Liquidity-Constrained Annuitization with Shifted-Power Preferences

We study optimal consumption, investment, and irreversible purchases of immediate life annuities when liquid wealth and consumption must remain nonnegative. Preferences have shifted-power form, with a nonnegative additive consumption parameter. A positive shift makes marginal utility at zero consumption finite and introduces an endogenous zero-consumption region. It cannot be removed by an unconstrained change of income. We obtain an explicit dual solution and a semi-explicit primal value function. A market-dependent constant gives the sharp threshold for finite expected utility. Above this threshold, the optimal annuitization boundary is increasing and concave: it is linear at low existing income, strictly concave at intermediate income, and affine at high income. We construct the optimal policy from a reflected diffusion on a fixed interval, resolving admissibility at the singular zero-wealth boundary. The solution recovers the CRRA annuitization ray as the shift vanishes and the exponential-utility boundary under a simultaneous curvature–shift limit. An exact no-annuity benchmark shows that annuities raise the large-wealth consumption slope when curvature exceeds one and lower it when curvature is below one. At the baseline parameters, access to annuities is worth up to 47.6 percent of initial wealth for a household with curvature two. Numerical examples illustrate the boundary geometry, consumption comparisons, and preference limits. Accompanying Python 3.13.16 code reproduces the figures and diagnostic calculations.