I believe the best way to understand production possibilities frontier, unit labor requirements and relative prices is with an example.
Home has 1,200 units of labor available. It can produce two goods apples and bananas. The unit labor requirement in apple production is 3, while in banana production it is 2.
The idea here is very simple. The economy consists of only workers and there are only two goods. We wish to look at what the price of goods will be, how much labor will be paid and so on.
The first important thing to cover is what are the assumptions of this model.
- To produce either good labor is the only input required to produce the good. We ignore the existence of capital (machines) or land.
- There are constant returns to scale (CRS). This means whether we scale up or scale down our production of either apples or bananas, we still produce the same amount of output.
- Labor is perfectly mobile. This means that workers can either produce apples or banana and can produce both goods equally as well.
- Workers only care about earning the maximum profit possible. They do not have preferences between working in the apple or banana industry.
Graph the production possibility frontier (PPF)
The production possibilities frontier shows the combination of goods that an economy can produce.
First step: let's work out how much the economy can produce when they produce only one good.
i) if the economy only produces apples
The economy has 1,200 units of labor (you can think of this as having 1,200 workers for simplicity) and it takes 3 units to produce one good (so it would take 3 people to produce one apple.. this is not a very good economy)
If they produce only apples they will produce 1,200/3 units of apples, so the economy will produce 400 apples.
ii) if the economy only produces bananas
The unit labor requirement is 2 so in this case they will be able to produce 1,200/2 so the economy will produce 600 bananas
With this information we can produce the production possibilities frontier.
From this we can see that the maximum apples we can produce is 400 and the maximum Bananas are 600. The slope is -(2/3) which shows the opportunity cost of production. It says that if we wish to produce another banana we need to give up 2/3 units of an apple.
Another way of putting it is suppose that we are producing 400 apples and wish to produce 3 bananas. Well to produce 3 units of bananas we would need 6 units of labor. Those 6 units of labor would need to come from apple farmers. If we reduce the labor picking apples by 6 units, we will produce 2 less apples (as it takes 3 units of labor to produce an apple). This is the same as the slope. When we increase our banana production by 3, our apple production decreases by 2.
In the absence of trade what would the price be? why?
Remember that we assumed that i) people can move freely between producing apples and oranges and ii) people only care how much profit they earn.
This means the following identity most hold:
Pa * A = Pb * B
Pa is the price of an apple
Pb is the price of a banana
A is how many units of an apple is produced with one unit of labor
B is how many units of banana is produced with one unit of labor
This can be intuitively thought of as the amount of profit earned from one unit of labor producing apples must equal the amount of profit earned from one unit of labor producing oranges.
If we substitute in how many units of a good we get with one unit we get
Pa * 1/3 = Pb * 1/2
with some re-arrangements we get
Pa/Pb = 3/2
This says that the relative price must equal 3/2. Since there are only two goods in the economy only the relative prices matter. We can let Pb = 1 and then conclude that Pa = 1.5
This leads to an important result: The relative price equals the inverse of the slope when you drop the negative sign