# American Institute of Mathematical Sciences

August  2020, 13(8): 2327-2346. doi: 10.3934/dcdss.2020093

## Explicit investment setting in a Kaldor macroeconomic model with macro shock

 1 School of Mathematics, South China University of Technology, Guangzhou, 510640, China 2 Guangzhou International Institute of Finance and Guangzhou University, Guangzhou, 510405, China

* Corresponding author: Shuanglian Chen

Received  July 2018 Revised  December 2018 Published  August 2020 Early access  June 2019

Fund Project: The work is supported by the National Natural Science Foundation (No.11701115) and Postdoctoral Science Foundation (No.2017610515)

As the inevitable attributes of macro shocks on macroeconomic system, in this paper, we develop a Kaldor macroeconomic model with shock. The shock is due to the investment uncertainty. We then provide an approach for macroeconomic control by calibrating the evolvement of the shocked Kaldor macroeconomic model with some expected benchmark process. The calibration is realized through the setting for investment. The benchmark process is usually the reflection of decisions or policies. An optimal investment setting associated with a five-dimensional nonlinear system of ordinary differential equations is presented. Through a logical modification for the boundary conditions, the nonlinear system is simplified to be linear and a completely explicit formula for the optimal investment setting is achieved. The rationality of the modification is supported by some stability condition. To cope with the systematic risk caused by the macro shock, we define a dynamic Value-at-Risk(VaR) as the risk measure capturing the risk level of the shocked Kaldor macroeconomic model and introduce a risk constraint into the programming of calibration. Then a constrained investment setting is presented. Finally, we carry out an application of the theoretical results by calibrating the evolvement of the shocked Kaldor macroeconomic model with the business cycle generated from the classical Kaldor model through the investment setting.

Citation: Zhenzhen Wang, Zhenghui Li, Shuanglian Chen, Zhehao Huang. Explicit investment setting in a Kaldor macroeconomic model with macro shock. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2327-2346. doi: 10.3934/dcdss.2020093
##### References:
 [1] I. Bashkirtseva, L. Ryashko and T. Ryazanova, Stochastic sensitivity analysis of the variability of dynamics and transition to chaos in the business cycles model, Comm. Non. Sci. Num. Simu., 54 (2018), 174-184.  doi: 10.1016/j.cnsns.2017.05.030. [2] C. Bernard, L. Rüschendorf, S. Vanduffel and J. Yao, How robust is the value-at-risk of credit risk portfolios?, Euro. J. Fina., 23 (2017), 507-534. [3] G. I. Bischi, R. Dieci, G. Rodano and E. Saltari, Multiple attractors and global bifurcations in a Kaldor-type business cycle model, J. Evo. Econ., 11 (2001), 527-554.  