American Institute of Mathematical Sciences

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January  2017, 13(1): 505-529. doi: 10.3934/jimo.2016029

Optimal consumption with reference-dependent preferences in on-the-job search and savings

Chi Zhou a,b, , Wansheng Tang a, and  Ruiqing Zhao a,

a. 

Institute of Systems Engineering, Tianjin University, Tianjin 300072, China
b. 

School of Management, Tianjin University of Technology, Tianjin 300384, China

Received  July 2014 Published  March 2016

This paper studies a model of on-the-job search and savings under reference-dependent preferences that implies loss aversion in a worker's consumption behaviors. The model analyzes how loss aversion affects the worker's consumption decisions in job search. The results demonstrate that the presence of loss aversion will lead to a set of high steady-state consumption levels and the range of steady-state consumption levels is wider if the worker is more loss averse. Nevertheless, we show that there is a unique steady-state consumption level, which is a lower bound of the set, in the absence of loss aversion. In addition, we also find that great loss aversion may reduce consumption level, while small loss aversion not only causes consumption to remain at a high level, but also induces that the worker's future consumption level goes down when the employment status changes.

Keywords: Stochastic dynamic programming, subdifferential, optimal consumption, reference-dependent preferences, loss aversion, job search.
Mathematics Subject Classification: Primary: 90C39, 91B06, 91B16; Secondary: 49L20, 90B40, 91B70.
Citation: Chi Zhou, Wansheng Tang, Ruiqing Zhao. Optimal consumption with reference-dependent preferences in on-the-job search and savings. Journal of Industrial & Management Optimization, 2017, 13 (1) : 505-529. doi: 10.3934/jimo.2016029
References:
[1]

M. Abdellaoui, H. Bleichrodt and C. Paraschiv, Loss aversion under prospect theory: A parameter-free measurement, Management Science, 53 (2007), 1659-1674.  doi: 10.1287/mnsc.1070.0711.  Google Scholar

[2]

S. R. Aiyagari, Uninsured idiosyncratic risk and aggregate saving, The Quarterly Journal of Economics, 109 (1994), 659-684.  doi: 10.2307/2118417.  Google Scholar

[3]

R. Alessie and A. Lusardi, Consumption, saving and habit formation, Economics Letters, 55 (1997), 103-108.  doi: 10.1016/S0165-1765(97)00061-X.  Google Scholar

[4]

Y. Algan, A. Chéron, J.-O. Hairault and F. Langot, Wealth effect on labor market transitions, Review of Economic Dynamics, 6 (2003), 156-178.  doi: 10.1016/S1094-2025(02)00013-3.  Google Scholar

[5]

V. Angelini, Consumption and habit formation when time horizon is finite, Economics Letters, 103 (2009), 113-116.  doi: 10.1016/j.econlet.2009.02.007.  Google Scholar

[6]

J. Apesteguia and M. A. Ballester, A theory of reference-dependent behavior, Economic Theory, 40 (2009), 427-455.  doi: 10.1007/s00199-008-0387-z.  Google Scholar

[7]

R. E. Bellman, Dynamic Programming, Princeton University Press, New Jersey, 1957.  Google Scholar

[8]

D. Bowman, D. Minehart and M. Rabin, Loss aversion in a consumption-savings model, Journal of Economic Behavior & Organization, 38 (1999), 155-178.  doi: 10.1016/S0167-2681(99)00004-9.  Google Scholar

[9]

M. Browning, T. F. Crossley and E. Smith, Asset accumulation and short-term employment, Review of Economic Dynamics, 10 (2007), 400-423.  doi: 10.1016/j.red.2006.12.002.  Google Scholar

[10]

R. Correa, A. Jofre and L. Thibault, Subdifferential characterization of convexity, in Recent Advances in Nonsmooth Optimization, World Scientific, River Edge, NJ, 1995, 18-23.  Google Scholar

[11]

E. G. De Giorgi and S. Legg, Dynamic portfolio choice and asset pricing with narrow framing and probability weighting, Journal of Economic Dynamics and Control, 36 (2012), 951-972.  doi: 10.1016/j.jedc.2012.01.010.  Google Scholar

[12]

E. G. De Giorgi and T. Post, Loss aversion with a state-dependent reference point, Management Science, 57 (2011), 1094-1110.   Google Scholar

[13]

R. A. Easterlin, Will raising the incomes of all increase the happiness of all?, Journal of Economic Behavior & Organization, 27 (1995), 35-47.  doi: 10.1016/0167-2681(95)00003-B.  Google Scholar

[14]

R. Foellmi, R. Rosenblatt-Wisch and K. R. Schenk-Hoppé, Consumption paths under prospect utility in an optimal growth model, Journal of Economic Dynamics and Control, 35 (2011), 273-281.  doi: 10.1016/j.jedc.2010.09.002.  Google Scholar

