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TEXTO PARA DISCUSSÃO No 1117 DYNAMIC OPTIMIZATION AND LEARNING: HOW SHOULD A MANAGER SET PRICES WHEN THE DEMAND FUNCTION IS UNKNOWN? Alexandre X. Carvalho Martin L. Puterman Brasília, setembro de 2005 TEXTO PARA DISCUSSÃO No 1117 DYNAMIC OPTIMIZATION AND LEARNING: HOW SHOULD A MANAGER SET PRICES WHEN THE DEMAND FUNCTION IS UNKNOWN?* Alexandre X. Carvalho** Martin L. Puterman*** Brasília, setembro de 2005 O presente texto, em língua inglesa, não foi objeto de revisão editorial. * Agradecimentos/Acknowledgments This research was partially supported by NSERC and the MITACS NCE (Canada). We wish to thank Dan Adelman for his helpful comments on an earlier draft of this manuscript. We are also thankful for the suggestions provided by participants at the Canadian Operations Research Society Annual Meeting 2003, Vancouver, and at the INFORMS Annual Meeting 2003, Atlanta. All errors are the authors'. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. ** Directorate of Regional and Urban Studies, Institute of Applied Economic Research (Ipea)/Diretoria de Estudos Regionais e Urbanos (Dirur), Instituto de Pesquisa Econômica Aplicada. *** Director of Centre for Health Care Management, Sauder School of Business, University of British Columbia. Governo Federal Ministério do Planejamento, Orçamento e Gestão Ministro – Paulo Bernardo Silva Secretário-Executivo – João Bernardo de Azeredo Bringel Fundação pública vinculada ao Ministério do Planejamento, Orçamento e Gestão, o Ipea fornece suporte técnico e institucional às ações governamentais – possibilitando a formulação de inúmeras políticas públicas e programas de desenvolvimento brasileiro – e disponibiliza, para a sociedade, pesquisas e estudos realizados por seus técnicos. Presidente Glauco Arbix Diretora de Estudos Sociais Anna Maria T. Medeiros Peliano Diretor de Administração e Finanças Celso dos Santos Fonseca Diretor de Estudos Setoriais João Alberto De Negri Diretor de Cooperação e Desenvolvimento Luiz Henrique Proença Soares Diretor de Estudos Regionais e Urbanos Marcelo Piancastelli de Siqueira Diretor de Estudos Macroeconômicos Paulo Mansur Levy Chefe de Gabinete Persio Marco Antonio Davison Assessor-Chefe de Comunicação Murilo Lôbo URL: http://www.ipea.gov.br Ouvidoria: http://www.ipea.gov.br/ouvidoria ISSN 1415-4765 JEL C44, C61, C63, D81 TEXTO PARA DISCUSSÃO Publicação cujo objetivo é divulgar resultados de estudos direta ou indiretamente desenvolvidos pelo Ipea, os quais, por sua relevância, levam informações para profissionais especializados e estabelecem um espaço para sugestões. As opiniões emitidas nesta publicação são de exclusiva e de inteira responsabilidade do(s) autor(es), não exprimindo, necessariamente, o ponto de vista do Instituto de Pesquisa Econômica Aplicada ou o do Ministério do Planejamento, Orçamento e Gestão. É permitida a reprodução deste texto e dos dados nele contidos, desde que citada a fonte. Reproduções para fins comerciais são proibidas. SUMÁRIO SINOPSE ABSTRACT 1 INTRODUCTION 7 2 MODEL FORMULATION 12 3 A FORMAL ANALYSIS OF THE TRADE-OFF BETWEEN OPTIMIZATION AND LEARNING 17 4 THE GENERAL MULTIPERIOD PROBLEM 20 5 MONTE CARLO SIMULATION 25 6 RANDOM PRICES AND ESTIMATION BIAS 32 7 CONCLUSIONS 36 APPENDIX 39 REFERENCES 43 SINOPSE Neste artigo, nós estudamos o problema de escolher preços sequencialmente, de forma a maximizar a receita esperada, em um ambiente onde os parâmetros da função de demanda são desconhecidos, e o horizonte de vendas é finito. Vários métodos de otimização seqüencial são discutidos, onde os preços e as vendas resultantes anteriores são utilizados para determinar o preço no período atual. Expansões de Taylor são empregadas para construir aproximações da função valor, explicitando a relação de compromisso entre maximização de receita no curto-prazo e maior ganho de informação para obter maior receita agregada no longo-prazo. A partir dessas expansões, nós derivamos estratégias promissoras, denominadas políticas de one-step look-ahead que combinam otimização e aquisição de informação dinamicamente. Simulações de Monte Carlo são apresentadas, onde constatamos a superioridade das políticas de one-step look-ahead, quando comparadas a diversas outras regras de otimização seqüencial. Finalmente, nós discutimos problemas de endogeneidade, onde fazemos um paralelo com a teoria de controle adaptativo, e discutimos a validade das regras de one-step look-ahead, mesmo quando endogeneidade é observada. ABSTRACT This paper considers the problem of changing prices over time to maximize expected revenues in the presence of unknown demand distribution parameters. It provides and compares several methods that use the sequence of past prices and observed demands to set price in the current period. A Taylor series expansion of the future reward function explicitly illustrates the tradeoff between short term revenue maximization and future information gain and suggests a promising pricing policy referred to as a one-step look-ahead rule. An in-depth Monte Carlo study compares several different pricing strategies and shows that the one-step look-ahead rules dominate other heuristic policies and produce good short term performance. The reasons for the observed bias of parameter estimates are also investigated. � ��������� ��� �������� ���������������fiffffifl����������! 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