Draft:Rational bubble

Rational bubble is a theoretical framework in economics to model speculative bubble, a situation in which the asset price exceeds the fundamental value defined by the present discounted value of future dividends of the asset. The qualifier "rational" refers to the fact that investors are willing to purchase the overpriced asset despite holding rational expectations, although common beliefs and information among investors are also implicitly assumed.

Definition

For simplicity, we define rational bubbles in a deterministic setting.^[1]

Consider an infinite-horizon, deterministic economy with a homogeneous good and time indexed by ${\displaystyle t=0,1,\ldots }$. Consider an asset with infinite maturity that pays dividend ${\displaystyle D_{t}\geq 0}$ and trades at ex-dividend price ${\displaystyle P_{t}}$, both in units of the time-${\displaystyle t}$ good. In the background, we assume the presence of rational, perfectly competitive investors. Let ${\displaystyle q_{t}>0}$ be the Arrow-Debreu price, that is, the date-0 price of the consumption good delivered at time ${\displaystyle t}$, with the normalization ${\displaystyle q_{0}=1}$. The absence of arbitrage implies

${\displaystyle q_{t}P_{t}=q_{t+1}(P_{t+1}+D_{t+1}).}$

Iterating forward and using ${\displaystyle q_{0}=1}$, we obtain

${\displaystyle P_{0}=\sum _{t=1}^{T}q_{t}D_{t}+q_{T}P_{T}.}$

Because ${\displaystyle q_{t}>0}$ and ${\displaystyle D_{t}\geq 0}$, the sequence ${\displaystyle {\Bigg \{}\sum _{t=1}^{T}q_{t}D_{t}{\Bigg \}}}$ is increasing in ${\displaystyle T}$ and bounded above by ${\displaystyle P_{0}}$. Therefore, by the monotone convergence theorem, the sequence converges and the infinite sum of the present value of dividends

${\displaystyle V_{0}:=\sum _{t=1}^{\infty }q_{t}D_{t}}$

exists, which is called the fundamental value of the asset. Letting ${\displaystyle T\to \infty }$ in the above summation, we obtain

${\displaystyle P_{0}=\sum _{t=1}^{\infty }q_{t}D_{t}+\lim _{T\to \infty }q_{T}P_{T}=V_{0}+B_{0},}$

where ${\displaystyle B_{0}:=\lim _{T\to \infty }q_{T}P_{T}}$ is the bubble component. When ${\displaystyle B_{0}=0}$, we say that the no-bubble condition^[2] holds, in which case we obtain ${\displaystyle P_{0}=V_{0}}$ and the asset price equals its fundamental value. If

${\displaystyle B_{0}=\lim _{T\to \infty }q_{T}P_{T}>0,}$

then ${\displaystyle P_{0}>V_{0}}$ and we say that the asset contains a (rational) bubble.

Interpretation

Because free disposal forces the asset price to be nonnegative (${\displaystyle P_{t}\geq 0}$), the limit ${\displaystyle B_{0}=\lim _{T\to \infty }q_{T}P_{T}}$ is always nonnegative. Therefore, ${\displaystyle P_{0}=V_{0}+B_{0}\geq V_{0}}$, so there can never be a "negative bubble".

The economic meaning of the bubble component ${\displaystyle B_{0}}$ is that it captures a speculative aspect, that is, investors buy the asset now for the purpose of resale in the future, rather than (in addition to) for the purpose of receiving dividends.^[3]

Bubble Characterization Lemma

The following Bubble Characterization Lemma due to Montrucchio^[4][5] is useful for determining the existence of bubbles.

Bubble Characterization Lemma—If ${\displaystyle P_{t}>0}$ for all ${\displaystyle t}$, the asset price exhibits a bubble if and only if ${\displaystyle \sum _{t=1}^{\infty }D_{t}/P_{t}<\infty }$.

Proof

Changing time from ${\displaystyle t}$ to ${\displaystyle t-1}$ in the no-arbitrage condition yields

${\displaystyle q_{t-1}P_{t-1}=q_{t}(P_{t}+D_{t}).}$

Dividing both sides by ${\displaystyle q_{t}P_{t}}$ yields

${\displaystyle {\frac {q_{t-1}P_{t-1}}{q_{t}P_{t}}}=1+{\frac {D_{t}}{P_{t}}}.}$

Multiplying from ${\displaystyle t=1}$ to ${\displaystyle t=T}$ yields

${\displaystyle {\frac {q_{0}P_{0}}{q_{T}P_{T}}}=\prod _{t=1}^{T}{\Big (}1+{\frac {D_{t}}{P_{t}}}{\Big )}.}$

Expanding terms and applying the inequality ${\displaystyle \exp(x)\geq 1+x}$ yields

${\displaystyle 1+\sum _{t=1}^{T}{\frac {D_{t}}{P_{t}}}\leq {\frac {q_{0}P_{0}}{q_{T}P_{T}}}\leq \exp {\Bigg (}\sum _{t=1}^{T}{\frac {D_{t}}{P_{t}}}{\Bigg )}.}$

Letting ${\displaystyle T\to \infty }$, it follows that ${\displaystyle \lim _{T\to \infty }q_{T}P_{T}>0}$ if and only if ${\displaystyle \sum _{t=1}^{\infty }D_{t}/P_{t}<\infty }$.

The Bubble Characterization Lemma states that there is a bubble if and only if future dividend yields ${\displaystyle D_{t}/P_{t}}$ are summable. In other words, the price-dividend ratio ${\displaystyle P_{t}/D_{t}}$ must grow sufficiently fast.

