Rational Bubbles

Rational Bubble

A rational bubble is a price component that can exist even when every investor is rational and the required return is correctly priced. It arises because the asset-pricing Euler equation
$P_t={\mathbb{E}}_t\left[\frac{D_{t+1}+P_{t+1}}{1+r}\right]$
does not have a unique solution. Iterating it forward and imposing only the dividend stream gives the fundamental price
$P_{F,t}={\sum }_{k=1}^{\infty }\frac{{\mathbb{E}}_t[D_{t+k}]}{(1+r{)}^k},$
but the recursion admits additional solutions of the form
$P_t=P_{F,t}+B_t,$
where the bubble component $B_t$ must satisfy the self-financing condition
${\mathbb{E}}_t[B_{t+1}]=(1+r)B_t.$
That is, the bubble must grow on average at exactly the required return. Under this condition, $P_t=P_{F,t}+B_t$ also satisfies the Euler equation, so a rational investor is indifferent between holding the asset at its fundamental value and holding it with the bubble attached.

Rational bubbles are mathematically permitted but heavily constrained. They cannot start — if $B_0=0$ then $B_t=0$ for all $t$, since the conditional expectation of a bubble that did not exist yesterday is zero. They cannot be negative for an asset with limited liability, because the expected price would eventually go negative. They cannot exist on assets in finite supply with a finite-horizon terminal payoff. And in dynamically efficient economies (those where $g<r$), they cannot exist at all on assets whose dividends grow no faster than the interest rate, because $B_t$ would eventually exceed total wealth.

The empirical relevance of rational bubbles is therefore limited. They serve as a benchmark: the rational-bubble framework shows that the standard pricing equation does not by itself rule out bubbles, so excluding bubbles from the data requires either an additional theoretical assumption (transversality) or a behavioral departure from rationality. Most of the modern bubble literature operates on the second route.

Intuition

The fundamental value of an asset is the discounted stream of its expected dividends. Rational pricing says today’s price should equal that stream. But the equation that enforces this — the one-period Euler equation — only ties today’s price to tomorrow’s price plus tomorrow’s dividend. It says nothing directly about what happens far in the future. If you believe the price tomorrow will be high enough, then a high price today is justified by the expected capital gain alone, without reference to dividends. The rational-bubble construction makes this self-referential possibility precise: a bubble component $B_t$ that grows at rate $r$ in expectation is exactly what tomorrow-justifies-today requires.

The condition ${\mathbb{E}}_t[B_{t+1}]=(1+r)B_t$ is restrictive because it forces the bubble to grow forever in expectation. Each period, the bubble must be larger than before (in expectation), which means it cannot stay bounded. In a finite economy, this growth eventually runs into a wall: the bubble exceeds the total resources available to buy the asset, and the no-Ponzi or transversality condition breaks. The technical theorems that “rule out” rational bubbles are all variants of this finiteness argument.

Properties

(Self-Financing Condition) The bubble must grow at the required return in expectation.

The defining condition ${\mathbb{E}}_t[B_{t+1}]=(1+r)B_t$ says the bubble component is a martingale with respect to the discounted measure. Equivalently, $B_t/(1+r{)}^t$ is a martingale. The condition is what makes the augmented price $P_t=P_{F,t}+B_t$ still satisfy the Euler equation: any deviation from this growth rate would violate the one-period pricing restriction and create an arbitrage opportunity.

(Cannot Start) A bubble that did not exist yesterday cannot exist today.

If $B_t=0$ at any date $t$, then ${\mathbb{E}}_t[B_{t+1}]=0$ by the self-financing condition. Since prices are bounded below (limited liability), $B_{t+1}\ge 0$ almost surely, and therefore $B_{t+1}=0$ almost surely. By the same argument going forward, the bubble remains zero in every state. A rational bubble must have been “born” at $t=-\infty$ or at the moment the asset was created.

(No Negative Bubbles) Limited liability rules out $B_t<0$.