[15]

L. Grüne and W. Semmler, Asset pricing with loss aversion, Journal of Economic Dynamics and Control, 32 (2008), 3253-3274.  doi: 10.1016/j.jedc.2008.01.002.  Google Scholar

[16]

D. Kahneman, J. L. Knetsch and R. H. Thaler, Anomalies: The endowment effect, loss aversion, and status quo bias, The Journal of Economic Perspectives, 5 (1991), 193-206.  doi: 10.1257/jep.5.1.193.  Google Scholar

[17]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica: Journal of the Econometric Society, 47 (1979), 263-291.   Google Scholar

[18]

B. Kőszegi and M. Rabin, A model of reference-dependent preferences, Quarterly Journal of Economics, 121 (2006), 1133-1165.  doi: 10.1093/qje/121.4.1133.  Google Scholar

[19]

B. Kőszegi and M. Rabin, Reference-dependent risk attitudes, The American Economic Review, 97 (2007), 1047-1073.   Google Scholar

[20]

B. Kőszegi and M. Rabin, Reference-dependent consumption plans, The American Economic Review, 99 (2009), 909-936.   Google Scholar

[21]

M. Lammers, The effects of savings on reservation wages and search effort Labour Economics, 27, (2014), 83-98. doi: 10.1016/j.labeco.2014.03.001.  Google Scholar

[22]

R. Lentz and T. Tranaes, Job search and savings: Wealth effects and duration dependence, Journal of Labor Economics, 23 (2005), 467-489.  doi: 10.1086/430284.  Google Scholar

[23]

J. Lise, On-the-job search and precautionary savings, The Review of Economic Studies, 80 (2013), 1086-1113.  doi: 10.1093/restud/rds042.  Google Scholar

[24]

J. Z. Liu, K.-F. C. Yiu and K. L. Teo, Optimal investment-consumption problem with constraint, Journal of Industrial and Management Optimization, 9 (2013), 743-768.  doi: 10.3934/jimo.2013.9.743.  Google Scholar

[25]

I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Operations Research, 55 (2007), 413-429.  doi: 10.1287/opre.1070.0393.  Google Scholar

[26]

S. Rendon, Job search and asset accumulation under borrowing constraints, International Economic Review, 47 (2006), 233-263.  doi: 10.1111/j.1468-2354.2006.00378.x.  Google Scholar

[27]

R. Rosenblatt-Wisch, Loss aversion in aggregate macroeconomic time series, European Economic Review, 52 (2008), 1140-1159.  doi: 10.1016/j.euroecorev.2007.12.001.  Google Scholar

[28]

J. Ruan, P. Shi, C.-C. Lim and X. Wang, Relief supplies allocation and optimization by interval and fuzzy number approaches, Information Sciences, 303 (2015), 15-32.  doi: 10.1016/j.ins.2015.01.002.  Google Scholar

[29]

A. Siegmann, Optimal saving rules for loss-averse agents under uncertainty, Economics Letters, 77 (2002), 27-34.  doi: 10.1016/S0165-1765(02)00113-1.  Google Scholar

[30]

J. Spinnewijn, Unemployed but optimistic: Optimal insurance design with biased beliefs, Journal of the European Economic Association, 13 (2015), 130-167.  doi: 10.2139/ssrn.1291566.  Google Scholar

[31]

N. L. Stokey and R. E. Lucas Jr., Recursive Methods in Economic Dynamics, Harvard University Press, Cambridge, MA, 1989.  Google Scholar

[32]

L. Sun and L. Zhang, Optimal consumption and investment under irrational beliefs, Journal of Industrial and Management Optimization, 7 (2011), 139-156.  doi: 10.3934/jimo.2011.7.139.  Google Scholar

[33]

S.H.M. Ting, C.-O. Ewald and W.-K. Wang, On the investment-uncertainty relationship in a real option model with stochastic volatility, Mathematical Social Sciences, 66 (2013), 22-32.  doi: 10.1016/j.mathsocsci.2013.01.005.  Google Scholar

[34]

A. Tversky and D. Kahneman, Loss aversion in riskless choice: A reference-dependent model, The Quarterly Journal of Economics, 106 (1991), 1039-1061.  doi: 10.2307/2937956.  Google Scholar

[35]

A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5 (1992), 297-323.  doi: 10.1007/BF00122574.  Google Scholar

[36]

A. Zhang and C.-O. Ewald, Optimal investment for a pension fund under inflation risk, Mathematical Methods of Operations Research, 71 (2010), 353-369.  doi: 10.1007/s00186-009-0294-5.  Google Scholar

[37]

