A Continuous-Time Utility Maximization Problem with Borrowing Constraints in Macroeconomic Heterogeneous Agent Models:A Case of Regular Controls under Markov Chain Uncertainty

This paper is concerned with the verification of a continuous-time utility max- imization problem frequently used in recent macroeconomics. By focusing on Markov chain uncertainty, the problem in this paper can feature many charac- teristics of a typical consumer’s problem in macroeconomics, such as...

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Bibliographic Details
Published inIDEAS Working Paper Series from RePEc
Main Author SHIGETA, Yuki
Format Paper
LanguageEnglish
Published St. Louis Federal Reserve Bank of St. Louis 01.01.2022
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Summary:This paper is concerned with the verification of a continuous-time utility max- imization problem frequently used in recent macroeconomics. By focusing on Markov chain uncertainty, the problem in this paper can feature many charac- teristics of a typical consumer’s problem in macroeconomics, such as borrowing constraints, endogenous labor supply, unhedgeable labor income, multiple asset choice, stochastic changes in preference, and others. I show that the value func- tion of the problem is actually a constrained viscosity solution to the associated Hamilton–Jacobi–Bellman equation. Furthermore, the value function is continu- ously differentiable in the interior of its domain. Finally, the candidate optimal control is admissible, unique, and actually optimal.