本文使用深度學習方法求解具有內生勞動供給的異質代理人不完全市場模型。不同於傳統數值方法在預先設定的狀態網格上逐點求解模型，本文學習到的政策函數能夠引導代理人從隨機初始化的狀態走向均衡，並透過 Euler 殘差最小化（Euler-residual minimization）來約束均衡一致性。
訓練完成後，本文固定神經網路，並將其作為一個模擬經濟環境，用以研究每個時間點下異質代理人之間的互動。這樣的架構使我們能夠檢視個體勞動反應如何影響總勞動供給、均衡工資，以及其他代理人的儲蓄決策。研究結果顯示，在所有制度中，高不確定性結合非線性稅制會產生最高的儲蓄率。事件時間分析的證據顯示，這個現象與兩類代理人的序列性反應相一致：第一類代理人在自身能力（ability）上升後增加勞動供給與儲蓄；而第二類代理人則因第一類代理人擴張勞動供給導致均衡工資下降，進而隨後提高自身的勞動供給與儲蓄。

This paper applies a deep learning approach to solve heterogeneous-agent incomplete-market models with endogenous labor supply. Unlike traditional numerical methods that solve the model pointwise over a predefined state grid, the learned policy function guides agents’ decisions toward equilibrium from randomly initialized states, while equilibrium consistency is disciplined by Euler-residual minimization.
After training, we fix the network and use it as a simulated economic environment to study interactions between heterogeneous agents at each time step. This structure allows us to examine how individual labor responses affect aggregate labor supply, equilibrium wages, and other agents’ saving decisions. We find that high uncertainty combined with non-linear taxation generates the highest saving ratios among all regimes. Event-time evidence suggests that this pattern is consistent with a sequential response between two types of agents: one group increases labor supply and saving after a rise in ability, while another group subsequently increases labor supply and saving in response to the lower equilibrium wage generated by the first group’s labor-supply expansion.

Aiyagari, S. R. (1994). Uninsured idiosyncratic risk and aggregate saving. The Quarterly Journal of Economics, 109(3), 659–684. https://doi.org/10.2307/2118417

Algan, Y., Allais, O., & Den Haan, W. J. (2010). Solving the incomplete markets model with aggregate uncertainty using parameterized cross-sectional distributions. Journal of Economic Dynamics and Control, 34(1), 59–68.

Bénabou, R. (2002). Tax and education policy in a heterogeneous-agent economy: What levels of redistribution maximize growth and efficiency? Econometrica, 70(2), 481–517. https://doi.org/10.1111/1468-0262.00248

Bewley, T. (1983). A difficulty with the optimum quantity of money. In J. H. Kareken (Ed.), Economic analysis of markets and games: Essays in honor of frank hahn (pp. 169–185). Academic Press.

Bewley, T. (1986). Stationary monetary equilibrium with a continuum of independently fluctuating consumers. In W. Hildenbrand & A. Mas-Colell (Eds.), Contributions to mathematical economics in honor of gérard debreu (pp. 79–102). North-Holland.

Bewley, T. F. (1977). The permanent income hypothesis: A theoretical formulation. Journal of Economic Theory, 16(2), 252–292. https://doi.org/10.1016/0022-0531(77)90009-6

Cybenko, G. (1989). Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems, 2(4), 303–314.

Den Haan, W. J. (1997). Solving dynamic models with aggregate shocks and heterogeneous agents. Macroeconomic dynamics, 1(2), 355–386.

Den Haan, W. J. (2010). Comparison of solutions to the incomplete markets model with aggregate uncertainty. Journal of Economic Dynamics and Control, 34(1), 4–27. https://doi.org/10.1016/j.jedc.2008.12.010

Den Haan, W. J., & Rendahl, P. (2010). Solving the incomplete markets model with aggregate uncertainty using explicit aggregation. Journal of Economic Dynamics and Control, 34(1), 69–78. https://doi.org/10.1016/j.jedc.2008.12.008

Heathcote, J., Storesletten, K., & Violante, G. L. (2017). Optimal tax progressivity: An analytical framework. Quarterly Journal of Economics, 132(4), 1693–1754. https://doi.org/10.1093/qje/qjx018

Hornik, K. (1991). Approximation capabilities of multilayer feedforward networks. Neural Networks, 4(2), 251–257.

