Market selection with learning and catching up with the Joneses

An agent is said to survive in the long run if the ratio of his consumption to the aggregate consumption stays positive with positive probability as time goes to infinity.

We assume that the mean growth rate follows an Ornstein–Uhlenbeck process.

This is a realistic assumption as correlations are extremely difficult to estimate empirically.

In our model, the effective risk aversion depends on the level of habit formation (see (4.1)).

Bhamra and Uppal [5], Dumas [13] and Wang [29] considered the same model, but with only two agent types and heterogeneity coming only from risk aversion.

In this setting, the model can be implemented by a complete securities market with a unique state price density derived in equilibrium (as for instance in Duffie and Huang [12]). More specifically, the filtration \(\mathcal{G}\) is generated by the Brownian motion s (which is interpreted as a public signal) and the aggregate endowment process D. Nevertheless, as explained in Remark 2.6, the filtration \(\mathcal{G}\) is also generated by the Brownian motions s and W ⁽⁰⁾. Thereby, the market can be completed by adding one additional security to S. However, since the price of this security would be determined endogenously, one would have to verify endogenous completeness. This can be done by using the techniques of Hugonnier et al. [17]. Otherwise, we can just assume that there are sufficiently many (derivative) assets, completing the market.

This paradigm of a utility function was first introduced in Abel [1], and is commonly referred to in the literature as a utility with exogenous habits. This specification describes a decision maker who experiences an impact of the ‘standard of living’ index.

One can check that the process (Z _it ) _t∈[0,∞) is a true martingale by verifying Novikov’s condition on a small interval and then applying a similar argument to the one used in Example 3 in Sect. 6.2 in Liptser and Shiryaev [23].