If $B_t<0$ then by the self-financing condition ${\mathbb{E}}_t[B_{t+1}]=(1+r)B_t<0$, so $B_{t+1}$ must be negative on average. Iterating, ${\mathbb{E}}_t[B_{t+k}]$ becomes arbitrarily negative as $k\to \infty$, eventually pushing the total expected price below zero. For an asset with limited liability ($P_t\ge 0$), this is impossible. Negative rational bubbles cannot exist on equity-like assets.

(Ruled Out in Finite-Horizon Settings) Backward induction kills bubbles when there is a terminal date.

If the asset has a known terminal payoff at date $T$, then $P_T$ is fixed at the terminal payoff and $B_T=0$. By backward induction using the self-financing condition, $B_t=0$ for all $t<T$. Rational bubbles can only exist on infinitely-lived assets where no terminal condition pins down the price.

(Tirole’s Dynamic Efficiency Condition) Bubbles need $r\le g$ to survive in general equilibrium.

Tirole (1985) shows that in a general equilibrium with a productive economy, a rational bubble can exist on an asset only if the economy is dynamically inefficient, meaning the interest rate is below the economy’s growth rate ($r<g$). In a dynamically efficient economy ($r>g$), the bubble would eventually exceed the economy’s total wealth and the transversality condition would force $B_t=0$. Since most empirical evidence suggests developed economies are dynamically efficient, rational bubbles are theoretically hard to sustain on standard assets.

(Why the Behavioral Literature Does Not Use Them) Rational bubbles cannot match the empirical patterns.

The data on bubble episodes shows high trading volume, extrapolative survey expectations, dispersion of beliefs, and concentration in specific asset classes. None of these features arise from a rational-bubble model — every investor in such a model agrees on the price and on the bubble’s growth rate, so there is no reason for trade. The behavioral literature replaces the rational-bubble construction with belief-disagreement models (Miller 1977, Harrison-Kreps 1978) precisely because those models reproduce the volume and disagreement patterns the rational story misses.

Examples

A bubble that grows at the riskless rate

Consider a riskless asset paying no dividend, with required return $r=5\%$ per year. The fundamental value is zero (no cash flows). A rational bubble on this asset would satisfy ${\mathbb{E}}_t[B_{t+1}]=1.05B_t$. Suppose B_0 = \1$.Then$\mathbb{E}_0[B_t] = (1.05)^t$—theexpectedbubblegrowswithoutbound.After$100$yearstheexpectedbubbleis$$131$,after$200 years \$$17{,}292, after $500$ years $$39$ billion. In a finite economy with bounded wealth, this growth eventually exceeds the resources available to buy the asset, and the transversality condition rules it out.

Stochastic bubbles à la Blanchard-Watson (1982)

A common rational-bubble specification allows the bubble to “burst” with some probability each period. Let
$B_{t+1}=\{\begin{array}{ll}\frac{1+r}{\pi }{B}_t& \text{with probability }\pi \\ 0& \text{with probability }1-\pi .\end{array}$
Conditional expectation: ${\mathbb{E}}_t[B_{t+1}]=\pi \cdot \frac{1+r}{\pi }{B}_t+(1-\pi )\cdot 0=(1+r)B_t$. The self-financing condition holds. The bubble grows faster than $r$ when it survives (rate $(1+r)/\pi >1+r$), compensating investors for the risk that it bursts. This construction generates the bubble-crash pattern observed in data while remaining internally rational, but it still inherits all the standard impossibility results — it cannot start, cannot be negative, and cannot survive in dynamically efficient economies on finite-supply assets.

See Also

• 8.7. Bubbles — parent hub.
• 8.7.2. Heterogeneous Beliefs and Overvaluation — the behavioral alternative built on belief disagreement and short-sale constraints.
• 8.7.3. Speculative Bubbles — the dynamic version where capital gains push the price beyond any single investor’s valuation.
• 8.3. Challenges to Rational Pricing — the empirical puzzles that motivate behavioral bubble models.