Central European Journal of Operations Research

, Volume 21, Issue 3, pp 507–522 | Cite as

Multiple equilibria and indifference-threshold points in a rational addiction model

  • Jonathan P. Caulkins
  • Gustav FeichtingerEmail author
  • Richard F. Hartl
  • Peter M. Kort
  • Andreas J. Novak
  • Andrea Seidl
Original Paper


Becker and Murphy (J Polit Econ 96(4):675–700, 1988) have established the existence of unstable steady states leading to threshold behavior for optimal consumption rates in intertemporal rational addiction models. In the present paper a simple linear-quadratic optimal control model is used to illustrate how their approach fits into the framework of multiple equilibria and indifference-threshold points. By changing the degree of addiction and the level of harmfulness we obtain a variety of behavioral patterns. In particular we show that when the good is harmful as well as very addictive, an indifference-threshold point, also known in the literature as a Skiba point, separates patterns converging to either zero or maximal consumption, where the latter occurs in the case of a high level of past consumption. This implicitly shows that an individual needs to be aware in time of these characteristics of the good. Otherwise, he/she may start consuming so much that in the end he/she is totally addicted.


Optimal control Indifference points History-dependence Rational addiction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Auld M, Grootendorst P (2004) An empirical analysis of milk addiction. J Health Econ 23(6): 1117–1133CrossRefGoogle Scholar
  2. Becker GS (1992) Habits, addictions, and traditions. Kyklos 45(3): 327–346CrossRefGoogle Scholar
  3. Becker GS, Murphy KM (1988) A theory of rational addiction. J Polit Econ 96(4): 675–700CrossRefGoogle Scholar
  4. Braun N, Vanini P (2003) On habits and addictions. J Inst Theor Econ 159: 603–626CrossRefGoogle Scholar
  5. Brock WA (1983) Pricing, predation and entry barriers in regulated industries. In: Evans DS (ed) Breaking up bell. North-Holland, New York, pp 191–229Google Scholar
  6. Brock WA, Dechert WD (1985) Dynamic Ramsey pricing. Int Econ Rev 26(3): 569–591CrossRefGoogle Scholar
  7. Brock WA, Malliaris AG (1989) Differential equations, stability and chaos in dynamic economics. North-Holland, AmsterdamGoogle Scholar
  8. Brock WA, Starrett D (2003) Nonconvexities in ecological managment problems. Environ Resour Econ 26(4): 575–624CrossRefGoogle Scholar
  9. Bultmann R, Feichtinger G, Tragler G (2010) Stochastic skiba sets: an example from models of illicit drug consumption. In: Lirkov I, Margenov S, Wasniewski J (eds) Large-scale scientific computing. Springer, Heidelberg, pp 239–246CrossRefGoogle Scholar
  10. Cass D. (1965) Optimum growth in an aggregative model of capital accumulation. Rev Econ Stud 32(3): 233–240CrossRefGoogle Scholar
  11. Caulkins JP, Feichtinger G, Grass D, Hartl RF, Kort PM, Seidl A (2010) Skiba points in free end time problems: the option to sell the firm (in submission)Google Scholar
  12. Clark C (1971) Economically optimal policies for the utilization of biologically renewable resources. Math Biosci 12(3–4): 245–260CrossRefGoogle Scholar
  13. Clark CW (1976) Mathematical bioeconomics, the optimal management of renewable resources. Wiley- Interscience, New YorkGoogle Scholar
  14. Dechert WD (1983) Increasing returns to scale and the reverse flexible accelerator. Econ Lett 13(1): 69–75CrossRefGoogle Scholar
  15. Dechert WD, Nishimura K (1983) A complete characterization of optimal growth paths in an aggregated model with a non-concave production function. J Econ Theory 31(2): 332–354CrossRefGoogle Scholar
  16. Deissenberg C, Feichtinger G, Semmler W, Wirl F (2004) Multiple equilibria, history dependence, and global dynamics in intertemporal optimization models. In: Barnett WA, Deissenberg C, Feichtinger G (eds) Economic complexity: non-linear dynamics, multi-agents economies and learning. Elsevier, Amsterdam, pp 91–122Google Scholar
