Every day a salesman visits N customers in order to sell the R identical items h

Question

Every day a salesman visits N customers in order to sell the R identical items he has in his van. Each customer is visited exactly once, and each customer buys zero
or one item. Upon arrival at a customer location, the salesman quotes one of the prices 0 < p 1 p 2 ... p m . Given that the quoted price is p i , a customer buys an item with probability r i . Naturally r i is decreasing in i. The salesman is interested in maximizing the total expected revenue for the day. Show that if r i p i is increasing in i, then it is always optimal to quote the highest price p m .

Explanation / Answer

Revenue from each customer = rj*pj where rj is the probability jth customer buys the product and pj is the price. j = 1 to N and rj and pj belong to (r1,r2,..,rm) and (p1,p2,..,pm) respectively

As ripi is increasing in i, we have r1p1<r2p2<..<rmpm.

So, revenue when price quoted is p1< revenue when price quoted is p2<..<revenue when price quoted is pm

Hence max revenue is when price is pm.

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