A street trader wishes to dispose of $k$ counterfeit Swiss watches. If he offers one for sale at price $u$ he will sell it with probability $a e^{-u}$. Here $a$ is known and less than 1 . Subsequent to each attempted sale (successful or not) there is a probability $1-\beta$ 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 $k$ watches.

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

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

Denote the optimal price by $u_{k}$ and show that

$$u_{k}=1+\beta V(k)-\beta V(k-1)$$

and that

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

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