Bubbles in Models of Stock and Money Markets
To reach a deeper understanding of unstable bubbles, it is worth examining some concrete economic examples. We shall look at stock market bubbles and bubbles in the price level and inflation.
22.2.1 Stock Market Bubbles
Assume a capital market in which investors are risk neutral. They choose between a stock and a safe asset with a certain rate of return r. In equilibrium, arbitrage will ensure that the expected rate of return of the stock will be equal to the rate of return of the safe asset:
where p is the price of the stock, and d is the dividend. The expected rate of return of the stock is equal to the expected capital gain plus the dividend as a proportion of the stock price. Equation (22.16) can be rearranged as
which has the form of (22.1), with 0 < a = b = 1/(1 + r) < 1.
The fundamental solution of (22.17) determines the stock price as a function of the expected future stream of dividends:
Under the fundamental solution, the equilibrium stock price is the present value of the expected future stream of dividends, discounted by the rate of return of the safe asset. Assuming that the expectation of future dividends grows at a rate less than r, the present value and hence the equilibrium stock price converges.
To find closed-form solutions, assume first that dividends follow a stationary AR(1) process of the form
alt=eq22-19.png>
where 
is the steady state dividend, 0 < λ < 1 is the degree of persistence of deviations of current dividends from the steady state dividend, and εt+1 is a white noise process.
From (22.18) and (22.19), the fundamental solution is given by
Thus, the stock market price based on fundamentals is a stationary stochastic process. Indeed it is an AR(1) process itself, with mean equal to the present value of the steady state dividend 
/r.
If dividends follow a nonstationary process, the stock market price is nonstationary as well. For example, assume that dividends follow a random walk with drift g > 0:
From (22.18) and (22.21), the fundamental solution is given by
Thus, the stock market price is a nonstationary variable. Indeed, it is a random walk with drift itself.
Consider now the solutions involving bubbles. The general solution takes the form
where z is a any extraneous variable that satisfies
which is equivalent to Etzt+1 = (1 + r)zt.
The general solution is thus equal to the fundamental solution plus the explosive variable z. Even if dividends follow a stationary AR(1) process, the stock market price will display explosive behavior. As long as investors continue investing in the stock, its price keeps increasing, and the expectations of investors are validated. This is what sustains the bubble.6
22.2.2 Money Market Bubbles, the Price Level, and Inflation
Consider next the Cagan [1956] model of money demand.
This is a model in which consumers decide between holding money and physical commodities. Money is a nominal asset whose value is affected by inflation. In this case, the demand for money is a negative function of expected inflation, and money market equilibrium requires that
where M is the nominal money supply, P the price level, and α > 0 the semi-elasticity of money demand with respect to expected inflation.
Taking the logarithm of both sides and denoting the logarithm of the nominal money supply by m and that of the price level by p, the model can be written as
Solving for pt yields
Equation (22.27) has the form of (22.1), with
The fundamental solution takes the form
The fundamental solution for the current price level depends on the discounted expectations for the evolution of the money supply in the future, with a discount factor equal to 0 < α/(1 + α) < 1.
Assume that the money supply follows a random walk with drift:
where μ is the mean rate of growth of the money supply, and ε is a white noise shock to the money supply.
Then the closed form of the fundamental solution for the level of prices is given by
The fundamental solution for the price level is a nonstationary stochastic process, but fundamental inflation is stationary, as it is given by
Consider now a bubble solution for the price level of the form
where
The price level displays explosive behavior, and inflation is determined by
Hence, in the presence of a bubble, both the price level and inflation display explosive behavior.
The economy eventually reverts to either hyperinflation or hyperdeflation, depending on whether the initial z is positive or negative. This would happen because of the bubble, even though the mean rate of growth of the money supply is constant. As long as households and firms believe that inflation will be increasing, they will reduce their money demand and buy goods at a higher rate, eventually causing an explosive rise in the price level. This may happen even though inflation is higher than its fundamental value, as defined by the rate of growth of the money supply.
22.3