Exercise 1.1 Suppose that chemical X is manufactured using a raw material B that is available from a location called the â€œmine.â€ Production of one ton of X requires 1/3 of a ton of B. A firm called X Enterprises, which has a contract to deliver 30 tons of X to a location called the â€œmarket,â€ is trying to decide where to locate its plant. The mine and the market are 50 miles apart. Overland shipment of both X and B costs $2 per ton per mile shipped. However, additional costs must be incurred because a river passes between the mine and the market, and the river has no bridge. Goods must be loaded onto barges to cross the river, which is located 16 miles from the mine. Barge operators charge $1 per ton of X shipped across the river. However, since the input B is highly toxic when mixed with water, barge operators must charge an extremely high price to transport B across the river. This price defrays the cost of insurance that the operators must carry to meet liability claims should they accidentally pollute the river with their cargo. 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 river. Assume that the width of the river is negligible, so that it can be ignored. Show your work.
(b) Illustrate your results in a carefully drawn diagram like that presented in figure 1.6 (use graph paper). 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). Note that the shipping-cost curves for this problem are straight lines with jumps at the river.
(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.
Exercise 1.2 Consider four cities, A, B, C, and D, located as follows: A D B C Suppose that the residents of these cities consume widgets, with consumption in each city equaling 100 widgets. The firm that produces widgets must decide how to arrange its production. It could set up four factories, one in each city, with each factory producing 100 widgets. In this case, the firm incurs no cost for shipping its output. Or the firm could locate its factory in the centrally located city, D. The single factory must then produce 400 widgets, 300 of which are shipped to cities A, B, and C. The shipping cost per widget is $2. A final assumption is that widget production exhibits economies of scale, with the cost per widget in a factory falling as output rises.
(a) Suppose the cost per widget varies with output as follows: cost is $4 if factory output is 100 widgets; cost is $1 if factory output is 400 widgets. In this situation, find the best arrangement of production for the firm (i.e., one central factory or four separate factories). The best arrangement leads to the lowest total cost for the firm, where the total is the sum of production and shipping cost.
(b) Repeat (a) if the cost per widget varies with output as follows: cost is $4 if factory output is 100 widgets; cost is $3 if factory output is 400 widgets.
(c) Explain intuitively any difference in the answers to (a) and (b). (d) Suppose production costs are those given in (a), and let shipping cost per widget be given by t. Although t = 2 in (a), what value of t would make the two arrangements for production (centralized vs. separate factories) equivalent in terms of cost?