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Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints
In this paper, we address the utility maximization problem of an infinitely lived agent who has the option to increase their income. The agent can increase their income at any time, but doing so incurs a wealth cost proportional to the amount of the increase. To prevent the agent from infinitely inc...
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| Published in: | Mathematics (Basel) 2024-11, Vol.12 (22), p.3536 |
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| creator | Kim, Geonwoo Jeon, Junkee |
| description | In this paper, we address the utility maximization problem of an infinitely lived agent who has the option to increase their income. The agent can increase their income at any time, but doing so incurs a wealth cost proportional to the amount of the increase. To prevent the agent from infinitely increasing their income and borrowing against future income, we additionally consider a non-negative wealth constraint that prohibits borrowing based on future income. This utility maximization problem is a mixture of stochastic control, where the agent chooses consumption and investment, and singular control, where the agent chooses a non-decreasing income process. To solve this non-trivial and challenging problem, we derive the Hamilton–Jacobi–Bellman (HJB) equation with a gradient constraint using the dynamic programming principle (DPP). Then, using the guess-and-verify method and a linearization technique, we obtain a closed-form solution to the HJB equation and, based on this, find the optimal strategy. |
| doi_str_mv | 10.3390/math12223536 |
| format | article |
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Then, using the guess-and-verify method and a linearization technique, we obtain a closed-form solution to the HJB equation and, based on this, find the optimal strategy.</description><identifier>ISSN: 2227-7390</identifier><identifier>EISSN: 2227-7390</identifier><identifier>DOI: 10.3390/math12223536</identifier><language>eng</language><publisher>Basel: MDPI AG</publisher><subject>Borrowing ; Closed form solutions ; Constraints ; Consumption ; consumption and investment ; Cost control ; Decision making ; Dynamic programming ; Earnings ; Financial planning ; Flexibility ; free boundary problem ; HJB equation ; Human capital ; Income ; income adjustment ; Investment analysis ; Investments ; linearization ; Maximization ; Optimal control ; Optimization ; Present value ; Securities markets ; singular control ; Skills ; Stochastic processes</subject><ispartof>Mathematics (Basel), 2024-11, Vol.12 (22), p.3536</ispartof><rights>COPYRIGHT 2024 MDPI AG</rights><rights>2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). 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The agent can increase their income at any time, but doing so incurs a wealth cost proportional to the amount of the increase. To prevent the agent from infinitely increasing their income and borrowing against future income, we additionally consider a non-negative wealth constraint that prohibits borrowing based on future income. This utility maximization problem is a mixture of stochastic control, where the agent chooses consumption and investment, and singular control, where the agent chooses a non-decreasing income process. To solve this non-trivial and challenging problem, we derive the Hamilton–Jacobi–Bellman (HJB) equation with a gradient constraint using the dynamic programming principle (DPP). Then, using the guess-and-verify method and a linearization technique, we obtain a closed-form solution to the HJB equation and, based on this, find the optimal strategy.</description><subject>Borrowing</subject><subject>Closed form solutions</subject><subject>Constraints</subject><subject>Consumption</subject><subject>consumption and investment</subject><subject>Cost control</subject><subject>Decision making</subject><subject>Dynamic programming</subject><subject>Earnings</subject><subject>Financial planning</subject><subject>Flexibility</subject><subject>free boundary problem</subject><subject>HJB equation</subject><subject>Human capital</subject><subject>Income</subject><subject>income adjustment</subject><subject>Investment analysis</subject><subject>Investments</subject><subject>linearization</subject><subject>Maximization</subject><subject>Optimal control</subject><subject>Optimization</subject><subject>Present value</subject><subject>Securities markets</subject><subject>singular control</subject><subject>Skills</subject><subject>Stochastic processes</subject><issn>2227-7390</issn><issn>2227-7390</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>PIMPY</sourceid><sourceid>DOA</sourceid><recordid>eNpNUU1rGzEQXUICCalv-QELudaJpNHH7tE1TWsI-NDmLLT6cGS8kquVG_rvO4lNsITQzJt5j3lM09xR8gDQk8fR1FfKGAMB8qK5wUjNFRYuz-LrZjZNW4Knp9Dx_qb5td7XOJpdu8xpOoyY5NSa5NpV-uunOvpU27dYXzG3efTtwm0PJ_i961suJb_FtPng12JiqtOX5iqY3eRnp_-2eXn6_nv5c_68_rFaLp7nlglR51wEga93VhmluOz6jg8hDJIoYQfHXM-4H4KkigDGVjJquFBKAQtCKAu3zeqo67LZ6n1BH-WfzibqDyCXjTalRrvzGobQ-U4FQqTkwrmB98JKQSR3SjDqUOv-qLUv-c8BnettPpSE42ugAMCUkBS7Ho5dG4OiMYWMli1e58doc_IhIr7oaAcglAAkfD0SbMnTVHz4HJMS_b42fb42-A983YmM</recordid><startdate>20241101</startdate><enddate>20241101</enddate><creator>Kim, Geonwoo</creator><creator>Jeon, Junkee</creator><general>MDPI AG</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7TB</scope><scope>7XB</scope><scope>8AL</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FR3</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>KR7</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0N</scope><scope>M7S</scope><scope>P62</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>Q9U</scope><scope>DOA</scope></search><sort><creationdate>20241101</creationdate><title>Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints</title><author>Kim, Geonwoo ; Jeon, Junkee</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c255t-45f545f9dc7a77468984bffb6075cbd2d924ebf61703d92c621a4577732f557c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Borrowing</topic><topic>Closed form solutions</topic><topic>Constraints</topic><topic>Consumption</topic><topic>consumption and investment</topic><topic>Cost control</topic><topic>Decision making</topic><topic>Dynamic programming</topic><topic>Earnings</topic><topic>Financial planning</topic><topic>Flexibility</topic><topic>free boundary problem</topic><topic>HJB equation</topic><topic>Human capital</topic><topic>Income</topic><topic>income adjustment</topic><topic>Investment analysis</topic><topic>Investments</topic><topic>linearization</topic><topic>Maximization</topic><topic>Optimal control</topic><topic>Optimization</topic><topic>Present value</topic><topic>Securities markets</topic><topic>singular control</topic><topic>Skills</topic><topic>Stochastic processes</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kim, Geonwoo</creatorcontrib><creatorcontrib>Jeon, Junkee</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Computing Database (Alumni Edition)</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>AUTh Library subscriptions: ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central</collection><collection>Engineering Research Database</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>Computer Science Database</collection><collection>Civil Engineering Abstracts</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Computing Database</collection><collection>Engineering Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>Publicly Available Content Database (Proquest) (PQ_SDU_P3)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><collection>DOAJ Directory of Open Access Journals</collection><jtitle>Mathematics (Basel)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kim, Geonwoo</au><au>Jeon, Junkee</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints</atitle><jtitle>Mathematics (Basel)</jtitle><date>2024-11-01</date><risdate>2024</risdate><volume>12</volume><issue>22</issue><spage>3536</spage><pages>3536-</pages><issn>2227-7390</issn><eissn>2227-7390</eissn><abstract>In this paper, we address the utility maximization problem of an infinitely lived agent who has the option to increase their income. 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| subjects | Borrowing Closed form solutions Constraints Consumption consumption and investment Cost control Decision making Dynamic programming Earnings Financial planning Flexibility free boundary problem HJB equation Human capital Income income adjustment Investment analysis Investments linearization Maximization Optimal control Optimization Present value Securities markets singular control Skills Stochastic processes |
| title | Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints |
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