doi: 10.1007/s191-001-8320-9. [4] M. K. Brunnermeier and Y. Sannikov, A macroeconomic model with a financial sector, Amer. Econ. Rev., 104 (2014), 379-421. [5] W. W. Chang and D. J. Smyth, The existence and persistence of cycles in a non-linear model: Kaldor's 1940 model re-examined, Rev. Econ. Stud., 38 (1971), 37-44.  doi: 10.2307/2296620. [6] S. Chen, Z. Li and K. Li, Optimal investment-reinsurance policy for an insurance company with VaR constraint, Insur. Math. Econ., 47 (2010), 144-153.  doi: 10.1016/j.insmatheco.2010.06.002. [7] A. A. Gaivoronski, S. Krylov and N. Van der Wijst, Optimal portfolio selection and dynamic benc-hmark tracking, Euro. J. opera. res., 163 (2005), 115-131.  doi: 10.1016/j.ejor.2003.12.001. [8] J. Greenwood, Z. Hercowitz and G. W. Huffman, Investment, capacity utilization, and the real business cycle, Amer. Econ. Rev., (1998), 402–417. [9] W. Hu, H. Zhao and T. Dong, Dynamic analysis for a kaldor-kalecki model of business cycle with time delay and diffusion effect, Complexity, 2018 (2018), Art. ID 1263602, 11 pp. doi: 10.1155/2018/1263602. [10] N. Kaldor, A model of the trade cycle, Econ. J., 50 (1940), 78-92.  doi: 10.2307/2225740. [11] M. Kalecki, A macrodynamic theory of business cycles, Econometrica. J. Econ. Soc., 3 (1935), 327-344.  doi: 10.2307/1905325. [12] M. Kalecki, Theory of Economic Dynamics, Routledge, 2013. doi: 10.4324/9780203708668. [13] N. Klimenko, S. Pfeil and J. C. Rochet, A simple macroeconomic model with extreme financial frictions, J. Math. Econ., 68 (2017), 92-102.  doi: 10.1016/j.jmateco.2016.04.002. [14] A. Krawiec and M. Szydłowski, The Kaldor-Kalecki business cycle model, Ann. Opera. Res., 89 (1999), 89-100.  doi: 10.1023/A:1018948328487. [15] A. Krawiec and M. Szydłowski, Economic growth cycles driven by investment delay, Econ. Model., 67 (2017), 175-183.  doi: 10.1016/j.econmod.2016.11.014. [16] Z. Li, Z. Wang and Z. Huang, Modeling business cycle with financial shocks basing on kaldor-kalecki model, Quan. Finan. Econ., 1 (2017), 44-66.  doi: 10.3934/QFE.2017.1.44. [17] Y. Liu and D. A. Ralescu, Value-at-risk in uncertain random risk analysis, Info. Sci., 391 (2017), 1-8.  doi: 10.1016/j.ins.2017.01.034. [18] G. Mircea, M. Neamtu and D. Opris, The Kaldor-Kalecki stochastic model of business cycle, Non. Anal. Model. Cont., 16 (2011), 191-205. [19] C. Nunes and R. Pimentel, Analytical solution for an investment problem under uncertainties with shocks, Euro. J. Opera. Res., 259 (2017), 1054-1063.  doi: 10.1016/j.ejor.2017.01.008. [20] G. Phelan, Financial intermediation, leverage, and macroeconomic instability, Amer. Econ. J. Macro., 8 (2016), 199-224.  doi: 10.1257/mac.20140233. [21] N. Zhang, Z. Jin, S. Li and P. Chen, Optimal reinsurance under dynamic VaR constraint, Insu. Math. Econ., 71 (2016), 232-243.  doi: 10.1016/j.insmatheco.2016.09.011. [22] Q. Zhang and Y. Gao, Portfolio selection based on a benchmark process with dynamic value-at-risk constraints, J. Comp. Appl. Math., 313 (2017), 440-447.  doi: 10.1016/j.cam.2016.10.001.

