Problem Set #5
Solutions
Textbook Chapter 18
Exercise 4
a.) Autarky: No trade
Each worker has one unit of labor
Income  Wage  w
Equal consumption shares:
Price of Apples  Consumptio n of Apples 
w
2
Price of Computers  Consumptio n of Computers 
Denote apples as the numéraire:
Price of Apples  1
Price of Computers  p
Labor market equilibrium says that wage is equal in both sectors
Wage must equal the number of apples a worker can produce
wNorth  160 Apples
wSouth  100 Apples
In the North a worker can produce 160 Apples or 20 Computers
Therefore, Cost of 160 Apples = Cost of 20 Computers:
p North  8 Apples per Computer
In the South a worker can produce 100 Apples or 2 Computers
Therefore, Cost of 100 Apples = Cost of 2 Computers:
pSouth  50 Apples per Computer
w
2
2
We can now calculate consumption and production
Denote:
cNorth  Computers produced in the North
aNorth  Apples produced in the North
Therefore:
a North 
wNorth 160 Apples

 80 Apples
2
2
wNorth
 80 Apples
2
80 Apples

 10 Computers
8 Apples/Com puter
p Northc North 
c North
Denote:
cSouth  Computers produced in the South
aSouth  Apples produced in the South
Therefore:
aSouth 
wSouth 100 Apples

 50 Apples
2
2
wSouth
 50 Apples
2
50 Apples
cSouth 
 1 Computer
50 Apples/Com puter
pSouthcSouth 
Employment in the North: 100 workers
Employment in the South: 400 workers
North
South
Wage, w
160 apples
100 apples
Price of a computer, p
8 apples
50 apples
Consumption of apples/person
80 apples
50 apples
Consumption of computers/person
10 computers
1 computer
Fraction of labor producing apples
50%
50%
Fraction of labor producing computers
50%
50%
Total production of apples
8,000 apples 20,000 apples
Total production of computers
1,000 computers 400 computers
3
b.) Free Trade
Free trade will equalize the prices in both parts of the world
What will the price be equal to?
Guess that all workers in the North produce computers
World Computer production  20 Computers/ worker  100 workers
 2,000 Computers
Guess that all workers in the South produce apples
World Apple production  100 Apples/wor ker  400 workers
 40,000 Apples
The equal expenditure shares assumption requires that:
Price of Apples  World Apple Productio n
 Price of Computers  World Computer P roduction
Since the price of apples has been set at unity:
1  40,000 Apples  p  2,000 Computers
p
40,000 Apples
 20 Apples/Com puter
2,000 Compters
In the South, workers produce apples, so wSouth  100 Apples
In the North, workers produce 20 computers
A computer is worth 20 apples, so wNorth  400 Apples
We can finally calculate consumption
Northern workers spend half their income on apples: 200 apples
Northern workers spend the other half on computers:
200 Apples  p  computer consumptio n
Computer consumptio n 
200 Apples
 10 Computers
20 Apples/com puter
Southern worker spend half their income on apples: 50 apples
Southern workers spend half their income on computers:
50 Apples  p  computer consumptio n
Computer consumptio n 
50 Apples
 2 12 Computers
20 Apples/com puter
4
North
South
Fraction of labor producing apples
0%
100%
Fraction of labor producing computers
100%
0%
Total production of apples
0 apples 40,000 apples
Total production of computers
2,000 computers
0 computers
Wage, w
400 apples
100 apples
Price of a computer, p
20 apples
20 apples
Consumption of apples/person
200 apples
50 apples
Consumption of computers/person
10 computers 2 12 computers
c.) In the original example, the North gained 120 apples/person from trade and the South
gained 1 computer/person from trade.
In the present case, the North gains 120 apples/person from trade and the South gains
1 12 computers/person.
Relative to the earlier example, there are an extra 400 computer produced in the
world. Half of the extra computers go to the North, half of the extra computers go to
the South. Worldwide apple production and consumption is unchanged from the
earlier free trade example.
Textbook Chapter 18
Exercise 5
a.) Autarky: No trade
Each worker has one unit of labor
Income  Wage  w
Equal consumption shares:
Price of Apples  Consumptio n of Apples 
w
2
Price of Computers  Consumptio n of Computers 
Denote apples as the numéraire:
Price of Apples  1
Price of Computers  p
Labor market equilibrium says that wage is equal in both sectors
w
2
5
Wage must equal the number of apples a worker can produce
wNorth  160 Apples
wSouth  160 Apples
In the North a worker can produce 160 Apples or 16 Computers
Therefore, Cost of 160 Apples = Cost of 16 Computers:
p North  10 Apples per Computer
In the South a worker can produce 160 Apples or 2 Computers
Therefore, Cost of 160 Apples = Cost of 2 Computers:
pSouth  80 Apples per Computer
We can now calculate consumption and production
Denote:
cNorth  Computers produced in the North
aNorth  Apples produced in the North
Therefore:
a North 
wNorth 160 Apples

