Question

A manufacturer produces some output called “Product” using a raw material called “Input”. To production of 1 ton of Product requires 3/5 ton of Input. The manufacturer has a contract to deliver 50 tons of Product to Store M (Store M is the “market” for Product). Input is available in Spot N (Spot N is the source of Input, equivalent to “mine” in the textbook). Store M is 40 miles away from the Spot N. The shipping cost of both Product and Input are $2 per ton per mile. We will help the manufacturer to find the total transport-cost minimizing factory location.

However, our manufacturer faces an additional issue. Store M and Spot N are in different countries, making the manufacturer subject to tariffs based on trade agreements between the two countries. The Border is 15 miles from Spot N. The width of the Border is negligible, placing Store M at 25 miles from the Border. The trade agreements are such that Product can be moved between the countries freely but transporting Input across the border involves a payment of $60 per ton in tariffs. Given this additional information, let us help the manufacturer re-calculate his shipping costs. We will go through similar steps as earlier, now following parts (d), (e) and (f).

(d) Now, consider four potential locations for the factory: Spot N, Spot N-side of Border (let’s label this location as N-side of Border), Store M-side of Border (label this location as M-side of Border) and Store M. Calculate shipping costs by filling in the cells of the table below. Show your calculations.

Location of Factory	Spot N	N-side of Border	M-side of Border	Store M
Input Shipping Cost
Product Shipping Cost
Total Shipping Cost

(e) Again, present a graphical illustration of your calculations. In addition to the previous locations on the x-axis, now mark out N-side of Border and M-side of Border. The y-axis represents shipping costs. Clearly label the axis, all three shipping cost lines and significant points on the graph.

(f) In this case, what is the total transport-cost minimizing location for the manufacturer? How do you make intuitive economic sense of this result? Explain in 1-2 sentences.

Answer 1

d )

A manufacturer produces some output called “Product” using a raw material called “Input”. To produce 1 ton of Product,the Manufacturer requires 3/5 ton of Input and he has a contract to deliver 50 tons of Product to Store M. So he requires (3/5)$\times$50 = 30 tons of input in order to produce 50 tons pf output. Now he faces shipping cost of both Product and Input are $2 per ton per mile. In addition,since Store M and Spot N are in different countries, it makes the manufacturer subject to tariffs based on trade agreements between the two countries such that Product can be moved between the countries freely but transporting Input across the border involves a payment of $60 per ton in tariffs. Given these information,the shipping costs of the Manufacturer for different locations of his factory are calculated as follows :

SPOT N :

Manufacturer does not need to transport his input as his factory is located at the very source of his input thus making his input shipping cost equal to $0.But from this point,he needs to transport 50 tons of the finished product to Store M which is located at 40 miles away from the location of his factory at Spot N and he does not face any tariff in moving his product through the border as per the trade agreements between the two countries. To transport final good he incurs cost of $[(2$\times$50)$\times$40] = $4000. So his product shipping cost turns out to be equal to $400 which is the same as his total shipping cost =Input shipping cost + product shipping cost= $4000 + $0 = $4000.

N-SIDE OF THE BORDER:

Since no exact location is stated about the factory, we assume that the factory is located at the distance of k miles from spot N at the N-side of the border where the 0<k<15 (the Border is 15 miles from Spot N and hence k spans the entire region between spot N and border denoting any location at the N-side of the border). Now, the Manufacturer need to transport 30 tons of his input from Spot N to the exact location of the factory for production purpose during which he incurs input shipping cost of $[(2$\times$k)$\times$30] = $60 k. On the other hand, he needs to transport 50 tons of the final product from the exact location of the factory to Store M which causes him to face product shipping cost worth of $[2(40-k)$\times$50] = $[100(40-k)]. Total shipping cost becomes equal to input shipping cost + product shipping cost = $60k + $[100(40-k)] = $(4000-40k) = $[40(100-k)] .

M-SIDE OF THE BORDER:

Similar to the above case,here also the exact location of the factory is not listed.Therefore,we assume that the factory is located at the distance of p miles from border at the M-side of the border where the 0<p<25 (the Border is 25 miles from Store M and hence p spans the entire region between Store M and border denoting any location at the M-side of the border).Now, the Manufacturer need to transport 30 tons of his input from Spot N to the exact location of the factory for production purpose during which he incurs input shipping cost of $[2$\times${(p+15)$\times$30}] + $(60$\times$30) = $60(p+15+30) = $60(p+45) as transport of input over the border makes the manufacturer subject to additional transport cost from tariff of $60 per ton of input. Since the 50tons of final product needs to be transported from exact location of the factory to the Store M spanning distance of (25-p)The product shipping cost becomes $[2$\times${(25-p)$\times$50] = $100(25-p). So the total cost becomes : $60(p+45) + $100(25-p) = $(5200-40p).

STORE M:

When the factory is located at Store M,the manufacturer incurs no product shipping cost but huge input shipping cost because he then needs to transport 30 units of his input from Spot N to Store M which causes cost = $[(2$\times$40)$\times$30] + $(60$\times$30) = $2400 +$ 1800 = $4200. So the total shipping cost becomes $4200.

Now we present the above calculations in a table form :

Location of Factory	Spot N	N-side of Border	M-side of Border	Store M
Input Shipping Cost	$ 0	$ 60k	$ (60p+2700)	$ 4200
Product Shipping Cost	$ 4000	$ (4000-100k)	$ (2500-100p)	$ 0
Total Shipping Cost	$ 4000	$ (4000-40k)	$ (5200-40p)	$ 4200

e) The graphical representation of the calculations is given as follows :

The lines marked similarly denote the three shipping cost lines respectively.The significant points are marked accordingly on the two axes.

129 干40 Co

f) In this case the total transport cost minimizing location will be anywhere on the side of Spot such that the location is approximately close to the Border from N side.

The intuitively as he moves his location of factory along the border, he has to incur significant amount of additional cost of tariff for transportation of his input.But as he moves closer to the border from N side, he total cost continues to decrease.So he would be at an economically optimal position when he moves very close to the border from N side but chooses to not cross it while choosing the location if his factory.