J. Zhang, Q. Gou, J. Zhang and L. Liang, Supply chain pricing decisions with price reduction during the selling season, International Journal of Production Research, 52 (2014), 165-187.  doi: 10.1080/00207543.2013.831189.  Google Scholar

[38]

C. Zhou, W. Tang and R. Zhao, An uncertain search model for recruitment problem with enterprise performance, Journal of Intelligent Manufacturing, (2014), 1-10.  doi: 10.1007/s10845-014-0997-1.  Google Scholar

[39]

C. Zhou, W. Tang and R. Zhao, Optimal stopping for dynamic recruitment problem with probabilistic loss of candidates, Sequential Analysis: Design Methods and Applications, 34 (2015), 187-210.  doi: 10.1080/07474946.2015.1030974.  Google Scholar

show all references

References:
[1]

M. Abdellaoui, H. Bleichrodt and C. Paraschiv, Loss aversion under prospect theory: A parameter-free measurement, Management Science, 53 (2007), 1659-1674.  doi: 10.1287/mnsc.1070.0711.  Google Scholar

[2]

S. R. Aiyagari, Uninsured idiosyncratic risk and aggregate saving, The Quarterly Journal of Economics, 109 (1994), 659-684.  doi: 10.2307/2118417.  Google Scholar

[3]

R. Alessie and A. Lusardi, Consumption, saving and habit formation, Economics Letters, 55 (1997), 103-108.  doi: 10.1016/S0165-1765(97)00061-X.  Google Scholar

[4]

Y. Algan, A. Chéron, J.-O. Hairault and F. Langot, Wealth effect on labor market transitions, Review of Economic Dynamics, 6 (2003), 156-178.  doi: 10.1016/S1094-2025(02)00013-3.  Google Scholar

[5]

V. Angelini, Consumption and habit formation when time horizon is finite, Economics Letters, 103 (2009), 113-116.  doi: 10.1016/j.econlet.2009.02.007.  Google Scholar

[6]

J. Apesteguia and M. A. Ballester, A theory of reference-dependent behavior, Economic Theory, 40 (2009), 427-455.  doi: 10.1007/s00199-008-0387-z.  Google Scholar

[7]

R. E. Bellman, Dynamic Programming, Princeton University Press, New Jersey, 1957.  Google Scholar

[8]

D. Bowman, D. Minehart and M. Rabin, Loss aversion in a consumption-savings model, Journal of Economic Behavior & Organization, 38 (1999), 155-178.  doi: 10.1016/S0167-2681(99)00004-9.  Google Scholar

[9]

M. Browning, T. F. Crossley and E. Smith, Asset accumulation and short-term employment, Review of Economic Dynamics, 10 (2007), 400-423.  doi: 10.1016/j.red.2006.12.002.  Google Scholar

[10]

R. Correa, A. Jofre and L. Thibault, Subdifferential characterization of convexity, in Recent Advances in Nonsmooth Optimization, World Scientific, River Edge, NJ, 1995, 18-23.  Google Scholar

[11]

E. G. De Giorgi and S. Legg, Dynamic portfolio choice and asset pricing with narrow framing and probability weighting, Journal of Economic Dynamics and Control, 36 (2012), 951-972.  doi: 10.1016/j.jedc.2012.01.010.  Google Scholar

[12]

E. G. De Giorgi and T. Post, Loss aversion with a state-dependent reference point, Management Science, 57 (2011), 1094-1110.   Google Scholar

[13]

R. A. Easterlin, Will raising the incomes of all increase the happiness of all?, Journal of Economic Behavior & Organization, 27 (1995), 35-47.  doi: 10.1016/0167-2681(95)00003-B.  Google Scholar

[14]

R. Foellmi, R. Rosenblatt-Wisch and K. R. Schenk-Hoppé, Consumption paths under prospect utility in an optimal growth model, Journal of Economic Dynamics and Control, 35 (2011), 273-281.  doi: 10.1016/j.jedc.2010.09.002.  Google Scholar

[15]

L. Grüne and W. Semmler, Asset pricing with loss aversion, Journal of Economic Dynamics and Control, 32 (2008), 3253-3274.  doi: 10.1016/j.jedc.2008.01.002.  Google Scholar

[16]

D. Kahneman, J. L. Knetsch and R. H. Thaler, Anomalies: The endowment effect, loss aversion, and status quo bias, The Journal of Economic Perspectives, 5 (1991), 193-206.  doi: 10.1257/jep.5.1.193.  Google Scholar

[17]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica: Journal of the Econometric Society, 47 (1979), 263-291.   Google Scholar

[18]

B. Kőszegi and M. Rabin, A model of reference-dependent preferences, Quarterly Journal of Economics, 121 (2006), 1133-1165.  doi: 10.1093/qje/121.4.1133.  Google Scholar

[19]

