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TD 1117 Dynamic Optimization and Learning How Should a Manager set Prices when the Demand Function is Unknown

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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. 
 
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