Question

2. Increasing returns to scale suggests that establishing a large factory that hires a large number of workers is efficient in production. But hiring a lot of workers in one place potentially drives up wages. Technology, transportation, and labor market interact and play an important role in determining the geography of production. This question illustrates these interactions and explains how the nature of production has evolved over time. Suppose a country has three cities, A, B, C. They are located on a straight line as below A-B-C Each city has 100 workers. Each worker demands 1 cellphone. There is one cellphone company that produces all the cellphones and sell them to the workers. Suppose there are two technologies available for the production of cellphones. The first tech- nology, which we call "backyard production", has the production function Q-5L, where Q is the number of cellphones produced, L is the number of workers hired. The second technology, which we call "assembly line production", has the production function Q-(L- 20)2. In each city,labor supply is wLo5, where w is the wage. Notice wage increases as the firm hires more workers. You can think this as there are other firms in the city. As the cellphone company hires more and more workers, it increases the competition with other firms, driving up the local wage. Shipping is costly. It costs the company $0.5 to ship one cellphone between A and B, S0.5 be- tween B and C, and $1 between A and C. The company can set up 3 factories, one in each city, that use backyard production. In this case, each factory produces 100 cellphones and does not incur any shipping cost. Alternatively, the com- pany can set up one large factory with assembly line in one city that produces 300 cellphones. The company needs to ship 100 cellphones to each of the two other cities. To save shipping cost, if the firm is to produce with an assembly line, it will set up the factory in city B. (You can verify by showing that setting up the factory in either A or C incurs higher shipping costs. You can also verify that it is not efficient to set up a factory in each city that uses assembly line production). The total cost to the compnay is the sum of cost of production and the cost of shipping. a. What is the total cost for producing and selling 300 cellphones if the company adopts backyard production? What is the total cost if the company adopts assembly line production? Which way of production will the company choose? (10 pts) b. Suppose now the shipping cost reduces to $0.2 to ship one cellphone between A and B, $0.2 between B and C, and S0.4 between A and C, how would your answer to part (a) change? (5 pts) c. Now suppose the company invents a new way of production, with somewhat misuse of the ter- minology, let's call this way of production "outsourcing Instead of making cellphones from raw inputs in one single factory, the firm realizes that it can split the production process into several steps. We assume now the production of a cellphone takes places in 3 factories. The first factory produces 300 units of part 1, the second factory produces 300 units of part 2, and the third factory produces 300 cellphones by assembling 300 units of part 1 and 300 units of part 2. The produc- tion function of the first factory is Q1 (4-15), where Q1 is the quantity of part 1, L1 is the number of workers in the first factory. Similarly, the production function of the second factory is Q2-(L2-15). Finally, the third factory hires 10 workers and assembles all the cellphones. If the company decides to use this technology, where would it put these three factories? How much does it cost to produce (not including shipping cost) 300 cellphones using oursourcing? (10 pts) d. Use the shipping cost schedule from part (b), and assume that the shipping cost of each part is the same as that for one cellphone. What is the total shipping cost if the company uses out- sourcing? Which way of production will the company choose? How would your answer change if technology of shipping further improves so shipping cost is negligible? (5 pts)

Answer 1

Ans (a) if the production function is backyard : Q= 5L

Q=300 then

300= 5L

L= 60

Labour used is 60 and the supply of labour at wage is given = √L then

√60= 7.74

7.74*60 ~ 465 ( because 7.74 is for only for one labour )

Including the shipping cost is 0.5+0.5+1= 2

For 300 cellphone it will be 2*300 = $600

Hence the total cost is $600+ 465

= 1065 ( this is cost if company is using backyard technology)

If company using assembly line technology then production function is Q= (L-20) ²

Q= 300 therefore

L= 17.32 ( solved step by step as above putting the value of Q )

7.74*37.32= 288.8~ $289

Including shipping cost then total cost will be = 289+600

= $889

From both the technology company will choose second one i.e the assembly line technology because its total cost is less then backyard technology.

Ans (b) if the shipping cost reduced then the total cost will also change or reduced by

Ist production function ie the backyard technology: shipping cost will be 0.3+0.3+0.6 = 1.2 per cellphone then the total cost is 300*1.2= 360

And total cost including labour cost = 465+ 360

= $865

Similarly we will calculate for assembly line technology

The total cost will be = $649

Ans (c) if outsourcing is used then the production function for one company or for part one is Q= (L1-15)⁴

Then again Q= 300

: L1= 19.161

And company used 10 labour then labor cost is 10*19.161

191.61 for part one and on adding 191.61( for part b )

Then total cost:

= 191.61+191.61 = 383.22 ( without shipping )

Then company will use outsourcing because from all the three technologies outsourcing cost is less then others.

Ans (d) if company includes shipping cost and suppose company used $0.6/ per cellphone then

0.6*300 = 180

And total cost including shipping cost= 383.22+ 180

= $563.22~ $563

If technology changes further and shipping cost is negligible then the total cost will be same as in part (c) ie not including shipping cost.