  The Merton problem is the well-known stochastic control problem of choosing
consumption over time, as well as an investment mix, to maximize expected
constant relative risk aversion (CRRA) utility of consumption. Merton
formulated the problem and provided an analytical solution in 1970; since then
a number of extensions of the original formulation have been solved. In this
note we identify a certainty equivalent problem, i.e., a deterministic optimal
control problem with the same optimal value function and optimal policy, for
the base Merton problem, as well as a number of extensions. When time is
discretized, the certainty equivalent problem becomes a second-order cone
program (SOCP), readily formulated and solved using domain specific languages
for convex optimization. This makes it a good starting point for model
predictive control, a policy that can handle extensions that are either too
cumbersome or impossible to handle exactly using standard dynamic programming
methods.
