# How can you interpret one of the parameters of optimal consumption at the Merton portfolio problem?

Statement: Let the dynamics of wealth of the agent satisfy $$dX_{t} = \pi_tX_t\Big(\mu dt+\sigma dB_{t}\Big)- c_t X_t dt, \qquad \textrm{with}\quad X_0=x_0 \in \mathbb{R},$$ where $$(\pi,c)$$ is an investment-consumption ($$\pi$$ - fraction of wealth to invest, $$c$$ - fraction of wealth to consume).

Under standard Merton optimization problem the agent is to maximize the expected utility $$J(\pi,c) =\mathbb{E}\Big[\int_0^TU(c_tX_t) dt + U(X_T)\Big],$$ under CRRA power utility $$U(x) = \frac{1}{1-\frac1\delta}x^{\frac1{1-\frac1\delta}}, \quad \delta > 0, \delta\neq 1,$$ so $$\delta$$ plays a role of risk-tolerance parameter.

The optimal plan is then given by $$\pi^* = \frac{\mu}{\sigma^2}\delta,\quad c^*_t=\Big( \frac1{\beta}-\big(1-\frac1\beta\big)e^{-\beta(T-t)}\Big)^{-1},$$ where $$\beta = \frac{\mu^2}{2\sigma^2}\delta(1-\delta).$$

Question: How can I interpret $$\beta$$ at this point? I have seen that $$\frac{\mu^2}{2\sigma^2}\delta$$ is sometimes referred to as expected portfolio return. But what is the meaning when I multiply it by $$1-\delta$$? If $$\delta > 1$$, it can be negative. I would call it effective expected portfolio return, but am not sure. If you can provide any reference, that would be perfect. Thanks in advance!

P.S. I was adapting my model to the standard Merton one, sorry for any discrepancies.

• Your parametrization of CRRA via $\delta$ seems a bit strange. In any case, the $\delta(1-\delta)$ expression in consumption share $\beta$ should reflect the elasticity of intertemporal substitution. Oct 28 '20 at 23:45
• @Michael, thanks for your answer. That is an interesting perspective and thanks for pointing this out. However, as I look at the definition of the elasticity of intertemporal substitution, it follows that it should be equal to $\frac1\delta$, or risk-aversion coefficient in other words. At this point I cannot see, how it can relate to $\beta$. If I am wrong, please, let me know. It would be nice as well, if you can refer me to some literature. Oct 30 '20 at 11:07
• First, I meant $\delta$, sorry. Do I understand correctly, that for running utility it essentially means the same. But that is interesting, as $\delta$ coming from terminal wealth optimization means risk-tolerance, but the one from the running integral optimization of consumption is an elasticity of intertemporal substitution and we require them to coincide for solvability. Am super strange seeing it for the first time. Again, would be glad if you refer me somewhere. Oct 30 '20 at 16:55