A mathematical approach to GDP
The concept of a nation's GDP is very intriguing, as it is just the sum of expenditures, such that this sum is earned by everyone in the nation.
It is simply calculated as
, where C represents Household Consumption, I represents Private Firms' Investment, G represents Government Purchases, and NX stands for Net Exports (which is the difference between exports and imports).
Yet, I think that this formula for GDP can be 'refined', using the matrix notions.
Introducing Aggregate Expenditure
I argue that GDP is easily calculated as the expenditure of all entities in the economy.
If every entity expends, then every entity earns. They earn as much as the sum of revenues from commerce, so we have that:
, where i stands for every item traded in the economy. This equation says that Aggregate Expenditure is the sum of the product the price and the quantity of each item.
This implies that GDP also has the same form as Aggregate Expenditure.
Letting price be represented as a row matrix, and quantity as a column matrix, we see their product is the equation given for AE.
Using Matrices to simplify Multiplication
Aggregate Expenditure is not GDP
Aggregate Expenditure in an economy consists of the trade of imported goods and/or services. But, this expenditure flows out to foreign firms, so it must be deducted from AE.
We can accomplish this by assuming the expenditure on imported goods and/or services is a fraction of Aggregate Expenditure. So that we'll have that:
, for each item i, where M represents imports.
We also know that local private firms are exporting, and earning revenue in the same form as AE.
So, we have that GDP is the sum of Non-Imported Aggregate Expenditure and Exports:
What if we don't want to lump all expenditures together?
I understand that the Expenditure formula's purpose for GDP is to distinguish household consumption from expenditures from firms and government. To refine the approach, we can write:
A (3 by n) dimensional AE matrix
Then we could add the net exports matrix.
This is how to calculate GDP using the matrices notion.