{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T08:27:45Z","timestamp":1765960065558,"version":"3.37.3"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,12,7]],"date-time":"2019-12-07T00:00:00Z","timestamp":1575676800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,12,7]],"date-time":"2019-12-07T00:00:00Z","timestamp":1575676800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P 30665-G27"],"award-info":[{"award-number":["P 30665-G27"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2020,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The paper presents a transformation of a multi-stage optimal control model with random switching time to an age-structured optimal control model. Following the mathematical transformation, the advantages of the present approach, as compared to a standard backward approach, are discussed. They relate in particular to a compact and unified representation of the two stages of the model: the applicability of well-known numerical solution methods and the illustration of state and control dynamics. The paper closes with a simple example on a macroeconomic shock, illustrating the workings and advantages of the approach.\n<\/jats:p>","DOI":"10.1007\/s10957-019-01598-5","type":"journal-article","created":{"date-parts":[[2019,12,7]],"date-time":"2019-12-07T06:02:28Z","timestamp":1575698548000},"page":"1065-1082","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Using Age Structure for a Multi-stage Optimal Control Model with Random Switching Time"],"prefix":"10.1007","volume":"184","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3891-5471","authenticated-orcid":false,"given":"Stefan","family":"Wrzaczek","sequence":"first","affiliation":[]},{"given":"Michael","family":"Kuhn","sequence":"additional","affiliation":[]},{"given":"Ivan","family":"Frankovic","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,12,7]]},"reference":[{"key":"1598_CR1","doi-asserted-by":"crossref","unstructured":"Wrzaczek, S., Kuhn, M., Frankovic, I.: Using age-structure for a multi-stage optimal control model with random switching time. TU Wien, Econ Working paper 06\/2019 (2019)","DOI":"10.1007\/s10957-019-01598-5"},{"key":"1598_CR2","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1007\/s00148-006-0068-9","volume":"20","author":"A Laporte","year":"2007","unstructured":"Laporte, A., Ferguson, B.S.: Investment in health when health is stochastic. J. Popul. Econ. 20, 423\u2013444 (2007)","journal-title":"J. Popul. Econ."},{"issue":"3","key":"1598_CR3","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1257\/mac.20160343","volume":"10","author":"J Laitner","year":"2018","unstructured":"Laitner, J., Silverman, D., Stolyarov, D.: The role of annuitized wealth in post-retirement behavior. Am. Econ. J. Macroecon. 10(3), 71\u2013117 (2018)","journal-title":"Am. Econ. J. Macroecon."},{"issue":"5","key":"1598_CR4","doi-asserted-by":"publisher","first-page":"1577","DOI":"10.1093\/jeea\/jvx036","volume":"16","author":"F van der Ploeg","year":"2018","unstructured":"van der Ploeg, F., de Zeeuw, A.: Climate tipping and economic growth: precautionary capital and the price of carbon. J. Eur. Econ. Assoc. 16(5), 1577\u20131617 (2018)","journal-title":"J. Eur. Econ. Assoc."},{"key":"1598_CR5","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s10640-018-0231-2","volume":"72","author":"F van der Ploeg","year":"2019","unstructured":"van der Ploeg, F., de Zeeuw, A.: Pricing carbon and adjusting capital to fend off climate catastrophes. Environ. Resour. Econ. 72, 29\u201350 (2019)","journal-title":"Environ. Resour. Econ."},{"key":"1598_CR6","doi-asserted-by":"crossref","unstructured":"Kuhn, M., Wrzaczek, S.: Rationally risking addiction: a two-stage approach. Mimeo (2019)","DOI":"10.1007\/978-3-030-54576-5_4"},{"issue":"6","key":"1598_CR7","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1080\/10556780310001639753","volume":"18","author":"VM Veliov","year":"2003","unstructured":"Veliov, V.M.: Newton\u2019s method for problems of optimal control of heterogeneous systems. Optim. Methods Softw. 18(6), 689\u2013703 (2003)","journal-title":"Optim. Methods Softw."