Huggett, M. (1993). The risk-free rate in heterogeneous-agent incomplete-insurance economies. Journal of Economic Dynamics and Control, 17(5-6), 953–969. https://doi.org/10.1016/0165-1889(93)90024-M

Judd, K. L. (1992). Projection methods for solving aggregate growth models. Journal of Economic Theory, 58(2), 410–452. https://doi.org/10.1016/0022-0531(92)90061-L

Judd, K. L. (1998). Numerical methods in economics. MIT Press.

Judd, K. L., Maliar, L., & Maliar, S. (2011). Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models. Quantitative Economics, 2(2), 173–210. https://doi.org/10.3982/QE14

Judd, K. L., Maliar, L., Maliar, S., & Valero, R. (2014). Smolyak method for solving dynamic economic models: Lagrange interpolation, anisotropic grid and adaptive domain. Journal of Economic Dynamics and Control, 44, 92–123. https://doi.org/10.1016/j.jedc.2014.03.003

Kim, S. H., Kollmann, R., & Kim, J. (2010). Solving the incomplete market model with aggregate uncertainty using a perturbation method. Journal of Economic Dynamics and Control, 34(1), 50–58. https://doi.org/10.1016/j.jedc.2008.11.011

Konda, V. R., & Tsitsiklis, J. N. (2000). Actor-critic algorithms. Advances in Neural Information Processing Systems, 12, 1008–1014.

Krusell, P., & Smith, J., Anthony A. (1998). Income and wealth heterogeneity in the macroeconomy. Journal of Political Economy, 106(5), 867–896. https://doi.org/10.1086/250034

LeCun, Y., Bengio, Y., & Hinton, G. (2015). Deep learning. Nature, 521(7553), 436–444.

Maliar, L., Maliar, S., & Winant, P. (2021). Deep learning for solving dynamic economic models. Journal of Monetary Economics, 122, 76–101.

Marcet, A., Obiols-Homs, F., & Weil, P. (2007). Incomplete markets, labor supply and capital accumulation. Journal of Monetary Economics, 54(8), 2621–2635. https://doi.org/10.1016/j.jmoneco.2006.12.011

McCulloch, W. S., & Pitts, W. (1943). A logical calculus of the ideas immanent in nervous activity. The Bulletin of Mathematical Biophysics, 5(4), 115–133.

Mi, Q., Xia, S., Song, Y., Zhang, H., Zhu, S., & Wang, J. (2023). TaxAI: A dynamic economic simulator and benchmark for multi-agent reinforcement learning. arXiv preprint arXiv:2309.16307.

Mnih, V., Badia, A. P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., & Kavukcuoglu, K. (2016). Asynchronous methods for deep reinforcement learning. International conference on machine learning, 1928–1937.

Perez, E., Strub, F., de Vries, H., Dumoulin, V., & Courville, A. (2018). FiLM: Visual reasoning with a general conditioning layer. Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18), 3942–3951.

Preston, B., & Roca, M. (2007). Incomplete markets, heterogeneity and macroeconomic dynamics (Working Paper No. 13260). National Bureau of Economic Research. https://doi.org/10.3386/w13260

Reiter, M. (2009). Solving heterogeneous-agent models by projection and perturbation. Journal of Economic Dynamics and Control, 33(3), 649–665. https://doi.org/10.1016/j.jedc.2008.08.010

Ríos-Rull, J.-V., et al. (1997). Computation of equilibria in heterogeneous agent models (tech. rep.). Federal Reserve Bank of Minneapolis.

Rosenblatt, F. (1958). The perceptron: A probabilistic model for information storage and organization in the brain. Psychological Review, 65(6), 386–408.

Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning representations by back-propagating errors. Nature, 323(6088), 533–536.

Sutton, R. S., McAllester, D., Singh, S., & Mansour, Y. (1999). Policy gradient methods for reinforcement learning with function approximation. Advances in Neural Information Processing Systems, 12, 1057–1063.