Explicit Consumption Functions with Borrowing Constraints: a Continuous Time Approach

Jordan Roulleau-Pasdeloup

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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.

Date: 2025-11
New Economics Papers: this item is included in nep-fdg
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