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Optimal Dynamic Choice of Durable and Perishable Goods

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  • Peter Bank
  • Frank Riedel

Abstract

We analyze the life cycle consumption choice model for multiple goods, focusing on the distinction between durables and perishables. As an approximation of the fact that rather high transaction costs and market imperfections prevail in markets for used durables, we assume that investment in durables is irreversible. In contrast to the additive model with one perishable good, the optimal consumption plan is not myopic. Instead, it depends on past as well as on (expected) future prices. The optimal stock level of the durable good is obtained by tracking a certain \emph{shadow level}: The household purchases just enough durables to keep the stock always above this shadow level. It is shown that this shadow level is given by a backward integral equation that replaces the Euler equation. For the perishable good, the `usual' Euler equation determines the optimal choice in terms of the optimal stock of durables. Since the optimal stock level aggregates past as well as future prices, the consumption of perishables ceases to be myopic as well. The solutions show that durables play an important part in intertemporal consumption decisions. In fact, major purchases of durables are being made early in life, whereas no durables are bought in the retirement years. Through substitution and complementarity effects, this has a significant impact on the consumption of perishable goods. On the technical side, the paper provides a new approach to singular control problems that might be widely applicable in other contexts like irreversible investment, price rigidities etc. We present a numerical algorithm that allows one to calculate the shadow level for arbitrary period utility functions and time horizons. Explicit solutions are given for the case of a homogeneous Markov setup with infinite time horizon and Cobb--Douglas type period utilities. This setup includes prices driven by Brownian motion and/or Poisson processes.

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Paper provided by UCLA Department of Economics in its series Levine's Bibliography with number 666156000000000402.

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Date of creation: 26 Nov 2003
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Handle: RePEc:cla:levrem:666156000000000402

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  1. Grossman, Sanford J & Laroque, Guy, 1990. "Asset Pricing and Optimal Portfolio Choice in the Presence of Illiquid Durable Consumption Goods," Econometrica, Econometric Society, vol. 58(1), pages 25-51, January.
  2. R. C. Merton, 1970. "Optimum Consumption and Portfolio Rules in a Continuous-time Model," Working papers 58, Massachusetts Institute of Technology (MIT), Department of Economics.
  3. Mankiw, N. Gregory, 1982. "Hall's consumption hypothesis and durable goods," Journal of Monetary Economics, Elsevier, vol. 10(3), pages 417-425.
  4. Detemple, Jerome B. & Giannikos, Christos I., 1996. "Asset and commodity prices with multi-attribute durable goods," Journal of Economic Dynamics and Control, Elsevier, vol. 20(8), pages 1451-1504, August.
  5. Peter Bank & Frank Riedel, 1998. "Non-Time Additive Utility Optimization - the Case of Certainty," GE, Growth, Math methods 9811002, EconWPA.
  6. Hindy, Ayman & Huang, Chi-fu & Kreps, David, 1992. "On intertemporal preferences in continuous time : The case of certainty," Journal of Mathematical Economics, Elsevier, vol. 21(5), pages 401-440.
  7. repec:wop:humbsf:2002-4 is not listed on IDEAS
  8. Harry Mamaysky, 2001. "Interest Rates and the Durability of Consumption Goods," Yale School of Management Working Papers ysm224, Yale School of Management, revised 01 Jan 2002.
  9. Hindy, Ayman & Huang, Chi-fu, 1993. "Optimal Consumption and Portfolio Rules with Durability and Local Substitution," Econometrica, Econometric Society, vol. 61(1), pages 85-121, January.
  10. Cuoco, Domenico & Liu, Hong, 2000. "Optimal consumption of a divisible durable good," Journal of Economic Dynamics and Control, Elsevier, vol. 24(4), pages 561-613, April.
  11. Damgaard, Anders & Fuglsbjerg, Brian & Munk, Claus, 2003. "Optimal consumption and investment strategies with a perishable and an indivisible durable consumption good," Journal of Economic Dynamics and Control, Elsevier, vol. 28(2), pages 209-253, November.
  12. Akerlof, George A, 1970. "The Market for 'Lemons': Quality Uncertainty and the Market Mechanism," The Quarterly Journal of Economics, MIT Press, vol. 84(3), pages 488-500, August.
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Cited by:
  1. Maria B. Chiarolla & Giorgio Ferrari & Frank Riedel, 2012. "Generalized Kuhn-Tucker Conditions for N-Firm Stochastic Irreversible Investment under Limited Resources," Working Papers 463, Bielefeld University, Center for Mathematical Economics.
  2. Maria B. Chiarolla & Giorgio Ferrari, 2011. "Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank-El Karoui Representation Theorem," Papers 1108.4886, arXiv.org, revised Dec 2013.
  3. Giorgio Ferrari & Frank Riedel & Jan-Henrik Steg, 2013. "Continuous-Time Public Good Contribution under Uncertainty," Papers 1307.2849, arXiv.org.

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