2025/11/05 by Jordan Roulleau-Pasdeloup, Roulleau-Pasdeloup, Jordan
Economics, Econometrics and Finance · Engineering · #Economic theories and models #FOS: Economics and business #Sports Dynamics and Biomechanics #Stochastic processes and financial applications #Theoretical Economics (econ.TH)
paper · pdf · doi:10.48550/arxiv.2511.03452
openalex publication_date 2025/11/05 · openalex created_date 2025/11/07 · openalex updated_date 2026/07/28
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∼ 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.