Effect of labour income on the optimal bankruptcy problem

Annals of Operations Research https://doi.org/10.1007/s10479-023-05166-z ORIGINAL RESEARCH Effect of labour income on the optimal bankruptcy problem Guodong Ding1 · Daniele Marazzina1 Accepted: 4 January 2023 © The Author(s) 2023 Abstract In this paper we deal with the optimal bankruptcy problem for agents who can optimally allocate their consumption rate, the amount of capital invested in the risky asset, as well as their leisure time. In our framework, the agents are endowed by an initial debt, and they are required to repay their debt continuously. Declaring bankruptcy, the debt repayment is exempted at the cost of a wealth shrinkage. We implement the duality method to solve the problem analytically and conduct a sensitivity analysis to the bankruptcy cost and benefit parameters. Introducing the flexible leisure/working rate, and therefore the labour income, into the bankruptcy model, we investigate its effect on the optimal strategies. Keywords Power utility optimization · Bankruptcy stopping time · Consumption-portfolio-leisure controls · Legendre–Fenchel transform · Variational inequalities 1 Introduction In this paper, we study an optimal stopping time problem, in which agents decide their consumption-portfolio-leisure strategy as well as the optimal bankruptcy time. Their utility is described by a power function concerning both consumption and leisure rates. The sum of labour and leisure rates is assumed to be equal to a constant L̄. The labour rate is lower bounded by a positive constant L̄ − L, L > 0, to retain the employment state. Relating to the labour rate, the agent earns the labour income with a fixed wage rate. By determining the continuous and stopping regions of the corresponding stopping time problem, we prove that the optimal bankruptcy time is the first hitting time of the wealth process downward to a critical wealth boundary. The idea is directly inspired by Jeanblanc et al. (2004), in which a stochastic control model is constructed to quantify the benefit of filing consumer bankruptcy in the perspective of complete debt erasure. Their research is a response to the sharp growth in bankruptcy cases between 1978 and 2003 due to the promulgation of the 1978 Bankruptcy Reform Act in American. The Act introduced two kinds of consumer bankruptcy mechanisms, which are reflected in its Chapter 7 and Chapter 13 separately: debtors following Chap- B Daniele Marazzina 1 Department of Mathematics, Politecnico di Milano, 20133 Milano, Italy 123 Annals of Operations Research ter 7 to file bankruptcy are granted the debt exemption, but must undertake the liquidation of non-exempt assets. Alternatively, the mechanism in Chapter 13 adopts the reorganization procedure instead of the liquidation. Debtors are permitted to retain assets, but the debt is required to be reorganized and paid continuously from future revenues. The statistical data shows that filing bankruptcy under Chapter 7 predominates in all consumer bankruptcy cases (1,156,274 out of 1,625,208 cases in 2003, accounting for 72%).1 Following Jeanblanc et al. (2004), an affine loss function, α(X (τ ) − F), is established to deal with both the fixed and variable costs of filing bankruptcy, which corresponds to the mathematical description for the bankruptcy mechanism under Chapter 7. Here X (τ ) is the wealth level at the moment of bankruptcy τ , and F represents the fixed cost of bankruptcy. Therefore, the bankruptcy option reduces the wealth from X (τ ) to X (τ + ) = α(X (τ ) − F), with a drop in wealth equal to F +(1−α)(X (τ )− F) = (1−α)X (τ )+α F. The loss proportion (1−α) of the remaining wealth after the bankruptcy liquidation, X (τ ) − F, is related, for example, to taxes. Compared to Jeanblanc et al. (2004), we make an extension in two aspects: firstly, a new control variable, the leisure rate, is inserted for a more realistic consideration: decreasing the leisure rate, the agent earns a larger labour income. For the introduction of the leisure as a control variable into optimal stopping time problems, the reader can refer to Choi et al. (2008) and Farhi and Panageas (2007), where authors studied the optimal retirement -from labour- model regarding the consumption-portfolio-leisure strategy. Different from these two researches, we consider the stopping time concerning the bankruptcy issue rather than the retirement: while the optimal retirement is the first hitting time of the wealth process to an upper critical wealth boundary (Barucci & Marazzina, 2012; Choi et al., 2008; Farhi & Panageas, 2007), the optimal bankruptcy is related to a lower boundary. This extension permits us to study the impact of the disutility from full work on the bankruptcy option. Secondly, in order to deal with a utility from consumption and leisure rate, we implement a different method from Jeanblanc et al. (2004), where the utility of the agent only depends on her consumption, solving the optimal problem with the duality method instead of the dynamic programming method, to deduce the solution analytically, as in Barucci and Marazzina (2012) and Choi et al. (2008). We would like to stress that in this work we deal with the duality method applied to intertemporal consumption; for terminal utility problem the reader can refer, for example, to Barucci et al. (2021), Colaneri et al. (2021) and Nicolosi et al. (2018). The duality method throughout this paper can be summarized into four steps. We first tackle the post-stopping time problem to deduce a closed form of the corresponding value function. Then we apply the convex dual transform to the utility function and to the value function of the post-bankruptcy time problem obtained in the first step. Afterwards, we construct the duality between the optimal control problem and the individual’s shadow price problem, by the aid of the liquidity and budget constraints and the dual transforms acquired before. Finally, we cast the dual shadow price problem as a free boundary problem, which leads to a system of variational inequalities and enables us to solve it analytically. The methodology discussed here refers to He and Pagès (1993) and Karatzas and Wang (2000): in Karatzas and Wang (2000) authors applied the duality method to solve a discretionary stopping time problem explicitly, while in He and Pagès (1993) authors used the duality approach to link the individual’s shadow price problem with the optimal control problem and investigated the impact of the liquidity constraint on the optimization. Other related literatures are Karatzas et al. (1997), where authors studied the general optimal control problem involving the consumption and investment, and offered the solution 1 Administrative Office of the U.S. Courts, Table F-2- Bankruptcy Filings (December 31, 2003) [Online]. Available: https://www.uscourts.gov/statistics/table/f-2/bankruptcy-filings/2003/12/31. 123 Annals of Operations Research in a closed-form (Sethi et al., 1995), where a general continuous-ti (...truncated)