36. Heterogeneous Beliefs

Sometimes people disagree about the fundamental value of an asset. This assumption is most appropriate when the asset is relatively new such that no enough information could possibly be gathered by any person to correctly update their prior beliefs. In reality, this kind of new asset is always associated with some technological innovations such as the .COM boom in late 1990s (internet stock bubble), the financial innovation (housing market bubble) around 2007 and the bitcoin boom relatively recently.

In this section, we think about a model where agents can have heterogeneous beliefs about the fundamental of an asset, and in that setting we can define and discuss asset price bubbles in a different way. We can see that the existence of bubbles is actually due to speculative strategy (resale option) based on heterogeneous beliefs.

36.1 Set-up

• Agents are indexed by \(i\in\mathcal{I}\), where \(\mathcal{I}\) is the set of all agents. Assume that agents are all risk neutral, and all agents have the same discount factor \(\beta\).
• An asset has dividends in each period that follows a first-order Markov process. Denote the current period dividends by \(d\in D\) and the next period dividends by \(d'\in D\). Suppose the true underlying transition probability from \(d\) to \(d'\) for any \(d,d'\in D\) is \(\pi(d'|d)\), which is never observed. Denote agent \(i\)'s subjective transition probability by \(\pi^i(d'|d)\).

36.2 The price of the asset and the bubble

The value of the asset ex dividend today is the discounted sum of the dividends of next period and the price of next period at which the holder can resell the asset. Since agents have disagreement on the fundamental of the stock (in particular, on the transition probability in this model), the agent who values the stock most will end up getting all the assets. Here we implicitly assume that no short-sell is allowed, and thus it is possible for all the supply of the asset to be completely absorbed by the agent who values the stock most.

• By such assumptions, we can define the equilibrium price \(P(d)\) as a function of current dividend \(d\), where

$$ P(d)=\max_i\beta\sum_{d'\in D}\pi^i(d'|d)\big(d'+P(d')\big) \tag{36.0a} $$

• Similarly, we can define \(q^i(d)\) as the highest price for agent \(i\) to buy and hold the asset forever, then

$$ q^i(d)=\beta\sum_{d'\in D}\pi^i(d'|d)\big(d'+q^i(d')\big) \tag{36.0b} $$

Note that the \(q^i(d)\) is the fundamental ex dividend value of the asset to agent \(i\) if the current dividend was \(d\).

36.3 An example

36.3.1 Heterogeneous beliefs

Suppose that there are only two types of dividends for all periods, \(0

$$ \begin{cases} \pi^a(d_0|d_0)=\rho^a & \pi^b(d_0|d_0)=\rho^b\\ \pi^a(d_1|d_1)=\rho^a & \pi^b(d_1|d_1)=\rho^b\\ \pi^a(d_1|d_0)=1-\rho^a & \pi^b(d_1|d_0)=1-\rho^b\\ \pi^a(d_0|d_1)=1-\rho^a & \pi^b(d_0|d_1)=1-\rho^b \end{cases}\qquad \text{where }\rho^a>\rho^b $$

36.3.2 Bellman equations

For type \(a\) agents:

$$ q^a(d_0)=\beta\big[\rho^a(d_0+q^a(d_0))+(1-\rho^a)(d_1+q^a(d_1))\big] \tag{36.1} $$

$$ q^a(d_1)=\beta\big[(1-\rho^a)(d_0+q^a(d_0))+\rho^a(d_1+q^a(d_1))\big] \tag{36.2} $$

For type \(b\) agents:

$$ q^b(d_0)=\beta\big[\rho^b(d_0+q^b(d_0))+(1-\rho^b)(d_1+q^b(d_1))\big] \tag{36.3} $$

$$ q^b(d_1)=\beta\big[(1-\rho^b)(d_0+q^b(d_0))+\rho^b(d_1+q^b(d_1))\big] \tag{36.4} $$

36.3.3 The existence of a price bubble due to heterogeneous beliefs

So \(q^a(d_0)+q^a(d_1)=q^b(d_0)+q^b(d_1)\), i.e. \(q^a(d_0)-q^b(d_0)=-(q^a(d_1)-q^b(d_1))\). From (36.1)–(36.4), we know that it cannot be that \(q^a(d_0)=q^b(d_0)\) and \(q^a(d_1)=q^b(d_1)\), so we can arbitrarily (WLOG) let

$$ q^a(d_1)>q^b(d_1),\qquad q^a(d_0)

By definition of equilibrium price (36.0a),

$$ \begin{aligned} P(d_1)&=\max_i\beta\big[\rho^i(d_1+P(d_1))+(1-\rho^i)(d_0+P(d_0))\big]\\ &\ge\max_i\beta\big[\rho^i(d_1+q^a(d_1))+(1-\rho^i)(d_0+q^a(d_0))\big]\\ &\ge\beta\big[\rho^a(d_1+q^a(d_1))+(1-\rho^a)(d_0+q^a(d_0))\big]=q^a(d_1) \end{aligned} $$

and similarly \(P(d_0)\ge\beta\big[\rho^b(d_0+q^b(d_0))+(1-\rho^b)(d_1+q^b(d_1))\big]=q^b(d_0)\), with strict inequality. So we can conclude that

$$ P(d_0)>q^b(d_0)=\max_i q^i(d_0),\qquad P(d_1)>q^a(d_1)=\max_i q^i(d_1) $$

which means that there is a bubble in the asset price (per definition of bubble in this model) at both dividend states \(d_0\) and \(d_1\).