  17. Dockner EJ, Feichtinger G (1993) Cyclical consumption patterns and rational addiction. Am Econ Rev 83(1): 256–263Google Scholar
  18. Feichtinger G, Steindl A (2006) DNS curves in a production/inventory model. J Optim Theory Appl 128(2): 295–308CrossRefGoogle Scholar
  19. Forster BA (1975) Optimal pollution control with a nonconstant exponential rate of decay. J Environ Econ Manag 2: 1–6CrossRefGoogle Scholar
  20. Gavrila C, Feichtinger G, Tragler G, Hartl RF, Kort PM (2005) History-dependence in a rational addiction model. Math Soc Sci 49(3): 273–293CrossRefGoogle Scholar
  21. Grass D, Caulkins JP, Feichtinger G, Tragler G, Behrens DA (2008) Optimal control of nonlinear processes: with applications in drugs, corruption and terror. Springer, HeidelbergCrossRefGoogle Scholar
  22. Hartl RF, Kort PM, Feichtinger G, Wirl F (2004) Multiple equilibria and thresholds due to relative investment costs. J Optim Theory Appl 123(1): 49–82CrossRefGoogle Scholar
  23. Iannaccone LR (1986) Addiction and satiation. Econ Lett 21(1): 95–99CrossRefGoogle Scholar
  24. Koopmans TC (1965) On the concept of optimal economic growth. Pontificiae Academiae Scientiarum Scripta Varia 28(1): 225–300Google Scholar
  25. Léonard D (1989) Market behaviour of rational addicts. J Econ Psychol 10(1): 117–144CrossRefGoogle Scholar
  26. Lewis TR, Schmalensee R (1982) Optimal use of renewable resources with nonconvexities in production. In: Mirman LJ, Spulber DF (eds) Essays in the economics of renewable resources. North-Holland, Amsterdam, pp 95–111Google Scholar
  27. Lucas RE (1988) On the mechanics of economic development. J Monet Econ 22(1): 3–42CrossRefGoogle Scholar
  28. Mäler KG (2000) Development, ecological resources and their management: a study of complex dynamic systems. Eur Econ Rev 44(4–6): 645–665CrossRefGoogle Scholar
  29. Melberg HO, Rogeberg OJ (2010) Rational addiction theory: a survey of opinions. J Drug Policy Anal 3(1): 5Google Scholar
  30. Orphanides A, Zervos D (1994) Optimal consumption dynamics with non-concave habit-forming utility. Econ Lett 44(1–2): 67–72CrossRefGoogle Scholar
  31. Orphanides A, Zervos D (1995) Rational addiction with learning and regret. J Polit Econ 103(4): 739–758CrossRefGoogle Scholar
  32. Orphanides A, Zervos D (1998) Myopia and addictive behaviour. Econ J 108(446): 75–91CrossRefGoogle Scholar
  33. Ryder HE, Heal GM (1973) Optimal growth with intertemporally dependent preferences. Rev Econ Stud 40: 1–33CrossRefGoogle Scholar
  34. Sethi SP (1977) Nearest feasible paths in optimal control problems: theory, examples, and counterexamples. J Optim Theory Appl 23(4): 563–579CrossRefGoogle Scholar
  35. Sethi SP (1979) Optimal advertising policy with the contagion model. J Optim Theory Appl 29(4): 615–627CrossRefGoogle Scholar
  36. Skiba AK (1978) Optimal growth with a convex-concave production function. Econometrica 46(3): 527–539CrossRefGoogle Scholar
  37. Stigler GJ, Becker GS (1977) De gustibus non est disputandum. Am Econ Rev 67(2): 76–90Google Scholar
  38. Wagener FOO (2003) Skiba points and heteroclinic bifurcations, with applications to the shallow lake system. J Econ Dyn Control 27(9): 1533–1561CrossRefGoogle Scholar
  39. Wirl F, Feichtinger G (2005) History dependence in concave economies. J Econ Behav Organ 57(4): 390–407CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Jonathan P. Caulkins
    • 1
  • Gustav Feichtinger
    • 2
    Email author
  • Richard F. Hartl
    • 3
  • Peter M. Kort
    • 4
    • 5
  • Andreas J. Novak
    • 3
  • Andrea Seidl
    • 2
  1. 1.Carnegie Mellon University, H. John Heinz III CollegePittsburghUSA
  2. 2.Department for Operations Research and Control SystemsInstitute for Mathematical Methods in Economics, Vienna University of TechnologyViennaAustria
  3. 3.Department of Business AdministrationUniversity of ViennaViennaAustria
  4. 4.Department of Econometrics and Operations Research and CenterTilburg UniversityTilburgThe Netherlands
  5. 5.Department of EconomicsUniversity of AntwerpAntwerpBelgium

Personalised recommendations