Explicit Consumption Functions with Borrowing Constraints: a Continuous Time Approach

TL;DR

This paper derives explicit global closed form solutions for the income fluctuation problem with borrowing constraints in continuous time, using Lambert W functions for the case r=0 and approximations for r>0, revealing supermodularity in consumption behavior.

Contribution

It provides the first explicit solutions for the problem with borrowing constraints in continuous time, including an approximation for positive interest rates.

Findings

Explicit solutions for r=0 using Lambert W function

Approximate solutions for r>0 near zero

Demonstration of supermodularity in consumption behavior

Abstract

There is no known explicit global closed form solution for the standard income fluctuation problem with a borrowing constraint and where wealth accumulates with a constant interest rate $r$. Using a continuous time formulation, I derive an explicit global closed form solution for the case $r=0$ using the Lambert W function. For the case $r>0$, I derive an explicit global closed form approximation that is valid for $r\sim 0$. I then use these to derive explicit expressions for the marginal propensity to consume out of assets and permanent income. I show that the cross-derivative between the two is strictly positive: the consumption consumption is supermodular.