B. Kőszegi and M. Rabin, Reference-dependent risk attitudes, The American Economic Review, 97 (2007), 1047-1073.   Google Scholar

[20]

B. Kőszegi and M. Rabin, Reference-dependent consumption plans, The American Economic Review, 99 (2009), 909-936.   Google Scholar

[21]

M. Lammers, The effects of savings on reservation wages and search effort Labour Economics, 27, (2014), 83-98. doi: 10.1016/j.labeco.2014.03.001.  Google Scholar

[22]

R. Lentz and T. Tranaes, Job search and savings: Wealth effects and duration dependence, Journal of Labor Economics, 23 (2005), 467-489.  doi: 10.1086/430284.  Google Scholar

[23]

J. Lise, On-the-job search and precautionary savings, The Review of Economic Studies, 80 (2013), 1086-1113.  doi: 10.1093/restud/rds042.  Google Scholar

[24]

J. Z. Liu, K.-F. C. Yiu and K. L. Teo, Optimal investment-consumption problem with constraint, Journal of Industrial and Management Optimization, 9 (2013), 743-768.  doi: 10.3934/jimo.2013.9.743.  Google Scholar

[25]

I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Operations Research, 55 (2007), 413-429.  doi: 10.1287/opre.1070.0393.  Google Scholar

[26]

S. Rendon, Job search and asset accumulation under borrowing constraints, International Economic Review, 47 (2006), 233-263.  doi: 10.1111/j.1468-2354.2006.00378.x.  Google Scholar

[27]

R. Rosenblatt-Wisch, Loss aversion in aggregate macroeconomic time series, European Economic Review, 52 (2008), 1140-1159.  doi: 10.1016/j.euroecorev.2007.12.001.  Google Scholar

[28]

J. Ruan, P. Shi, C.-C. Lim and X. Wang, Relief supplies allocation and optimization by interval and fuzzy number approaches, Information Sciences, 303 (2015), 15-32.  doi: 10.1016/j.ins.2015.01.002.  Google Scholar

[29]

A. Siegmann, Optimal saving rules for loss-averse agents under uncertainty, Economics Letters, 77 (2002), 27-34.  doi: 10.1016/S0165-1765(02)00113-1.  Google Scholar

[30]

J. Spinnewijn, Unemployed but optimistic: Optimal insurance design with biased beliefs, Journal of the European Economic Association, 13 (2015), 130-167.  doi: 10.2139/ssrn.1291566.  Google Scholar

[31]

N. L. Stokey and R. E. Lucas Jr., Recursive Methods in Economic Dynamics, Harvard University Press, Cambridge, MA, 1989.  Google Scholar

[32]

L. Sun and L. Zhang, Optimal consumption and investment under irrational beliefs, Journal of Industrial and Management Optimization, 7 (2011), 139-156.  doi: 10.3934/jimo.2011.7.139.  Google Scholar

[33]

S.H.M. Ting, C.-O. Ewald and W.-K. Wang, On the investment-uncertainty relationship in a real option model with stochastic volatility, Mathematical Social Sciences, 66 (2013), 22-32.  doi: 10.1016/j.mathsocsci.2013.01.005.  Google Scholar

[34]

A. Tversky and D. Kahneman, Loss aversion in riskless choice: A reference-dependent model, The Quarterly Journal of Economics, 106 (1991), 1039-1061.  doi: 10.2307/2937956.  Google Scholar

[35]

A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5 (1992), 297-323.  doi: 10.1007/BF00122574.  Google Scholar

[36]

A. Zhang and C.-O. Ewald, Optimal investment for a pension fund under inflation risk, Mathematical Methods of Operations Research, 71 (2010), 353-369.  doi: 10.1007/s00186-009-0294-5.  Google Scholar

[37]

J. Zhang, Q. Gou, J. Zhang and L. Liang, Supply chain pricing decisions with price reduction during the selling season, International Journal of Production Research, 52 (2014), 165-187.  doi: 10.1080/00207543.2013.831189.  Google Scholar

[38]

C. Zhou, W. Tang and R. Zhao, An uncertain search model for recruitment problem with enterprise performance, Journal of Intelligent Manufacturing, (2014), 1-10.  doi: 10.1007/s10845-014-0997-1.  Google Scholar

[39]

C. Zhou, W. Tang and R. Zhao, Optimal stopping for dynamic recruitment problem with probabilistic loss of candidates, Sequential Analysis: Design Methods and Applications, 34 (2015), 187-210.  doi: 10.1080/07474946.2015.1030974.  Google Scholar

Figure 1.  Steady-state consumption levels under loss-neutral
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Figure 2.  An example of kinked gain-loss utility function $v(z)$
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Figure 3.  Steady-state consumption levels under loss-neutral and loss aversion
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Figure 4.  The range of steady-state consumption levels under loss aversion
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