},{"key":"1598_CR8","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1016\/j.jmaa.2008.05.012","volume":"346","author":"VM Veliov","year":"2008","unstructured":"Veliov, V.M.: Optimal control of heterogeneous systems: basic theory. J. Math. Anal. Appl. 346, 227\u2013242 (2008)","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"1598_CR9","doi-asserted-by":"publisher","first-page":"137","DOI":"10.2307\/2296058","volume":"32","author":"ME Yaari","year":"1965","unstructured":"Yaari, M.E.: Uncertain lifetime, life insurance, and the theory of the consumer. Rev. Econ. Stud. 32(2), 137\u2013150 (1965)","journal-title":"Rev. Econ. Stud."},{"key":"1598_CR10","doi-asserted-by":"publisher","first-page":"B495","DOI":"10.1287\/mnsc.17.8.B495","volume":"17","author":"MI Kamien","year":"1971","unstructured":"Kamien, M.I., Schwartz, N.L.: Optimal maintenance and sale age for a machine subject to failure. Manag. Sci. 17, B495\u2013B504 (1971)","journal-title":"Manag. Sci."},{"issue":"3","key":"1598_CR11","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1007\/BF00939419","volume":"64","author":"EK Boukas","year":"1990","unstructured":"Boukas, E.K., Haurie, A., Michel, P.: An optimal control problem with a random stopping time. J. Optim. Theory Appl. 64(3), 471\u2013480 (1990)","journal-title":"J. Optim. Theory Appl."},{"key":"1598_CR12","first-page":"633","volume-title":"Analysis and Optimization of Systems","author":"EK Boukas","year":"1988","unstructured":"Boukas, E.K., Haurie, A.: Optimality conditions for continuous-time systems with controlled jump Markov disturbances: application to an FMS planning problem. In: Bensoussan, A., Lions, J.L. (eds.) Analysis and Optimization of Systems, pp. 633\u2013676. Springer, Berlin (1988)"},{"issue":"3","key":"1598_CR13","doi-asserted-by":"publisher","first-page":"607","DOI":"10.1007\/BF00941305","volume":"70","author":"G Sorger","year":"1991","unstructured":"Sorger, G.: Maximum principle for control problems with uncertain horizon and variable discount rate. J. Optim. Theory Appl. 70(3), 607\u2013618 (1991)","journal-title":"J. Optim. Theory Appl."},{"key":"1598_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-76755-5","volume-title":"Infinite Horizon Optimal Control: Deterministic and Stochastic Systems","author":"DA Carlson","year":"1991","unstructured":"Carlson, D.A., Haurie, A.B., Leizarowitz, A.: Infinite Horizon Optimal Control: Deterministic and Stochastic Systems. Springer, Berlin (1991)"},{"issue":"3","key":"1598_CR15","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1016\/0165-1889(85)90010-7","volume":"9","author":"K Tomiyama","year":"1985","unstructured":"Tomiyama, K.: Two-stage optimal control problems and optimality conditions. J. Econ. Dyn. Control 9(3), 317\u2013337 (1985)","journal-title":"J. Econ. Dyn. Control"},{"issue":"3","key":"1598_CR16","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/0165-1889(89)90027-4","volume":"13","author":"K Tomiyama","year":"1989","unstructured":"Tomiyama, K., Rossana, R.J.: Two-stage optimal control problems with an explicit switch point dependence: optimality criteria and an example of delivery lags and investment. J. Econ. Dyn. Control 13(3), 319\u2013337 (1989)","journal-title":"J. Econ. Dyn. Control"},{"issue":"12","key":"1598_CR17","doi-asserted-by":"publisher","first-page":"1935","DOI":"10.1016\/S0165-1889(00)00009-9","volume":"25","author":"M Makris","year":"2001","unstructured":"Makris, M.: Necessary conditions for infinite-horizon discounted two-stage optimal control problems. J. Econ. Dyn. Control 25(12), 1935\u20131950 (2001)","journal-title":"J. Econ. Dyn. Control"},{"key":"1598_CR18","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1007\/BF00250793","volume":"54","author":"ME Gurtin","year":"1974","unstructured":"Gurtin, M.E., MacCamy, R.C.: Nonlinear age-dependent population dynamics. Arch. Ration. Mech. Anal. 54, 281\u2013300 (1974)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"1598_CR19","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1016\/0025-5564(81)90015-8","volume":"55","author":"ME Gurtin","year":"1981","unstructured":"Gurtin, M.E., Murphy, L.F.: On the optimal harvesting of age-structured populations: some simple models. J. Math. Biosci. 55, 115\u2013136 (1981a)","journal-title":"J. Math. Biosci."},{"key":"1598_CR20","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/BF00275209","volume":"13","author":"ME Gurtin","year":"1981","unstructured":"Gurtin, M.E., Murphy, L.F.: On the optimal harvesting of persistent age-structured populations. J. Math. Biol. 13, 131\u2013148 (1981b)","journal-title":"J. Math. Biol."