26
S. BASCO
We now want to analyze whether the bubble we have just described can be obtained in equilibrium. Tirole (1985) derives two conditions under which rational bubbles may emerge in equilibrium: (i) arbitrage and (ii) rationality.
Bt+1 /Bt ≥ 1 + π,
(3.1)
eLt ≥ Bt .
(3.2)
Equation (3.1) is the arbitrage condition. It means that investors will not purchase the bubble unless the return on the bubble is higher (or equal) than the alternative investment option. In our model, the alternative to investing in the bubble is the storage technology, which has a return equal to π. Equation (3.2) is the rationality condition. It means that the price of the bubble is constrained by the amount of money in the economy. In the most optimistic case, in which all young agents choose to put their endowment in the bubble, the total amount of money that they would put together would be eLt. Equation (3.2) means that the price of the bubble cannot be higher than this quantity. In other words, we want to avoid situations in which Bt > eLt. How could the price of the piece of paper be higher than eLt? As we said, if all young agents purchase the bubble, the price is eLt. The price of the asset could be above eLt if investors from another model (or Planet) were purchasing the asset. This is why we label this inequality, the rationality condition. In other words, we cannot justify the emergence of a bubble with an asset price appreciation that requires a demand higher than the total amount of money in the economy. It is convenient to rewrite Eqs. (3.1) and (3.2) using per capita terms. Denoting bt ≡ Bt /Lt, these conditions become, 1+π 1+n bt if bt ≤ e bt+1 = 1+π (3.3) 1+n , ∞ bt if bt = e
Equation (3.3) describes the evolution of the bubble for any given initial bubble. For example, if we start with an initial bubble equal to b0 < e, the size of the bubble next period will be b1 = 1+π 1+n b0. As the reader can see, the bubble will be growing over time if π > n, and it will shrink over time if π < n. Which is the equilibrium size of the bubble? Before answering this question, let us introduce the notion of steady-state equilibrium. We say that b∗ is the steady-state equilibrium if bt+1 = bt = b∗ . Intuitively, if the size of the bubble is b∗, the bubble will no longer move
