3.3 A Model of Rational Housing Bubbles

investors may change. Note that, on average, behavioral and rational investors agree that the value of the payoff is σ. However, these behavioral agents are like Waermondt and they form their expectations given the signals they receive.

To assess the effect of behavioral agents, let us frst consider the case in which all investors are rational. In this case, the price of the asset is determined by the arbitrage condition between the two assets. That is, the nonbehavioral price is given by,

This equation means that agents are indifferent between investing in the storage technology (it has a gross return of 1 + π) and purchasing the risky asset (it has a gross return of σ p ).

We now introduce behavioral agents in our economy. In this case, we need to fnd two equilibrium objects: (i) the price of the risky asset and (ii) the agents who purchase this asset. This type of model is characterized by a threshold equilibrium. Let us assume that agents with a signal si above ¯ s, which will be determined in equilibrium, purchase the risky asset. The rest of agents purchase the riskless asset. We make a parametrical assumption that guarantees that only behavioral investors purchase the risky asset.3 The assumption states that the endowment of behavioral agents is large enough. Given our setup, the two equilibrium objects, the signal of the marginal investor ¯ s and the price pB, are given by

The frst equation characterizes the marginal investor. The investor with signal ¯ s is indifferent between investing in the storage technology (the return is 1 + π) and purchasing the risky asset. According to her belief, the payoff of this asset will be ¯ s, and, thus, the return on the risky asset is

p NB =

σ 1 + π

¯ s pB = 1 + π ,

p B = eµ 1 − ¯ s 2σ .

3 In our simple model, the assumption that guarantees that only behavioral investors purchase the risky asset is e > 2σ/µ(1 + π). In words, we assume that behavioral agents have enough wealth.

¯ s/pB. The second equation defnes the price of the risky asset as a function of the marginal investor. Since the supply of the risky asset is one, the price of this asset is given by the demand. In other words, the price of the risky asset is the amount of money used to purchase this asset. Given the uniform distribution and the fact that behavioral agents are a mass µ, it means that there is a mass µ 1 − ¯ s 2σ of investors with a signal above ¯ s. These are the optimistic investors that will purchase the risky asset. Since each of these investors have an endowment of e, the price of the risky asset is eµ 1 −

¯ s 2σ .

Finally, if we combine the two equations, we can fnd an expression for the equilibrium price of the risky asset in the presence of behavioral investors that depends only on the parameters of the model,

We want to highlight three features of this behavioral price. First, in the presence of behavioral investors, the price of the asset is higher (i.e., pB > pNB). This is not surprising given that the behavioral investors who purchase the asset are the optimistic ones. Second, the price increases with the mass of behavioral investors, µ. As it can be seen from the equation, when µ increases, the price rises. The interpretation is that when there are more behavioral investors, the demand for the risky asset increases, which raises the price of the asset. Third, the price of the risky asset increases with the endowment of the behavioral agents, e. The intuition is the same as above. If behavioral agents are richer, the demand for the risky asset increases, which pushes up its price.

We can use this behavioral model to explain the origin of housing bubbles. Let us imagine that in the beginning there are no behavioral investors in the housing market. That is, in the housing market there are only investors who want to purchase the house to live in it. These rational agents know the payoff of purchasing the house, so that the price of housing is given by pNB. Nonetheless, for some exogenous reason, maybe the stock market bubble has collapsed, behavioral investors become interested in the housing market. As they become interested in the market, they receive a signal about the payoff of the house. Note that it means that there are now two types of investors in the housing market (i.e., µ> 0). Some behavioral investors will have received a good signal

p B = σ

σ µe + 1+π 2

and, thus, they will invest in housing. That is, the presence of behavioral agents has increased the price of housing (i.e., pB > pNB). Note also that the larger is the weight of behavioral agents in the housing market, the larger is the house price appreciation. Indeed, if a small part of the population gets interested into the housing market, very few people will get positive signals and the size of the housing bubble will be low. In contrast, if everyone gets interested in the housing market, a lot of people will receive good signals and the housing bubble will be large. In other words, we will observe a large housing bubble when a huge fraction of the housing market is formed by behavioral agents. Moreover, remember that the effect of behavioral agents on the price is magnifed with their endowment (e). That is, let us imagine that the entry of behavioral agents coincides with enhanced access to capital (e.g., international capital infows or low interest rates), which enables behavioral investor to leverage their investment. In this case, the increase in house prices caused by the presence of behavioral agent will be exacerbated.

