, 4th ed.Manchester: Manchester University Press.

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