Suppose that chemical X is manufactured using a raw material B that is available from a location called the “mine”. Production of one ton of X requires 1/3 of aton of B. A firm called X Enterprises, which has a contract to deliver 30 tons ofX to a location called the “market”, is trying to decide where to locate its plant.The mine and the market are 50 kilometres apart. Overland shipment of both X and B costs $2 per ton per km. shipped. However, additional costs must beincurred because a river passes between the mine and the market, and the riverhas no bridge. Goods must be loaded onto barges to cross the river, which islocated 16 kilometres from the mine. Barge operators charge $1 per ton of X shipped across the river. However, since the input B is highly toxic when mixedwith water, barge operators must charge an extremely high price to transport Bacross the river. This price defrays the cost of insurance that the operators mustcarry to meet liability claims should they accidentally pollute the river with theircargo. The cost of shipping one ton of B across the river is $195.
a) Using the above information, find the transport-cost-minimizing location for X Enterprises. The answer can be found by computing transport costs at four locations: mine, market, mine side of the river, and market side of the river.Assume that the width of the river is negligible, so that it can be ignored. Show your work.
2b) Illustrate your results in a drawn diagram like that presented in Figure 1.6. Plot the input shipping-cost curve by plotting the input shipping costs at the same four locations as in (a) and then connecting the dots (the curve is drawn backward). Similarly, plot the output shipping cost at the four locations, and then connect the dots to generate the output shipping-cost curve. Then, plot the total shipping-cost curve by adding the input and output shipping costs at each of the four locations, plotting the points and connecting the dots. Using the diagram, identifying the best location for X Enterprises, which should be the same as your answer in (a).
c) Explain your results intuitively.
d) Suppose that a bridge were built across the river, which would eliminate the cost of crossing it. Repeat (a), (b), and (c) under this assumption.