The Steel Factory

1. The steel factory makes steel from iron ore and coal. Both iron and coal mines are near city
A, while the demand for steel is in city B, which is 100 kilometers away. The factory makes 1 ton of
steel from 1 ton of coal and 2 tonnes of iron ore. Shipping is costly and exhibits increasing returns
to scale. Shipping 1 ton of goods (either raw materials or final products) incurs a terminal cost
of $100 (the fixed cost incurred as long as one needs to ship something, no matter how far it is
shipped) and $10 per kilometer.
a. Is the production of steel weight-losing or weight-gaining? (5 pts)
b. Determine where the factory will be located in order to save as much as possible in shipping
cost? Suppose that if the factory is built in city A, it has immediate access to coal and iron ore, no
shipping cost involving raw materials is required. if the factor is built in city B, it has immediate
access to the market, no shipping cost involving the final product is required. (10 pts)
c. Now suppose that the coal mine near city A is depleted. Now coal is shipped to city B, which is

also a major port city, from overseas. Suppose now if the factory is located in city B, it has imme-
diate access to coal, no shipping cost is incurred. Determine where the facotry will be located in

this new situation. (5 pts)
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

1

supply is w = L

0.5, 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, $0.5 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 $0.4 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 = (L1 − 15)

4
, 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)
4
. 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)

3. Consider the advertisement industry, where firms use models (workers) to shoot adver-
tisements. Each model works for only one firm. Each worker (model) comes in with a different

hair style, which can be divided into 24 types. In other words, the hair style wheel is like a 24-
hour clock. Each firm enters the market and makes one advertisement. Each advertisement has
a specific requirement for hair style. If a model’s hair style does not match the advertisement’s
requirement, the model must spend money redoing the model’s hair in order to close the gap. The

2

cost of re-styling is $2 for each unit of hair style shift. For example, to go from style #3 to style
#2, the cost is $2; to go from style #5 to style #2, the cost is $6. Each advertisement requires two
models. There are two cities in the region: Smallville has 4 models and Bigburg has 12. The gross
wage for each model is $50.
Assume that models are evenly distributed around the hair style wheel. Compute the net wage
for models in each of the two cities. Show steps for partial credit. (10 pts)
4. Cosider a firm that produces a certain good. The demand is uncertain. There is 50% chance
the demand is high, and 50% chance the demand is low. When the demand is high, its demand
for labor is w = 12 − 0.5L. When the demand is low, its demand for labor is w = 8 − 0.5L. Here w
is wage for each worker and L is the total number of workers it hires. When the demand is high,
the profit of the firm can be expressed as (12 − w)×L/2; when the demand is low, the profit of the
firm can be expressed as (8 − w)×L/2 (review slides for why this is the case).
The firm considers to locate in one of the two cities. City A is small and only has 8 workers.
These 8 workers have perfectly inelastic supply of labor (employment is constant, but wage can
change). City B is big and has many workers. The labor supply is perfectly elastic, which means
that all workers work at a given wage no matter whether the demand is high or low.
a. If the firm decides to locate in city A. If the demand is high, how many workers does the firm
hire? What is the wage? How much is firm’s profit? How about when the demand is low? (10 pts)
b. What is the firm’s expected profit in city A? What is the worker’s expected wage? (5 pts)
c. Workers are freely mobile between the two cities. The fact that both cities have some workers
means that the expected wage is the same in both cities. What is the wage in city B? (5 pts)
d. If the firm decides to locate in city B. If the demand is high, how many workers does the firm
hire? What is the wage? How much is firm’s profit? How about when the demand is low? (10 pts)
e. What is the firm’s expected profit in city B? Which city will the firm choose to locate? (10 pts)

 

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