In a remote area of the American Midwest before the railroads arrived, cast iron cookstoves were much desired, but people lived far apart, roads were poor, and heavy stoves were expensive to transport. Stoves could be shipped by river boat to the town of Bouncing Springs, Missouri. Ben Kinmore was the only stove dealer in Bouncing Springs. He could buy as many stoves as he wished for $20 each, delivered to his store. Ben’s only customers were farmers who lived along a road that ran east and west through town. There were no other stove dealers along the road in either direction. No farmers lived in Bouncing Springs, but along the road, in either direction, there was one farm every mile. The cost of hauling a stove was $1 per mile. The owners of every farm had a reservation price of $120 for a cast iron cookstove. That is, any of them would be willing to pay up to $120 to have a stove rather than to not have one. Nobody had use for more than one stove. Ben Kinmore charged a base price of $p for stoves and added to the price the cost of delivery. For example, if the base price of stoves was $40 and you lived 45 miles west of Bouncing Springs, you would have to pay $85 to get a stove, $40 base price plus a hauling charge of $45. Since the reservation price of every farmer was $120, it follows that if the base price were $40, any farmer who lived within 80 miles of Bouncing Springs would be willing to pay $40 plus the price of delivery to have a cookstove. Therefore at a base price of $40, Ben could sell 80 cookstoves to the farmers living west of him. Similarly, if his base price is $40, he could sell 80 cookstoves to the farmers living within 80 miles to his east, for a total of 160 cookstoves.
(a) If Ben set a base price of $p for cookstoves where p < 120, and if he charged $1 a mile for delivering them, what would be the total number of cookstoves he could sell? _________. (Remember to count the ones he could sell to his east as well as to his west.) Assume that Ben has no other costs than buying the stoves and delivering them. Then Ben would make a profit of p − 20 per stove. Write Ben’s total profit as a function of the base price, $p, that he charges: _________.
(b) Ben’s profit-maximizing base price is _________. Ben’s most distant customer would be located at a distance of 50 miles from him. Ben would sell _________ cookstoves and make a total profit of _________.
(c) Suppose that instead of setting a single base price and making all buyers pay for the cost of transportation, Ben offers free delivery of cookstoves. He sets a price $p and promises to deliver for free to any farmer who lives within p−20 miles of him. (He won’t deliver to anyone who lives further than that, because it then costs him more than $p to buy a stove and deliver it.) If he is going to price in this way, how high should he set p? _________. How many cookstoves would Ben deliver? _________. How much would his total revenue be? _________. How much would his total costs be, including the cost of deliveries and the cost of buying the stoves? _________. How much profit would he make? _________. Can you explain why it is more profitable for Ben to use this pricing scheme where he pays the cost of delivery himself rather than the scheme where the farmers pay for their own deliveries?__________.