 80 Apples
2
2
wNorth
 80 Apples
2
80 Apples

 8 Computers
10 Apples/Com puter
p Northc North 
c North
Denote:
cSouth  Computers produced in the South
aSouth  Apples produced in the South
Therefore:
a South 
wSouth 160 Apples

 80 Apples
2
2
6
wSouth
 80 Apples
2
80 Apples

 1 Computer
80 Apples/Com puter
pSouthcSouth 
cSouth
Employment in the North: 100 workers
Employment in the South: 400 workers
North
South
Wage, w
160 apples
160 apples
Price of a computer, p
8 apples
80 apples
Consumption of apples/person
80 apples
80 apples
Consumption of computers/person
8 computers
1 computer
Fraction of labor producing apples
50%
50%
Fraction of labor producing computers
50%
50%
Total production of apples
8,000 apples 32,000 apples
Total production of computers
800 computers 400 computers
b.) Free Trade
Free trade will equalize the prices in both parts of the world
What will the price be equal to?
Guess that all workers in the North produce computers
World Computer production  16 Computers/ worker  100 workers
 1,600 Computers
Guess that all workers in the South produce apples
World Apple production  160 Apples/wor ker  400 workers
 64,000 Apples
The equal expenditure shares assumption requires that:
Price of Apples  World Apple Productio n
 Price of Computers  World Computer P roduction
Since the price of apples has been set at unity:
1  64,000 Apples  p  1,600 Computers
p
64,000 Apples
 40 Apples/Com puter
1,600 Compters
7
In the South, workers produce apples, so wSouth  160 Apples
In the North, workers produce 16 computers
A computer is worth 40 apples, so wNorth  640 Apples
We can finally calculate consumption
Northern workers spend half their income on apples: 320 apples
Northern workers spend the other half on computers:
320 Apples  p  computer consumptio n
Computer consumptio n 
320 Apples
 8 Computers
40 Apples/com puter
Southern worker spend half their income on apples: 80 apples
Southern workers spend half their income on computers:
80 Apples  p  computer consumptio n
Computer consumptio n 
80 Apples
 2 Computers
40 Apples/com puter
North
South
Fraction of labor producing apples
0%
100%
Fraction of labor producing computers
100%
0%
Total production of apples
0 apples 64,000 apples
Total production of computers
1,600 computers
0 computers
Wage, w
640 apples
160 apples
Price of a computer, p
40 apples
40 apples
Consumption of apples/person
320 apples
80 apples
Consumption of computers/person
8 computers 2 computers
c.) In the original example, the North gained 120 apples/person from trade and the South
gained 1 computer/person from trade.
In the present case, the North gains 240 apples/person from trade and the South gains
1 computers/person.
Relative to the original example, there are an extra 24,000 apples produced in the
world. Half of the extra apples go to the North, half of the extra apples go to the
South. Worldwide computer production and consumption are unchanged from the
earlier free trade example.
8
Textbook Chapter 18
Exercise 8
In the text example, Southern workers consume 50 apples and 2 computers under free
trade. If one worker migrates to the North, he could consume 200 apples and 8
computers. Therefore, the first immigrant could be charged as much as 150 apples and 6
computers. If all of the Southern workers migrated to the North, they would each
consume 80 apples and 8 computers. Therefore they would pay as much as 30 apples and
6 computers to migrate.
The workers in the North consume 200 apples and 8 computers under free trade. They
only consume 80 apples and 8 computers if all Southern workers move to the North.
There are 400 workers originally in the South and only 100 workers originally in the
North. Therefore each Northern worker could collect the entrance fees from 4 of the
migrants from the South. Therefore, just the apple fees from 4 Southern workers would
give them 120 extra apples, so they would be as well off as they would be with free trade
(200 apples and 8 computers). If they also receive even one computer from the migrants,
they would be unambiguously better off.