A cautionary note on linear aggregation in macroeconomic models under the RINCE preferences

Farmer (1990) formulated the Risk-Neutral Constant Elasticity (RINCE) preferences and obtained a closed-form solution to the consumption-saving problem in the presence of idiosyncratic shocks to flow income and investment return. Both the value and policy functions are linear in wealth, and their linearity has been exploited to facilitate aggregation in macroeconomic models. However, Farmer's solution implicitly assumed that the natural borrowing limit never binds. A counterexample is provided herein in which the correct solution is nonlinear because the natural borrowing limit binds. To resurrect the linearity, one needs to restrict carefully the shock process of flow income and investment return so that the natural borrowing limit never binds. By doing so, however, one could throw away some important classes of problems with income shocks. Two models with temporary and persistent income shocks are used to show that the linear solution violates the natural borrowing limit with non-negligible probability in realistic settings.