},{"key":"1598_CR21","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/BF00276559","volume":"23","author":"M Brokate","year":"1985","unstructured":"Brokate, M.: Pontryagin\u2019s principle for control problems in age-dependent population dynamics. J. Math. Biol. 23, 75\u2013101 (1985)","journal-title":"J. Math. Biol."},{"key":"1598_CR22","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1016\/j.jmaa.2003.07.001","volume":"288","author":"G Feichtinger","year":"2003","unstructured":"Feichtinger, G., Tragler, G., Veliov, V.M.: Optimality conditions for age-structured control systems. J. Math. Anal. Appl. 288, 47\u201368 (2003)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"1598_CR23","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1007\/s10957-014-0680-x","volume":"167","author":"B Skritek","year":"2015","unstructured":"Skritek, B., Veliov, V.M.: On the infinite-horizon optimal control of age-structured systems. J. Optim. Theory Appl. 167(1), 243\u2013271 (2015)","journal-title":"J. Optim. Theory Appl."},{"key":"1598_CR24","first-page":"1094","volume":"11","author":"V Krastev","year":"2013","unstructured":"Krastev, V.: Arrow-type sufficient conditions for optimality of age-structured control problems. Cent. Eur. J. Math. 11, 1094\u20131111 (2013)","journal-title":"Cent. Eur. J. Math."},{"key":"1598_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-77647-5","volume-title":"Optimal Control of Nonlinear Processes: With Applications in Drugs. Corruption and Terror","author":"D Grass","year":"2008","unstructured":"Grass, D., Caulkins, J.P., Feichtinger, G., Tragler, G., Behrens, D.: Optimal Control of Nonlinear Processes: With Applications in Drugs. Corruption and Terror. Springer, Berlin (2008)"},{"key":"1598_CR26","doi-asserted-by":"publisher","first-page":"2005","DOI":"10.1126\/science.1081056","volume":"299","author":"RB Alley","year":"2003","unstructured":"Alley, R.B., Marotzke, J., Nordhaus, W.D., Overpeck, J.T., Peteet, D.M.: Abrupt Climate Change. Science 299, 2005\u20132010 (2003)","journal-title":"Science"},{"issue":"5","key":"1598_CR27","doi-asserted-by":"publisher","first-page":"544","DOI":"10.1257\/aer.104.5.544","volume":"104","author":"ML Weitzman","year":"2014","unstructured":"Weitzman, M.L.: Fat Tails and the Social Cost of Carbon. Am. Econ. Rev. Pap. Proc. 104(5), 544\u2013546 (2014)","journal-title":"Am. Econ. Rev. Pap. Proc."},{"key":"1598_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1111\/ecca.12117","volume":"82","author":"RJ Barro","year":"2015","unstructured":"Barro, R.J.: Environmental protection, rare disasters and discount rates. Economica 82, 1\u201323 (2015)","journal-title":"Economica"},{"issue":"10","key":"1598_CR29","doi-asserted-by":"publisher","first-page":"2947","DOI":"10.1257\/aer.20140806","volume":"105","author":"IWR Martin","year":"2015","unstructured":"Martin, I.W.R., Pindyck, R.S.: Averting catastrophes: the strange economics of scylla and charybdis. Am. Econ. Rev. 105(10), 2947\u20132985 (2015)","journal-title":"Am. Econ. Rev."},{"key":"1598_CR30","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1146\/annurev-economics-080511-110932","volume":"4","author":"RJ Barro","year":"2012","unstructured":"Barro, R.J., Ursua, J.F.: Rare macroeconomic disasters. Annu. Rev. Econ. 4, 83\u2013109 (2012)","journal-title":"Annu. Rev. Econ."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-019-01598-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-019-01598-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-019-01598-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,6]],"date-time":"2020-12-06T00:28:16Z","timestamp":1607214496000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-019-01598-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,7]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,3]]}},"alternative-id":["1598"],"URL":"https:\/\/doi.org\/10.1007\/s10957-019-01598-5","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2019,12,7]]},"assertion":[{"value":"18 October 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 October 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 December 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}