Question

To produce output, a firm in the United States uses a series of inputs indexed by x, where x ranges from 0 to 1. These labels describe the "sophistication" of an input, such that x=0 is the least sophisticated" input, x = 0.5 is the input of median sophistication, and x = 1 is the most sophisticated" input. To produce one unit of output, the US firm requires one unit of each input. There are two countries that can supply inputs, the US and China. One hour of US labor can produce one unit of each input. That is, American labor productivity in input x is Ax.US = 1 for every x. One hour of Chinese labor can produce 1 - x units of input x. That is, Chinese labor productivity in input x is Ax.China = 1 - x. The Chinese wage is $1 per hour. Inputs are supplied competitively. The US firm incurs no trade costs when purchasing inputs from US suppliers. a) Suppose the US wage were $4 per hour and importing an input from China cost $1 per unit imported. Which inputs would the firm purchase from China? Which from US suppliers?

Answer 1

Input-Output function for the US firm producing the good:

1 unit of input needed for 1 unit of output

Input label based on the degree of sophistication:

x=0 (least sophisticated)

x=0.5 (median sophistication)

x=1 (most sophisticated)

US input supplier labor productivity:

Ax,US = 1 for every x (all kinds of sophistication)

US wage rate = $4 per hour; all kinds of input (based on sophistication) will cost $4 per unit since it requires 1 hour to make every input (based on labor productivity)  

Chinese input supplier labor productivity:

Ax, China = 1-x

The productivity will vary depending on the value of x, so we have 3 scenarios here since x has 3 values (0, 0.5, 1)

Productivity when x = 0 (least sophisticated input), Ax,China = 1 - 0 = 1

Productivity when x = 0.5 (median sophistication input), Ax,China = 1- 0.5 = 0.5

Productivity when x = 1 (most sophisticated input), Ax,China = 1 - 1 = 0

Chinese wage rate = $1 per hour:

To produce the least sophisticated input in china it will cost $1

To produce median sophisticated input in china it will cost $2 (per hour productivity for this kind of input is 0.5; 2 hours needed to produce one input so $1 * 2 = $2)

There is also an import cost of $1 per unit from china so it is a type of fixed cost for every unit of input imported from Chinese suppliers:

The total cost to produce and import inputs of least sophistication (x=0) the cost is = $1 + $1(import) = $2

The total cost to produce and import inputs of median sophistication (x=0.5) the cost is = $2 + $1(import) = $3

Both the cost of inputs (of least and median sophistication) mentioned above are less than the US hourly wage rate of $4 so both of these inputs will be imported from China by the US manufacturing firm.

The most sophisticated goods can't be produced in China since its productivity is 0, so it will be produced by US input suppliers.

Inputs with the least and moderate degree of sophistication (x=0, 0.5) will be purchased from Chinese suppliers and most sophisticated inputs (x=1) will be purchased from the US suppliers.