3.2 tHeory of rAtionAl BuBBles

After discussing a behavioral explanation of asset price bubbles, we now turn to a model of rational bubbles. In this model, all agents will have the same beliefs. This discussion is based on Samuelson (1958). Even though the word “bubble” does not appear in the work of Samuelson, his work is widely considered as the frst model of rational bubbles. To summarize his model in one line, Samuelson shows that when there is a shortage of assets, assets without fundamental value may be priced in equilibrium (i.e., bubbles can emerge) and this will be good for the economy.

Agents in this economy live for two periods: (i) young and (ii) old. It is an overlapping generation economy, which means that at any point in time, t, young agents (which are born at time t) live along with old agents (which were born at time t − 1). We assume that population grows at the rate n. It means that the number of agents at time t + 1 is Lt+1 = (1 + n)Lt. We further assume, without loss of generality, that only old agents derive utility from consumption. As the reader will see, this assumption is made to simplify the savings decision of young agents. However, it does not affect the main results of the model. Thus, we can write the lifetime utility of an agent born at time t as,

Ut = u(ct+1 ),

where ct+1 is the consumption that the agent will make when she will be old (time t + 1).

We assume that the timing of events of an agent born at time t is as follows. At time t, the agent is born and receives and endowment (e). This endowment is constant over time. Agents have access to a storage technology. If the agent invests a units in this technology at time t, she will obtain (1 + π)a units the next period. At time t + 1, agents do not receive any income. They consume the returns on the investment they made at time t and they die. Given this timing of events, we can write the problem faced by an agent born at time t as,

Maxu(ct+1 ) subject to e ≥ at , (1 + π )at ≥ ct+1. As the reader can see, agents in this economy do not really have any option. When the agent is young, she needs to put all her endowment into the storage technology (at = e). When the agent becomes old, she consumes the returns on her investment, ct+1 = (1 + π )e, and she dies.

Let us now imagine that old agents are unhappy with the returns they get from the storage technology and want to increase their consumption. One option may be that old agents ask for a loan to young agents. The representative old agent may say to the representative young agent: “lend us your endowment today and in the next period we will return your endowment plus an interest rate above π”. What would the reader reply to this offer if she were the representative of young agents? At frst sight, it seems like a great deal. Before this offer, young agents had to invest in the storage technology, which had a return equal to π. Now, the old agent is promising a higher return. However, after second thoughts, the representative of young agents would shout “… but they will be dead!”. Indeed, the problem with the offer of the representative of old agents is that old agents will not be around the next period to pay the credit back. Therefore, this loan market between generations is not possible.

After young agents reject the offer, old agents keep thinking on how to increase their consumption and they come up with the following idea. The representative old agent picks a piece of paper and writes the words “Bubble certifed”. She writes these letters with a very rare pen, which she destroys after writing these words. After doing that, the

representative old agent goes again to meet the representative young agent and tells her: “look at this piece of paper with words ‘Bubble certifed’ written in a very rare pen. The pen has been destroyed after writing these words. Would you like to purchase this piece of paper?”. If the reader were the representative of young agents, what would she answer? First, she would notice that this is indeed a very special piece of paper, which cannot be reproduced nor forged. Next, she would ask herself which the value of the paper will be the next period. To answer this question, she needs to wonder whether the next generation of young agents will be interested in this specifc piece of paper. If the next generation does not purchase the piece of paper, the young agent will end up the ensuing period with a worthless piece of paper. However, if the next generation does purchase the asset, the return on this piece of paper will be Pt+1/Pt , where Pt is the price of the piece of paper at time t. That is, the return on this investment is the change in the price of the piece of paper. The same reasoning of the current generation of young agents applies to the next generation of young agents, which also applies to the following generation and so on and so forth. Therefore, the piece of paper will only have value if all future generations of young agents think that they will be able to sell the piece of paper to the next generation of young agents.

Let us now assume that, indeed, all young agents think that they will be able to sell the piece of paper to the next generation of young agents. Would the purchase of this piece of paper be a good investment opportunity? It depends. If all young agents use their endowment to purchase the piece of paper, the price of the paper at any time t is Pt = eLt. The reason is that each agent has an endowment of e and there are Lt young agents at time t. This implies that the return on investing in this piece of paper is Pt+1 /Pt = Lt+1 /Lt = 1 + n. In other words, the return on this investment is the increase in the demand for this piece of paper (we have kept the supply fxed), which is the population growth (remember that the endowment is fxed). Therefore, young agents will prefer to purchase the piece of paper instead of investing in the storage technology if n > π. In this case, young agents only invest in the piece of paper and they will consume c = (1 + n)e. Note that what we have described is the formation of a rational bubble. This piece of paper is an asset without any fundamental value, which is purchased at a positive price. In this example, the price of the paper is exactly the bubble component (i.e., P = B).