Example

The following example is perhaps the simplest example of a rational bubble attached to dividend-paying assets.^[6]

Consider a two-period overlapping generations model with Cobb Douglas utility

${\displaystyle (1-\beta )\log c_{t}^{y}+\beta \log c_{t+1}^{o},}$

where ${\displaystyle c_{t}^{y},c_{t+1}^{o}}$ are consumption of generation ${\displaystyle t}$ when young and old. Assume generation ${\displaystyle t}$ has endowment ${\displaystyle a_{t}>0}$ when young but nothing when old. Suppose the initial old are endowed with a dividend-paying asset in unit supply. Let ${\displaystyle P_{t},D_{t}}$ be the price and dividend of the asset at time ${\displaystyle t}$.

Letting ${\displaystyle x_{t}}$ be the asset holdings of the young at time ${\displaystyle t}$, the budget constraints of generation ${\displaystyle t}$ are

${\displaystyle {\begin{aligned}&{\text{Young:}}&c_{t}^{y}+P_{t}x_{t}&=a_{t},\\&{\text{Old:}}&c_{t+1}^{o}&=(P_{t+1}+D_{t+1})x_{t}.\end{aligned}}}$

Solving the utility maximization problem yields ${\displaystyle c_{t}^{y}=(1-\beta )a_{t}}$. Using the market clearing condition ${\displaystyle x_{t}=1}$ yields the asset price

${\displaystyle P_{t}=P_{t}x_{t}=a_{t}-c_{t}^{y}=\beta a_{t}.}$

The dividend yield is therefore

${\displaystyle {\frac {D_{t}}{P_{t}}}={\frac {D_{t}}{\beta a_{t}}}.}$

Consequently, by the Bubble Characterization Lemma, we obtain the following result.

Proposition—In this overlapping generations model, there is a rational bubble if and only if ${\displaystyle \sum _{t=1}^{\infty }D_{t}/a_{t}<\infty }$.

Notable works

The 1958 paper of Samuelson^[7] that introduced money in the overlapping generations model is generally considered to be the first example of a rational bubble model (with a zero-dividend asset (money), which is often called a pure bubble). Bewley (1980) constructed a rational bubble model with infinitely-lived agents.^[8] Tirole (1985) showed that the emergence of a bubble resolves capital overaccumulation (dynamic inefficiency).^[9] Kocherlakota (1992) showed that whenever a bubble arises, the present value of the aggregate endowment is infinite.^[10] Santos and Woodford (1997) showed that when dividends comprise a non-negligile fraction of the aggregate endowment, then bubbles are impossible.^[11] Hirano and Toda (2025a) proved that when the dividend growth rate exceeds the natural interest rate but is lower than the economic growth rate (perhaps due to unbalanced growth between different sectors),^[12] then bubbles inevitably emerge.^[13]

Notes

1. Hirano & Toda 2024, Section 2.
2. Giglio, Stefano; Maggiori, Matteo; Stroebel, Johannes (2016). "No-Bubble Condition: Model-Free Tests in Housing Markets". Econometrica. 84 (3): 1047–1091. doi:10.3982/ECTA13447.
3. Hirano & Toda 2025b, p. 6.
4. Montrucchio 2004, Proposition 7.
5. Hirano & Toda 2025a, Lemma1.
6. Hirano & Toda 2025a, Section III.A.
7. Samuelson, Paul A. (December 1958). "An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money". Journal of Political Economy. 66 (6): 467–482. doi:10.1086/258100.
8. Bewley, Truman (1980). "The Optimum Quantity of Money". In Kareken, John H.; Wallace, Neil (eds.). Models of Monetary Economies. Federal Reserve Bank of Minneapolis. pp. 169–210.
9. Tirole, Jean (1985). "Asset Bubbles and Overlapping Generations". Econometrica. 53 (6): 1499–1528. doi:10.2307/1913232. JSTOR 1913232.
10. Kocherlakota, Narayana R. (1992). "Bubbles and constraints on debt accumulation". Journal of Economic Theory. 57 (1): 245–256. doi:10.1016/S0022-0531(05)80052-3.
11. Santos, Manuel S.; Woodford, Michael (1997). "Rational Asset Pricing Bubbles". Econometrica. 65 (1): 19-57. doi:10.2307/2171812. JSTOR 2171812.
12. Hirano, Tomohiro; Toda, Alexis Akira (2025). "Unbalanced growth and land overvaluation". Proceedings of the National Academy of Sciences. 122 (14) e2423295122. arXiv:2307.00349. Bibcode:2025PNAS..12223295H. doi:10.1073/pnas.2423295122. PMC 12002243. PMID 40178889.
13. Hirano & Toda 2025a, Theorem 2 and Section V.

References

• Hirano, Tomohiro; Toda, Alexis Akira (2024). "Bubble economics". Journal of Mathematical Economics. 111 102944. doi:10.1016/j.jmateco.2024.102944.
• Hirano, Tomohiro; Toda, Alexis Akira (2025a). "Bubble Necessity Theorem". Journal of Political Economy. 133 (1): 111–145. arXiv:2305.08268. doi:10.1086/732528.
• Hirano, Tomohiro; Toda, Alexis Akira (2025b). "Toward bubble clarity: A comment on Miao and Wang". Econ Journal Watch. 22 (1): 1–17.
• Martin, Alberto; Ventura, Jaume (2018). "The Macroeconomics of Rational Bubbles: A User's Guide" (PDF). Annual Review of Economics. 10 (1): 505–539. doi:10.1146/annurev-economics-080217-053534.
• Montrucchio, Luigi (2004). "Cass transversality condition and sequential asset bubbles". Economic Theory. 24 (3): 645–663. doi:10.1007/s00199-004-0502-8.