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##### References:
 [1] I. Bashkirtseva, L. Ryashko and T. Ryazanova, Stochastic sensitivity analysis of the variability of dynamics and transition to chaos in the business cycles model, Comm. Non. Sci. Num. Simu., 54 (2018), 174-184.  doi: 10.1016/j.cnsns.2017.05.030. [2] C. Bernard, L. Rüschendorf, S. Vanduffel and J. Yao, How robust is the value-at-risk of credit risk portfolios?, Euro. J. Fina., 23 (2017), 507-534. [3] G. I. Bischi, R. Dieci, G. Rodano and E. Saltari, Multiple attractors and global bifurcations in a Kaldor-type business cycle model, J. Evo. Econ., 11 (2001), 527-554.  doi: 10.1007/s191-001-8320-9. [4] M. K. Brunnermeier and Y. Sannikov, A macroeconomic model with a financial sector, Amer. Econ. Rev., 104 (2014), 379-421. [5] W. W. Chang and D. J. Smyth, The existence and persistence of cycles in a non-linear model: Kaldor's 1940 model re-examined, Rev. Econ. Stud., 38 (1971), 37-44.  doi: 10.2307/2296620. [6] S. Chen, Z. Li and K. Li, Optimal investment-reinsurance policy for an insurance company with VaR constraint, Insur. Math. Econ., 47 (2010), 144-153.  doi: 10.1016/j.insmatheco.2010.06.002. [7] A. A. Gaivoronski, S. Krylov and N. Van der Wijst, Optimal portfolio selection and dynamic benc-hmark tracking, Euro. J. opera. res., 163 (2005), 115-131.  doi: 10.1016/j.ejor.2003.12.001. [8] J. Greenwood, Z. Hercowitz and G. W. Huffman, Investment, capacity utilization, and the real business cycle, Amer. Econ. Rev., (1998), 402–417. [9] W. Hu, H. Zhao and T. Dong, Dynamic analysis for a kaldor-kalecki model of business cycle with time delay and diffusion effect, Complexity, 2018 (2018), Art. ID 1263602, 11 pp. doi: 10.1155/2018/1263602. [10] N. Kaldor, A model of the trade cycle, Econ. J., 50 (1940), 78-92.  doi: 10.2307/2225740. [11] M. Kalecki, A macrodynamic theory of business cycles, Econometrica. J. Econ. Soc., 3 (1935), 327-344.  doi: 10.2307/1905325. [12] M. Kalecki, Theory of Economic Dynamics, Routledge, 2013. doi: 10.4324/9780203708668. [13] N. Klimenko, S. Pfeil and J. C. Rochet, A simple macroeconomic model with extreme financial frictions, J. Math. Econ., 68 (2017), 92-102.  doi: 10.1016/j.jmateco.2016.04.002. [14] A. Krawiec and M. Szydłowski, The Kaldor-Kalecki business cycle model, Ann. Opera. Res., 89 (1999), 89-100.  doi: 10.1023/A:1018948328487. [15] A. Krawiec and M. Szydłowski, Economic growth cycles driven by investment delay, Econ. Model., 67 (2017), 175-183.  doi: 10.1016/j.econmod.2016.11.014. [16] Z. Li, Z. Wang and Z. Huang, Modeling business cycle with financial shocks basing on kaldor-kalecki model, Quan. Finan. Econ., 1 (2017), 44-66.  doi: 10.3934/QFE.2017.1.44. [17] Y. Liu and D. A. Ralescu, Value-at-risk in uncertain random risk analysis, Info. Sci., 391 (2017), 1-8.  doi: 10.1016/j.ins.2017.01.034. [18] G. Mircea, M. Neamtu and D. Opris, The Kaldor-Kalecki stochastic model of business cycle, Non. Anal. Model. Cont., 16 (2011), 191-205. [19] C. Nunes and R. Pimentel, Analytical solution for an investment problem under uncertainties with shocks, Euro. J. Opera. Res., 259 (2017), 1054-1063.  doi: 10.1016/j.ejor.2017.01.008. [20] G. Phelan, Financial intermediation, leverage, and macroeconomic instability, Amer. Econ. J. Macro., 8 (2016), 199-224.  doi: 10.1257/mac.20140233. [21] N. Zhang, Z. Jin, S. Li and P. Chen, Optimal reinsurance under dynamic VaR constraint, Insu. Math. Econ., 71 (2016), 232-243.  doi: 10.1016/j.insmatheco.2016.09.011. [22] Q. Zhang and Y. Gao, Portfolio selection based on a benchmark process with dynamic value-at-risk constraints, J. Comp. Appl. Math., 313 (2017), 440-447.  doi: 10.1016/j.cam.2016.10.001.
Evolvement of the gross production $Y$. The blue line depicts the evolvement in the shocked Kaldor model without calibration and the red line depicts the cycle fluctuation. The intensity of shock is set as $\epsilon = 0.5$
Evolvement of the gross production $Y$. The blue line depicts the evolvement in the shocked Kaldor model without calibration and the red line depicts the cycle fluctuation. The intensity of shock is set as $\epsilon = 2$
Evolvement of the gross production $Y$. The blue line depicts the calibrated evolvement in the shocked Kaldor model and the red line depicts the cycle fluctuation. The intensity of shock is set as $\epsilon = 0.5$
Evolvement of the gross production $Y$. The blue line depicts the calibrated evolvement in the shocked Kaldor model and the red line depicts the cycle fluctuation. The intensity of shock is set as $\epsilon = 2$
Evolvement of the gross production $Y$. The green line depicts the calibrated evolvement in the shocked Kaldor model with risk constraint and the blue line depicts the one without risk constraint. The intensity of shock is set as $\epsilon = 0.5$
Evolvement of the gross production $Y$. The green line depicts the calibrated evolvement in the shocked Kaldor model with risk constraint and the blue line depicts the one without risk constraint. The intensity of shock is set as $\epsilon = 2$
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