B2.15

Optimization and Control | Part II, 2001

A street trader wishes to dispose of kk counterfeit Swiss watches. If he offers one for sale at price uu he will sell it with probability aeua e^{-u}. Here aa is known and less than 1 . Subsequent to each attempted sale (successful or not) there is a probability 1β1-\beta that he will be arrested and can make no more sales. His aim is to choose the prices at which he offers the watches so as to maximize the expected values of his sales up until the time he is arrested or has sold all kk watches.

Let V(k)V(k) be the maximum expected amount he can obtain when he has kk watches remaining and has not yet been arrested. Explain why V(k)V(k) is the solution to

V(k)=maxu>0{aeu[u+βV(k1)]+(1aeu)βV(k)}V(k)=\max _{u>0}\left\{a e^{-u}[u+\beta V(k-1)]+\left(1-a e^{-u}\right) \beta V(k)\right\}

Denote the optimal price by uku_{k} and show that

uk=1+βV(k)βV(k1)u_{k}=1+\beta V(k)-\beta V(k-1)

and that

V(k)=aeuk/(1β)V(k)=a e^{-u_{k}} /(1-\beta)

Show inductively that V(k)V(k) is a nondecreasing and concave function of kk.

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